polymers physics - yale university

Polymers Physics Michael Rubinstein

University of North Carolina at Chapel Hill

1. “Real” Chains

2. Thermodynamics of Mixtures

3. Polymer Solutions

Outline

Summary of Ideal Chains Ideal chains: no interactions between monomers separated by many bonds

Mean square end-to-end distance of ideal linear polymer 22 NbR

Mean square radius of gyration of ideal linear polymer

6

22 Nb

Rg

2

22/3

232

3exp

2

3,

Nb

R

NbRNP d

Probability distribution function

Free energy of an ideal chain 2

2

2

3

Nb

RkTF

Entropic Hooke’s Law RNb

kTf

2

3

Pair correlation function 2

13)(

rbrg

Real Chains Include interactions between all monomers

Short-range (in space) interactions

Probability of a monomer to be in contact with another monomer

for d-dimensional chain

d

d

R

Nb*

If chains are ideal 2/1bNR 2/1* dN small for d>2

Number of contacts between pairs of monomers that are far along

the chain, but close in space 2/2* dNN small for d>4

For d<4 there are many contacts between monomers in an ideal chain.

Interactions between monomers change conformations of real chains.

Life of a Polymer is a Balance of entropic and energetic parts of free energy

Fent Feng

Entropic part “wants”

chains to have ideal-like

conformations

Energetic part typically

“wants” something else

(e.g. fewer monomer-

monomer contacts

in a good solvent).

Chain has to find a compromise between these two desires

and optimize its shape and size.

R

Flory Theory Number density of monomers in a chain is N/R3

Probability of another monomer being within excluded

volume v of a given monomer is vN/R3

2

2

3

2

Nb

R

R

NvkTF

Excluded volume interaction energy per monomer kTvN/R3

Excluded volume interaction energy per chain kTvN2/R3

Entropic part of the free energy is of order kTR2/(Nb2)

Flory approximation of the total free energy of a real chain

Free energy is minimum at 5/35/25/1 NbvR

Universal relation NR ~ - Flory scaling exponent

More accurate estimate 588.0

R

M w ( g / m o l e )

1 0 4 1 0 5 1 0 6 1 0 7 1 0 8

Rg (

nm

)

1

1 0

1 0 0

1 0 0 0

1 / 2 0 . 5 9

Radius of gyration of polystyrene chains in a q-solvent (cyclohexane

at 34.5oC) and in a good solvent (benzene at 25oC).

Fetters et al J. Phys. Chem. Ref. Data 23, 619 (1994)

Polymer Under Tension

Rf

x

Unperturbed size 2/1

0 bNR 5/3bNRF

Tension blobs (Pincus blobs) of size x

contain g monomers

2/1bgx5/3bgx

Chains are almost unperturbed on length scales up to x

On larger length scales they are stretched arrays of Pincus blobs

xxx

20

2 RNb

g

NR f

3/2

3/5

3/2

3/5

xxx F

f

RNb

g

NR

Size of Pincus blobs

fR

R20x

2/3

2/5

f

F

R

Rx

Ideal chain Real chain

Polymer Under Tension

Size of Pincus blobs

fR

R20x

2/3

2/5

f

F

R

Rx

Free energy cost for stretching a chain is on order kT per blob 2

0

R

RkT

RkT

g

NkTF

ff

x

2/5

F

ff

R

RkT

RkT

g

NkTF

x


Tension force is on order kT divided by blob size x

0020

R

R

R

kTR

R

kTkTf

ff

x

2/32/3

2/5

F

f

Ff

FR

R

R

kTR

R

kTkTf

x

1 3/2

kT/b f

R0 RF Rmax

Rf

linear elasticity non-linear elasticity

even at low forces

0 For b = 1nm at room T

kT/b = 4pN Log – log plot

Polymer Under Tension Ideal chain Real chain

fRRkTf 2

0 2/32/5

fF RRkTf 3/2

RF

1

kT/b

f

R0 Rmax

Rf linear elasticity non-linear elasticity

Challenge Problem 1: Pulling a Ring Consider a pair of forces applied to monomers 1 and 1+N/2

i. Show that the modulus of an ideal ring

is twice the modulus of a linear N/2-mer

ii. How does modulus of a ring in a good solvent Gr

compare to twice the modulus of linear N/2-mer GN/2?

f

f

2f

2f

2f

2f

a. Gr = 2GN/2 similar to ideal case

b. Gr < 2GN/2 because the entropy of sections of a ring is lower

c. Gr > 2GN/2 because ring sections reinforce each other

Biaxial Compression

D

R||


D

R||

On length scales smaller than compression blob of size D chain is almost

unperturbed 2

b

Dg

3/5

b

Dg

Occupied part of the tube

2/12/1

|| bNg

NDR

Nb

D

b

g

NDR

3/2

||

Free energy of confinement 2

D

bkTN

g

NkTFconf

3/5

D

bkTN

g

NkTFconf

20

D

RkTFconf

3/5

D

RkTF F

conf

Challenge Problem 2:

Biaxial Confinement of a Semiflexible Chain

Consider a semiflexible polymer – e.g. double-

stranded DNA with Kuhn length b=100nm and

contour length L=16mm.

Assume that excluded volume diameter of double

helix is d=3nm (larger than its actual diameter

2nm due to electrostatic repulsions).

Calculate the size RF of this lambda phage DNA in dilute solution

and the length R|| occupied by this DNA in a cylindrical channel

of diameter D (for d<D<RF).

D

R||

RF

Uniaxial Compression

D

Free energy of confinement in a slit is the

same as in the cylindrical pore

Longitudinal size of an ideal chain in a slit is the same as for an

unperturbed ideal chain. 2/1|| bNR

2-dimensional Flory theory for real chain confined in a slit

2

2||

2||

22

)/(

)/(

DgN

R

R

gNDkTF

D2 is the excluded area

of a confinement blob

with g monomers

Longitudinal size of a real chain in a slit

4/14/3

4/3

||

D

bbN

g

NDR

Fractal dimension of real chains in 2-d is D = 4/3

Scaling Model of Real Chains Thermal blob - length scale at which excluded

volume interactions are of order kT kTg

vkTT

T 3

2

x

Chain is ideal on length scales smaller than thermal blob 2/1

TT bgx

Number of monomers in a thermal blob 26 vbgT

Size of a thermal blob v

bT

4

x

xT R

xT

Rgl

5/35/1

3

5/3

Nb

vb

g

NR

TT

x 3/1

3/1

23/1

Nv

b

g

NR

T

T

x

good solvent v>0

poor solvent v<0

Flory Theory of a Polymer in a Poor Solvent

2

2

3

2

Nb

R

R

NvkTF In poor solvent v < 0 and 0R

Cost of Confinement

2NbR Compression blob of size R

with g monomers

2

b

Rg

Confinement free energy 2

2

R

NbkT

g

NkTFconf

2

2

2

2

3

2

R

Nb

Nb

R

R

NvkTF For v < 0 0R

Three Body Repulsion

6

3

3

2

2

2

2

2

R

Nw

R

Nv

R

Nb

Nb

RkTF

Size of a globule 3/1

v

wNRgl

gT

xT

N 1

b

R

3/5

3/5

good

1/3

poor 1/2 q-solvent

1/3 non-solvent

athermal

5/3bNRathermal

3/1bNR solventnon

2/1bNR q

5/3

TTgood

g

NR x

3/1

TTpoor

g

NR x

xT

xT

Real Chains in Different Solvents

Temperature Dependence of Chain Size

Mayer f-function

1exp

kT

rUrf

-1 for r<b where U(r)>>kT { kT

rU for r>b where |U(r)|<kT

Excluded volume

32

0

2

0

2 14

44 bT

drrrUkT

drrdrrrfvb

b

q

Interaction parameter z is related to number of thermal blobs per chain

2/12/1

3N

T

TN

b

v

g

Nz

T

q

Chain contraction in a poor solvent

3/1

2/1

z

bN

R

Chain swelling in a good solvent

12

2/1

zbN

R

2

6

v

bgT

N1/2

(1 - q/T)

-40 -30 -20 -10 0 10 20 30

Rg

2/R

q

2

0.0

0.5

1.0

1.5

2.0

2.5

N1/2

(1 - q/T)

-5 0 5 10 15 20

Rg

2/R

q

2

0.0

0.5

1.0

1.5

2.0

2.5

Monte-Carlo simulations

Graessley et.al., Macromolecules

32, 3510, 1999 & I. Withers

Polystyrene in decalin

Berry, J. Chem. Phys.

44, 4550, 1966

Universal Temperature Dependence

of Chain Size

Summary for Real Chains

2/1bNR q

Size of Linear Chains

nearly ideal in q-solvents

588.018.0

3

12

3N

b

vbN

b

vbRgood

swollen in good solvents

588.0bNbNRathermal in athermal solvents

3/123/1NbvRpoor

collapsed into a globule in poor solvents

In good solvents Stretching a real chain

43.2)1/(1

FF R

RkT

R

RkTF

non-Hookean elasticity

Confinement of a real chain

7.1/1

D

RkT

D

RkTF FF

Excluded volume

T

Tbv

q3

1. Real Chains



Outline

Assume, for simplicity, no volume change on mixing

Consider a mixture with total number n of monomers (sites)

and total volume v0n where v0 – volume of a lattice site

Assume that a monomer of each species occupies the same volume v0

Binary Mixtures Homogeneous – if components are intermixed on a molecular scale

Heterogeneous – if there are distinct phases

Volume fractions

BA

AA

VV

V+ =

VA VB VA+VB

1B

A n = n monomers of type A and B nB = (1- n monomers of type B

Entropy of Mixing

+ =

- volume fraction of A

nv0 1nv0 nv0 v0 – volume of a lattice site

Number of translational states of a molecule

in a mixture is the number of sites n n

v

VV BAAB

0

Entropy change upon mixing of a molecule A

Number of states of molecule A in a pure A phase nv

VAA

0

ln1

lnlnlnln kkkkkSA

ABAABA

nA/NA and nB/NB – number of A and B molecules

Total entropy change upon mixing

1lnln

B

B

A

AB

B

BA

A

A

N

n

N

nkS

N

nS

N

nS

S

0 1

Infinite

slopes

1 - volume fraction of B

z – coordination number

(number of neighbors)

uij – interaction of i with j

Average energy per monomer of pure components

before mixing BBAAi uuz

u 12

Mean Field Theory Average energy per A monomer

ABAAA uuz

u 12

B monomer with probability 1 – A ?

? ? ?

? ?

A monomer with probability

Average energy per B monomer BBABB uuz

u 12

Average energy per monomer in homogeneous mixture

BABBAA

f uun

ununu 1

Energy of mixing ifm uuu

U

BBu

z

2

AAuz

2

1 0

um

um BBABAA uuu

z 22 1122

Energy of a Homogeneous Mixture

Energy of Mixing

kTnuu

uznnuU BBAAABmm

1

21

0 1

Um

Energy change on mixing per site

Probability of AB contact 1

)1( kTum

per site

)1(1ln

1ln

BA

m

NNkT

n

F

Free Energy of Mixing

mmm STUF

2

BBAAAB

uuu

kT

z

Flory interaction parameter

Regular solution: NA = NB = 1 Polymer solution: NA = N >>1, NB = 1

Polymer blend: NA >>1, NB >> 1

At lower T for >0 repulsive

interactions are important and

there is a composition range

with thermodynamically stable

phase separated state.

Flory-Huggins Free Energy of Mixing

11ln

1ln

BA

mix

NNkT

n

F

At high T entropy of mixing dominates,

Fmix is convex and homogeneous

mixture is stable at all compositions.

'''

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 0.2 0.4 0.6 0.8 1

250o K

” ’

F

mix/(

kn)

300o K

350o K

NA = 200 NB = 100 T

Ko5

This composition range, called

miscibility gap, is determined by

the common tangent line.

Phase Diagrams

11ln

1ln

BA

mix

NNkT

n

F

For cr mixture is stable at all

compositions. 2

11

2

1

>

BAcr

NNFor

there is a miscibility gap for '''

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

N

Single Phase

Two Phases

Ncr

cr

For a symmetric blend NA=NB=N

cr=2/N cr=1/2

For polymer solutions NA = N, NB = 1

BA

B

crNN

N

Critical composition

NNcr

2

11

2

1

Ncr

1

-0.1

0

’ ” sp1 sp2

N

Fm

ix

kT

n N=2.7

NA = NB

Phase Diagram of Polymer Solutions

M=53.3kg/mole

Polyisoprene in dioxane

Takano et al., Polym J. 17, 1123, 1985

Single Phase

Two Phases

spinodal

binodal

Polymer solutions phase separate upon decreasing

solvent quality below q-temperature

Upper critical solution temperature B>0

Solution phase separates below

the binodal in poor solvent

regime into a dilute supernatant

of isolated globules at ` and

concentrated sediment at ``. ` ”

`

dilute

supernatant

`` concentrated

sediment

T

BA

metastable

(nucleation

and growth)

unstable

(spinodal

decomposition)

membrane

h

solvent solution

10

6 P

/cR

T (m

ole

s/g

)

c (g/ml)

0.00 0.01 0.02

0

5

10

15

20

25

30

M n = 70800 g/mole

M n = 200000 g/mole

M n = 506000 g/mole

Intermolecular Interactions

Osmotic pressure

P ...2

2cAM

cRT

n

Osmometer

Poly(a-methylstyrene)

in toluene at 25 oC

Noda et al, Macromol.

16, 668, 1981

A2 – second virial coefficient

Mixtures at Low Compositions

11ln

1ln

BA

mix

NNkT

n

F

...

62

1

2

1ln

32

BBBA

mix

NNNNkT

n

F

Expand ln(1-) in powers of composition

Osmotic pressure

P ...

322

1 32

33

2

BBA

mix

n

mix

NNNb

kTn

F

bV

F

A

Virial expansion in powers

of number density cn = /b3

P ...

2

32nn

A

n wccv

N

ckT

Excluded volume 32

1b

Nv

B

3-body interaction

BN

bw

3

6

In polymer solutions NB = 1 AvNb

MA

T

T

b

v3

202

3

221

q

Polymer Melts Consider a blend with a small concentration of NA chains in a melt

of chemically identical NB chains.

No energetic contribution to mixing = 0.

Excluded volume BB N

bb

Nv

332

1

is very small for NB >> 1

Thermal blob 2

2

6

BT Nv

bg BTT bNgb x

Chains smaller than thermal blob NA < NB2 are nearly ideal.

In monodisperse NA = NB and weakly polydisperse

melts chains are almost ideal.

In strongly asymmetric blends NA > NB2

long chains are swollen 10/1

2

2/1

5/3

2

5/3

B

AA

B

AB

T

ATA

N

NbN

N

NbN

g

NR x

RA

NA

bNA

NB2

1/2

3/5

b 1

Flory Theorem

M. Lang

RA

2/(

b2N

A)

NA/NB2

NB

NB

NB

NB

NB

NB

NB

Long NA-mer in a 3-d Melt of NB-mers

Why doesn’t Flory Theorem work?

5/1

22

2

B

A

A

A

N

N

Nb

R



v0 – volume of a lattice site = volume of a small B monomer

mv0 – volume of an A monomer m times larger than B monomer

Mixing of Polymers with Asymmetric Monomers

NA – monomers

per A chain

NB – monomers

per B chain

Derive the free energy of mixing

Fmix of A and B polymers.

Calculate the size RA of dilute A-

chains in a 2-d of B-chains for

m>>1 in the case of =0.

Summary of Thermodynamics of Mixtures

Free energy of mixing consists of entropic and energetic parts.

Entropic part per unit volume (translational entropy of mixing Smix )

1ln

1ln

BA

mix

vvkT

V

ST

Energetic part per unit volume

10v

kTV

Umix

Many low molecular weight liquids are miscible

Some polymer – solvent pairs are miscible

Very few polymer blends are miscible

vA – volume of A chain

vB – volume of B chain

v0 – volume of a lattice site

T

BA Flory interaction parameter

Chains are almost ideal in polymer melts as long as they are

shorter than square of the average degree of polymeriztion NA<N2

1. Real Chains



Outline

Which of These Chains are Ideal?

Quiz #1

C. Chain in a sediment of a polymer solution

in a poor solvent

A. Polymer in a solution of its monomers

B. Polymer dissolved in a melt of identical chains

D. None of the above sediment

Polymer Solutions

T

2-phase

Tc

` ``

1/2 c 0

poor solvent

Theta solvent

021 33

bT

Tbv

q

Chains are nearly ideal

at all concentrations NbR

Overlap concentration

NR

Nb 13

3* q

good solvent

q dilute q *

* * *1

semidilute

qsolvent

Chain is ideal if it is smaller than

thermal blob

v

bT

4

x

Boundaries of dilute q-regime

N

bb

T

Tbv

33321

q

NT

11qTemperatures at which chains begin

to either swell or collapse

v

Poor Solvent

Critical composition

NNcr

2

11

2

1

Ncr

1

Critical interaction

parameter

M = 43.6 kg/mole

Solution phase separates below

the binodal into a dilute supernatant

of isolated globules at ` and

concentrated sediment at ``.

` ``

Sediment concentration `` is determined by the balance of second and

third virial term (similar to the concentration inside globules).

3

3

312''

glR

Nb

b

v

3/1

3/12

3/1

3/1

12 v

NbbNRgl

Polystyrene in cyclohexane ( ) and

polyisobutylene in diisobutyl ketone ( )

Shultz and Flory, J. Am. Chem. Soc.

74, 4760, 1952

M = 1,270 kg/mole

Poor Solvent Dilute Supernatant of Globules

Globules behave as liquid droplets with size

3/1

3/12

v

NbRgl

Surface tension is of order kT per thermal blob

2

2

2

8212

x

b

kTv

b

kTkT

T

v

bT

4

x

Total surface energy of a globule 3/2

4

3/4

2

22 N

b

vkTkTRR

T

glgl

x

Concentration of a dilute supernatant

3/2

4

3/4

3

2

expexp''' Nb

v

b

v

kT

Rgl

is balanced by its translational entropy kTln’

is different from the

mean field prediction.

Rgl

xT

Good Solvent

* dilute

good

(swollen)

semidilute good

conce

ntr

ated

T q

2-phase

dilute q semidilute q c

Tc

` `` dilute

poor

(globules)

v

0

Swollen chain size

in dilute solutions

5/35/1

3N

b

vbR

Overlap concentration

5/4

5/33

3

3*

N

v

b

R

Nb

Semidilute solutions *1

x

Correlation length x

Distance from a monomer to nearest

monomers on neighboring chains.

Correlation Length

x

R

For r < x monomers are surrounded by

solvent and monomers from the same chain.

The properties of this section of the chain

of size x are the same as in dilute solutions.

g – number of monomers inside a correlation

volume, called correlation blob.

For r > x sections of neighboring chains overlap and screen each other.

On length scales r > x polymers are ideal chains - melt

with N/g effective segments of size x.

Correlation blobs are at overlap

5/35/1

3g

b

vb

x

3

3

x

gb 4/3

4/13

x

v

bb

2/1

8/1

3

2/1

Nb

vb

g

NR

x

Semidilute Solutions

R

N g

x

xT

gT 1

b 1/2

1/2

3/5 x

R

On length scales less than xT chain is ideal because excluded volume

interactions are weaker than kT.

r

n

r ~ n1/2

On length scales larger than xT but smaller than x excluded volume

interactions are strong enough to swell the chain. r ~ n3/5

On length scales larger than x excluded volume interactions are

screened by surrounding chains. r ~ n1/2

xT

Scaling Theory of Semidilute Solutions

At overlap concentration 5/4

5/33*

N

v

b

chain has its dilute solution size 5/3

5/1

3N

b

vbRF

In semidilute solutions R is a power of that matches dilute size at *

xxxx

F Nb

vbRR

5/)43(5/)31(

3*

Chains in semidilute solutions are random walks

2

1

5

43

xx = -1/8

2/1

8/1

3

8/1

*N

b

vbRR F

Similarly correlation length yyyy

F Nb

vbR

x 5/)43(

5/)31(

3*

is independent of N in

semidilute solution 3 + 4y = 0 y = -3/4 4/3

4/134/3

*

x

v

bbRF

Concentrated Solutions Correlation length x decreases with concentration, while thermal blob

size xT is independent of concentration.

At concentration ** the two length are equal and

intermediate swollen regime disappears. Txx

Tv

b

v

bb xx

44/3

4/13

3

**

b

v

This concentration is analogous to ’’ in poor solvent at

which two- and three-body interactions are balanced.

In concentrated solutions chains are ideal at all length scales.

On length scales less than x chains are ideal because excluded volume

interactions are weaker than kT.

On length scales larger than x chains are ideal because excluded

volume interactions are screened by surrounding chains.

Polymer Solutions

T q

2-phase

*


Tc

` `` dilute

poor

(globules)

dilute

good

(swollen)

semidilute good

conce

ntr

ated

v

0

**

R

-1/8 RF

R0

** *

-3/4

-1

xT

1

b

8/1

**0

RR

for * < < **

In athermal solvent 3bv bT x

1**

4/3x

b

8/1

2/1

bNR

Osmotic Pressure

In dilute solutions < * - van’t Hoff Law Nb

kT 3

P

Osmotic pressure P in semidilute solutions > * is a stronger

function of concentration

P

*3

f

Nb

kT

*

f { 1 for < *

(/*)z for > *

15/45/3

3

1

3*3

P z

zz

z

Nb

v

b

kT

Nb

kT

Osmotic pressure in semidilute solutions

is independent of chain length (4z/5-1=0). z=5/4

3

4/94/3

33 x

kT

b

v

b

kT

P

Neighboring blobs repel each other with energy of order kT.

3.1

**

3

1

P

f

kT

Nb

Poly(a-methylstyrene) in toluene at 25 oC

Noda et al, Macromol. 16, 668, 1981

Concentration Dependence of Osmotic Pressure

Semidilute Theta Solutions

T q

2-phase

*


Tc

` `` dilute

poor

(globules)

dilute

good

(swollen)

semidilute good

conce

ntr

ated

v

0

** Chains are almost ideal 2/1bNR

x 2

bg

x

Correlation blobs are space-filling

xx

x

x

bbbgb

3

23

3

3 /

NbR 0xNR

Nb 130

3* at

2/1

*

2/1 xx

x

NbbN

x

Correlation length in semidilute solutions

is independent of chain length (1+x)/2 = 0 x=-1

Scaling assumption for > *

x

b

x

b

What is the meaning of

correlation length in semidilute

theta solutions?

Quiz # 2

How different are semidilute theta

solutions from ideal solutions of

ideal chains?

Osmotic Pressure in Semidilute Theta Solutions

Mean-field prediction

P ...3

63

b

w

Nb

kT

First term (van’t Hoff law) is important in dilute solutions

Three-body term is larger than linear

in semidilute solutions > 1/N1/2 3

3

b

kTP

Scaling Theory

P

*3

h

Nb

kT

*

h { 1 for < *

(/*)y for > *

12/1

3*3

P yy

y

Nb

kT

Nb

kT

Osmotic pressure in semidilute qsolutions

is independent of chain length (y/2-1=0). y=2 3

3

3 x

kT

b

kTP

0.001 0.01 0.1 1

P (

Pa

)

102

103

104

105

3

1

2.31

Osmotic Pressure

Polyisobutylene in benzene

at q24.5 oC (filled circles),

in benzene at 50 oC (filled

squares), in cyclohexane at

30 oC (open circles) and at

8 oC (open squares)

Flory and Daoust J. Polym.

Sci. 25, 429, 1957

Correlation length in semidilute q-solvents is of the order

of the distance between 3-body contacts.

Number density of n-body contacts ~ n/b3

Distance between n-body contacts in 3-dimensional space 3/nn br

x

b

3

33

x b

kTkTP

Summary of Polymer Solutions

T q

2-phase

*


Tc

` `` dilute

poor

(globules)

dilute

good

(swollen)

semidilute good

conce

ntr

ated

v

0

**

1/2

In poor solvent part of the

diagram binodal separates

2-phase from 2 single phase

regions:

Dilute globules at low

concentrations < ’

Concentrated solutions

with overlapping ideal

chains at > ”

Near qtemperature there are dilute and semidilute q-regimes

with ideal chains.

Dilute good solvent regime with swollen chains at < * and v > 0.

Semidilute good solvent regime at * < < ** with chains swollen

at intermediate length scales shorter than correlation length x.

Osmotic pressure in semidilute solutions is kT per correlation volume x3.

Challenge Problem 5: Confinement of Polymers in Solutions & Melt

Calculate the pressure between two solid plates fully immersed into

a semidilute polymer solution or melt as a function of separation D

between plates for separations smaller than chain size R.

Assume no attraction between plates and polymer.

Separately consider cases with correlation length x<D<R and b<D<x

semidilute polymer

solution or melt

D

x

Single Chain Adsorption xads

Volume fraction in a chain section of size xads containing g monomers


adsads

bgb

xx

3

3

adsorption blob 2/1bgads x5/3bgads x

adsorption blob

3/4

3

3

adsads

bgb

xx

Number of monomers per adsorption blob in contact with the surface

bb

bads

xx

2

3

3/22

3

bb

bads

xx

Energy gain per monomer in contact with the surface is -ekT

Energy gain per adsorption blob

kTb

kT ads x

e kTb

kT ads

3/2

xe

Single Chain Adsorption xads Ideal chain Real chain

Size xads and number of monomers g in an adsorption blob

1badsx

e 1

3/2

badsx

e

ex

bads

2/3ex

bads

Free energy of an adsorbed chain

2ekTNg

NkTF

2/5ekTNg

NkTF

22

e

x

bg ads 2/5

3/5

e

x

bg ads

Flory Theory of Adsorption

D

b Fraction of monomers in direct contact

with the surface is b/D

Number of monomers in direct contact

with the surface is Nb/D

Energy gain per monomer contact with

the surface is -kTe

Energy gain from surface interactions per chain D

bkTNF eint

Total free energy is a sum of interaction and confinement parts

D

bNkT

D

bkTNF e

2

D

bNkT

D

bkTNF e

3/5

intFFF conf Ideal chain Real chain

Optimal thickness

e

bD

2/3e

bD

xads

x(z) z

z de Gennes’ self-similar carpet

Multi-Chain Adsorption

First layer is dense packing of adsorption blobs

Polymer concentration decays from the high value in the first layer.

The correlation length x(z) corresponding to concentration (z) at

distance z from the surface is of the order of this distance.

xads

Single-chain adsorption

2/3ex

bads

3/4

b

zz

z

z

bz

4/3x Coverage is controlled by

the first layer of blobs.

22

2/13/12

3/4

3

3ads

ads

ads

R g

b

bbdz

b

zbdz

b

z

adsx

e

x

x

2/5 eadsg

x

H

s

z

Alexander – de Gennes Brush

Grafting density s number of chains per

unit area.

Distance between sections of chains

x s1/2

Number of monomers per blob

g ~ x1/ ~ s-1/(2)

Thickness of the brush H ~ x N/g ~ Ns(1-)/(2)

Energy per chain Echain ~ kT N/g ~ kTNs1/(2)

Energy per unit volume P3xx

sss kTkT

Hg

NkT

HE

V

Echain

chain

102

103

10-5

10-4

10-3

10-2

10-1

Grafting density s (nm-2)

Bru

sh h

eight

H (

nm

) Polymer Brush Height depends on Grafting Density

If the grafting density is high, chains

repel each other and stretch away from

the surface, forming a polymer brush.

Mushroom Regime

unperturbed size

H ≃ R0

H ~ N s1/3

polymers physics - yale university

Documents