Title: Dynamic compression of asphaltic glasses
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Title: Dynamic compression of asphaltic glasses
Physical Description: xii, 189 leaves. : illus. ; 28 cm.
Language: English
Creator: Busot, Jesus Carlos, 1938-
Publication Date: 1970
Copyright Date: 1970
 Subjects
Subject: Asphalt concrete   ( lcsh )
Compressibility   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida, 1970.
Bibliography: Bibliography: leaves 187-189.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00097709
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000953419
oclc - 16943473
notis - AER5875

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DYNAMIC COMPRESSION OF ASPHALTIC GLASSES















By
J. CARLOS BUSOT















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA

1970




































A Ml ESPoSA EESPrIRANZA













ACK!O

The author wishes to thank Dr. H. E. Schvayer for his 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


Page


ACKNOWLEDGMENTS . .

LIST OF TABLES . .

LIST OF FIGURES

ABSTRACT . . .

CHAPTER

I. INTRODUCTION


A. Glassiness
B. Objective of


this Research . .


II. CRITICAL ANALYSIS OF CONTEE;PORARY THINKING ON
THE QUANTITATIVE DESCRIPTION OF GLASSINESS

A. Description . . . . . . . .

Inducement of Glassiness . . . . .
Single Point Measurements . . . . .
Thermodynanic Description of Glassiness

B. Explanation . . . . . . .

Theory of the Glass State, WLF Equation .
Volumetric Viscosity . . . . . .

C. E>. rimental . . . . . . .. .

The Ehrenfest Relations, Pressure Dependence
of the Glass Transition Temperature
Applications . . . . . . . .

11I. INTERPRETATION OF APPLICABLE THEIR iODYNAMICS


Definition of Glass . . . . . 35
Thermodynamics oF 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


Iii


viii


xi


4
. . . 1
. . . 4


6

6


20

20
27

30










TABLE OF CONTENTS (cont.)


CHAPTER Page

Equations . . . . . . . . 46
Dissipative Terms . . . . . ... 50

IV. PROPOSED GENERAL DESCRIPTION OF GLASSINESS . 55

A. Glassiness and Excess Thermodynamic Functions 57

Changes in Volume in Unconstrained Materials,
"Constant" Pressure Experiments . . 57
Excess Thermodynamic Functions . . .. 58
Graphical Description of Glassiness . .. 63
Isobaric Transitions . . . . ... .70

V. DEVELOPMENT OF EXPERIMENTAL TECHNIQUE . . . 73

A. Materials and Apparatus . . . . . 77
B. Preliminaries . . . . . . . . 80

Age, Preparation of the Sample and Presenta-
tion of Results . . ....... 82
Machine Characteristics. . . . ... 86

C. Homogeneous Deformation . . . . .. 92

Independence of Dynamic Response on Sample
Length in Relation to H-- --.. :ty ... 93
Study of Dynamic Effects, Relating to Homo-
geneous Deformation . . . . . 99

VI. RESULTS, CONCLUSIONS AND RECOMMENDATIONS . .. 119

A. Results . . . . . . . . . 120

General Procedure . . . . . ... 120
Density versus Pressure . . . . . 123
Recoil and Relaxation . . . . ... 134

B. Summary of Conclusions . . . . .. 146
C. Recommendations . . . . . . .. 149










TABLE OF CONTENTS (cont.)


APPENDIX


A. HISTO.iCAL DEVELOPMENT OF EXPERIMENTAL
METHOD . . . . . . . . 152

1. Thermostatics . . . . . 152
2. Preliminary Dynamic Studies . .. 155
3. Development of the Experimental
Technique . . . . ... .158
4. Preliminaries . . . . . 160

B. CONTINUUM MECHANICS . . . . .. 171

1. Phenomenological Concepts . . . 171
2. Thermodynamics ... . . . . 177

REFERENCES . . . . . . . . .. . 187













LIST OF TABLES


Table Page

1. Properties of the Four Selected Asphalt Cements 78

2. Effect of Aging . . . . . . .... .. 83

3. illustration of the Effect of Sampl Length on
Dynamic Determination of Density. Asphalt
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


viii










LIST OF FIGURES (cent,)


Figure Table

17. Illustration of Acceleration Effect . . 91

18. Deformaticn of Two Samples of Different
Lengths without Lubrication . . .. 94

19. Decompression of Two Samples of Different
Lengths Using Lubrication . . .. 96

20. Assembly for Studies on Effectiveness cf
Lubrication . . . . . . 103

21. 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.)


Figure Page

35. Relaxation of All Selected Asphalts at 250C 139

36, Relaxation of All Selected Asphalts at 00C 140

37. Relaxation of All Selected Asphalts at
-300C . . . . . . . . .. 141

38. Recoil of All Selected Asphalts at 00C . 142

39. Recoil of All Selected Asphalts at -300C 143

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

By

J. Carlos Busot

August, 1970


Chairman: Dr. H. E. Schweyer
Major Department: Chemical Engineering

It was the purpose of this research to clarify the pheiomeno-

logical description of the behavior of asphalt glasses, to conduct a

critical review of the literature concerning the meaning of glaZsiiess

as a material characteristic and to develop an experimental approach to

study quantitatively glassiness in asphalts.

The 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-

l ization.

This research generalizes that glassiness be defined as the

phenomena observed when the properties of a material depend on the

past history of its thermodynamic state.

The experimental approach proposed for the more general scudy of

glassiness was to observe the pressure and the entropy responses of a ma-

terial when subjected to homogeneous changes in volume. The thermo-

dynamics of 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-

like behavior.

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.













CHAPTER I

INTRODUCTION


A. Glassiness

Asphalts in service or at ordinary ambient temperature often

exhibit a behavior which is neither that of a viscous liquid nor

that of a crystalline sold. Phenomenologicaily, this behavior may

be ascribed to a glassy state. In the case of asphalt, glassiness

is manifested by hardness, by cracking and conchoidal fractures

under sudden stress, by a glassy surface appearance and by a very

high viscosity. These properties, which are undesirable from the

standpoint of road performance, are accentuated as the temperature

decreases.

The onset of these properties with temperature is determined

not only by the chemical composition of the asphalt but also by

the physical and spacial interactions among its numerous components.

The relationshipsdescribing these thermodynamic phenomena are dif-

ficilt to resolve. However, the thermodynamic theories for poly-

mers and organic glasses that have been developed to explain thsir

theological properties and their temperature dependence would ap-

pear to be applicable to asphalt.

Work in this laboratory by Shoor (46) has shown that glassi-

ness in asphalt can be detected by freezing the sample and determin-

ing the change in hardness (penetration) as the temperature increases.

An abrupt change is noticed on the change of hardness with temperature










as shown in Figure i. The temperature obtained at the intersect io

of the lines presenting the !ow temperature and high temperature

behavior is defined as the glass transition temperature of the

material; T

This empirical determination of T9, by the intersection of uwo

straight lines representing high and low temperacurke behavior.

constitutes the generally accepted phenomenological definition of

glass transition. This graphical method is also applied to plots

of viscosity, volume (and many other properties) vs. temperature.

Shoor, Majidzadeh, and Schweyer (47) used the glass transi-

tion temperature (T ) determined with the penetrometer method to

correlate the temperature dependence of the viscosity of eight cif-

ferent asphalt cements. The correiation scheme used by Shoor et.

al. consisted of shifting the data obtained at different temperatures

along a logarithmic time axis and determining the value~ of the

shift factor required at each temperature to form a composite

curve. The values of the shift factor were related to temperature

through an empirical equation involving a characteristic temperature.

This temperature was the penetration related Tg. This scheme is

kno.'n as the 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.










2; 1


0.03 -











I I
O k

I 7 I










:0-
-9 /













Figure .-TypIcal Glass Transition Temperatre Determination f
0.02a Paving Asp by Penetroter
I / 1-








I 0 2I








Temperature, OF
Figure l.--TypIcal Glass Transition TemperatL re Determination bf
a Paving Asphalt by Penetronate-,









Wada and Hirose (51) used a dilat:metrically measured glass transi-

tion temperature to correlate the 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

extremely difficult.


B. Objectives of this Research

A comprehensive study on the thermodynamic background control-

ling the rheological properties of asphalt and how they vary with

temperature may be fruitful in understanding the physical behavior

of these complex organic materials. This study should provide the










basis to understand and describe, with generality, the observed

glass phenomena.

The research described herein is intended to elaborate on

this thermodynamic background, through the following specific

object ives :

1. A critical analysis of the contemporary thinking on

the quantitative description of glassiness.

2. An interpretation of the applicable thermodynamics.

3. A proposed general description of glassiness.

4. An experimental technique for the study of the

proposed description in 3 above.

A separate chapter will be devoted to each of the above

objectives.













CHAPTER II

CRITICAL ANALYSIS OF CONTEMPORARY THINKING ON
THE QUANTITATIVE DESCRIPTION OF GLASSINESS


Some statement about the distinction between description

and explanation of glassiness should precede any critical analysis

of this subject.

The description of the glass transition must be clarified

before attempting the interpretation of the experimental results;

the description of glassiness constitutes its phenomenological

definition. Many phenomena of diverse nature may be involved in

the transition from a 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.


A. Description

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.

















I






A
< A y






I H G


-r 'I
S/








ST T2 T9 T
TB -- TA
S' -". .-. "- .-- --..... t

Temperat u re
Figure 2.--Effect of Temperature History on Deteminat:ion of Tg










Single Point Measurements

The topic of representative measurements of material properties

during dynamic transitions is of foremost importance for a proper

description of glassiness. This topic and its relation to the

phenomenological definition of glassiness will be discussed in

Chapter IV. However, because of its relation to the state of the

art, the implications of "single point measurements" will be discussed

here.

A review of the literature of glass transition temperature

reveals that the effects of sample shape on the spatial distribu-

tion of the responses to external changes are usually neglected.

The assumption is generally made that a single point measurement

is representative of the state of the sample. This assumption is

only true if either the mal.~rial is at equilibrium or if it is under-

going a ho.'og.j2nous deformation. This last statement needs qualifi-

cation and is discussed furT':er in Appendix B, section Id. Never-

theless, it is presented here to direct the attention of the

reader to the importance of a priori theoretical considerations about

the experimental conditions. The most common justification for

single point measurements given by experimentalists studying glass

transition phenomena is based on the size of the sample. It is said

that if the size of the sample is small enough, a uniform temperature

distribution will be established "fast enough" or "rapidly." The

possibility is ignored that regardless of the size of the sample

and the uniform temperature, enormous spatial gradients in density

and pressure ra be present if a "volume viscosity" exists.










No general statement can be made about the correlation,

thermodynamic or otherwise, between stimuli and their response,

unless the responses and stimuli are known at each point of the

material. For example, a uniform state of stress and strain through-

out an elastic material is required before the Young modulus can be

determined experimentally. Design of specimens, such that applica-

tion of external forces will produce a describable state of stress

and strain, is always an important consideration when testing solid

materials. Also, the velocity at each point of a flowing Newtonian

fluid is required before its viscosity can be established. Visco-

meters are designed to produce describable velocity profiles, so

that measurements at boundaries will allow calculation of the

response at any and every point of the fluid.


Thermodynamic Description of
Glassiness

When dealing with the experimental conditions necessary for

determination of thermodynamic parameters near a transition tempera-

ture one has to consider the relaxational or irreversible character

of the process. The difficulties of describing an irreversible

process are accentuated when trying to reconcile classic concepts

of thermostatics, i.e., specific heat, thermal expansion, pressure,

internal pressure, etc., with classic concepts of rheology like

viscosity and viscous stress. This will be illustrated by comment-

ing on the widely recognized work of Davies and Jones (13).

Davies and Jones are the only investigators known to this

writer who have attempted an irreversible thermodynamic approach










to describe glassiness and establish relationships among the

measured thermodynamic parameters. In the opinion of this writer,

the most important contribution of Davies and Jones was to define

a "volume" viscosity to quantify the "time effects" observed during

glass transitions. Regretably, their work has been consider

superficially in the literature to provide background for discus-

sion of the effect of pressure upon glass transition. A brief

presentation of the work of Davies and Jones follows. It is intended

to illustrate tho involved and far reaching conclusions drawn by

these authors, and others, by their manipulation of a phenomeno-

logical description of glassiness.

Figure 3 illustrates Davies and Jones' phenomenological concept

of glassiness. The line LAC represents the enthalpy vs temperature

idealized equilibrium line of a 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).





13












L
I I








B
Observat ion i
/ Y
I i












T T I To
Tempera tioure

Figure 3.--Davies and Jones Concept of Glassiness. Observation
at Constant Temperature.
Li p








at Constant Timoeratjre.











This simplified description, in this writer's opinion, led

Davies and Jones to overlook the influence of the rate at which

the irreversible path AB is traveled. A glass once at a point away

from AC, point G along AB for instance, cannot be brought back

reversibly to the original point in the equilibrium line. The paradox

in their description is that a glass can be "cooled" reversibly from

the liquid (point of departure from the equilibrium line) but can

not be restored to a liquid state reversibly.

Davies and Jones did not recognize the importance of the

"preparation time" of the glass (time along AG) and the time involved

in the sensing of the glass properties. These authors considered

the preparation time sufficiently fast and assumed the sensing time

to be of no importance. The latter assumption is not consistent

with their observation that the rate of cooling would affect the

"fictive" tempperature.

For a theoretical development of the equations necessary to

describe their phenorenological concept of glassiness, Davies and

Jones introduced a ne: state variable (Z). This variable is defined

to be continuous and to remain constant when the pressure (p) and

the temperature (T) are changed "rapidly." These authors attributed

to Z the significance of being a measure of the configurational

order, 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
-- a
as the entropy at p and T. Their final results state:

cp_ = aT AV A
p A Aa (1)
dT Sp AV AS


AS AC P (2)
Sp AS TVAa

where AI Aa and AC are the discontinuities in compressibility,

thermal expansion, and specific heat at the point where Z = constant

- Zo, Although these authors only presented data on glass transi-

tion with temperature, ACp vs. T and Aa vs. T, the theory was

intended to be general and to describe the phenomenological behavior

of glassiness with pressure.

The authors' objective in introducing the pressure dependence

on Tg was to be able to define different types of driving forces

for the delayed volume changes. This was accomplished by using

equation I and by defining a fictive pressure, p, given by:


aThe symbol x will b- used in the text to represent thei
partial derivative c,.-oratorr tf.) where y and z represenr the
other independent variables 3x / y,z











S= (3)
T TTVAc


This result of their theory proposed that a sudden

isobaric change in te.: ,,rature AT which leaves T and p unaltered

is equivalent thermodynamically to a pressure increment of:

Ap : (ACp/TVAa)AT." This calculated Ap has been used throughout

the literature to replace AT as an equivalent driving force on a

thermally produced change in volume. This "excess" pressure is

then related to the volume rate of change through a volume viscosity.

The basis for this concept is an equation expressing the change of

irreversible structural entropy,

TdS;,r r AdZ (4)

where the affinity A is


A = (p p) T,Z (5)
9p

The value of A is zero when the system is in equilibrium. Equation

5 presupposes that the system is "close" to equilibrium. Under these

conditions and assuming that dZ can be represented by dp, the ir-

reversible change in entropy and the rates of production of entropy

are given by these authors after some manipulation by:

a
More detailed and rigorous treatments of the classic theory of
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 -= -
V TI


where r is the proportionality constant and represents a volume

viscosity. Equation 8 would be approximately valid if a process

were devised where the phenomenological description of glassiness

given in Figure 3 is true.

The authors used their description of glassiness to estimate

the relaxation tirn: (and viscosity) of thermal relaxation of glycerol,

and volume relaxation of glucose. The experimental methods consisted

of cooling a sample of glycerol below its glass temperature inside

an adiabatic calorimeter, and studying the approach of the temperature

of the sample to equilibrium at constant enthalpy.


aA dot above a variable indicates time rate of change; ;.e.
drx .
xdt
dt










Figure 4 is a schematic representation of this process. The

driving forces For the irreversible adiabatic recovery of the

sample after it is cooled to To are 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

glassiness, ttien























'i


*I

Drivring force = T T
:i
bi
(.


.1


c:*

i I
Ai












..-!.
q7
B-l
i
ri


t:
-j
?> j


!i
Sri








I1
l.i
i:


l i

r ^"
l J '
['i T


i 2


T3


Te mpe rat u re


Figure 4.---Davies F rcl Jcn-e Concept of Driving
Aaiabatic Pcovrv, of a Glss,.


Forces du


Forces during


Cool ing











_:13 TV >c,
L% a Cp


The nature of this second parameter, however, would have the

same character as Z. It would represent a difference from a

"fictive" state and would not include the time effect during prepara-

tion oF the glass.

In addition to Davies and Jones. and Goldstein (25), the

fictive state approach has been used by Kovacs (29), (30), O'Reilly

(34), Passaglia and Martin (35), and in general by all experimenters
a
interested on the variation of T. with pressure.


B. Explrant ion


Theory_ of the Glass State, ILF
Euat ion

The theories of models of the glass state which have been

proposed are intended to explain the success found in using

empirical glass transition temperatures to express the tempera-

ture dependence of relaxation processes, more specifically

viscosity.

The cornerstone of the theories developed to explain glassi-

ness is the empirical equation proposed by Vogel (50) to express

the d:clnd n: of viscosity upor, temperature. This equation was

modified and used for the first time for the development of a

theory for the glass state by 'Williams, Landel and Ferry (54).


aThis subject wil be discussed in more detail on p. 30.









The equation in its most common form has the expression

n CI (T'"Ts)
log (10)
rs C2 (T-Ts)

where

l/r5s is the ratio of viscosities at
temperatures T and T respectively

C1 = 8.86

C2 = 101.60C

Ts = reference temperature


Equation 10 is known as the WLF equation. Williams and co-

workers (5.2. (53), (54) found that if a separate reference

temperature Ts is suitably chosen for each system, equation 10

expresses the temperature dependence of viscosity for a wide

variety of glass -forming liquids over a temperature range of ap-

proximately 1000C above the glass transition point. Most signif-

icantly they found that the temperature Ts lies 500C above the Tg,

with a standard deviation of + 50C. They also showed that C1 was

proportional to the expansion coefficient of the liquid.


Free volume theories

Williams, Landel, and Ferry used Doolittle's experimental

findings (15) that viscosity depended exponentially on "free

volur-e" (vf); which they proposed as:


Vf vg [0.025 + LA (T Tg)]


(li)










where v is the volume at the glass transition temperature, and

is the difference between the thermal coefficients of the liquid

and the glass. The original free volume theory of Williams, Landel

and Ferry assumed that the ratio of free volume to total volume

would remain fixed below Tg and that its pressure and temperature

variation above T9 would be given by


AS = g (12)


Aa = a1 o (13)

where 3 and a are the compressibility and thermal expansion coef-

ficient of the liquid (1) and the glass (g).

Cohen and Turnbull (10) developed a more complete theory of

diffusion and no;nentum transfer based on the basic concepts proposed

by Williams, Landni and Ferry. Cohen and Turnbull did not intend,

however, to include associated liquids whose viscosity varies

markedly with temperature at constant volume. Rather. these authors

proposed a description of glassiness and molecular transport in

liquids and glasses in which no potential energy barriers among

molecules existed (hard sphere model). The potential energy of a

molecule was assumed constant except upon intermolecular contact.

Interactions among molecules were not necessary to explain glassiness,

and viscosity would only depend on temperature through volume.

Cohen and Turnbull visualized flow as a process involving

molecules jumping over barriers created by the need of formation

and redistribution of holes in a liquid quasilat ice. These










authors assumed that these barriers stemmed from the need of free

voiume (v) to be greater than some value (v"'). This idea of flow

is quite similar to the activation state or kinetic theory of Eyring

(20) and (23). in Cohen's theory, flow occurs when "there is a

fluctuation in density which opens up a hole within a cage large

enough to permit a considerable displacement of the molecules

contained by it. Such a displacement gives rise to diffusive motion

only if another molecule jumps into the hole before the first can

return to its original position" (10). The transport coefficient

for a molecule, D, according to this picture of flow is:

D(v) = D(v*) P(v*-) (14)

where D is the diffusion coefficient which is a function of the

volume, v, of the cage, v* is the critical or "activated" volume

just large enough to allow a molecule to displace itself, and P(v*)

is the probability of finding a hole of volume larger than v*.

This probability is given by Cohen and Turnbull (10) as:

P(v'-) = exp[- Yv*/vf] (15)

where vf is the average free volume, defined below, and Y is a

numerical factor introduced to correct for overlap of free volume.

The final expression becomes:

D(v) = D(v)* exp[ yv*/vf] (16)

where D(v*) is a function only of the molecular diameter, the

temperature dependent velocity of the molecules, and a geometric

factor. Equation 16 has the form found by Doolittle for the

viscosity of hydrocarbons. In order to test their theory Cohen

and Turnbull defined free volume as:











vo exp.. a dT (17)
L J
where vo is the van der Waals volume which is assumed independent

of temperature, a= is the coefficient of thermal expansion, and To

is the temperaLure at which the free volume vanishes. This defini-

tion assumes that the free volume is given by the total therrnal

expansion at constant pressure and is zero at a temperature To.

The temperature and pressure dependence of viscosity will be

determined, according to this theory, by the pressure and temperature

dependence of free volume as follows:

Vf =" v (T To) vp AP (18)


where a and v are the mean values of the expansivity and van der

Waals voluic-, over the temperature range (T To), evaluated at A P = 0.

B and Vp are the mean compressibility and volume over the pressure

increment A P.

The definition of free volume as a function of thermal expan-

sion alone (equations 17 and 11) is insufficient for satisfactory

description of the pressure and temperature dependence of viscosity

of polyr.eric liquids. Other modifications of the free volume

theory are designed to provide thn theory with the flexibility of

a free volume wh:ch would show a temperature dependence below T (7), (S

(40). The basic consideration of all these theories is the falling

of the free voluren below soe critical value where the high

viscosities would r- ke reltAation tir;s of the order of days.

These theories !Fha' been used to explain the rheological behavior

of asphalt by 11j idzadh and Sc- r (31).










1, ( -., 0 e t jvg arra qe^- nt
confilurat ional entr.ovy

Gibbs and coworkers (22), (1 ) have defined T as "the quasi-

static glass temperature below which molecular relaxation times are

too long to permit establishment of equilibrium in the duration of

even the slowest experiments ('time scale' of hours to days)" (1 ).

Their concern was to relate by 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

liquid.

A cooperatively rearranging region is defined by Adam and

Gibbs ( ) as "a subsystem of the sample which, upon a sufficient

fluctuation in energy (or, more correctly, enthalpy), can rearrange

into another configuration independently of its environment."

By assuming a partition function, Q, for a system containing

a number of 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,

Ap by


ZAl = AG = kT In Q


(19)










This implies the existence of a uniform structural unit, e.g.,

molecules or a number of molecular segments of a polymer chain. The

average transition probability W(T) of a cooperative region as a

function of its size is given by:

W(T) A exp[ Z* Ap/kT] (20)

where Z* is now the smallest critical size. Cooperative regions of

Z < Z-: yield zero transition probabilities. These small size regions

remain in the same configuration when energy is supplied to the

system. The frequency factor A is nearly independent of temperature.

Finally ;\ is expressed as


W = A exp TSc



W = A exp -
I TS c


(21)


(22)


where C is independent of temperature, Sc is the entropy of the macro-

scopic sample, and s c is the smallest critical entropy. It is the

entropy of configuration per molecule corresponding to the minimum

number of configurations determined by Z- at a given temperature.

The minimum value of configurations corresponding to Z* as 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:











A s

C = 2.303 k and (23)
T2
ACp % In-


Ts
Ts In ---

C2 2 (24)
in r + 1 + In -
T2 s Ts





where T2 is the temperature where s ---:- 0, aid AC is the

specific heat difference between the liquid and the glass at T9.

The cooperative rearrangement theory predicts that the

"universal" parameters of the WLF equation will depend upon: the

ratio of the reference temperature to the equilibrium temperature

T2, a free energy barrier restricting transitions, a critical

configurational entropy, and the difference between specific heats.


Volumetric Viscosity

According to the general contemporary opinion the time

dependence or relaxational character of the glass transition makes

a theory of glassiness a particular case of a theory of liquid

viscosity (24). The two previous theories represent the two most

widely discussed viewpoints on the origin of viscosity of liquids

near their glass transition.

Summarizing the previous paragraphs it is seen that the free

volume theory relies on a kinetic argument b;sed on the dependence










of flow upon the availability of holes or free spaces for the mole-

cules to move into. Glassiness is considered a 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

decreases.

Both theories relate more closely to the dependence of

viscosity on teinperature and pressure, than to the actual mechanism

of viscous flow. Both consider local rearrangements (hole distribu-

tion or cooperative molecular rearrangement) necessary conditions

for flow. However, these rearrangements are not sufficient; they

also occur on the fluid at rest. The questions unanswered by these

theories about the nature of flow are, paraphrazing Goldstein (24):

What is the relationship betwv'een the local rearrangements and

the microscopic deformation?

How does the external stress bias the local rearrangements for

a jump to occur?

Why does not the deformed state reverse to the original state

when the biasing stress is removed?

These questions are subjected to extensive analysis by Goldstein.

The answer to the third question is of particular importance in order

to establish the relationship [btw.en shear and volume viscosities.

For flow to occur irreversibly under a free volume mechanism, two

conditions must Lb met. First, molecules must jul? into a hole,

and second, holes must vaiish r:n reform. If the r.'-]axatiorn time of










the second step is smaller than the one for the first step, random-

ization takes place after the jump and the flow becomes irreversible

under the biasing stress (visccus flow). The appearance and dis-

appearance of holes could be thought of as a volume relaxation

process and, thus, the importance of the volumetric viscosity.

According to the theory of cooperative rearrangements, the

relaxation time involved in the creation of holes (second step)

should be larger than the relaxation time for a simple redistribu-

tion (first step). This appreciation is based on the consideration

of the apparently larger cooperative character for the creation of

a hole. If this picture of flow is correct, the volumetric viscosity

should limit the rate of shear relaxation. In acoustic experiments

volume relaxation times are almost equal to shear relaxation times

implying no essential differences on the nature of volume and shear

viscosities (26).

Determination of the volumetric viscosity through acoustic

experiments, however, assumes that the shear and volume effects are

additive. The additiveness assumption implies that the absorption

unaccounted for by shear viscosity can be attributed to volume

relaxation. This is complicated by the fact that shear viscosity

itself may not be represented by a simple New.tonian model as assumed,

but probably it is frequency dependent. In this case, the calculated

volumetric effect would be affected by the frequency dependence.

The additiveness assumption may or may not be justified on

the basis of the infinitesimal amplitude (or magnitude) of the

deformation involved in acoustic vibrations, but it leaves the










question open as to the phenomenological behavior of materials

undergoing finite changes in volume. In a finite volume change

structural rearrangements may be triggered which would not have

been detected or induced by the acoustic deformations.


C. Experimental


The Ehrenfest Relations. Pressure
Dependence of the Glass
Transition Temrerature

The Ehrenfest relations state (8):


dT2/dP = AS/Ao (25)


= T VM/ACp (26)

These equations express the change in the temperature at which a

transition occurs T2, with a change in pressure; 3, Ao. and Cp

are the discontinuities in compressibility, thermal expansion, and

specific heat at the transition. The volume V is measured at P and

T2*

Davies and Jones (see p. 16) used the Ehrenfest relations to

substitute dP for dT as the driving fo:ce in volume changes during

glass transition. These authors did not discuss the theoretical

implications of the relations, neither did they present any data

on the change of Tg with pressure. However, Goldstein (25) used

Davies and Jones (13) phenomenological description of glassiness

and interpreted equations 25 and 26 as criteria to test the validity

of the free volu e anid the cooperative rearrangerent theories.










The free volume theory proposed that glassiness occurs because

the free volume approaches zero at the transition temperature.

Goldstein indicated that this condition was similar to the condition

For tile validity of equation 25.

The cooperative rearrangement theory implies that the structural

entropy becomes small as the sample is cooled towards T2. According

to Goldstein, equation 26 requires that a condition similar to this

be met. i.e., that the entropy of transition be zero.


A6plicat ons

The experiments of O'Reilly (34), and Passaglia and Martin

(35), will be discussed to illustrate typical past efforts to

elucidate the meaning of equations 25 and 26.

O'Reilly studied the effect of pressure on the T of polyvinyl

acetate (PVA). This author measured the dielectric relaxation of

PVA at different temperatures, at constant pressure. By observing

the temperature at which the dialectric characteristics changed

abruptly, a transition temperature T was defined. Repeating the
9 AT
T9
experiment at different pressures allowed calculation of
AP
O'Reilly found this ratio to be independent of pressure and equal

to 0.021C/atm.

This author also determined the force required to compress

PVA at different temperatures. Ho'.:ver, some of the results of

these experiments should be accepted with caution. O'Peilly did not

determine the effect of compression rate upon the pressure vs.

volume plots. Furthermore, no precautions were taken by this author










to eliminate the drag at the walls of the container.

Nevertheless, O'Reilly defined a transformation pressure P9,

"as the pressure at which molecular rearrangements can no longer fol-

low the applied pressure and the polymer exhibits a glass-like compres-

sibility." It should be noted that this definition of the transition

pressure contains two different and perhaps opposite phenomenological

concepts.

A transition pressure at which relaxation phenomena "can no

longer follow the applied pressure" implies a dynamic situation where

viscous dissipations play a foremost important role; they cause glas-

siness. Thus, the importance of defining the effect of compression

rates on P and the importance of eliminating the viscous dissipa-

tion at the walls. A transition pressure defined as that at which

"the polymer exhibits 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
ATg 9
values of --, obtained with the dielectric experiments.

However, attempts to prove or disprove the application of

equations 25 or 26 to glass transition were inconclusive. The main

difficulty being the curvature of the plots of volume vs. pressure.

In contrast to volume vs. te peraiure plots (Figure 2), the volume

vs. pressure plots do not show rcasonpble straight lines anywhere.

There is no ur .ivoca'l 2mthoJ tc cate irce A.






33



Arguments were presented by O'Reilly in favor of the co-

operative rearrangement theories. These arguments were based on

the decrease of volume with increasing temperature well above the

transition pressure. It is the opinion of this writer, that because

of the high pressures and because "measurements were usually taken
a
at increasing temperatures," the decrease in volume may have been

affected by leakage.

Passaglia and Martin (35) determined the variations of Tg

with pressure on polypropylene. These authors used a direct

experimental procedure. The changes in density with temperature

were studied at several constant pressures. These plots approximated

straight lines at high and low temperature ranges. The transition

temperature was defined at the intersection of the extrapolated

straight lines as shown in Figure i.
AT
By plotting the values of Tg vs. pressure, a value 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,













CHAPTER li

INTERPRETATION OF APPLICABLE THERMCDYNAMICS


The complexity of the problem involving the quartitative defini-

tion of glassiness was illustrated by previous comments on the work

of Davies and Jones. Difficulties in the past have originated

mainly from the lack of general definitions of "time scale," volume

viscosity, and other dynamic parameters not related directly to the

thermodyna:ic theories used to correlate the experimental data.

However, many of the elements necessary for a comprehensive

definition of glassiness already exist, for example: Bondi's

indication that a consistent phenomenological definition is the

key to the problem (5), the general observation of the influence

of "the prior history" upon the properties of a material and

Coleman's comprehensive description of thermodynamic processes

(11), (1 .

In this chapter a rational approach to the study of glas-

siness, comprising all the elements presented above, will be at-

tempted. First, a quantitative definition of glassiness will be

proposed; sec.-ndly, an explicit description of the type of time

dependent processes where this definition cculd bc applied will

be presented; and thirdly, a therrodynamic theory will be used

to develop the equations necessary for the description of glass;nes3.










A. Definition of Glass

It is the opinion of this writer that a glass must be defined

as a material whose thermodynamic properties depend upon the co!-

lection of all past thh r(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

"thermostatic transitions."

Figure 5 is a mechanical representation of the concepts of

instantaneous dissipation, and delayed or relaxa>tional dissipation.

The dashpot Di, illustrates the concept of instantaneous dissipation;

the body mrrkcd :"solid," because of the lVIcc of a d .-:; ipaivc element

Di in seric: with the elastic elcr .t K ; ,iil not c ::,''i'


























.1



r



ii I~
:i i,, F.


<1K


L'


d
L


"fluid"


K< D
Cl d



sl:. i. "

''SOl T U


ad he ied


Figur-e 5.-Machclanical ,:~
Dissi ipations.rs


OF InAZIM MOLci i










instantaneous dissipation and can eventually stand external stresses

withoijt deforming. However,! the delayed dissipation Dd indicates

that this body will show relaxational effects.

instantaneous 'viscous dissipation may exist only in Flow

situations; e.g., in the case of shear or elongational deformations

of fluids, and at the roving boundary of a solid. However, as il-

lustrated in Figure 5, a body can be deformed without exhibiting

instantaneous dissipation. For example, the dissipations occurring

in either the isotropic change in volume of a fluid, or the shear

deformation of a solid are generally delayed by elastic elements;

these are the bulk and shear moduli respectively. These types of

deformations will be referred as 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

experimental step.











Processes where thermodynamic variables are changed at a

given rate, and processes where flow occurs at similar rates must

be compared experimentally, e.g., by compressing a 10 cc sample of

a material at 1 cc per sec and forcing another sample through a
-1
capillary at a rate of shear of 0.1 sec In both cases thz rate
-1
of deformation is 0.1 sec This should establish if the "time

scale" has meaning per se or only when related to flow processes.

Particularly, it will be important to establish whether or not a
a
homogeneous volume change occurs with dissipative effects.


B. Th:-,' :d -,-,ics cf 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

effects.


aGoldstein's viewpoint on the subject based primarily on
Litovitz aco- ic e. -rin nt wts briefly discussed on page 29.










As discussed in the previous section, processes where no

instantaneous viscous dissipation are present may have special

significance for the study of glassiness; if no instantaneous

viscous effects are present, the theory of thermodynamic processes

of materials with fading memory or quasi-elastic materials,

constitutes the most general thermodynamic theory available. (See

Appendix B.)

A majority of the published experimental evidence indicates

the existence of a viscous response of glasses to changes in volume.

However, it is not an instantaneous viscous dissipation; the dis-

sipation is always delayedd" by the compressibility of the material

(29), see Figure 6. Therefore, the theory of thermodynamic processes

of 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



where


aAl equations de'eloped in Appendix B aie labeled with a 3
preceding the number.





40










LX
? AX 4-. ~



I I

! /'////i/// Constrai t
1 L
-- -- /







s ~ Figure 6a

1 I





Y I


I ./,/ ,/ t ,


' in Volume or Te, .rature


Figure G.- i Co.istrai e<









a
P = p F+ (B-19)
T,t


S - ) (B-20)
F,t


D = p (B-21a)

F,T
where

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

B-21a.


The

of entropy

the entree

div h r..--

heat sorc

mater als

The effect

effects o0

process D


si':.iOcn D presents the excess in the rate of producLior

'r the entropy fiow du; to grad'ients in temperature pilus

rovlded by hcr ogeneous external heat sources. The term

-ts th heat flux, and the teri q a homogeneous external

The significance of D to t th theory of 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

written.

pT ,t Shist = D = pTS div h pq> 0 (27)


where atShist is the instantaneous entropy production due to relaxa-

tion of structural changes.

The term, Shist, would represent the difference in entropy

between the structures present at time t, and those present if the

system had always been at rest at T and V. The structures present

at time t are only a function of the past values of V and T. The

system is always tending towards the order represented by the

relaxed state at T and V; this accounts for the fact that tShist

is always positive. This interpretation of the dissipation term

is not intended here as a quantitative theory, but rather a quali-

tative model which may provide some aid for the understanding of

glass i ness.

Equation 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

transitions.


C. D. l.., -n: of Lh- Ecr.... ; rns to, h. UI d
fri; t r i, i, GL ;


Grarhical 1 1 f S ,
9" L. L r---

According to the definitions and discussions of the previous

section it is suggested that a study of the riraning of "Lim'i scale"

and "dissipation" in 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

two displacements.


AX = X, + aXr (28)


The elastic displacement can be conceived as the contribution

of numerous relaxational processes with negligibly small relaxa-

tional times. The delayed changes could also be conceived as a

composite response. In the latter case, the relaxation character-

istic would be given by a spectrum of relaxational times. The

minimum relaxational times contributing to an experimental observa-

tion will be l ir ited by the response tir.c of the sens;n9 i nstrrment

and the duration of the imposed chan3o. The process contributing










to the maxiinum relaxational tiires on the other hand, will be limited

by the sensitivity of the recording instrument or by the patience of

the experimenter.


E c "io '

For the process represented i; Figure 6, the thermodynaTic

equations expressing the change in free energy A will be reduced to:


A = f(V; T) + HA(Vt TL)


(29a)


(29b)


For quasi-elastic materials A is a potential for pressure and entropy.

Therefore, these properties could b' used to specify the response of

the system.

The equations expressing the pressure of the system as a

function of thE voiu .-i rerature history are:


= g(V, T, t)


P p(V, T) Hp(Vt, Tt)


= T(V, T, t)

d ( 4)


dt \
Vt


dt /i

VT


The tern

they represent

tir, a: V and

the pest volur<

EquatIo s 30 &

interpreter iot

energy c '':


p(V, T) is

the respon:er

Siril ia l



: 3 ar
o' i press


analogous to f(V, T) of equation 29a;

of the system after an infinitely long

. '" T ) r, nrcsents the effect 01

history from the relaxed state.

'essios of smooth'ness and p:'cvide

uC res rse terys as functions of fcC


(30a)


(30b)


(31)









For a 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


Furthermore:


3(3A/ OV)t I
-V----- t
IV,t


2I
S 2A 2i A/ TIL Vt
TdV 3 VI T, t
T ;t


T,t


a(A/3 V)T.t
- --1-1-._-_-


V,T


_ 32A
at V


because of the properties of exact differentials. Therefore,

equation 31 can be rewritten as


d P .P 2( J )


dV +
d l


dt DV T
T,


(36)


aThe r i us sign is due to the face that pressures are now
considered itive; while in previous discussions the stress 'as
considered positive wan exerting a traction.


(32)


(33)


(34)


(35)


a V
T,


S(A/ Vt I










or in more familiar terms,


V +
P= -K -+DV (37)

where the instantaneous icchain;cal bulk modulus for the process K

equals -V(aP/aV), The process instantaneous latent heat of expan-

sion (thermal bulk modulus) qV equals T(S/DV); and DV represents

the instantaneous loss of rate of work (power) for isothermal volume

changes, DV --3((A/t) /3V.

It should be noted that all the coefficients in equations 36

and 37 represent rates of change with respect to the bulk (volume)

of the system determined while holding the temperature constant at

a particular history. These "process" coefficients are material

properties determined by the constitutive equation of the material.

In addition to pressure, entropy is needed to express the

overall response of th process depicted in Figure 6. The equations

expressing the instanane us rate of change of entropy are:


S + T + (38)
T,t V,t V,T

where

as ai- (^ _- iA\
Sv v T I V'/
T, t Tt V, t


t(3

V,t









SC (40)
a T/ T
V,t


and



t) ; (41)
'V,T IV, T V,


therefore, the rate of change of entropy can be expressed in terms

of coefficients which represent changes with respect to the temper-

ature of the system evaluated while holding the volume constant, at

a particular history


S ( ) T V TJ- T + ) (42)
V,t V,t V,t

or


S = + -C. V + CV D (43)
B T

Where a : Q and CV are the instantaneous, 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.











Dissipative Terms

The dissipati/e terms Dy. and DT are defined by equations 31

and 38 respectively. Explicitly:


(44)


DT
tC;


However, these parameters were rewritten in terms of the rate of

sipation of total work (free energy). Equations 35 and 41 state


Dv = _- A\






DT : A 8 A
T -\t
T T t


(45)


dis-


(35a)






(41a)


where


aA - D ( 0
at


The interpretation of 3 and ^ needs some comments. The develop-

ment of equations 35a and 41a required the interchange of partial

operators, e.g.;


D r\
v;, ( -
0*- k *1l


2A -
:* uVi


fa "-c
NVVty


and










Therefore, in order to provide a better understanding of the instan-

taneous dissipation of the response variables, their relation to the

history term must be discussed. Pressure will be chosen to illustrate

the discussion. Differentiating equation 30:


d- P(V, T; V Tt) = d P(V, T) 4 L Hp(Vt, T ) (46)
dt dt dt


or



tV T tVt t



Differentiating a functional, e.g., Hp, is not a common operation

and no tabulation of derivatives can be found. However, conceptually,

differentiation of a functional implies the same steps required for

differentiating a function, i.e., incrementing the functional by

incremernting one of its arguments and evaluating the ratio of these

two increments as the increment of the argument approaches zero (2 .

For example,



9HP(Vt Tt) d Hp(Vt Tt ) E- 0 (48)
-qt d E



where E represents a finite arbitrary increment in the history Vt.

Equation 48 must be evaluated st a constant temperature history.

The meaning of the dissipative response terms can be now

established, e.g.,











/cP V- + v- T-
Sty' V (49)
V,T T V. V,,T


since at constant V and T the first two terms of equation 47 vanish.

However, because of the smoothness condition (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)









or


D 3HA t (__'A t
-- D (53)



An apparent result from this equation is that the change of free

energy of the material with respect to the history of a state variable

is always opposite in sign to the rate of change of that variable

during the process; the material is not expected to gain free energy

as a result of remembering the past. This result could have been

established directly as a consequence of the 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.













CHAPTER iV

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

process.










If the "ordering" of the above material occurs without ap-

preciable chan3r in volume, no significant mechanical effect may

be noticed; i.e., the stress required for ordering or crystalliza-

tion may be negligible. However, the entropic effects of ordering

may be important. Furtherr:ore, these effects (energy exchange wich

internal degr:c s of freedom) may be delayed. If this were the case,

temperature equilibration would not be instantaneous. For such a

material, glsssiness would be observable only by these temperature

delays.

The discussion on the preceding paragraphs is ir'ended to bring

to the reader's attention the possibility that mechanical effects

observed in isochiic flow processes may not have a mechanical

counterpart in homogeneous 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

relaxation.

In this chapter a general description of glassiness will be

proposed which, hopefully, will allow determination of the thermo-

dynamic data necessary for the corroboration of existing theories of

glassiness, or for the pr,. .-.l of new ones. The contribution

intended in this chapter is to grc.'iically describe glassiness in a

way consistent to the thermodynamic equations and concepts discussed

in Chapter III.










A. Glassiness and Excess Thermodynamic Functions


Changes in Voiume in Unconstrained
Materials. "Constant" Pressure
Ex er imTnts

The usual type of experiment encountered in determination of

glass transition temperature consists of rapid cooling of the sample,

and recording of its volume as a function of temperature at a

constant arrbient pressure.

The description of the deformation undergone by a material

contracting in such an experiment may be quite complicated, even

if a uniform temperature throughout the sample is approximated.

For a liquid, whose viscosity may be negligible, uniform

temperature insures a local density in equilibrium with the tempera-

ture at the pressure of the experiment. However, as the liquid turns

into a glass and relaxational phenomena become important, a change

in temperature will not be followed by the equilibrium change in

volume. More importantly, a pressure in excess to the pressure cor-

respording to the equilibrium situation (infinitely slow changes)

will appear. This excessa pressure is caused by the now significant

opposition to volume changes. The pressure in general will be

distributed nonhonogeneously throughout an unconstrained sample.

Force balances must be satisfied at the free boundaries, causing

shattering of the glass in so:r extreme cases.


aA more specific definition of "excess" will be g;ven in the
next section.










If the purpose of a 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

and T.


P = p(V, T) + Hp(V -, Tt) (30)

t t
S s(V, T) + HS(V-, T-) (55)


P =(- v +DV (31)
T,t T,t



S : ) V T+DS (L42)
V,t V,t





Equations 30 and 55 indicate that the response of the system

to the history of its thermodynamic state (Vt. Tt) is th'. sum, of

two contributions: the pressure or entropy of the static state

(V, T), and tl ccnt ribt ion of 11 the th dynii c states prior










to the present one (V-, T ), The contribution of the past states

may be conceived as an "excess" thermodynamic function, an excess

over the values of tht thermodynamic state had the system been at

the present values of V and T, always. This static state will be

called the rest state.

The rest state is, of course, a hypothetical or fictive state.

However, it could be conceived as either an equilibrium state or as

a relaxed state. Therefore, the rest state should be treated

experimentally according to the choice made.

If the system had always been at V and T, it would have never

experienced an excess thermodynamic function, in this respect, if

the thermodynamic state at V and T were approached without ever

experiencing an "excess," i.e, by an infinitely slow process

(thermostatic), the rest state could be conceived as an equilibrium

state. On the other hand, if a material had arrived at V and T

through a dynamic process, it may have experienced excess thermo-

dynamic functions. In this case, if the material were held at V

and T indefinitely, the excess functions would dissipate. This,

experimentally obtainable, relaxed state could be conceived as

the rest state. The rest state would be the present state (V, T,

t) as t->- .

The equilibrium state and the relaxed state are not neces-

sarily the same. The changes induced in the material during a

dynamic process may be qualitatively different from the changes

occurring had the process been reversible. Furthermore, each

history may ihve caused per r nent c,:,rn.s to occur within th;.










material, in which case the relaxed state may be different at V

and T.

However, a cc'-ronise is possible. For the purpose of il-

lustration, it v ill be assumed that the relaxed state at V and T

will be the sape regardless of the history, had there been a history.

This is similar to the principle of static continuation or stress

relaxation of Coleman and Noll and is implied in the thermodynamic

theory of 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

a
the concepts oF the relaxed and the equilibrium states. The curve

ABC in Figure 7 represents an isothermal homogeneous compression, at

a finite rate. The compression process has been stopped at Vf and

Pf and the material kept at a constant Vf, and temperature. Curve

CDE represents the relaxation of the excess thermodynamic functions

(e.g., excess pressure). Note that the 'sciss3 has been changed

from a volume scale to an arbitrary time scale to illustrate the

relaxation phenomenon. The pressure at t -- oc tends to a value

Pr. This value represents the relaxed value. The value AP is

the contribution of the history to the thermodynamic state of the

system at Vf and T. Curve A'B'C' represents a process similar to

ABC, but at a higher compression rate; AP' is now the contribution

a
For illustration purposes pressure will be chosen as the
response of the s' tem.

















I.''



I t:
SI i3


i 11

Iii
*1r








'N.I


C.. I n

N --


N Cl
N.C

~1....,
cr N.- -, t

IC)


4'I C-




I,1




1:4

i IL


oC. ar lJ (,, n/) =) :LU















- ~ .~ -


Q)
*- Il
-w
DD
0J tI


I.-

a-
Cr


Cr'
a-


u- \



K


N

2.


KN ~T\\




S \ \


t



LI
'4


I





1u
I

















r,
C)





11
iU
r, -;-


^


\\



S1r
a- a-
a" '
V -- C
P -


- fl, uj L" vC: &C -U


--.. -









of the history represented by A'B'C' to the thermodynamic state at

Vf and T. It should be noted that Pr has been assumed the same

for both processes.

Figure 8 illustrates the concept of the equilibrium state.

The curve ABC represents again a dynamic isothermal, homogeneous

compression. The relaxation processes are new represented by the

lines at constant volume. The points Vf i Pf., Pr where i -1,

2, 3 represent the values of the final volume, final pressure, and

relaxed pressure for three processes at the same compression rate,

but ending at arbitrary values of V. The line uniting the final

relaxed states (Vfi, Pri) represents the collection of reference

states. The line XYZ represents the thermodynamic states of the

system if the compression proceeds infinitely slowly. The "equi-

librium" process represented by XYZ will not necessarily coincide

with the relaxed states (Vf., Pr.) The "equilibrium state" for

a glass may not be real. the relaxed state is.

The concept of values of the thermodynamic functions in excess

to the values of the thermodynamic real relaxed state will be the

basis for the description of glassiness, in accord with this

author's proposed definition.


Graohical Description of Glassiness

Pressure will be chosen to illustrate the proposed description

of glassiness. it must be recalled that a history which did not

affect the volume dependence of the dissipation rtLes wili not

produce an/ excess pressure. Hoe-.'ever, such a history r,'y produce










an excess entropy if it affected the temperature dependence (see

equation 47).

A glass is represented in Figure 9 as any point above or

below the plane H = 0. The plane Hp = 0 represents the relaxed

state of solids and liquids. Two processes are illustrated in

Figure 9. The path I S A in the zero excess pressure plane is an

infinitely slow process allowing crystallization at Tc; a AV of

crystallization is shown. The path LGI G2 represents a prepara-

tion process of a glass G2, whose relaxation processes have become

very slow, aic could be considered an unrelaxed "solid" at G2.

The increase in Hp with time along LG1 G2 is a consequence of

memory. As the glass moves along LGIG2 it remembers its past. If

a path of increasing pressure such as LGIG2 is chosen, the material

will "reremberr" state of looer pressure than the present one. When

the process is 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


















4h
2 (2,"2 te






att

T, Terp.-
> IA
ci


/ x
:1 C)



I /l
t 1 N

u, 6 'n
_ _ _ _





ii 1/
i? __I I~--------, `
'


Figure F-r-txce~; ucctions an- 0 1as '. -as





















,4





*i
t]


G



-- T, Temperature---


Sr (o., T0)


/
/
/
/
/


Figure lO.--Decompress ion anc Heati of a Glass


-I
'.


~1


-4

/ i




-j '


S



I










glass G. The path L Sc A is represented again to provide the reader

with a graphical reference to the previous graph. The effect of

history is represented as a negative parameter going through a

minimum and disappearing as the liquid is approached. The negative

effect of history can be explained by remembering what was said

about memory. In this case, as the glass moves along a decompress-

ing path it will remember a state of higher pressures, and if the

process were stopped, the glass tend towards these states. The

minimum in the GL path and the disappearance of the excess pressure

as the liquid is approached are consequences of assumptions about

the nature of memory. It is generally accepted that the relaxation

processes become faster as the temperature and volume are increased.

However, the faster relaxation processes do not necessarily mean a

decay of the excess pressure. It is only when the excess free

energy, brought about by the finite rate of change of the process

variables, can be dissipated faster than can be accumulated, that

the excess terms will start to decrease. As the liquid is ap-

proached both factors contribute to the disappearance of the

history term; the relaxation rates are faster, and there are less

significant structural changes contributing to the excess free

energy.

Figure 11 illustrates a simple type of experiment which can

be used to study the influence of volune changes upon glass transi-

tion; these are isothermal, finite, dynamic compressions. The

importance of these experimerrs is the possibility of direct









i' Vt r2 tp
Vv, T, !t
Vf T' P

I /t ,


ST em .--

d, p/
i / / Vi n '

o i / 2/o. 0 /
I--, T j
I/p I'
S! // I i
!< ,

.i i
Ui i
!i i / z


/ / / I/ i;


r 1/ ..






Coripres es onis.










determination of a volume viscosity. A volume viscosity could be

defined by using equation 30 restricted to isothermal processes,
Hp
i.e.. = - -
viri "
V/V
The viscosity n is directly related to the thermodynamic

parameters used to describe glassiness.

Furthermore, by performing these experiments at different

temperatures, the temperature dependence of the mechanical response

(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

effects.










Isobaric Transitions

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-

cesses.

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.























Creep
0 DE
u Ci
XI











1 i :4,
Lu iP."













/i /
Relaxation .-TerpI C~ie pera tur e ,












//
Delayed,-I C
V ~ i M , v o u e Ji n cr ea s i ngg r a







B I


Ii



~1



-j
Ui
g`



4.I
4.












Li




fj.


Figure 12.--Glassiness during Ischaric Processes.


(I-~~b?:


Liquid


I


ii

~11
44





ri


*.1



;I


Ii
'1


I ;-i










Davies and Jones indicated that as the cooling rate increased,

the glasses formed at higher temperatures. This observation is rep-

resented, using the description proposed by this writer, by the

paths ABD and ABE. It must be recognized that as the cooling rate

increases the excess pressure also increases; these effects have

been considered irrelevant to the observation of glassiness. It

is this writer's opinion that the cause of the observed rate

dependence of glassiness at constant pressure is the delay in

volume caused by dissipative processes. The increased "sharpness"

of the curvature induced by higher cooling rates is illustrated in

Figure 12. Mathematically, the delay in volume, for a cooling

process at constant pressure (P = 0) represented by T = T(t) is

given using equation 31 by:


V = ( VS T + DV) / 3VP (56)


The first term in this equation represents the instantaneous

(equilibrium) changes in volume. The term DV/ 9VP represents the

delay in the rate of change in volure due to the presence of dis-

sipative processes. This delayed volume change is made evident if

the sample is held at constant pressure and temperature after

deformation. The phenomenon is called creep, see Figure 12.














CHAPTER V

DEVELOPMENT OF EXPERIMENTAL TECHNIQUE


The contribution of this research in the experimental field

consisted in the development of a method to study homogeneous changes

in voiumens of asphalts without the interference of shear viscous

forces. This technique can be used to study the transition of a

substance from a 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

asphalts.

Muller (33) in a recent review on the thermodynamics of

deformation and the calorimetric investigations of deformation

processes, discussed the current state of the art on experimental

methods. The author presented numerous examples of efforts to

understand the causes of the mechanical behavior of materials by

measuring 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

73










the different thermostatic effects which could be studied with an

elongational calorimeter designed by Engelter and himself (17).

Dynamic effects are Jiscussed only qualitatively since this calori-

meter method shows a considerable delay between the time a deforma-

tion is induced and the time when heat effects are recorded. How-

ever, this author recognized the importance of irreversible effects

and proposed the use of the calorimeter to determine permanent

final effects (state differences) caused by deformation, e.g., the

degree of crystallization induced by the deformation.

Muller also describes cc'-,ro;sive experiments where temperature

variations have been recorded in order to study thermonrechanical

effects. However, this author dismissed this approach because all

att --ts to place a thermocouple inside the sample were unsuccess-

ful and because, "in addition, the degree of deformation during

compression of a cylinder is 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__





SUPPORT
PLATE J


BARRL -- -- ---------- ^



~161
BARREL ~6'

I I

PLUNGER --




ASPHALT

PLUG L

CLAMPI NG NUT .


Figure 13.--Line Sketch of C-


session Equ ipnment.










considerations, and finally, a third section where the technique is

analyzed and data are presented in support of the premise that the

deformation obtained is indeed homogmner.us, and that shear inter-

ference is e lmi.nated.


A. Materials and Awoaratus

Four asphalt cerents were selected to test the applicability

of the dynamic compression technique: Texas 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-

pendix A).

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










Table 1

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. -
o
Glass Transition Point, C
(Penetro eLer method) -13.9 -10.8

Viscosity (250C), poise 1.02010 1.05x10 0.66x06 --

Viscosity (60C), poise i.70x10 1.73x10 1.11x103 5.22x103

Molecular eichc 943 939 76 --
(iNumber avcr )
a
Generic Groups
(Per cent by weight)

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.




































1*4


Figure l1-. --lnstrcn Universial Testing IHaclinie.










Capillary Rheoreter Asserbly (Figures 13 and 15). The compression

rates used ranged from 0.5 in/min to 0.0002 in/min. The travel of

the crosshead and therefore the actual length to the sample can be

determined .-ithin four tenthojsandths of an inch. The force ap-

plied with the piston to compress the simple is measured by a

calibrated strain- gaje cell which provides an accuracy of 0.25 per

cent of the scale used. There are six scales ranging from 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

negligible (39).


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-
































-0.3753
0.12

















1.125



0. 7D0 A I d;mens ions
i










I inch






o .146
V I








I 0.3 65
0.11.12525
I I2 I '
I i








16 -



Di720 enion of its Parts.
1 iin inchc!s
i 1 021465

4 I L I _


\ i n 16 --




Figure 15.- "Diss3sebebled Modi ifi Capfllary Rheometcr- and
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










Table 2

Effect of Aging


Sample S6309

Run No. R70-15

Reference (XLI, 99-100-104)


Date
1/31/70


p

0.

185

615

923

1000


p

1.0303

1.0390

1.0562

1.0666

1.0690


B

4.49

4.16

3.42

2.90

2.78


P pressure (atm.)
(atm-l x 105)


Temp: 250C



Velocity of Crosshead VXH:
0.1 in/min


Date
2/5/70


p

1.0294

1.0380

1.0553

1.0660

1.0686


Date
2/10/70


B

4.48

4.18

3.50

3.03

2.91


p

1.0297

1.0382

1.0557

1.0662

1.0686


B

4.59

4.25

3.48

2.95

2.82


p : density (gm./cm3), = compressibility


-~----~--- --'------~------ -LIII--


---------. ----------_f--------------_----I~---~ -











three types of data: (1) the weight of the sample determined

directly, (2) the length of the sample determined with the Instron

machine, and (3) the change in length versus load requirements,

obtained from the recorder chart.

The weight of the sample can be determined at the beginning

from the weight of the balloon and the weight of the sample plus

the balbon. The sample weight can also be determined- at the end

since the simple and balloon are easily recovered in mot ca~es.

Weights are determined with an analytical balance and are considered

accurate to within 0.1 per cent.

The length of the sample is obtained by direct measurement

with the Instron after the sample is in place. 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
-4
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

used.

The practical importance of the results of this section is

that a sample f asphalt can be poured, and be ready for testing

within a few minutes or stored for study. This permits preparation

of several samples simultaneously a,-,d saving of time if the

procedure were used routinely.













A .1 -for tic

Ronk esti mted the deformation of the rheorneter assembly when

applying a pressure (39). However, the effect of different ccrores-

sion rates was not determined.

Calibration of the machine consists on compressing a piece of

steel and recording the deformation versus the load required. No

differences in the calibration curves ware noticed when compressing

steel samples at rates ranging from 0.005 in/min to 0.2 in/min

(::108).a Figure 16 is an actual tracing of load versus deforma-

tion at two different rates of compression. The recorder chart was

rolled back to match the starting point, the chart speeds were ad-

just"-d so that for both cases ane inch traveled by the chart rep-

resented 0.01 in of travel by the rovin- c> -osshd The two curves

coincided. Coating the barrel with a lubricant did not affect the

ca i brat i -ns

The correction due to the deformation of the asserbiy was

calculated from the deformation of steel cylinders at different

lengths (XL I.34 47). The correction was calculated by assuming

that deformation of the assembly and that of the steel samples were

additive. The deformation of the steei ser''ples -.'ere calculated

directly from the applied load and the bulk ic Siu us of ste-'. By


References designated by a roman nurieral folice.-'d by an
arabic nir',,ber refer to pages (arabic) in the research notebooks
(roman) of different investigators working in the- Asphair Laboratory
of the University of Florida.









I -- VXH 0.2 in/min
I /
/
/



6I











1 2
10ormat n (inches x 10

/
81

1 /






at Two tes of i
21- /

S/
i i4'
D Deformmtaon (inches x 1bao

Figure 16,--Tracing of L-o.,U rsus Deformadin for Ca !Sration
ai: Two *ates of .iiiprese ion.




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