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Structure and Thermodynamics of Polyelectrolyte Complexes

  • Johannes FruehEmail author
  • Meiyu Gai
  • Simon Halstead
  • Qiang HeEmail author
Chapter
Part of the Engineering Materials book series (ENG.MAT.)

Abstract

Polyelectrolytes (PEs) find applications in many fields of modern life starting from food additives to flocculation and solubility enhancers, down to viscosity adjusting agents in cosmetics and subterranean gelling or drug delivery agents. Most of these properties are related to the PE charge density, structure, counterions, temperature or counter PE. This book chapter gives an overview of the current state of understanding of the thermodynamical properties of PEs in solution. The theoretical predictions and results are compared with current state of the art computer simulations (with a focus on molecular dynamics) as well as experiments on PE structure, complex formation and viscosity properties.

Keywords

Counterion Condensation Zimm Model Entanglement Concentration Polyampholytes Charged Monomers 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

e

Valence of PE monomer group

\(k_{B} T\)

Thermal energy

\(k_{B}\)

Boltzmann constant

\(\varepsilon\)

Dielectric constant

\(\varepsilon_{L}\)

Local dielectric constant

\(l_D\)

Debye length

k

Inverse Debye length

\(l_0\)

Persistence length

\(l_B\)

Bjerrum length

\(l_M\)

Length between two monomers

\(l_d\)

Length of dipole

L

Length of fully elongated PE

\(L_C\)

Length of collapsed PE

\(L_{\xi }\)

Correlation length

\(L_M\)

Distance between 2 PE chain centers

\(L_str\)

Length of string between globules

\(L_{\xi }\)

Size of a globule/bead

\(c_S\)

Salt concentration

\(c_P\)

PE concentration

\(c^{*}\)

Crossover concentration

\(c_{I}^{loc}\)

Local concentration of counterions next to PE

f

Fraction of charged monomers

a

Monomer length/diameter

K

Reduced coupling constant (Based on thermal energy)

\(K^{\prime}\)

Reduced coupling constant (based on excluded volume)

v

Excluded volume per monomer

\(v_{0}\)

Local volume close to monomer

\(\bar{a}\)

Elongation per monomer

\(\zeta\)

Size of electrostatic blob

\(V_{\zeta }\)

Volume of electrostatic blob

R

Total length of PE

\(R_H\)

Hydrodynamic radius

N

Number of monomers

\(n^{*}\)

Number of charged groups in PE

\(N_M\)

Number of monomers in the diameter D of a rodlike molecule

\(N_Msr\)

Number of monomers in string

\(N_Mbead\)

Number of monomers in a bead

\(N_E\)

Number of monomers in entangled region

\(N^{\prime}\)

Number of P-bonds in strong PE

\(N_2\)

Amount of PE molecules

\(N_1\)

Amount of solvent molecules

\(N_1i\)

Amount of solvent molecules inside \(v_{0}\)

\(N_1o\)

Amount of solvent molecules outside \(v_{0}\)

\(N_C\)

Number of counter ions

\(n_{\Psi }\)

Number of ψ bindings

D

Effective diameter of the rod-like molecule

d

Degree of dissociated charges

M

Mol

L

Liter

b

Charge density

n

Valence of counterions

k

Reaction constant

\(v_{1}\)

Volume fraction of solvent

\(v_{2}\)

Volume fraction of PE

\(v_{3}\)

Volume fraction of counterion

\(v_{l}\)

Molecular volume of solvent

\(v_{c}\)

Molecular volume of counterion

\(v_{PE}\)

Molecular volume of PE

\(V_{l}\)

Molar volume of solvent

\(V_{2}\)

Molar volume of PE

x

Number of gratings used in Flory lattice

\(\Delta H\)

Free enthalpy

\(H_{e}\)

Electrostatic enthalpy

\(\Delta S\)

Free entropy

\(S_{M}\)

Entropy of mixing

\(S_{K}\)

Kuhn entropy

\(S_{C}\)

Counter ion entropy

\(F_{e}\)

Free energy of electrolyte

\(F_{A}\)

Free energy of ionic atmosphere

\(F_{0}\)

Non electrostatic part of free energy

F

Free energy

\(\chi\)

Flory Huggins solution parameter

E

Electrostatic based energies

\(E_{D}\)

Debye electrostatic energy of the PE chain

\(E_{a}\)

Energy of the counterion adsorbed on the PE

\(E_{{_{ela} }}\)

Elastic energy of the stretched, charged PE

\(\beta\)

Fraction of free counter ions

\(\beta_{c}\)

Fraction of condensed counter ions

u

Charge density parameter

\(k_{1}\)

Constant related to \(k_{2}\)

\(k_{2}\)

Concentration parameter ~\(- \log c_{P}\)

\(\lambda\)

Charge density parameter

\(z_{e}\)

Euler number ex

\(e_{e}\)

Electron charge

\(Z_{1}\)

Effective expansion factor

W

Strength parameter for short ranged effects

A

Å

\(V_{ + }\)

Positive ion cloud scaling factor

\(V_{ - }\)

Negative ion cloud scaling factor

\(R_{G}\)

Ideal gas constant

\(F_{1}\)

Entropy of condensed counterions

\(F_{2}\)

Translational entropy of ions

\(F_{3}\)

Fluctuations between PE and ions

\(F_{4}\)

Ion pair energy

\(F_{5}\)

Free energy term for partly charged PE

\(F_{6}\)

Correlation term (ion pair PE)

\(\eta_{s}\)

Viscosity of the solvent

\(\eta_{{}}\)

Viscosity of the solution

\(\tau_{RZ} , \tau_{RR}\)

Zimm and Rouse relaxation time

\(\tau\)

Time

\(D_{i}\)

Diffusion coefficient

\(D_{Z} , D_{R}\)

Zimm and Rouse diffusion coefficient

\({\rm Z}_{P}\)

Friction coefficient of the polymer

\({\rm Z}_{B}\)

Friction coefficient of the beads

\(\sigma_{s}\)

Shear stress

\(\dot{\gamma }\)

Shear rate

\(\beta_{R}\)

Reduced shear rate

\(\phi\)

Volume fraction

\(L_{a}\)

Tube diameter

\(T_{R}\)

Reduced temperature

\(\theta\)

Theta temperature of solvent/PE

X

Concentration dependent parameter, which depends on concentration (values used for X are shown below the equations)

Notes

Acknowledgments

Acknowledgments: This work was supported by the National Nature Science Foundation of China (91027045), 100-talent Program of HIT, China Postdoctoral Science Foundation (2013M531019) and New Century Excellent Talent Program (NCET-11-0800) and Harbin Institute of Technology.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Key Laboratory of Microsystems and Microstructures Manufacturing, Ministry of Education, Micro/Nano Technology Research CentreHarbin Institute of TechnologyHarbinChina
  2. 2.School of Chemical Engineering and TechnologyHarbin Institute of TechnologyHarbinChina

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