1

Semidilute Solutions

2

Overlap ConcentrationOverlap Concentration

10-4 10-3 10-2 10-1 2x10-1

5

10

100

N = 94, f = 1

<Re

2>1/2

Rcm

c*

R/

c 3

-1/3

At the overlap concentration

cme RR 2/12

Logarithmic corrections to N-dependence of overlap concentration c*.

Scaling theory predicts

23

* ~ NR

Nc

e

123

* ln~ NNR

Nc

e 10 100 40010-5

10-4

10-3

10-2

10-1

f = 1

f = 1/3

f e2 c* 3

N

3

Dependence of Overlap Concentration Dependence of Overlap Concentration on Degree of Polymerizationon Degree of Polymerization

-22

3* ~ N

R

Nc

e

Long polyelectrolytechains are almostfully stretched

4

Semidilute Polyelectrolyte SolutionsSemidilute Polyelectrolyte Solutions

De

z

/2

De

De

z

/2

22*

2* ln

cflDD

gfl

Tk

UB

ee

eB

B

el

Electrostatic energy of a polyelectrolyte blob

Electrochemical potential of a monomer

22*

2*

22

2

ln cfl

DD

gfl

gb

D

Tk Bee

eB

e

e

B

Elasticdeformation

Interaction withother blobs

Interaction withfree counterions

f* -fraction of free counterions

Electrostatic interactions

Minimizing with respect to De and

*3/1* ln

eee D

eDD

Electrostatic blob size Correlation length

2/16/12/1 ln

cc

ecc b

where 3/12*

* ufbDe

5

Semidilute Polyelectrolyte SolutionsSemidilute Polyelectrolyte Solutions

De

z

/2

De

De

z

/2

In average the net charge of eachcorrelation volume is equal to zero. The charge of polyelectrolyte chain is compensated by surrounding counterionbackground.

At length scales smaller than thesolution correlation length chainsare strongly stretched due to electrostaticinteractions between similarly charged monomers, similar tochain conformations in dilute solutions.

Comments:

6

Correlation LengthCorrelation Length

2 10 300.01

0.1

1

10

gintra

(r)g

inter(r)

gtotal

(r)

c = 0.015 -3

N=40 N=94 N=187

g(r)

r/

gintra) ≈ ginter()

At correlation length from a given monomerit is equally likely to find monomers belonging to the same and to different chains.

7

Concentration Dependence Concentration Dependence of Correlation Lengthof Correlation Length

10-5 10-4 10-3 10-2 10-1

1

10

100

-1/2

f=1 N=25 N=40 N=60 N=94 N=187 N=300

c 3

1 10 100 1000 100000.01

0.1

1

-1/2

f=1 N=25 N=40 N=60 N=94 N=187 N=300

/R

e(c*)

c/c*

Finite size effects are important for short chains that contain only few correlation blobs.

/Re(c*) ~ (number of correlation blobs per chain)-1

Scaling theory predicts

~ c-1/2

8

Concentration Dependence of the Correlation Concentration Dependence of the Correlation LengthLength

-1/2

9

Persistence LengthPersistence Length

0 50 100 150 200 250

0.1

1

N=300, f=1 c3

0.15 0.05 0.015 0.005 0.0015 0.0005 0.00015

<co

sk>

k

10-5 10-4 10-3 10-2 10-1

1

10

100

-1/2

N = 300

l

p

l/

c 3

Persistence length ≈ correlation length

bs bs+k

pkss

kssk k

k

bb

bbexpφcos

kp monomers

lp

lp ≈ ~ c-

1/2

10

Polyelectrolyte Chain in a Semidilute Polyelectrolyte Chain in a Semidilute SolutionSolution

1 10 100 2001

10

20

1/2 f = 1 N = 300 N = 187 N = 94 N = 60 N = 40

f = 1/3 N = 187 N = 94 N = 61 N = 40

Re/

N/g

In a semidilute solutionchain is a random walk of correlation blobs

2/1

g

NRe

11

End-to-End Distance for Polyelectrolytes End-to-End Distance for Polyelectrolytes in Semidilute Solutionsin Semidilute Solutions

1 10 100 10005

10

100

200

<R

e2 / 2>

1/2

c/c*

f=1 N=25 N=40 N=60 N=94 N=187 N=300

0.00 0.01 0.02 0.03 0.040.05

0.10

0.15

0.20

0.25

f = 1

f = 1/3

1/N

Scaling theory predicts:

Re ~ c-

4/12/1

2/1

~

cN

g

NRe

12

Osmotic Pressure in Semidilute SolutionsOsmotic Pressure in Semidilute Solutions

1 10 100 100010-6

10-5

10-4

10-3

10-2

10-1

100

f = 1 k

BT/ 3

(k

BT/3 )

c/c*

3* ξ

1πcf

TkB

Osmotic pressure insemidilute solution hastwo contributions:

counterioncontribution

polymeric contribution

13

Cell Model in Semidilute SolutionsCell Model in Semidilute Solutions

1 10 100 10000

1

2

Flexible chains N = 300, f = 1 Simulation Results Cell Model

c/c*

Osmotic coefficient in cell model

α

1γarctan

α

1arctan

σlnα 0R

πξcξg

R

R

Cell Model

0

2

cell 2γ

α1φ

where parameter is a solution of the equation

Cell radius:

14

10-7 10-6 10-5 10-4 10-3 10-2 10-1 1000.1

1 Solvent, f = 1 N=25 N=40 N=60 N=94 N=187 N=300

c (-3)

Non-monotonic Concentration Non-monotonic Concentration Dependence of Osmotic CoefficientDependence of Osmotic Coefficient

Polymeric contribution to osmotic pressure is important only at highconcentrations.

10-4 10-3 10-2 10-110-4

10-3

10-2

10-1

f=1/3 f=1

c min(

-3)

c*(-3)

Minimum of osmotic coefficient is close to overlap concentration in agreement with 2-zone model.

15

Osmotic Coefficient in Salt SolutionsOsmotic Coefficient in Salt SolutionsIn ionic systems, the Donnan equilibrium requires the charge neutrality on bothsides of a membrane across which the osmotic pressure is measured.

ssB

ion cccfTk

24 22*

from Dobrynin, A.V., Colby,R.H. & Rubinstein,M. Macromolecules 28, 1859-1871 (1995).

Osmotic pressure of polyelectrolyte solutions is controlled by its ionic part.

cs is a salt concentration

16

Semidilute Solutions ofSemidilute Solutions ofNecklacesNecklaces

17

Semidilute String Controlled RegimeSemidilute String Controlled Regime

Correlation length:

Chain is strongly stretched on the length scales smaller than correlation length

~ /c 1 2 R N c~ / /1 2 1 4

Chain size:

c*< c< cstr

18

Semidilute Bead Controlled RegimeSemidilute Bead Controlled Regime

Db <cstr < c< cb

Dobrynin&Rubinstein 99

Colloidal fluid of beads

Beads on neighboring chains screen electrostatic repulsion of beads on the same chain reducing thelength of strings to the distance between beads .

m c m cb b 3 1 3 1 3 / /

~ c-1/3 f -2/3

Chain size: R N m N cb / / /1 2 1 3

Correlation length

19

Correlation Length

NaPSS, MW(PS)=68 000

-0.33

-0.66

20

Single Chain Form FactorSingle Chain Form Factor

(NaPSS, MW(PSH)=68 000, MW(PSD)=73 000 )

Theory: Db~ f -2/3

Experiment: Db~ f-0.7

Bead size vs fraction of charged monomers

21

Effect of Added Salt

For charge fraction f=0.64 at polymer concentration C = 0.34 M

Spitery &Boue ‘97

R c M A

R c M A

R c M A

g s

g s

g s

( )

( . )

( . )

0 97 5

0 34 73 8

0 68 66 5

22

Correlation LengthCorrelation Length

cstr c

-1/2

lstr

Db

-1/3

cb

23

Correlation LengthCorrelation Length

String Controlled

Bead Controlled

Concentrated

24

Nonmonotonic Dependence of the Chain Nonmonotonic Dependence of the Chain Size on Polymer ConcentrationSize on Polymer Concentration

Polymer concentration increases

25

cstrc

R

b N

-1/4

-1/3

1

f1/3

cb

Dependence of the Chain Size Dependence of the Chain Size on Polymer Concentrationon Polymer Concentration

String controlledregime

Bead controlledregime

26

Dependence of the Chain Size Dependence of the Chain Size on Polymer Concentrationon Polymer Concentration

10-6 10-5 10-4 10-3 10-2 10-1 100

10

100

<R

e2 /2 >

1/2

c3

f = 1/3 N = 25 N = 40 N = 61 N = 94 N = 124 N = 187

Poor solvent -solvent

10-6 10-5 10-4 10-3 10-2 10-1

7

10

100

<Re2 /

2 >1/2

c 3

f=1/3 N=25 N=40 N=61 N=94 N=187

27

Chains in Concentrated SolutionChains in Concentrated Solution

N=187, f=1/3, LJ =1.5, u=3, c3 = 10-1