Excluded Volume Effects in Polymer Solutions

•i

Table of Contents

1.

Power Laws, Scaling, Universal Crossover — a Survey of Phenomenology

1

P a r t I- B a s i c T h e o r e t i c a l T e c h n i q u e s 2.

3.

4.

5.

F r o m a Microscopic Description to Simple Models

13

2.1

T h e model

13

2.2

R e l a t i o n to a realistic microscopic description

16

T h e Noninteracting Gaussian C h a i n

21

3.1

E n d p o i n t correlations

21

3.2 3.3

Density correlations Calculation of the Greensfunction

23 26

A 3

Appendix

29

A 3.1 A 3.2

The δ-Function T h e Fourier transform

21) 30

A 3.3

T h e Gaussian integral and integration i n d - d i m e n s i o n a l space

31

T h e Cluster Expansion

33

4.1 4.2

F i r s t order corrections t o endpoint correlations T h eFeynman rules

35 38

4.3

Cumulants

45

4.4 A 4

A n e x a m p l e : t h e s e c o n d v i r i a l coefficient Appendix

47 49

A 4.1

F o r m a l e x p r e s s i o n for t h e s e c o n d v i i i a l coefficient

49

A 4.2

E v a l u a t i o n of some segment s u m m a t i o n s

49

G r a n d Canonical Description of Solutions at F i n i t e C o n c e n t r a t i o n 5.1

B a s i c features o f t h e g r a n d c a n o n i c a l f o r m a l i s m 5.1.1 Definitions 5.1.2 L i n k e d cluster theorem and t h e r m o d y n a m i c limit

53 56 56 57

X

Table of Contents

5.2

7*.

Diagrammatics

58

5.1.4

Correlation functions

62

L o o p expansion, vertex irreducible graphs a n d screened i n t e r a c t i o n

65

5.2.1 5.2.2

T h e notion of 'loops' Vertex reducibility

65 66

5.2.3

T h e screened i n t e r a c t i o n

69

5.3

P o l y d i s p e r s i t y : N o t a t i o n a n d b a s i c features

72

5.4

T r e e a p p r o x i m a t i o n , s c r e e n i n g , a n d a one l o o p e x a m p l e . . 5.4.1 Tree a p p r o x i m a t i o n

A 5

6.

5.1.3

.

76 76

5.4.2

A n a l y s i s o f t h e screened i n t e r a c t i o n

78

5.4.3

O n e l o o p c o r r e c t i o n t o the F l o r y - H u g g i n s e q u a t i o n

79

Appendix A 5.1 L i n k e d cluster theorem

82 82

A 5.2

V e r t e x r e d u c i b i l i t y for t h e g r a n d p o t e n t i a l

84

A 5.3

Derivation of the loop expansion by f o r m a l m a n i p u l a t i o n of f u n c t i o n a l i n t e g r a l s . . .

86

A 5.3.1

T h e m e t h o d of a u x i l i a r y fields

86

A 5.3.2

F o r m a l d e r i v a t i o n of t h e l o o p e x p a n s i o n

89

A 5.4

A short discussion of the canonical formalism - . .

91

A 5.4.1

M a n i p u l a t i o n of closed f o r m expressions

91

A 5.4.2

Diagrammatic perturbation theory

94

A 5,4.3

C o m p a r i s o n t o the g r a n d c a n o n i c a l f o r m a l i s m . , .

96

Notes

97

B r e a k d o w n of the Expansions in the Excluded Volume Region

99

Continuous Chain

Model

and Naive Two Parameter Theory

103

7.1

C o n t i n u o u s c h a i n l i m i t a n d the E d w a r d s m o d e l

105

7.2

E x i s t e n c e of the c o n t i n u o u s c h a i n l i m i t

109

A 7

Appendix

118

A 7.1

118 122

F o r m a l relation t o field theory

Notes

P a r t II. R e n o r m a l i z a t i o n G r o u p a n d S c a l i n g 8,

T h e Renormalization Group: Fundamental Aspects 8.1 8.2 8.3 Notes

B a s i c ideas E x p l i c i t construction of an R G m a p p i n g F l o w e q u a t i o n s a n d crossover

....

127 128 133 136 141

Table of Contents

9.

XI

Scaling Theory

143

9.1

Concentration blobs and screening

141

9.2

The temperature-concentration diagram

149

9.3 Some words of caution Notes

151 154

10*. More Complicated Interactions and the General Structure of the Renormalization Group

155

10.1

Higher interactions and additive renormalization

157

10.2

The general structure of the renormalization group

162

10.2.1

Dilatations

162

10.2.2

Comparison to our previous results

165

10.2.3

The general structure of the renormalization group

10.3

167

Field theoretic renormalization group and universality

Notes

. . . 174 176

Part III. Field Theoretic Formulation of the Theory 11.

Renormalized Perturbation Theory and Field-Theoretic Renormalization Group

179

11.1 Illustration of renormalized perturbation theory

182

11.2

The theorem of

185

11.3

Renormalization group equations

11.4

Global form of the R G

11.5

The nonuniversal parameters and two parameter theory . . 192

11.6

Sealing laws in the dilute limit

194

11.7

Renormalization and scaling at finite overlap

197

11.7.1

Renormalized expressions

197

11.7.2

Scaling laws

190

11.8

rcnormalizability

187 flow

A comment on the universality of the scaling functions

and the status of our results A 11 Appendix A "11,1

Remarks on renormalizability

Notes 12.

189

200 202 202 205

Dimensional Regularization and Minimal Subtraction . . 207 12.1

Equivalence of discrete or continuous chains 209 12.1.1 Renormalizability of the continuous chain model . 209 12.1.2

12.2 12.3

Existence of the equivalent continuous chain model Minimal subtraction Final form of renormalized perturbation theory 12.3.1 General procedure

210 212 214 214

XII

13.

Table of Contents 12.3.2 A simple example 12.4 Perturbation theory in three dimensions vs ε-expansion . . 12.4.1 V i r t u e s and disadvantages of the ε-expansion . . . 12.4.2 Direct evaluation in three dimensions Notes .

216 218 218 221 223

Quantitative F o r m of the R e n o r m a l i z a t i o n G r o u p M a p p i n g

225

13.1 13.2

T h e renormalization group m a p p i n g in three dimensions . . Choice of the renormalized length scale 13.2.1 Qualitative considerations 13.2.2 D e t e r m i n a t i o n of the theoretical parameters . . . . 13.2.3 Some warning 13.3 T h e R G mapping in different regions of parameter space . . 13.3.1 General form (d = 3) 13.3.2 Special limits 13.3.3 Crossover diagram for ≤ 1 13.3.4 T h e strong coupling region A 13 A p p e n d i x A 13.1 E l i m i n a t i o n of bu(1) A 13.2 Parameter dependence of renormalized perturbation theory A 13.2.1 Choice of A 13.2.2 Choice of

14,

q0 c0

220 228 228 230 232 233 233 234 236 238 240 240 241 241 242

Notes

243

T h e R e n o r m a l i z e d Tree A p p r o x i m a t i o n : a Simple Illustration of the T h e o r y

245

14.1

Density autocorrelations 14.1.1 General expression 14.1.2 R a d i u s of gyration 14.1.3 M o m e n t u m dependence 14.2 Osmotic pressure 14.3 Density cumulant 14.4 S u m m a r y of important aspects

246 216 216 249 251 253 256

P a r t I V . O n e L o o p A p p r o x i m a t i o n : Detailed Results a n d C o m p a r i s o n to E x p e r i m e n t 15.

D i l u t e L i m i t : Density Correlations, V i r i a l Coefficients, a n d Interpenetration Ratio, 15.1

T h e radius of gyration 15.1.1 General expressions 15.1.2 E x c l u d e d volume l i m i t

259 261 261 262