Research Progress of the Ion Activity Coefficient of Polyelectrolytes: A Review

Polyelectrolyte has wide applications in biomedicine, agriculture and soft robotics. However, it is among one of the least understood physical systems because of the complex interplay of electrostatics and polymer nature. In this review, a comprehensive description is presented on experimental and theoretical studies of the activity coefficient, one of the most important thermodynamic properties of polyelectrolyte. Experimental methods to measure the activity coefficient were introduced, including direct potentiometric measurement and indirect methods such as isopiestic measurement and solubility measurement. Next, progress on the various theoretical approaches was presented, ranging from analytical, empirical and simulation methods. Finally, challenges for future development are proposed on this field.

[1]  K. Song,et al.  The solubility behavior of sodium arsenate in NaOH solution based on the Pitzer model , 2023, Heliyon.

[2]  Gaigai Duan,et al.  Nanocellulose and Its Derived Composite Electrodes toward Supercapacitors: Fabrication, Properties, and Challenges , 2022, Journal of Bioresources and Bioproducts.

[3]  Zhibin He,et al.  Sources, production and commercial applications of fungal chitosan: A review , 2022, Journal of Bioresources and Bioproducts.

[4]  E. Gálvez,et al.  Estimating the Shear Resistance of Flocculated Kaolin Aggregates: Effect of Flocculation Time, Flocculant Dose, and Water Quality , 2022, Polymers.

[5]  R. Kalinova,et al.  Functional Polyion Complex Micelles for Potential Targeted Hydrophobic Drug Delivery , 2022, Molecules.

[6]  Y. Levin,et al.  Widom insertion method in simulations with Ewald summation. , 2022, The Journal of chemical physics.

[7]  J. F. Waters,et al.  Chemical speciation models based upon the pitzer activity coefficient equations, including the propagation of uncertainties: Artificial seawater from 0 to 45 °C , 2022, Marine Chemistry.

[8]  Susan K. Kozawa,et al.  Fundamentals and mechanics of polyelectrolyte gels: Thermodynamics, swelling, scattering, and elasticity , 2021, Chemical Physics Reviews.

[9]  Honglei Guo,et al.  In Situ Evaluation of the Polymer Concentration Distribution of Microphase-Separated Polyelectrolyte Hydrogels by the Microelectrode Technique , 2021, Macromolecules.

[10]  Xiangdong Liu,et al.  Calculation Methods of Solution Chemical Potential and Application in Emulsion Microencapsulation , 2021, Molecules.

[11]  Chau‐Chyun Chen,et al.  Modeling aqueous multivalent polyelectrolytes systems with polyelectrolyte NRTL model , 2021 .

[12]  S. Mathew,et al.  Extraction and characterization of myofibrillar proteins from different meat sources: A comparative study , 2021 .

[13]  Shaohua Jiang,et al.  3D printing hydrogels for actuators: A review , 2021 .

[14]  Hongbing Deng,et al.  Chitin derived nitrogen-doped porous carbons with ultrahigh specific surface area and tailored hierarchical porosity for high performance supercapacitors , 2021 .

[15]  H. Abruña,et al.  Organic electrode materials for fast-rate, high-power battery applications , 2021 .

[16]  K. Lian,et al.  A review of neutral pH polymer electrolytes for electrochemical capacitors: Transitioning from liquid to solid devices , 2021 .

[17]  Die Hu,et al.  Chitosan-Based Biomimetically Mineralized Composite Materials in Human Hard Tissue Repair , 2020, Molecules.

[18]  A. Panagiotopoulos.  Simulations of activities, solubilities, transport properties, and nucleation rates for aqueous electrolyte solutions. , 2020, The Journal of chemical physics.

[19]  W. D. de Vos,et al.  Multiple Approaches to the Buildup of Asymmetric Polyelectrolyte Multilayer Membranes for Efficient Water Purification , 2020 .

[20]  J. Pleiss,et al.  Analysis of Thermophysical Properties of Deep Eutectic Solvents by Data Integration , 2019, Journal of Chemical & Engineering Data.

[21]  Jeffrey M. Young,et al.  Activity Coefficients and Solubility of CaCl2 from Molecular Simulations , 2020, Journal of Chemical & Engineering Data.

[22]  C. Holm,et al.  Poly(sodium acrylate) hydrogels: synthesis of various network architectures, local molecular dynamics, salt partitioning, desalination and simulation. , 2019, Soft matter.

[23]  Chau‐Chyun Chen,et al.  Nonrandom two-liquid activity coefficient model for aqueous polyelectrolyte solutions , 2019, Fluid Phase Equilibria.

[24]  Jonas Landsgesell,et al.  Cell Model Approaches for Predicting the Swelling and Mechanical Properties of Polyelectrolyte Gels , 2019, Macromolecules.

[25]  Polymer Thermodynamics , 2019, Chemical Thermodynamics for Process Simulation.

[26]  B. Cabane,et al.  Osmotic pressure in polyelectrolyte solutions: cell-model and bulk simulations. , 2018, Soft matter.

[27]  J. Dufrêche,et al.  Activity Coefficients of Aqueous Sodium, Calcium, and Europium Nitrate Solutions from Osmotic Equilibrium MD Simulations. , 2018, The journal of physical chemistry. B.

[28]  William R. Smith,et al.  Recent progress in the molecular simulation of thermodynamic properties of aqueous electrolyte solutions , 2018 .

[29]  H. Orland,et al.  Dielectric constant of ionic solutions: Combined effects of correlations and excluded volume. , 2018, The Journal of chemical physics.

[30]  D. Heyes,et al.  Chemical potential of a test hard sphere of variable size in hard-sphere fluid mixtures. , 2018, The Journal of chemical physics.

[31]  Kurt Kremer,et al.  Spatially Resolved Thermodynamic Integration: An Efficient Method To Compute Chemical Potentials of Dense Fluids. , 2018, Journal of chemical theory and computation.

[32]  Jeffrey M. Young,et al.  System-Size Dependence of Electrolyte Activity Coefficients in Molecular Simulations. , 2018, The journal of physical chemistry. B.

[33]  Kenneth S. Pitzer,et al.  Activity Coefficients in Electrolyte Solutions , 2017 .

[34]  S. Maset,et al.  Interaction between Charged Cylinders in Electrolyte Solution; Excluded Volume Effect. , 2017, The journal of physical chemistry. B.

[35]  T. Richter,et al.  A self-consistent mean-field model for polyelectrolyte gels. , 2017, Soft matter.

[36]  G. Odriozola,et al.  Competition between excluded-volume and electrostatic interactions for nanogel swelling: effects of the counterion valence and nanogel charge. , 2017, Physical chemistry chemical physics : PCCP.

[37]  J. Simonin,et al.  Thermodynamic properties of ring polyelectrolytes in the binding mean spherical approximation , 2017 .

[38]  B. Freeman,et al.  Predicting Salt Permeability Coefficients in Highly Swollen, Highly Charged Ion Exchange Membranes. , 2017, ACS applied materials & interfaces.

[39]  Peter V Coveney,et al.  Rapid, Accurate, Precise, and Reliable Relative Free Energy Prediction Using Ensemble Based Thermodynamic Integration. , 2017, Journal of chemical theory and computation.

[40]  A. Erbaş,et al.  Ionic Conductivity in Polyelectrolyte Hydrogels , 2016 .

[41]  H. Hasse,et al.  Activities in Aqueous Solutions of the Alkali Halide Salts from Molecular Simulation , 2016 .

[42]  T. Kurokawa,et al.  Molecular structure and properties of click hydrogels with controlled dangling end defect , 2016 .

[43]  Honglei Guo,et al.  Quantitative Observation of Electric Potential Distribution of Brittle Polyelectrolyte Hydrogels Using Microelectrode Technique , 2016 .

[44]  J. Dzubiella,et al.  Swelling of ionic microgel particles in the presence of excluded-volume interactions: a density functional approach. , 2016, Physical chemistry chemical physics : PCCP.

[45]  B. Freeman,et al.  Ion Activity Coefficients in Ion Exchange Polymers: Applicability of Manning’s Counterion Condensation Theory , 2015 .

[46]  T. Richter,et al.  Modeling of Polyelectrolyte Gels in Equilibrium with Salt Solutions , 2015 .

[47]  Athanassios Z Panagiotopoulos,et al.  Mean ionic activity coefficients in aqueous NaCl solutions from molecular dynamics simulations. , 2015, The Journal of chemical physics.

[48]  D. Boda,et al.  Unraveling the behavior of the individual ionic activity coefficients on the basis of the balance of ion-ion and ion-water interactions. , 2014, The journal of physical chemistry. B.

[49]  B. Ghalami-Choobar,et al.  Mean activity coefficients measurement of sodium chloride and thermodynamic modeling of sodium chloride and polysodium acrylate aqueous mixtures at T =(298.2 and 303.2)K , 2014 .

[50]  S. Mattedi,et al.  A new approach for the thermodynamic modeling of the solubility of amino acids and β-lactam compounds as a function of pH , 2013 .

[51]  A. Safronov,et al.  Activity of counterions in hydrogels based on poly(acrylic acid) and poly(methacrylic acid): Potentiometric measurements , 2012, Polymer Science Series A.

[52]  G. Kontogeorgis,et al.  Comparison of the Debye−Hückel and the Mean Spherical Approximation Theories for Electrolyte Solutions , 2012 .

[53]  D. Paschek,et al.  Activity Coefficients of Complex Molecules by Molecular Simulation and Gibbs-Duhem Integration , 2012 .

[54]  V. Vlachy,et al.  Correlation between flexibility of chain-like polyelectrolyte and thermodynamic properties of its solution , 2012, 1202.4282.

[55]  G. Gor,et al.  Interactions of phosphororganic agents with water and components of polyelectrolyte membranes. , 2011, The journal of physical chemistry. B.

[56]  A. Klamt.  The COSMO and COSMO‐RS solvation models , 2011 .

[57]  J. Hermans,et al.  The dimensions of charged long chain molecules in solutions containing electrolytes , 2010 .

[58]  J. Dzubiella,et al.  Ionic force field optimization based on single-ion and ion-pair solvation properties. , 2009, The Journal of chemical physics.

[59]  G. Maurer,et al.  Aqueous Solutions of Polyelectrolytes: Vapor–Liquid Equilibrium and Some Related Properties , 2010 .

[60]  J. D. de Pablo,et al.  Swelling and collapse of polyelectrolyte gels in equilibrium with monovalent and divalent electrolyte solutions. , 2009, The Journal of chemical physics.

[61]  Chau‐Chyun Chen,et al.  Symmetric Electrolyte Nonrandom Two-Liquid Activity Coefficient Model , 2009 .

[62]  G. Maurer,et al.  A model for the Gibbs energy of aqueous solutions of polyelectrolytes , 2009 .

[63]  P. A. P. Filho,et al.  An extension of the Pitzer equation for the excess Gibbs energy of aqueous electrolyte systems to aqueous polyelectrolyte solutions , 2008 .

[64]  S. Lamperski,et al.  The individual and mean activity coefficients of an electrolyte from the inverse GCMC simulation , 2007 .

[65]  R. Car,et al.  Dipolar correlations and the dielectric permittivity of water. , 2007, Physical review letters.

[66]  P. Linse,et al.  Monte Carlo simulation of polyelectrolyte gels: Effects of polydispersity and topological defects , 2007 .

[67]  D. Antypov,et al.  Osmotic coefficient calculations for dilute solutions of short stiff-chain polyelectrolytes , 2007 .

[68]  J. Gong,et al.  Polyelectrolyte Gels-Fundamentals and Applications , 2006 .

[69]  V. Vlachy,et al.  Theoretical aspects and computer simulations of flexible charged oligomers in salt-free solutions. , 2006, The Journal of chemical physics.

[70]  D. Antypov,et al.  The Osmotic Behavior of Short Stiff Polyelectrolytes , 2006 .

[71]  P. Linse,et al.  Monte Carlo simulations of cross-linked polyelectrolyte gels with oppositely charged macroions. , 2006, Langmuir : the ACS journal of surfaces and colloids.

[72]  F. Crea,et al.  Modelling of natural and synthetic polyelectrolyte interactions in natural waters by using SIT, Pitzer and Ion Pairing approaches , 2006 .

[73]  D. Antypov,et al.  Optimal cell approach to osmotic properties of finite stiff-chain polyelectrolytes. , 2006, Physical review letters.

[74]  Q. Yan,et al.  Molecular dynamics simulation of discontinuous volume phase transitions in highly-charged crosslinked polyelectrolyte networks with explicit counterions in good solvent. , 2005, The Journal of chemical physics.

[75]  R. Schwarzenbach,et al.  Activity Coefficient and Solubility in Water , 2005 .

[76]  T. Nishio,et al.  Effects of Ion Size and Valence on Ion Distribution in Mixed Counterion Systems of a Rodlike Polyelectrolyte Solution. 2. Mixed-Valence Counterion Systems† , 2003 .

[77]  Qi Liao,et al.  Molecular dynamics simulations of polyelectrolyte solutions: Nonuniform stretching of chains and scaling behavior , 2003 .

[78]  Qi Liao,et al.  Molecular dynamics simulations of polyelectrolyte solutions: Osmotic coefficient and counterion condensation , 2003 .

[79]  Kurt Kremer,et al.  Computer simulations for macromolecular science , 2003 .

[80]  R. Schwarzenbach,et al.  Environmental Organic Chemistry: Schwarzenbach/Environmental , 2002 .

[81]  I. Jano,et al.  Activity Coefficients of Individual Ions from Titration Data , 2002 .

[82]  Marshall Rafal,et al.  Electrolyte solutions: from thermodynamic and transport property models to the simulation of industrial processes , 2002 .

[83]  M. Deserno,et al.  Theory and simulations of rigid polyelectrolytes , 2002, cond-mat/0203599.

[84]  M. Deserno,et al.  Cell model and Poisson-Boltzmann theory: A brief introduction , 2001, cond-mat/0112096.

[85]  Honglai Liu,et al.  Activity coefficients of NaCl in polyelectrolyte solutions from EMF measurements , 2001 .

[86]  T. Nishio,et al.  Effects of ion size and valence on ion distribution in mixed counterion systems of rodlike polyelectrolyte solution. I. Mixed-size counterion systems with same valence , 2000 .

[87]  M. Deserno,et al.  Fraction of Condensed Counterions around a Charged Rod: Comparison of Poisson−Boltzmann Theory and Computer Simulations , 1999, cond-mat/9906277.

[88]  Y. Chiew,et al.  Analytical integral equation theory for a restricted primitive model of polyelectrolytes and counterions within the mean spherical approximation. I. Thermodynamic properties , 1999 .

[89]  T. Nishio,et al.  Monte Carlo simulations of ion activities in rodlike polyelectrolyte solutions , 1999 .

[90]  P. Dave,et al.  Dilute solution behaviour of polyacrylamides in aqueous media , 1999 .

[91]  D. Theodorou,et al.  Monte Carlo simulations of a single polyelectrolyte in solution : Activity coefficients of the simple ions and application to viscosity measurements , 1998 .

[92]  Jianwen Jiang,et al.  A molecular-thermodynamic model for polyelectrolyte solutions , 1998 .

[93]  S. Plimpton,et al.  The effect of added salt on polyelectrolyte structure , 1998 .

[94]  R. Danner,et al.  Colligative properties of polyelectrolyte solutions , 1998 .

[95]  G. Maurer,et al.  Partitioning of Some Amino Acids and Low Molecular Mass Peptides in Aqueous Two-Phase Systems of Poly(ethylene glycol) and Dextran in the Presence of Small Amounts of K2HPO4/KH2PO4-Buffer at 293 K: Experimental Results and Correlation , 1997 .

[96]  N. N. Medvedev,et al.  Calculation of Chemical Potentials by a Novel Delaunay-Simplex Sampling Technique for Particle Insertion , 1995 .

[97]  Kurt Kremer,et al.  The nature of flexible linear polyelectrolytes in salt free solution: A molecular dynamics study , 1995 .

[98]  G. Maurer,et al.  Aqueous Two‐Phase Systems of Poly(ethylene glycol) and Di‐Potassium Hydrogen Phosphate with and without partitioning Biomolecules – Experimental Results and Modeling of Thermodynamic Properties , 1995 .

[99]  G. Maurer,et al.  Aqueous two-phase systems of poly(ethylene glycol) and dextran — experimental results and modeling of thermodynamic properties , 1995 .

[100]  D. Dolar,et al.  Activity Coefficient of a Polyelectrolyte from Solubility Measurements , 1995 .

[101]  Wilfred F. van Gunsteren,et al.  A molecular dynamics simulation study of chloroform , 1994 .

[102]  T. Nishio.  Monte Carlo simulations on potentiometric titration of cylindrical polyelectrolytes: Introduction of a method and its application to model systems without added salt , 1994 .

[103]  T. Pačes,et al.  Activity coefficients in electrolyte solutions: 2nd ed., edited by K. S. Pitzer. CRC Press, 1991, 542p., US $195.00 (ISBN 0-8493-5415-3) , 1993 .

[104]  G. Orkoulas,et al.  Chemical potentials in ionic systems from Monte Carlo simulations with distance-biased test particle insertions , 1993 .

[105]  M. Molero,et al.  Individual ionic activity coefficients from a symmetric Poisson–Boltzmann theory , 1992 .

[106]  Koichi Ito,et al.  Sodium ion activity and electrical conductivity of poly(maleic acid) and poly(isobutylene-alt-maleic acid) in aqueous salt-free solution , 1991 .

[107]  D. Beveridge,et al.  Grand canonical Monte Carlo simulations on aqueous solutions of sodium chloride and sodium DNA: excess chemical potentials and sources of nonideality in electrolyte and polyelectrolyte solutions , 1991 .

[108]  Ivo Nezbeda,et al.  A New Version of the Insertion Particle Method for Determining the Chemical Potential by Monte Carlo Simulation , 1991 .

[109]  E. Ochiai.  Paradox of the activity coefficient , 1990 .

[110]  G. Grest,et al.  Dynamics of entangled linear polymer melts: A molecular‐dynamics simulation , 1990 .

[111]  R. Danner,et al.  An excess Gibbs free energy model for polyelectrolyte solutions , 1989 .

[112]  C. Woodward,et al.  Widom's method for uniform and non-uniform electrolyte solutions , 1988 .

[113]  Ye,et al.  Mean spherical approximation (MSA) for a simple model of electrolytes. I. Theoretical foundations and thermodynamics , 1988 .

[114]  M. Record,et al.  Grand canonical Monte Carlo calculations of thermodynamic coefficients for a primitive model of DNA-salt solutions , 1986 .

[115]  G. Vesnaver,et al.  Entropies of dilution of strong polyelectrolyte solutions , 1986 .

[116]  V. Vlachy,et al.  A grand canonical Monte Carlo simulation study of polyelectrolyte solutions , 1986 .

[117]  Kremer,et al.  Molecular dynamics simulation for polymers in the presence of a heat bath. , 1986, Physical review. A, General physics.

[118]  Donald A. McQuarrie,et al.  A theory of cylindrical polyelectrolyte solutions , 1985 .

[119]  C. Rha,et al.  Counter-ion activity in a cationic polyelectrolyte solution , 1984 .

[120]  M. Lozada-Cassou.  Hypernetted chain theory for the distribution of ions around a cylindrical electrode , 1983 .

[121]  P. Laszlo,et al.  Calculation of activity coefficients from a novel numerical solution of the Poisson–Boltzmann equation and application to 23Na NMR of sodium polystyrene sulfonate , 1982 .

[122]  H. Wennerström,et al.  The cell model for polyelectrolyte systems. Exact statistical mechanical relations, Monte Carlo simulations, and the Poisson–Boltzmann approximation , 1982 .

[123]  Paul Zema,et al.  Interactions of sodium ions with the sodium salts of poly(acrylic acid/acrylamide) copolymers of varying charge density , 1981 .

[124]  J. Valleau,et al.  Primitive model electrolytes. I. Grand canonical Monte Carlo computations , 1980 .

[125]  I. Snook,et al.  The grand canonical ensemble Monte Carlo method applied to electrolyte solutions , 1980 .

[126]  A. Delville.  A simple solution to the poisson-boltzmann equation for ion binding to a polyelectrolyte , 1980 .

[127]  G. Weisbuch,et al.  Polyelectrolyte theory. 2. Activity coefficients in Poisson-Boltzmann and in condensation theory. The polarizability of the counterion sheath , 1979 .

[128]  S. Oman.  Osmotic coefficients of aqueous polyelectrolyte solutions at low concentrations, 3. Dependence on macroion molecular weight†‡ , 1977 .

[129]  Charles H. Bennett,et al.  Efficient estimation of free energy differences from Monte Carlo data , 1976 .

[130]  S. Oman.  Osmotic coefficients of aqueous polyelectrolyte solutions at low concentrations, 2. dependence of polystyrenesulfonate osmotic coefficients on macroion molecular weight , 1974 .

[131]  S. Oman.  Osmotic coefficients of aqueous polyelectrolyte solutions at low concentrations, 1. Polystyrenesulfonates with mono‐ and bivalent counterions , 1974 .

[132]  G. Boyd.  Thermodynamic Properties of Strong Electrolyte — Strong Polyelectrolyte Mixtures at 25°C [1] , 1974 .

[133]  R. W. Hockney,et al.  A 10000 particle molecular dynamics model with long range forces , 1973 .

[134]  J. D. Wells,et al.  Thermodynamics of polyelectrolyte solutions. An empirical extension of the manning theory to finite salt concentrations , 1973 .

[135]  Kenneth S. Pitzer,et al.  Thermodynamics of electrolytes. I. Theoretical basis and general equations , 1973 .

[136]  B. Preston,et al.  Model connective tissue system: The effect of proteoglycan on the diffusional behavior of small non‐electrolytes and microions , 1972, Biopolymers.

[137]  Kazutoshi Suzuki,et al.  Potentiometric titration of polyelectrolytes having stiff backbones , 1972 .

[138]  A. Katchalsky.  Polye1ectrolytes , 1971 .

[139]  J. Marinsky,et al.  Further investigation of the osmotic properties of hydrogen and sodium polystyrenesulfonates , 1970 .

[140]  T. Okubo,et al.  Mean activity coefficient of polyelectrolytes. VIII. Osmotic and activity coefficients of poly(styrenesulfonates) of various gegenions , 1968 .

[141]  N. Ise,et al.  Mean activity coefficient of polyelectrolytes. IX. Activity coefficients of poly(ethylenesulfonates) of various gegenions , 1968 .

[142]  T. Okubo,et al.  Mean activity coefficient of polyelectrolytes. X. Activity coefficients of polyphosphates of various gegenions , 1968 .

[143]  T. Okubo,et al.  Mean activity coefficient of polyelectrolytes. V. Measurements of polyvinyl sulfates of various gegenions , 1967 .

[144]  T. Okubo,et al.  Mean activity coefficient of polyelectrolytes. IV. Isopiestic measurements of sodium polyacrylates , 1967 .

[145]  T. Okubo,et al.  Mean Activity Coefficient of Polyelectrolytes. III. Measurements of Hydrochlorides of Polyethylenimine and Its Low Molecular Weight Analogs1 , 1966 .

[146]  T. Okubo,et al.  Mean Activity Coefficient of Polyelectrolytes. II. Measurements of Sodium Salts of Polyvinyl Alcohols Partially Acetalized with Glyoxylic Acid1 , 1966 .

[147]  T. Okubo,et al.  Mean Activity Coefficient of Polyelectrolytes. I. Measurements of Sodium Polyacrylates1 , 1965 .

[148]  B. Widom,et al.  Some Topics in the Theory of Fluids , 1963 .

[149]  R. B. Huff,et al.  Isopiestic vapor pressure apparatus: A physical chemistry experiment , 1961 .

[150]  M. Nagasawa,et al.  Colligative properties of polyelectrolyte solutions. IV. Activity coefficient of sodium ion , 1957 .

[151]  I. Kagawa,et al.  Activity of counter ion in polyelectrolyte solutions , 1955 .

[152]  M. Nagasawa,et al.  Statistical thermodynamics of polyelectrolyte solutions. I. Activity coefficient of counter ion in salt‐free system , 1955 .

[153]  A. Katchalsky,et al.  Polyelectrolyte gels in salt solutions , 1955 .

[154]  Fumi Osawa,et al.  Theory of strong polyelectrolyte solutions. I. Coiled macro ions , 1954 .

[155]  S. Lifson,et al.  The electrostatic free energy of polyelectrolyte solutions. II. Fully stretched macromolecules , 1954 .

[156]  A. Katchalsky,et al.  The electrostatic free energy of polyelectrolyte solutions. I. Randomly kinked macromolecules , 1953 .

[157]  H. Gregor,et al.  Studies on Ion Exchange Resins. VI. Water Vapor Sorption by Polystyrenesulfonic Acid , 1953 .

[158]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[159]  I. Kagawa,et al.  Activity coefficient of byions and ionic strength of polyelectrolyte solutions , 1952 .

[160]  R. Robinson,et al.  The Activity Coefficients of the Alkali Chlorides and of Lithium Iodide in Aqueous Solution from Vapor Pressure Measurements , 1934 .

[161]  D. A. Sinclair.  A Simple Method for Accurate Determinations of Vapor Pressures of Solutions. , 1932 .