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Aqueous Solutions of Polyelectrolytes: Vapor–Liquid Equilibrium and Some Related Properties

  • G. MaurerEmail author
  • S. Lammertz
  • L. Ninni Schäfer
Chapter
Part of the Advances in Polymer Science book series (POLYMER, volume 238)

Abstract

This chapter reviews the thermodynamic properties of aqueous solutions of polyelectrolytes, concentrating on properties that are related to phase equilibrium phenomena. The most essential phenomena as well as methods to describe such phenomena are discussed from an applied thermodynamics point of view. Therefore, the experimental findings concentrate on the vapor–liquid phase equilibrium phenomena, and the thermodynamic models are restricted to expressions for the Gibbs energy of aqueous solutions of polyelectrolytes.

Keywords

Aqueous solutions Counterion condensation Excess Gibbs energy Osmotic coefficient Polyelectrolytes Salt effects Thermodynamics Vapor–liquid equilibrium 

Abbreviations

A

Inverse length (in model of Lifson and Katchalsky)

A

Repeating unit (in model of Lammertz et al.)

\( {A_{\phi}} \)

Debye-Hückel Parameter

As,s

Second osmotic virial coefficient for interactions between solutes S in water

As,s,s

Third osmotic virial coefficient for interactions between solutes S in water

a

Anion

a

Radius

\( a_i^{(k)} \)

Activity of species i normalized according to composition scale k

aw

Activity of water

\( a_{i,L}^{(0)} \)

Binary interaction parameter between species (groups) i and L

\( a_{i,L}^{(1)} \)

Binary interaction parameter between species (groups) i and L

\( a_{MX}^{(0)} \)

Binary interaction parameter between cations M and anions X

\( a_{MX}^{(1)} \)

Binary interaction parameter between cations M and anions X

\( {a_{p,p}} \)

Binary interaction parameter between repeating units

\( a_{p,\text{Cl}} \)

Binary interaction parameter between repeating units and the chloride ion

b

Distance between two electrolyte groups in a polyelectrolyte backbone

b

Numerical value in Pitzer’s model (b = 1.2)

\( {b^*} \)

Configurational parameter

\( {b_{i,\,L,\,k}} \)

Ternary interaction parameter between groups i, L and k

BaPSS

Poly(barium styrene sulfonate)

C

Repeating unit that will never dissociate (in model of Lammertz et al.)

c

Cation

ci

Concentration of species i

ci

Molarity of species i

CI

Counterion

ck,b

Molarity of monomeric groups saturated with counterion k

cm

Molarity of repeating units

cP

Concentration of polyelectrolyte P

cs

Molarity of salt S

\( {\tilde{c}_i} \)

Mass density of solute i

CaPAM

Calcium salt of copolymer of acrylic acid and acrylamide

D

Repeating unit undergoing a chemical reaction (in model of Lammertz et al.)

DMO

Differential membrane osmometry

DS

Degree of substitution

e

Proton charge

EMF

Electromotive force measurement

EQDIA

Equilibrium dialysis

F

Dissociated repeating unit (in model of Lammertz et al.)

F

Free energy

f

Short-range parameter

fel

Function in the theory of Lifson and Katchalsky

fi

Functions (in model of Lammertz et al.); i = 1, 2

FPD

Freezing point depression

f(M)

Molecular mass distribution function

G

Gibbs energy

\( {G_{ji}} \)

Binary interaction parameter (in model of Nagvekar and Danner)

\( {G_{ji, ki}} \)

Interaction parameter (in model of Nagvekar and Danner)

\( {g_{ji}} \)

Energy parameter (in model of Nagvekar and Danner)

GDM

Gel deswelling method

h

Length of a polyion

HPAA

Poly(acrylic acid)

HPAMS

Poly(2-acrylamido-2-methyl-1-propane sulfonic acid)

HPAS

Poly(anethole sulfonic acid)

HPES

Poly(ethylene sulfonic acid)

HPMAA

Poly(methacrylic acid)

HPMSS

Poly(methyl styrene sulfonic acid)

HPP

Poly(phosphoric acid)

HPVB

Poly(vinyl benzoic acid)

HPVS

Poly(vinyl sulfuric acid)

HPVSA

Poly(vinyl sulfonic acid)

HPSS

Poly(styrene sulfonic acid)

I

Ionic strength

\( {I_{{m}}} \)

Ionic strength (on molality scale)

\( {I_{m, MX}} \)

Ionic strength (on molality scale) of an aqueous solution of MX

\( {I_s} \)

Ionic strength (on molarity scale)

ISO

Isopiestic experiments

j

Abbreviation

j

Component

K

Chemical reaction constant (in model of Lammertz et al.)

k

Boltzmann’s constant

k

Component

k

Concentration scale

k

Degree of counterion condensation at infinite dilution (in model of Lammertz et al.)

KPA

Poly(potassium acrylate)

KPAM

Potassium salt of copolymer of acrylic acid and acrylamide

lB

Bjerrum length

LiCMC

Lithium carboxymethylcellulose

M

Cation

M

Molecular mass

Mr

rth moment of distribution function for molecular mass

Mn

Number-averaged molecular mass

Mw

Mass-averaged molecular mass

\( M_{{w}}^* \)

Relative molecular mass of water divided by 1,000

mi

Molality of species i

\( {m^{\circ} } \)

Unit of molality \( {m^{\circ}} = 1 \ \text{mol/(kg \ water)} \)

\( m_j^* \)

Modified molality of species j (in model of Pessoa and Maurer)

MgPAM

Magnesium salt of copolymer of acrylic acid and acrylamide

MO

Membrane osmometry

MX

Salt (cations M and anions X)

n

Mole number

NA

Avogadro’s number

\( {n_{p,\text{diss}}} \)

Number of moles of dissociated repeating units

\( n_{\text{free CI}}^{(p)} \)

Number of moles of counterions originating from P (in Manning’s theory)

\( n_{\text{free CI}}^{(s)} \)

Number of moles of counterions originating from S (in Manning’s theory)

\( {n_{\text{T}}} \)

Total mole number

NaCMC

Sodium carboxymethylcellulose

NaDS

Sodium dextran sulfate

NaPA

Poly(sodium acrylate)

NaPAM

Sodium salt of copolymer of acrylic acid and acrylamide

NaPAMA

poly(sodium acrylamido-co-trimethyl ammonium methyl methacrylate)

NaPAMS

Sodium salt of HPAMS

NaPES

Poly(sodium ethylene sulfate)

NaPMAA

Poly(sodium methacrylate)

NaPP

Poly(sodium phosphate)

NaPSS

Poly(sodium styrene sulfonate)

NaPVAS

Poly(sodium vinyl sulfate)

NH4PA

Poly(ammonium acrylate)

NMR

Nuclear magnetic resonance

P

Polyelectrolyte

P

Polydispersity (M w/M n)

p

Pressure

pw

Vapor pressure of water

PAAm

Poly(allylamino hydrochloride)

PDADMAC

Poly(diallyldimethyl ammonium chloride)

PEI

Poly(ethyleneimine)

PMETAC

Poly(2-(methacryloyloxy) ethyl trimethyl ammonium chloride)

PTMAC

Poly(trimethyl ammonium methyl methacrylate)

PVA

Poly(vinyl alcohol)

PVAm

Poly(vinyl amine)

PVBTMAC

Poly(vinyl benzene trimethyl ammonium chloride)

q

Number of charges

\( {q_{\text{gl}}} \)

Surface parameter of the globular form of the polyelectrolyte

qi

Surface parameter of species i (in model of Lammertz et al.)

qmax

Maximum number of charges

\( {q_{\text{st}}} \)

Surface parameter of the stretched polyelectrolyte

R

Universal gas constant

R

Radius of a cylindrical cell around a polyion

r

Exponent

r

Distance

ri

Volume parameter of species i (in model of Lammertz et al.)

T

Temperature

UV/VIS

Ultraviolet/visible light

V

Volume

Vp

Volume in Manning’s theory

VO

Vapor pressure osmometry

xi

Mole fraction of species i

X

Anion or counterion

\( {X_j} \)

Modified mole fraction of component j

X-DP

Salt (with counterion X) of dextran phosphate

X-DS

Salt (with counterion X) of dextran sulfate

\( {z_{\text{CI}}} \)

Absolute valency of counterion

\( {z_j} \)

Absolute valency of ion j

\( z_j^* \)

Modified absolute valency of ions j (in model of Pessoa and Maurer)

\( {z_M} \)

Absolute valency of cation M

\( {z_X} \)

Absolute valency of anion X

\( {z_p} \)

Absolute valency of a repeating unit of polyelectrolyte P

Greek Symbols

\( \alpha \)

Constant (\( \alpha \) = 2) in Pitzer’s model

\( \alpha \)

Total degree of dissociation of the repeating units (in model of Lammertz et al.)

\( {\alpha_{ji}} \)

Nonrandomness parameter (in model of Nagvekar and Danner)

\( {\alpha_{ji, ki}} \)

Nonrandomness parameter (in model of Nagvekar and Danner)

\( \beta \)

Dimensionless parameter

\( \Gamma \)

Salt-exclusion parameter

\( {\Gamma_i} \)

Activity coefficient (on molality scale) of species i (in model of Lammertz et al.)

\( {\Gamma_{l,Z}} \)

Activity coefficient (on molality scale) of species l in state Z (in model of Lammertz et al.)

\( \gamma \)

Activity coefficient

\( {\gamma_{\text{LK}}} \)

Dimensionless parameter in the theory of Lifson and Katchalsky

Δ\( \Delta \)

Difference

ΔTFP

Freezing point depression

ε

Relative permittivity of pure water

ε0

Permittivity of vacuum

\( \Phi \)

Osmotic coefficient

\( \Phi_p \)

Osmotic coefficient (for pressure)

\( \Phi_p^0 \)

Osmotic coefficient (for pressure) at infinite dilution

\( {\Phi_{{T}}} \)

Osmotic coefficient (for temperature)

\( \Phi_s^{(c)} \)

Osmotic coefficient (for pressure) on molarity scale due to salt S

\( \Phi_{p + s}^{(c)} \)

Osmotic coefficient (for pressure) on molarity scale due to salt S and polyion P

\( {\varphi_p} \)

Volume fraction of the polyelectrolyte

\( \phi (r) \)

Electrostatic potential that depends on radius r

\( {\theta_k} \)

Degree of condensation of a counterion k

\( {\theta_z} \)

Ratio in Manning’s theory

\( \theta_z^{(0)} \)

Limit for \( {\theta_z} \) in Manning’s theory

\( \kappa \)

Inverse radius of the ionic cloud (Debye–Hückel theory)

\( \lambda \)

Charge density parameter

\( {\lambda_{ij}} \)

Binary interaction parameter (in model of Pessoa and Maurer)

\( \lambda_{ij}^{(0)} \)

Binary interaction parameter (in model of Pessoa and Maurer)

\( \lambda_{ji}^{(1)} \)

Binary interaction parameter (in model of Pessoa and Maurer)

\( {\mu_i} \)

Chemical potential of component i

\( \Xi \)

Volume fraction of polyelectrolyte (in model of Lammertz et al.)

\( \nu \)

Number of repeating units of a polyelectrolyte molecule

\( {\nu^*} \)

Number of dissociated repeating units of a polyelectrolyte molecule

\( {\nu_M} \)

Stoichiometric coefficient for cation M in salt MX

\( {\nu_X} \)

Stoichiometric coefficient for anion X in salt MX

\( \pi \)

Osmotic pressure

\( {\Theta_{\text{gl}}} \)

Surface fraction of the polyelectrolyte in its globular shape

\( {\Theta_{{L}}} \)

Surface fraction of group L

\( {\Theta_{\text{st}}} \)

Surface fraction of the polyelectrolyte in its stretched shape

\( \rho_i^* \)

Specific density of pure solvent i

\( \bar{\rho }_i^* \)

Molar density of pure solvent i

\( {\sigma_i} \)

Parameter of species i (in model of Pessoa and Maurer)

\( {\tau_{ji}} \)

Binary interaction parameter (in model of Nagvekar and Danner)

\( {\tau_{ji, ki}} \)

Interaction parameter (in model of Nagvekar and Danner)

\( \upsilon_p^* \)

Molar volume in Manning’s theory

\( \bar{\upsilon }_{s,\text{pure}} \)

Molar volume of pure solvent s

\( {\varpi^{(0)}} \)

Configurational parameter

\( {\varpi^{(1)}} \)

Configurational parameter

Subscripts

A

Repeating unit (in model of Lammertz et al.)

a

Anionic component

c

Cationic component

C

Repeating unit that will never dissociate (in model of Lammertz et al.)

CI

Counterion

Cl

Chloride ion

COI

Coion

cond.

CI contribution due to condensed counterions

D

Repeating unit undergoing a chemical reaction (in model of Lammertz et al.)

F

Dissociated repeating unit (in model of Lammertz et al.)

Free CI

Free counterions

Free COI

Free coions

id.liq.mix.

Ideal liquid mixture

id.mix.

Ideal mixture

H

Hydrogenium ions

K

Potassium ion

k

Contribution

local

Local

LK

Lifson and Katchalsky

M

Cations

m

Solvent component

Ma

Manning’s theory

Mg

Magnesium ion

MX

Salt (cations M and anions X)

Na

Sodium ions

p

Polyelectrolyte

Pb

Lead ions

pure liquid

Pure liquid component

pure water

Pure water

rp

Repeating unit of polyelectrolyte

sym

Symmetrical convention

w

Water

(w + s)

In an aqueous solution of the salt

(w + s + p)

In an aqueous solution of (salt + polyelectrolyte)

X

Ion X

Superscripts

(c)

On molarity scale

Comb.

Combinatorial

E

Excess

el

Contribution from electrostatics

fv

Free volume

id.mix.

Ideal mixture

(k)

Characterizes the concentration scale

LR

Long-range

(m)

On molality scale

ref

Reference state

SLE

Solid–liquid equilibrium

SR

Short-range

vdW

Van der Waals

(x)

On mole fraction scale

Infinite dilution

Δconf

Caused by a difference in the configuration (in model of Lammertz et al.)

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Thermodynamics, Department of Mechanical and Process EngineeringUniversity of KaiserslauternKaiserslauternGermany

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