79. Dynamics of Polyelectrolyte Solutions

80. Tutorial: Rouse and Zimm Dynamics
Rouse Model: In this model polymer chain is represented by a bead-spring chain
Model assumptions:
beads have no excluded volume;
there are no hydrodynamic interactions between beads.
Chain friction coefficient of the chain in Rouse model
Chain diffusion coefficient
Longest chain relaxation time
Terminal modulus
Viscosity of solution of Rouse chains

81. Zimm Model
Zimm model takes into account hydrodynamic interactions between
monomers inside polymer coil. The hydrodynamic interactions are
long-ranged and and decay as 1/r. Thus motion of polymer segments is
accompanied motion of the solvent inside polymer coil. Polymer drags
all solvent as it moves. R
Chain friction coefficient of the chain in Zimm model

82. Scaling of Zimm Model Chain diffusion coefficient
Longest chain relaxation time
Terminal modulus
Solution viscosity

83. Semidilute Unentangled Polyelectrolyte Solutions
The hydrodynamic interactions between sections of a chain in semidilute solutions
are screened at the length scales larger than the solution correlation length x.
The dynamics of chain sections at the length scales smaller than the solution
correlation length is strongly hydrodynamically coupled – Zimm-like.
The relaxation time of section of a chain with gx monomers inside correlation blob
Each chain consists of N/gx correlation blobs.
The hydrodynamic interactions between blobs are screened and their motion is described by Rouse dynamics x x D D e e

84. Semidilute Unentangled Polyelectrolyte Solutions
The chain self-diffusion coefficient is concentration independent
The terminal modulus G of solution of Rouse chains is kBT per chain
The viscosity of polyelectrolyte solution in this regime is

85. Tutorial: Reptation motion
In semidilute and concentrated polymer solutions the topological constraints imposed on a given by neighboring chains restrict its motion to tube-like region.
entanglements Ne
Number of monomers
between entanglements
Localizing tube around a chain.
Tube diameter

86. Tutorial: Reptation motion
Curvilinear diffusion of a chain along the tube
In a melt chain diffusion coefficient is Rouse diffusion coefficient
The time which requires for the chain to diffusive out of the original
tube of length L is the reptation time

87. Semidilute Entangled Polyelectrolyte Solutions
Entanglements are characterized by a tube diameter a (the mesh of temporary entanglement network)
The conformation of a polymer
strand between entanglements is
a random walk of correlation blobs x. x
where Ne is the number of monomers
between entanglements.
The relaxation time of the entangled
chain is a
Relaxation time
of a strand between
entanglements
Reptation

88. Semidilute Entangled Polyelectrolyte Solutions
The plateau modulus of the entangled solutions is
where ne is the density of entanglement strands. The volume per entanglement
is equal to Therefore
n-number of entanglement
strands inside a3 volume
Solution viscosity in the entangled regime
Chain self-diffusion coefficient

89. Summary of Scaling Relations for Polyelectrolyte Solutions
Unentangled
Entangled t G h
Dself
where
and

90. Modulus of Polyelectrolyte Solutions
Concentration dependence of the terminal modulus calculated from steady shear relaxation time (filled symbols) and oscillatory shear relaxation time (open symbols) at 250C in ethylene glycol solvent of random copolymer 2-vinyl pyridine and
N-methyl-2-vinyl pyridinium chloride (PMVP-Cl) with various charge densities and uncharged neutral parent poly (2-vinyl pyridine) (P2VP) of Mw=
(Provided by R. H. Colby)

91. Viscosity of Polyelectrolyte Solutions
Dependence of the solution specific viscosity on polymer concentration for semidilute salt-free
solutions of NaPSS. Filled squares are data of Boris and Colby for M=1.2x106; filled triangles
are data of Prini and Lagos for M=3x105; and data of Oostwal for M= (filled circles) and
M= (inverted filled triangles).
from Boris,D.C. & Colby,R.H. Macromolecules 31, (1998).)

92. Phase Separation in Polyelectrolyte Solutions

93. Mean-Field Approach
Flory-Huggins lattice model of salt-free polyelectrolyte solutions
Counterion
contribution
small
Solution stability region is determined from the following equation
Location of a critical point

94. Mean-Field Approach
Phase Diagram
N=1000

95. Mean-Field Approach
Flory-Huggins lattice model at high salt concentrations
Equation for the spinodal line of the phase diagram
Critical point

96. Mean-Field Approach
Comments: The Flory-Huggins lattice consideration of the polyelectrolyte solutions incorrectly describes dilute polyelectrolyte solutions. In the Flory-Huggins approach the monomers are uniformly distributed over the whole volume of the system leading to underestimation of the effect of short-range monomer-monomer interactions and of the intrachain electrostatic interactions. A similar problem appears in the Flory-Huggins theory of phase separation of polymer solutions. This oversimplification leads to the incorrect expression for the low polymer density branch of the phase diagram.

97. Necklace Model of Phase Separation
In the necklace model the phase separation is driven by counterion condensation and bead formation on polymer backbone.
(uf2)1/3
(f/u)-1/3
cb3
q-solvent
-(uf2)1/3
Bead
Controlled
Concentrated
Solution
-(f/u)1/3
String-controlled
Dilute necklaces
Phase Separation t
Phase diagram of polyelectrolytes in poor solvent. Logarithmic scales.
from Dobrynin,A.V. & Rubinstein,M. Macromolecules 34, (2001).

98. Acknowledgements Collaborations Financial Support J. Jeon (Uconn)
V. Panchagnula (Uconn)
Prof. M. Rubinstein (UNC-CH)
Dr. Q. Liao (Institute of Polymer Science, China)
Dr. I. Withers (University of Bristol, England)
Prof. S. Obukhov (UF)
Prof. R. Colby (PennState)
Prof. J.-F Joanny (Institute Curie, France)
Financial Support
The Petroleum Research Fund
The National Science Foundation,
University of Connecticut Research Foundation
Eastman Kodak Company.