DYNAMIC COMPRESSION OF ASPHALTIC GLASSES
By
J. CARLOS BUSOT
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1970
A Ml ESPoSA EESPrIRANZA
ACK!O
The author wishes to thank Dr. H. E. Schvayer for his hp
Ful guidance in directing this research. The auwtho is lso irn
debted to his Supervisory Committee for their counsel and criticism
and to Dr. M. A. Ariet for his motivating discussions during the
development of this research. The economic assistance of the E, i.
Du Poni de eemours Compan, is gratefully acknowledged.
TABLE OF CONiTENTS
Page
ACKNOWLEDGMENTS . .
LIST OF TABLES . .
LIST OF FIGURES
ABSTRACT . . .
CHAPTER
I. INTRODUCTION
A. Glassiness
B. Objective of
this Research . .
II. CRITICAL ANALYSIS OF CONTEE;PORARY THINKING ON
THE QUANTITATIVE DESCRIPTION OF GLASSINESS
A. Description . . . . . . . .
Inducement of Glassiness . . . . .
Single Point Measurements . . . . .
Thermodynanic Description of Glassiness
B. Explanation . . . . . . .
Theory of the Glass State, WLF Equation .
Volumetric Viscosity . . . . . .
C. E>. rimental . . . . . . .. .
The Ehrenfest Relations, Pressure Dependence
of the Glass Transition Temperature
Applications . . . . . . . .
11I. INTERPRETATION OF APPLICABLE THEIR iODYNAMICS
Definition of Glass . . . . . 35
Thermodynamics oF QuasiElastic Materials. 38
Develop rnt of thle Eautions to be Used for
the Description of Glassiness . . 44
GrCohicai ticdZ c f a Systcr, Sltowing Quasi
Elastic ' pornss . . . . . . 41I
Iii
viii
xi
4
. . . 1
. . . 4
6
6
20
20
27
30
TABLE OF CONTENTS (cont.)
CHAPTER Page
Equations . . . . . . . . 46
Dissipative Terms . . . . . ... 50
IV. PROPOSED GENERAL DESCRIPTION OF GLASSINESS . 55
A. Glassiness and Excess Thermodynamic Functions 57
Changes in Volume in Unconstrained Materials,
"Constant" Pressure Experiments . . 57
Excess Thermodynamic Functions . . .. 58
Graphical Description of Glassiness . .. 63
Isobaric Transitions . . . . ... .70
V. DEVELOPMENT OF EXPERIMENTAL TECHNIQUE . . . 73
A. Materials and Apparatus . . . . . 77
B. Preliminaries . . . . . . . . 80
Age, Preparation of the Sample and Presenta
tion of Results . . ....... 82
Machine Characteristics. . . . ... 86
C. Homogeneous Deformation . . . . .. 92
Independence of Dynamic Response on Sample
Length in Relation to H .. :ty ... 93
Study of Dynamic Effects, Relating to Homo
geneous Deformation . . . . . 99
VI. RESULTS, CONCLUSIONS AND RECOMMENDATIONS . .. 119
A. Results . . . . . . . . . 120
General Procedure . . . . . ... 120
Density versus Pressure . . . . . 123
Recoil and Relaxation . . . . ... 134
B. Summary of Conclusions . . . . .. 146
C. Recommendations . . . . . . .. 149
TABLE OF CONTENTS (cont.)
APPENDIX
A. HISTO.iCAL DEVELOPMENT OF EXPERIMENTAL
METHOD . . . . . . . . 152
1. Thermostatics . . . . . 152
2. Preliminary Dynamic Studies . .. 155
3. Development of the Experimental
Technique . . . . ... .158
4. Preliminaries . . . . . 160
B. CONTINUUM MECHANICS . . . . .. 171
1. Phenomenological Concepts . . . 171
2. Thermodynamics ... . . . . 177
REFERENCES . . . . . . . . .. . 187
LIST OF TABLES
Table Page
1. Properties of the Four Selected Asphalt Cements 78
2. Effect of Aging . . . . . . .... .. 83
3. illustration of the Effect of Sampl Length on
Dynamic Determination of Density. Asphalt
S6313 . . . . . . . . .. . 98
4. Characteristic Time of Recoil Curves for Several
Previous Histories . . . . . . . 117
5. Coefficients of the Equation p= Ao + AIP + A2P2
and Compressibilities at 0 and 1000 atms tor
S6309 . . . . . . . .... .. 129
6. Coefficients of the Equation p= Ao + AIP + A2P2
and Compressibilities at 0 and 1000 atrrs for
S6313 . . . . . . . . .. . 130
7. Coefficients of the Equation P= Ao + AP + A2P2
and Compressibilities at 0 and 1000 atms for
S6320 . . . . . . . .. .. . 131
8. Coefficients of the Equation P= Ao + AlP + A2P2
and Compressibilities at 0 and 1000 atms for
S6447 . . . . . . . . .. . 132
9. Dynamic Parameters of Selected Asphalts ..... 147
Al Preliminary Data at 32F for Rheology of Twelve
Florida Selected Asphalts . . . . .... 157
A2 Effect of Aging on Compressibility of Asphalt
S639 . . . . . . . . ... . 162
A3 Deformation Readings with Special Assembly for
Determination of Drag Effects . . . .. 166
LIST CF FIGURES
1. Typical Glass Transition Temperature Deter
mination of a Paving A:phalt by Penetro
meter . . . . . . . . . 3
2. Effect of Temperature History on Deteri;na
tion of T . . . . . . . 9
3. Davies and Jones Concept of GCassiness.
Observation at Constant Te:perature 13
4. Davies and Jones Concept or Driving Forces
during Adiabatic Recovery of a Glass 19
5. Mechanical Model of Instantaneous and Delayed
Dissipations . . . . . . . 36
6. Constrained Changes in Volume or Temperature LO
7. Excess Pressure during Compression, Relaxed
State .... ...... . .... . 61
8. Equilibrium State versus Relaxed State . 62
9. Excess Functions and Glassine.s ... . 65
10. Decompression and Heating of a Glass ... . 66
11. Classiness during Isothermal, Finite, Dynamic
Compressions . . . . . ... .68
12. Glassiness during Isobaric Processes . 71
13. Line Sketch of Compression Equipment . .. 7
14. Instron Univcrsel Testinc Machine .... 79
15. Disassembled MIh oded Capillary Rheoieter
and Dirmensios of its Parts .. ... . 81
16, Tracing of Load versus Df cr atici for Cli
bra.ioni at T0wo Rates o C,; precs ion . 87
viii
LIST OF FIGURES (cent,)
Figure Table
17. Illustration of Acceleration Effect . . 91
18. Deformaticn of Two Samples of Different
Lengths without Lubrication . . .. 94
19. Decompression of Two Samples of Different
Lengths Using Lubrication . . .. 96
20. Assembly for Studies on Effectiveness cf
Lubrication . . . . . . 103
21. DecompressionRecoil Experiment . . .. 105
22. Total Recoil as a Function cf Deformation
Time . . . . . . . . . 107
23. Recoil after Decompression Recorded by the
Two Cells . . . . . . . . 109
24. Recoil on S6313 after Different Histories
(Samne ', Different Deformation Ties) Ill
25. Recoil on S6313 after Different Histories
(Different Y, Same Deformation Time) . 112
26. Recoil on S6313 and S6447 after Different
Histories at Different Temperatures . 114
27. Thermocouple and Load Readings during Compres
sion Cycles . . . . . . . 115
28. Illustration of Experimental Procedures . 121
29. Density versus Pressure for S639 . . 124
30. Density versus Pressure for S6313 . .. 125
31. Density versus Pressure for S6320 .... .126
32. Density versus Pressure for S6347 . .. 127
33. Recoil and Relaxation of S6320 at 25C . 136
34. Recoil of All Selected Asphalts at 250C . 137
LIST OF FIGURES (cont.)
Figure Page
35. Relaxation of All Selected Asphalts at 250C 139
36, Relaxation of All Selected Asphalts at 00C 140
37. Relaxation of All Selected Asphalts at
300C . . . . . . . . .. 141
38. Recoil of All Selected Asphalts at 00C . 142
39. Recoil of All Selected Asphalts at 300C 143
A1. Specific Heat versus Temperature for Asphalt
Cement (S6320) . . . . . ... .154
A2. Pressure Required for Compression with and
without Lubrication . . . . . 156
A3. Assembly for Studies on Effectiveness on
Lubrication . . . . . . . . 164
A4. Failure of Silicone Lubrication at 250C . 169
B1. Motion and Configurations . ... . ...... 173
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fuifillment
of the Requirements for the Degree of Doctor of Philosophy
DYt4MAMIC COMPRESSIONf OF ASPHALTIC GLASSES
By
J. Carlos Busot
August, 1970
Chairman: Dr. H. E. Schweyer
Major Department: Chemical Engineering
It was the purpose of this research to clarify the pheiomeno
logical description of the behavior of asphalt glasses, to conduct a
critical review of the literature concerning the meaning of glaZsiiess
as a material characteristic and to develop an experimental approach to
study quantitatively glassiness in asphalts.
The indepth review of the applicable literature revealed that
the molecular theories of glassiness and its definition centered around
a phenomenological description limited to the changes undergone by most
substances when cooled at a sufficiently rapid rate Lo prevent crystal
l ization.
This research generalizes that glassiness be defined as the
phenomena observed when the properties of a material depend on the
past history of its thermodynamic state.
The experimental approach proposed for the more general scudy of
glassiness was to observe the pressure and the entropy responses of a ma
terial when subjected to homogeneous changes in volume. The thermo
dynamics of quasiclast:c materials was used to develop the formLn_uL
for the analysis of this experimental approach.
The experimental technique developed consisted of compressing or
decompressing an asphalt sample encapsulated in a latex membrane, and
confined in a lubricated steel barrel. Extensive studies showed that
this technique allowed homogeneous deformation of the sample without
interference from shear disturbances at the wall.
Results on four selected asphalts indicate that these materials
show glasslike behavior under c.:' session. The results also indicate
that this technique can be used to differentiate the mechanical (e.g.,
free volume) from the caloric (e.g., entropy of rearrangement) contribu
tions to the observed delayed changes in pressure at constant volume and
temperature. In fact it was found that the experimental arrangement cou
be used as a thermodynamic calorimeter by insertion of a thermistor with
out appreciable effect upon the homogeneity of the deformation.
The proposed technique and experimental approach developed in this
research should provide infor.tion to 'etrrnmir the effect of asphalt
composition upon its transition from a liquidlike behavior to a solid
like behavior.
Since inservice behavior of asphaltic materials subjects them
to a compressiondecc oression environment at different temperatures,
it is expected that studies of glassiness such as in this dissertation
will be of value in e lainil: their performance.
CHAPTER I
INTRODUCTION
A. Glassiness
Asphalts in service or at ordinary ambient temperature often
exhibit a behavior which is neither that of a viscous liquid nor
that of a crystalline sold. Phenomenologicaily, this behavior may
be ascribed to a glassy state. In the case of asphalt, glassiness
is manifested by hardness, by cracking and conchoidal fractures
under sudden stress, by a glassy surface appearance and by a very
high viscosity. These properties, which are undesirable from the
standpoint of road performance, are accentuated as the temperature
decreases.
The onset of these properties with temperature is determined
not only by the chemical composition of the asphalt but also by
the physical and spacial interactions among its numerous components.
The relationshipsdescribing these thermodynamic phenomena are dif
ficilt to resolve. However, the thermodynamic theories for poly
mers and organic glasses that have been developed to explain thsir
theological properties and their temperature dependence would ap
pear to be applicable to asphalt.
Work in this laboratory by Shoor (46) has shown that glassi
ness in asphalt can be detected by freezing the sample and determin
ing the change in hardness (penetration) as the temperature increases.
An abrupt change is noticed on the change of hardness with temperature
as shown in Figure i. The temperature obtained at the intersect io
of the lines presenting the !ow temperature and high temperature
behavior is defined as the glass transition temperature of the
material; T
This empirical determination of T9, by the intersection of uwo
straight lines representing high and low temperacurke behavior.
constitutes the generally accepted phenomenological definition of
glass transition. This graphical method is also applied to plots
of viscosity, volume (and many other properties) vs. temperature.
Shoor, Majidzadeh, and Schweyer (47) used the glass transi
tion temperature (T ) determined with the penetrometer method to
correlate the temperature dependence of the viscosity of eight cif
ferent asphalt cements. The correiation scheme used by Shoor et.
al. consisted of shifting the data obtained at different temperatures
along a logarithmic time axis and determining the value~ of the
shift factor required at each temperature to form a composite
curve. The values of the shift factor were related to temperature
through an empirical equation involving a characteristic temperature.
This temperature was the penetration related Tg. This scheme is
kno.'n as the timetemoerature suoIjrosition _rircioDle.
Brodnyn (6), Gaskins and others (21) ,were among the first
investigators to use the superposition principle for asphaltic
materials. They suggested the ASTh Ring and Ball softening point
as the characteristic temperature and concluded that, in general,
asphalts behave as low. molecular weight viscoelastic poly.rrs.
2; 1
0.03 
I I
O k
I 7 I
:0
9 /
Figure .TypIcal Glass Transition Temperatre Determination f
0.02a Paving Asp by Penetroter
I / 1
I 0 2I
Temperature, OF
Figure l.TypIcal Glass Transition TemperatL re Determination bf
a Paving Asphalt by Penetronate,
Wada and Hirose (51) used a dilat:metrically measured glass transi
tion temperature to correlate the temperaturetime dependence of
asphalt retardation times. Sukanoue (48) correlated the shear
modulus of asphalt in the same manner. Barrall ( 3) used a dif
ferential thermal apparatus and indicated a dependence of this Tg
on the asphaltene content of the asphalt. Schmidt and coworkers
(41), (42), (43) have measured glass transition temperatures of
numerous asphalt by noting the changes in volume on cooling and/or
heating (20C/min) on a specially designed dilatometer. They also
succeeded in obtaining a fair correlation of viscositytemperature
data using their T as the correlating parameter.
The moderate success of the investigations mentioned above
illustrates that the glass transition is a fundamental phenomenon.
It provides an empirical method for the correlation of physical
properties with temperature. However, the lack of a uniform
definition of Tg as well as the need for an accepted method to
determine rheological properties of asphalt at low temperatures
is apparent. This lack oF generality makes comparison of data
among different investigators and theoretical considerations
extremely difficult.
B. Objectives of this Research
A comprehensive study on the thermodynamic background control
ling the rheological properties of asphalt and how they vary with
temperature may be fruitful in understanding the physical behavior
of these complex organic materials. This study should provide the
basis to understand and describe, with generality, the observed
glass phenomena.
The research described herein is intended to elaborate on
this thermodynamic background, through the following specific
object ives :
1. A critical analysis of the contemporary thinking on
the quantitative description of glassiness.
2. An interpretation of the applicable thermodynamics.
3. A proposed general description of glassiness.
4. An experimental technique for the study of the
proposed description in 3 above.
A separate chapter will be devoted to each of the above
objectives.
CHAPTER II
CRITICAL ANALYSIS OF CONTEMPORARY THINKING ON
THE QUANTITATIVE DESCRIPTION OF GLASSINESS
Some statement about the distinction between description
and explanation of glassiness should precede any critical analysis
of this subject.
The description of the glass transition must be clarified
before attempting the interpretation of the experimental results;
the description of glassiness constitutes its phenomenological
definition. Many phenomena of diverse nature may be involved in
the transition from a liquidlike to a solidlike behavior, or
more importantly, the liquid to solid transition may be sensed and
recorded differently depending upon the experimental parameters
used to observe and define it. Thus, the phenomenological defini
tion must consider these experimental factors.
Conversely, the explanation or theoretical definition of
glassiness must follow the acquisition of experimental data.
This involves the proposal of molecular models in order to explain
the observed behavior.
A. Description
Inducement of Glassiness
In general most liquids can be transformed to a noncrystal
line solid state, if they are cooled through the crystallization
temperature range fast enough to prevent the formation of crystal
nuclei. It is possible to supercool many liquids; organic poly
mers (18), organic liquids such as glycerine and glucose (13),
fused salts (2), and metals (4) demonstrate this phenomenon.
The prevention of crystallization can be understood if one
considers the two steps involved in this process. In crystal
lizations a nucleus must form, and then it must grow. Nuclei forma
tion is opposed by a free energy barrier because of the fact that
the melting point of very small crystals is lower than that of
large ones. Thus, in a supercooled liquid, crystals smaller than
a certain size are unstable, i.e., the nuclei tend to redissolve.
In addition, crystal growth is hampered by viscous flow. As the
temperature is reduced the rate of formation of nuclei may increase
but the rate of crystal growth is reduced because of the increased
time required for molecular motions. If the increase of viscosity
is large enough the supercooled liquid (glass) acts effectively
as a solid.
The glass transition temperature is determined experimentally
as the temperature at which second order transitions (change in
slope) are observed in the values of the thermodynamic state
variables. However, in the temperature region above and below
Tg, the viscosity of the liquid increased very rapidly. Volume
changes with temperature often show significant delays; this delay
is also observed for most physical properties. Therefore, the
glass transition temperature Tg as generally measured in the
laboratory depends on the temperature history of the sample.
(See Figure 2.)
Curve ABC represents schematically the equilibrium curve for
the volume of a material as it is cooled or heated through the
glass transition region very slowly to TB. The recorded transition
temperature, if any, would be T91. The curve AHD represents the
temperature relationship when the sample is cooled rapidly from
TA to TB. If the sample remained long enough at TB its volume
would approach the equilibrium value at C. Heating at this time
would proceed along a curve similar to EFA. However, if the sample
had been heated immediately after quenching it would have followed
a curve similar to DGA. The values of Tg recorded would have
depended on the heating and cooling rates through the transition
region and the time held at TB after freezing.
Bondi (5), discussing a description of glassiness similar
to the above, notes that the rapid increase in viscosity, character
istic or even the cause of glass transition, may be due to dif
ferent mechanisms depending upon the system considered. This author
further states the need for a consistent phenomenological definition
where "the rate and amplitude of deformation as well as the thermal
history of the sample and the instantaneous temperature T" are taken
into consideration. Bondi also notes that the lack of such consider
ation makes analysis of published data of a qualitative rather than
a quantitative character.
I
A
< A y
I H G
r 'I
S/
ST T2 T9 T
TB  TA
S' ". .. " . ..... t
Temperat u re
Figure 2.Effect of Temperature History on Deteminat:ion of Tg
Single Point Measurements
The topic of representative measurements of material properties
during dynamic transitions is of foremost importance for a proper
description of glassiness. This topic and its relation to the
phenomenological definition of glassiness will be discussed in
Chapter IV. However, because of its relation to the state of the
art, the implications of "single point measurements" will be discussed
here.
A review of the literature of glass transition temperature
reveals that the effects of sample shape on the spatial distribu
tion of the responses to external changes are usually neglected.
The assumption is generally made that a single point measurement
is representative of the state of the sample. This assumption is
only true if either the mal.~rial is at equilibrium or if it is under
going a ho.'og.j2nous deformation. This last statement needs qualifi
cation and is discussed furT':er in Appendix B, section Id. Never
theless, it is presented here to direct the attention of the
reader to the importance of a priori theoretical considerations about
the experimental conditions. The most common justification for
single point measurements given by experimentalists studying glass
transition phenomena is based on the size of the sample. It is said
that if the size of the sample is small enough, a uniform temperature
distribution will be established "fast enough" or "rapidly." The
possibility is ignored that regardless of the size of the sample
and the uniform temperature, enormous spatial gradients in density
and pressure ra be present if a "volume viscosity" exists.
No general statement can be made about the correlation,
thermodynamic or otherwise, between stimuli and their response,
unless the responses and stimuli are known at each point of the
material. For example, a uniform state of stress and strain through
out an elastic material is required before the Young modulus can be
determined experimentally. Design of specimens, such that applica
tion of external forces will produce a describable state of stress
and strain, is always an important consideration when testing solid
materials. Also, the velocity at each point of a flowing Newtonian
fluid is required before its viscosity can be established. Visco
meters are designed to produce describable velocity profiles, so
that measurements at boundaries will allow calculation of the
response at any and every point of the fluid.
Thermodynamic Description of
Glassiness
When dealing with the experimental conditions necessary for
determination of thermodynamic parameters near a transition tempera
ture one has to consider the relaxational or irreversible character
of the process. The difficulties of describing an irreversible
process are accentuated when trying to reconcile classic concepts
of thermostatics, i.e., specific heat, thermal expansion, pressure,
internal pressure, etc., with classic concepts of rheology like
viscosity and viscous stress. This will be illustrated by comment
ing on the widely recognized work of Davies and Jones (13).
Davies and Jones are the only investigators known to this
writer who have attempted an irreversible thermodynamic approach
to describe glassiness and establish relationships among the
measured thermodynamic parameters. In the opinion of this writer,
the most important contribution of Davies and Jones was to define
a "volume" viscosity to quantify the "time effects" observed during
glass transitions. Regretably, their work has been consider
superficially in the literature to provide background for discus
sion of the effect of pressure upon glass transition. A brief
presentation of the work of Davies and Jones follows. It is intended
to illustrate tho involved and far reaching conclusions drawn by
these authors, and others, by their manipulation of a phenomeno
logical description of glassiness.
Figure 3 illustrates Davies and Jones' phenomenological concept
of glassiness. The line LAC represents the enthalpy vs temperature
idealized equilibrium line of a glassforming liquid. On cooling at
a finite rate, of the order of degrees per minute, the liquid will
depart from the equilibrium curve at point A and move towards B.
At lower rates of cooling the glass would move along XY instead
of AB. The point A represents according to the authors, a thermo
dynamic "fictive" state. The fictive state is that at which the
glass would find itself in equilibrium if brought there rapidly
from its actual state. According to these authors, in the case of
an isobaric experiment the state of the glass can be described
completely by giving its actual temperature T, and the temperature
at point A (fictive temperature, T0).
13
L
I I
B
Observat ion i
/ Y
I i
T T I To
Tempera tioure
Figure 3.Davies and Jones Concept of Glassiness. Observation
at Constant Temperature.
Li p
at Constant Timoeratjre.
This simplified description, in this writer's opinion, led
Davies and Jones to overlook the influence of the rate at which
the irreversible path AB is traveled. A glass once at a point away
from AC, point G along AB for instance, cannot be brought back
reversibly to the original point in the equilibrium line. The paradox
in their description is that a glass can be "cooled" reversibly from
the liquid (point of departure from the equilibrium line) but can
not be restored to a liquid state reversibly.
Davies and Jones did not recognize the importance of the
"preparation time" of the glass (time along AG) and the time involved
in the sensing of the glass properties. These authors considered
the preparation time sufficiently fast and assumed the sensing time
to be of no importance. The latter assumption is not consistent
with their observation that the rate of cooling would affect the
"fictive" tempperature.
For a theoretical development of the equations necessary to
describe their phenorenological concept of glassiness, Davies and
Jones introduced a ne: state variable (Z). This variable is defined
to be continuous and to remain constant when the pressure (p) and
the temperature (T) are changed "rapidly." These authors attributed
to Z the significance of being a measure of the configurational
order, assumd constant for rapid changes of pressure and temperature.
A glass of fixed structure," Z0, is thus proposed, when cooling
aHow this structure is fixed by a finite rate of cooling is
not explained. Furthermore, these authors reFecrad to other in
vesLigators who attributed the freezing to a second order transi
tion, as "misunderstanding the nature of the phe nol non." (Davies
and Jones, o,__ct_. 2.)
through the fictive temperature at a particular rate. The thermo
dynamic state of Davies and Jones glass, Z = Zo, is represented b/
a curve Z(p, T) = Zo, where Zo satisfies the equation A(Zo, p, T) = 0,
where A, defined by equation 4, equals zero when the system ;s in
equilibrium. This implies that the equilibrium volume for a glass,
V(Zo, p, T), will be constant along the AB curve. The volume at p
and T is the same as the volume at p and T; AV equals zero. The
same is true for the entropy; the entropy S at p and T is the same
 a
as the entropy at p and T. Their final results state:
cp_ = aT AV A
p A Aa (1)
dT Sp AV AS
AS AC P (2)
Sp AS TVAa
where AI Aa and AC are the discontinuities in compressibility,
thermal expansion, and specific heat at the point where Z = constant
 Zo, Although these authors only presented data on glass transi
tion with temperature, ACp vs. T and Aa vs. T, the theory was
intended to be general and to describe the phenomenological behavior
of glassiness with pressure.
The authors' objective in introducing the pressure dependence
on Tg was to be able to define different types of driving forces
for the delayed volume changes. This was accomplished by using
equation I and by defining a fictive pressure, p, given by:
aThe symbol x will b used in the text to represent thei
partial derivative c,.oratorr tf.) where y and z represenr the
other independent variables 3x / y,z
S= (3)
T TTVAc
This result of their theory proposed that a sudden
isobaric change in te.: ,,rature AT which leaves T and p unaltered
is equivalent thermodynamically to a pressure increment of:
Ap : (ACp/TVAa)AT." This calculated Ap has been used throughout
the literature to replace AT as an equivalent driving force on a
thermally produced change in volume. This "excess" pressure is
then related to the volume rate of change through a volume viscosity.
The basis for this concept is an equation expressing the change of
irreversible structural entropy,
TdS;,r r AdZ (4)
where the affinity A is
A = (p p) T,Z (5)
9p
The value of A is zero when the system is in equilibrium. Equation
5 presupposes that the system is "close" to equilibrium. Under these
conditions and assuming that dZ can be represented by dp, the ir
reversible change in entropy and the rates of production of entropy
are given by these authors after some manipulation by:
a
More detailed and rigorous treatments of the classic theory of
thermodynaic relaxation are given by Herzfeld and Litovitz (26), pp.
159170; by Patterson (37); and by Prigogine et al.
This equation is the source of the paradox. A glass can be
cooled at a finite rate followir Z Zo and therefore. dSirr = 0
Ho wver, it cannot be reverted reversihiy to liquid through a path
of constant Z.
TdSirr = V(pp) [AcdT A3dp]
Si =ir r v a
irr r T "
where cand $ar, the instantaneous values of the temperature
expansivity and the compressibility.
A further assumption is required before a viscosity can be
defined. In the same way of other irreversible phenomena close to
equilibrium, "forces" are assumed proportional to "flows"; the
expression in brackets in equation 7 can be interpreted as a "flow"
and its coefficients as "forces." Therefore, finally, the kinetic
equation is given by
CT p" v = 
V TI
where r is the proportionality constant and represents a volume
viscosity. Equation 8 would be approximately valid if a process
were devised where the phenomenological description of glassiness
given in Figure 3 is true.
The authors used their description of glassiness to estimate
the relaxation tirn: (and viscosity) of thermal relaxation of glycerol,
and volume relaxation of glucose. The experimental methods consisted
of cooling a sample of glycerol below its glass temperature inside
an adiabatic calorimeter, and studying the approach of the temperature
of the sample to equilibrium at constant enthalpy.
aA dot above a variable indicates time rate of change; ;.e.
drx .
xdt
dt
Figure 4 is a schematic representation of this process. The
driving forces For the irreversible adiabatic recovery of the
sample after it is cooled to To are represened by T T. Davies
and Jones' phenomenological interpretation of glassiness in:plies
that during cooling of the glass the instantaneous volurre of the
sample was at all times in pace with the temperature and pressure
of the sample; therefore all the volume change occurred adiabatically
after cooling and was caused by the T To parameter. Their pioneer
ing results and experimental conclusions proved that there were
relaxation phenor:.ena associated with volume changes on glasses.
Numerically, these results are of limited value, because their
viscosity includes more than volumetric effects, as indicated by
equation 8. The parameter T in equat;cn 8, just happens to have
dimensions of viscosity.
Goldstein (25) proved that if the phenomenological description
of Davies and Jones is true and general, their parameter Z which
controls the relaxational phenomena after glass preparation can be
interpreted as either an excess volume or an excess entropy. A
process described by, AZ = 0 is equivalent to processes described
by AV = 0, or AS = 0. Goldstein derived equations 1 and 2 as
results of these observations, and indicated that only if a AV = 0
process is completely equivalent to a AS 0 process, would these
equations be valid.
If another parafreter in addition to Z i; rcuuired to describe
glassiness, ttien
'i
*I
Drivring force = T T
:i
bi
(.
.1
c:*
i I
Ai
..!.
q7
Bl
i
ri
t:
j
?> j
!i
Sri
I1
l.i
i:
l i
r ^"
l J '
['i T
i 2
T3
Te mpe rat u re
Figure 4.Davies F rcl Jcne Concept of Driving
Aaiabatic Pcovrv, of a Glss,.
Forces du
Forces during
Cool ing
_:13 TV >c,
L% a Cp
The nature of this second parameter, however, would have the
same character as Z. It would represent a difference from a
"fictive" state and would not include the time effect during prepara
tion oF the glass.
In addition to Davies and Jones. and Goldstein (25), the
fictive state approach has been used by Kovacs (29), (30), O'Reilly
(34), Passaglia and Martin (35), and in general by all experimenters
a
interested on the variation of T. with pressure.
B. Explrant ion
Theory_ of the Glass State, ILF
Euat ion
The theories of models of the glass state which have been
proposed are intended to explain the success found in using
empirical glass transition temperatures to express the tempera
ture dependence of relaxation processes, more specifically
viscosity.
The cornerstone of the theories developed to explain glassi
ness is the empirical equation proposed by Vogel (50) to express
the d:clnd n: of viscosity upor, temperature. This equation was
modified and used for the first time for the development of a
theory for the glass state by 'Williams, Landel and Ferry (54).
aThis subject wil be discussed in more detail on p. 30.
The equation in its most common form has the expression
n CI (T'"Ts)
log (10)
rs C2 (TTs)
where
l/r5s is the ratio of viscosities at
temperatures T and T respectively
C1 = 8.86
C2 = 101.60C
Ts = reference temperature
Equation 10 is known as the WLF equation. Williams and co
workers (5.2. (53), (54) found that if a separate reference
temperature Ts is suitably chosen for each system, equation 10
expresses the temperature dependence of viscosity for a wide
variety of glass forming liquids over a temperature range of ap
proximately 1000C above the glass transition point. Most signif
icantly they found that the temperature Ts lies 500C above the Tg,
with a standard deviation of + 50C. They also showed that C1 was
proportional to the expansion coefficient of the liquid.
Free volume theories
Williams, Landel, and Ferry used Doolittle's experimental
findings (15) that viscosity depended exponentially on "free
volure" (vf); which they proposed as:
Vf vg [0.025 + LA (T Tg)]
(li)
where v is the volume at the glass transition temperature, and
is the difference between the thermal coefficients of the liquid
and the glass. The original free volume theory of Williams, Landel
and Ferry assumed that the ratio of free volume to total volume
would remain fixed below Tg and that its pressure and temperature
variation above T9 would be given by
AS = g (12)
Aa = a1 o (13)
where 3 and a are the compressibility and thermal expansion coef
ficient of the liquid (1) and the glass (g).
Cohen and Turnbull (10) developed a more complete theory of
diffusion and no;nentum transfer based on the basic concepts proposed
by Williams, Landni and Ferry. Cohen and Turnbull did not intend,
however, to include associated liquids whose viscosity varies
markedly with temperature at constant volume. Rather. these authors
proposed a description of glassiness and molecular transport in
liquids and glasses in which no potential energy barriers among
molecules existed (hard sphere model). The potential energy of a
molecule was assumed constant except upon intermolecular contact.
Interactions among molecules were not necessary to explain glassiness,
and viscosity would only depend on temperature through volume.
Cohen and Turnbull visualized flow as a process involving
molecules jumping over barriers created by the need of formation
and redistribution of holes in a liquid quasilat ice. These
authors assumed that these barriers stemmed from the need of free
voiume (v) to be greater than some value (v"'). This idea of flow
is quite similar to the activation state or kinetic theory of Eyring
(20) and (23). in Cohen's theory, flow occurs when "there is a
fluctuation in density which opens up a hole within a cage large
enough to permit a considerable displacement of the molecules
contained by it. Such a displacement gives rise to diffusive motion
only if another molecule jumps into the hole before the first can
return to its original position" (10). The transport coefficient
for a molecule, D, according to this picture of flow is:
D(v) = D(v*) P(v*) (14)
where D is the diffusion coefficient which is a function of the
volume, v, of the cage, v* is the critical or "activated" volume
just large enough to allow a molecule to displace itself, and P(v*)
is the probability of finding a hole of volume larger than v*.
This probability is given by Cohen and Turnbull (10) as:
P(v') = exp[ Yv*/vf] (15)
where vf is the average free volume, defined below, and Y is a
numerical factor introduced to correct for overlap of free volume.
The final expression becomes:
D(v) = D(v)* exp[ yv*/vf] (16)
where D(v*) is a function only of the molecular diameter, the
temperature dependent velocity of the molecules, and a geometric
factor. Equation 16 has the form found by Doolittle for the
viscosity of hydrocarbons. In order to test their theory Cohen
and Turnbull defined free volume as:
vo exp.. a dT (17)
L J
where vo is the van der Waals volume which is assumed independent
of temperature, a= is the coefficient of thermal expansion, and To
is the temperaLure at which the free volume vanishes. This defini
tion assumes that the free volume is given by the total therrnal
expansion at constant pressure and is zero at a temperature To.
The temperature and pressure dependence of viscosity will be
determined, according to this theory, by the pressure and temperature
dependence of free volume as follows:
Vf =" v (T To) vp AP (18)
where a and v are the mean values of the expansivity and van der
Waals voluic, over the temperature range (T To), evaluated at A P = 0.
B and Vp are the mean compressibility and volume over the pressure
increment A P.
The definition of free volume as a function of thermal expan
sion alone (equations 17 and 11) is insufficient for satisfactory
description of the pressure and temperature dependence of viscosity
of polyr.eric liquids. Other modifications of the free volume
theory are designed to provide thn theory with the flexibility of
a free volume wh:ch would show a temperature dependence below T (7), (S
(40). The basic consideration of all these theories is the falling
of the free voluren below soe critical value where the high
viscosities would r ke reltAation tir;s of the order of days.
These theories !Fha' been used to explain the rheological behavior
of asphalt by 11j idzadh and Sc r (31).
1, ( ., 0 e t jvg arra qe^ nt
confilurat ional entr.ovy
Gibbs and coworkers (22), (1 ) have defined T as "the quasi
static glass temperature below which molecular relaxation times are
too long to permit establishment of equilibrium in the duration of
even the slowest experiments ('time scale' of hours to days)" (1 ).
Their concern was to relate by statisticalmechanical arguments the
relaxation properties of glassforming liquids to their "quasistatic"
properties. These authors proposed that the cause for increased
relaxation tine, T, with decreasing temperatures is the reduced
probability, W. of a cooperative rearrangement of the parts of the
liquid.
A cooperatively rearranging region is defined by Adam and
Gibbs ( ) as "a subsystem of the sample which, upon a sufficient
fluctuation in energy (or, more correctly, enthalpy), can rearrange
into another configuration independently of its environment."
By assuming a partition function, Q, for a system containing
a number of pats which could rearrange cooperatively, Adam and
Gibbs define a free energy, AG. This potential represents the
energy hindering the rearrangement at constant pressure and temper
ature. The size (Z) of a rearranging region is defined by assuming
that AG can be expressed using a potential energy per unit size,
Ap by
ZAl = AG = kT In Q
(19)
This implies the existence of a uniform structural unit, e.g.,
molecules or a number of molecular segments of a polymer chain. The
average transition probability W(T) of a cooperative region as a
function of its size is given by:
W(T) A exp[ Z* Ap/kT] (20)
where Z* is now the smallest critical size. Cooperative regions of
Z < Z: yield zero transition probabilities. These small size regions
remain in the same configuration when energy is supplied to the
system. The frequency factor A is nearly independent of temperature.
Finally ;\ is expressed as
W = A exp TSc
W = A exp 
I TS c
(21)
(22)
where C is independent of temperature, Sc is the entropy of the macro
scopic sample, and s c is the smallest critical entropy. It is the
entropy of configuration per molecule corresponding to the minimum
number of configurations determined by Z at a given temperature.
The minimum value of configurations corresponding to Z* as TT is
two. The smallest size Z* must be large enough to have two possible
configurations; the region where it resides before rearrangement,
and another configuration to transform into.
In terms of Gibbs parameters the coefficients of the WLF
equation (see equation 10) become:
A s
C = 2.303 k and (23)
T2
ACp % In
Ts
Ts In 
C2 2 (24)
in r + 1 + In 
T2 s Ts
where T2 is the temperature where s : 0, aid AC is the
specific heat difference between the liquid and the glass at T9.
The cooperative rearrangement theory predicts that the
"universal" parameters of the WLF equation will depend upon: the
ratio of the reference temperature to the equilibrium temperature
T2, a free energy barrier restricting transitions, a critical
configurational entropy, and the difference between specific heats.
Volumetric Viscosity
According to the general contemporary opinion the time
dependence or relaxational character of the glass transition makes
a theory of glassiness a particular case of a theory of liquid
viscosity (24). The two previous theories represent the two most
widely discussed viewpoints on the origin of viscosity of liquids
near their glass transition.
Summarizing the previous paragraphs it is seen that the free
volume theory relies on a kinetic argument b;sed on the dependence
of flow upon the availability of holes or free spaces for the mole
cules to move into. Glassiness is considered a nonspecific process
outweighing specific effects of chemical structure.
Conversely, Adam and Gibbs present viscosity as the result of
specific structural interactions. These interactions are represented
by the increasing size of the rearranging regions as the temperature
decreases.
Both theories relate more closely to the dependence of
viscosity on teinperature and pressure, than to the actual mechanism
of viscous flow. Both consider local rearrangements (hole distribu
tion or cooperative molecular rearrangement) necessary conditions
for flow. However, these rearrangements are not sufficient; they
also occur on the fluid at rest. The questions unanswered by these
theories about the nature of flow are, paraphrazing Goldstein (24):
What is the relationship betwv'een the local rearrangements and
the microscopic deformation?
How does the external stress bias the local rearrangements for
a jump to occur?
Why does not the deformed state reverse to the original state
when the biasing stress is removed?
These questions are subjected to extensive analysis by Goldstein.
The answer to the third question is of particular importance in order
to establish the relationship [btw.en shear and volume viscosities.
For flow to occur irreversibly under a free volume mechanism, two
conditions must Lb met. First, molecules must jul? into a hole,
and second, holes must vaiish r:n reform. If the r.']axatiorn time of
the second step is smaller than the one for the first step, random
ization takes place after the jump and the flow becomes irreversible
under the biasing stress (visccus flow). The appearance and dis
appearance of holes could be thought of as a volume relaxation
process and, thus, the importance of the volumetric viscosity.
According to the theory of cooperative rearrangements, the
relaxation time involved in the creation of holes (second step)
should be larger than the relaxation time for a simple redistribu
tion (first step). This appreciation is based on the consideration
of the apparently larger cooperative character for the creation of
a hole. If this picture of flow is correct, the volumetric viscosity
should limit the rate of shear relaxation. In acoustic experiments
volume relaxation times are almost equal to shear relaxation times
implying no essential differences on the nature of volume and shear
viscosities (26).
Determination of the volumetric viscosity through acoustic
experiments, however, assumes that the shear and volume effects are
additive. The additiveness assumption implies that the absorption
unaccounted for by shear viscosity can be attributed to volume
relaxation. This is complicated by the fact that shear viscosity
itself may not be represented by a simple New.tonian model as assumed,
but probably it is frequency dependent. In this case, the calculated
volumetric effect would be affected by the frequency dependence.
The additiveness assumption may or may not be justified on
the basis of the infinitesimal amplitude (or magnitude) of the
deformation involved in acoustic vibrations, but it leaves the
question open as to the phenomenological behavior of materials
undergoing finite changes in volume. In a finite volume change
structural rearrangements may be triggered which would not have
been detected or induced by the acoustic deformations.
C. Experimental
The Ehrenfest Relations. Pressure
Dependence of the Glass
Transition Temrerature
The Ehrenfest relations state (8):
dT2/dP = AS/Ao (25)
= T VM/ACp (26)
These equations express the change in the temperature at which a
transition occurs T2, with a change in pressure; 3, Ao. and Cp
are the discontinuities in compressibility, thermal expansion, and
specific heat at the transition. The volume V is measured at P and
T2*
Davies and Jones (see p. 16) used the Ehrenfest relations to
substitute dP for dT as the driving fo:ce in volume changes during
glass transition. These authors did not discuss the theoretical
implications of the relations, neither did they present any data
on the change of Tg with pressure. However, Goldstein (25) used
Davies and Jones (13) phenomenological description of glassiness
and interpreted equations 25 and 26 as criteria to test the validity
of the free volu e anid the cooperative rearrangerent theories.
The free volume theory proposed that glassiness occurs because
the free volume approaches zero at the transition temperature.
Goldstein indicated that this condition was similar to the condition
For tile validity of equation 25.
The cooperative rearrangement theory implies that the structural
entropy becomes small as the sample is cooled towards T2. According
to Goldstein, equation 26 requires that a condition similar to this
be met. i.e., that the entropy of transition be zero.
A6plicat ons
The experiments of O'Reilly (34), and Passaglia and Martin
(35), will be discussed to illustrate typical past efforts to
elucidate the meaning of equations 25 and 26.
O'Reilly studied the effect of pressure on the T of polyvinyl
acetate (PVA). This author measured the dielectric relaxation of
PVA at different temperatures, at constant pressure. By observing
the temperature at which the dialectric characteristics changed
abruptly, a transition temperature T was defined. Repeating the
9 AT
T9
experiment at different pressures allowed calculation of
AP
O'Reilly found this ratio to be independent of pressure and equal
to 0.021C/atm.
This author also determined the force required to compress
PVA at different temperatures. Ho'.:ver, some of the results of
these experiments should be accepted with caution. O'Peilly did not
determine the effect of compression rate upon the pressure vs.
volume plots. Furthermore, no precautions were taken by this author
to eliminate the drag at the walls of the container.
Nevertheless, O'Reilly defined a transformation pressure P9,
"as the pressure at which molecular rearrangements can no longer fol
low the applied pressure and the polymer exhibits a glasslike compres
sibility." It should be noted that this definition of the transition
pressure contains two different and perhaps opposite phenomenological
concepts.
A transition pressure at which relaxation phenomena "can no
longer follow the applied pressure" implies a dynamic situation where
viscous dissipations play a foremost important role; they cause glas
siness. Thus, the importance of defining the effect of compression
rates on P and the importance of eliminating the viscous dissipa
tion at the walls. A transition pressure defined as that at which
"the polymer exhibits glasslike compressibilityj' however, implies
an arbitrary definition of glessiike compressibility and the notion
of an equilibrium volume vs. pressure experiment. O'Reilly used
this last aspect to define several arbitrary transition pressures
at a given temperature. The plots of these transition pressures vs.
temperature gave values of TT in excellent agreement with those
ATg 9
values of , obtained with the dielectric experiments.
However, attempts to prove or disprove the application of
equations 25 or 26 to glass transition were inconclusive. The main
difficulty being the curvature of the plots of volume vs. pressure.
In contrast to volume vs. te peraiure plots (Figure 2), the volume
vs. pressure plots do not show rcasonpble straight lines anywhere.
There is no ur .ivoca'l 2mthoJ tc cate irce A.
33
Arguments were presented by O'Reilly in favor of the co
operative rearrangement theories. These arguments were based on
the decrease of volume with increasing temperature well above the
transition pressure. It is the opinion of this writer, that because
of the high pressures and because "measurements were usually taken
a
at increasing temperatures," the decrease in volume may have been
affected by leakage.
Passaglia and Martin (35) determined the variations of Tg
with pressure on polypropylene. These authors used a direct
experimental procedure. The changes in density with temperature
were studied at several constant pressures. These plots approximated
straight lines at high and low temperature ranges. The transition
temperature was defined at the intersection of the extrapolated
straight lines as shown in Figure i.
AT
By plotting the values of Tg vs. pressure, a value ofp
0.0200C/atm. was obtained for polypropylene. Passaglia and Martin
proved that in static experiments like theirs, equation 25 is an
algebraic consequence of being consistent when defining Tg and
when assigning values to AS and Aa.
aO Reilly, o i. p. 432,
CHAPTER li
INTERPRETATION OF APPLICABLE THERMCDYNAMICS
The complexity of the problem involving the quartitative defini
tion of glassiness was illustrated by previous comments on the work
of Davies and Jones. Difficulties in the past have originated
mainly from the lack of general definitions of "time scale," volume
viscosity, and other dynamic parameters not related directly to the
thermodyna:ic theories used to correlate the experimental data.
However, many of the elements necessary for a comprehensive
definition of glassiness already exist, for example: Bondi's
indication that a consistent phenomenological definition is the
key to the problem (5), the general observation of the influence
of "the prior history" upon the properties of a material and
Coleman's comprehensive description of thermodynamic processes
(11), (1 .
In this chapter a rational approach to the study of glas
siness, comprising all the elements presented above, will be at
tempted. First, a quantitative definition of glassiness will be
proposed; sec.ndly, an explicit description of the type of time
dependent processes where this definition cculd bc applied will
be presented; and thirdly, a therrodynamic theory will be used
to develop the equations necessary for the description of glass;nes3.
A. Definition of Glass
It is the opinion of this writer that a glass must be defined
as a material whose thermodynamic properties depend upon the co!
lection of all past thh r(iiodynamic states (prior history).
In order to complete this definition, it is necessary to estab
lish a measure of glassiress, e.g., Daies tnd Jones ordering para
meter Z. The measure of glassiness proposed here is that the
intensity of a glassy state should be defined by its characteristic
time; measured in a relaxation process where instantaneous viscous
dissipations are absent. The bas' for this seemingly arbitrary
requirement is the convenience of differentiating between the two
basic types of dissipative pheno:,ena occurring in glosses. These
are the dissipation caused by "flow" and the dissipation caused by
molecular rearrangemenats. These dissipations are the origin of the
dynamic measurements recorded during glass transition.
To the category of flow dissipation phernorena belongs the
instantaneous dissipation characteristic of flows caused by spatial
gradients; temperature gradients causing entropy floi, and
velocity gradients causing momentum flow. To the rearrangermnts
category belongs the relaxation of structures characteristic of
"thermostatic transitions."
Figure 5 is a mechanical representation of the concepts of
instantaneous dissipation, and delayed or relaxa>tional dissipation.
The dashpot Di, illustrates the concept of instantaneous dissipation;
the body mrrkcd :"solid," because of the lVIcc of a d .:; ipaivc element
Di in seric: with the elastic elcr .t K ; ,iil not c ::,''i'
.1
r
ii I~
:i i,, F.
<1K
L'
d
L
"fluid"
K< D
Cl d
sl:. i. "
''SOl T U
ad he ied
Figure 5.Machclanical ,:~
Dissi ipations.rs
OF InAZIM MOLci i
instantaneous dissipation and can eventually stand external stresses
withoijt deforming. However,! the delayed dissipation Dd indicates
that this body will show relaxational effects.
instantaneous 'viscous dissipation may exist only in Flow
situations; e.g., in the case of shear or elongational deformations
of fluids, and at the roving boundary of a solid. However, as il
lustrated in Figure 5, a body can be deformed without exhibiting
instantaneous dissipation. For example, the dissipations occurring
in either the isotropic change in volume of a fluid, or the shear
deformation of a solid are generally delayed by elastic elements;
these are the bulk and shear moduli respectively. These types of
deformations will be referred as nonflow processes.
The convenience of coceptually differentiating between instan
taneous and relaxational dissipations in glasses is based on the
connection of relaxational dissipations with changes in temperature
and density. Isotropic changes in volume and homogeneous changes
in temperature are delayed and rciaxational in nature. Glassiness
is generally described by the phenomena observed when changing
volume and temperature.
The definition proposed here calls for observation and study
of glasses in nonflow processes. Observation and study of glas
siness during isothermal and isochoric (constant volume) flow
processes is quite possible and convenient. However, this writer
believes that in order to connect glassiness to any kind of thermo
dynamic theory, processes where vclv's andr temperature changes are
studied in the absence of flo' represent the first rational
experimental step.
Processes where thermodynamic variables are changed at a
given rate, and processes where flow occurs at similar rates must
be compared experimentally, e.g., by compressing a 10 cc sample of
a material at 1 cc per sec and forcing another sample through a
1
capillary at a rate of shear of 0.1 sec In both cases thz rate
1
of deformation is 0.1 sec This should establish if the "time
scale" has meaning per se or only when related to flow processes.
Particularly, it will be important to establish whether or not a
a
homogeneous volume change occurs with dissipative effects.
B. Th:,' :d ,,ics cf OCJasiEl st i ri. l.
As mentioned in the introduction of this chapter, a thermo
dynamic theory is needed to develop the equations required for the
description and study of glassiness. This writer's contribution
in this and the next section consists on adapting an existing
general theory to provide these equations.
The usual theory of irreversible processes (Appendix B) may
yield useful guidelines for the quantitative analysis of glassiness,
and may even render results formally equivalent to those obtained
from a more complex theory. However, the explicit omission in
this theory of the effect of the past values of the thermodynamic
state upon the present values may lead the investigator using this
theory to define arbitrary terms to account for these history
effects.
aGoldstein's viewpoint on the subject based primarily on
Litovitz aco ic e. rin nt wts briefly discussed on page 29.
As discussed in the previous section, processes where no
instantaneous viscous dissipation are present may have special
significance for the study of glassiness; if no instantaneous
viscous effects are present, the theory of thermodynamic processes
of materials with fading memory or quasielastic materials,
constitutes the most general thermodynamic theory available. (See
Appendix B.)
A majority of the published experimental evidence indicates
the existence of a viscous response of glasses to changes in volume.
However, it is not an instantaneous viscous dissipation; the dis
sipation is always delayedd" by the compressibility of the material
(29), see Figure 6. Therefore, the theory of thermodynamic processes
of quasielastic simple materials should be applicable to glasses.
The most important results of the theory of materials with
fading memory by Coleman and Noll (l1), ,.12) are summarized by
the following equations from Appendix B.
A f(F, T) + HA (Ft Tt) (B18a)a
A = g(F T, t) (B18b)
diA dF + () dT + 2() dt (B18c)
T,t F,t FT
where
aAl equations de'eloped in Appendix B aie labeled with a 3
preceding the number.
40
LX
? AX 4. ~
I I
! /'////i/// Constrai t
1 L
  /
s ~ Figure 6a
1 I
Y I
I ./,/ ,/ t ,
' in Volume or Te, .rature
Figure G. i Co.istrai e<
a
P = p F+ (B19)
T,t
S  ) (B20)
F,t
D = p (B21a)
F,T
where
D =PTS div h pq (B21b)
The nomenclature as well as the meaning of these equations will
be reviewed in the following paragraphs.
Equation B18a expresses the principle of determinism in
materials with memory. The deformation F, the temperature T, and
the present instant t are taken as the independent variables (see
Appendix B). This principle is equivalent to the assumption that
the present state of the system is determined by its past. Specific
ally, equation B 18a indicates that for quasielastic materials, the
free energy A is given by the sum of two terms: f(F, t), the value
of the free energy if the material had been at its present state
forever, and HA(F, T) the contribution of the collection of
states occupied throughout the entire past of the material (F, T).
The influence of this past history upon the present will depend on
the material and is expressed by a functional relationship. A
functional, e.g., HA, assigns a number to the influence of the col
lection of past valucs.
aF+ is the transpose of F.
impied i n equation B1Sa is the assumption that the state of
the material does not depend explicitly upon past histories of the
time derivatives of the state variables, i., there are no instan
taneous viscous dissipation effects.
The transition from the past
to the prsert. is always smooth with no discontinuities of the
stress P, or the entr py of the system S (equations B19 and B20)
Equatio El18b is the instantaneous equivalent of equation
B18a. The form of the function g(F, T, t) will depend on the past
history. Hc ,ve:r, since the history has already impressed its ef
Fects upon t i material, g(F, T, t) becomes a smooth function of
time and of te instantaneous state of the system (V, T).
Equ3tI"on B18c is the mathematical statement of the sought
pcten~tiac' .::'tions B19, B20, and B21 are = sessionsns for t:h press
sure P, cr. S, and dissipation D in terms of the thermodynamic
potcntiala A. Th:3s, ions are valid at any i:;scIt thrc:. '.
the process. Of particular importance is the term 'tg of equation
B21a.
The
of entropy
the entree
div h r..
heat sorc
mater als
The effect
effects o0
process D
si':.iOcn D presents the excess in the rate of producLior
'r the entropy fiow du; to grad'ients in temperature pilus
rovlded by hcr ogeneous external heat sources. The term
ts th heat flux, and the teri q a homogeneous external
The significance of D to t th theory of quasielastic
that it rer'es.ent the effect of imsory upon th system.
moiy tn. free en .. given by D, deterrmir s all
o3 on P and S. For a "rev rs ible': tI;ie index ri:.i:
If time dependence on a nonflow process is assumed caused
by the delayed relaxation of structures within the material, the
dissipation will acquire the meaning of the rate of change of free
energy due to these relaxation processes. Equation B21b could be
written.
pT ,t Shist = D = pTS div h pq> 0 (27)
where atShist is the instantaneous entropy production due to relaxa
tion of structural changes.
The term, Shist, would represent the difference in entropy
between the structures present at time t, and those present if the
system had always been at rest at T and V. The structures present
at time t are only a function of the past values of V and T. The
system is always tending towards the order represented by the
relaxed state at T and V; this accounts for the fact that tShist
is always positive. This interpretation of the dissipation term
is not intended here as a quantitative theory, but rather a quali
tative model which may provide some aid for the understanding of
glass i ness.
Equation B21b explicitly states one of the most important
limitations for obtaining meaningful data. The material must be
either a perfect conductor or a perfect insulator. Otherwise, a
temperature gradient will exist throughout the sample, and a heat
balance at each point will be required in order to establish the
contribution of entropy flow to the dissipation.
If the body could be deformed homogeneously, the dissipation
could be determined by holding the deformation and temperature
constant at a given instant and studying the behavior of the system.
In this manner the contribution of memory to the thermodynamic
properties of the material could be found. This is another reason
for proposing that equivalence of glasses be defined through an
experiment where V and T remain constant. Such an experiment is
called a relaxation experiment.
Relaxation times and their measurement are the subject of an
enormous quantity of literature, from nuclear relaxation times in
nuclear magnetic resonance experiments to the rheological relaxa
tion time of viscoelasticity. Any of these methods could be used to
determine a relaxation time for the material as glassinesss" is
induced by cooling or c ,essing. However, one must keep in mind
that the condition of homogeneity must be approximated before any
of these relaxation tires can bc considered as an indication of
the structural changes. Also, the meaning of glassinesss" will be
related to the time scale of the experiment used to study the
transitions.
C. D. l.., n: of Lh Ecr.... ; rns to, h. UI d
fri; t r i, i, GL ;
Grarhical 1 1 f S ,
9" L. L r
According to the definitions and discussions of the previous
section it is suggested that a study of the riraning of "Lim'i scale"
and "dissipation" in noflo. processes clarify the nature of
glassiness in certain materials. Such a process is illustrated
in simplified manner in Figure 6. The process represents the
constrained onedimensional change in volume depicted in Figure 6a.
The entropy S, and the onedimensional stress P (pressure) are
chosen as the response to changes in the process variables: volume
V(t), and temperatures T(t). These state variables are represented
by the vector X; the response variables S and P are represented by
the vector Y.
The response resulting from a change in either of the state
variables is assumed homogeneous throughout the body. The changes
or displacement capable of producing mechanical responses in the
system are balanced at any instant. The pressure produced by
instantaneous elastic displacement LXe is the same as the pres
sure produced by the delayed relaxational change, AXr. The observ
able change in the therr~odynamic state, however, is the sum of these
two displacements.
AX = X, + aXr (28)
The elastic displacement can be conceived as the contribution
of numerous relaxational processes with negligibly small relaxa
tional times. The delayed changes could also be conceived as a
composite response. In the latter case, the relaxation character
istic would be given by a spectrum of relaxational times. The
minimum relaxational times contributing to an experimental observa
tion will be l ir ited by the response tir.c of the sens;n9 i nstrrment
and the duration of the imposed chan3o. The process contributing
to the maxiinum relaxational tiires on the other hand, will be limited
by the sensitivity of the recording instrument or by the patience of
the experimenter.
E c "io '
For the process represented i; Figure 6, the thermodynaTic
equations expressing the change in free energy A will be reduced to:
A = f(V; T) + HA(Vt TL)
(29a)
(29b)
For quasielastic materials A is a potential for pressure and entropy.
Therefore, these properties could b' used to specify the response of
the system.
The equations expressing the pressure of the system as a
function of thE voiu .i rerature history are:
= g(V, T, t)
P p(V, T) Hp(Vt, Tt)
= T(V, T, t)
d ( 4)
dt \
Vt
dt /i
VT
The tern
they represent
tir, a: V and
the pest volur<
EquatIo s 30 &
interpreter iot
energy c '':
p(V, T) is
the respon:er
Siril ia l
: 3 ar
o' i press
analogous to f(V, T) of equation 29a;
of the system after an infinitely long
. '" T ) r, nrcsents the effect 01
history from the relaxed state.
'essios of smooth'ness and p:'cvide
uC res rse terys as functions of fcC
(30a)
(30b)
(31)
For a quasielastic material:
P = (A/ V)Tt a
therefore, by substituting equation 33 into 32:
dP = (3 A/ AV) dV j /1l./_L dT 12AA/ nV'
dt aV dt 5T dt 3t
Furthermore:
3(3A/ OV)t I
V t
IV,t
2I
S 2A 2i A/ TIL Vt
TdV 3 VI T, t
T ;t
T,t
a(A/3 V)T.t
 11.__
V,T
_ 32A
at V
because of the properties of exact differentials. Therefore,
equation 31 can be rewritten as
d P .P 2( J )
dV +
d l
dt DV T
T,
(36)
aThe r i us sign is due to the face that pressures are now
considered itive; while in previous discussions the stress 'as
considered positive wan exerting a traction.
(32)
(33)
(34)
(35)
a V
T,
S(A/ Vt I
or in more familiar terms,
V +
P= K +DV (37)
where the instantaneous icchain;cal bulk modulus for the process K
equals V(aP/aV), The process instantaneous latent heat of expan
sion (thermal bulk modulus) qV equals T(S/DV); and DV represents
the instantaneous loss of rate of work (power) for isothermal volume
changes, DV 3((A/t) /3V.
It should be noted that all the coefficients in equations 36
and 37 represent rates of change with respect to the bulk (volume)
of the system determined while holding the temperature constant at
a particular history. These "process" coefficients are material
properties determined by the constitutive equation of the material.
In addition to pressure, entropy is needed to express the
overall response of th process depicted in Figure 6. The equations
expressing the instanane us rate of change of entropy are:
S + T + (38)
T,t V,t V,T
where
as ai (^ _ iA\
Sv v T I V'/
T, t Tt V, t
t(3
V,t
SC (40)
a T/ T
V,t
and
t) ; (41)
'V,T IV, T V,
therefore, the rate of change of entropy can be expressed in terms
of coefficients which represent changes with respect to the temper
ature of the system evaluated while holding the volume constant, at
a particular history
S ( ) T V TJ T + ) (42)
V,t V,t V,t
or
S = + C. V + CV D (43)
B T
Where a : Q and CV are the instantaneous, historydependent values
of the expansivity, compressibility, and heat capacity respectively.
The last term DT represents the instantaneous loss of rate of work
during isochoric temperature changes. Both the thermal dissipa
tion DT and the voluretric dissipation DV will be discussed in more
details in the next section.
Summarizing, equation 36 indicates that the volume dependence
of the responses of the system determines its pressure; equation 42
indicates that tihc teerperaLure c :,cdence of the response dter'rins
the entropy of the system.
Dissipative Terms
The dissipati/e terms Dy. and DT are defined by equations 31
and 38 respectively. Explicitly:
(44)
DT
tC;
However, these parameters were rewritten in terms of the rate of
sipation of total work (free energy). Equations 35 and 41 state
Dv = _ A\
DT : A 8 A
T \t
T T t
(45)
dis
(35a)
(41a)
where
aA  D ( 0
at
The interpretation of 3 and ^ needs some comments. The develop
ment of equations 35a and 41a required the interchange of partial
operators, e.g.;
D r\
v;, ( 
0* k *1l
2A 
:* uVi
fa "c
NVVty
and
Therefore, in order to provide a better understanding of the instan
taneous dissipation of the response variables, their relation to the
history term must be discussed. Pressure will be chosen to illustrate
the discussion. Differentiating equation 30:
d P(V, T; V Tt) = d P(V, T) 4 L Hp(Vt, T ) (46)
dt dt dt
or
tV T tVt t
Differentiating a functional, e.g., Hp, is not a common operation
and no tabulation of derivatives can be found. However, conceptually,
differentiation of a functional implies the same steps required for
differentiating a function, i.e., incrementing the functional by
incremernting one of its arguments and evaluating the ratio of these
two increments as the increment of the argument approaches zero (2 .
For example,
9HP(Vt Tt) d Hp(Vt Tt ) E 0 (48)
qt d E
where E represents a finite arbitrary increment in the history Vt.
Equation 48 must be evaluated st a constant temperature history.
The meaning of the dissipative response terms can be now
established, e.g.,
/cP V + v T
Sty' V (49)
V,T T V. V,,T
since at constant V and T the first two terms of equation 47 vanish.
However, because of the smoothness condition (quasielasticity),
\9t/ VT \t V,T (50)
By equating equations 49 and 50 it is evident that the dis
sipation terms represent the instantaneous rate of change of the
response due to the influence oF the history of the thermodynamic
state, at the particular instant under consideration.
The signs of the dissipative rates may be of particular interest
for interpretation of experimental data. In order to establish the
parar eters which may co~nroi t cse signs in a particular process the
relationship of DV and DT to fre: energy must be studied.
Using equation 49 and substituting the definition of P for
a quasielastic material the following equation is obtained
IVro r bt cn t T (51
Furthermore, by changing the order of partial operators;
(,_ .". /, Q. (5\)',
\ ; T (52)
or
D 3HA t (__'A t
 D (53)
An apparent result from this equation is that the change of free
energy of the material with respect to the history of a state variable
is always opposite in sign to the rate of change of that variable
during the process; the material is not expected to gain free energy
as a result of remembering the past. This result could have been
established directly as a consequence of the ClausiusDuhem in
equality (see equation B31b).
In order to obtain information on the instantaneous dissipative
change in pressure, it is necessary to determine the effect the
history of the material has had upon the volume dependence of the
rate of dissipation of free energy.
An equation similar to equation 53 can be obtained for the
entropy dissipation term.
D T FT V (54)
This equation indicates that the dissipative entropy term will depend
on the manner the history affected the temperature dependence of the
ra;e of dissipation.
Both the "freevolh '" and the "cooperative" theories predict
thLat terpsraLures and volu, wiill increase the rate of dissipative
processes. However, the concern here is not the effect of tempera
ture or vaoumr upon the dissipation rates in general, but specific
ally, the temperature and volume dependence, after a given history,
of the dissipation rates.
Up to this point, the thermodynamic theory of quasielastic
materials has been of help to describe and analyze the responses
of materials undergoing a specific process. in order to progress
further into the explanation of the magnitude and sign of the para
meters (partial derivatives evaluated at V, T, and t) describing
the process response, a materials theory is required; perhaps, many
materials theories will be required. However, before a materials
theory is proposed it may be convenient to determine experimentally
the behavior of the material of interest.
CHAPTER iV
PROPOSED GE 1AL DESCRIPTION OF GLASSINESS
In Chapter Ill, pp. 353, it was stated that glassiness could
be detected and studied on flcw processes. However, the quanti
tative definition given in that same secLion required that the
measurement of the characteristic time be performed in the absence
of instantaneous dissipation. This requirement implies that in
order to be certain that the anomalous rise of shear viscosity of
a certain material when cooled is caused by the same phenomena
causing volume delays, nonflow relaxational experiments must he
perFormed. These experiments are necessary to establish the relative
magnitude of the glassy phenomenon versus other type of interactions
which may be present in flow processes.
It is this writer's opinion that a material may exist which
exhibits a large glasslike dependence of shear viscosity upon
temperature but does not show marked glassy mechanical effects in
a nonflow process. For example, polymer melts possessing a wide
molecular weight distribution, or a wide spectrum of chemical
species, may behave differently in flow and nonflow processes. In
a flow process, a large shear viscosity may be caused by steric op
position of large molecules which may be partially crystallized (or
in a solidlike partially ordered structure). This same polymer melt
may not show a correspondingly large volume viscosity in a nonflow
process.
If the "ordering" of the above material occurs without ap
preciable chan3r in volume, no significant mechanical effect may
be noticed; i.e., the stress required for ordering or crystalliza
tion may be negligible. However, the entropic effects of ordering
may be important. Furtherr:ore, these effects (energy exchange wich
internal degr:c s of freedom) may be delayed. If this were the case,
temperature equilibration would not be instantaneous. For such a
material, glsssiness would be observable only by these temperature
delays.
The discussion on the preceding paragraphs is ir'ended to bring
to the reader's attention the possibility that mechanical effects
observed in isochiic flow processes may not have a mechanical
counterpart in homogeneous nonflow processes. Large viscosities
do not necessarily c use glassiness; they may be indirect manifesta
tions of the phenoir na causing glassiness. Glassiness, as defined
previously by this writer, is the manifestation of thermodynamic
relaxation.
In this chapter a general description of glassiness will be
proposed which, hopefully, will allow determination of the thermo
dynamic data necessary for the corroboration of existing theories of
glassiness, or for the pr,. ..l of new ones. The contribution
intended in this chapter is to grc.'iically describe glassiness in a
way consistent to the thermodynamic equations and concepts discussed
in Chapter III.
A. Glassiness and Excess Thermodynamic Functions
Changes in Voiume in Unconstrained
Materials. "Constant" Pressure
Ex er imTnts
The usual type of experiment encountered in determination of
glass transition temperature consists of rapid cooling of the sample,
and recording of its volume as a function of temperature at a
constant arrbient pressure.
The description of the deformation undergone by a material
contracting in such an experiment may be quite complicated, even
if a uniform temperature throughout the sample is approximated.
For a liquid, whose viscosity may be negligible, uniform
temperature insures a local density in equilibrium with the tempera
ture at the pressure of the experiment. However, as the liquid turns
into a glass and relaxational phenomena become important, a change
in temperature will not be followed by the equilibrium change in
volume. More importantly, a pressure in excess to the pressure cor
respording to the equilibrium situation (infinitely slow changes)
will appear. This excessa pressure is caused by the now significant
opposition to volume changes. The pressure in general will be
distributed nonhonogeneously throughout an unconstrained sample.
Force balances must be satisfied at the free boundaries, causing
shattering of the glass in so:r extreme cases.
aA more specific definition of "excess" will be g;ven in the
next section.
If the purpose of a nonhomogeneous unconstrained experiment
were to determine a volume viscosity, it would be necessary to obtain
a description of such a chaotic phenomenon. Otherwise, shear resist
ances present in all nonisotropic volume deformations will be
included in the volumetric viscosity.
E_ C.e TI " .,' J cx F,r.,i ions
A glass was defined as a material whose thermodynamic properties
depended upon the prior history of its thernodyncmic sLate. The
dependence on the prior history implies a smooth transition from
the past to the present. The following equations express this
continuity of the response of the system on the past values of V
and T.
P = p(V, T) + Hp(V , Tt) (30)
t t
S s(V, T) + HS(V, T) (55)
P =( v +DV (31)
T,t T,t
S : ) V T+DS (L42)
V,t V,t
Equations 30 and 55 indicate that the response of the system
to the history of its thermodynamic state (Vt. Tt) is th'. sum, of
two contributions: the pressure or entropy of the static state
(V, T), and tl ccnt ribt ion of 11 the th dynii c states prior
to the present one (V, T ), The contribution of the past states
may be conceived as an "excess" thermodynamic function, an excess
over the values of tht thermodynamic state had the system been at
the present values of V and T, always. This static state will be
called the rest state.
The rest state is, of course, a hypothetical or fictive state.
However, it could be conceived as either an equilibrium state or as
a relaxed state. Therefore, the rest state should be treated
experimentally according to the choice made.
If the system had always been at V and T, it would have never
experienced an excess thermodynamic function, in this respect, if
the thermodynamic state at V and T were approached without ever
experiencing an "excess," i.e, by an infinitely slow process
(thermostatic), the rest state could be conceived as an equilibrium
state. On the other hand, if a material had arrived at V and T
through a dynamic process, it may have experienced excess thermo
dynamic functions. In this case, if the material were held at V
and T indefinitely, the excess functions would dissipate. This,
experimentally obtainable, relaxed state could be conceived as
the rest state. The rest state would be the present state (V, T,
t) as t> .
The equilibrium state and the relaxed state are not neces
sarily the same. The changes induced in the material during a
dynamic process may be qualitatively different from the changes
occurring had the process been reversible. Furthermore, each
history may ihve caused per r nent c,:,rn.s to occur within th;.
material, in which case the relaxed state may be different at V
and T.
However, a cc'ronise is possible. For the purpose of il
lustration, it v ill be assumed that the relaxed state at V and T
will be the sape regardless of the history, had there been a history.
This is similar to the principle of static continuation or stress
relaxation of Coleman and Noll and is implied in the thermodynamic
theory of quasielastic materials. The validity of this assumption
could be determined experirmntally,
Experimentally, the relaxed state is chosen as the reference
(static) state for equations 30 and 55. Figures 7 and 8 illustrate
a
the concepts oF the relaxed and the equilibrium states. The curve
ABC in Figure 7 represents an isothermal homogeneous compression, at
a finite rate. The compression process has been stopped at Vf and
Pf and the material kept at a constant Vf, and temperature. Curve
CDE represents the relaxation of the excess thermodynamic functions
(e.g., excess pressure). Note that the 'sciss3 has been changed
from a volume scale to an arbitrary time scale to illustrate the
relaxation phenomenon. The pressure at t  oc tends to a value
Pr. This value represents the relaxed value. The value AP is
the contribution of the history to the thermodynamic state of the
system at Vf and T. Curve A'B'C' represents a process similar to
ABC, but at a higher compression rate; AP' is now the contribution
a
For illustration purposes pressure will be chosen as the
response of the s' tem.
I.''
I t:
SI i3
i 11
Iii
*1r
'N.I
C.. I n
N 
N Cl
N.C
~1....,
cr N. , t
IC)
4'I C
I,1
1:4
i IL
oC. ar lJ (,, n/) =) :LU
 ~ .~ 
Q)
* Il
w
DD
0J tI
I.
a
Cr
Cr'
a
u \
K
N
2.
KN ~T\\
S \ \
t
LI
'4
I
1u
I
r,
C)
11
iU
r, ;
^
\\
S1r
a a
a" '
V  C
P 
 fl, uj L" vC: &C U
.. 
of the history represented by A'B'C' to the thermodynamic state at
Vf and T. It should be noted that Pr has been assumed the same
for both processes.
Figure 8 illustrates the concept of the equilibrium state.
The curve ABC represents again a dynamic isothermal, homogeneous
compression. The relaxation processes are new represented by the
lines at constant volume. The points Vf i Pf., Pr where i 1,
2, 3 represent the values of the final volume, final pressure, and
relaxed pressure for three processes at the same compression rate,
but ending at arbitrary values of V. The line uniting the final
relaxed states (Vfi, Pri) represents the collection of reference
states. The line XYZ represents the thermodynamic states of the
system if the compression proceeds infinitely slowly. The "equi
librium" process represented by XYZ will not necessarily coincide
with the relaxed states (Vf., Pr.) The "equilibrium state" for
a glass may not be real. the relaxed state is.
The concept of values of the thermodynamic functions in excess
to the values of the thermodynamic real relaxed state will be the
basis for the description of glassiness, in accord with this
author's proposed definition.
Graohical Description of Glassiness
Pressure will be chosen to illustrate the proposed description
of glassiness. it must be recalled that a history which did not
affect the volume dependence of the dissipation rtLes wili not
produce an/ excess pressure. Hoe.'ever, such a history r,'y produce
an excess entropy if it affected the temperature dependence (see
equation 47).
A glass is represented in Figure 9 as any point above or
below the plane H = 0. The plane Hp = 0 represents the relaxed
state of solids and liquids. Two processes are illustrated in
Figure 9. The path I S A in the zero excess pressure plane is an
infinitely slow process allowing crystallization at Tc; a AV of
crystallization is shown. The path LGI G2 represents a prepara
tion process of a glass G2, whose relaxation processes have become
very slow, aic could be considered an unrelaxed "solid" at G2.
The increase in Hp with time along LG1 G2 is a consequence of
memory. As the glass moves along LGIG2 it remembers its past. If
a path of increasing pressure such as LGIG2 is chosen, the material
will "reremberr" state of looer pressure than the present one. When
the process is storld the material will tend towards its past states
of lower pressures. Therefore, the excess pressure over the relaxed
state must be represented as a positive value. Mathematically,
the path LGlt represents the sams process except the preparative
process ends at a higher temperature and volume than before. At
these conditions the relaxation tiras are small enough to allow
observation of the relaxation process along GCt. The relaxing glass
at t will eventually rest at the plane Hp = 0. The corresponding
relaxed solid is represented by S1
Figure 10 illustrates the dec pressing and heating of a glass.
The rela ed solid Sr. (0, T) represents the relaxed siate of tl.e
4h
2 (2,"2 te
att
T, Terp.
> IA
ci
/ x
:1 C)
I /l
t 1 N
u, 6 'n
_ _ _ _
ii 1/
i? __I I~, `
'
Figure Frtxce~; ucctions an 0 1as '. as
,4
*i
t]
G
 T, Temperature
Sr (o., T0)
/
/
/
/
/
Figure lO.Decompress ion anc Heati of a Glass
I
'.
~1
4
/ i
j '
S
I
glass G. The path L Sc A is represented again to provide the reader
with a graphical reference to the previous graph. The effect of
history is represented as a negative parameter going through a
minimum and disappearing as the liquid is approached. The negative
effect of history can be explained by remembering what was said
about memory. In this case, as the glass moves along a decompress
ing path it will remember a state of higher pressures, and if the
process were stopped, the glass tend towards these states. The
minimum in the GL path and the disappearance of the excess pressure
as the liquid is approached are consequences of assumptions about
the nature of memory. It is generally accepted that the relaxation
processes become faster as the temperature and volume are increased.
However, the faster relaxation processes do not necessarily mean a
decay of the excess pressure. It is only when the excess free
energy, brought about by the finite rate of change of the process
variables, can be dissipated faster than can be accumulated, that
the excess terms will start to decrease. As the liquid is ap
proached both factors contribute to the disappearance of the
history term; the relaxation rates are faster, and there are less
significant structural changes contributing to the excess free
energy.
Figure 11 illustrates a simple type of experiment which can
be used to study the influence of volune changes upon glass transi
tion; these are isothermal, finite, dynamic compressions. The
importance of these experimerrs is the possibility of direct
i' Vt r2 tp
Vv, T, !t
Vf T' P
I /t ,
ST em .
d, p/
i / / Vi n '
o i / 2/o. 0 /
I, T j
I/p I'
S! // I i
!< ,
.i i
Ui i
!i i / z
/ / / I/ i;
r 1/ ..
Coripres es onis.
determination of a volume viscosity. A volume viscosity could be
defined by using equation 30 restricted to isothermal processes,
Hp
i.e.. =  
viri "
V/V
The viscosity n is directly related to the thermodynamic
parameters used to describe glassiness.
Furthermore, by performing these experiments at different
temperatures, the temperature dependence of the mechanical response
(pressurevolumne) of the system can be determined. This temperature
dependence should establish in part the contribution of entropy to
the r.echanical responses. This contribution versus the contribution
of free volume to relaxation at constant temperature constitute the
essence of the dispute between the existing theories of glassiness.
The contribution of free volume to the rate of relaxstional processes
at constant temperature can be determined by studying the relaxation
of the excess pressure at different volumes. These relaxational
processes are illustrated in Figure 11 by the processes at temperature
TI. The broken line connects states at different volumes and which
have been relaxing for the same time after preparation. Again,
faster relaxation is expected towards the liquid at higher volumes.
The processes illustrated at temperature T2, represent a direct
manner of studying the effects of history. Varying the preparation
time tp, the initial volume Vo, and the direction from which the
final state is approached are different manners of studying history
effects.
Isobaric Transitions
The last section dealt with general cons iderat ions and results
of an interpretation of giassiness limited to "nonHlow processes."
This is not the type of process used in practice to obtain informa
tion about glass transition. In general nonhonogeneous flow proc
esses are ust:d to obtain such information. A detailed description
of such a process is not available. Furtheri.ore, because of the
difficulties in describing the deformation mathematically, and the
complexities of the boundary value problem necessary to interpret
the experincntal results, a detailed decripticn may never be avail
able (Appendix B).
However, most experiments of glass transition are carried out
isobarically in the sense that the samples arc constrained only by
the ambient pressure. Therefore, the study of the behavior of quasi
elastic materials in he; :ous isobaric processes nay help to
understand sorc of the pheno ~ana present in. more co pl icated pro
cesses.
Following Davies and Joies in their twodin.'nsIonal description
of glassiness, the first characteristic to note is the dependence of
the onset of glassiness on the rate of cooling. If a liquid is
cooled at constant pressure and at a rate sufficient to avoid
crystallization, it follow, the path ABC, see Figure 12. The
pressure at C' would gnerraily be lower than the pressure at C.
It must be noted that although the process pressure P is constant
in an isobaric process, the excess Hp, in gcn erai, is not.
Creep
0 DE
u Ci
XI
1 i :4,
Lu iP."
/i /
Relaxation .TerpI C~ie pera tur e ,
//
Delayed,I C
V ~ i M , v o u e Ji n cr ea s i ngg r a
B I
Ii
~1
j
Ui
g`
4.I
4.
Li
fj.
Figure 12.Glassiness during Ischaric Processes.
(I~~b?:
Liquid
I
ii
~11
44
ri
*.1
;I
Ii
'1
I ;i
Davies and Jones indicated that as the cooling rate increased,
the glasses formed at higher temperatures. This observation is rep
resented, using the description proposed by this writer, by the
paths ABD and ABE. It must be recognized that as the cooling rate
increases the excess pressure also increases; these effects have
been considered irrelevant to the observation of glassiness. It
is this writer's opinion that the cause of the observed rate
dependence of glassiness at constant pressure is the delay in
volume caused by dissipative processes. The increased "sharpness"
of the curvature induced by higher cooling rates is illustrated in
Figure 12. Mathematically, the delay in volume, for a cooling
process at constant pressure (P = 0) represented by T = T(t) is
given using equation 31 by:
V = ( VS T + DV) / 3VP (56)
The first term in this equation represents the instantaneous
(equilibrium) changes in volume. The term DV/ 9VP represents the
delay in the rate of change in volure due to the presence of dis
sipative processes. This delayed volume change is made evident if
the sample is held at constant pressure and temperature after
deformation. The phenomenon is called creep, see Figure 12.
CHAPTER V
DEVELOPMENT OF EXPERIMENTAL TECHNIQUE
The contribution of this research in the experimental field
consisted in the development of a method to study homogeneous changes
in voiumens of asphalts without the interference of shear viscous
forces. This technique can be used to study the transition of a
substance from a liquidlike to a solidlike material, and the
influence of the volume and temperature histories upon this phenomenon.
This chapter will deal with the details involved in the develop
ment of the experimental technique. The comparative discussion of
some experimental results on the selected asphalts will be presented
in the next chapter. The analysis of this general type of data
will be presented in Chapter VI. This analysis will point to ranges
of temperatures, pressures and densities where further studies should
provide more specific information about the behavior of the different
asphalts.
Muller (33) in a recent review on the thermodynamics of
deformation and the calorimetric investigations of deformation
processes, discussed the current state of the art on experimental
methods. The author presented numerous examples of efforts to
understand the causes of the mechanical behavior of materials by
measuring thermomechanical effects. The work, however, concentrate
on the discussion of elor.gation calorimetry and other experinrelts
at constant volu e. This author provides extensive discussion of
73
the different thermostatic effects which could be studied with an
elongational calorimeter designed by Engelter and himself (17).
Dynamic effects are Jiscussed only qualitatively since this calori
meter method shows a considerable delay between the time a deforma
tion is induced and the time when heat effects are recorded. How
ever, this author recognized the importance of irreversible effects
and proposed the use of the calorimeter to determine permanent
final effects (state differences) caused by deformation, e.g., the
degree of crystallization induced by the deformation.
Muller also describes cc',ro;sive experiments where temperature
variations have been recorded in order to study thermonrechanical
effects. However, this author dismissed this approach because all
att ts to place a thermocouple inside the sample were unsuccess
ful and because, "in addition, the degree of deformation during
compression of a cylinder is knom to vary considerably from point
to point (flow cone formation is a wellknown phenomenon)" (33).
The compression technique used consisted of compressing a sample
cylinder (approximately 2 cm. in height and 2 cm. in diameter)
between two plates. The sample was not confined or restricted to
flow axially; the cylinder was defcrr:ed to the shape of a barrel
by compressing up to about 20 per cent compression. Shear viscous
forces must have baen present on this nonho ogeneous deformation.
The experimental method proposed herein to study the thermo
dyna ric properties of asphalts in nonflow processes consists of
co rn essing a cylindrical s~ :;le oF approxirateiy 1 to 4 inches
long and 3/8 iichs in diarc er in a confini steel barrel. The
asphalt specimen is enclosed in a rubber (latex) balloon which in
turn is coated with silicone oil to reduce the drag between the
balloon and the steel barrel. The balloon enclosure is leak proof.
A calibrated thermistor or thermocouple can be inserted to determine
the increase or decrease in temperature due to the heat of compres
sion or decompression. Figure 13 shows a line sketch of compression
equipment. A great deal of work was involved in the development of
the above technique (see Appendix A).
The improvement to Muller's (33) and others contributions
consisted on implementing through persistent experimentation a
method of reducing the effect of shear forces in the study of dynamic
compression. The motivation behind this effort was the firm belief
that foremost to any conclusion on the causes of the socalled glass
like phenomena discussed in previous chapters was the necessity to
describe the changes undergone by the material. This writer suspected
that some of the effects attributed to delays in volume were caused
by the shear forces present on any nonhomogeneous volume change,
e.g, those causing the "Wjellknown flow cone formation" mentioned
by Muller. Therefore, reduction if not elimination of shear forces
was considered the first step towards the description and study of
processes involving volume changes.
This chapter will consist of three sections. A section
containing a brief description of the apparatus used to implement
the technique and the list of physical properties of asphaltic
materials employed. A second section ; containing sorO p2iiliminary
COMPRESS ION LOAD t
CELL (FR) _ !I
MOVING CROSSHEA.D__ L 
LOAD CELL EXTENSf1 ON__
SUPPORT
PLATE J
BARRL    ^
~161
BARREL ~6'
I I
PLUNGER 
ASPHALT
PLUG L
CLAMPI NG NUT .
Figure 13.Line Sketch of C
session Equ ipnment.
considerations, and finally, a third section where the technique is
analyzed and data are presented in support of the premise that the
deformation obtained is indeed homogmner.us, and that shear inter
ference is e lmi.nated.
A. Materials and Awoaratus
Four asphalt cerents were selected to test the applicability
of the dynamic compression technique: Texas SteamRefined inter
mediate (S639), LowSulfur AirBlo',n Naphthenic (S6313), Los
Angeles Basin (S6320), and a United States Bureau of Public Roads
asphalt, especially chosen because it had a considerably lower
penetration than the other three. The USBPR identification number
of this sample is B 3057, the identification number given by the
University of Florida Asphalt Laboratory is S6447. Table 1 suim
marizes the physicochemical properties of these asphalts.
The apparatus used to compress the materials and to register
the required forces was an Instron Universal Testing Instrument
(Model TT L C, Modification M31). A detailed description of
the parts and capabilities of this machine can be found in a thesis
by Ronk (39). Ronk's work was part of the concerted effort being
made at the asphalt Laboratory of the University of Florida to
establish the basic characteristics of these materials (see Ap
pendix A).
Figure 14 shows the Instron Universal Testing machine.
Compression of the so ple is accoiplishacd by lower;ng th: traal
ing crossheaj towards a fixed tbLe sL' *,ortin a irncif ;n.1tron
Table 1
Properties of the Four Selected i'phalt Ce: ents
Identification S639 S6313 S6320 S6447
Density (250C), c/cm 1.033 0.M88 1.015 1.025
Penetration (250C) 85 89 89 14
Ductility (250C) cm. 150 150 150 150+
Softening Point, OC
(Ring and Ball Mhthod) 46.1 48.3 46. 
o
Glass Transition Point, C
(Penetro eLer method) 13.9 10.8
Viscosity (250C), poise 1.02010 1.05x10 0.66x06 
Viscosity (60C), poise i.70x10 1.73x10 1.11x103 5.22x103
Molecular eichc 943 939 76 
(iNumber avcr )
a
Generic Groups
(Per cent by weight)
ParaffinicNapht' hnic 13.1 23.9 10.3
Nahth nicAiro Lic 28.3 23.6 24.2
Heavy Aro atic 43.7 37.6 52.3
Hexasph ltenes 15.8 12.1 11.8
Pecrolcncs 85.1 87.3 90.0
Based on Sch, *erChiplcy s araticn proce:ure.
1*4
Figure l1. lnstrcn Universial Testing IHaclinie.
Capillary Rheoreter Asserbly (Figures 13 and 15). The compression
rates used ranged from 0.5 in/min to 0.0002 in/min. The travel of
the crosshead and therefore the actual length to the sample can be
determined .ithin four tenthojsandths of an inch. The force ap
plied with the piston to compress the simple is measured by a
calibrated strain gaje cell which provides an accuracy of 0.25 per
cent of the scale used. There are six scales ranging from 0200
to 010 :'0 pounds. The modified Instron Capillary Rheometer Assembly
was enclosed in an environmental chamber for temperature control.
The operating temperature range of this chamber is from about 40
to 50 degrees Centigrade. Cooling capacity is provided by two
standard refrigeration units, and by liquid N2 injection. Excess
cooling capacity is balanced by a variable 1000 watt heater. Precision
control of the temperature is achieved with a De:otensky sensing
element, an onoff controller, and a 100 watt light bulb. A high
velocity air recirculating system (100 cuft/min) provides a uniform
temperature (0.10C) throughout the chamber. The temperature of the
barrel can be controlled to 0.10C. A thermocouple inserted in the
barrel indicated that the temperature of th' asphalt when in equi
librium was independent of the axial distance indicating that heat
transfer through. the bottom plug and the compressing plunger was
negligible (39).
B. Prelimin rie
This section describes stUedies i F :o bs ic questions concern
ing th,; e. rir 1 ntal lsthoc: (1) tih e'f: ci oC '"ey; .: o tj re
0.3753
0.12
1.125
0. 7D0 A I d;mens ions
i
I inch
o .146
V I
I 0.3 65
0.11.12525
I I2 I '
I i
16 
Di720 enion of its Parts.
1 iin inchc!s
i 1 021465
4 I L I _
\ i n 16 
Figure 15. "Diss3sebebled Modi ifi Capfllary Rheometcr and
D0rresi ons of its Parts.
temperature history from the time the sample is prepared to the time
testing begins; and (2) the effect upon the results of the machine
characteristics (e.g., elastic deformation, tim responses, etc. of
the metal components).
Age. Pr a tion of the Sanmo
_a . .= .  = 1
Studi:s by Ste'wat and this writer during the development of
the final experimental method proved that the timeteinperature
history of the asphalt samples from the time of preparation of the
sample to the time the testing began was of no relevance, provided
that the asphalt samples were not heated above temperatures of 300 F
for extended periods of tire (see Appendix A). Heating for 15
minutes at 320F has been found satisfactory to melt the samples
without affecting the physical properties.
The treatrrnt of the sample prior to testing consisted in
pouring thr molten sample into the latex balloons, sealing the
balloons, and setting the sale for compression (see Figure 13).
The specific rnner in which the balloons were sealed depended
upon th: objective of the run, and will be discussed later. The
samples were sor etines held at room temperature for various lengths
of time before running with no apparent change in their compressive
characteristics. Table 2 summrizes the result of three runs on
the same s<.. ,le ten days apart. Before discussing the results
presented in Table 2 the origin of these nurim rs Jmust be discussed.
The results p rscned in Tc 1', 2 were obai by calculations using
Table 2
Effect of Aging
Sample S6309
Run No. R7015
Reference (XLI, 99100104)
Date
1/31/70
p
0.
185
615
923
1000
p
1.0303
1.0390
1.0562
1.0666
1.0690
B
4.49
4.16
3.42
2.90
2.78
P pressure (atm.)
(atml x 105)
Temp: 250C
Velocity of Crosshead VXH:
0.1 in/min
Date
2/5/70
p
1.0294
1.0380
1.0553
1.0660
1.0686
Date
2/10/70
B
4.48
4.18
3.50
3.03
2.91
p
1.0297
1.0382
1.0557
1.0662
1.0686
B
4.59
4.25
3.48
2.95
2.82
p : density (gm./cm3), = compressibility
~~ '~ LIII
. _f_I~~ 
three types of data: (1) the weight of the sample determined
directly, (2) the length of the sample determined with the Instron
machine, and (3) the change in length versus load requirements,
obtained from the recorder chart.
The weight of the sample can be determined at the beginning
from the weight of the balloon and the weight of the sample plus
the balbon. The sample weight can also be determined at the end
since the simple and balloon are easily recovered in mot ca~es.
Weights are determined with an analytical balance and are considered
accurate to within 0.1 per cent.
The length of the sample is obtained by direct measurement
with the Instron after the sample is in place. Althoh the
travel of the crosshead is accurate to within foLu tenthousandLhs
of an inch, it is believed that determination of the initial length
is the most important source of error in the determinLio of
densities. Improper tightening of the clamping nut and other
threading attachments have been found a common source of error.
However, after the sample is in place and an initial read
ing has been accepted the deformation of a sample is reproducible
to within four tenthousandths of an inch. For example, a typical
asphalt sample may show a deforvration when compressed between the
4
load limits of the ass mbly (02r13 Ibs.) of 1000 x 10 inches.
The load measuring device provided an accuracy of 0.25 per cent of
the scale used as rcentioned before. Densities are calculated
directly From tho data isLetirncd above after correcting for machine
deforation (see next section). Compressibilities are calculated
from an analytical expression obtained by fitting a polynomial to
the deformation versus load data obtained from the recorder chart.
After trying polynomials of the first, second, third, and fourth
orders it was found that the second order polynomial gave the best
fitting with an error never larger than 0.01 per cent between the
calculated values and the recorded values.
The results in Table 2 clearly illustrate that the age of the
sample had no effect on the results. The variations on density and
ccmpressibilities are well within the experimental recording errors.
The correspondence of the values of the compressibility is a
sensitive test of reproducibii L s;in these values are obtained
from the dcirvaT:ive of the experimental curves. P~srticuLarly the
values at 0 and 1000 atmospheres which are calculated by extra
polation are very sensitive indicators; small differences in the
data would be a!mr :ied when comparing the slopes at these extra
polated points. Although data are presented in Table 2 for one
asphalt similar results were observed for the other three samples
used.
The practical importance of the results of this section is
that a sample f asphalt can be poured, and be ready for testing
within a few minutes or stored for study. This permits preparation
of several samples simultaneously a,,d saving of time if the
procedure were used routinely.
A .1 for tic
Ronk esti mted the deformation of the rheorneter assembly when
applying a pressure (39). However, the effect of different ccrores
sion rates was not determined.
Calibration of the machine consists on compressing a piece of
steel and recording the deformation versus the load required. No
differences in the calibration curves ware noticed when compressing
steel samples at rates ranging from 0.005 in/min to 0.2 in/min
(::108).a Figure 16 is an actual tracing of load versus deforma
tion at two different rates of compression. The recorder chart was
rolled back to match the starting point, the chart speeds were ad
just"d so that for both cases ane inch traveled by the chart rep
resented 0.01 in of travel by the rovin c> osshd The two curves
coincided. Coating the barrel with a lubricant did not affect the
ca i brat i ns
The correction due to the deformation of the asserbiy was
calculated from the deformation of steel cylinders at different
lengths (XL I.34 47). The correction was calculated by assuming
that deformation of the assembly and that of the steel samples were
additive. The deformation of the steei ser''ples .'ere calculated
directly from the applied load and the bulk ic Siu us of ste'. By
References designated by a roman nurieral folice.'d by an
arabic nir',,ber refer to pages (arabic) in the research notebooks
(roman) of different investigators working in the Asphair Laboratory
of the University of Florida.
I  VXH 0.2 in/min
I /
/
/
6I
1 2
10ormat n (inches x 10
/
81
1 /
at Two tes of i
21 /
S/
i i4'
D Deformmtaon (inches x 1bao
Figure 16,Tracing of Lo.,U rsus Deformadin for Ca !Sration
ai: Two *ates of .iiiprese ion.

PAGE 1
DYNAMIC COMPRESSION OF ASPHALTIC GLASSES By J. CARLOS BUSOT A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1970
PAGE 2
iigii"
PAGE 3
A Ml ESPOSA ESPERANZA
PAGE 4
ACKNOWLEDGMENTS The author wishes to thank Dr. H. i . Schweyer for his helpful guidance in directing this research. The author is also indebted to his Supervisory Committee for their counsel and criticism and to Dr. M. A. Ariel for his motivating discussions during the development of thi^> research. The economic assistance of the EÂ„ i. Du Pone da Nemours Company i:gratefully acknowledged. i i
PAGE 5
TABLE OF CONTENTS Page i i i vi i vi ii xi ACKNOWLEDGMENTS . LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER I. INTRODUCTION 1 A. Glassiness ..... 1 B. Objective of this Research 4 II. CRITICAL ANALYSIS OF CONTEMPORARY THINKING ON THE QUANTITATIVE DESCRIPTION OF GLASS INESS . 6 A. Description 5 Inducement of Glassiness 6 Single Point Measurements 10 Thermodynamic Description of. Glass i ness . . 11 B. Explanation o 20 Theory of the Glass State, V/LF Equation . . 20 Volumetric Viscosity . 27 C. Experimental 30 The Ehrenfest Relations, Pressure Dependence of the Glass Transition Temperature ... 30 Applications 31 III. INTERPRETATION OF APPLICABLE THERMODYNAMICS . . 3k A. Definition of Glass 35 B. Thermodynamics of Quas i Elast ?c Materials . 38 C. Development of the Equations to be Used for the Description of Glass inesr. kk Graphical Modal of a System Slewing Quas i Elastic Responses . ; 4 L 1 v
PAGE 6
TABLE OF CONTENTS (cent.) CHAPTER Page Equations k6 Cissipative Terms 50 IV. PROPOSED GENERAL DESCRIPTION OF GLASS I MESS ... 55 A.. Glassiness and Excess Thermodynamic Functions 57 Changes in Volume in Unconstrained Materials. "Constant" Pressure Experiments ..... 57 Excess Thermodynamic Functions 58 Graphical Description of Glassiness . . , . . 63 Isobaric Transitions 70 V. DEVELOPMENT OF EXPERIMENTAL TECHNIQUE 73 A. Materials and Apparatus 77 BÂ„ Preliminaries , 80 Age, Preparation of the Sample and Presentation of Results . Â„ 82 Machine Characteristics 86 C. Homogeneous Deformation 92 Independence of Dynamic Response on Sample Length in Relation to Hor "ty .... 93 Study of Dynamic Effects, Relating to Homogeneous Deformation 9 VI. RESULTS, CONCLUSIONS AND RECOMMENDATIONS .... 119 A. Results 120 General Procedure 120 Density versus Pressure 123 Recoil and Relaxation 13^ B. Summary of Conclusions 1^6 C. Recommendations ..... 1^9
PAGE 7
TABLE OF CONTENTS (cont.) APPENDIX Page A. HISTORICAL DEVELOPMENT OF EXPERIMENTAL METHOD 152 1. Thermostatics 152 2. Preliminary Dynamic Studies 155 3. Development of the Experimental Technique . . 158 4. Preliminaries 160 B. CONTINUUM MECHANICS ..... 171 1. Phenomenological Concepts 171 2. Thermodynamics 177 REFERENCES , 187 VI
PAGE 8
LIST OF TABLES Table Page 1. Properties of the Four Selected Asphalt Cements . 78 2. Effect of Aging 83 3. Illustration of the Effect of Sample Length on Dynamic Determination of Density. Asphalt S6313 98 k. Characteristic Time of Recoil Curves for Several Previous Histories ... 117 5. Coefficients of the Equation p= A Q + AjP f A~P and Compressibilities at and 1000 atms for S6309 129 6. Coefficients of the Equation P= A Q + AjP + A 2 P and Compressibilities at and 1000 atms for S6313 130 7. Coefficients of the Equation P = A Q + AjP + A 2 P 2 and Compressibilities at and 1000 atms for S6320 131 8. Coefficients of the Equation P= A Q + AjP + A 2 P and Compressibilities at and 1000 aims for S64J*7 132 9. Dynamic, Parameters of Selected Asphalts ..... 1^7 Al Preliminary Data at 32Â°F for Rheology of Twelve Florida Selected Asphalts 157 A2 Effect of Aging on Compressibility of Asphalt S639 162 A 3 Deformation Readings with Special Assembly for Determination of Drag Effects 166 vt 1
PAGE 9
Li ST OF FIGURES Figure Pace 1 Â„ Typical Glass Transition Temperature Determination of a Paving Asphalt by Penetrometer 3 2. Effect of Temperature History on Determination of T q 9 3. Davies and Jones Concept of Glassiness. Observation at Constant Temperature ... 13 k. Davies and Jones Concept of Driving Forces during Adiabatic Recovery of a Glass . . 19 5. Mechanical Model of Instantaneous and Delayed Dissipations 36 6. Constrained Changes in Volume or Temperature ^0 7. Excess Pressure during Compression, Relaxed State 61 8. Equilibrium State versus Relaxed State ... 62 9. Excess Functions and Glassiness ...... 65 10. Decompression and Heating of a Glass .... 66 11. Glassiness during Isothermal, Finite, Dynamic Compressions , 68 12. Glassiness during Isobaric Processes .... 71 13. Line Sketch of Compression Equipment .... 76 \k. Instron Universal Testing Machine 79 15. Disassembled Modified Capillary Rheometer and Dimensions of its Parts ....... 81 16 C Tracing of Load versus H. tion for Calibration at Two Rates of Compression ... o7 VIM
PAGE 10
LIST OF FIGURES (cert J Figure Table 17. Illustration of Acceleration Effect 91 18. Deformation of Two Samples of Different Lengths without Lubrication 94 19. Decompression of Two Samples of Different Lengths Using Lubrication 96 20. Assembly for Studies on Effectiveness of Lubrication 103 21. DecompressionRecoil Experiment 105 22. Total Recoil as a Function of Deformation Time 107 23. Recoil after Decompression Recorded by the Two Cells 109 24. Recoil on S 63 ~ 1 3 after Different Histories (Samet j Different Deformation Tires) . . Ill 25. Recoil on S63~13 after Different Histories "(Different"', Same Deformation Time) ... 112 26. Recoil on S 63 1 3 and S6447 after Different Histories at Different Temperatures ... 114 27. Thermocouple and Load Readings during Compression Cycles 115 23. Illustration of Experimental Procedures . . . 121 29. Density versus Pressure for S 63 ~S 124 30. Density versus Pressure for S63 _ 13 125 31. Density versus Pressure for S63~20 126 32. Density versus Pressure for S63~47 127 33. Recoil and Relaxation of S6320 at 25Â°C ... 136 34. Recoil of All Selected Asphalts at 25Â°C ... 137 iX
PAGE 11
LIST OF FIGURES (cont.) Figure Page 35. Relaxation of All Selected Asphalts at 25Â°C . 139 36. Relaxation of AM Selected Asphalts at 0Â°C . 140 37. Relaxation of All Selected Asphalts at 30Â°C 1/fl 38. Recoil cf Ail Selected Asphalts at 0Â°C ... 1 if 2 33. Recoil of All Selected Asphalts at "30Â°C . . 143 Al. Specific Heat versus Temperature for Asphalt Cement (S63"20) ]$k A2. Pressure Requirea for Compression with and without Lubrication 156 A~3. Assembly for Studies on Effectiveness on Lubrication 164 A k. Failure of Silicone Lubrication at 25Â°C . . 169 Bl. Motion and Configurations . . . 173
PAGE 12
Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Phi lose; DYNAMIC COMPRESSION OF ASPHALT! C GLASSES By J. Carlos Busot August, 1970 Chairman: Dr. H. E. Schweyer Major Department: Chemical Engineering It was the purpose of this research to clarify the phenomenoIogical description of the behavior of asphalt glasses, to conduct a critical review of the literature concerning the meaning of glass iness as a material characteristic and to develop an experimental approach to study quantitatively glassiness in asphalts. The indepth review of the applicable literature revealed that the molecular theories of glassiness and its definition centered around a phenomenological description limited to the changes undergone by most substances when cooled at a sufficiently rapid rate to prevent crystal 1 izat ion. This research generalizes that glassiness be defined as the phenomena observed when the properties of a material depend on the past history of its thermodynamic state. The experimental approach proposed for the more general study of glassiness was to observe the pressure and the entropy responses of a ma' terial when subjected to homogeneous changes in volume. The thermodynamics of quasi elastic 1 aterials was used to develop the formulas
PAGE 13
for the analysis of this exper imental approach. The experimental technique developed consisted of compressing or decompressing an asphalt sample encapsulated in a latex membrane, and confined in a lubricated steel barrel. Extensive studies showed that this technique allowed homogeneous deformation of the sample without interference from shear disturbances at the wall. Results on four selected asphalts indicate that these materials show glasslike behavior under compression. The results also indicate that this technique can be used to differentiate the mechanical (e.g., free volume) from the caloric (e.g., entropy of rearrangement) contributions to the observed delayed changes in pressure at constant volume and temperature. In fact it was found that the experimental arrangement cou be used as a thermodynamic calorimeter by insertion of a thermistor with out appreciable effect upon the homogeneity of the deformation. The proposed technique and experimental approach developer! in this research should provide information to determine the effect of as r composition upon its transition from a liquidlike behavior to a solid1 i ke behavior . Since inservice behavior of asphaltic materials subjects them to a compressiondecompression environment at different temperatures, it is expected that studies of glassiness such as in this dissertation will be of value in explaining their performance. XI I
PAGE 14
CHAPTER I INTRODUCTION A . G lass i ness Asphalts in service or at ordinary ambient temperature often exhibit a behavior which is neither that of a viscous liquid nor that of a crystalline solid. Phenorrv: no logical 1 y } this behavior may be ascribed to a glassy state. In the case of asphalt, glassiness is manifested oy hardness, by cracking and conchoidal fractures under sudden stress., by a glassy surface appearance and by a veryhigh viscosity. These propert ies> which are undesirable from the standpoint of road performance, are accentuated as the temperature decrease i . The onset of these properties with temperature is determined not only by the chemical composition of the asphalt but also by the physical and special interactions among its numerous components. The. rel at i onsh 5 ps descr i bi ng these thermodynamic phenomena are difficilt to resolve. However, the thermodynamic theories for polymers and organic glasses that have been developed to explain their Theological properties and their temperature dependence would appear to be applicable to asphalt. Work in this laboratory by Shoor ('4c) has shown that glass iness in asphalt can be detected by freezing the sample and determining the change in hardness (penetration) as the temperature increases. An abrupt change is noticed on the change of hardness with temperature !
PAGE 15
as shown in Figure i. The temperature obtained at the intersection o f the lines representing the low temperature and high temperature behavior is defined as the glass transition temperature of the mater ial . T_ . S This empirical determination of J , by the intersection of two straight lines representing high and lev/ temperature behavior,, constitutes the generally accepted phenomenol og ica 1 definition of glass transition. This graphical method is also applied to plots ct viscosity, volume (and many other properties) vs. temperature. Shoor, Majidzadehj and Schweyer (h7) used the glass transition temperature (T ) determined with the penetrometer method to correlate the temperature dependence of the viscosity of eight different asphalt cements. The correlation scheme used by Shoor et. Â§J_. consisted of shifting the data obtained at different temperatures along a logarithmic time axis and determining the values of the shift factor required at each temperature to form a composite curve. The values of the shift factor were related to temperature through an empirical equation involving a characteristic temperature. This temperature was the penetration related 1 This scheme is known as t he t ? me t e mj a r a t u r e s u perp o sjt j on _ g r inc? pja . Brodnyn ( 6 ) , Gask T ns and others (21) were among the first investigators to use the superposition principle for asphaltic materials. They suggested the ASTM Ring and Ball softening point as the characteristic temperature and concluded that, in general, asphalts behave as low molecular weight viscoelast ic poly 3rs.
PAGE 16
.:.'..... x Â— i .T.r. Â»j"'s^t;;.."' .,.., kk ... Â• .... aa^ ssaeas^g^ w^vawi^^ttsaa gggK Â«*} 0.03 0.02 1! 0.01 " 20 1 " 20 ' J 30 temperature, r Figure 1. "Typical Glass Transition Temperature Determinat i or of a Paving Asphalt by Penetrometer,
PAGE 17
5 Wada and Hirose (51) used a di latometrical Iy measured glass trans!' tion temperature to correlate the temperaturet ime dependence of asphalt retardation times. Sukanoue (kS) correlated the shear modulus of asphalt in the sane manner. Barrail ( 3) used a differential thermal apparatus and indicated a dependence of this T q on the asphaltene content of the asphalt. Schmidt and coworkers (M), (^2),, (^3) nave measured glass transition temperatures of numerous asphalt by noting the changes in volume on cooling and/or heating (2 C/min) on a specially designed di latometer . They also ucceeded in obtaining a fair correlation of viscositytemperature data using their T as the correlating parameter. The moderate success of the investigations mentioned above illustrates that the glass transition is a fundamental phenomenon. It provides an empirical method fcr the correlation of physical properties with temperature. However, the lack of a uniform definition of T g as well as the need for an accepted method to determine rheological properties of asphalt at low temperatures is apparent. This lack of generality makes comparison of data among different investigators and theoretical considerations extremely diff icul t. B . Objective of this R esearch A comprehensive study on t he thermodynamic background control ling the rheological properties of asphalt and how they vary with temperature may be fruitful in understand: ng the physical behavior of these complex organic materials. This study should provide the
PAGE 18
basis to understand and describe, with generality, the observed glass phenomena. The research described herein is intended to elaborate on this thermodynamic background, through the following specific object ives : 1. A critical analysis of the contemporary thinking on the quantitative description of glassiness. 2. An : nterpretat ion of the applicable thermodynamics. 3. A proposed general description of glassiness. h. An experimental technique for the study of the proposed description in 3 above. A separate chapter will be devoted to each of the above object ives .
PAGE 19
CHAPTER I I CRITICAL ANALYSIS OF CONTEMPORARY THINKING ON THE QUANTITATIVE DESCRIPTION OF GLASS I NESS Some statement about the distinction between description and explanation of glassiness should precede any critical analysis of this s u b j e c t . The description of the glass transition must be clarified before attempting the interpretation of the experimental results; the description of glassiness constitutes its phencrr.enological definition. Many phenomena of diverse nature may be involved in the transition from a liquidlike to a solidlike behavior, or more importantly, the liquid to solid transition may be sensed and recorded differently depending upon the experimental parameters used to observe and define it. Thus, the phenomenological definition must consider these experimental factors. Conversely, the explanation or theoretical definition of glassiness must follow the acquisition of experimental data. This involves the proposal of molecular models in order to explain the observed behavior. A . Descr i pt ion Inducement of Glassiness In general most liquids can be transformed to a noncrystalline solid state, if they are cooled through the crystallization
PAGE 20
temperature range fast enough to prevent the formation of crystal nuclei. It is possible to supercool many liquids; organic polymers (18) , organic liquids such as glycerine and glucose (13);, fused salts (2)., and metals (k) demonstrate this phenomenon. The prevention of crystallization can be understood if one considers the two steps involved in this process. !n crystallizations, a nucleus must form, and then it must grow. Nuclei formation is opposed by a free energy barrier because of the fact that the melting point of very small crystals is lower than th3t of large ones. Thus, in a supercooled liquid, crystals smaller than a certain size are unstable, i.e., the nuclei tend to redissclve. In addition, crystal growth is hampered by viscous flow. As the temperature is reduced the rate of formation of nuclei may increase but the rate of crystal growth is reduced because of the increased time required for molecular motions. If the increase of viscosity is large enough the supercooled liquid (glass) acts effectively as a sol id. The glass transition temperature is determined experimentally as the temperature at which second order transitions (change in slope) are observed in the values of the thermodynamic state variables. However, in the temperature region above and below Tg, the viscosity of the liquid increased very rapidly. Volume changes with temperature often show significant delays; this delay is also observed for most physical properties. Therefore, the glass transition temperature T as generally measured in the
PAGE 21
laboratory depends on the temperature history of the sample. (See Figure 2.) Curve ABC represents schematically the equilibrium curve for the volume of a material as it is cooled or heated through the glass transition region very slowly to Tn. The recorded transition temperature, if any, would be Tgi. The curve AHD represents the temperature relationship when the sample is cooled rapidly from Ta to Tg. If the sample remained long enough at Tg its volume would approach the equilibrium value at C. Heating at this time would proceed along a curve similar to EFA. However, if the sample had been heated immediately after quenching it would have followed a curve similar to DGA. The values of T q recorded would have depended on the heating and cooling rates through the transition region and the time held at Tg after freezing. Bond i (5); discussing a description of glassiness similar to the above, notes that the rapid increase in viscosity, characteristic or even the cause of glass transition, may be due to different mechanisms depending upon the system considered. This author further states the need for a consistent phenomenological definition where "the rate and amplitude of deformation as vie]] as the thermal history of the sample and the instantaneous temperature T" are taken into consideration. Bondi also notes that the lack of such consideration makes analysis of published data of a qualitative rather than a quantitative character.
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sSSSS^?JsasszsezÂ£ jrr:.. Â— : si Â• v_r ....::_;.~ i : r;_~iVS.Ji. . . '._'Â• 1 I ? : o > I i i I I I ! T gl T g2 T 93 T g^ ..._. Â• . Temperature Figure 2. Â—Effect of Temperature History on Determination cf T q
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10 Single Point M ea surement s The topic of representative measurements of material properties during dynamic transitions is of foremost importance for a proper description of glassiness. This topic and its relation to the phenomenologi cal definition of glassiness will be discussed in Chapter IV. However, because of its relation to the state of the art, the implications of "single point measurements" will be discussed here. A review of the literature of glass transition temperature reveals that the effects of sample shape on the spatial distribution of the responses to external changes are usually neglected. The assumption is generally made that a single point measurement is representative of the state of the sample. This assumption is only true if either the material is at equilibrium or if it is undergoing a homogeneous deformation. This last statement needs qualification and is discussed further in Appendix B, section Id. Nevertheless, it is presented here to direct the attention of the reader to the importance of a prior ? theoretical considerations about the experimental conditions. The most common justification for single point measurements given by experimentalists studying glass transition phenomena is based on the size of the sample. It is said that if the size of the sample is small enough, a uniform temperature distribution will be established "fast enough" or "rapidly." The possibility is ignored that regardless of the size of the sample and the uniform temperature, enormous spatial gradients in density and pressure may bs present if a "volume viscosity" exists.
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11 No general statement can be made about the correlation, thermodynamic or otherwise, between stimuli and their response, unless the responses and stimuli are known at each point of the material. For example, a uniform state of stress and strain through' out an elastic material is required before the Young modulus can be determined experimentally. Design of specimens, such that application of external forces will produce a describable state of stress and strain, is always an important consideration when testing solid materials. Also, the velocity at each point of a flowing Newtonian fluid is required before its viscosity can be established. Viscometers are designed to produce describable velocity profiles, so that measurements at boundaries will allow calculation of the response at any and every point of the fluid. Thermodynamic Descript ion of Glass iness When dealing with the experimental conditions necessary for determination of thermodynamic parameters near a transition temperature one has to consider the relaxational or irreversible character of the process. The difficulties of describing an irreversible process are accentuated when trying to reconcile classic concepts of thermostatics, i.e., specific heat, thermal expansion, pressure, internal pressure, etc., with classic concepts of rheology like viscosity and viscous stress. This will be illustrated by commenting on the widely recognized work of Davies and Jones (13). Davies and Jones are. the only investigators known to this writer who have attempted an irreversible thermodynamic approach
PAGE 25
12 to describe glassiness and establish relationships among the measured thermodynamic parameters. In the opinion of this writer, the most important contribution of Davies and Jones was to define a "volume" viscosity to quantify the "time effects" observed during glass transitions. Regretably, their work has been consider superficially in the literature to provide background for discussion of the effect of pressure upon glass transition. A brief presentation of the work of Davies and Jones follows. It is intended to illustrate the involved and far reaching conclusions drawn by these authors, and others, by their manipulation of a phenomenological description of glassiness. Figure 3 illustrates Davies and Jones' phenomenological concept of glassiness. The line LAC represents the enthalpy vs temperature idealized equilibrium line of a gl ass f ormi ng liquid. On cooling at a finite rate, of the order of degrees per minute, the liquid will depart from the equilibrium curve at point A and move towards B. At lower rates of cooling the glass would move along XY instead of AB. The point A represents according to the authors, a thermodynamic "fictive" state. The fictive state is that at which the glass would find itself in equilibrium if brought there rapidly from its actual state. According to these authors, in the case of an isobaric experiment the state of the glass can be described completely by giving its actual temperature T, and the temperature at point A (fictive temperature, T ) .
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13 ~~ (ESPStfie^KS " . : ;Â« EK?xreii.*Si2 5 . . 2 " " .' >faro x: c til Preparat i on Observat ion Temperature Figure 3.Davies and Jones Concept of Glassiness. Observation at Constant Temperature.
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14 This simplified description, in this writer's opinion, led Davies and Jones to overlook the influence of the rate at which the irreversible path AB is traveled. A glass once at a point away from AC, point G along AS for instance, cannot be brought back reversibly to the original point in the equilibrium line. The paradox in their description is that a glass can be "cooled" reversibly from the liquid (point of departure from the equilibrium line) but can not be restored to a liquid state reversibly. Davies and Jones did not recognize the importance of the "preparation time" of the glass (time along AG) and the time involved in the sensing cf the glass properties. These authors considered the preparation time sufficiently fast and assumed the sensing time to be of no importance. The latter assumption is not consistent with tiieir observation that the rate of cooling would affect the "f ict ive" temperature . For a tneoretical development of the equations necessary to describe their phenomenological concept of glassiness, Davies and Jones introduced a new state variable (Z) . This variable is defined to be continuous and to remain constant when the pressure (p) and the temperature (T) are changed "rapidly." These authors attributed to Z the significance of being a measure of the conf igurat ional order, assumed constant for rapid changes of pressure and temperature. A glass of fixed structure, u Z Q , is thus proposed, when cooling a How this structure is fixed by a finite rate of cooling is not explained. Furthermore, these authors referred to other investigators who attributed the freezing to a second order transition, as "misunderstanding the nature of the phenomenon." (Davies and Jones, op. cit ., j>. 29.)
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15 through the fictive temperature at a particular rate. The thermodynamic state of Davies and Jones glass, Z = Z c , is represented by a curve Z(p f T) = Z , where Z satisfies the equation A(Z , p. T) Â• 0, where A, defined by equation k, equals zero when the system is in equilibrium. This implies that the equilibrium volume for a glass, V(Z , p, T), v:ill be constant along the AB curve. The volume at p and T is the same as the volume at p and T; AV equals zero. The same is true for the entropy; the entropy 5 at p and T is the same Â— a as the entropy at p and T. Their final results state: dp = il_T V = Aa (1) dT 3 p AV A3 3 T AS r A C p (2) dÂ„ AS iVAa where A3 Aa and Ac are the discontinuities in compressibility, thermal expansion, and specific heat at the point where Z = constant r Z , Although these authors only presented data on glass transition with temperature, Â£C p vs. T and Aa vs. T, the theory was intended to be general and to describe the phenomenological behavior of glassiness with pressure. The authors' object ive in introducing the pressure dependence on T n was to be able to define different types of driving forces for the delayed volume changes. This was accomplished by using equation 1 and by defining a fictive pressure, p, given by: a The symbol ? x will be used in the text to represent the partial derivative operator f&L \ , where y and z represent the her independent var iables: 9x / y,z ot
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16 B^ * J& => _ _^E (3) T T A6 TVAa This result of their theory proposed that "... a sudden isobaric change in temperature AT which leaves T and p unaltered is equivalent thermodynami cal 1 y to a pressure increment of: Ap z (AC p /TVAa)AT. u This calculated Ap has been used throughout the literature to replace AT as an equivalent driving force on a thermally produced change in volume. This "excess" pressure is then related to the volume rate of change through a volume viscosity. The basis for this concept is an equation expressing the change of irreversible structural entropy, TdS ifr E AcZ (k) where the affinity A is A = (P P) I*" T,2 (5) 3P The value of A is zero when the system is in equilibrium. Equation 5 presupposes that the system is "close" to equilibrium. Under these conditions and assuming that dZ can be represented by dp, the irreversible change in entropy and the rates of production of entropy are given by these authors after some manipulation by: a More detailed and rigorous treatments of the classic theory of thermodynamic relaxation are given by Herzfeld and Litovitz (26), pp. 1 591 70 ; by Patterson (37); and by Prigogine et al . This equation is the source of the paradox. A glass can be cooled at a finite rate following Z = Z c and therefore, dS: rr = C. However, it cannot be reverted reversibiy to liquid through' a path of constant Z.
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17 TdS irr V(pp) [AadT Agdp] (6) S ?rr tlfiEL [aT _ 3p . L v] (?) where a and pare the instantaneous values of the temperature expansivity and the compressibility. A further assumption is required before a viscosity can be defined. in the same way of other irreversible phenomena close to equilibrium, "forces" ate assumed proportional to "flows"; the expression in brackets in equation 7 can be interpreted as a "flow" and its coefficients as "forces." Therefore, finally, the kinetic equation is given by a j _ 3 p . L (} = B_^E ( 8 ) where n, is the proportionality constant and represents a volume viscosity. Equation 8 would be approximately valid if a process were devised where the phenomenological description of glassiness given in Figure 3 is true. The authors used their description of glassiness to estimate the relaxation time (and viscosity) of thermal relaxation of glycerol, and volume relaxation of glucose. The experimental methods consisted of cooling a sample of glycerol below its glass temperature inside an adiabatic calorimeter, and studying the approach of the temperature of the sample to equilibrium at constant enthalpy. A dot above a variable indicates time rate of change; i.e, ~ A Â• dt
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18 Figure k is a schematic representation of this process. The driving forces for the irreversible adiabatic recovery of the sample after it is cooled to T are represented by T T. Davies and Jones' phenomenological interpretation of glassiness implies that during cooling of the glass the instantaneous volume of the sample was at all tires in pace with the temperature and pressure of the sample; therefore all the volume change occurred adiabat ical ly after cooling and was caused by the T T c parameter. Their pioneering results and experimental conclusions proved that there were relaxation phenomena associated with volume changes on glasses. Numerically, these results are of limited value, because their viscosity includes more than volumetric effects, as indicated by equation 8. The parameter n in equation 8, just happens to have dimensions of viscosity. Goldstein (25) proved that If the phenomenological description of Davies and Jones is true and general, their parameter Z which controls the relaxational phenomena after glass preparation can be interpreted as either an excess volume or an excess entropy. A process described by, AZ = is equivalent to processes described by AV = 0, or AS = 0. Goldstein derived equations 1 and 2 as results of these observations, and indicated that only if a AV = process is completely equivalent to a AS process, would these equations be valid. If another parameter in addition to Z is required to describe glass'ness, then
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19 . . ,, . . a. ra c UJ & < 5 . Driving force / / / A > Cool I ng . ; ^ / / 1 i i i ; Temperature Figure h. Â— Davies ' Jcnes Concept of Driving Forces during Aa;3b.:iLic Rec of a Glass.
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20 j^P / TV Ac;. Act AC n The nature of this second parameter, however, would have the same character as Z, It would represent a difference from a "fictive" state and would not include the time effect during preparation of the glass. In addition to Davies and Jones, and Goldstein (25), the fictive state approach has been used by Kovacs (29),. (30), O'Reilly (34), Passaglia and Martin (35), and in general by all experimenters a interested on the variation of T a with pressure. B . Explanation Theory of the Glass State, WLF Equat Ion The theories of models of the glass state which have been proposed are intended to explain the success found in using empirical glass transition temperatures to express the temperature dependence of relaxation processes, more specifically viscos i ty. The cornerstone of the theories developed to explain glass jness is the empirical equation proposed by Vogel (50) to express the dependence of viscosity upon temperature. This equation was modified and used for the first time for the development of a theory for the glass state by Williams, Landel and Ferry (5*0. This subject will be discussed in more detail on p. 30.
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21 The equation in its most common form has the expression where .og JO= C ' (T T Â»> (,0) n C 2 + Tc) ri/ri s is the ratio of viscosities at temperatures T and T respectively C] = 8.86 C 2 101. 6Â°C T s = reference temperature Equation 10 is known as the WLF equation. Williams and coworkers ( 5$ Â• (53), (5*+) found that if a separate reference temperature T is suitably chosen for each system, equation 10 expresses the temperature dependence of viscosity for a wide variety of glass ~formi ng liquids over a temperature range of approximately 1 00Â°C above the glass transition point. Most significantly they found that the temperature T s lies 50Â°C above the 1Â„, with a standard deviation of t 5 C. They also showed that C^ was proportional to the expansion coefficient of the liquid. Free volume theories Williams, Landel, and Ferry used Doolittle's experimental findings (15) that viscosity depended exponentially on "free volume" (vr) ; which they proposed as: V, S v c [O.O.'iS + Aa (T TJ] (11)
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22 where v is the volume at the glass transition temperature, and is the difference between the thermal coefficients of the liquid and the glass. The original free volume theory of Williams, Landel and Ferry assumed that the ratio of free volume to total volume would remain fixed below T q and that its pressure and temperature variation above Tq would be given by AS = {?Â•] 3 g (12) Aa a, <^ (13) where 8 and a are the compressibility and thermal expansion coefficient of the liquid (1) and the glass (g) . Cohen and Turnbull (10) developed a more complete theory of diffusion and momentum transfer based on the basic concepts proposed by Williams, Landel and Ferry. Cohen and Turnbull did not intend, however, to include associated liquids whose viscosity varies markedly with temperature at constant volume. Rather, these authors proposed a description of glassiness and molecular transport in liquids and glasses in which no potential energy barriers among molecules existed (hard sphere model). The potential energy of a molecule was assumed constant except upon intermolecuiar contact. Interactions among molecules were not necessary to explain glassiness, cr.'j viscosity would only depend on temperature through volume. Cohen and Turnbull visualized flow es a process involving molecules jumping over barriers created by the need of formation and redistribution of holes in a liquid quasilatt ice. These
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23 autho r s assumed that these barriers stemmed from the need of free volume (v) to be greater than some value (v*) . This idea of flow is quite similar to the activation state or kinetic theory of Eyring (20) and (23). in Cohen's theory, flow occurs when ''there is a fluctuation in density which opens up a hole within a cage large enough to permit a considerable displacement of the molecules contained by it. Such a displacement gives rise to diffusive motion only if another molecule jumps into the hole before the first can return to its original position" (10). The transport coefficient for a molecule, D, according to this picture of flow is: D(v) = D(vv) P(v*) (1'+) where D is the diffusion coefficient which is a function of the volume, v, of the cage, v* is the critical or "activated" volume just large enough to allow a molecule to displace itself, and P(v') is the probability of finding a hole of volume larger than v*. This probability is given by Cohen and Turnbull (10) as: P(vv) = exo[Yv*/v f ] (15) where v, is the average free volume, defined below, and Y is a numerical factor introduced to correct for overlap of free volume. The final expression becomes: D(v) = D(y)* exp[ yvVvf] (16) where D (v) is a function only of the molecular diameter, the temperature dependent velocity of the molecules, and a geometric factor. Equation 16 has the form found by Doolittle for the viscosity of hydrocarbons. In order to test their theory Cohen and Turnbull defined free volume as:
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r . T e~v G i expj a dT 1 (17) where v is the van der Waals volume which is assumed independent of' temperature; \ is the coefficient of thermal expansion, and T is the temperature at which the free volume vanishes. This definition assumes that the free volume is given by the total thermal expansion at constant pressure and is zero at a temperature T . The temperature and pressure dependence of viscosity will be determined, according to this theory, by the pressure and temperature dependence of free volume as follows: v f a v (T T Q ) Bv" p AP (18) where a ar.d v are the mean values of the expansivity and van der Waals volume over the temperature range (T T ) , evaluated at A P = 0, 3 and v p are the mean compressibility and volume over the pressure increment A P. The definition of free volume as a function of thermal expansion alone (equations 17 and 11) is insufficient for satisfactory description of the pressure and temperature dependence of viscosity of polymeric liquids. Other modifications of the free volume theory are designed to provide the theory with the flexibility of a free volume which would show a temperature dependence below T (7), (9 (40) . The basic consideration of all these theories is the falling of the free volume below some critical value where the high viscosities would rake relaxation times of the order of days. These theories have been used to explain the rheological behavior of asphalt by Majidzad ind Scl r (30.
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25 Coope rative rearrangement co n f iaurat io nal entropy Gibbs and coworkers (22), (1 J have defined T as "the quasi static glass temperature below which molecular relaxation times are too long to permit establishment of equilibrium in the duration of even the slowest experiments ( s time scale 1 of hours to cays)" (1 ). Their concern was to relate by stat ist leal mechanical arguments the relaxation properties of gla^sform? ng liquids to their "quasi static" properties. These authors proposed that the cause for increased relaxation time, x, with decreasing temperatures Is the reduced probability, I,', of a cooperative rearrangement of the parts of the 1 iquid. A cooperatively rearranging region is defined by Adam and Gibbs ( 1 ) as "a subsystem of the sample which, upon a sufficient fluctuation in energy (or, more correctly, enthalpy), can rearrange into another configuration independently of its environment." By assuming a partition function, Q., for a system containing a number of parts which could rearrange cooperatively, Adam and Gibbs define a free energy, AG. This potential represents the energy hindering the rearrangement at constant pressure and temperature. The size (Z) of a rearranging region is defined by assuming that AG can be expressed using a potential energy per unit size, Au , by ZAy AG a kT In d (19)
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26 This implies the existence of a uniform structural unit, e.g., molecules or 3 number of molecular segments of a polymer chain. The average transition probability W(T) of a cooperative region as a function of its size is given by: W(T) = A exp[ Z* Ay/kT] (20) where I* is now the smallest critical size. Cooperative regions of Z < Z* yield zero transition probabilities. These small size regions remain in the same configuration when energy is supplied to the system. The frequency factor A is nearly independent of temperature. Finally W is expressed as L" k TS c J W = A exp Â£Â— or, (21) [4] " " A " p L ^7 J (22) where C is independent of temperature, S c is the entropy of the macroscopic sample, and s c is the smallest critical entropy. It is the entropy of configuration per molecule corresponding to the minimum number of configurations determined by Z* at a given temperature. The minimum value of configurations corresponding to 1* as T * T is y two. The smallest size Z* must be large enough to have two possible configurations; the region where it resides before rearrangement, and another configuration to transform into. In terms of Gibbs parameters the coefficients of the WLF equation (see equation 10) becon .:
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27 Au s_ 2.303 k AC p T s ln T l and (23) T\ 1 n T 2 In is. + i + riiim T T 2 L T T sJ T : (24) where TÂ„ is the temperature where s .0, and A C D is the specific heat difference between the liquid and the glass at T q . The cooperative rearrangement theory predicts that the "universal" parameters of the V/LF equation will depend upon: the ratio of the reference temperature to the equilibrium temperature 7y, a free energy barrier restricting transitions, a critical conf igurat ional entropy^ and the difference between specific heats Vol time tr ic Vise psity According to the general contemporary opinion the time dependence or re Taxational character of the glass transition makes a theory of glassiness a particular case of a theory of liquid viscosity (20 . The two previous theories represent the two most widely discussed viewpoints on the origin of viscosity of liquids near their glass transition. Summarizing the previous paragraphs it is seen that the free volume theory relies on a kinetic argument based on the dependence
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of f'ow upon the availability of holes or free spaces for the molecules to move into. Glassiness is considered a nonspecific process outweighing specific effects of chemical structure. Conversely., Adam and G i bbs present viscosity as the result of specific structural interactions. These interactions are represented by the increasing size of the rearranging regions as the temperature decreases . Both theories relate more closely to the dependence of viscosity on temperature and pressure, than to the actual mechanism of viscous flow. Both consider local rearrangements (hole distribution cr cooperative molecular rearrangement) necessary conditions for flow. However, these rearrangements are not sufficient; they also occur on the fluid at rest. The questions unanswered by these theories about the nature of flow are, paraphrazing Goldstein (2^): " What is the relationship between the local rearrangements and the microscopic deformation? How does the external stress bias the local rearrangements for a jump to occur? Why does not the deformed state reverse to the original state when the biasing stress is removed? These questions are subjected to extensive analysis by Goldstein, The answer to the third question is of particular importance in order to establish the relationship between shear and volume viscosities. For flow to occur irreversibly under a free volume mechanism, two conditions mi met. First, molecules must jump into a hole, and second, holes must vanish and reform. If the relaxation time of
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s 23 the second step is smaller than the one for i:he first step, randomization takes place after the jump and the flow becomes irreversible under the biasing stress (viscous flow). The appearance and disappearance of holes could be thought of as a volume relaxation process and, thus, the importance of the volumetric viscosity. According to the theory of cooperative rearrangements, the relaxation time involved in the creation of holes (second step) hould be larger than the relaxation time for a simple redistribution (first step). This appreciation is based on the consideration of the apparently larger cooperative character for the creation of a hole. If this picture of flow is correct, the volumetric viscosity should limit the rate of shear relaxation. In acoustic experiments volume relaxation times are almost equal to shear relaxation times implying no essential differences on the nature of volume and shear viscos it ies (26) . Determination of the volumetric viscosity through acoustic xperiments, however, assumes that the shear and volume effects are dditive. The additiveness assumption implies that the absorption unaccounted for by shear viscosity can be attributed to volume relaxation. This is complicated by the fact that shear viscosity itself may not be represented by a simple Newtonian model as assumed, but probably It is frequency dependent. In this case, the calculated volumetric effect would be affected by the frequency dependence. The additiveness assumption may or may not be justified on the basis of the infinitesimal amplitude (or magnitude) of the deformation involved in acoustic vibrations, but il leaves the e a
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30 question open as to the phenonie no logical behavior of materials undergoing fir.ite changes in volume. In a finite volume change structural rearrangements may be triggered which would not have been detected or induced by the acoustic deformations. C . Experimental The Ehrenfest Re lations. Pressure Dependence of the Glass Transit ion T emperature The Ehrenfest relations state (8): dT 2 /d? = AB/Aa. (25) T 2 VAc/ACp (26) These equations express the change in the temperature at which a transition occurs T 2; with a change in pressure; Aj3 , Aa and A C are the discontinuities in compress i bi 1 ity, thermal expansion, and specific heat at the transition. The volume V is measured at P and T 2Davies and Jones (see p. 16) used the Ehrenfest relations to substitute dP for c'T as the driving force in volume changes during glass transition. These authors did not discuss the theoretical implications of the relations, neither did they present any data on the change of T g with pressure. However, Goldstein (25) used Davies and Jones ( 13) phenomenolog lea ! description of glassiness and interpreted equations 25 and 26 as criteria to test the validity of the free volume and the cooperative rearrangei nt theories.
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31 The free volume theory proposed that glassiness occurs because the free volume approaches zero at the transition temperature. Goldstein indicated that this condition was similar to the condition for the validity of equation 25. The cooperative rearrangement theory implies that the structural entropy becomes small as the sample is cooled towards To. According to Goldstein, equation 26 requires that a condition similar to this be met. i.e., that the entropy of transition be zero. Ap pi icat ions The experiments of O'Reilly (3^), and Passaglia and Martin (35), will bs discussed to illustrate typical past efforts to elucidate the meaning of equations 25 and 26. O'Reilly studied the effect of pressure on the T of polyvinyl acetate (PVA) . This author measured the dielectric relaxation of PVA at different temperatures, at constant pressure. By observing the temperature at which the dialectric characteristics changed abruptly, a transition temperature T was defined. Repeating the. 9 At 9 experiment at different pressures allowed calculation of Â— Â— Â— r AP O'Reilly found this ratio to be independent of pressure and equal to 0.021 C/atrn. This author also determined the force required to compress PVA at different temperatures. However, some of the results of these experiments should be accepted with caution. O'Reilly did not determine the effect of compression rstt upon the pressure vs. volume plols. Furthermore, no precautions were taken by this author
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32 to eliminate the drag at the walls of the container. Nevertheless., O'Reilly defined a transformation pressure ?Â„, "as the pressure at which molecular rearrangements can no longer follow the appl ieo pressure and the polymer exhibits a glass"! ike compres Sibil! ty. " It should be noted that this definition of the transition pressure contains two different and perhaps opposite phenomenological concepts . A transition pressure at which relaxation phenomena "can no longer follow the applied pressure" implies a dynamic situation where viscous dissipations play a foremost important role; they cause glassiness. Thus, the importance of defining the effect of compression rates on P , and the importance of eliminating the viscous dissipation at the walls. A transition pressure defined as that at which "the polymer exhibits glasslike compress i bi 1 ity" however, implies an arbitrary definition of glasslike compressibility and the notion of an equilibrium volume vs. pressure experiment. O'Reilly used this last aspect to define sevsral arbitrary transition pressures at a given temperature. The plots of these transition pressures vs. temperature gave values of 775Â— in excellent agreement with those . AT g 9 values of Â— Â— , obtained with the dielectric experiments. AP However, attempts to prove or disprove the application of equations 25 or 26 to glass transition were inconclusive. The main difficulty being the curvature of the plots of volume vs. pressure. In contrast to volue. vs. temperature plots (Figure 2), the volume vs. pressure plots do not she/ reasonable straight lines anywhere. There is no unequivocal method tc deti in
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33 Arguments were presented by O'Reilly in favor cf the cooperative rearrangement theories. These arguments were based on the decrease of volume with increasing temperature well above the transition pressure. It is the opinion of this writer, that because of the high pressures and because "measurements were usually taken at increasing temperatures/' the decrease in volume may have been affected by leakage. Passaglia and Martin (35) determined the variations oT T q with pressure on polypropylene. These authors used a direct experimental procedure. The changes in density with temperature were studied at several constant pressures. These plots approximated straight lines at high and low temperature ranges. The transition temperature was defined at the intersection of the extrapolated straight lines as shown in Figure i. 5y plotting the values of Tg vs. pressure, a value of AT c Ap" 0.020 C/atm. was obtained for polypropylene. Passaglia and Martin proved that in static experiments like theirs, equation 25 is an algebraic, consequence of being consistent when defining T and when assigning values to A3 and Ao:. a 0'Reilly, op. cit .. p. *02.
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CHAPTER I I i INTERPRETATION OF APPLICABLE THERMODYNAMICS The complexity of the problem involving the quantitative defini tion of glassiness was illustrated by previous comments on the work of Davies and Jones. Difficulties in the past have originated mainly from the lack of general definitions of "time scale," volume viscosity, and other dynamic parameters not related directly to the thermodynamic theories used to correlate the experimental data. However, many of the elements necessary for a comprehensive definition of glassiness already exist, for example: Dondi's indication that a consistent phenomenol og i cal definition is the key to the problem (5); the general observation of the influence of "the prior history" upon the properties of a material and Coleman's comprehensive description of thermodynamic processes (11), (12. In this chapter a rational approach to the study of glassiness, comprising all the elements presented above, will be attempted. First, a quantitative definition of glassiness will be proposed; secondly, an explicit description of the type of time dependent processes where this definition ecu Id be applied will be presented; and thirdly, a thermodynamic theory will be used to develop the equations necessary for the description o f glassinssu 3',
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^ Â• Definition of Glass it is the opinion of this writer that a glass must be defined as a material whose thermodynamic properties depend upon the collectior: or ail pas" thermodynamic spates (prior history}. In order to complete this definition, it is necessary to establish a measure of gl iness, e.g., Davies and Jones ordering parameter Z. The measure of glass iness proposed here is that the intensity of a glassy state should be defined by its characteristic time, measured in a relaxation process where instantaneous viscous dissipations are absent. The basis for this seemingly arbitrary requirement is the convenience of differentiating between the two basic types of dissipative phenomena occurring in glasses. These are the dissipation caused by "flow" and the dissipation caused by molecular rearrangements. These dissipations are the origin of the dynamic measurements recorded during glass transition. To the category of flow dissipation phenomena belongs the instantaneous dissipation characteristic of flows caused by spatial gradients; temperature gradients causing entropy flow, and velocity gradients causing momentum flow. To the rearrangements category belongs the relaxation of structures characteristic of "thermostat i c trans it ions . " Figure 5 is a mechanical representation of the concepts of instantaneous dissipation, and delayed or relaxational dissipation. The dashpot Oj, illustrates the concept of instantaneous dissipation; the body marked "solid," because of the lack of a c.; r .;.' Ipative element Dj in series with tl slastic element K;, will no;: exhibit
PAGE 49
36 .Â». j .. . ,_ Â• "Â•Â• i =^Ks Â«., Id, j P !:, I "fluid" Â„ "Â•.:<: i K d ; . : Â». L "solid" Figure 5, Mechanical Mode' Diss i pat ions . of Instantaneous and Del a
PAGE 50
37 instantaneous dissipation and can eventually stand external stresses without deforming. However, the delayed dissipation Dj indicates that this body will show relaxational effects. instantaneous viscous dissipation may exist only in flow situations; e.g., in the case of shear or elongational deformations of fluids, and it the moving boundary of a solid. However, as illustrated in Figure 5, a body can be deformed without exhibiting instantaneous dissipation. For example., the dissipations occurring in either the isotropic change in volume of a fluid, or the shear deformation of a solid are generally delayed by elastic elements; these are the bulk and shear moduli respectively. These types of deformations will be referred as nonflow processes. The convenience of conceptually differentiating between instan* taneous and relaxational dissipations in glasses is based on the connection of relaxational dissipations with changes in temperature and density. Isotropic changes in volume and homogeneous changes in temperature are delayed and relaxational in nature. Glass iness is generally described by the phenomena observed when changing volume and temperature. The definition proposed here calls For observation and study of glasses in nonflow processes. Observation and study of glassiness during isothermal and isochoric (constant volume) flow processes is quite possible and convenient. However, this writer believes that in order to connect glassiness to any kind of thermodynamic theory, processes where volume and temperature changes are studied in the absence of fl > r pr sent the first rational exper imental step.
PAGE 51
38 Processes where thermodynamic variables are changed at a given rate, and processes where flow occurs at similar rates must be compared exper imental 1 y, e.g., by compressing a 10 cc sample of a material at 1 cc per sec and forcing another sample through a capillary at a rate of shear of 0.1 sec . In both cases the rate I of deformation is 0.1 sec . This should establish if the "time scale" has meaning per se or only when related to flow processes. Particularly, it will be important to establish whether or not a a homogeneous volume change occurs with dissipative effects. Â®Thermodyna mics of Oua s i Elasl ic Ma terials As mentioned in the introduction of this chapter, a thermodynamic theory is needed to develop the equations required for the description and study of glassiness. This writer's contribution in this and the next section consists on adapting an existing general theory to provide these equations. The usual theory of irreversible processes (Appendix B) may yield useful guidelines for the quantitative analysis of glassiness, and may even render results formally equivalent to those obtained from a more complex theory. However, the explicit omission in this theory of the effect of the past values of the thermodynamic state upon the present values may lead the investigator using this theory to define arbitrary terms to account for these history effects . Goldstein's viewpoint on the subject based primarily on Litovitz acoustic experiments was briefly discussed on page 23.
PAGE 52
39 As discussed in the previous section, processes where no i ns tantaneous viscous dissipation are present may nave special significance for the study of glassiness; if no instantaneous viscous effects are present, the theory of thermodynamic processes of materials with fading memor/ or quas 'Â• elast ic materials, constitutes the most general thermodynamic theory available. (See Appendix B.) A majority of the published experimental evidence indicates the existence of a viscous response of glasses to changes in volume. However, it is not an instantaneous viscous dissipation; the dissipation is always "delayed" by the compressibility of the materia 1 (25), see Figure 6. Therefore, the theory of thermodynamic processes of quas i elast ic simple materials should be ope] [cable to glasses, The most important results of the theory of materials with fading memory by Coleman and Noll (11), 12^ ere summarized by the following equations from Appendix B. A = f(F, T) + H A (A, T^) (BI8a) a A = g(F, T, t) (BI8b) *(*) v(Â§) Mf) dt (B ,&> T,t F,t F,T where a, All equations developed in Appendix B are labeled with a B preceding the number.
PAGE 53
ko i AX n / r ///////// AX ~e r AX r AX ! Â• 7 <""/ A Constrai Forces Y / / Figure 6a r W, / '//////////////if/////' . . . _ . Figure 6.Constrained Change in Volume or Temperature
PAGE 54
M (*) a T,t F,T (B19) (B2G) (B21a) where D = p TS d i v h pq (B2lb) The nomenclature as wall as the meaning or these equations will be reviewed in the following paragraphs. Equation Bl8a expresses the principle of determinism in materials with memory. The deformation F, the temperature T, and the present instant t are taken as the independent variables (see Appendix B) . This principle is equivalent to the assumption that the present state of the system is determined by its past. Specific ally, equation B 1 Sa indicates that for quas i elast I c materials, the free energy A i? given by the sum of two terms: f(F, t), the value of the free energy if the material had been at its present state forever, and Ha(F, T), the contribution of the collection of states occupied throughout the entire past of the material (F, T) . The influence of this past history upon the present will depend on the material and is expressed by a functional relationship. A functional, e.g., Hn, assigns a number to the influence of the collection of past values. F + is the transpose of F
PAGE 55
hi Implied in equation Bl8a is the assumption thai the state of the mater' loes not depend explicitly upon past histories of the time derivatives of the state variables, i.e, there are no instantaneojs viz ..; dissipation effects. The transition from the past to the present is always smooth with no discontinuities of the stress P, or the entropy of the system S (equations B 1 9 2nd B~20) . Equation B~l8b is the instantaneous equivalent of equation Bl&a. T cr the function g(F, T, t) will depend on the past history. However, since the history has already impressed its effects upon the material, g(F, T, t) becomes a smooth function of time and of the instantaneous state of the system (V, T) . Equi Bl8c is the mathematical statement of the sought potenti [uations B19, S20, and B~2l are expressions for the pres sure P, entropy S, and dissipation D in terms of the thermodynamic potential A. These equations are valid at any instant throughout the process. Of particular importance is the term 3 t g of equation B2la. The dissipation D represents the excess in the rale of product ior of entropy over the entropy flow due to gradients in temperature plus the entropy provided by homogeneous external heat sources. The term div h represents the heat flux, and the term q a homogeneous external heat source. Trie significance of D to the theory of quasi elastic materials is that it represents the effect of memory upon the systei . The effect of i >ry upon free energy, given by D, determines all effects of m : , on P and 5. For a "reversible" time inde process D r 0.
PAGE 56
hi If time dependence on a nonflow process is assumed caused by the delayed relaxation of structures within the material, the dissipation will acquire the meaning of the rate of change of free energy due to these relaxation processes. Equa:ion B 21 b could be wr It ten. pT & t S h . st = D z pTS div h pq> (27) where cLSLÂ« st is the instantaneous entropy production due to relaxation of structural changes. The term. St,., would represent the difference in entropy ' h i st' ' ' ' between the structures present at time t, and those present if the system had always been at rest at T and V. The structures present at time t are only a function of the past values of V and T. The system is always tending towards the order represented by the relaxed state at T and V: this accounts for the fact that 3 . S, Â• . t hist is always positive. This interpretation of the dissipation term is not intended here as a quantitative theory, but rather a qualitative model which may provide some aid for the understanding of glass i ness . Equation B 2lb explicitly states one of the most important limitations for obtaining meaningful data. The material must be either a perfect conductor or a perfect insulator. Otherwise, a temperature gradient will exist throughout the sample, and a heat balance at each point will be required in order to establish th contribution of entropy flow to the dissipation.
PAGE 57
M* If the body could be deformed homogeneously, the dissipation could be determined by holding the deformation and temperature constant at a given instant and studying the behavior of the system. In this manner the contribution of memory to the thermodynamic properties of the material could be found. This is another reason for proposing that equivalence of glasses be defined through an experiment where. V and T remain constant. Such an experiment is called a relaxation experiment. Relaxation times and their measurement are the subject of an enormous quantity of literature, from nuclear relaxation times in nuclear magnetic resonance experiments to the rheological relaxation time of viscoe last icity. Any of these methods could be used to determine a relaxation time for the material as "glass iness" is induced by cooling or compressing. However, one must keep in mind that the condition of homogeneity must be approximated before any of these relaxation tines can be considered as an indication of the structural changes. Also, the meaning of "glassiness" will be related to the time scale of the experiment used to study the trans it ions . C Â• Developmen t of t he Equations to be Used for the Description of Glass i ness Graphical Model of a System Sh o wing Q,uas i Elast ic Re sponses According to the definitions and discussions of trie previous section it is suggested that a study of the meaning of "time scale" and "dissipation" in nonflow processes may clarify the nature of
PAGE 58
a. me 45 glassinass in certain materials. Such a process is illustrated in simplified manner in Figure 6. The process represents the constrained one di mens ional change in volume depicted in Figure Â£, The entropy S, and the onedimensional stress P (pressure) are chosen as the response to changes in the process variables: vclu V(t), and temperatures T(t). These state variables are represented by the vector X; the response variables S and P are represented by the vector Y. The response resulting from a change in either of the state variables is assumed homogeneous throughout the body. The changes or displacement capable of producing mechanical responses in the system are balanced at any instant. The pressure produced by instantaneous elastic displacement A X e is the same as the pressure produced by the delayed relaxational change, A X r . The observable change in the thermodynamic state, however, is the sum of thess two displacements. AX = Ax s + Ax r (28) The elastic displacement can be conceived as the contribution of numerous relaxational processes with negligibly small relaxational times. The delayed changes could also be conceived as a composite response. In the latter case, the relaxation characteristic would be given by a spectrum of relaxational times. The minimum relaxational times contributing to an experimental observation will be lii ited by the response time of the sensing instrui and the duration of the imposed change. The process contributing
PAGE 59
he to the maximum relaxational times on the other hand, will be limited by the sensitivity of the recording instrument or by the patience of the exper in enter. Equat io For the process represented in Figure 6,, the thermodynamic equations expressing the change in free energy A will be reduced to: A f(V ; T) :H A (V t , T) (29a) = g(V, T, t) (2%) For quasi elastic materials A is a potential for pressure and entropy Therefore, these properties could be used to specify the response of the system. The equations expressing the pressure of the system as a function of the volumetemperature history are: P : P(V, T) + H p 0A, T) (30a) = tt(V, T, t) (30b) t [ovj dt Ur ) dt "'" I 3t i (3D T,t V y V,t V,J The term p (V, T) is analogous to f(V, T) of equation 29a; they represent the respc of the system after an infinitely long time at V and T. Similarly, H p (V'% T w ) represents the effect of the past volumetemperature history from the relaxed state. Equations 30 : 31 are expressions of smoothness and p.'cvi ' interpretation of ths pressure response terms as functions o r , energy chanc 25 .
PAGE 60
47 For a quas i e I ast i c material: P "(3 A/ 3V) T ^ t (32) therefore^ by substituting aquation 53 into 32: dÂ£ _ afeA/ i\t) dV _ a.feA/ 3V) dT .. afr.A/ Vl , . rft 3V dt 3T dt 3t U ' ; Furthermore : and 3(3A/ 3v) T>t o I 3 2 A = 3(3 A/ 3T) 3TdV 3 V V,t M T,t 3S 3V T,t (34) 3(A/3 V) Tt 3t V,T 3t 3V 3 D 3 V _3(jA/3_t) v t _____ T,t T,t (35) because of the properties of exact differentials. Therefore, equation 31 can ba rewritten as E. _ _. \ \3V j T,t _. + [ 3 _s ] _L dt I 3 Vy dt (36) The sign is due Lc ths fact that pressures are now considered positive, while in previous discussions the stress was considered positive when exerting a traction.
PAGE 61
k8 or In more familiar terms, P = K I + q v * + P y (37) where the instantaneous mechanical bulk modulus for the process K equals VCgP/sV), The process instantaneous latent heat of expansion (thermal bulk modulus) q.. equals T(3S/3V) ; and D>. represents the instantaneous loss cf rate of work (power) for isothermal volume changes, D y =3(3A/9t) /8V. It should be noted that all the coefficients in equations 36 and 37 represent rates of change with respect to the bulk (volume) of the system determined while holding the temperature constant at a particular history. These "process" coefficients are material properties determined by the constitutive equation of the material. In addition to pressure, entropy is needed to express the overall response of the process depicted in Figure 6. The equations expressing the instantaneous rate of change of entropy are: where ( ) T,t 3S 3 V 'ft) u T + V,t T,t (it) V,T i__ (Â„ A ) d Â— (*A 3 V \ cT J ~3 T \ J V j T,t ( ' \ a V,t v,t (38) (33)
PAGE 62
h9 3 S \ _ Â£v 3 1} T V,t (**0) and ' V/l lv,T V/ (M) therefore, the rate, of change of entropy can be expressed in terms of coefficients which represent changes with respect to the temperature of the system evaluated while holding the volume constant, at a particular history v,t v.t (**) V,t or S r a. v 3 c v L+ d t (^3) Where , q } and Cy are the instantaneous, historydependent values of the expansivity, compressibility, and heat capacity respectively. The last term Dj represents the instantaneous loss of rate of work during isochoric temperature changes. Both the thermal dissipation Dy and the volumetric dissipation Dw will be discussed in more details in the next section. Summarizing, equation 36 indicates that the volume dependence of the responses of the system determines its pressure; equation hi indicates that the temperature dependence of the response determines the entropy of the system.
PAGE 63
CO Dissipative Terms The dissipative terms Dy, and D T are defined by equations 3 1 and 38 respectively. Explicitly: and > v 5 (^ (kk) D T 42JU (45) o t V,T However, these parameters were rewritten in terms of the rate of d i s sipation of total work (free energy). Equations 35 and k] state , D V = 3_(3a\ (35a) 3V\3t/ and D T = " L_f ?A.l (^1a) where 3A r D ^ 3t The interpretation of 3.. and 3 needs some comments. The development of equations 35a and k]a required the interchange of partial operators, e.g., 7 * ' at UvJ v ;atj
PAGE 64
51 Therefore, in order to provide a better understanding of the instantaneous dissipation of the response variables, their relation to the history term must be discussed. Pressure will De chosen to illustrate the discussion. Differentiating equation 30: d_ p(v, T; M% T 1 ) = iP(V, T) 4
PAGE 65
52 \I,T Â„ 1 ('49) V . T sines at constant V and T the first two terms of equation h~] vanish. However, because of the smoothness condition (quasi elasticity) } W V,T \3tj V; T (50) By equating equations kS and 50 it is evident that the dissipation terms represent the instantaneous rate of change of the response due to the Influence of the history of the thermodynamic state, at the particular instant under consideration. The signs of the dissipative rates may be of particular interest for interpretation of experimental data. In order to establish the parameters which may control these signs in a particular process the relationship of Dy and D_. to free energy must be studied. Using equation ^9 and substituting the definition of P for a quas i el ast i c material the following equation is obtained 3(SLV 3t \3V/ "t 9 31" 3H Ai (51) Furthermore, by changing the order of partial operators; MS( ; ; (Â®* <#>
PAGE 66
53 or d\> k 3V* (53) An apparent result from this equation is that the change of free energy of the material with respect to the history of a state variable is always opposite in sign to the rate of change of that variable during the process; the material is not expected to gain free energy as a result of remembering the past. This result could have been established directly as a consequence of the Claus ius Duhem inequality (see equation B~31b). In order to obtain information on the instantaneous dissipative change in pressure, it is necessary to determine the effect the history of the material has had upon the volume dependence of the rale of dissipation of free energy. An equation similar to equation 53 can be obtained for the entropy dissipation term. 3Â— D 3T 3_ 9T CM, 3VV (5*0 This equation indicates that the dissipative entropy term will depend on the marfher the history affected the temperature dependence of the ra, L e of diss ipat ion. Both the "freevolu :" and the "cooperative" theories predict that te peral ires and \ 1 will increase the rate of dissipative
PAGE 67
54 processes. However, the concern here is not the effect of temperature or volume upon the dissipation rates in general, but specifically, the temperature and vol una dependence, after a given history, of the d is;; I pat ion rates. Up to this point, the thermodynamic theory of quasi elast ic materials has been of help to describe and analyze the responses of materials undergoing a specific process. In order to progress further into the explanation of the magnitude and sign of the parameters (partial derivatives evaluated at V, T, and t) describing the process response, a materials theory is required; perhaps, many materials theories will be required. However, before a materials theory is proposed it may be convenient to determine experimentally the behavior of the material of interest.
PAGE 68
CHAPTER iV PROPOSED GENERAL DESCRIPTION OF GLASS I NESS in Chapter l!i ; pp. 358, it was stated that glassiness could be detected and studied on flew processes. However, the quantitative definition given in that same section required that the measurement o^ the characteristic tirre be performed in the absence of instantaneous dissipation. This requirement implies that in order to be certain that the anomalous rise of shear viscosity of a certain material when cooled is caused by the same phenomena causing volume delays, nonflow relaxational experiments must be performed. These experiments are necessary to establish the relative magnitude of the glassy phenomenon versus other type of interactions which may be present in flow processes. It is this writer's opinion that a material may exist which exhibits a large glasslike dependence of shear viscosity upon temperature but does not show marked glassy mechanical effects in a non~flow process. For example, polymer melts possessing a wide molecular weight distribution, or a wide spectrum of chemical species, may behave differently in flow and nonflow processes. In a flow process, a large shear viscosity may be caused by steric opposition of large molecules which may be partially crystallized (or in a solidlike partially ordered structure). This same polymer melt may not show a correspondingly Id's' volume viscosity in a nonflow proc55
PAGE 69
56 If the "ordering" of the above materia] occurs without appreciable change in volume, no significant mechanical effecc may be noticed; i.e., the stress required for ordering or crystallization may be negligible. However, the entropic effects of ordering may be important. Furthermore, these effects (energy exchange with internal degrees of freedom) may be delayed. If this were the case, temperature equilibration would not be instantaneous. For such a material, glassiness would be observable only by these temperature delays. The discussion on the preceding paragraphs is intended to bring to the reader's attention the possibility that mechanical effects observed in isochric flow processes may not have a mechanical counterpart in homogeneous nonflow processes. Large viscosities do not necessarily cause glassiness; they may be indirect manifestations of the ph 1a causing glassiness. Glassiness, as defined previously b> this writer, is the manifestation of thermodynamic relaxat ion. In this chapter a general description of glassiness will be proposed which, hopefully, will allow determination of the thermodynamic data necessary for the corroboration of existing theories of glassiness, or for the proposal of new ones. The contribution intended in this chapter is to graphically describe glassiness in a way consistent to the thermodynamic equations and concepts discussed i n Chapter III.
PAGE 70
57 A Â• Glassiness and Exce ss Thermodyn amic Functions Changes, in Volume in Unconstrained Material s_. "Constant" Pressure Experiments Tlia usua! cype of experiment encountered in determination of glass transition temperature consists of rapid cooling of the sample, and recording of i;:s volume as a function of temperature at a constant ambient pressure. The description of the deformation undergone by a material contracting in such an experiment may be quite complicated, even if a uniform temperature throughout the sample is approximated. For a liquid, whose viscosity may be negligible, uniform temperature insures a local density in equilibrium with the temperature at the pressure of the experiment. However, as the liquid turns into a glass and relaxational phenomena become important, a change in temperature will roc be followed by the equilibrium change in volume. More importantly, a pressure in excess to the pressure corresponding to the equilibrium situation (infinitely slow changes) will appear. This excess pressure is caused by the now significant opposition to volume changes. The pressure in general will be distributed nonhomogeneous ly throughout an unconstrained sample. Force balances must be satisfied at the free boundaries, causing shattering of the glass in sor ne extreme cases. a A more specific definition o? "excess" will be given in the next sect i on.
PAGE 71
58 if the purpose of a non homogeneous unconstrained experiment were to determine a volume viscosity, it would be necessary to obtain a description of such a chaotic phenomenon. Otherwise, shear resistances present in all nonisotropic volume deformations will be included in the volumetric viscosity. Excess Thermodynamic Functions A glass was defined as a material whose thermodynamic properties depended upon the prior history of its thermodynamic state. The dependence on the prior history implies a smooth transition from the past to the present. The following equations express this continuity of the response of the system on the past values of V and T. F p(V, T) + H P (V^ Tt) (30) S s(V, T) + H S (V, T) (55) '>(#) . *$ i + D \/ (3D T,t T,t = (f) \r) T + D s (42) ."7 v.t v,t Equations 30 and 53 indicate that the response cf the system to the history oP its thermodynamic state (V . T ) is the sum of two contributions: the pressure or entropy of the static state (V, T), and the contribution of ell the thermodyi ; states prior
PAGE 72
59 to the present one (V, T) . The contribution of the past states may be conceived as an "excess" thermodynamic function., an excess over the values of the thermodynamic state had the system been at the present values of V and T, always. This static state wi I 1 be called the rest state. The rest state is, of course, a hypothetical or fictive state. However, it could be conceived as either an equilibrium state or as a relaxed state. Therefore, the rest state should be treated experimentally according to the choice made. If the system had always been at V and T, it would have never experienced an excess thermodynamic function. In this respect, if the thermodynamic state at V and T were approached without ever experiencing an "excess," i .e, by an infinitely slow process (thermostatic), the rest state could be conceived as an equilibrium state. On the other hand, if a material had arrived at V and T through a dynamic process, it may have experienced excess thermodynamic functions. In this case, if the material were held at V and T indefinitely, the excess functions would dissipate. This, experimentally obtainable, relaxed state could be conceived as the rest state. The rest state would be the present state (V, T, t) as t Â— >Â• <Â». The equilibrium state and the relaxed state are not necessarily the same. The changes induced in the material during a dynamic process may be qualitatively different from the chan< occurring had the process been reversible. Furthermore, each history may have caused p i :nt ch s to occur within tl
PAGE 73
60 material, in which case the relaxed state may be different at V and T. However, a compromise is possible. For the purpose of illustration, it will be assumed that the relaxed state at V and T will be the same regardless of the history, hed there been a history, This is similar to the principle of static continuation or stress relaxation of Coleman and Noll and is implied in the thermodynamic theory of quasi elastic materials. The validity of this assumption could be determined experimentally, Experimentally, the relaxed state is chosen as the reference (static) state for equations 30 and 55. Figures 7 and 8 illustrate the concepts of the relaxed and the equilibrium states. The curve ABC in Figure 7 represents an isothermal homogeneous compression, at a finite rate. The compression process has been stopped at V f and P f and the material kept at a constant V f , and temperature. Curve CDE represents the relaxation of the excess thermodynamic functions (e.g., excess pressure). Note that the abscissa has been changed from a volume scale to an arbitrary time scale to illustrate the relaxation phenomenon. The pressure at t Â— *oc tends to a value P r . This value represents the relaxed value. The value aP is the contribution of the history to the thermodynamic state of the system at V f and T. Curve A'B'C represents a process similar to ABC, but at a higher compression rate; aP ' is now the contribution a For illustration purposes pressure will be chosen a; ths response of the system.
PAGE 74
61 0. a: ijj to in zd cÂ± ui
PAGE 75
,'.:. I m t/j o X ;> a) Â•M ra CO I J . . .... .
PAGE 76
63 of the history represented by A'B'C 1 to the thermodynamic state at Vf and T. It should be noted that P r has bean assumed the same for both processes. Figure 8 illustrates the concept of the equilibrium state. The curve ABC represents again a dynamic isothermal, homogeneous compression. The relaxation processes are new represented by the lines at constant volume. The points Mr. , Pc , P where i 1, i i ' i 2, 3 represent the values of the final volume, final pressure, and relaxed pressure for three processes at the same compression rate, but ending at arbitrary values of V. The line uniting the final relaxed states {\'f, P r .) represents the collection of reference states. The line XYZ represents the thermodynamic states of the system if the compression proceeds infinitely slowly. The "equilibrium" process represented by XYZ will not necessarily coincide with the relaxed states (Vf , P r ) . The "equilibrium state" for i i a glass may not be real, the relaxed state is. The concept of values of the thermodynamic functions in excess to the values of the thermodynamic real relaxed state will be the basis for the description of glassiness, in accord with this author's proposed definition. G raphical De s cription of G lass iness Pressure will be chosen to illustrate the proposed description of glassiness. It must be recalled that a history which did not affect the volume depandence of the dissipation rates will not produce any excess pressure. However, such a history may produce
PAGE 77
64 an excess entropy if it affected the temperature dependence (see equat ion kj) . A glass is represented in Figure 9 as any point above or below the plane H_ = 0. The plane H = represents the relaxed state of solids and liquids. Two processes are illustrated in Figure 9. The path I. S c A in the zero excess pressure plane is an infinitely slow process allowing crystallization at T c ; a AV of crystallization is shown. The path LG^ G2 represents a preparation process of a glass G~, whose relaxation processes have become very slow, and could be considered an unrelaxed "solid" at G2. The increase in H p with time along LG] G2 is a consequence of memory. As the glass moves along LG1G2 it remembers its past. If a path of increasing pressure such as LG]G2 is chosen, the material will "remember" state of lower pressure than the present one. V/hen the process is stopped the material will tend towards its past states of lower pressures. Therefore, the excess pressure over the relaxed state Hp must be represented as a positive value. Mathematically, the path LG^t represents the same process except the preparative process ends at a higher temperature and volume than before. At these conditions the relaxation times are small enough to allow observation of the relaxation process along Git. The relaxing glass at t will eventually rest at the plane H p = 0. The corresponding relaxed solid is represented by S,. Figure 10 illustrates the decompressing and heating of a glass. The relaxed solid S r (V , T c ) represents the relaxed stat the
PAGE 78
65 \ rf if t 1 Â» V : .' . . . .''.. ^L(V L ,T L ) Figure 9. Â— Excess Fun :tfons and CI ass inoss ,
PAGE 79
66 ' Â• Â•" ' * Â• ' ' Â• Â• .. . . Â•: : Figure 10. Â—Decompress ion and Heating of a Glass.
PAGE 80
67 glass G. The path L S A is represented again to provide the reader with a graphical reference to the previous graph. The effect of history is represented as a negat ive parameter going through a minimum and disappearing as the liquid is approached. The negative effect of history can be explained by remembering what was said about memory. In this case, as the glass moves along a decompressing path it will remember a state of higher pressures, and if the process were stopped, the glass tend towards these states. The minimum in the GL path and the disappearance of the excess pressure as the liquid is approached are consequences of assumptions about the nature of memory. It is generally accepted that the relaxation processes become faster as the temperature and volume are increased. H iver, the faster relaxation processes do not necessarily mean a decay of the excess pressure. It is only when the excess free energy, brought about by the finite rate of change of the process variables, can be dissipated faster than can be accumulated, that the excess terms will start to decrease. As the liquid is approached both factors contribute to the disappearance of the history term; the relaxation rates are faster, and there are less significant structural changes contributing to the excess free energy. Figure 11 illustrates a simple type of experiment which can be used to study the influence of volume changes upon glass transition; these are isothermal, finite, dynamic compressions. The importance of these experiments is the possibility of direct
PAGE 81
68 Figure 11. Glossiness during Isothermal, Finite, D yn ami, Compressions
PAGE 82
69 determi nat ion of a volurre viscosity. A volume viscosity could be defined by using equation 30 restricted to isothermal processes, The viscosity n. is directly related to the thermodynamic parameters used to describe glassiness. Furthermore, by performing these experiments at different temperatures, the temperature dependence of the mechanical response (pressure vol urr.?.) of the system can be determined. This temperature dependence should establish in part the contribution of entropy to the mechanical responses. This contribution versus the contribution of free volume to relaxation at constant temperature constitute the essence of the dispute between the existing theories of glassiness. Tne contribution of free volume to the rate of relaxstional processes at constant temperature can be determined by studying the relaxation of the excess pressure at different volumes. These relaxational processes are illustrated in Figure 11 by the processes at temperature T] , The broken line connects states at different volumes and which have been relaxing for the same time after preparation. Again, faster relaxation is expected towards the liquid at higher volumes. The processes illustrated at temperature TÂ£, represent a direct manner of studying the effects of history. Varying the preparation time t , the initial volume V , and the direction from which the final state is approached are different manners of studying history effects.
PAGE 83
70 Isobaric Transitions The last section dealt with general considerations and results of an interpretation of glassiness limited to "nonflow processes." TnU is not the type of process used in practice to obtain information about glass transition. In general non homogeneous flow processes are used to obtain such information. A detailed description of such a process is not available. Furthermore, because of the difficulties in describing the deformation mathematically, and the complexities of the boundary value problem necessary to interpret the experimental results, a detailed description may never be available (Appendix B) . However, most experiments of glass transition are carried out isobar ically in the sense that the samples are constrained only by the ambient pressure. Therefore, the study of the behavior of quasi elastic materials in homogeneous isobaric processes may help to understand some of the phenomena present in more complicated processes . Following Davies and Jones in their twodimensional description of glassiness, the first characteristic to note is the dependence of the onset of glassiness on the rate of cooling. If a liquid is cooled at constant pressure and at a rate sufficient to avoid crystallization, it may follow the path ABC, see Figure 12. The pressure at C would generally be lower than the pressure at C. It must be noted that although the process pressure P is constant in an isobaric process, the excess H p , in genera., is not.
PAGE 84
,,,.... , ' .".'..1' 'i ] Creep Temperature. i ncreas i nq rate". J ! Figure 1 2. Glass i ness during Isobaric Processes.
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72 Davies and Jones indicated that as the cooling rate increased, the glasses formed at higher temperatures. This observation is represented, using the description proposed by this writer, by the paths ABD and ABE. It must be recognized that as the cooling rate increases the excess pressure also increases; these effects have been considered irrelevant to the observation of glassiness. It is this writer's opinion that the cause of the observed rate dependence of glassiness at constant pressure is the delay in volume caused by dissipative processes. The increased "sharpness" of the curvature induced by higher cooling rates is illustrated in Figure 12. Mathematically, the delay in volume, for a cooling process at constant pressure (P = 0) represented by T = T(t) is given using equation 31 by: V = (3 V S T + D v ) / 3 V P (56) The first term in this equation represents the instantaneous (equilibrium) changes in volume. The term DJ 3yP represents the delay in the rate of change in volume due to the presence of dissipative processes. This delayed volume change is made evident if the sample is held at constant pressure and temperature after deformation. The phenomenon is called creep, see Figure 12.
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s CHAPTER V DEVELOPMENT OF EXPERIMENTAL TECHNIQUE The contribution of this lesearch in the exper imental field consisted in the development of a method to study homogeneous changes in volumes of asphalts without the interference of shear viscous forces. This technique can be used to study the transition of a substance from a liquid like to a solidlike material, and the influence of the volume and temperature histories upon this phenomenon, This chapter will deal with the details involved in the development of the experimental technique. The comparative discussion of ome experimental results on the selected asphalts will be presented in the next chapter. The analysis of this general type of data will be presented in Chapter VI. This analysis will point to ranges of temperatures, pressures and densities where further studies should provide more specific information about the behavior of the different asphalts. Muller (33) in a recent review on the thermodynamics of deformation and the calorimetric investigations of deformation processes, discussed the current state of the art on experimental methods. The author presented numerous examples of efforts to understand the causes of the mechanical behavior of materials by measuring thermomechanical effects. The work, however, concentrates on the discussion of elongation calorimetry and other experiments at constant volume. This author provides extensive discussion of 73
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e lh the different thermostati c effect;which could be studied with an elongational calorimeter designed by Engelter and himself (17). Dynamic effects are discussed only qualitatively since this calorimeter method shows a cons iderable delay between the time a deformation is induced and the time when heat effects are recorded, Howver, this author recognized the importance of irreversible effects and proposed the use of the calorimeter to determine permanent final effects (state differences) caused by deformation, e.g., the degree of crystallization induced by the deformation. Muller also describes compressive experiments where temperature variations have been recorded in order to study thermomechani cal effects. However, this author dismissed this approach because all attempts to place a thermocouple inside the sample were unsuccessful and because, "in addition, the degree of deformation during compression of a cylinder is known to vary considerably from point to point (flow cone formation is a well known phenomenon)" (33). The compression technique used consisted of compressing a sample cylinder (approximately 2 cm. in height and 2 en. in diameter) between two plates. The sample was not confined or restricted to flow axially; the cylinder was deformed to the shape of a barrel by compressing up to about 20 per cent compression. Shear viscous forces must have been present on this ncn homogeneous deformation. The experimental method proposed herein to study the thermodynamic properties of asphalts in nonflow processes consists of c .essing a cylindrical sample of approximately 1 to k inches long and 3/8 inche in diameter in a confining steel barrel. The
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75 asphalt specimen is enclosed in a rubber (latex) balloon which in turn is coated with silicone oil to reduce the drag between the balloon and the steel barrel. The balloon enclosure is leak proof. A calibrated thermistor or thermocouple can be inserted to determine the increase or decrease in temperature due to the heat of compression or decompression. Figure 13 shows a line sketch of compression equipment. A great deal of work was involved in the development of the above technique (see Appendix A). The improvement to Muller's (33) and others contributions consisted on implementing through persistent experimentation a method of reducing the effect of shear forces in the study of dynamic compression. The motivation behind this effort was the firm belief that foremost to any conclusion on the causes of the socalled glass 1 i ke phenomena discussed in previous chapters was the necessity to describe the changes undergone by the material. This writer suspected that some of the effects attributed to delays in volume were caused by the shear forces present on any nonhomogeneous volume change, e.g, those causing the "wellknown flow cone formation" mentioned by Muller. Therefore, reduction if not elimination of shear forces was considered the first step towards the description and study of processes involving volume changes. This chapter will consist of three sections. A section containing a brief description of the apparatus used to implement the technique and the list of physical properties of asphaltic materials employed. A second section containing some preliminary
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7b COMPRESSION LOAD CELL (FR) MOVING CROSS HEAD LOAD CELL EXTENSION SUPPORT PLATE BARREL PLUNGER ASPHALT PLUG In r CLAMPING NU1 16 ! Figure 13.~~L?ne Sketch of Compression Equipment
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11 considerat ions, and finally, a third section where the technique is analyzed and data are presented in support of the premise that the deformation obtained is indeed homogeneous, and that shear interference is eliminated. A . Mate rials a nd Aopara'. i , ._ Four asphalt cements were selected to test the applicability of the dynamic compress ion technique: Texas SteamRefined Intermediate (5639)., LowSulfur AirBlown Naphthenic (S6313), Los Angeles Basin (S63~20), and a United States Bureau of Public Roads asphalt, especially chosen because it had a considerably lower penetration than the other three. The USBPR identification number of this sample is B 3057, the ident if icat icr. nu ber given by the University of Florida Asphalt Laboratory is SGkkl . Table 1 summarizes the physicochemical properties of these asphalts. The apparatus used to compress the materials and to register the required forces was an instron Universal Testing instrument (Model TT L C, Modification M3l). A detailed description of the parts and capabilities of this machine can be found in a thesis by Ronk (39). Rorik's work was part of the concerted effort being made at the asphalt Laboratory of the University of Florida to establish the basic characteristics of these materials (see Appendix A) . Figure 1^ shows the Instron Universal Testing machine. Compression o^ the : . is accomplished by lowering the traveling crosshead towards a fixed table supporting a ir if : ron
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73 Table 1 Properties of the Four Selected Asphalt Cements Identification S639 S c3 ~1 3 S63~20 SGkky Density (25Â°C),gm/cm 3 1.033 O.988 1.015 1.025 Penetration (25Â°C) 85 S9 83 )k Â„ + Ductility (25 C) cm. 15C 150 150 150 Softening Pointy Â°C (Ring and Ball Method) *f6.1 48.3 46.1 o Glass Transition Point, C (Penetrometer Method) "13.9 10.8 Viscosity (25Â°C), poise 1.02x10 1.05xl0 6 0.66xl0 6 Viscosity (60Â°C), poise 1.70x10 1.73xl0 3 l.llxlO 3 5.22x10 3 Molecular Weight 3^3 339 769 (Number average) a Generic Groups (Per cent by weight) ParaffinicNaphthenic 13.1 23.9 10.3 NaphthenicAromatic 28.3 23.6 24.2 Heavy Aromatic 43.7 37.6 52.3 Hexasphaltenes 15.8 12.1 11.8 Petroler.es 85.1 87.3 90.0 Based on Schwsyer Chi pley separation procedure.
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79 Fioure ]k. I nstron Uriivei,a] Testing Machine.
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80 Capillary Rheometer Assembly (Figures 13 and 15). The compression rates used ranged from 0.5 in/min to 0.0002 fn/min. The travel of the crosshead and therefore the actual length co the sample can be determined within four tenthousandths of an inch. The force applied with ths piston to compress the sample is measured by a calibrated straingage cell which provides an accuracy of 0.25 per cent of the scile used. There are six scales ranging from 0200 co 010000 pounds. The modified instron Capillary Rheometer Assembly was enclosed in an environmental chamber for temperature control. The operating temperature range of this chamber is from about ~kQ to 50 degrees Centigrade. Cooling capacity is provided by two standard refrigeration units., and by liquid N2 injection. Excess cooling capacity is balanced by a variable 1000 watt heater. Precision control of the temperature is achieved with a Demote nsky sensing element, an onoff controller, and a 100 watt light bulb. A high velocity air recirculating system (100 cuft/min) provides a uniform temperature (0.1 C) throughout the chamber. The temperature of the barrel can be controlled to 0.1 C. A thermocouple inserted in the barrel indicated that the temperature of the asphalt when in equilibrium was independent of the axial distance indicating that heat transfer through the bottom plug and the compressing plunger was negligible (39). B . P rel iminar ies This section describes studies o r : two basic questions concerning the experimental method: (1) the e of "aging," 01 t T me
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81 m ..... . Â• . Â• : " I 0.125^ 1 k0.375 E ' 0.200" [ i k; 20 0.1468 M 0.374Q pm >Â£ _JL_ 0.125 ' / 1 1 fr 1.125 All d i mens ions in inches 0.365
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82 temperature history from the time the sample is prepared to the time testing begins; and (2) the effect upon the results of the machine characteristics (e.g., elastic deformation, time responses., etc. of the metal components) . Age, Prepa ration of the Sample and Presentation of Re su l ts Studies by Stewart and this writer during the development of the final experimental method proved that the time temperature history of the asphalt samples from the time of preparation of the sample to the time the testing began was of no relevance, provided that the asphalt samples were not heated above temperatures of 300 F for extended periods of time (see Appendix A). Heating for 15 minutes at 320Â°F has been found satisfactory to melt the samples without affecting the physical properties. The treatment of the sample prior to testing consisted in pouring the molten sample into the latex balloons, sealing the balloons, and setting the sample for compression (see Figure 13). The specific manner in which the balloons were sealed depended upon the objective of the run, and will be discussed later. The samples were sometimes held at room temperature for various lengths of time before running with no apparent change in their compressive characteristics. Table 2 summarizes the result of three runs on the same sample ten days apart. Before discussing the results presented in Table 2 tiie origin of these numbers must be discussed. The results p ted in 1 2 were obtained by calculations using
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83 Table 2 Effect of Aging Sample $6309 Temp: 25Â°C Run No. R7015 Reference (XLI, 95100104) Velocity of Crosshead VXH 0.1 in /in i n Date Date Date 1/31/70 2/5/70 2/10/70 p Â° P 3 P 3 p 3 0. 1.0303 4.49 1.0294 4.48 1.0297 4.59 185 1.0350 4.16 1.0380 4.13 1.0382 4.25 615 1.0562 3.42 1.0553 3.50 1.0557 3.48 923 1.0666 2.90 1.0660 3.03 1.0662 2.95 1000 1.0690 2.78 1.0636 2.91 1.0686 2.82 P ~ pressure (atm.), p density (gin. /cm ),3= compressibility (atm~l x 105)
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8h three types of data: (1) the weight of the sample determined directly, (2) the length of the sample determined with the Instron machine, and (3) the change in length versus load requirements, obtained from the recorder chart. The weight of the sample can be determined at the beginning from the weight of the balloon and the weight of the sample plus the balloon. The sample weight can also be determined at the end since the sample and balloon are easily recovered in most cases. Weights are determined with an analytical balance and are considered accurate to within 0.1 per cent. The length of the sample is obtained by direct measurement with the instron after the sample is in place. Although the travel of the crosshead is accurate to within four tenthousandths of an inch, it is believed that determination of the initial length is the most important source of error in the determi no:, i on of densities. Improper tightening of the clamping nut and other threading attachments have been found a common source of error. However, after the sample is in place and an initial reading has been accepted , the deformation of a sample is reproducible to within four tenthousandths of an inch. For example, a typical asphalt sample may show a deformation when compressed between the load limits of the assembly (02000 lbs.) of 1000 x 10 inches. The load measuring device provided an accuracy of 0.25 per cent of the scale used as mentioned before. Densities are calculated directly from the data mentioned above after correcting for machine
PAGE 98
85 deformation (see next section). Compressibilities are calculated from an analytical expression obtained by fitting a polynomial to the deformation versus load data obtained from the recorder chart. After trying polynomials of the first, second, third, and fourth orders it was found that the second order polynomial gave the best fitting with an error never larger than 0.01 per cent between the calculated values and the recorded values. The results in Table 2 clearly illustrate that the age of the sample had no effect on the results. The variations on density and compress i bi litres are well within the experimental recording errors, The correspondence of the values of the compressibility is a sensitive test of reproducibility, since these values are obtained from the derivative of the experimental curves. Particularly the values at and 1000 atmospheres which are calculated by extrapolation are very sensitive indicators; small differences in the data would be amplified when comparing the slopes at these extrapolated points. Although data are presented in Table 2 for one asphalt similar results were observed for the other three samples used. The practical importance of the results of this section is that a sample cf asphalt can be poured, and be ready for testing within a few minutes or stored for study. This permits preparation of several samples simultaneously and saving of time if the procedure were used routinely.
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86 Machine Charac teristics, Assembl y deformation Ronk estimated the deformation of the rheorneter assembly when applying a pressure (39). However, the effect of different compression rates was not determined. Calibration of the machine consists on compressing a piece of steel and recording the deformation versus the load required. No differences in the calibration curves were noticed when compressing steel samples at rates ranging from 0.005 in/min to 0.2 in/min (XL I ^ 1 08) . Figure 16 is an actual tracing of load versus deformation at two different rates of compression. The recorder chart was rolled back to match the starting point, the chart speeds were adjusted so that for both cases one inch traveled by the chirt represented 0.01 in of travel by the moving crosshead The two curves coincided. Coating the barrel with a lubricant did not affect the cal i brat i ens . The correction due to the deformation of the assembly was calculated from the deformation of steel cylinders at different lengths (XL 1 ,3^, ^7) . The correction was calculated by assuming that deformation of the assembly and that of the steel samples were additive. The deformation of the steel sam i re calculated directly from the applied load and the bulk modulus of steel. By References designated by a roman numeral followed by an arabic number refer to pages (arabic) in the research notebooks (roman) of different investigators working in the Asphalt Laboratory of the University of Florida.
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87 ZI*&itSUXZ?aV&*rX:*jKk '.:i.w*x%jriwcv.<^W 16 L 14 r 12 lor81i I 1 VXH 0.2 in/mi n ...^. .v.,..^,^ vxj; c.005 in/min / I 2 2 Defcrmr.t'on (inches x 10 ) 3 ^ . 5 Figure 16. ; racing of Load versus Deformation for Calibration at Two Rates of C >mpression.
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88 subtraction the def ormat i on of the assembly was found. The deformation of the assembly was used to correct the readings indicating the total travel of the machine. The total travel of the machine minus the correction is the deformation of the sample. This correction is given by the following equation: AL = 1.973 x 10~9 F 2 + 1.8^0 x 10~ 5 F 2.608 x 10" 3 (57) where AL is the assembly deformation in inches and F represents the applied force in pounds. Periodic checks on the validity of this equation indicate that the assembly deformation remains the same. The values of machine deformation determined at three temperatures 25Â°Cj 0Â°C and 30Â°C indicated that only small corrections had to be included to account for the change in temperature in the environmental chamber. A correction of approximately 5x10' inches per each degree centigrade below room temperature must be added to the length of the sample (XL, 81). Equation 57 remains the same at the different temperatures; the only correction is to the total length of the sample. The redial deformation of the confining barrel has been estimated very conservatively at less than 0.08 per cent of the diameter (>'LI, 130133). This small deformation will be considered negligible and no correction will be used for it. Cross h ead f ollo wup., gea r tol erance An extensive experimental study (XL I ^ 99~100) of the control mechanisms of the Instron testii , machine corroborated the manufacturers
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39 claim of the ability of the crosshead to move at the desired constant velocities (27). The followup errors caused by the inability of the machine to stop or start moving instantaneously at a constant rate are negligible, less than 0.0001 in at 0.2 in/min and 0.0004 in at 2.0 in/min. The limit to how fast a sample can be deformed and still obtain reliable load versus deformation data seems to be the response characteristics of the recorder. For accurate force data the recorder pen should not travel at a speed faster than that required to traverse the chart in less than three seconds (39). A crosshead speed oF 0.5 in/min is recommended as a practical upper limit for the compression experiments. However, when the direction of travel of the crosshead is reversed, a period of acceleration of the crosshead is noticed (XL 1 , 99) This acceleration is accentuated by the fact that the Â• "play" or tolerance of the gears driving the crosshead is in the order of 4 x 10 inches. i he velocity at which the crosshead moves is proportional to the difference (error) between the pos i ~ tion of the crosshead at any instant and the position at which it should be according to a synchronous motor used for setting the reference position (27). When the command is given to the machine to reverse direction the crosshead will not move until the 4x10 inches of "play" have been reduced to zero and the gears start driving the crosshead in the opposite direction. However, the synchronous motor would have reversed direction immediately and therefore, the error would have been increasing until the crosshead
PAGE 103
90 had started to move. This large error demands an increased speed of the crosshead until the followup error is brought back to normal and the crosshead keeps F ace with the synchronous motor. This effect is mentioned here because it limits the usefulness of cycling experiments and because the sudden change observed in the load versus deformation relationship may be confused with a material effect. Because of the above phenomenon, whenever preparing for a compression or decompression prec/Cvat ions must be taken. To this effect, it is recommended that the last direction in which the crosshead had moved before the Lest be the same as the direction in which it is to move for the test. Figure 17 illustrates the "acceleration" effect. Curve 1 is an actual decompression test (crosshead upwards) ran after the sample had been compressed (crosshead downwards) to a stating load and waited for a period of time enough to establish equilibrium. Curve 2 is a decompression run (crosshead upwards) of the same material which had been compressed to a load slightly higher than the intended starting load for the decompression test and then the crosshead reversed to bring the load back (crosshead upwards) to this starting load. Enough time was allowed for equilibrium. As it is apparent in Figure 17, no "acceleration" effect can be noticed when the above recommendation was observed. At crosshead rates of 0.1 in/mlr end higher the small "acceleration" effects caused by the normal followup error somet irres observed even though precautions .re taken to eliminate the r ree play of the gears.
PAGE 104
. . drx.sn , T. c o +j o (0 o L. Â— CD sÂ— CD a> Qo o f0 o o o 0) Â•ft L. 0) > ^
PAGE 105
92 C . Homogeneous Deformation The importance of homogeneous deformation has been discussed in Appendix B, part Id and in Chapter III. A homogeneous deformation allows assignment, to the v;hole body, of the response measured at a single point of the material. In the case under consideration if the drag or the sample on the walls of the barrel were completely eliminated the load registered by the sensing element at the surface in contact with the compressing piston could be assumed to be the same throughout the sample. Furthermore, if the drag were eliminated completely, a sensing element (load cell) placed in contact with the bottom end of the sample would read exactly the sane as a load cell in contact with the moving piston. Also; another method to check the validity of the assumption of homogeneity of the dynamic changes in volume is to study the effect of sample length upon the values of density versus pressure determined dynamically, particularly the effect upon the values of the initial slope of the density versus pressure curves. Both methods have been used to determine the extent of drag reduction obtained with different lubricants (see Appendix A). In this section a detailed analysis of the results obtained with these two methods will be presented. It should be pointed out that this writer die not concc"ve these methods at the onset of this research. Instead, they became necessary ?z complements of each other. Effects which appeared to be manifestation of ron
PAGE 106
93 homogeneities had either to be explained or the whole effort abandoned. The basic working assumption taken by this author in order to develop an experimental procedure which would be acceptable to himself was, that the response of asphaltic materials to dynamic compression was perfectly elastic. With this working assumption any time or rate dependent effect observed was attributed to viscous dissipation at the surfaces or to unknown characteristics of the machine until proven otherwise. Some of the observations described in the next sections may be considered irrelevant, however, it is believed that they may be of help if a lubrication procedure had to be developed for other substances under conditions net contemplated in this research. I ndepend en ce of Dy namic Response on Sample Length in Relation to Homogeneity Figure 18 illustrates the results obtained by compressing two samples of different length without any lubrication. An initial almost linear portion is obtained followed by a curve. This linear portion corresponds almost exactly to the calibration line for the assembly deformation. Therefore, almost no deformation of the sample occurs during this timeonly the assembly is deformed. The load sustained by the ]zrcer sample before actually deforming is larger than the load supported by the smaller sample fcr the same velocity of the crosshead. This clearly indicated mat these effects were due to drag on the alls; the larger the surface available the larger the losd which could be supported before
PAGE 107
LA O fi Â• Â— O ! vT> X O to > o <_> ^ o ! 00 VD ! fN J vD , .*>
PAGE 108
95 deforming. Also, had it been a homogeneous effect the ratio of crosshead velocity to sample length should have determined the effective rate of deformation, in which case the smaller sample should have shown a larger initial increase in lead. This conclusion was definitely corroborated by removing the bottom plug and repeating the experiment. The same initial effects were, observed when there was nothing to stop the samples from flowing except the drag on the walls. Actually, no flow was observed out of the open end of the barrel until the load had build up to approximately the same load corresponding to the deviation from the linear behavior. These results were expected, however, it must be remembered that some works had been published (3't) , (35) which neglected these effects a pr ior i . The relevance of these non homogeneous experiments was that they suggested a fast quantitative method to judge the effectiveness of a given lubrication procedure. Previous to the idea of using a latex balloon to enclose the sample and avoid adhesion of the asphalt to the walls of the confin ing cylinder silicone stopcock grease end other lubricants hsd been used for this purpose (see Appendix A). Figure 15 shows decompression curves for two samples of different length using the best lubricating method found. Namely, enclosing the sample in a balloon and coating the outside of the balloon with Dow Corning Silicone 200 fluid. The runs illustrated are at 30Â°C. Similar curves were obtained for each of the four selected asphalts at ~30, 0, 25, and l )5Â°C with similar results.
PAGE 109
9b " ' . ' .
PAGE 110
97 !n none of the casas was there an initially larger slope which could be relate d to sample length . In some perimeters a small initially larger slope was observed but it was attributed to the "acceleration" effect discussed in section B of this chapter. Table 3 summarizes some of the results illustrating the influence of sample length on density and compressibility. The results shown in Table 3 are from decompression experiments. Decompression experiments are more critical since lubrication is more likely to fail at the higher pressures. The results summarized in Table 3 and the shape of the curves illustrated in Figure 19 indicate very strongly that the deformation occurring during compression and decompression when using proper lubrication are independent of sample size and possibly homogeneous. There is a disturbing aspect concerning the results in Table 3. The results are the same for the same VXH fcross'head velocity) and do not seem to depend on the rat T o of velocity to length, which is supposed to represent the homogeneous deformation rate. This aspect was the object of a thorough analysis leading to interesting and important applications cf the experimental techniques The experimental results presented in this section, however, demonstrate the advantages of the present lubrication techniques over the point by point non~l ubr i cat i on method used by Ronk (Appendix A) in determining the P,V,T thermostatic relationship. Ronk's method because of the drag required up to sixteen hours tc determine sufficient V vs. F points at one temperature to establish an equation. The method proposed here at a crosshead speed of
PAGE 111
93
PAGE 112
99 0.005 in/min requires less than 30 minutes to provide a continuous P vs. V curve. Study of Dynamic Effects, Rela ting to Homogeneous Deformation As illustrated in Table 3 the values of the compressibilities 3 depended upon duration of the experiment, crosshead velocity (VXH) of 0.005 vs. 0.100 inches per minute. However, these values did not seem to depend (for a given VXH) on the size of the sample. The dependence of 3 on VXH seemed more pronounced than its dependence on the rate of deformation y. This observation seems to indicate the presence of nonhomogeneous dynamic effects. An analysis of this phenomenon will be attempted in Lhe following paragraphs . The compressibility 3 is given by: v U> J" L \? p ) (58) N T T, C where V is the volume of the sample, L its length, and C its crosssectional area, P is the pressure. However, the values in Table 3 are the overall and not the partial changes with respect to pressure. Therefore, equation 37, of Chapter IV, expressing the overall rate of change of pressure should be used; the overall rate of change of pressure P is related directly to the overall compressibility reported on Table 3,
PAGE 113
100 P = dP 6P_ dL _ L dt dL dt dÂ£. d L dL dt I L I VXH Y S L 3 (59) Equation 37 expressed P as a function of the process variables, P K X + q v I + D v (37) Substituting 59 into 37 2nd observing that; V/V = L./L = y, r y K Y + Q 'V I + Â» D. (60) as can be noticed none of the terms of equation 60 depends on VXH alone but on Y . The dissipation term Dm is given by: \ a ,t . p) r+ 4 ) L t L 4 (61) Equation Gl is direct consequence of equation 53 of Chapter IV. The rate change of the length history Lis given by t dLJLs = dÂ„ /L (t) + VXH * s\ dt ds { L(t) J (62) s Â»o VXH and the homogeneous dissipation term Dw should also depend on y rather then VXH.
PAGE 114
10] Therefore, the dependence of the values of 3 reported in Table 3 do indicate that there is a non homogeneous dissipative effect . As discussed in Chapter 111 non homogeneous dissipative effects are caused by flow processes: momentum flew (instantaneous viscous dissipations) and entropy flow (heat transfer). In simpler words, the nonhomogeneous dissipation occurs because of the delay in dissipation of the heat of compression, i.e., q,, Â— Â£ 0, because of viscous flow dissipation, or both. V j An experimental program had to be developed to establish the contributions of these two non homogeneous sources of dissipation, !t was hoped that these effects could be either eliminated or be described quantitatively. Otherwise, there would be no opportunity to determine reliably the contribution of the homogeneou s dissipation term to the process. Without a reliable measurement of the contribution of the homogeneous dissipation term the technique would be of no use to study glassiness or nonflow dynamic processes. Summarizing, the following three sources of dissipation had to be isolated experimentally: Instantaneous viscous responses due to poor lubrication. Heat transfer due to temperature changes, Homogeneous nonflow relaxations! processes. Each of these phenomena has a particular characteristic which could be used experimentally to identify its contribution.
PAGE 115
02 Instantaneous viscous dissipation at the walls will produce forces which will be larger at the end of the sample where the confining cylinder is in relative motion with the sample. These forces will be always opposite to this relative motion. To differentiate between the two remaining dissipations it suffices to realize that heat dissipation must occur through temperature gradients, Therefore, by measuring the temperature difference between the sample and the cylinder as a function of time the effect of heat transfer could be identified. Additional information could be obtained by remembering that the homogeneous relaxational dissipation term Dy is determined by the vol ume temperature history (equation 61). For example, this additional information can be obtained by studying pressure relaxational effects after compression and comparing these with recoil effects after decompression. EllSSiiJ. venes s of lubrication Figure 20 illustrates an assembly used to identify the load required to overcome the drag on the walls. This assembly permitted the choice of which of the two ends of the sample would be in relative motion with the walls of the barrel. It was established that the force required to move the barrel by hand when maintaining a compressive pressure of 1000 atmospheres was negligible (XLI,^7). The barrel was forced to rest against the fixed crosshesd by properly positioning very strong springs. Thus, wren supported in this manner the sample end in contact with the DR cell would not move relative to the barrel during cc sion or decompression.
PAGE 116
103 FR CE!.L MOVING CROSS HEAD EXTENSION SPRING BARREL PLUNG :. ASPHALT SUPPORT FOR BARREL EXTENSION DR CELL FIXED CROSSHEAD i NT " L LjCE: i I i 1 r a it r 111 [\\\^ hS s *. i_ iFigure 20.*.:: ibly foe Studies on Effectiveness of Lubrication,
PAGE 117
\ok With the barrel supported against the fixed crosshead during decompression the FR cell senses the force C minus the force F. The force C is the sum of the compressive forces resulting from the sampl change in volume and F is the frictional force exherted by the barrel on the sample. However, under the same circumstances the DR cell would sense C plus F, that is, the frictional forces on t he sample are directed downward against the relative movement of the sampl e, with respect to the barrelin this case upward. If any time during decompression the machine is stopped a sudden jump in the recorded load should be noted because F vanishes. The FR cell should record a "sudden" increase in load when the negative effect of F ceases. The opposite should be true for the DR cell. in order to record the contribution of the viscous dissipation at the walls it was convenient to run "recoi I" experiments after decompression. Sy decompressing, e.g., from 1000 lbs to 20 lbs, the more sensitive scale of the recorder could be used to record the recoil (increase) in pressure resulting when decompression was stepped. Figure 21 illustrates a typical decompression experiment, followed by the recoil or increase in pressure. Th s sensitivity of the lead recorder was changed as the losd decreased from a minimum sensitivity of 200 pounds per inch of chart to a maximum of 20 lbs/ in. A set of these experiments was conducted to determine the magnitude and the duration of the supposedly instantaneous viscous dissipation at the walls. e a Actually, the maximum recommended load for the use of the DR cell is 1000 lbs; for the equivalent pressures in Ibs/sq in multiply loads by nine,.
PAGE 118
105 Â• La ! ! Â— . (S9LJDUI) pGO"!
PAGE 119
os A sample of asphalt was compressed to 1000 lbs and after equilibration, decompressed at 0.1 in/min to about 20 lbs. rhe time required for the decompression and the change in length were determined. The recoil was recorded until steady state was reached. The same sample was compressed again, now to a smaller load and the same steps repeated. The decompression time, change in length, and recoil were now smaller. The procedure was repeated for several initial loads covering the range from 20 to 1000 lbs. Since the available recorder had one pen, only the response from one of the cells could be recorded at one time. The above procedure was repeated, recording alternatively with each of the cells. The results with a crosshead velocity of 0.1 in/min are illustrated in Figure 22. The values plotted are the total recoil from stop to steady state versus the decompression tine, I.e., the time the frictional forces had to build up. The difference between the values of the two cells is twice the frictional forces at the walls. It can be seen that this difference is almost constant regardless the duration of the decompression. This indicates thai: the frictional force F builds up almost instantaneously. Also, illustrating this point is the observation that extrapolation to zero would indicate a negative lead :eg T stered by the DR cell. This extrapolated negative reading points out to the sudden decrease in load due to the almost instantaneous fading of F when the machine is stopped.
PAGE 120
107 c o E o <+ci a c o to o u 0) ro o II \ \ \ (sqij [looay ib^oi 
PAGE 121
108 Figure 23 shows the recoil after decompression recorded by the two cells. The chart speed was very rapid (10 inches per minute) so that the time it takes for the wall dissipation to fade afte r stopping could be visualized. This illustration corroborates the previous conclusion that the time it takes for the wall dissipation to appear and disappear is negligible. This effect can be easily identified and taken into account by recording the recoil using a fast chart speed and extrapolating to zero time. Extrapolation to zero time of the DR cell recording shews the positive sign of the effect of F on the readings of the DR cell. it can be seen, as expected, that for the same extrapolated compressive forces at the time of stop, the lead recorded by the DR cell was larger than the load recorded by the FR cell. Di fferentiat i on betwee n hom ogeneou s and no n~homoqeneo us delay effects As was indicated in the discussion presented during the introduction to "Independence of Dynamic Response on Sample Length in Relation to Homogeneity," a direct manner to distinguish between ncn homogeneous heat transfer and homogeneous relaxational effects was to record the temperature of the sample versus the temperature of the surroundings. This was dene and positive results ware obtained. However, the presentation of these results will be delayed and considered as only secondary evidence because it is feared that introduction of a sensing device into the sample could by itself affect the homogeneity of the deformation. Instead, an indirect
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109 i Â• Â»o o 3 JO 0) a l_ o (J a) cÂ£ o in in Â« O O O Ufv J" c"i i I \ I (saipu 5) peon .
PAGE 123
110 me thod was employed. This indirect method was based on the results of equations 61 and 62 that the homogeneous dissipation term D y depended on the prior history which in turn is a function of y and not of the crosshead speed directly. Other decompressionrecoil experiments similar to the ones previously described were run. The immediate objective of these new experiments was to obtain the same total amount of recoil by submitting the asphalt to different histories, then studying the recoil as a function of time. If the recoil at constant temperature was due to the history of the length of the sample as indicated by equation 61, it should be determined by Y a nd the duration of the run, Figure 2'+ is an actual tracing of the recoil versus time on the same asphalt. The total recoil and the force versus time tracing are almost identical in spite of the different length histories. In both cases the sample was compressed at the same Y but in one case the total deformation time was twice as much as in the other. Figure 25 represents again actual tracings of experimental data. In this case the decompression times were the same but the rate of deformation of experiment one was twice that of experiment two. The fact that the same "excess" (recoil) thermodynamic pressure verSLS time p'c 4 :<. were produced by different length histories indicates that these were not the cause of the apparent excess cressure.
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Ill c
PAGE 125
112 c
PAGE 126
113 Figure 26 illustrates even more dramatically the nondependence of the rel c?xat ional phenomenon (recoil) on the prior lengthhistory. The superimposed curves have been obtained by making all steady state pressure after recoil the same. This proves that given a pressure difference from this steady state it will dissipate following the same curve regardless which manner the pressure was obtained, Analyzing equations 60 and 61 and remembering that instantaneous dissipation at the walls have been taken into account, it must be concluded that a change in temperature must have occurred which is the cause of the apparent excess pressure. Given a temperature difference between the sample and the confining barrel, the time dependent process by which the temperature returns to the temperature of the environment is independent of whatever process motivated it. . Thus, the results of Figures 2 l \, 25, and 26. These experiments were repeated with the S6U:7 asphalt sample which is the one with the highest shear viscosity and lower penetration. Similar results were obtained. One of the curves obtained with S&kkj is shown in Figure 26. The use of a pair of matched thermocouples placed inside the sample and on a minute hole drilled into the barrel close to the inside wall demonstrated that the re'exat ional effects observed after compression and decompress ion were related to the return to normal of the temperature of the sample. Figure 27 shows in the same graph and in arbitrary scales the readings of the thermocouples
PAGE 127
]\k ::;^*>u.^ C *i .NO co LA c<\ O O CO LT\ CO O LT\ O Â— O O CJ
PAGE 128
ns Figure 27. Â— Thermocouple and Load Readings during Compression Cycles.
PAGE 129
116 and the force being recorded simultaneously. It can be noted that the sample heats up during compression and cools during decompression. The important point to note is that the change in temperature with time after the deformation process is stopped is the same as the change in time of the excess pressure. At this time, the idea of introducing a temperature sensing device is being developed. Thermistors have been used very satisfactorily and because of their sizes seem to be the answer to a sensing device which would not interfere with the dynamic effects. Another important observation made during these homogeneity studies is that as temperature is decreased relaxational effects are observed which seem to be created by history dependent phenomena. These will be discussed in Chapter VI. Table h summarizes the characteristic tines of the recovery experiments depicted in Figures 2k through 27. As can be seen these are similar and of the order of forty seconds. This characteristic time is of importance to determine the magnitude of the characteristic time of the relaxational processes that can be studied. It can be seen that unless the characteristic time of the relaxational times of 'nterest are larger than at least ninety seconds, no clear cut observation of relaxational effects will be possible with the proposed experimental technique. Fortunately the reported characteristic time of glasses rear their transition temperature exceed fifteen minutes (5). It is believed that the experiments illustrated and discussed in this section prove that effects other the', homogeneous re'laxat ionai
PAGE 130
117 Table k Characteristic Time of Recoil Curves for Several Previous Histories Sample
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118 effects can be quantitatively taken into account, the phenomena related to glassiness can be studied. The main conclusion on this regard is that the three main causes of excess pressures, apparent cr otherwise, can be distinguished by their characteristic time: (a) viscous dissipation at the v.'alls were proven to be almost instantaneous; (b) temperature dependent apparent relaxations have a characteristic time of approximately forty seconds regardless of the asphalt; and (c) homogeneous rtlaxational exper iments concerning glassiness are expected to have a characteristic time of the order of fifteen minutes.
PAGE 132
CHAPTER VI RESULTS, CONCLUSIONS, AND RECOMMENDATIONS The results of the general experiments conducted on the four selected asphalts illustrate the use of the finite dynamic compression technique for the study of glassiness. The manifestation of glassiness in these asphalts was not observed as a discontinuity in its physical properties. That is, at a given temperature the density versus pressure curves were smooth second order polynomials, regardless of the rate of deformation used to compress or decompress. However, as the temperature of the experiment was decreased the history effects were noticeable; the density versus pressure curves determined during compression compared to those determined during decompression differed more at the lower temperatures. Also, an excess pressure which may have been caused by the volume history was noticed. This effect was small at 25 C ar.d increasingly more pronounced at the lower temperatures. It should be pointed out that research efforts on areas related to homogeneous dynamic compression are currently being pursued at the Asphalt Laboratory of the University of Florida. One definite area of interest is the development of temperature measurements schemes which will not affect the homogeneous character of the deformation. This will allow the dynamic compression technique to become a thermodynamic calorimeter. Another zrea is the investigation of the possible effect on shear viscosity of the structural changes occurring 119
PAGE 133
120 during compression of these thermally active asphalts. A. Results General Proc edure Figure 28 is an illustration of the actual experimental procedure followed during the compression and decompression experiments, Detailed, step by step procedures can be found on pages 3 and 10 of research notebook XL.VII! of the Asphalt Laboratory at the University of Florida. These procedures are not presented here because their details were not considered essential to the results obtained. Instead it is believed that a discussion of Figure 28 will properly illustrate the fundamentals cf the procedure. Three stages were common to both the compression and decompression procedures: (1) preparation, (2) deformation, and (3) recovery. The preparation stage consisted of all the operations from the time the sample was obtained to the point where it was ready to be tested. As discussed in Chapter V the time temperature history of the sample prior to insertion in the machine at room temperature (25 C) had negligible effect on the properties observed during actual testing. After the sample was placed in the machine at 25 C it was deformed until a ioad of 1620 lbs (1000 atms) was reached and held at this defcrmaticn while the lead relaxed. After a steady load was observed the sample was recycled between 1620 and 162. lbs f orthree o,three and onehalf cycles; three cycles if the testing vaÂ« to bz decompression, three and onehalf cycles if the
PAGE 134
121 '
PAGE 135
122 sample was to be compressed. The purpose of this treatment was to eliminate any excess silicone lubricant or air trapped between the balloon and the barrel. It was also intended by this pret reatment that the parts of the assembly became properly seated. It must be noted that at the end of the last cycle the cycle was extended beyond either 1620 or 162 lbs force and then the machine direction reversed and stopped before deformation. This step is important and is performed to avoid an initial "acceleration" period of the moving crosshead (Chapter V, p. SO) . The deformation stage consisted of increasing or decreasing the volume of the sample by moving the crosshead at different rates of speed. The abscissa in Figure 28 during the deformation process is represented as change in length. This can be accomplished by adjusting the chart speed to be a constant multiple of the crosshead speed, e.g., one hundred times. Thus, by rolling the chart paper back and starting the deformation procedures at the same point in the chart the relative difference on the pressure versus deformation curves due to different rates could be observed. Finally the last stage during the testing procedures was the study of the relaxation processes after deformation was stopped. These processes are illustrated in Figure 28 as changes in pressure with respect to time. In the actual testing procedure the relaxation and recoil experimentswere recorded separately from the deformation stage so that appropriate chart speeds and load sensitivities could be used (Chapter V, Figure 21). At least three runs for compression and three runs for d> coi pression were made with
PAGE 136
e 123 each sample of asphalt at each temperature and at least two different samples length were run for each asphalt following the procedure outlined above. Experiments at different temperatures on a given sample wer always made immediately after the sample had been checked at 25 C for compressibility and density. For example, if a sample was to be run at '30 C, prior to bringing the temperature to ~30 C the sample had to be compressed to 1620 lbs Force at 25Â°C, checked for compressibility and density and then the cooling of the assembly proceeded. No permanent deformation was noticed on the samples after treatment at the lew temperatures, so that a sample run at ~30Â°C could be reheated to room temperature, and the original density versus pressure at 25 C obtained, This effect was also noticed by Ronk (39) and Stewart (Appendix A), Dersity versus, Pressure. Figures 29, 30, 31 > and 32 summarize the density versus pressure results obtained on the four selected asphalts. These curves were obtained in experiments of approximately 30 minutes duration and which according to the studies described in the last section of Chapter V must have proceeded isothermical ly. That is, if the characteristic time of the heat transfer process is of the order of kO seconds, a steady temperature should have resulted in approximately two minutes after deformation started. Insertion of calibrated thermistors proved thai the increase in temperature at the low rates of deformation did not cicceo 0.1Â°C (XLVIM.3).
PAGE 137
Mh Â• " 1.0200 1.0300 L 1.0700 1.0600 $ 1.0500 I/) c v c 1.0*400 ''/1.0200 1.0300 I 5 k 5 6' 7 PRESSURE (atrn) Figure 29." Density versus Pressure for So.5
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125 1.0700: l.o6oo 1.0500'_ 1.0400"' E u CD g 1.0300, 1.0200: 1.000 25Â°C 1.0100;; * 3 k 5 PRF.SSURZ (atm) sasssKiSrii.. Â•Â•Â• .. : .' Figure 30. Density versus Pressure for 563 13
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26 1.0700 1.0600 I 1.0500 1 . 0400 u E o >C5 1.0300 ;'/ 1.0200 \r i 1.0100 1.0000 t Figure 31 7 8 _L i PRESSURE (atm) Â•Density versus Pressure for S63 ?.0 S . . j 3
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127 ., 1.0300 1.0800  1.0700 . 1.0600 cr g ^osoo 1.0400 I 1.0300 PRESSURE (atm) 1 2 3^56789 1.0200 _ ^ J ! ! ..L. ,J._^J^J,:.J . Figure 32. Density versus Pressure for 5o3"47. . 
PAGE 141
128 In addition to Figures 29 through 32, Tables 5 through 8 summarize the coefficients of the density versus pressure relationship, and the results on the compressibilities at the extremes of the pressure range. Data are pressnted for the isothermal case as well as for an arbitrarily fast rate of crosshead speed. The coefficients presented in Tables 5 through 8 are for a second degree polynomial. Three other polynomials were tried to fit the data: first, third, and fourth order polynomials. In every case presented in these tables, the second degree polynomial fitted the data as well or better than the other polynomials. The error between the calculated densities and the experimental values was less than 0.01 per cent in every case. No point of inflection was noted on the calculated or experimental curves. Transition of density and compressibility from liquidlike values at low pressures to solidlike values at higher pressures proceeded smoothly for these asphalts at all temperatures tested. However, compress ibi 1 ity and density differences between the values obtained during compression and those obtained during decompression for the slow isothermal experiments illustrate the effect of different volumehistories. These effects were noticeable at "30 C and high pressures. The values of the compressibility ^nc 1 coefficients of the densitypressure relationship at the higher crosshead speed reported in Tables 5 through 8 are of particular value in connection with the increase in temperature as compression proceeds. Results of current research on implementation of calorimetry using the
PAGE 142
129 o
PAGE 143
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131
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132
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133 la o o o o ca LA o o co. o CM < LA O X (A
PAGE 147
M developed dynamic compression technique indicate that a temperature rise of 20.6 C/rain takes place when compressing a 3.30 grams sample of S6320 at 25 C at 0.1 in/mi n. The initial power required to compress the sample at this rate over that required at the slow isothermal rate was determined by the difference in initial slopes of the deformation versus load curves (the area between the curves represents the work). This excess power input amounted to 52.7 Ibffoot per minute. By simple calculations the instantaneous heat capacity of this asphalt at 25 C and 60 atm (the starting compression point) was apparently 0.26 cal/gm/Â°C. The values of the heat capacity of asphalt S63"20 found by Breton (XLVII,63) by standard calorimetric methods at constant pressure ranged from 0.216 at 40Â°C to 0.425 at 100Â°C (Appendix A). The value of 0.26 cai/gm/Â°C would correspond to the value of the specific heat at 16Â°C. These data are offered only for comparison and motivation. Further analysis or discussion should wait the final implementation of the calorimetric techniques. Recoil and R el a x ? t .? on The recoil experiments were described in Chapter V,, p. )0k, when discussing the effect cf the temperature change during decc: press ion. Relaxation experiments consisted of compressing the sample at a constant ambient temperature and then, recording the decrease in pressure versus time after compression. The reason recoil had been used for the homogeneity studies of Chapter V was
PAGE 148
135 because the more sensitive scales of the recorder chart could be used at the end of the decompression test. However, careful examination of the recorder capabilities revealed that an electric signal equivalent to 1024 lbs of force could be subtracted from the actual reading on the recorder by the turn of a knob. Effectively, this allowed recording of forces between 1024 and 1224 lbs on the most sensitive scale. The detailed procedure can be found in the notebook reference given above. Essentially, compression tests were run at 0.1 in/min up to 1224 lbs and the relaxation, generally smaller than 200 lbs recorded on the sensitive scale. In these experiments prevention of leakage was necessary. In order to guarantee that no leaks occurred when studying the relaxation curves the balloons were sealed by tying their open ends with surgical thread. This allowed removal of the encapsulated sample for inspection after each run. Figure 33 illustrates the results of the relaxation and recoil experiments on S&3"20 at 25 C. It can be noted that no essential differences exist between the two curves. However, these differences are considered significant. The relaxation curve shows a delayed effect which is not observed in the recoil curve. Figure 34 summarizes the recoil effect on all four asphalt samples at 25 C. It can be noted that except for small sh if tings all four curves superimposed, The purpose of the shifting is to correct for the initial magnitude of the recoil. The results depicted in Figure 34 seem to corroborate the conclusions of Chapter V, chat the recoil effect at 25Â°C >"Â£s the product of the return to normal of the excess
PAGE 149
136 jC . I o CM Â— I O < Q ..
PAGE 150
137 ~^Â— "..i Â• : . aa . .. L3 ON I CO vD 1 f ,v\ o CM I i \0 \D H r Â• o 8 a> P J I
PAGE 151
138 temperature due to the heat of compression, and that its characteristic time did not depend on the material. However, Figure 35 depicts the relaxation curves for the four selected asphalts at 25Â°C. The dotted line represents the recoil versus time results of Figure 3k. The differences between any of the fcur relaxation curves and the recoil one indicates a delayed effect. This possibly is the result of the different volume histories when compressing and when decompressing (see Chapter IV, p. 68). The differences among the four asphalts are small. However, it is believed that together with the results at the other temperatures these differences may be significant. Figures 36 and 37 illustrate the results at and 30Â°C respectively. The differences among asphalts now become more apparent. These results correspond to those for the relaxation stud ies . Figures 38 and 39 illustrate the results of the recoil experiments. In this case the increase in pressure after decompression had been attributed to the return of the sample to the temperature of the surroundings. This may not be the only significant contributing cause at the lower temperatures. It should be noted that the recoils are faster, at the lower temperatures. Furthermore, these curves are no longer the same "or the four asphalts. In order to explain the results observed during the relaxation and recoil experiments at low temperatures it is postulated that delayed structural changes take place when Finite homogeneous
PAGE 152
139 .
PAGE 153
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PAGE 157
lâ€¢t changes in volume are imposed on the selected asphalts. It is further postulated that these relaxational changes dissipate or supply free energy in the form of entropy resulting in a delayed relative increase in pressure and/or temperature regardless of whether the deformation was a compression or decompression. The delayed free energy dissipation should always increase the pressure of the system as a result of the entropy release, by increasing the mechanical energy term (PV at constant volume) or by increasing the temperature of the system. The above postulates explain: (1) the larger initial relaxational times compared with the initial recoiling times; (2) the reduced recoiling times after decompression as the test temperature is reduced; (?.) the difference in delayed characteristic times for recoil and decompression; and (h) the relative differences among the asphalts. The larger initial relaxational times would be explained by the opposition to pressure relaxation, caused by normal cooling of the sample after compression, of the significant delayed contribution of the free energy dissipated by the relaxing structures. Free a See equation B~31b and its explanation in Chapter 111. The rubber elasticity effect where an increase in entropy (temperature) increases the tens i on at constant deformation is not expected here. However, there are liquids which show a similar effect (e.g., water below hÂ°C) , that is they show a decrease in pressure with an increase in entropy at constant volume.
PAGE 158
l';5 a energy decreases as the system tends towards the relaxed state, at constant volume. The free energy released by the relaxing structures is reflected in increased temperatures and pressures before dissipating into the surroundings. The reduced recoiling times can be explained under the same postulate. The delayed free energy production of the structures tending towards the relaxed state would in this case also increase the temperature and pressure of the system. This collaborates in bringing the low temperatures due to the normal heat demanded during decompression back to the temperatures of the surroundings. At the low temperatures where the delayed effects are more marked the delayed heating up of the sample can only contribute to reduce the tiros to bring the sample back to normal temperatures. This explains the second observation presented above. The third observation about the significantly faster change of pressure with time for the recoil experiment may be a consequence of the expected faster relaxational tires at higher volumes (free volume theories) . The relative effects of volume and temperature upon the rite of relaxation determine the relative magnitude of D^ or Dj upon the dynamic effects observed. Regrettably, not enough data are available for a quantitative study. However, both effects seem significant. The effect of temperature on the rats of relaxation (Oj) can. be appreciated by comparing the characteristic times at ! Fcr a discussion of the relaxed state refer to ChapterIV,
PAGE 159
146 the different temperatures. The effect of volume is evident by comparing the rate of relaxation versus the rate of recoil at constant temperature. Table 9 summarizes these results. The reader must be warned that the characteristic times presented in Table 9 are only offered as empirical indications. These times were obtained by plotting the pressure versus time data on semi log paper and approximately fitting two exponentials by the method of successive approximation. The smaller exponential time constants were in the neighborhood of kO seconds, which is the time corresponding to the heat transfer process unaided by internal changes . Relative comparison of the four asphalts indicates that glassiness as judged by the relaxational times at any of the temperatures is more pronounced on 56^^7 followed next by S63~20 and S639, and lost by S63"13. B . Surr na ry of Concl u s ion s The following conclusions are reached after study of the results presented in ( ;h^ previous section: (1) The dynamic compression technique is a suitable method for the study of glassiness on asphalt. It should provide thermodynamic data which may elucidate many of the questions concerning the real world behavior of these rrtc aerials.
PAGE 160
147 Table 9 Dynamic Parameters of Selected Asphalts Sample
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143 (2) With the aid of this technique the onset of excess thermodynamic properties could be quantitatively described. The excess thermodynamic properties contributed by mechanical effects, e.g.,, free volume, could be distinguished from entropy effects. (3) Relaxation versus recoil experiments should provide one specific manner to study the effect of volume history. These experiments carried out at different temperatures should allow establishment of the relative effect of temperature and volume on the rate of dissipation. Thus allowing estimate of Dw and Dj. (0 At low enough temperatures, e.g., 30 C, experiments can be conducted at low enough rate to allow dissipation of the compression heat; the experiment could be considered isothermal and still one could observe glassiness phenomena, e.g., change in compressibility. By increasing the rate of deformation ar;d determining directly the temperature changes a thermodynamic calorimeter could be implemented. In this manner volume dependent and temperature dependent dissipations may be observed directly. (5) The changes in density and compressibility of asphalts were smooth functions of pressure and rate of deformation at a given temperature. No second order transition was noticed when compressing or decompressing the four silected asphalts.
PAGE 162
149 (6) The 56^^7 sample showed glasslike behavior more markedly than the other three samples. Since this sample was a i; harder" sample (lower penetration) this result is not entirely surprising. C . R ecornmendat i ons The following recornmendat ions are based in the previous conclusions and are suggested in order to gain more understanding of the nature of the thermomechanical behavior of asphalts: (1) Research must be directed, in general, towards obtaining a reliable method of registering temperature changes without affecting the homogeneous character of the deformat ion. (2) In order to establish the contribution of different asphalt parameters upon glass iness it is recommended that testing be done at constant temperatures and the degree of glass iness of the material be determined by its reaction to a set of volume histories. Dynamic compression may be used to study the contribution of the different generic components, e.g., paraffins and aromatic compounds. It is suspected that some of the manifestations of glassiness may be caused by mutual solubility end "miscibi 1 ity" of the generic groups in asphalt. (3) After perfecting t u .e thermodynamic calorimetry techniques, it is recommended that heat effects (heat of compression,
PAGE 163
150 instantaneous heat Capacity, entropy, etc.) be studied as a function of pressure at the pressure ranges used for capillary rheometry. It is believed that some of the scatter of the shear rate versus shear stress data with different capillaries may be caused by the significantly different pressures at the entrance of the different capillaries.
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APPENDICES
PAGE 165
APPENDIX A HISTORICAL DEVELOPMENT OF EXPERIMENTAL METHOD in early 1 967 it was this writer's opinion, inspired by the work of Majidzadeh and Schweyer (31); that thermodynamics would provide the means to explain certain aspects of the behavior of asphalts cements at low temperature (XLI,,8). a The compressive technique presented above was not the result of a premeditated rational analysis of this early opinion. Instead it is the product of a laborious development and follow up of earlier empirical methods. The following account is presented here with the hope that it will aid future investigators in expanding this research. 1 . Thermosta t ics Since thermostatic data on asphalts were scarce, it was the consideration of the writer that calorimetric data as well as P. V,T, data for an "equation of state," had to be gathered in order to establish a firm basis for the study of the dynamic effects. Breton (XLVII, 11130) completed preliminary work by Scott and Busot (XLI, 1117) determining quantitatively the specific heat versus temperature data for twelve asphalts and some of their References designated by a roman numeral followed by an arabic number, refer to pages (arable) in the research notebooks (roman) of different investigators working in the Asphalt Laboratorv of the University of Florida. 152
PAGE 166
153 chromatographic fractions. Breton's work is contained : n a report. for the Master oi' Engineering degree to the Chemical Engineering Department, University of Florida ( 1 969) . This author concluded that no unique second order transition cculd he observed in the Cp vs. T curve between room temperature and kOÂ°C. Figure Al illustrates a typical result from Breton's work. The "glass iness" (or discontinuity) observed in volume; versus temperature and penetration versus temperature discussed in Chapter I was not detected in these experiments. Instead, several changes in slope were noticed in the temperature range shown in Figure Al. However, a large change in specific heat with temperature was noticed for all asphalts tested. The determination of density as a function of pressure and temperature for some of the asphalts used by Breton had been previously assigned to D. E. Konk who completed a Master of Science Thesis on this subject (1368) (39). Measurements of volume were made at different pressures, at constant temperature. The procedure was time consuming; the sample was poured directly into the barrel and compressed stepwise to different densities after cooling to the desired temperature. Leakage was practically eliminated by the use of Teflon "0" rings. However, in order to obtain the pressure P in equilibrium with a certain density P tine was required for the pressure to equilibrate. The time required for the pressure to become steady after a change in density was of the order of 30 minutes to one hour at the lower temperatures used. Ten to fifteen pressure den's ity points were, determined at e?c ! i t temperature, A second degree polynomial was used to e^pre?^. the
PAGE 167
\5k ,/ . 0.350! I 0.30$ ./ / / / / / / / J fi ! J'Â— 0.250 Y / f xko 30 20 I em. :ure ( C) 10 JO 50 I 1.210, ' J    j j" Figure AI . Specific Heat vers, e for As h Cement (S6320). 30 ! ho 1
PAGE 168
355 density as a function of pressure at each temperature. Ronk'i work covered the temperature range from G C to 50 C in intervals of 5 C. The coefficients of the density versus pressure second order polynomials obtained at constant temperatures were expressed as a function of temperature. Thus, "P, V,T" relations were available to compliment the calorimetric data of Breton. Ronk did not notice any discontinuity in the volume versus pressure plots obtained at constant temperature neither has this author noted any abrupt change with temperature of the coefficients of the P vs. P equations. 2 . Preliminary, U/namic S tu dies Concurrent with P.onk's static experiments this writer undertook the study of the excess pressure developed during compression of the sample (see Figure A2) . It was proposed then, that a study of the relation of equilibrium versus excess stress would be related to the. changes in entropy and energy within a material undergoing the dynamic compression (XL 1, 22). However, emphasis in these early experiments was placed upon the measurement of a "consistency" or resistance to deformation which could be used to describe the asphalts rheologically. This emphasis was motivated by this writer's background on continuum mechanics, and the current need for mechanical specifications of asphalts at low temperatures. Most of this work has been published in a paper presented to a Highway Research Board meeting and in a report to the Florida Department of Transportation ( ; '5; . Table Al summarizes the results of the consistency and characteristic time at C for
PAGE 169
156 lubricated samp], Tims , u > 1 .2 secÂ— j Figure. A2 Â—P ^essure Required for Compression with and Â• i cnout Lubr i c
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157 Table Al Preliminary Data at 32Â°F for' Rheology of Twelve Florida Selected Asphalts I dent i f i cat ion Compressibility Consistency % def/psi Modulus(MU) (x 10' 4 ) Smackover Venezuelan Florida AC 8 I ntermediate Steam Refined Gulf Coast Naph, Steam Refined Air Blown Naph. Low Sul f ur Air Blown High Sul fur S. Texas Heavy Steam Refined Merey (Venezuela) Panuco (Mexico) Los Angeles Basin Kern River S634
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153 twelve asphalts of similar pt.net rat ion. The consistency and characteristic time were obtained by a method described in Appendix A of reference (iÂ»5) . These consistencies and characteristic times were highly empirical in nature. Their values were obtained from graphical evaluation of the slope of the relaxation curve (decaying force versus time) after arbitrary deformations at arbitrary rates. However, the similarities observed between the ratings of the asphalt according to their consistency and according to their viscosities were remarkable. 3 . Development of the Ex perimenta l Technique Encouraged by the results of these preliminary experiments, it was decided that the development of a method of testing based on finite compression of the asphaltic samples would be fruitful in order to provide meaningful rheological parameters and thermodynamic data. One of the shortcomings of the preliminary studies was tl fact that the deformation of the sample was not homogeneous and therefore, it was very difficult to describe. The interaction of the sample with the confining cylinder was considered the main reason for the lack of homogeneity. Coating the inside of the barrel with a 0.0005 in film of Silicone stopcock crease was found to reduce the excess pressure during compression significantly ('Â•>Â«+) . Figure A2 clearly illustrates the differences in excess pressure required to compress two asphalt samples of the same material with and without lubrication.
PAGE 172
1 Â• Â• . The magnitude of the drag of the materials against the wall of the containing vessel war. a surprise to this writer. Previous investigators had assumed the drag negligible (3 /j ), (3.6). The discovery that most of the excess pressure observed previously was due to viscous shear dissipation at the walls changed the direction and emphasis of this research. The emphasis had been to obtain rheological parameters. Instead, the emphasis shifted to perfecting the experimental technique to allow accurate measurement of thermodynamic parameters. Two main reasons motivated this shift in emphasis. A pragmatic reason, and an intuitive theoretical reason. The pragmatic reason was that with lubrication the relatively large excess pressure (ba^is for all rheological comparisons) had been reduced to a small percentage of the total load required for compression and therefore, its measurement difficulted. Also, more important was the fact that P, V,T data could be obtained in a matter of minutes rather than hours as with the method used by Ronk (39), facilitating gathering of thermodynamic data. The intuitive theoretical reason for the reverse in emphasis back to thermodynamics, was the belief that a description of the thermodynamic states at low temperatures and high pressures would establish a relationship for the reported glassy behavior of asphalts.
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ICO Equat Ions p (ft) Â° + fe) T + (ft) 'T,t V ' V,t V / V,T (30 S =/ii\ v + /JLi\ T.. 3 V) T , ^Tiy.t and their equivalent., equations 37 or 43 could be used to establish the influence of different factors on the dynamic behavior of the samples . ^ . Pre! imlnar ies With the use of lubrication and the reduction of the magnitude of the dynamic effects several factors which had been assumed "relatively small" were no longer negligible. Preliminary work on reproducibility; leaks, machine calibration, etc., had to be repeated, now with a more critical attitude. The development of an experimental procedure, rather than its immediate use, became the primary goal of the experimental part of the research. Stewart ( 1 969) working under this writer's supervision established an experimental procedure to determine the dynamic compressibility of asphalt cements in the range of 0Â°C to 30Â°C. Essentially, this procedure consisted in compressing a sample from about 200 to 2000 lbs. at different rates of compressions. The temperature of the steel confining barrel was kept at constant temperature (0.1 'C)
PAGE 174
161 by controlling the temperature of a calibrated thermometer attached directly to the barrel. a. Age. Stewart corroborated Ronk's result that the "age" cF the sample did not affect the response of the system, i.e., the materials showed complete fading memory above room temperature. !n the sense mentioned above, "age" is understood to be the timetemperature history from the time the sample is prepared to the time testing starts. Stewart and Ronk kept sample? for weeks at room temperature and obtained the same results when running these aged samples as when running the freshly prepared samples as shown in Table A2. The table illustrates the negligible effect of "aging" on the values of the secant compressibility. The secant compressibility is defined as the ratio of the total deformation divided by the pressure causing it. The fresh sample did not show any difference from the aged sample. Table A~2 also illustrates the reproducibility cf this o c method by comparing values for different runs at 25 and 25 C. Stewart's work included determination of compressibilities at different rates. However, it was found after this work was completed that the silicone stopcock grease used for lubrication failed to provide effective lubrication below 25 C, and perhaps even at temperatures up to 0Â°C at the higher pressures (XLI, 6785). Because a For a detailed description of the procedure adopted by Stewart, the reader is referred to notebook XLVi and report or, "Procedure for Study of Asphaltic Materials below Zero Degrees Centigrade," August, i c 69, Asphalt Laboratory, College of Engineei i nq , Un i vers ; ty of Fl or i da ,
PAGE 175
162 ai >Â• o X i CO a. < o CM
PAGE 176
163 of this, Stewart's results, although reproducible, may not be of quantitative value. Other major questions leftunanswered by this preliminary work were: (l) the effectiveness of lubrication, (2) the extent of the drag of the piston and l! 0" rings, and (3) the extent of leaks at high pressures. The answer to these questions are of primary importance to establish the influence of irreversible processes occurring within the material. The effects of poor lubrication, strong drag by the "0" rings, and significant leakage may be confused with irreversible material changes. h . Effect ivene ss o f silicon e grease lubri catio n i. Room temperature. In order to answer the question about the effectiveness of lubrication the set up illustrated in Figure A 3 was used (XL. I, 46~57) . The main difference between this assemblage and the standard rheometer assembly is that the barrel is not suspended and the bottom plug is not held in place by a clamping nut. Instead, the barrel is supported by the fixed crosshead and the bottom plug, which is no longer clamped, is resting against the sensing element of another load cell, DR. When compressing the FR cell would sense the drag on the walls, which would contribute to apparently larger loads required for a given deformation. Conversely, contribution of the drag to the readings on the DR cell would result on a lower force required for the sane deformation. When decompress ing the barrel is no longer in a completely fixed position wich respect to either of
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I6*t PR CELL MOVING CROSSHEAD Â— EXTENSION SPRING '. D BARREL PLUNG ASPHALT SUPPORT FOR BARREL EXTENSION i j FIXED CROSSHEAD Figure A 3.Â— Assembly for Studies c fectfve Lubrication
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165 the eel lb a;;.:! no predictions can be made art the effect of the drag. o Compressions were run at 25 C with two pieces of steel, one and two inches long for calibration purposes. Two compression rates were chosen 0.05 and 0.005 in/min. These two compression rates were known to give appreciable viscous wall effects with an asphalt of moderate viscosity when run without lubrication. An asphalt sample was also compressed at these same rates. It was a sample 0.843 inches long of ar. asphalt recovered from an actual pavement. The reason for choosing "his asphalt is that it had a viscosity of 100 megapoises at 25 C which was in the same order of magnitude of the viscosity of the "fresh" paving asphalts at temperatures close to C. it must be pointed, out that the arrangement shown in Figure A~3 is outside the environmental chamber where the colder tests are performed., and therefore test temperatures when using this arrangement were limited to room temperature. The results of these experiments are sui r ed in Table A~3 (XL. 1, 55). The values of the total change in length caused by changes in load were usee! to establish the effect of the drag. The changes in force are equivalent to 300 atmospheres pressure changes. The effect of the drag of the sample on the walls of the barrel was negligible; the changes in length caused by the pressure changes were essentially the same when using either cell to record the For ^.ore details on the contribution of th c assembly to the apparent deformation of the sample see Chapter V.
PAGE 179
166 Table A 3 Deformation Readings with Special Assembly for Determination of Drag Effects Cell FR DR Compression rate, 0.005 0.05 0.005 0.05 in/ml n deformation, inches x 10 from 50 to 55 Ibf Â£pJB5Â£e ss inq Total 178 185 179 176 Assembly _8l_ _8l_ JÂ± JJ_ Sample 97 \Qk 1 02 59 Decompress i.nq Total Assembly Sample deformation, inches x 10 from 4 15 to 865 Ibf 188
PAGE 180
167 pressure. Also, it Is seen that this sample exhibited the same value of the deformation whether compressing or decompressing, and approximately the same values regardless of the compression rates used. Other observations made using the assembly shown i r: Figure A 3 concerned the resistance due to the "0" rings. !n the procedure used, before compressing the sample, the sample and bottom steel plug were pushed back into the barrel away from the DR cell, Then the reading of the deformation was taken as either of the cells started to record any load. The FR cell started to record about iyOxlO in before the DR cell would. These 0.019 inches were the distance from the bottom plug to the DR cell. This arrangement allowed determination of the total force required for moving the "0" rings and lubricated sample before compressing it against the DR cell. The load required to overcome this resistance was less than 50 lbs at the lower rates (0.005 in/min) and decreased sharply to less than 10 lbs at the higher rates (0.05 in/min) . Although it was not the objective of these experiments to determine, the pressure dependence of the compressibility or compressibility modulus of this sample a commonly used equation was fitted to the data in Table A 3 K K Q + K,P (Al) where K = 'is the secant compressibility modulus, P = pressure. The results K = 2.18 x 10 ,J dyne/cm and Kj = S.hh are in good agreement for the values of K Q (2
PAGE 181
s T6G was the fact that the confining barrel could be turned and moved vertically by hand without disturbing the sample when in the as~ embly shown in Figure A 3 and,, while the sample was supporting a load of 10C0 lb force or approximately 3000 lb/so in pressure (XLI,47). ii. L ow temperatures . Although the stopcock grease is supposed to maintain its effectiveness from ^0 C C to 200Â°C according to label specifications it was found that at _ 30 C the lubrication properties failed (XL I ^ 67) Figure i\k illustrates the manifestation of the "failure" or the lubrication characteristics of the silicone stopcock grease used to coat the inside of the barrel. Figure A 4 shows the decrease in load when decompressing from about 1000 aim pressure at 25 C. The discontinuity in the force versus time curve created some expectation when first observed. It was thought that this discontinuity was an indication of a glass like transition of the sample (XL 1 ^ 6768). Careful study of this effect by running different asphalts and several sample lengths demonstrated that genera! shape of the curve and the load and deformation at which the discontinuity appeared were independent of the sample length and the kind of asphalt used (XL!, 6785). It was concluded that the cause for the initial increased resistance to deformation followed by a "failure" and the return of the load versus time curve to a more "normal" behavior illustrated in Figure Ah could bo ai rib led to a solid like transition of the lubricant. This transition may
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]6S . . . .....'. ... I O 1/1 .a Â•o O 18 p I 16 IÂ— 14 8. 12 10 Assembly de format ion S63
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T70 have been induced by the low temperatures and high pressures used in these experiments. d . Other lubricants A search for lubricants which would work at low temperatures included graphite powder, graphite suspension in various liquids, ethylene g!y ccl ; silicone 200 Dow Corning fluid, and Texaco low temperature greases for aviation, none of these gave batter lubrication than silicone stopcock grease. Most of the lubricants, including graphite and the aviation greases did not provide smooth durable coatings; after a few compressiondecompression cycles, even at room temperature, lubrication seemed to be reduced. Inspection of the samples after removal from the barrel shewed that the liquids used had partially dissolved or mixed with the asphalt. The graphite particles were coated by the asphalt, losing their lubricative properties (XLI, 75, 85, 37). The idea of enclosing the asphalt sample in a rubber balloon occurred to this writer on January 23, 1970 (XLI, 85). This was a breakthrough, for it also allowed elimination of the dragcausing "0" rings and guaranteed an almost leak proof capsule. The use. of the encapsulating technique allowed the use of lubricants which otherwise would partially mix or dissolve the asphalt. The experimental work done to develop the encapsulating technique into its final form is described in Chapter V.
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APPENDIX B CONTINUUM MECHANICS 1 . Phenomenoloqi ca'i Concepts In this section some basic concepts of continuum mechanics will be presented. These concepts will be necessary for a quant itatlve general description of glassiness. Continuum mechanics is a theory of materials in the same sense that geometry is a theory of figures; it is a language. Bodies , motions, and forces are the primitive elements of mechanics. Bodies undergo motions, which are sustained and/or opposed by forces. The differences in the motionforce relationship shown by different bodies are described mathematically by constitutive equations. They define ideal materials, and ere formulated to abstract most of the phenomenol og ical behavior of real materials. a Â• Conf_i..gurat io ns and motions A body conceived as a set of particles can be represented by a collection of points from Euclidean space, E . A collection of points representing a body at a particular value of time is called a configuration. A sequence of configurations of a 3 designated region of E fulfilling the axiom of continuity is called a motion. The region chosen to none each of the particles Textbooks by Eringen (16), Jaunzemis (23), and Middleman (32) are very satisfact Tor genera] background. The theory of Simple Materials has been extracted from Truesdell (49). 17
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172 of the body, is called the reference conf igurat ion. R. Mathemati cally, the motion is expressed by r r X(R, t) (Bl) Where r expresses the position of a particle P. at time t. X is a mapping (linear transformation) of the reference configuration (body) onto space. It is supposed to bo invertible, single valued and as many times continuously different iable as required. This last statement, axiom of continuity, guaranteethe intuitive notions one has about matter: permanence and impenetrability. b . Changes in shape A general motion on a plane, E^, is represented in Figure Bl. In order to describe what happens to a neighborhood of the body about a particle, a pair of vectors dri and dr~ are chosen in the "deformed' configuration. Knowing what has happened to the whole body (motion X) one trie; to determine what changes have undergone the corresponding reference vectors, dR] and dRp_. More specifically, since onlychanges in "length" and "angles" within the material are considered to cause stress, the product dri and dr~ expressed as a function of dRj. dRp will provide all the information required. Remembering that dri . dror dr, coz a (B2) This is a referential or Lagra resentation of . mot i on.
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173 Â«Â•. Â• . . . Figure B1 . Â— Motion and Configurations
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\7k and that r X(R, t), then dr. = (3X/dR) . dR. i = 1,2 (B3) where 3r,X is the gradient of the motion in the reference configuration. Substituting into Equation B 1 2 , one obtains dr, dr. cos a = F . dR, . F , dPvÂ£ : F . dR . dR 2 . F F . F : dR] dR 2 C : dRj dR 2 (B4) + where C = F . F is a measure of the changes in "length" and "angles" of an arbitrary small neighborhood of R. !t is also seen that enough information is contained in F to describe the change in shape. The causes of stress in materials^ are thus described by ,F(R,t). c . Histor y o f a function, det ermi nism, s imple m ater ials The history of a function is the portion of a function of all time which corresponds to the present and past times only. Let f (t) These materials are called unstructured materials, and rep' res r i at are known as homogeneous real materials.
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175 be the function of x, where 0. The history of f(x) is the collection of values of f(x) counting from the present time backwards . The principle of determinism expresses the most general experi mental observation of the mechanics of bodies: The stress, P, at the particle Rj in the body R at time t is determined by the history of the motion, X : P(R], t) = H p (X*, R]) (B6) Equation B6 is a constitutive equation, H p is a functional, a rule of correspondence which assigns a number to a function. A more restricted constitute equation would result if the assumption is made that only changes in shape determine the stress at a particle of a body. The resulting constitutive equation wi 1 1 be general enough to include all known variations of the theory of v iscosl ast icity, from linear viscous fluids to elastic solids. This equat Ion wi 11 be : I = "*}Â£> ^ (37) Equation B~7 expresses that the stress P at R, will be known if the complete collection of deformations' undergone by particle kj is given. Equation B~7 is the constitutive equation of a simple
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176 material. In general^ Equation 7 is not enough to correlate experimental data. !n order to determine the values of the deformation at an internal points as a function of external stimuli, a boundary value problem must be solved. This requires prior knowledge of the constitutive equation; or. conversely, by experimental determination of the values throughout the body, the boundaryvalue problem can be solved to determine the constitutive equation. d . rlornoqere oL's defo r mat ions The need to measure the values of deformation throughout a sample, and then solve a complex mathematical problem can be eliminated, if the experimental material is a simple material. All there is to know about a motion to determine the response of a simple material is F; the strain or change in shape as previously shown is given by the gradient of the motion. Motions with arbitrary F can be produced very simply by motions of the general form: r(t) = F(t) (R R Q ) + r Q (t) (B8) where r (t) is a place in the deforming configuration which corresponds to R Q in the reference configuration, r(t) and R are correspond ing general points in the body. A motion of the type expressed by this equation is called a homogeneous, motion. The experimental importance of this motion is that F (t) dees not depend un the position of th , rticle.
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177 Therefore, measurement of the stress at the boundary of a homogeneously deforming simple materia] wl 1! give the response of the sample to the deformation. The need for complicated experimental techniques and involved mathematical calculations is thus obviated. 2 . Thermodynamic s The difficulties encountered in establishing a phenomenological definition of glass transition are intrinsically related to the proper application of thermodynamics. Thermostatic relationships are used as general and extraneous factors are introduced to reduce the gap between observation and "theory," A poor stateof affairs was described in Chapter I!, pace 10, concerning the consequences of assuming a homogeneous deformation when cooling an increasingly viscous liquid. However, the assumption of homogeneity has its basis on the state of the art concerning thermodynamic theories of "irreversible" processes. Most thermodynamic treatments were "developed" for homogeneous processes, that is, tiie quantities defined such as "entropy production" were functions o\ time only, a . Homogeneous linear irrever sible thermod yn a mj_cs a Standard treatments of irreversible thermodynamics start with the first and second laws of thermodynamics expressed as Gibbs cr d This discipline has been given a wide variety of names* irreversible thermodynamics, nonequ i 1 ibr ium thermodynamics, thermodynamics of flow systems, Onsager tl Â•' mics, etc. Se DeGroot and Mazur (\k) and Fitts (19) for genera] jnd .
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178 Helmholtz relations n TdS = dE + pdV Z V C I< ^ E " 9 ^ k=l where the internal energy E, specific volume V, and the mass fraction Ck, determine the "state of the system." The assumption is made that equation B~9 applies locally to a system undergoing an Irreversible process. The result may be considered a universal constitutive relation for materials undergoing dynamic changes: n TS =E+pVi \ C k (B10) k=l where the dots indicate time rate of change at a particle of the material, and the temperature and pressures are instantaneous "equilibrium" values. Further development of the subject includes some linear constitutive relations to account for flow of the "state" properties caused by gradients in temperature, pressure., and mass concentration. The theory goes into molecular arguments as well as macroscopic tensoral considerations to establish useful correlations among the coefficients of the flow equations. It must be pointed out that these theories are very useful for the interpretation of experimental observations in a molecular scale. However, in the opinion of this writer, the introduction of equation B10 without explicit indi 11 ior that it constitutes a theory of mate, ials in addition to an ass ... process equilibrium may lead to erroneous application of the ry.
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179 As a process equation, equation B10 is valid for deviations from equilibrium which are not "too large" (see De Croot, op. cit . , p. 23). As a constitutive, equation it implies that the history of the state variables is not contributing to the present state of the system. Changes in shape are altogether ignored as thermodynamic variables. b . Therm ody na mics of simple mate rials More recently, research on description of processes involving materials sensitive to changes in shape and temperature have provided a more, diaphanous thermodynamic description of the experimental reality. In particular, the works of Coleman (11) (12) on simple materials should prove appropriate to establish a clearer definition of "time scale" and, therefore, of glassiness. it was stated that the primitive concepts of mechanics were: bodies, motions, and forces. The state of the body in mechanics was the place occupied at a given time (configuration). A thermodynamic, state must be accepted in the same "primitive" way. This thermodynamic state is a set of C + 1 functions ef time where C is the number of components of the system, Temperature, 7, and configuration X. a are two of the parameters selected to describe the thermodynamic state. The other C'l parameters are usually taken as the mass oF Cl constituents. a The concept of X generalizethe concept o<" volume used in thermostatics. ~
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180 in order to calculate, or even define thermodynamic quantities, one must proceed with caution. For example, pressure may not have an equivalent concept in thermodynamics, as it is, the properties of stress P will indeed resemble those of the thermostatic concept of pressure for some kind of material. An enumeration of the quantities involved in the most general momentum and energy balances will indicate the existence of eight functions of time and position to be considered in a thermodynamic process. These eight fields are: A motion x, stress P, and external body force field b, coming from the momentum balance equation, p x : div P + pb (BIl) where p is the density and x (t) is the acceleration. A temperature T, energy E, entropy S, heat supply q, and heat flux h, necessary to express the concept involved in the first and second laws of thermodynamics pE z W + div h + pq balance of energy (Bllb) where W is the rate of mechanical working, i.e., the rate at which the stress P, does work. pTS Â£ T div(h/T) + co ClausiusDuhem inequality (B12) This inequality applies at a point, and generalizes the macroscopic observation that the n Â•: of heat supplied ! body does not accounl : or the total rate or increase of entropy.
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18] The eight functions of position and time defined to express the phenomenologi cal laws, equations Bila, B Â— 1 1 b and B12, constitute a thermodynamic process. The external force field b, and the external supply of heat q are usually given. This leaves six fields to satisfy two balance equations (Blla and B 1 1 b) subject to one constraint (b12). The principle of determinism can now be generalized to include thermodynamic processes. The thermodynamic constitutive assumption is that the histories of the temperature and the deformation at a point of a material determine all other independent thermodynamic fields, i.e. stress P, heat flux h, energy E, and entropy 3. = Htf (F*, T*) (B13) where represents the collection (P, h, E, and S) . Instead of E, a specific Helmholtz free energy A is defined to use as a possible thermodynamic potential for P and 5. S3 A E E TS (B~U:) Based on this more general theory of materials, speciiic assumptions sometimes implicit in other theories can be proven to be only consequences of more generally accepted experimental laws. An example of this is Coleman's result that the specific thermodynamic properties of a simple material (equation B13) do not depend on the temperature gradients within it. This result is a consequence of the generalized second law of thermodynamics
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18.2 (equation 512). Also, the conditions to be satisfied, by material and process, for the generalized free energy to be a thermodynamic potential can be explicitly stated. These conditions are very important in any consideration of application of thermodynamics to glass transition. They are indeed the object of the search for a phenomenologi cal definition of glass iness . c . Thermodynamic potentials, irreversible thermody nami cs The existence of a state function which determines the pressure and entropy is the cornerstone of thermostatics, and in genera^ useful thermodynamics. Pressure and entropy can be used as state variables, only because they are "proven" to be the "gradients of a potent i a 1 ." For a material described by equation B~13, Coleman has proven that such a potential exists in the following cases: 1. For all processes of very special kind of materials, usually the object of molecular thermodynamics. 2. For all processes, if the material has fading memory or quas i elast ic response. 3. For all simple materials in the limit of slow processes. k. For all simple materials in the limit of fast processes. In the first case, it is assumed that the rate of work of the stress tensor, P, is given by
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183 k f n wdV r I Pi ( 17,0 y . (e15) [=1 p Â— ,_, "I ~ , where in turn k P, = P, (T, y) + I w. b (T, y) Y, (B16) 1 b=1 The thermodynamic pressures, Pj, are intensive properties which do work when the process represented by the vector y changes. The rate of change of the state variables is given by y . Equation B16 represents the assumption that the rate of flow or "fluxes" are linearly related to the "driving forces," The excess mechanical work vi is presented integrated over the body because of the homogeneous character of these theories. Under these conditions it is shown that a thermodynamic potential is indeed related to the "pressures." P,(T, Y) = JA (B17) 3Y: where A is Helmholtz free energy, and Pj ! s represent the equilibrium values of pressure and chemical potentials when y is chosen as the volume and the concentration of the constituents of the system. it must be noted that only pj's, and not P j ' s are state variables independent of the "path." The phenomenolcgica] Â•Â• fficients w.^
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18'+ are not determined by the state of the system; however, trey must satisfy certain constraints due to the ClauslusH i! =n inequality. These systems are in equilibrium when at constant 7; histories are not considered. The second case where a thermodynamic potential exists is when dealing with quas i elast ic materials. Thermodynami cally, a material is said to be quasi elastic or to show fading memory, if the state of the system (F*, T*) at t is entirely determined by the state at tdt and the elapsed time dt (see equation Bl8c, next page). This definition implies that a small change En the past history causes a proportionate change in the thermodynamic functions 0. This is a postulate of smoothness of these functions with time, it has be proven that this smoothness condition is satisfied by H if the material shows fading memory. Fading memory in general is the observed property of real materials whereby materials respond less to long past deformations than to recent ones. In particular, the thermodynamic properties of a material which has been at rest for a sufficiently long time should be the "static" properties of the material. This idea is expressed specifically by: H(F\ T l ) = f(F(t), T(t) ) + H (FS T) (BI8a) (F, T, t) (B1
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183 d0 = / \ f \ _a \ dF + ( jU) dT +( i9) F .' T (BI8c) where f is a function of the present values of F, and T and Hg is a functional of the past histories (present values not included, t t (F,T) ) of F and T. Equation 81 Sb indicates that for a given materia] the function g wi 1 1 depend upon what the past history is in Equation Bl8a. The assumption of quasi elast icity or feding memory imposes a restriction on the material or process, namely that purely viscous stresses are not allowed. Quasi elast ic materials do not include the behavior of materials where instantaneous viscous dissipation is present. In this case if F and T are held constant while changing t, equation B18 will not ft! express the changes in P or any other of the responses. The viscous stress could be discontinuous at t, and Hg(F, T) would not. include its effects because the history (F~, T*) only contains values of F and T for the past. The most important result of the theory of quasi elasticity is that the effect of memory upon the free energy A determines all the effects of memory upon P and S. Using the expression Bl8c and the Claus ius Duhem inequality the stress, entropy and free energy dissipation, D, are given, at any instant, by
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1 86 P = p(3g/8F) F + (B19) S = (8g/3T) F,t (B20) D = (3g/3f) F T (B21a) = PTS div h pq (B2lb) where, the internal dissipation, D, represents the local excess of entropy production, D>0.
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REFERENCES 2. 3. 4. 5, 6. 7. 8. 9. II, 12. 13. 14. 15. 16. Adam, G., end Gibbs, J. H., J. Chem. Phvs. 43. 139 (1965). Angel 1, C. A., J. Chem. Phvs. 46_, ifÂ£73 (1967). Barrall, E. M., li, Schmidt, R. J., and Johnson, J. F., Presented a t Am. Chem . _S oc. on Asphalt : Com position, Cheml stry and Ph ysics , April (1964). Cinder, G., and Muller, F. H. , KolloidZ. . 177. 129 (1961). Bondi, A., "Physical Properties of Molecylar Crystals, Liquids, and Glasses," John Wiley and Sons, inc., New York, Chapter 13 (1968). Brodnyn, J . G . , Highwa y Resea rch Board, Bu 11 et i n 192 ( 1 958) . Bueche, F., J. Chem. Phvs ., 16, 2940 (1962). Callen, H. BÂ„, "Thermodynamics," John V/iley and Sons, Inc., New York (I960) . Chen, H. S., and Turnbull, D., J. Chem. Phvs., 48, 2560 (1968). Cohen, M. H., and Turnbull, D., J. Chem. Phvs., 3_L H64 (1959). Coleman, B. D., Arch. Ra tion. Mech. Anal ., 17, 1 (1964). Coleman, B. D., ibid .. 230 (196% Davies, P.. 0., and Jones, G. 0., Proc. Roy r Soc. (London). A 217, 26 (1953). DeGroor, S. R., Mazur, P., "NonEquilibrium Thermodynamics," North Hoi land Publishing Co., Amsterdam (1963). Doo 1 i 1 1 } e , A. K . , J. Ap pl Phvs . 71, 1 47 1 (1951). Er'ngen, A. C., "Nonlinear Theory of Continuous Media," McGrawHill Book Co., Inc., Nov.. York (1962). 17. Engelter, A., and Muller, F. H., Rheol [ Acta. Â±, 39 (1958) 8. Ferry, J, D., "Viscoelast ic Properties of Polymers," J hn Wiley and Sons, Inc., New York (196i). 187
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183 19. Fitts, D. D., "Nonequi 1 ibrium Thermodynamics," McGrawHill Book Co., Inc., New York (1962). 2C. Frenkel, J., "Kinetic Theory of Liquids," Oxford Univ. Press. London and New York (1946). 21. Gaskins, F. H., Brodnyn, J. G., Philippoff, W. , and Thelen, E., Trans. Soc. Rheol . , 4 (1900). 2.2. Gibbs, J. H., and Dimarzio, E. A., J. Chem. Phys ., 28, 373 (1958) 23. Glasstone, S., Laidler, K. J., and Eyring, H., "The Theory of Rate Processes," McGrawHill Book Co., inc.. New York (1941). 24. Goldstein, M., J, Chem. Phys., 51, 3728 (1969). 25. Goldstein, M., J. Chem. Phys.. IS... 3369 (1963). 26., Herzfeld, K. F., and Litovitz, T. A. "Absorption and Dispersion of Ultrasonic Waves," Academic Press, London and New York (1959). 27. Instron Engineering Corporation, Maintenance instructions Manual, Manual No. 102913 "(A) , Massachusetts (1966). 28. Jaunzemis, W., "Continuum Mechanics," The Mac'millan Company, New Yo k (1967) . 29. Kovacs, A. J., Trans. Soc. Rheol . ^ 285 (1961). 30. Kovacs, A. J., "Rendiconti della Scuola I nternazionale di Fisica Enrico Fermi XXV!! Corso," Sette, D., ed., Academic Press' York (1963). 31. Maj idzadeh, K., and Schweyer, H. E., Mat R es_,__S t; d . , 6, 617, (1966). 32. Middleman, S., "The Flow of High Polymers," I nterscience Publishers, John Wiley and Sons, Inc., Mew York (1968). 33. Muller, F. H., "Rheology," Vol. ^ Chapter 8, Eirich, F. R., ed., Academic Press, New York (1969). 34. O'Reilly, J. M., J. Poly. Sc?.. 51, 429 (1962). 35. Passaglia, E., and Martin, G. M., J, Res,. Natl. Bur. ..St., 6 8A . 273 (1964). 36. Passaglia, E., and Mevorkian, H. K., J^ Aj ' . Phyj.. 3Jb ^Â° (1963),
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i S3 37. Patter son j, D., Rubber Chem, Techno! . . 40, 1 (1967). 38. Prigogine, I., Trappeniers, N., and Mathot, V., Discussions Faraday See.. JJj, 93 (1953). 39. Ronk, D. c"., "Compressibility and Expansivity of Asphalts," Thesis, University of Florida, Gainesville, Florida ( 1 968) . 40. Rush, K. C., J. Macromo l. Ssf . Phys.. B2, 421 (1968). 41. Schmidt, R. J. and Barrel!, E. M., J. Inst, of Pet .. iL 162 (1965). 42. Schmidt, R. J., Boynton, R. F., and Santucci, L. E., Ai em. Sjje, P.rroleurn Division Preprints. K) : (1965). 43. Schmidt, R. J., and Santucci, L. E., Proceedings of the Associa t j on of Asphalt, Payjng Techno! og [stSj 3_5_, 61 (1966). 4'!, Schweyer, H. E., and Busot, J. C., Highway Research Board, Bulletin 273 (1969). 45. Schweyer, H. E., and Busot, J. C., Annual Report Research on Asphalt ic Materials, Project CR5949. Florida Engineering and Industrial Experiment Stati Report 4 .c Florida State Department of Transportation (1968). 46. Shoo.,, S. K., "Studies of ViscosityTen,;"... re Relations for Asphalt Cements," Thesis, University of Florida, Gainesvlll Florida (1965). 47. Shoor, S. K., Majtdzadeh, K., and Schweyer, H. E., HUgl R .Bulletin JJit (1966) . 48. Sukanoue, S., Nippon Kagaku Zasshi. 84. 38^ (1963). 49. Truesdell, C., "The Elements of Continuum Mechanics," SpringerVerlag, New York, Inc., New York (1966), 50 Vogel, H., Phvsik. Z ., 22, 645 (1921). 51. Wada, Y.j and Hi rose, H. J., J. Phys. Soc, Japan, J5_, 1885 (I960) , 52. Williams, M. L., J. Appl . Phys ., 2Â£, 1395 (1958). 53. Williams, M. L , J. Phvs. Chem. , 5J), 95 (1955). 54. Williams, M. L., Landel, R, F., and Ferry, J. D,, J. Am. CI Soc. 7J . 3701 (1955).
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BI0GRAPH1 ;AL SKETCH J, Carlos Busot was bum in Havana, Cuba, or, August 2h, 1 33^. He was a senior in Chemical Engineering at the University of Villanueva in Marianao, Cuba, when he and his wife had to leave their country for political reasons. J. Carlos Busot became a political refugee in July, 1961. fie worked as a laborer in a paint factory in New York for a short period of tine. After this initial contact with the American way of life he decided to enroll at the University of Florida t.o continue his studies. In June, 1962, after he received the degree of Bachelor in Chemical Engineering, with honors, he continued with his s t u d ,' e s . In December, 1962, he receive: the degree of Master of Science in Chemical Enc n i ng. From December, 1962, to January, 1967, he worked with the Du Pont Company at tl D Technical Laboratory in Kinston, North Carolina, as a Research Engineer. His work dealt primarily with the manufactui ing of polyethylene terephthalate. He holds a p nt in this area. In January, I967, he obtained a leave of absence from his employer to cc ' his studies. After working as a Research Associate at the Asphalt Laboratory of the University of Florida, h .2 full student in the Fall of 1967.
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In AprN ; 1S70, he and his wife, Esperanza, pledged allegiance to the United States of America and became citizens of this nation, J. Carlos Busot is a member of the American Institute of Chemical Engineers and the Society of the Sigma Xi. He is a registered Professional Engineer in the State of Florida and a member of the National Society of Professional Engineers. He is the father of three children, Jose Carlos, Esther Mar fa and Francisco Luis.
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This dissertation was prepared under the directi >n 3f the chairman of the Candida : supei /isory committee and has been approved by all bers oF that committee. It was submitted no the Dean of the Engineering ; id to the Graduate Council, end was approved as partial fulfillment oF the requirements for the ee of Doctor of Philosopl A ;st, 19 Dean,, College of Enginee i ; i sory Committee: Chai . Dear., Gradu= te School (y
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