-, .‘.,,.....-.....~-----o~

DIMENSTONS 0F LONG CHAIN
MOLECULES IN DILUTE SOLUTTONS:

THERMODYNAMIC EFFECTS AND ‘
INTRINSIC VISCOSITY MEASUREMENTS

Thesis for the Degree of M. S.
MICHIGAN STATE UNIVERSITY
BAKULESH NAVRANGLAL SHAH

1972 .

 

 
 

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L .1 3 R A R Y
Michigan State
University

 

ABSTRACT

DIMENSIONS OF LONG CHAIN MOLECULES
IN DILUTE SOLUTIONS: THERMODYNAMIC EFFECTS
AND INTRINSIC VISCOSITY MEASUREMENTS
By

Bakulesh Navranglal Shah

The purpose of this work was to study different theoretical
approaches which have been proposed to explain the ”excluded
volume" effect which is observed when studying the properties of
dilute solutions of polymer molecules. In theory the polymer
molecule is described statistically as a random coil of connected
beads. The excluded volume effect may be thought Of as an osmotic
swelling of the randomly coiled chain by the polymer—solvent inter-
actions.

This work is divided into two parts. Part one contains a
discussion of excluded volume theories and the behavior of the
equations resulting from theories at very large (asymptotic) and
very low (near to unity) values Of O<, a factor expressing the linear

expansion of a polymer molecule owing to polymer-solvent

Bakulesh Navranglal Shah

interaction compared with that at the Flory (theta) temperature
condition. At the Flory (theta) temperature, the polymer-solvent
interaction vanishes, the excluded volume effect is not present, and
the random coil end —to -end distance is a minimum similar to its
dimensions in the bulk, undissolved polymer. At the theta condition

the expansion factor, (X is unity. The equations are

Flory: O<5 - 0(3 = (4/3) Z

KSR: (0(3 -O()(1+1/30(2)3/2 = (32/35/2) z

Kurata: (0(3 - 1) + 3/8 (C)(5 -O(3) = (5/2) 2

Fixman: 0(3 - 1 == (2) z

Ptitsyn: (4. 68 O<2 — 3. 68)3/2 - 1 = (9.36) z

FON: O<5 - 0.49310? - 0.2499CX‘1'332 sin (1.073 InCX)

- 0. 5069051332 cos (1. 073 111% .-_- (2.603) 2
where

z = (3/:2TT)3/2 Ba'B N“2

for a chain of N links of the effective chain length a. ,8 represents
the binary cluster integral, an integral function of the polymer-

solvent pair potential energy function, e. g. Lennard Jones potential

Bakulesh Navranglal Shah

for nonpolar systems. These equations were expanded in the form
of power series and each equation was compared to the exact power
series obtained by the perturbation theory Of the excluded volume

effect. This exact series is
2 2
O< =1+(4/3)Z+2.0752 +..

Part two contains the comparison of expansion factor cal—
culated from these equations with experimental data for expansion
factor obtained from dilute solution viscosity data from the literature.
The experimental observation of dilute solution viscosity for several
polymers of various molecular weights in a number of solvents
along with that at the Flory (theta) temperature yields the expansion
factor DC for these solutions. Thus the theory and data may be
compared and the influence of polymer molecular weight and polymer-
solvent thermodynamics on molecular size in solution may be studied.

The conclusions as summarized are that at least for the
nonpolar polymer-nonpolar solvent systems, Flory' s thermodynamic
equation predicts expansion factor in reasonable agreement with
intrinsic viscosity measurements. The discrepancy between the
experimental expansion factor function (0(5 - 0(3) and the theoretical

calculation from thermodynamic parameters at low values of (I

Bakulesh Navranglal Shah

(near to 1) could be reduced if not eliminated by the use of a function
(1 - BEM) where B is a constant which might be related to the
binary cluster integral B. For the polar systems exact values

of Flory thermodynamic parameters, XS and XH’ are very
important in the success of the expansion factor predictions and

the use of a universal entropy parameter, XS’ equal to 0. 34 in
general warrants further investigation. In spite of the better
success of some of the newer theories in correlating molecular
weight dependence of (X with experimental Q values, Flory' s
theory offers the maximum applicability to the study of the polymer-

solvent thermodynamic inter-actions, because the thermodynamic

parameters are available from nonviscometric data sources.

DIMENSIONS OF LONG CHAIN MOLECULES
IN DILUTE SOLUTIONS: THERMODYNAMIC EFFECTS

AND INTRINSIC VISCOSITY MEASUREMENTS

By

Bakulesh Navranglal Shah

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemical Engineering

1972

A I}
b)

To my parents

ii

ACKNOWLEDGMENTS

The author would like to express his gratitude to his
academic adviser, Dr. R. F. Blanks, for his guidance, encourage-
ment and assistance. The author also wishes to thank other faculty
members and fellow students in the Departments Of Chemistry and
Chemical Engineering.

For financial support for future work, the author is
indebted to the Division of Engineering Research and Department of

Chemical Engineering, Michigan State University.

iii

TAB LE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

INTRODUCTION

CHAPTER

I.

II.

IH.

IV.

PART I. THEORETICAL CONSIDERATIONS

FLORY THEORY .
OTHER THEORIES .

Kurata, Stockmayer and Roig
(KSR) Theory

Fixman Theory .

Kurata Theory . . . . . .

Fujita, Okada and Norisuye
(FON) Theory

Ptitsyn Theory .

ASYMPTOTIC BEHAVIOR OF DIFFERENT
THEORIES

COMPARISON OF DIFFERENT THEORIES
WITH ACTUAL SERIES .

RELATION BETWEEN EXPANSION
FACTOR O<, INTRINSIC VISCOSITY [7?],
AND INTRINSIC VISCOSITY AT THE
FLORY (THETA) TEMPERATURE [7716

iv

Page

vii

. viii

12

12
13
13

14

16

17

21

26

CHAPTER

PART II. COMPARISON OF DIFFERENT THEORIES
WITH EXPERIMENT

VI. SYSTEMS AND SCHEME OF COMPARISON

Systems Studied
Scheme of Comparison

VII. FLORY THEORY .

Plot of the Expa sion Factor Function
Against M1 3 with
(X: (ml/[77] )1/3
Plot of (Expansion actor Function)/M
Against M with
o<- ([771/[77] )1/3
Plot of (Expansion Factor Function)/M
Against (X with
_ 1/3
(—[7'] I /[ )
Plots o<withO(= (7/ ]/ .'7.7]IQ)1/2 43
Plots with 0( Values fro Light
Scattering .

VIII. OTHER THEORIES .

Kurata, Stockmayer and Roig (KSR)
and Fixman Theories . .

Kurata Theory .

Fujita, Okada and NOriSIIye (FON)
Theory . . . . .

Ptitsyn Theory

IX. DISCUSSION OF DIFFERENT THEORIES

Flory Theory

Kurata, Stockmayer and ROig (KSR)
and Fixman Theories . .

Kurata Theory .

Ptitsyn Theory. .

Fujita, Okada and NoriSUye (FON)
Theory

112'

1/2'

Page

29

29
30

34

41
48
53
53
58
80
8O

83

84
85

89
89
93

94
95

CHAPTER

X. CONCLUSIONS AND RECOMMENDATIONS

Conclusions

Future Work Recommended

NOME NC LA TUR E

REFERENCES .

vi

Page
97

97

. 100

104

109'

LIST OF TAB LES

Table Page
. 2 1/2
I-l. Unperturbed end —to —end dlstance (< r > /M)
for the polymers polystyrene and polymethyl
methacrylate in different 6 solvents . . . . . . . . 8
VII-1. Comparison of Flory' 8 equation with experiment
Atactic polystyrene in benzene at 25° C. . . . 36
VII -2. Comparison of Flory' 5 equation with experiment
Anionic polystyrene in benzene at 30 C. . . . 37
VII -3. Comparison of Flory' 3 equation with experiment
Polystyrene in toluene at 25° C.
Polystyrene in toluene at 30°C. . . . . . . . 38
VII -4. Comparison of Flory' 8 equation with experiment
Polymethyl methacrylate in methylisobutyrate
at30°C. 39

vii

Figure

LIST OF FIGURES

Page
, _ _ . 1/2
ExpanSIOn factor functions 15. M for
polystyrene -benzer.e at 30° C. with

(X: ([77] Iin16II/3. 42

1/2

Expansion factor functions YE. M for
polystyrene —toluene at 30° C. with

O<=([7)]/[7}]6)1/3. 45

1/2

Expansion factor functions E M for
polymethyl methacrylate -toluene at 27° C. with

04: (ml/[7716)1/3.

Expansion factor functions :s_ M”2 for
polymethyl methacrylate -methyl isobutyrate

at 27°C. with (X = ([fl] /[77]6)1/3

Expansion factor functions/IVIU‘2 YE M for
polystyrene —benzene at 30" C. with
1/3

(X=([77]/[77]9)................49

2
Expansion factor functions/MU \_r_s_ M for
polystyrene -toluene at 30° C. with

04 = (mi/[7719]”.

46

47

50

Expansion factor functions/MN2 is: M for
polymethyl methacrylate -toluene at 27° C. with
1/3

O(=([77]/[77]6)................51

viii

Figure

10.

11.

12.

13.

14.

15.

16.

Page

Expansion factor functions/MHZ y_s_ M for
polymethyl methacrylate -methyl isobutyrate
at 27°C. with (X ([77] mpg)“

Expansion factor functions/MU2 E (X for
polystyrene -benzene at 300 C. with

(X: (ml/[7716)“

52

54

Expansion factor functions/MHZ YE (X for
polystyrene —toluene at 30° C. with

o<=(mI/I7)19)”3

Expansion factor functions/MU2 v_s_O( for
polymethyl methacrylate -toluene at 27° C. with

O< = (mi/{7716)1/3

55

56

Expansion factor functions/.‘i/Il/Z XS. (X for
polymethyl methacrylate -methyl isobutyrate

at 27° C. with (X : ([77] /[77]e)1/3
Expansion factor functions vs M1/'2 for
polystyrene ~benzene at 30° C. with

O( = ([7)1/17719)1/2'43

Expansion factor functions vs M
polystyrene -toluene at 300 C. with

O(= (mi/[77191343

57

59
1/2

for

60
1/7

Expansion factor functions v_§_ M L' for
polymethyl methacrylate in toluene at 27° C. with

O<= ([771/[77]9)1/2'43 62

1/2

Expansion factor functions YE. M for
polymethyl methacrylate in methyl isobutyrate

at 27°C. with O( .—. (Tm/[7716)1/2'43. 63

ix

Figure

17.

18.

19.

20.

21.

22.

23.

24.

25.

Expansion factor functions/Mu2 __
polystyrene —benzene at 30° C. with

(X = ([771/[7)]6)1/2' 43

Expansion factor functions/MU2 __
polystyrene -toluene at 30° C. with

O<= ([771/[77]6)1/2'43
1/2

Expansion factor functions/M

O<= ([nl/[n]6)1/2'43

1/2

Expansion factor functions / M

at 27° C. with (X = (FRI/[7716)

Expansion factor functions/MU2 vs (X for

polystyrene —benzene at 30° C. with—
(X: (ml/[7716)1/2'43

2
Expansion factor functions/MU vs (X for

polystyrene —toluene at 30° C. with—
O(= ([nllln]6)1/2'43

Expansion factor functions/MHZ ‘13. O( for
polymethyl methacrylate -toluene at 270 C. with

oz: (Ml/[7716)1/2'43

Expansion factor functions/MU2 Ea for
polymethyl methacrylate -methy1 isobutyrate
1/2. 43

at 27° C. with O< = ([77] “7716)
Expansion factor functions YE Ml/2
polystyrene -toluene at 20° C. with
(X from light scattering

vs M for

vs M for

vs; M for
polymethyl methacrylate -toluene at 27° C. with

vs M for

polymethyl methacrylate -methy1 i-s—Obutyrate
1/2. 43

for

Page

65

67

68

69

70

71

72

73

74

Figure

26.

27.

28.

29.

30.

31.

Expansion factor functions XE. M”2 for
polyvinylchloride ~methyl ethyl ketone at
25° C. with 0( from light scattering

Expansion factor functions/MU2 E M for
polystyrene —toluene at 20° C. with
04 from light scattering

1
Expansion factor functions/M /° ‘13 M for
polyvinylchloride ~methy1 ethyl ketone at
25° C. with O< from light scattering

Expansion factor functions/MHZ \_f_s_ O( for
polystyrene -toluene at 20 C. with
OC from light scattering .

Expansion factor functions/M1/2 v_s_ O( for
polyvinylchloride -methyl ethyl ketone at
25° C. with (X from light scattering

Expansion factor function of the Ptitsyn
equation vs M1/2 for polystyrene -benzene

at 30°C. with O( from ([77] /[7}]6)1/3
and ([‘01/[7]16)1/“'°3

xi

Page

75

76

77

78

79

86

INTRODUCTION

During the last three decades much effort has been devoted
to the development of a quantitatively adequate statistical mechanical
theory of the large deviations found in polymer solutions from the
ideal theormodynamic behavior represented by Raoult' s law.

Despite the considerable success achieved by the lattice theory of
Flory1 and Huggins, 2 the inherent complexity of the problem has so
far prevented the achievement of a rigorous and complete treatment
for sufficiently detailed models. For this reason the comparison

of theory and experiment is still a matter of importance in apprising
the adequacy of approximations made. in the various derivations.

The "excluded volume" effect which is observed when
studying the properties of dilute solutions of polymer molecules may
be thought of as an osmotic swelling of the randomly coiled chain by
the polymer -solvent interactions. In theory the polymer molecule
is described statistically as a random coil of connected beads. The
problem of the "excluded volume" effect has essential importance
for the structural interpretation of the solution properties of linear

polymer molecules because the molecular weight, temperature,

and intermolecular thermodynamic interaction dependencies of
solution properties are markedly influenced by the existence of this
effect. Thus the study of the volume effects has been a central
problem of polymer solution theory since its discovery by Flory.
The exact statistical mechanical calculation of the average
dimensions of a linear polymer molecule in nonideal solvent is one
of the ultimate goals of the theory of dilute polymer solutions.
Unfortunately this calculation is a formidably difficult problem,
even with a greatly simplified model of polymer chain such as the
pearl necklace model. Since the pioneering work of Flory, a
number of investigators have attempted to solve this problem by
introducing various simplifying assumptions. The results so far
reported are at variance with each other and with experiment,
depending mainly on the assumptions used to evaluate the interaction

energy of chain elements in the molecule.

PART I

THEORETICAL CONSIDERATIONS

4—H

CHAPTER I

FLORY THEORY

The dimensions of long chain molecules in solution are
greatly influenced by the interactions between chain elements.
The interactions may be divided into two classes:1 The "short-
range” interactions and the "long -range” interactions. Short-range
interactions are those between atoms or groups separated by only
a small number of valence bonds; and because of them, there is an
effective constancy of bond angles. ° The "long -range" interactions
are those between nonbonded groups which are separated in the
basic chain structure by many valence bonds. 2 The discussion here
is limited to nonbranched chains. Polyelectrolytes are not considered.

In the absence of both types of interaction, long chain
molecules would obey Gaussian or random flight statistics. 1 Of
course covalent binding forces are present which fix the lengths of the
chain links. Under these conditions the mean square value of the
spherical radius of gyration of the random flight macromolecule is

given by1

U1

2 2
> = . ..
< S 00 ln/6 (I 1)

where n is the number Of bonds in the chain, 1 is the bond length,
and the double zero subscript denotes lack of both kinds of inter—
action.

The chain without long -range interaction may be called the
"unperturbed" chain, 1 and the mean square radius of gyration may

be written as°

< S2 >O = sl°n/6 (1—2)

where s is a "structural" parameter independent of n and accounting
for the effects of the "short—range" interactions. "Long —range"
interactions, on the other hand, give rise to "excluded volume"
effect1 which can be thought of as an osmotic swelling of the randomly
coiled chain by the polymer—solvent interactions. 2 As a result of
both types of interaction, the mean square radius of gyration of a

. . . . . . 1
real linear macromolecule In dilute solution 13 written as

<S°> : OC°<S° >O : (C(28) 1°n/6 (I—3)

where.O( is the linear expansion factor which depends on the number
of bonds n as well as s and temperature. The factor 5 primarily

represents the molecular structural features of the polymer chain

while (X depends upon the effects of interactions of the polymer with
the solvent. For a clear understanding of the conformational
properties of chain molecules in solution, separate determination of
the factor OC and the factor 5 is necessary.

It is well known that the excluded volume effect vanishes
under a special condition of temperature or solvent, which is usually
known as the Flory (theta) condition. Certain measurements per-
formed on solutions under theta conditions can furnish direct
knowledge of the "unperturbed" dimensions and the factor s. With
the aid of Flory -Fox°’ 4 equations, viscosity measurements can
furnish information on long --range interaction and the factor 04 .

These equations are

63/2¢< s2 >3/2/M (I--4)

[7)]

[77] [7219 0(3 (1-5)
[7719 =¢KMl/° (145)

K s 63/“¢ (< 8° >0/M)°/° (1-7)
where [7]] and [7) 16 represent intrinsic viscosities in ordinary

and theta solvents respectively. M is the molecular weight of the

polymer and Cl) and K are constants independent of M.

-J

The ratio between the efflux times of solvent (to) and of
solution (t) from a capillary viscometer for the same densities of
the two gives relative viscosity nrel' Specific viscosity
T7sp : 7(rel -1'

The concentration dependence of the viscosity of polymer

solutions is eliminated by the extrapolation of measured values of

 

’77
the reduced viscosity Csp or of the logarithm of the reduced
viscosity Crel , to infinite dilution. The extrapolated value is

called the intrinsic viscosity [77 ] . Since CD is supposed to be a
universal constant1 for all the flexible linear polymers, the

. 2 .
unperturbed mean square radius of gyration < S > can readlly

0
be calculated from K. The unperturbed dimensions of various
chain molecules thus determined are found in very many cases to
be practically independent of the particular theta solvent employed
and are thus characteristic of the chain under consideration apart
from a usually slight downward trend with increasing temperature. 5
The unperturbed dimensions of various polymers in different theta
solvents are listed in reference 23 and one can see that for the
same polymer there is a slight variation in unperturbed dimension
in different theta solvents. Table I- 1 shows, from reference 23,

/2

sample entries for unperturbed end -to —end distance (< r° >0/M)1 .

2
The quantity < r >O/M is independent of molecular weight.

Table I—1
. . 2 1/2 .
Unperturbed End t-O -I:nd Distance (< r > 0/l\/I) for the
Polymers Polystyrene and Poly(methyl Methacrylate)
in Different 6 Solvents

 

 

 

Polymer Solvent 90 Temp. (< r2 >O/M)1/°
C.

Polystyrene Ethyl or Methyl

Atactic Cyclohexane °°° 7° 650 i 15
Cyclohexane 34 690 i 10
73%—trans- 18 655
decalin
_100%Ttrans- 24 670
decahn

Poly(methyl _ . _

methacrylate) Butyl Chlorlde 35. 4 537
3 —Octanone 72 560 :t 10

 

 

 

 

-)
Once the values of K or < S > O/M are known, the expansion

)1/3

factor O( can be evaluated from the ratio ( [77 ] H77 16) or from

1/21/2

2 2 _.
(< S2 >/< S2 >0) or from (< r >/< r° >0) where < r > is the

mean square end ~to -end distance of a chain at the temperature of
2 , ,
measurement and < r >O Is the same at the theta temperature.

For Gaussian chains the mean radius Of gyration and mean square

end -to-end distance are related by

<S°>=<r° >/6 (1-8)

Thermodynamic analysis of the polymer-solvent system may then
be accomplished with the aid of an appropriate theory such as that

of Flory. 1

CX° —O(° 2CM(1/2 -X)M1/2 (1-9)

01" (XS-a3 /2

2CM Wu — e/TJM1 (I-9a)

where CM is an essentially numerical factor for a particular polymer-

solvent pair.

/2 2 /2

CM = (27/25 7T3/2) (-v—°/NAV1) (6 < s >O/M)'3 (1-10)

where 7 is the partial specific volume of polymer, V1 is the molar

volume of solvent and NA is the Avogadro number. According to
Guggenheim's6 more refined lattice treatment, X is a parameter
related to the local free energy of the polymer rather than simply a
heat—of-mixing parameter as used by Flory. 1 The parameter X is
a dimensionless quantity which includes the interaction energy

characteristic of a given solvent-solute pair. Thus X is often

written as
X=XH+XS (I-11)

where the two terms refer to a heat parameter and entropy parameter

respectively. X is a parameter characterizing the entropy of

10

dilution of polymer with solvent. 9 is defined as

9 = K T/(fl (1-12)

where K is a heat parameter such that

— , 2
AH1-RTRV2 (1-13)

A H1 is partial molar heat of dilution; and v 2 is volume fraction

of polymer. Similarly
— 2
ASI=RWv2 (1-14)

where ASl is partial molar entropy of solution.

K, w and X are related as
Kl-t/j=X-1/2 (1-15)

In a poor solvent, where both K and K/w generally are positive,
6 also will be positive. At the temperature T = 6 , the chemical
potential due to polymer segment-solvent interactions is zero,
and deviations from ideality vanish.

Equation (1- 9) directs attention to a number of important
characteristics of the molecular expansion factor (X . In the first
place 01 is predicted to increase slowly with molecular weight

(assuming T > 6 ) and without limit even when the molecular weight

11

becomes very large. Secondly, 0( depends on the intensity of the
thermodynamic interactions between solvent and polymer. The
larger the factor (1/2 -X), the greater the value of CC for a given
M and thus, the better the solvent, the greater the "swelling" of the
molecule. X is less than 1/2 for soluble system, approaches 1/2 at
the theta condition where 0( tends to 1.0, and decreases when the
strength of polymer solvent interaction increases, i. e. by hydrogen
bonding. Ordinarily for positive 9 , therefore, 045 -O(3 may be
expected to decrease as temperature is decreased in a poor solvent.
Phase separation sets a practical limit on how poor a solvent may be
and yet permit the existence of a stable homogeneous dilute solution. 1
6 represents the lowest temperature for complete miscibility in
the given poor solvent.

Several assumptions were made in arriving at Equation (1- 9).
Tompa24 discusses these assumptions quite thoroughly. Guggenheim°5
and Tompa24 have discussed the refinements to Flory' s lattice theory
in detail. Most of the refinements are toward improvement of the
theory. However, their approach does not remove inherent imper-

fections in the lattice model on which the whole theory is based.

CHAPTER 11

OTHER THEORIES

Kurata, Stockmayer and Roig (KSR) Theory

 

Kurata, Stockmayer and Roig26 proposed a new approximate
theory of the excluded volume effect in an attempt to resolve some
of the discrepancies Observed in Flory‘ s equation (I- 9). Their

equation is

3 1 -32 52 32 12 3
O( -O<=(I+3_&§) / (4/3)’ (3/27T)/,8N//a (II-1)

where a is an effective bond length, and 8 is the binary cluster
integral (see Chapter IV) of each chain element. This treatment
uses an equivalent ellipsoid model for the linear polymer molecule in
deriving Equation (11- 1). The closed expression for excluded volume

effect in terms of parameters used by Flory can be written as

6

T

/2

(0(3—01) (1 + —4—2)3/2 = (4/3)5/2 ch — )M1 (11-2)

30(

where C is a constant different from that of Flory' s in Equation (1— 9).

12

13

Fixman Theory

 

Fixman11 derived an approximate differential equation to
get a closed form of excluded volume effect on the basis of quite

different statistical considerations. His differential equation is
2
ZO((dO(/d2)= I-OC +2(Z/O() (II—3)

where 0(* refers to the change in end ~to—end distance of a
macromolecule from that of the unperturbed dimension. Fixman

did not solve or analyze the above equation. When the above equation
is solved with proper boundary condition (CX (Z) = 1 for Z = 0

at T = 6 ), one obtains
0(3 -1=2z (II-4)

Kurata Theory

 

An approximate closed expression for the excluded volume
effect has been developed by Kurata°° with the help of the uniform

expansion model of perturbed chains. His equation is

<O<3 - 1) + (3/8) (0&5 -O<3) = (5/2) 2 (11-5)

 

*See Chapter IV for difference between O( and (XL.

14

The uniform expansion approximation consists in replacing the
perturbed chain by an equivalent Gaussian chain whose average
dimension is chosen so as to coincide with that of the original chain.
Undoubtedly the uniform expansion model becomes increasingly poor
as Z is increased toward infinity.

Kurata' 8 equation was derived in closed form from the basic
equation developed by Fujita et a1. 30 This equation is a simple

hybrid form between the Flory equation
(X5 -O(3 = (4/3) 2
and the Fixman equation
(Oé3 - 1) = 22

The proportion Of hybridization is 73% for the Fixman character and

27% for the Flory character. 2

Fujita, Okada and Norisuye (FON) Theory

 

Fujita, Okada and Norisuye30 derived a closed expression
for excluded volume effect subject to the condition that interaction
of a pair of chain segments is characterized by the binary cluster
integral/8. Starting with this expression and estimating the total

number of segmental collisions in such a chain by use of the

15

approximation of uniform distribution of segments and of the ellipsoid
approximation to the overall shape of the molecule, an approximate

closed expression for CX was obtained. The result is

015 - 0. 49310(3 .- o. 2499O<'1' 332-sin (1.073 InOC)

- 0. 5069O(°1' 33° cos (1.073 Ina) = 2. 630 Z (II-6)

The validity of the above relation is attributed almost entirely
to the appropriateness of the physical assumptions used to estimate
the total number of two—body collisions of chain segments in the
perturbed polymer molecule. Those assumptions were:

a. The approximation of uniform distribution of chain segments
in the molecule considered,

b. The use of an ellipsoid model to represent the average
shape of such a molecule, and

c. The uniform expansion approximation to obtain expressions
for the major and minor axis of the ellipsoid.

In fact all the treatments of the excluded volume effect
which are approximate have had recourse to approximations similar

to those mentioned above.

16

Ptitsyn Theory

 

The approximate theory of Ptitsyn, 33 which takes interaction
between segments of the chain roughly into account, leads to an
equation differing substantially from that of Flory. Ptitsyn' 3

equation is
(4.80(° -3.68)°/° -1= 9.36Z (II-7)

It was derived by means of a somewhat artificial modification of the
approach which was used by Fixman for the derivation of his
equation. Ptitsyn took into account that the volume effects lead to a
non-Gaussian distribution function w(h) for the end -to-end distance.
The non-Gaussian character of the function w(h) implies that the
influence of the volume effects on the chain dimensions cannot be
considered as a simple increase in the length of a statistical element

of the chain.

CHAPTER III

ASYMPTOTIC BEHAVIOR OF DIFFERENT THEORIES

The behavior of an equation in the limit of large Z and

hence large 0( is considered. In summary the appropriate equations

 

are:
Flory:31 (X5 -CX3 = (2. 60) Z (III-1)
KSRz°° (0(3 —O() (1 + 12 )3/2 = (32/35/2) z (III-2)
30¢

Kurataz°° (013 - 1) + 34015 - 0(3) = (5/2) z (III-3)
Fixman:3° (0(3 - 1) = (2) Z (III—4)
Ptitsynz°° (4. 68C><2 — 3.68)°/° - 1 = (9. 36) 2 (111-5)
FON:°° 0(5 - .4931O(3 - .24990<'1‘ 332 sin (1.0731nOC)

- .50690é‘1'332 cos (1.0731nCX)

= (2.630) Z (III—6)

In the limit of large Z, the well known equation of Flory

yields (X5 proportional to Z whereas the KSR equation yields 0(3

l7

18

proportional to Z. All the equations fall in either of the above two
categories. Equation (III-4) of Fixman, Equation (III-5) of Ptitsyn
and Equation (III-2) of KSR belong to the same category in the sense
that all these yield 0(3 proportional to Z in the limit of indefinitely
large Z, and hence they are called the third power type. On the
other hand, in the asymptotic limit Equation (III- 1) of Flory,
Equation (III — 6) of FON and Equation (III- 3) of Kurata yield (X5
proportional to Z and hence they are called fifth power type. A
new equation of Flory and Fisk34 also belongs to this latter type,
but it will not be considered here because it refers to the expansion
factor defined in terms of the root mean square radius of gyration
of the molecule.

Comparison of Equations (III- 3) and (111-6) with Equation
(III-'1) indicates that asymptotic behavior is reached at a smaller
value of Z by Equation (III-3) than by Equations (III- 1) and (111-6).

In the asymptotic limit

a5 = (6. 67) Z for Equation (III-3)
(XS = (2. 630) Z for Equation (III—6)
(X5 = (2. 60) Z for Equation (III- 1)

Thus Equations (III-6) and (III- 1) have a remarkable similarity in

19

the asymptotic limit. Similarly, asymptotic relations for the other

category are

04 3 = (2.05) Z for Equation (111-2)
0(3 = (2) Z for Equation (III-4)
0(3 == (. 92) Z for Equation (III -5)

Kurata Equation (III-3) is numerically closer to Ptitsyn Equation
(111-5) in the ordinary range of O( (i. e. for 1 5 O( 3 2) as found by
Kurata. 2° Again Kurata equation has been found29 to be practically
closer to the equations of Fixman and Ptitsyn in behavior than to that
of Flory, in the ordinary range of O( values, but it is in contrast
to the equations of Fixman and Ptitsyn as far as asymptotic behavior
is concerned. This shows that it is practically impossible to argue
from the experimental data concerning the asymptotic dependence of
04 on Z, because in the ordinary range of O( values, some theories
show similar behavior to each other even though they do not belong to
the same category.

The asymptotic behavior of KSR equation has a remarkable
resemblance to that of Fixman' 3 equation. In fact it has been found2
that the KSR equation is also in good agreement with numerical solution

of Fixman differential equation (II- 3) for intermediate values of Z.

20

This is reflected in almost coinciding positions of plots from KSR
and Fixman equations, both showing the quantity (CX 5 - OC 3)/Z to
be an increasing function of Z rather than constant as expected
from Flory' 5 equation. This agreement between the KSR and Fixman
equations is all the more remarkable in view of their apparently
totally different methods of derivation; and it is therefore tempting
to regard this agreement as additional support for their validity.

The really accurate experimental determination of (X as
a function of Z, actually of molecular weight, with a well defined
system of polymer and solvent is one of the most crucial tasks required
of the experimentalist in the field. On the other hand, since the
range of Z accessible to experiment is rather limited, it is beyond
the problem of experimental work to determine what is the correct
asymptotic form of O< in the limit of large Z. Edwards35 has
developed a sophisticated theory on this form and reached a conclu-
sion which favored the fifth -power type mentioned above. However,
it must be noted that this fact does not necessarily imply the correct—

ness of Flory's equation.

CHAPTER IV

COMPARISON OF DIFFERENT THEORIES

WITH ACTUAL SERIES

The long -range interactions between chain elements, i. e.
interactions between nonbonded segments of polymer chains, are
represented in sufficiently dilute solutions by the binary cluster

integral,
(>0
3‘ f 1 -exp[—w(l)/k T] 47Tl°d1 (1v-1)
0

provided that pair potential of average force w(l) as a function of
the intersegmental distance 1 is assumed to be of short-range

nature as required by the inequality
B< < < r 2 > 3/2

Here k T has its usual meaning and r is the displacement length of
chain considered. The expansion factor O( is then a function of a

single parameter which may be written

2 =(3/2TT)3/2,8a'3N1/2 (IV-2)

21

22

for a chain of N links of the effective length a. In the absence of
this type of interaction, i. e. at T = 9 , where B = O, the mean-

square dimensions of the chain are simply written

<r°>0=6<S°>0=a°N (IV—3)

which is identical to Equation (1-2).

For small values of Z, exact expressions for expansion
factors can be Obtained by means of perturbation treatments of the
interactions (see Zimm, stockmayer and Fixman9 and Yamakawa and

39

Kurata ). These are, for example, for mean square radius of

gyra tion

0(2 2 2

L =<s >/<S > =1+(134/105)z-. .. (IV-4)

0

and for mean square end -to ~end distance

2 2
O( =<r >/<r2> =l+(4/3)Z-2.07SZ2

O
+6.459Z°+. .. (IV—5)
The difference between (X and (XL is caused by the non -Gaussian
character of the chain, but it is rather small and negligible for most
purposes. These equations can be effectively applied to systems in

poor solvents near the theta temperature, but not to systems in good

solvents because of slow convergence of the series. The expansion

23

factor 04 for long chains is a function of single variable Z provided
that clusters of three or more segments can be ignored.
When notations are brought into correspondence, 10 we find

that for Flory' 3 equation
B=2vltfl1<i= 9m (IV-6)

and finally we have
0(5—O(3=CZ (IV-7)

where constant C has the value 33/2/2 = 2. 60 or about twice that
needed to secure agreement with the linear term in Equation (IV - 5).

It is found13 that if C in Flory' 3 equation is set equal to
(4/3), thus forcing agreement with the linear term in Equation (IV - 5),
the coefficient of Z2 in the expansion of Flory's equation is in fair
agreement (-2. 667 compared to -2. 075) with the correct value;
and coefficient of Z3 is 9. 778 compared with 6. 459 in Equation (IV -5).

The proposal to adopt this new value of C, in general was
not accepted by Orofino and Flory14 who supposed that at higher values
of (X and Z, the original constant would be superior.

Fixman11 proposed to generalize the Flory theory by using
more realistic segment density distribution. But Casassa and

Orofino12 have shown that the coefficient of Z is made even larger

24

by the use of the more realistic segment density distribution. This
lends support to Flory's surmise that the use of a single Gaussian
function has a relatively minor effect in comparison with other
approximations made in the derivation.

The KSR equation can be written as
3
OC -O(=Cg(O()Z (IV-8)
where

g(O()=80C°/(30(°+1)°/° (IV-9)

In deriving this equation, the constant C was set equal to 4/3 to make
an agreement with the coefficient of Z in the exact series expansion.
The KSR equation then gives a value of -O. 2 for the coefficient of Z2.

The Fixman equation when expanded in the series form for
small values of Z corresponds to the linear term of the precise
equation but it leads to the coefficient of Z2 considerably lower than
the precise coefficient (-0. 67 instead of -2. 075).

The Ptitsyn equation when expanded gives the coefficients
4/3 for Z, —2.075 for Z2 and 8. 611 for Z3 in which first and second
coefficients are given correctly but third is slightly overestimated
in comparison with exact values.

The Kurata equation has the three coefficients as 4/3, -8/9

and 88/81. Thus here there is a large deviation of second and third

25

coefficients from actual values, -2. 075 and 6.459 for second and
third coefficients respectively as in Equation (IV - 5).

The FON equation gives the first coefficient as 4/3.

Thus if we judge the theories by the values of the coefficients,
the Ptitsyn theory should be ranked highest, followed by the Flory
theory and then the other theories. However, this kind of criterion
is practically meaningless because it should be remarked that the
inferiority of these equations to Flory' s in predicting the numerical
coefficients in expansion series need not be regarded as a serious
matter, since the convergence of the series is so slow that for all
but the smallest value of Z, terms well beyond the second play quite

a dominant role.

CHAPTER V

RELATION BETWEEN EXPANSION FACTOR Oé,
INTRINSIC VISCOSITY [77 ] , AND INTRINSIC VISCOSITY
AT THE FLORY (THETA) TEMPERATURE [7716'
In 1949, Flory31 developed an approximate theory of the
excluded volume effect in a polymer chain and indicated that the
mean square radius of gyration < 82 > of a chain of N links becomes
asymptotically proportional to a power of N higher than first. It
has been concluded that the dependence of intrinsic viscosity on

molecular weight arises from the excluded volume effect rather

18, 38

 

than the draining effect of solvent molecules. The essential

 

correctness of this View is now widely accepted, but there remain
several deviations of the theory from experiments. For example,

in the case of solutions of polystyrene in cyclohexane, if the theory

is tested by plotting the intrinsic viscosity against the radius of
gyration using a log log scale, the points fall on a line of slope 2. 2,
instead of three as anticipated. 20 This implies that the hydrodynamic
radius of a polymer coil is not always proportional to the statistical

radius.

26

27

According to Flory and Fox4’ 1°

[hi/[n]6= o<3 (v-1)

But Kurata and Yamakawa17 have developed a perturbation theory
of the intrinsic viscosity, taking the non -Gaussian character of

chains into account and derived
n
[DI/IUIQ=CX ,n=2,43 (V-2)

That the exponent n is lower than 3 has also been confirmed by the
experiment of Schulz, Kirste and Inagaki40 for the polymethyl
methacrylate -butylchloride sys'tem .

Since the equation of Kurata and Yamakawa is now known
to be theoretically correct at least for values of (X close to unity,
this difference implies that Equation (V - 1) cannot be a generally
valid relation. Also there exists no theory which justifies the
validity Of Equation (V - 1) for larger values of (X. Thus at present
no general relation is available which permits unambiguous calcula—
tion of (X from measurements of limiting viscosity number alone.
Nevertheless, numerous existing data appear to support the empirical
validity of the Flory-Fox viscosity relation given by Equation (V — 1).

In view of this somewhat confusing situation, all the proposed
equations for excluded volume effect will be tested by calculating 04

from Equation (V - 1) as well as Equation (V —2),

PART II

COMPARISON OF DIFFERENT THEORIES WITH EXPERIMENT

CHAPTER VI

SYSTEMS AND SCHEME OF COMPARISON

Systems Studied

 

Broadly the systems of polymers and solvents can be
classified into two categories —-polar and nonpolar. Again the
sub -classification may be made as,

a. nonpolar polymer and nonpolar solvent,

b. nonpolar polymer and polar solvent,

c. polar polymer and nonpolar solvent,

d. polar polymer and polar solvent.
The data for all the systems studied in this work were obtained from
the literature. The systems studied are:

1. Nonpolar polymer and nonpolar solvent

Polystyrene-benzene (P-B) at 30° C. (Reference 44),

same at 25° C. (Reference 51).

Polystyrene —toluene (P— T) at 25° C. (References 45

and 52), same at 30° C. (Reference 53).

29

30

2. Polar polymer and nonpolar solvent
Polymethyl methacrylate — toluene (PMMA- T) at 27° C.
(Reference 46).
3. Polar polymer and polar solvent
Polymethyl methacrylate - methyl isobutyrate (PMMA -
MIB) at 27° C. (Reference 47).
All the polymers studied were quite narrow -molecular -weight-
distribution polymers. An attempt was made to get as wide a
range of molecular weights as possible. Accurate intrinsic viscosity
data at temperatures very near to the Flory temperature are hard
to find.
Two sets of data for 04 from light scattering were used as
"control data. " They are polystyrene-toluene (PS- T) at 20° C.
(Reference 48) and polyvinyl acetate-methyl ethyl ketone (PVA-
MEK) at 25° C. (Reference 36).
The Xvalues used for comparing Flory' s theory with

experiment were obtained from References 49 and 50.

Scheme of Comparison

 

The equations are

Flory: (X5 -O(° = C Z (VI—1)

31

KSR: (O(3 -O() (1 + 1/3 0(2)3/2 = C Z (VI-2)
3 5 3
Kurata: ((X —1)+3/8(O( —O( )=CZ (VI-3)
Fixman: 0(3 - 1 = C Z (VI-4)
Ptitsyn: (4. 680<2 - 3. 68)°/° - 1 = C 2 (VI—5)
FON: (X5 - 0.493IO<3 — 0.24996X'1'332 sin (1.073 InOO
- 0. 5069 O(°1'°°° cos (1.073 lnCX) = C Z (VI-6)

Whe re the constant C for each equation is different. Flory' 3

equation may be written as

0(5 - (X3 = 2 CM (1/2 -X) M”2 (VI-7)

For comparison of this equation with experiment, one must calculate
the expansion factor function (O( 5 - 04°) from experimental O( values
and the theoretical calculation from the known theoretical or
expe rimental (other than viscosity measurements) values of X.

The most straightforward manner to examine these equations
is to make the following plots and study the deviations from the
eXpe cted shapes of the plots. The plots are made using the 04 values

fr 0m

Mam/17716

32

and

0(2'43=1771/17)19

to test these latter two equations also.

1. Expansion factor functions of all the equations against MHZ.
It is expected that these plots should give straight lines
passing through the coordinate origin because the form of
each equation is y = mx although there is no physical mean-
ing to the extrapolation of the line to the zero molecular
weight.

2. Expansion factor function/MU2 against M, molecular
weight. These plots should give horizontal lines because
the quantities on the right hand side are independent of
molecular weight and characteristic of a particular polymer-
solvent system.

3. Expansion factor function/MU2 against (X, the expansion
factor. These plots also should give horizontal lines for
the same reason as in 2 above.

It was thought desirable to have some ”control plots. "
These plots were made using above equations (VI- 1 to VI-6) using

the (X values from light scattering, thus avoiding the doubtful

CXvaluesfromO(°=[’Y}]/[77]6 orO(°'4°=[7}]/[77]6 . Thus

33

one can make the comparison of different equations with experiment
irrespective of the above O( values since the exact value of the

exponent n in an = [’77 ] /[77 la is unknown.

CHAPTER V II

FLORY THEORY

Flory' 3 equation is

0(5 -013 = 2CM Wu - 6/T)M1/2 (VII-1)

or C15 —O(3=2CM(1/2 -X)M1/2 (VII-2)
5 3 1/2

or O< -O< = 2CM (1/2 - XS - XH) M (VII-3)

For nonpolar systems according to Hildebrand -Scatchard43 regular

solution theory
V
1 2
XHfl—R-T—(61-62) (VII-4)

where XH is the heat parameter, XS is the entropy parameter of

the total parameter X (X is fully described in Chapter I), V s

1 i
the molar volume of the solvent, and CS is the solubility parameter,
subscript 1 and 2 referring to solvent and solute respectively. For

the system of nonpolar polymer and nonpolar solvent the Flory

equation is

34

35

V
5 3_ 1 2 1/2
O< -o< -2cM 1/2-Xs-fi.(51-62) M (VII-5)

A similar equation for polar systems is also available49 but it will
not be considered here. For nonpolar systems XS equal to 0. 34
was taken as suggested by Blanks5O and V1 and 6 values were
obtained from Reference 49. From CX values from the literature
expansion factor function was calculated and thus both sides of the
equation were compared. Tables VII-1, VII -2 and VII -3 show this
for nonpolar systems. It was observed that the theoretical calcula-
tion of the Flory equation was very sensitive to X values. It can be
seen that in most of the nonpolar systems of a polymer and a solvent,
the theoretical calculation is greater than the expansion factor
function. Also this difference is found to decrease in general as the
value of (X increased. Tables VII-1, VII-2 and VII-3 show this.

The general trend, at least for the nonpolar systems, is that below
the value of O< equal to about 1. 2, the theoretical calculation is
twice or more than twice the expansion factor function. However,
this must be determined carefully in the future by running a series
of experiments at different values of (X. This sort of experimental
data at low values of 04 is hard to find and also it is liable to serious
experimental errors.

For polar systems (Table VII -4) the theoretical calculation

of the equation was done by finding the X values as shown in

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40

Reference 49 using XS equal to 0.34 as suggested therein. It was
observed that the theoretical calculation was consistently throughout
the range of (X values far greater than (4 to 5 times) the experimental
value of the expansion factor function. Also the theoretical calcula-
tion of the equation was extremely sensitive to the X values. This
implies that XS may not be 0. 34 for the systems where the above
fact was observed. It should be kept in mind that the calculation of
the XH value is subject to the limitations of the Hildebrand-
Scatchard theory. From the above observations one may suggest
that the correction terms which may be incorporated in Flory' 3
equation to improve it may be different in different cases depending
on the type of the polymer -solvent system. It was observed that a
small change in the value of X altered the value of the theoretical
calculation to a large extent. This observation warrants a new
investigation in the value of XS, since XS is a function of the size
and shape of the molecules in solution.

Stockmayer10 has strongly'suggested that the factor C

M

in Flory' 3 equation should be replaced by 0. 49 C for temperatures

M

near the Flory (theta) temperature on the basis of the calculation

 

of X from Flory' 3 equation and by some method other than the

viscosity measurements. He found that Flory's equation gave the

value of (1/2 - X) exactly half the value of (1/2 - X) obtained by the

41

other method. He7 has also shown that the Monté Carlo calculations
of Wall and his co-workers15 are in distinctly better agreement
with the lower value of the constant in Flory' s equation. This
seems to be quite valid as can be observed from Tables VII-1,

VII -2 and VII -3. One cannot say that the same is true for the polar
systems also without having the correct values of X to compare the
value of the expansion factor function with theoretical calculations.
Thus no conclusion can be made for the nonpolar systems regarding
the value of C without having the correct values of X.

M

The proposal to adopt the new value of the constant CM in
general was not accepted by Orofino and Flory14 who supposed that
at higher values of the expansion factor O(, the original constant
would be superior. Tables VII-1, VII-2 and VII -3 support the
conclusion of Flory and Orofino. It seems, however, that over a
certain range of O< values, the original constant would be superior

and the determination of this range (X values for different systems

requires exhaustive experimentation.

Plot of the Expansion Factor Function Against M1/2

WithO(=([;T]]/[‘Q]€)1/§

Figure 1 shows this plot for P-B with (X calculated from

[7]] H7716 = (X3. The data with the Flory equation reasonably

42

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43

fall on a straight line; however, the line does not pass through the
coordinate origin. The observation is similar for P-T in Figure 2.
Here higher CX values are not available. For PMMA -T in Figure 3
also the observation is similar except that the point with the highest
value of (X falls outside of the straight line. This could be due to

an experimental error. For PMMA -MIB in Figure 4, there is a slight
scatter of the points from the straight line. In all but the last case,
there is a small positive intercept on the abscissa if the lines are
extended towards the coordinate origin. The lines have not been
extended towards the origin because very low values of OC (near to 1)
are not available. Okada et a1. 41 have observed downward as well
as upward curvature at extremely low values of DC The positive
intercepts on the abscissa axis indicate that Flory' 3 equation is
adequate only in the region of molecular weights above a certain
value which depends on the kind of Solvent (and probably on tempera -
ture also). Below the molecular weight corresponding to the intercept
on the abscissa axis, the factor (X is sensibly equal to unity indepen—
dent of molecular weight. For PMMA -MIB (Figure 4), the intercept
on abscissa axis is negative. It is not clear whether this is due to
experimental error or whether it reflects a certain specific inter—
action between methyl methacrylate monomer and methyl isobutyrate

molecule. One has to note that in the region of (X close to unity,

44

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Figure 4. --Expansion factor functions vs M

47

_ 1,2 for polymethyl
methacrylate -methyl isobutyrate at 27° C. with

 

1 3
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48

plots of this type are very sensitive to small errors in the
measurement of [77 ] . It appears therefore a little hazardous to
extend the curves to the coordinate origin in absence of data at

values of (X close to unity.

Plot of (Expansion Factor Function)/M1/2

Against M with O( = ([31]/[7l]%’)1/3

 

Flory' 3 equation predicts that if one plots (0(5 - 0(3)/M1/2

against M, the quantity should be independent of M at every given
temperature. However, a marked dependence of ((15 - CX3HM1/2

on M has been observed for several polymer solvent systems by
Krigbaum and Flory. 42 Figures 5, 6, 7 and 8 show these plots for
P—B, P—T, PMMA—T and PMMA-MIB respectively. Figure 5 for
P-B shows that the points are widely scattered. Figure 6 for P-T
shows that there is a continuous rise in the value of (0(5 - 0(3)/M1/2
as M increases. Figure 7 for PMMA -T shows that there is a uniform

marked dependence of (0(5 - C)(3)/M1/2

on M as observed by Krig-
baum and Flory. 42 Figure 8 for PMMA —MIB shows a somewhat
different behavior than that observed above. Here data show a
drooping line as molecular weight increases. However there is a

scatter of points about, this line. It appears that X is a function of

M for nonpolar-nonpolar systems but not for polar polymers.

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53

Plot of (Expansion Factor Function)/M1/ 2

Against O< with (X = ([m/[UJQHIB

 

A plot of this kind should give a horizontal line. Figures 9,
10, 11, and 12 show these plots for all the four systems. For P-B
(Figure 9) there is a wide scatter of points while for P-T (Figure 10),
the points show an increase in value as (X increases. For PMMA -T
(Figure 11), first the plot increases gradually and then decreases
gradually, thus showing a distinct maximum as in the case of a plot
of (0(5 - CX3)/M1/2 E M. For PMMA—MIB (Figure 12), the plot
shows points falling downward as 0( increases. Thus in all cases,
these plots are similar to the plots of (O(5 —O(3)/M1/2 against M
as expected.

All the three kinds of plots show deviations from the expected
shapes. This discrepancy may be attributed first of all to the
incorrectness of (X—values obtained by CC: ([77] /[77]6)1/3 from the
viscosity data and secondly to the imperfection of Flory' 3 equation.
This imperfection could be corrected by obtaining a correct (X value
from [77] and [7719 or by incorporating a molecular weight dependence

on X since XS is a function of size and shape of a molecule.

Plots with (X = ([71] HTZLéDI/Z. 43

The expansion factor function with the above value of (X is

far different from the theoretical calculations with X values as

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compared to the calculations with (X = ([77] /[77]Q)1/3 (this is quite
obvious), thus Opposing the new value of (X. Figures 13, 14, 15,
and 16 show plots of (O(5 -O(3) against M1,2 and it is observed that
the scatter of the points is much more than the case where

0(3 = [77] ”7719. The same is true-with all the other plots (Figures 17 -

24). The points are more widely scattered.

Plots with (X Values from Light Scattering

 

Figures 25 and 26 show the plots of the expansion factor
function against M1,2 for PS -T and PVA -MEK respectively with
O( values from light scattering obtained from the literature. Un-
fortunately the data are not available over a wide range. In both
the cases, the points are_f_a_r_ from a straight line passing through the

coordinate origin. Figures 27 and 28 show (0(5 -0( 3)/M1/2

against
M for the two systems. The plots do not show any horizontal line.
Thus Flory's equation does not show any good correlation with O(
values obtained from the light scattering. This is also the case

/2

1
with the plots of ((X5 - 0(3)/M against 0( as shown in the

Figures 29 and 30.

59

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Figure 15. --Expansion factor functions v_s_ M”2 for polymethyl
methacrylate in toluene at 27°C. with

o< = ([771/[7719>1’2'43.

Expansion Factor Functions

 

 

 

62
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63

 

 

1 2
Figure 16. —-Expansion factor functions v_s_ M l for polymethyl
methacrylate in methyl isobutyrate at 27° C. with

(X: (Ml/[7716)1/2'43.

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methacrylate -methyl isobutyrate at 27° C. with

0(= ([77] llflle)

U1

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CA)

N

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69

 

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Figure 22. -- Expansion factor functions/MHZ vs DC for polystyrene -

toluene at 30° C with O( = (“VJ/[7716)

1/2.43

72

 

 

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75

 

 

 

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methyl ethyl ketone at 25° C. with 0( from light scattering.

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2 _ O
Q
t Q
o 9 b
1 h (5 b o
0 W— l l
0.2 1 —6 2 3
MX 10

Figure 28. -- Expansion factor functions/M1,2 YE. M for polyvinylchloride-
methyl ethyl ketone at 25° C. with 0( from light scattering.

78

 

 

mo 5” o— FON o-
H 0 Kurata
X
N" ‘3 Flory
: 4
2 1 <? Ptitsyn ¢
\
2 x KSR
.2 0
E3 _ ‘ Fixman
:3
[:4
Lu
3 o-
321 0-
Ln 0 t.
8 0 r5
0% a Q
“3 1~ 0- i 9.
g o
m a 9 ‘9
‘P
0 r r l ' f I '
l 1.1 1.2 1.3 1.4 1.5 1.6 1.65
(X
1/2

Figure 29. -- Expansion factor functions] M 33 0( for polystyrene-
toluene at 20° C. with O( from light scattering.

79

 

 

6 1
o- FON
m 0 Kurata
O
;5 _ 9' Flory 0_
EA 5 Ptitsyn
HE " KSR
\4 _ A Fixman
m 91
‘5
a .. ..
5 0— °
E143 _
‘5
*5 a 8
as o
In ¢,
§ Q * a
a. o
m a, 6 6
v1 -
O F ? I T fl ‘
1.2 1.3 1.4 (X 1.5 1.6 1.7

Figure 30. -- Expansion factor function’s / M1, 2 1’3 (X for polyvinylchloride -
methyl ethyl ketone at 25 C. with (X from light scattering.

CHAPTER VIII

OTHER THEORIES

Kurata, Stockmayer and RoiJgJKSR)
and Fixman Theories

 

 

The KSR and Fixman equations will be discussed together
since it can be seen from the figures that both equations give points

very close to each other.

Plots of Expansion Factor Functions

lflgiiinst M1/2 with o< = (ml /[11]<4}>1

 

/3

In the cases of P-B and PMMA -T, as shown in Figures 1
and 3, it can be observed that at high values of (X or for higher
molecular weights, the points give a curve. From this it seems
that the KSR and Fixman equations are appr0priate for relatively
low molecular weights. The data for P-T (Figure 2) and PMMA-
MIB (Figure 4) fall reasonably on a straight line. The lines do not
pass through the coordinate origin and in the cases of PMMA -T

(Figure 3) and PMMA -MIB (Figure 4) they show negative intercept

80

81

on the abscissa axis. Other observations are similar to those

observed in plots from Flory' 3 equation.

Plots of Expansion Factor

1/2_1§gainst M

 

Functions / M

 

Figures 5-8 show these plots for P-B, P-T, PMMA -T and
PMMA -MIB respectively. In these plots the points are much less
scattered above the horizontal lines than the points in plots from
Flory's equation. In the plot for P-B (Figure 5), for small values
of molecular weight, one can observe an upward curve. This may be
due to an error in the values of intrinsic viscosities at low values
of molecular‘weight. As the molecular weight increases, there is
a slight fall in the curve which finally becomes a horizontal line at
higher molecular weights. This observation supports the validity of
these equations at higher molecular weights which is $13123 in contrast
to the observation made for plots of expansion factor function against
MHZ. In the plot for P-T (Figure 6), the deviation of points is much
less as compared to Flory' s theory, although the points do not fall
on a horizontal line. . Here the line slowly rises with molecular
weight with a slight scattering of the points about this line. For

PMMA -T (Figure 7) and PMMA -MIB (Figure 8), the line slowly

falls with an increase in the molecular'weight.

82

Plot of (Expansion Factor
Function)/M172 égainst
o< with Q: ([73] H7119)“

 

 

Figures 9-12 show these plots. For P-B (Figure 9) one
observes a slight upward curvature about the horizontal line and
then a fall with a final horizontal line. For P-T (Figure 10) one
observes a rise in the plot with increasing 0C values. For PMMA -T
and PMMA -MIB (Figures 11 and 12) there is a uniform fall in the
plot with increasing (X values.

From all the above plots, it can be seen that the KSR and
Fixman equations represent the-experimental observation better

than the Flory equation.

Plots with O( =“711/I711m1’2'43

No noticeable improvement in the plots is observed with

the above (X values (Figures 13 -2 4).

Plots with 04 Values Direct
from Light Scattering_

 

 

Figures 25—30 show different plots for both the systems.
Figures 25 and 26 show the plots of eXpansion factor function against
M1,2 for PS-T and PVA -MEK respectively. The plots are far

from expected shapes and are no better than those obtained with

83

Flory's equation. Figures 27 and 28 show the plots of (expansion

1/2

factor function)/M against M. The representation is much better

than Flory' 3 equation. Similar is the case with the plots of (expansion

1/

factor function)/M 2 against 0( as shown in Figures 29 and 30.

Kurata Theory

 

The Kurata equation shows a slight upward curvature for
P-B and PMMA -T in the plots of expansion factor function against
M”2 as shown in Figures 1 and 3 respectively. This is similar to
that observed in the same plots for the KSR and Fixman equations
(Figures 1 and 3). In other plots (Figures 2 and 4) the shape is
similar to the plots of KSR and Fixman equations (Figures 2 and 4),
but the scattering of points is slightly more. From most of the
plots one can see that the Kurata plots are similar to the KSR and
Fixman plots with slightly magnified error.

The Kurata and KSR -Fixman plots have much different

”2 against M is plotted

shapes when (expansion factor function)/ M
for PVA -MEK (Figure 28) and the same against (X is plotted for
PS -T (Figure 29) and PVA -MEK (Figure 30) with (X calculated

from light scattering. In most of the other plots one observes a

little more deviation in the Kurata plots than in the KSR -Fixman

plots .

84

Fujita, Okada and Norisuye (FON) Theory

 

Figure 1 shows a plot of expansion factor function against

1/2

M for P—B with (X = ([77] /[77]6)1/3. It can be seen that at

higher values of molecular weight, the points are so scattered that

it is hard to draw a line through them. While for the same plot,

a considerable improvement is observed with O( = ([77] ”7716)“? 43

as shown in Figure 13.

Other plots shown in Figures 2, 14, 15, and 16 show shapes
similar to those obtained before.

The plot of (expansion factor function)/M1/2 against M

(Figure 5) shows a wide scatter about a horizontal line and this is

1/2' 43, as can be seen in Figure 17.

even worse when (X = ([77] /[77]6)
for P-B. Similar is the case for the same plots for P-T as shown

in Figures 6 and 18. For PMMA -T, the plot shows a good horizontal

line as shown in Figure 7; but the same plot with O< .—. ([72] [[n16)1/2. 43

shows an upward curve with a distinct maximum (Figure 19). The

plot for PMMA -MIB (Figure 20) shows a downward curve.
1/2

Figure 9 shows a plot of (expansion factor function)/ M

,3. There is an upward

against CX for P-B with (X = ([771/[77]6)1

curve with the points widely scattered about the curve. The same

1/2.43

plot when made with O( = ([77] /[77]9) gives points scattered

at random as shown in Figure 21. Figures 10 and 22 show plots

85

for P-T with both values of 0( respectively. Both plots show an
increase in the plot with increasing O(.

Figure 11 shows a surprisingly good horizontal line with
(X: ([77] /[7}]6)1/3, with the point with highest (X falling out of the
line. Figure 23 shows the same plot with CX = ([77] /[77]6)1/2' 43.
There is an upward curve. Figures 12 and 24 show plots for PMMA-
MIB with both values of (X respectively. There is a fall in the line
with increasing (X.

Figures 25-30 show plots of FON equation with (X values

from light scattering. It can be seen that the FON equation shows

the maximum deviation from the expected shapes.

Ptitsyn Theory

 

Expansion Factor Function
Against Mllz with

1 3
o< = “TH/[121917

 

 

Figure 31 shows this plot for P-B. At the low values of
M, the plot is a straight line, but not passing through the coordinate
origin. At high values, there is a smooth curve, with points slightly
scattered about this curve. Figure 2 shows the plot for P-T. The
behavior here is similar to that observed with other equations.
Figure 3 shows the plot for‘PMMA -T. The plot is a straight line

at low values of M, almost passing through the coordinate origin.

86

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87

At high values, it shows a curve. Figure 4 shows the plot for
PMMA -MIB. Here the points are too scattered to draw any line

OI" curve.

(Expansion Factor

Function)/M1/ 2
with (X = (ml/[7.1163

 

[Against M
1/3

 

Figures 5-8 show these plots. For P-B and P-T (Figures 5
and 6), the deviation from a horizontal line is the least for Ptitsyn
equation in comparison with all the other equations. Figure 7 shows
this plot for PMMA -T where the curve continuously falls as molecular
weight increases. Similar is the case with PMMA -MIB (Figure 8),

where there is a downward curve.

(Expansion Factor Function)/M1/ 2

Against Oé

 

Figure 9 shows an almost horizontal line for P-B. In the
case of P-T (Figure 10), there is a very gradual rise with increasing
“values. Figure 11 shows practically a horizontal line for
PMMA -T, the point with the highest value of (X falling out slightly.
For all the above cases, the behavior of the Ptitsyn equation is
closest to the expected behavior. But in the case of PMMA -MIB,
as shown in Figure 12, the behavior of the Ptitsyn equation is the

worst. It is not clear why it is so.

88

No improvement is observed when all the plots are made

with (X values from 0(2' 43 = ([77] H7716) (Figures 14, 24, and 31).

Plots with 0( from
Light Scatterg

 

 

Figures 25-30 show these plots. Plots of (expansion

”2 against M and the same against (X (Figures 27-

factor function) / M
30) show much less deviation from a horizontal line as compared to

other equations except the equations of KSR and Fixman where the

behavior is pretty much the same.

CHAPTER IX

DISCUSSION OF DIFFERENT THEORIES

Flory Theory

 

The comparison of the values of the expansion factor function
with the experimental (X values and the theoretical calculations with
the X values shows that for the nonpolar polymer -nonpolar solvent
systems, the theoretical calculation is usually greater than the
expansion factor function by a factor of 2 or more for the (X values
roughly less than 1. 2.

In the light of the above observations and other defects
observed in the Flory equation when compared with the experiment,
one has the following remarks and discussion when examining this
equation.

Flory and Orofinol4 observed the values of the quantity
(1/ 2 - X ) exhibiting a slight dependence on the molecular weight,
although no such account is taken care of in this equation. In most
cases (1/2 - X) decreases as the molecular weight increases.

This could be due to:

89

90

(a) deviation of the spatial distribution of the molecule from
the assumed Gaussian form,

(b) failure to consider the contiguity of molecular segments
in space,

(c) neglect of higher terms in derivation of the expression

for excluded volume effect,

(d) effects of molecular heterogeneity which may increase with
molecular weight of fraction.

(a) is investigated by Flory14 and he believes it to be an
unlikely source of the variation observed. No satisfactory theory
is available for dealing with (b). Possibility (c) may be responsible
at least in part for the apparent molecular weight dependence of the
quantity (1/2 -X ). With regard to possibility (d), if the molecular
heterogeneity of the polymer fraction increases with molecular
weight owing to the decreasing efficiency in separation, then
accordingly this would be reflected in values of (1/2 - X).

As is well known, Fox and Flory16 have pointed out that the
constant 95 takes a common value, 2. 1 X 1021, not only for suf-
ficiently large values of M, but also down to low values of M.
Kurata and Yamakawa17 have shown that the counter contributions
of the draining effect and the volume effect make q) approximately

independent of M over wide ranges insofar as the argument X ,

91

the draining parameter of Kirkwood and Riseman18 theory takes a
value larger than about ten. However, they17 say that as X
decreases beyond ten, the balance inclines toward the draining
effect and a decrease of the Cb values becomes significant. Kurata
and Yamakawal'7 further predict two kinds of deviations of ¢ from
the constancy; one is related to the volume effect and the other is
related to the draining effect and often is observed for relatively
short chains or rigid molecules such as the cellulose derivatives.
They concluded that the draining effect is important for quantitative
interpretation of intrinsic viscosity as well as the excluded volume

effect.

According to Flory and Fox16

3/2 2 3/2 3
6 gb (<S >0) a

[77] = M

 

Numerical Value of?

 

Kurata, Yamakawa and Utiyama19 predict the value of
as 2. 87 X 1021 instead of 2. 1 X 1021. About half this difference,
according to them, is explainable from the polydisperity effect,

but the other half is not by this means.

92

Dependence of C25 on Temperature

 

2
Krigbaum and Carpenter 0 showed that the Flory constant
d) should be a decreasing function of temperature, at least in the
vicinity of the theta temperature.

Dependence of ¢ on the
Solvent Nature

 

 

The value of 95 decreases with the increasing expansion of
the polymer coil or the increasing solvent power.

In the vicinity of the Flory temperature, the parameter Z,
hence O(, increases rather markedly with the temperature. There-
fore the constant CD is expected to decrease with the temperature
and/or the solvent power. 21 On the other hand, at temperatures
far above the 9 point, this tendency generally becomes weak, although
not completely vanished; and accordingly, the 9b would behave, at
least approximately, like a constant as claimed by Flory and Fox.

Krigbaumzz made an attempt to explain the variation in

<o<5 -0(3)/M1’2

by means of a theory that assumes that < r2 > and
< S2 > differ to a different extent on swelling. In conclusion it may
be said that it may be possible to improve Flory' 3 equation, at

least for the nonpolar systems, so that it can represent the experi-

mental observations better by multiplying ZCM (1/2 - X) M”2 by

93

(1 - BEM) where B is a constant and M is molecular weight, such
that at high values of M, 133M would be much less than 1 while at
low values of M, (1 - BEM) would tend to 1/2.

Kurata, Stockmayer and Roig (KSR)
and Fixman Theories

 

 

In all the plots it is observed that the points due to equations
from the above two theories nearly coincide. In fact, the KSR
equation is also in good agreement with the numerical solution of
Fixman differential equation, Equation (II -3). Both the equations show
(0(5 -0( 3)/M1/2 to be an increasing function of Z unlike Flory' 3
equation, as observed experimentally. In comparison with the Flory
plots, the plots from the above two equations show much less
scattering of points. There is no definite evidence in the improvement
of the plots when (X was calculated from O( = ([77] /[77]6)1/2' 43,
as strongly suggested by Ohyanagi and Matsumoto. 28

In the derivation of the KSR equation, instead of a spherical

symmetrical model, an ellipsoidal model with three radii of inertia

—1—Na2+l—<r2>, R2=R2= —Na2

R 36 12 y z 36

2

x
2 . .

was used, where < r > is the mean square distance between the ends

of the chain, N is the number of segments of the chain, a is the

effective length of the link joining neighboring segments, and x, y,

94

and 2 denote the three axes. (The axis x coincides with the vector
joining the ends of the chain.) The KSR equation was derived for the
Gaussian chain and its use for chains with volume effects means that
the latter increase only the distance between the ends of the chain
and have no effect on its cross sectional dimensions. This would
lead to a strong dependence of the shape of the chain on volume
effects, which is contrary to the strict theory and experiment. Thus
KSR equation does not in fact involve a more precise model of the
chain (ellipsoidal instead of spherical) but introduces a physically
unjustified assumption that in the limit, the swelling of the chain is
anisotropic. But Kurata, Stockmayer and Roigz6 claim that their
equation is in excellent agreement with the Monte Carlo calculations
of Wall and Erpenbeck for a diamond lattice chain and also with

viscosity data for various polymer solutions.

Kurata Theory

 

The Kurata equation shows characteristics somewhere
between the Flory equation and the Fixman equation, as can be
seen from the plots, leaning more toward the Fixman equation.
It shows more scattering than Fixman' s equation but less than
. . . a 1/2 . 43
Flory's equation, and scattering 18 more for = ([77] /[77]6) ,

Since this equation is a hybrid of Flory's and Fixman' 3 equations,

95

it has inherent weaknesses of both and does not show much
improvement on either of the other two. On the contrary, it seems
Fixman' s equation is closer to the experiment than Kurata' s

equation.

Ptitsyn Theory

 

The plots of (expansion factor function)/M1/2 against M
and the same against O< show that the Ptitsyn equation shows
characteristics very much similar to those of KSR and Fixman
equations. (The only difference being that the Ptitsyn equation
shows much less scatter of the points. In fact, the Ptitsyn equation
gave the best plots in most of the cases.) This is because the
Ptitsyn equation is an artificial modification of the Fixman equation.
It seem, the allowance for the non ~Gaussian character of the
function W(h) has led to considerably improved agreement between
this theory and experiment. Kurata29 has observed that in a plot
of (X3 against Z, his equation is closer to that of FWan and Ptitsyn
rather than to that of Flory. This is quite evident since the Ptitsyn

equation is an artificial modification of the Fixman equation.

FLL'Lita, Okada and Norisuye (FON) Theory

 

The FON equation deviates the most from the experiment.

Fujita et al. 30 observed that in the region of 0(3 up to about 3, their

96

equation when plotted for (X3 against Z gives a curve which nearly
coincides with that predicted by the Ptitsyn equation; but beyond
this, the latter yields 0(3 which increases more rapidly with

increasing Z.

CHAPTER X

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

 

Flory' s Equation

 

The most important factors that focus one' s attention on
Flory' s equation (XS - (X3 =2CM (1/2 -X) M“2 for improvement
are CM and X . For the nonpolar polymer-nonpolar solvent
systems, the theoretical calculations using X values changed from
nearly the same (in fact a little less) value as that of the expansion
factor function calculated from the experimental values of (X to twice
or more than twice the value of the expansion factor function as 0(
decreased. Accepting Flory's theory, the theoretical calculation
requires an (X dependent adjustment. This adjustment could either
depend on M or X or both.

The expansion factor function (O(5 - 0(3) when plotted
against MI/ 2 gives in most cases nearly a straight line. From

/2

this one can conclude that 2CM (1/2 - X) M1 is not a function

of M.

97

98

The plot of (CX5 - 0(3)/M1/2 against M deviated from the
horizontal line at comparatively low values of M. 2CM (1/2 - X) Mllz
is not independent of M for polystyrene if M is less than about

3 X 105 but it seems independent of M for M greater than 3 X 105.
It may be possible to rectify this situation by multiplying

”2 by (1 - 133M) where B is a constant such that

2CM (1/2 _X ) M
at high values of M, BeM would be muchless than 1, while at low
values of M (1 - BEM) would tend to 1/2. It may be possible to
relate the constant B to the binary integral/8.

For polar systems no clear conclusion can be drawn as in
the case of nonpolar systems unless the independent values of X
are available to make a comparison between the experimental value
of the expansion factor function (Oi5 - 0(3) and the theoretical

1/2. Complete data for only one

calculation of 2CM (1/2 -X) M
system were obtained from the literature. The X values were
obtained from Reference 49. The value of the entropy parameter
X S is suggested as O. 34 and a method of calculation of the heat
parameter XH is shown in the same reference. The use of this
X value made the theoretical calculation of 2CM (1/2 -X) M”2
four to five times larger than the experimental expansion factor

function (as - 0(3) for the system investigated. Also it was

observed that a small variation in the X value altered the value of

99

the theoretical calculation to a large extent. This observation
warrants a new investigation in the value of X 8' Since X S is a
function of the size and shape of the molecules in solution, it seems

this dependence is very strong in the polar systems.

 

Relation Between [7)] , [7)][3 andCX
a. . T

Many workers have strongly suggested that the value of n
in O(n = [771/[7716 should not be 3 but a little less than that. One
suggested value for n is 2.43. With this value of n, (X was
calculated and the different plots were made, but there was hardly
any improvement observed in the plots. On the contrary, the plots
with n = 3 were closer to the experiment. This shows that n = 3 is
the best so far.

The evaluation of n by light scattering is not strictly pos —
sible because light scattering gives the-change in the radius of
gyration from that at the theta temperature, i. e. one gets

2

2 2
O(L=<S >/<S >0

0(n = [’77] /[77]6 refers to the change in the end -to -end distance of

whereas the expansion factor (X in the equation

a chain, i. e. 0(2 = < r2 >/< r2 >0. (XL and (X are slightly different

as can be seen from the exact series expansion of the two

0(2.=<r2>/<r2> =1+(4/3)Z -. ..

0

0(2 =<S2 >/< 82 >'

L =1+(134/105)z-. ..

0

100

This shows the limitation of the light scattering data in evaluation

of n.

Ranks of Different Theories

 

If the various theories are ranked on the basis of the plots
made to compare the molecular weight dependence of the expansion
factor with the experiment, then the old theory of Flory is certainly
not the best. Some of the newer theories like the Ptitsyn theory
and the KSR -Fixman theories are closer to the experiment than the
Flory theory, the Kurata and FON theories deviating the most from
the experiment. In spite of this little shortcoming of the Flory
theory, it is the only theory which relates the expansion factor O(
to indepe\ndently determined thermodynamic parameters of polymer-
solvent interactions, thus rendering it the mosewidely applicable

theory currently available.

Future Work Recommended

 

Reliable experimental data is very important to test the
different equations of excluded volume. For this purpose data over
a wide range of (X values at different temperatures for all kinds of
systems are necessary. The basic measurement is intrinsic
viscosity for different systems at different temperatures. Measure -

ment of intrinsic viscosity is very sensitive to slightest experimental

101

error. Good temperature control is also very important. Viscosity
of polymer solutions, as determined in capillary viscometers,
depends on the rate of flow of the solution. Depending on the rate
of shear, the viscosity of the polymer solution will be higher or
lower. The viscosities of polymer solutions of the same molecular
weight in the same solvent and at the same temperature, as given
in the literature, show unsatisfactory agreement. This is because
the viscometers used by the different workers had different efflux
times for the same solvent, because the shear rate-was different

in each case. The shear rate, however small, has an effect on
intrinsic viscosity. Therefore the use of a multiple bulb Cannon-
Ubbelhode type viscometer is strongly recommended. All the bulbs
have different shear rates. From this, the viscosity at zero shear
rate can be found out by extrapolation. There are anomalies in the
concentration dependence of viscosity at high dilutions. Rafikov

et al. 27 have described them elaborately. These must be carefully
studied before extrapolation to zero concentration.

Light scattering data for different systems are necessary
over a wide range along with the intrinsic viscosity data to find the
best value of the exponent n in [’77] /[7716 = O(n.

For the comparison of both sides of the Flory equation,

values measured by some method other than intrinsic viscosity

102

measurements are necessary, and for this osmometer measurements

are highly recommended.

NOMENCLATURE

NOME NC LA TUR E

Effective length of a link
Constant
Concentration in gms/lOO c. c.

Constant in the Flory, KSR, Kurata, Fixman,
Ptitsyn and FON equations (Chapter VI)

Constant in Flory' 3 equation

Base of natural logarithm

Partial molar heat of dilution

Boltzman constant

Constant in the theoretical intrinsic viscosity

relationship [7716 = KMll2

Intersegmental distance

Molecular weight

104

El
2!

2
(r >,

<r2>

105

Number and weight average molecular weights
respectively

Exponent on (X in O(n = [77] ”7716
Number of bonds in a chain
Number of links or segments
Avagadro' 3 number

Mean square and unperturbed mean square distance
between ends of a chain

Gas constant

Radii of inertia in-x, y, and z directions respectively
Structural parameter accounting short

Partial molar entropy of dilution

Mean square radius of gyration, the same in absence
of long range interactions, the same in absence of
both long range and short range interactions

Efflux time of solution and solvent respectively out
of a capillary viscometer

Absolute temperature

<|

w(h)

w(r)

106

Partial specific volume of polymer

Volume fraction of polymer

Molar volume of solvent

Radial distribution function for end -to -end
coordinates of a polymer chain (usually Gaussian)

Pair potential of average force

A parameter of which expansion factor (X is a
single valued function

Greek Symbols

 

Factor expressing the linear deformation of a

polymer molecule owing to solvent interaction,

2 >0)1/2, and the same for

1/2

i.e. O( = (< r2 >/< 1‘
. . . 2 2
radius of gyration, 1. e. (XL = (< S >/< S <0)

Binary cluster integral representing the mutually
excluded volume per segment pair.

Solubility parameters of solvent and polymer
respectively

Viscosity of solution and solvent respectively

Relative viscosity, 77/ 770

[771

[7716

107

Specific viscosity, nrel _ 1

Intrinsic viscosity, dl/g
Intrinsic viscosity at the Flory temperature, dl/ g

Temperature at which chemical potential due to
alymer segment-solvent interactions is zero and
= 1

Parameter expressing the energy, divided by kT,
of interaction between a solvent molecule and

polymer
Flory' s constant

Flory interaction parameter, heat parameter and
entropy parameter of the same

Parameter characterizing the entropy of dilution
of polymer with solvent

REFERENCES

10.

11.

12.

13.

REFERENCES

Flory, P. J. "Principles of Polymer Chemistry, " Cornell
University Press, New York (1953).

Kurata, M., and W. H. Stockmayer, Forschr. Hochpolymer.
Forsch. , Z_3_, 196 (1963)..

Flory, P. J., and T. G. Fox, J. Am. Chem. Soc. 73, 1904
(1951). _

Fox, T. G., and P. J. Flory, J. Phys. Colloid. Chem. 53,
197 (1949).

Outer, P., C. I. Carr, and B. H. Zimm, J. Chem. Phys.
1__8_, 830 (1950).

Guggenheim, E. A., Trans. Faraday Soc., 31:, 1007 (1948).

Stockmayer, W. H., Makromol. Chem. _3_§, 54 (1960).

. Hill, T. L. , ”Statistical Mechanics, " Chapter 6, McGraw-

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