Semidilute Solutions

1
Semidilute Solutions
2
Overlap Concentration
At the overlap concentration
Scaling theory predicts
Logarithmic corrections to N-dependence of
overlap concentration c.
3
Dependence of Overlap Concentration on Degree of
Polymerization
Long polyelectrolyte chains are almost fully
stretched
4
Semidilute Polyelectrolyte Solutions
Electrostatic energy of a polyelectrolyte blob
Interaction with other blobs
Interaction with free counterions
Electrochemical potential of a monomer
Electrostatic interactions
Elastic deformation
f -fraction of free counterions
Minimizing m with respect to De and x
Electrostatic blob size
Correlation length
where
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Semidilute Polyelectrolyte Solutions
Comments
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 x chains are strongly stretched due to
electrostatic interactions between similarly
charged monomers, similar to chain conformations
in dilute solutions.
6
Correlation Length
At correlation length from a given monomer it is
equally likely to find monomers belonging to the
same and to different chains.
x
gintra(x) ginter(x)
x
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Concentration Dependence of Correlation Length
8
Concentration Dependence of the Correlation Length
-1/2
9
Persistence Length
kp monomers
lp
Persistence length correlation length
lp x c-1/2
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Polyelectrolyte Chain in a Semidilute Solution
In a semidilute solution chain is a random walk
of correlation blobs
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End-to-End Distance for Polyelectrolytes in
Semidilute Solutions
Scaling theory predicts
Re c-n
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Osmotic Pressure in Semidilute Solutions
Osmotic pressure in semidilute solution has two
contributions
polymeric contribution
counterion contribution
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Cell Model in Semidilute Solutions
Cell Model
Osmotic coefficient in cell model
Cell radius
where parameter a is a solution of the equation
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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
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 p is
measured.
cs is a salt concentration
Osmotic pressure of polyelectrolyte solutions is
controlled by its ionic part.
from Dobrynin, A.V., Colby,R.H. Rubinstein,M.
Macromolecules 28, 1859-1871 (1995).
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Semidilute Solutions ofNecklaces
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Semidilute String Controlled Regime
Chain is strongly stretched on the length
scales smaller than correlation length
clt c lt cstr
Chain size
Correlation length
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Semidilute Bead Controlled Regime
Dobrynin Rubinstein 99
cstr lt c lt cb
Db ltx
x
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 x.
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Correlation Length
-0.33
NaPSS, MW(PS)68 000
-0.66
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Single Chain Form Factor
Bead size vs fraction of charged monomers
Theory Db f -2/3 Experiment Db
f-0.7
(NaPSS, MW(PSH)68 000, MW(PSD)73 000 )
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Effect of Added Salt
Spitery Boue 97
For charge fraction f0.64 at polymer
concentration C 0.34 M
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Correlation Length
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Correlation Length
Bead Controlled
String Controlled
Concentrated
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Nonmonotonic Dependence of the Chain Size on
Polymer Concentration
Polymer concentration increases
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Dependence of the Chain Size on Polymer
Concentration
String controlled regime
Bead controlled regime
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Dependence of the Chain Size on Polymer
Concentration
Poor solvent
q-solvent
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Chains in Concentrated Solution
N187, f1/3, eLJ 1.5, u3, cs3 10-1