1. 52 Semidilute Solutions

2. 53 Overlap Concentration -1/3 At the overlap concentration Logarithmic corrections to N-dependence of overlap concentration c*. Scaling theory predicts

3. 54 Dependence of Overlap Concentration on Degree of Polymerization -2 Long polyelectrolyte chains are almost fully stretched

4. 55 Semidilute Polyelectrolyte Solutions Electrostatic energy of a polyelectrolyte blob Electrochemical potential of a monomer Elastic deformation Interaction with other blobs Interaction with free counterions f * -fraction of free counterions Electrostatic interactions Minimizing  with respect to D e and  Electrostatic blob sizeCorrelation length where

5. 56 Semidilute Polyelectrolyte Solutions In average the net charge of each correlation volume is equal to zero. The charge of polyelectrolyte chain is compensated by surrounding counterion background. At length scales smaller than the solution correlation length  chains are strongly stretched due to electrostatic interactions between similarly charged monomers, similar to chain conformations in dilute solutions. Comments:

6. 57 Correlation Length  g intra  ) ≈ g inter (  ) At correlation length from a given monomer it is equally likely to find monomers belonging to the same and to different chains. 

7. 58 Concentration Dependence of Correlation Length Finite size effects are important for short chains that contain only few correlation blobs.  /R e (c * ) ~ (number of correlation blobs per chain) -1 Scaling theory predicts  ~ c -1/2

8. 59 Concentration Dependence of the Correlation Length -1/2

9. 60 Persistence Length Persistence length ≈ correlation length bsbs b s+k k p monomers lplp l p ≈  ~ c -1/2

10. 61 Polyelectrolyte Chain in a Semidilute Solution In a semidilute solution chain is a random walk of correlation blobs 

11. 62 End-to-End Distance for Polyelectrolytes in Semidilute Solutions Scaling theory predicts: R e ~ c -

12. 63 Osmotic Pressure in Semidilute Solutions Osmotic pressure in semidilute solution has two contributions: counterion contribution polymeric contribution

13. 64 Cell Model in Semidilute Solutions Osmotic coefficient in cell model R  Cell Model where parameter  is a solution of the equation Cell radius:

14. 65 Non-monotonic Concentration Dependence of Osmotic Coefficient Polymeric contribution to osmotic pressure is important only at high concentrations. Minimum of osmotic coefficient is close to overlap concentration in agreement with 2-zone model.

15. 66 Osmotic Coefficient in Salt Solutions In ionic systems, the Donnan equilibrium requires the charge neutrality on both sides of a membrane across which the osmotic pressure  is measured. 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. c s is a salt concentration

16. 67 Semidilute Solutions of Necklaces

17. 68 Semidilute String Controlled Regime Correlation length: Chain is strongly stretched on the length scales smaller than correlation length  Chain size: c * < c  < c str

18. 69 Semidilute Bead Controlled Regime D b <  c str < c  < c b  Dobrynin& Rubinstein 99 Colloidal fluid of beads Beads on neighboring chains screen electrostatic repulsion of beads on the same chain reducing the length of strings to the distance between beads .  ~ c -1/3 f -2/3 Chain size: Correlation length

19. 70 Correlation Length NaPSS, M W (PS)=68 000 -0.33 -0.66

20. 71 Single Chain Form Factor (NaPSS, M W (PS H )=68 000, M W (PS D )=73 000 ) Theory: D b ~ f -2/3 Experiment: D b ~ f -0.7 Bead size vs fraction of charged monomers

21. 72 Effect of Added Salt For charge fraction f=0.64 at polymer concentration C = 0.34 M Spitery & Boue ‘97

22. 73 Correlation Length  c str c -1/2 l str DbDb -1/3 cbcb

23. 74 Correlation Length String Controlled Bead Controlled Concentrated

24. 75 Nonmonotonic Dependence of the Chain Size on Polymer Concentration Polymer concentration increases

25. 76 c str c -1/4 -1/3 1 f 1/3 cbcb Dependence of the Chain Size on Polymer Concentration String controlled regime Bead controlled regime

26. 77 Dependence of the Chain Size on Polymer Concentration Poor solvent  -solvent

27. 78 Chains in Concentrated Solution N=187, f=1/3,  LJ =1.5, u=3, c  3 = 10 -1