DYNAMIC COMPRESSION OF ASPHALTIC GLASSES
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
A Ml ESPoSA EESPrIRANZA
The author wishes to thank Dr. H. E. Schvayer for his h-p-
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
ACKNOWLEDGMENTS . .
LIST OF TABLES . .
LIST OF FIGURES
ABSTRACT . . .
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 Quasi-Elastic Materials. 38
Develop r--nt of thle Eautions to be Used for
the Description of Glassiness . . 44
GrCohicai ticdZ c f a Systcr, Sltowing Quasi-
Elastic -' pornss- . . . . . . 41I
. . . 1
. . . 4
TABLE OF CONTENTS (cont.)
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.)
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
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
S63-13 . . . . . . . . .. . 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
S63-09 . . . . . . . .... .. 129
6. Coefficients of the Equation p= Ao + AIP + A2P2
and Compressibilities at 0 and 1000 atrrs for
S63-13 . . . . . . . . .. . 130
7. Coefficients of the Equation P= Ao + AP + A2P2
and Compressibilities at 0 and 1000 atms for
S63-20 . . . . . . . .. .. . 131
8. Coefficients of the Equation P= Ao + AlP + A2P2
and Compressibilities at 0 and 1000 atms for
S64-47 . . . . . . . . .. . 132
9. Dynamic Parameters of Selected Asphalts ..... 147
A-l Preliminary Data at 32F for Rheology of Twelve
Florida Selected Asphalts . . . . .... 157
A-2 Effect of Aging on Compressibility of Asphalt
S63-9 . . . . . . . . ... . 162
A-3 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 D-f cr atici for C-li-
bra.ioni at T0wo Rates o C,; precs ion . 87
LIST OF FIGURES (cent,)
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. Decompression-Recoil 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 S63-13 after Different Histories
(Samne ', Different Deformation Ti-es) Ill
25. Recoil on S63-13 after Different Histories
(Different Y, Same Deformation Time) . 112
26. Recoil on S63-13 and S64-47 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 S63-9 . . 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 S63-20 at 25C . 136
34. Recoil of All Selected Asphalts at 250C . 137
LIST OF FIGURES (cont.)
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
A-1. Specific Heat versus Temperature for Asphalt
Cement (S63-20) . . . . . ... .154
A-2. Pressure Required for Compression with and
without Lubrication . . . . . 156
A-3. Assembly for Studies on Effectiveness on
Lubrication . . . . . . . . 164
A-4. Failure of Silicone Lubrication at -250C . 169
B-1. 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
J. Carlos Busot
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 in-depth 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-
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 quasi-clast: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 glass-like 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 liquid-like behavior to a solid-
Since in-service behavior of asphaltic materials subjects them
to a compression-decc -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.
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
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
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 time-temoerature 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 visco-elastic poly.rrs.
I 7 I
Figure .-TypIcal Glass Transition Temperatre Determination f
0.02a Paving Asp by Penetroter
I / 1-
I 0 2I
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 temperature-time 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 viscosity-temperature
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
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
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
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 liquid-like to a solid-like 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.
Inducement of Glassiness
In general most liquids can be transformed to a non-crystal-
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.
< A y
I H G
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
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
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 glass-forming 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).
Observat ion i
T T I To
Figure 3.--Davies and Jones Concept of Glassiness. Observation
at Constant Temperature.
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
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, assum-d 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
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
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)
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:
More detailed and rigorous treatments of the classic theory of
thermodyna-ic relaxation are given by Herzfeld and Litovitz (26), pp.
159-170; 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(p-p) [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 -= -
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.
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 represen-ed 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
Drivring force = T T
l J '
Te mpe rat u re
Figure 4.---Davies F rcl Jcn-e Concept of Driving
Aaiabatic Pcovrv, of a Glss,.
_: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
interested on the variation of T. with pressure.
B. Explrant ion
Theory_ of the Glass State, ILF
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
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)
rs C2 (T-Ts)
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
volur-e" (vf); which they proposed as:
Vf vg [0.025 + LA (T Tg)]
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)
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 statistical-mechanical arguments the
relaxation properties of glass-forming 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
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 pa-ts 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,
ZAl = AG = kT In Q
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
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 T--T 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:
C = 2.303 k and (23)
ACp % In-
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.
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 non-specific 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
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
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.
The Ehrenfest Relations. Pressure
Dependence of the Glass
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
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.
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
experiment at different pressures allowed calculation of
O'Reilly found this ratio to be independent of pressure and equal
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 glass-like compres-
sibility." It should be noted that this definition of the transition
pressure contains two different and perhaps opposite phenomenological
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 glass-like compressibilityj' however, implies
an arbitrary definition of gless-iike 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 T--T in excellent agreement with those
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.
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
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.
By plotting the values of Tg vs. pressure, a value of---p
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,
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(i-iodynamic 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 "flo-w" 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
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'
:i i,, F.
sl:. i. "
''SOl T U
ad he ied
Figur-e 5.-Machclanical ,:~
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 non-flow 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 non-flow 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
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
capillary at a rate of shear of 0.1 sec In both cases thz rate
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
homogeneous volume change occurs with dissipative effects.
B. Th:-,' :d -,-,ics cf OCJasi-El 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
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 quasi-elastic materials,
constitutes the most general thermodynamic theory available. (See
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 quasi-elastic 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) (B-18a)a
A = g(F T, t) (B-18b)
diA dF + () dT + 2() dt (B-18c)
T,t F,t FT
aAl equations de'eloped in Appendix B aie labeled with a 3
preceding the number.
? AX 4-. ~
! /'////i/// Constrai t
-- -- /
s ~ Figure 6a
I ./,/ ,/ t ,
' in Volume or Te, .rature
Figure G.- i Co.istrai e<
P = p F+ (B-19)
S - ) (B-20)
D = p (B-21a)
D =PTS div h pq (B-21b)
The nomenclature as well as the meaning of these equations will
be reviewed in the following paragraphs.
Equation B-18a 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 quasi-elastic 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.
impi-ed i n equation B-1Sa 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 B-19 and B-20)
Equatio E-l18b is the instantaneous equivalent of equation
B-18a. 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 t-e instantaneous state of the system (V, T).
Equ3tI"on B-18c is the mathematical statement of the sought
pcten~tiac' .-::'tions B-19, B-20, and B-21 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
div h r..--
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 quasi-elastic
that it rer'es.ent the effect of imsory upon th- system.
moi-y t--n-. 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 non-flow 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 B-21b could be
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 B-21b 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
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 no-flo. 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 one-dimensional change in volume depicted in Figure 6a.
The entropy S, and the one-dimensional 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
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
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)
For quasi-elastic materials A is a potential for pressure and entropy.
Therefore, these properties could b' used to specify the response of
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)
tir, a: V and
the pest volur<
EquatIo s 30 &
energy c '':
p(V, T) is
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
For a quasi-elastic 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
3(3A/ OV)t I
S 2A 2i A/ TIL Vt
TdV 3 VI T, t
because of the properties of exact differentials. Therefore,
equation 31 can be rewritten as
d P .P 2( J )
dt DV T
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.
S(A/ Vt I
or in more familiar terms,
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
as ai- (^ _- iA\
Sv v T I V'/
T, t Tt V, t
a T/ T
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
S = + -C. V + CV D (43)
Where a : Q and CV are the instantaneous, history-dependent 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 d--ter'rins
the entropy of the system.
The dissipati/e terms Dy. and DT are defined by equations 31
and 38 respectively. Explicitly:
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
aA - D ( 0
The interpretation of 3 and ^ needs some comments. The develop-
ment of equations 35a and 41a required the interchange of partial
v;, ( -
0*- k *1l
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
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 .
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
/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 (quasi-elasticity),
\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 quasi-elastic material the following equation is obtained
IVro r b-t cn t T- (51
Furthermore, by changing the order of partial operators;
(,_ .". /, Q. (5\)',
\ ; T- (52)
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 Clausius-Duhem in-
equality (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
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 "free-volh '" and the "cooperative" theories predict
thLat te-rpsraLures 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 quasi-elastic
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.
PROPOSED GE -1AL DESCRIPTION OF GLASSINESS
In Chapter Ill, pp. 35-3, 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, non-flow 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 glass-like 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 non-flow 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 solid-like partially ordered structure). This same polymer melt
may not show a correspondingly large volume viscosity in a non-flow
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
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 non-flow 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
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
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 resist-
ances present in all non-isotropic 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
P = p(V, T) + Hp(V -, Tt) (30)
S s(V, T) + HS(V-, T-) (55)
P =(- v +DV (31)
S : ) V T+DS (L42)
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
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 quasi-elastic 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
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
For illustration purposes pressure will be chosen as the
response of the s' tem.
C.. I n
cr N.- -, t
oC. ar lJ (,, n/) =) :LU
- ~ .~ -
S \ \
V -- C
- 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
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 stor-ld 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
2 (2,"2 te
t 1 N
u, 6 'n
_ _ _ _
i? __I I~--------, `
Figure F-r-txce~; ucctions an- 0 1as '. -as
-- T, Temperature---
Sr (o., T0)
Figure lO.--Decompress ion anc Heati of a Glass
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
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 .--
i / / Vi n '
o i / 2/o. 0 /
I--, T j
S! // I 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,
i.e.. = - -
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-volumne) 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
The last section dealt with general cons iderat ions and results
of an interpretation of giassiness limited to "non-Hlow processes."
This is not the type of process used in practice to obtain informa-
tion about glass transition. In general non-honogeneous 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 de--cripticn 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-
Following Davies and Joies in their two-din.'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.
1 i :4,
Relaxation .-TerpI C~ie pera tur e ,
V ~ i M , v o u e Ji n cr ea s i ngg r a
Figure 12.--Glassiness during Ischaric Processes.
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.
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 liquid-like to a solid-like 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
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 thermo-mechanical 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
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 kno-m 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 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 non-ho ogeneous deformation.
The experimental method proposed herein to study the thermo-
dyna ric properties of asphalts in non-flow 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 so-called 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 non-homogeneous volume change,
e.g, those causing the "Wjell-known 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__
BARRL -- -- ---------- ^
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 Steam-Refined inter-
mediate (S63-9), Low-Sulfur Air-Blo',n Naphthenic (S63-13), 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 identification number given by the
University of Florida Asphalt Laboratory is S64-47. Table 1 suim-
marizes the physico-chemical 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 M-3-1). 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-
Figure 14 shows the Instron Universal Testing machine.
Compression of the so ple is accoiplishacd by lower;ng th: tra-al-
ing crossheaj towards a fixed tb-Le sL' *,ortin a irncif ;n.1tron
Properties of the Four Selected i'phalt Ce: ents
Identification S63-9 S63-13 S63-20 S64-47
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. -
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 )
(Per cent by weight)
Paraffinic-Napht' hnic 13.1 23.9 10.3
Nahth nic-Airo 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, *er-Chiplcy s araticn proce:-ure.
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 0-200
to 0-10 :'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 on-off 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
B. Prelimin rie
This section describes stUedies i F -:o bs ic questions concern-
ing th,; e. rir 1 ntal lsthoc: (1) ti-h e'f: ci oC '"ey; .: o- tj re-
0. 7D0 A I d;mens ions
I 0.3 65
I I2 I '
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
D0rres-i 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 time-teinperature
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 r-nner in which the balloons were sealed depended
upon th: objective of the run, and will be discussed later. The
samples were sor etin-es held at room temperature for various lengths
of time before running with no apparent change in their compressive
characteristics. Table 2 su-mmrizes 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 rscn-ed in Tc 1', 2 were obai by calculations using
Effect of Aging
Run No. R70-15
Reference (XLI, 99-100-104)
P pressure (atm.)
(atm-l x 105)
Velocity of Crosshead VXH:
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. Altho--h 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 determinL-io- 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
load limits of the ass mbly (0-2r1-3 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
defo-ration (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 w-ould 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
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
10ormat n (inches x 10
at Two tes of i
D Deformmtaon (inches x 1bao
Figure 16,--Tracing of L-o.,U rsus Deformadin for Ca !Sration
ai: Two *ates of .iiiprese ion.