Discretization dependence of criticality in model fluids: a hard-core electrolyte

Ion Partition in Polyelectrolyte Gels and Nanogels

Polyelectrolyte gels provide a load-bearing structural framework for many macroscopic biological tissues, along with the organelles within the cells composing tissues and the extracellular matrices linking the cells at a larger length scale than the cells. In addition, they also provide a medium for the selective transportation and sequestration of ions and molecules necessary for life. Motivated by these diverse problems, we focus on modeling ion partitioning in polyelectrolyte gels immersed in a solution with a single type of ionic valence, i.e., monovalent or divalent salts. Specifically, we investigate the distribution of ions inside the gel structure and compare it with the bulk, i.e., away from the gel structure. In this first exploratory study, we neglect solvation effects in our gel by modeling the gels without an explicit solvent description, with the understanding that such an approach may be inadequate for describing ion partitioning in real polyelectrolyte gels. We see that this type of model is nonetheless a natural reference point for considering gels with solvation. Based on our idealized polymer network model without explicit solvent, we find that the ion partition coefficients scale with the salt concentration, and the ion partition coefficient for divalent ions is higher than for monovalent ions over a wide range of Bjerrum length (lB) values. For gels having both monovalent and divalent salts, we find that divalent ions exhibit higher ion partition coefficients than monovalent salt for low divalent salt concentrations and low lB. However, we also find evidence that the neglect of an explicit solvent, and thus solvation, provides an inadequate description when compared to experimental observations. Thus, in future work, we must consider both ion and polymer solvation to obtain a more realistic description of ion partitioning in polyelectrolyte gels.

Theory of polyelectrolyte complexation-Complex coacervates are self-coacervates

The complexation of mixtures of cationic and anionic polymers to produce complex-coacervate phases is a subject of fundamental importance to colloid and polymer science as well as to applications including drug delivery, sensing technologies, and bio-inspired adhesives. Unfortunately the theoretical underpinnings of complex coacervation are widely misunderstood and conceptual mistakes have propagated in the literature. Here, a simple symmetric polyelectrolyte mixture model in the absence of salt is used to discuss the salient features of the phase diagram, including the location of the critical point, binodals, and spinodals. It is argued that charge compensation by dimerization in the dilute region renders the phase diagram of an oppositely charged polyelectrolyte mixture qualitatively and quantitatively similar to that of a single-component symmetric diblock polyampholyte solution, a system capable of "self-coacervation." The theoretical predictions are verified using fully fluctuating field-theoretic simulations for corresponding polyelectrolyte and diblock polyampholyte models. These represent the first comprehensive, approximation-free phase diagrams for coacervate and self-coacervate systems to appear in the literature.

The Influence of Polymer and Ion Solvation on the Conformational Properties of Flexible Polyelectrolytes

The study of the coupling between the conformational properties of a polyelectrolyte chain and the distribution of counter-ions surrounding the chain is important in developing predictive theories for more complex polymer materials, such as polyelectrolyte gels. We investigated the influence of solvent affinity to counter-ions and the polyelectrolyte backbone on the conformational properties of highly charged flexible polymer chains using molecular dynamics simulations that include both ions and an explicit solvent. We find that the solvation of the polyelectrolyte backbone can be achieved by either increasing the solvent affinity for the polyelectrolyte segments or by increasing the solvent affinity for the counter-ions. However, these two mechanisms influence the conformational properties of the polyelectrolyte chain in rather different ways, suggesting the inadequacy of polyelectrolyte solution models that treat the solvent as a continuum medium.

Thermal and structural properties of ionic fluids

The electrostatic interaction in ionic fluids is well known to give rise to a characteristic phase behavior and structure. Sometimes its long range is proposed to single out the electrostatic potential over other interactions with shorter ranges. Here the importance of the range for the phase behavior and the structure of ionic fluids is investigated by means of grandcanonical Monte Carlo simulations of the lattice restricted primitive model (LRPM). The long-ranged electrostatic interaction is compared to various types of short-ranged potentials obtained by sharp and/or smooth cutoff schemes. Sharply cutoff electrostatic potentials are found to lead to a strong dependence of the phase behavior and the structure on the cutoff radius. However, when combined with a suitable additional smooth cutoff, the short-ranged LRPM is found to exhibit quantitatively the same phase behavior and structure as the conventional long-ranged LRPM. Moreover, the Stillinger-Lovett perfect screening property, which is well known to be generated by the long-ranged electrostatic potential, is also fulfilled by short-ranged LRPMs with smooth cutoffs. By showing that the characteristic phase behavior and structure of ionic fluids can also be found in systems with short-ranged potentials, one can conclude that the decisive property of the electrostatic potential in ionic fluids is not the long range but rather the valency dependence.

Simulation of transport around the coexistence region of a binary fluid

We use Monte Carlo and molecular dynamics simulations to study phase behavior and transport properties in a symmetric binary fluid where particles interact via Lennard-Jones potential. Our results for the critical behavior of collective transport properties, with particular emphasis on bulk viscosity, is understood via appropriate application of finite-size scaling technique. It appears that the critical enhancements in these quantities are visible far above the critical point. This result is consistent with an earlier report from computer simulations where, however, the authors do not quantify the critical singularity.

When is a conductor not perfect? Sum rules fail under critical fluctuations

Perfect screening of all charges characterizes a conductor, a fact embodied in the Stillinger-Lovett sum rule: namely, the charge-charge correlation or structure factor, S(ZZ)(k), varies with momentum transfer k→0 as ξ(D)(2)k(2) where the Debye length ξ(D) is a universal function, √k(B)T/ρq(D)(2), of T and the ion density ρ, with a scaled charge q(D). For a charge-symmetric hard-sphere electrolyte our grand canonical simulations, with a new finite-size scaling device, confirm the Stillinger-Lovett rule except, contrary to current theory, for its failure at criticality. Furthermore, the k(4) term in the S(ZZ)(k) expansion is found to diverge like the compressibility when T→T(c) at ρ(c).

Liquid-liquid phase separation in solutions of ionic liquids: phase diagrams, corresponding state analysis and comparison with simulations of the primitive model

Phase diagrams of ionic solutions of the ionic liquid C(18)mim(+)NTF(2)(-) (1-n-octadecyl-3-methyl imidazolium bistrifluormethylsulfonylimide) in decalin, cyclohexane and methylcyclohexane are reported and compared with that of solutions of other imidazolium ionic liquids with the anions NTF(2)(-), Cl(-) and BF4(-) in arenes, CCl(4), alcohols and water. The phase diagrams are analysed presuming Ising criticality and taking into account the asymmetry of the phase diagrams. The resulting parameters are compared with simulation results for equal-sized charged hard spheres in a dielectric continuum, the restricted primitive model (RPM) and the primitive model (PM) that allows for ions of different size. In the RPM temperature scale the critical temperatures vary almost linearly with the dielectric permittivity of the solvent. The RPM critical temperatures of the solutions in non-polar solvents are very similar, somewhat below the RPM value. Correlations with the boiling temperatures of the solvents and a dependence on the length of the side chain of the imidazolium cations show that dispersion interactions modify the phase transition, which is mainly determined by Coulomb forces. Critical concentrations, widths of the phase diagrams and the slopes of the diameter are different for the solutions in protic and aprotic solvents. The phase diagrams of the solutions in alcohols and water get a lower critical solution point when represented in RPM variables.

Charge fluctuations and correlation lengths in finite electrolytes

Fluctuations of the charge Q_{Lambda} inside a subdomain Lambda embedded in an electrolyte contained in a finite cubical box of dimensions LxLxL with periodic boundary conditions are investigated. When Lambda is an LxL "slab" of width W , asymptotically exact expressions for the mean-square fluctuation Q_{Lambda};{2} are obtained in terms of the Lebowitz length xi_{L}(T,rho) and of the "true" or asymptotic screening/decay length xi_{Z,infinity}(T,rho) together, when the charge correlation decay is oscillatory, with the characteristic wavelength lambda_{Z}(T,rho) . In finite systems, the normalized charge fluctuations exhibit threefold scaling behavior in the ratios xi_{Z,infinity}W , Wlambda_{Z} , and WL . This enables one to estimate all the correlation lengths away from criticality quite precisely from finite-size grand canonical Monte Carlo simulations. The results for xi_{Z,infinity},lambda_{Z} , and xi_{L} are presented for the restricted primitive model or hard-sphere 1:1 electrolyte for densities rho less, similar1.3rho_{c} and T greater, similar4T_{c} . The fitted values compare favorably with the expectations of generalized Debye-Hückel theory [Lee and Fisher, Europhys. Lett. 39, 611 (1997)]; specifically, if xi_{D} proportional, variant(Trho);{12} is the Debye length, we find xi_{Z,infinity}<xi_{D}<xi_{L} , although, since ion-pairing is neglected, the charge oscillations set in only for densities approximately 1.9 times larger than predicted.

Ising analogue to compact-star matter

By constructing an Ising analogue of compact-star matter at subsaturation density we explored the effect of Coulomb frustration on the nuclear liquid-gas phase transition. Our conclusion is twofold. First, the range of temperatures where inhomogeneous phases form expands with increasing Coulomb-field strength. Second, within the approximation of uniform electron distribution, the limiting point upon which the phase-coexistence region ends does not exhibit any critical behavior. Possible astrophysics consequences and thermodynamical connections are discussed.

Planar electric double layer for a restricted primitive model electrolyte at low temperatures

Monte Carlo simulation and the modified Poisson-Boltzmann theory are used to investigate the planar electric double layer for a restricted primitive model electrolyte at low temperatures. Capacitance as a function of temperature at low surface charge is determined for 1:1, 2:2, 2:1, and 3:1 electrolytes. Negative adsorption can occur for 1:1 electrolytes at low surface charge with low electrolyte concentration. The 1:1 electrolyte diffuse layer potential as a function of surface charge displays a maximum at low densities. At high densities, the diffuse layer potential is negative with a negative slope. The Gouy-Chapman-Stern theory fails in this low-temperature regime, whereas the modified Poisson-Boltzmann theory is fairly successful in this regard.

Static and dynamic critical behavior of a symmetrical binary fluid: A computer simulation

A symmetrical binary, A+B Lennard-Jones mixture is studied by a combination of semi-grand-canonical Monte Carlo (SGMC) and molecular dynamics (MD) methods near a liquid-liquid critical temperature T(c). Choosing equal chemical potentials for the two species, the SGMC switches identities (A-->B-->A) to generate well-equilibrated configurations of the system on the coexistence curve for T<T(c) and at the critical concentration, x(c)=12, for T>T(c). A finite-size scaling analysis of the concentration susceptibility above T(c) and of the order parameter below T(c) is performed, varying the number of particles from N=400 to 12 800. The data are fully compatible with the expected critical exponents of the three-dimensional Ising universality class. The equilibrium configurations from the SGMC runs are used as initial states for microcanonical MD runs, from which transport coefficients are extracted. Self-diffusion coefficients are obtained from the Einstein relation, while the interdiffusion coefficient and the shear viscosity are estimated from Green-Kubo expressions. As expected, the self-diffusion constant does not display a detectable critical anomaly. With appropriate finite-size scaling analysis, we show that the simulation data for the shear viscosity and the mutual diffusion constant are quite consistent both with the theoretically predicted behavior, including the critical exponents and amplitudes, and with the most accurate experimental evidence.

Critical dynamics in a binary fluid: simulations and finite-size scaling

We report comprehensive simulations of the critical dynamics of a symmetric binary Lennard-Jones mixture near its consolute point. The self-diffusion coefficient exhibits no detectable anomaly. The data for the shear viscosity and the mutual-diffusion coefficient are fully consistent with the asymptotic power laws and amplitudes predicted by renormalization-group and mode-coupling theories provided finite-size effects and the background contribution to the relevant Onsager coefficient are suitably accounted for. This resolves a controversy raised by recent molecular simulations.

Phase behavior of the lattice restricted primitive model with nearest neighbor exclusion

The global phase behavior of the lattice restricted primitive model with nearest neighbor exclusion has been studied by grand canonical Monte Carlo simulations. The phase diagram is dominated by a fluid (or charge-disordered solid) to charge-ordered solid transition that terminates at the maximum density rho*(max)= sqrt 2 and reduced temperature T* approximately equal to 0.29. At that point, there is a first-order phase transition between two phases of the same density, one charge-ordered, and the other charge-disordered. The liquid-vapor transition for the model is metastable, lying entirely within the fluid-solid phase envelope.

Size-asymmetric primitive model at low temperature: description of ion pairing and location of the critical point

We argue that Bjerrum's approach to ion pairing is inappropriate for the size-asymmetric primitive model in the neighborhood of its critical point, and propose a new approach based on the Stillinger-Lovett pairing procedure. The new approach recursively scales up the ion size until linear approximations are suitable for analyzing such a model. To locate the critical point, a residual van der Waals interaction between pairs is added, with an energy cutoff adjusted to match the critical temperature of the restricted primitive model. The locations and downward trends of T(c) and rho(c) with asymmetry are found to compare favorably with simulations.

Universality of ionic criticality: size- and charge-asymmetric electrolytes

Grand-canonical simulations designed to resolve critical universality classes are reported for z:1 hard-core electrolyte models with diameter ratios lambda=a+/a- less than or approximately equal 6. For z=1 Ising-type behavior prevails. Unbiased estimates of Tc(lambda) are within 1% of previous (biased) estimates but the critical densities are approximately 5% lower. Ising character is also established for the 2:1 and 3:1 equisized models, along with critical amplitudes and improved Tc estimates. For z=3, however, strong finite-size effects reduce the confidence level although classical and O (n>or=3) criticality are excluded.

Criticality in charge-asymmetric hard-sphere ionic fluids

Phase separation and criticality are analyzed in z:1 charge-asymmetric ionic fluids of equisized hard spheres by generalizing the Debye-Hückel approach combined with ionic association, cluster solvation by charged ions, and hard-core interactions, following lines developed by Fisher and Levin for the 1:1 case (i.e., the restricted primitive model). Explicit analytical calculations for 2:1 and 3:1 systems account for ionic association into dimers, trimers, and tetramers and subsequent multipolar cluster solvation. The reduced critical temperatures, Tc* (normalized by z), decrease with charge asymmetry, while the critical densities increase rapidly with . The results compare favorably with simulations and represent a distinct improvement over all current theories such as the mean spherical approximation, symmetric Poisson-Boltzmann theory, etc. For z not equal to 1, the interphase Galvani (or absolute electrostatic) potential difference, Deltaphi(T), between coexisting liquid and vapor phases is calculated and found to vanish as absolute value (T-Tc) beta when T-->Tc-with, since our approximations are classical, beta = (1/2). Above Tc, the compressibility maxima and so-called k-inflection loci (which aid the fast and accurate determination of the critical parameters) are found to exhibit a strong z dependence.

Screening in ionic systems: simulations for the Lebowitz length

Simulations of the Lebowitz length, xiL (T, rho), are reported for the restricted primitive model hard-core (diameter a) 1:1 electrolyte for densities rho approximately < 4rho(c) and T(c) approximately < T approximately < 40T(c). Finite-size effects are elucidated for the charge fluctuations in various subdomains that serve to evaluate xiL. On extrapolation to the bulk limit for T approximately > 10T(c) the exact low-density expansions are seen to fail badly when rho > 1/10 rho(c) (with rho(c)a3 approximately = 0.08). At higher densities xiL rises above the Debye length, xiD proportional to square root(T/rho), by 10%-30% (up to rho approximately =1.3rho(c)); the variation is portrayed fairly well by the generalized Debye-Hückel theory. On approaching criticality at fixed rho or fixed T, xiL (T, rho) remains finite with xiL(c) approximately = 0.30a approximately = 1.3xiD(c) but displays a weak entropylike singularity.

How multivalency controls ionic criticality

To understand how multivalency affects criticality in z:1 ionic fluids, we report an ion-cluster association theory embodying ionic solvation and excluded volume for equisized hard-sphere models with z=1-3. In accord with simulation but contradicting integral equation and field theories, the reduced critical temperature falls when z increases while the density rho(c) rises steeply. These trends can be explained semiquantitatively by noting that 80%-90% of the ions near T(c) are bound in neutral or charged clusters, depleting the ionic strength. For z not equal 1, predicted interphase Galvani potentials vanish at T(c).

Thermodynamic properties of lattice hard-sphere models

Thermodynamic properties of several lattice hard-sphere models were obtained from grand canonical histogram- reweighting Monte Carlo simulations. Sphere centers occupy positions on a simple cubic lattice of unit spacing and exclude neighboring sites up to a distance sigma. The nearestneighbor exclusion model, sigma = radical2, was previously found to have a second-order transition. Models with integer values of sigma = 1 or 2 do not have any transitions. Models with sigma = radical3 and sigma = 3 have weak first-order fluid-solid transitions while those with sigma = 2 radical2, 2 radical3, and 3 radical2 have strong fluid-solid transitions. Pressure, chemical potential, and density are reported for all models and compared to the results for the continuum, theoretical predictions, and prior simulations when available.

Critical point of electrolyte mixtures

The critical behavior of electrolyte mixtures was studied using grand canonical Monte Carlo simulations. Mixtures consist of large multivalent macroions and small monovalent co- and counterions. The system can be viewed as a binary mixture of macroions (with their counterions) and salt (co- and counterion pair). The primitive model description was used, in which the ions are point charges with a hard core and the solvent is treated as a uniform dielectric continuum. The grand canonical simulations are based on insertions and removals of neutral molecules: macroion with its counterions or coions and a counterion. We propose a distance biasing method that enables direct grand canonical simulations up to charge asymmetry of 10:1. We calculated the critical loci that connect the salt-free state, which consists of only macroions and counterions, with the pure salt state using mixed-field finite-size scaling with no pressure mixing. The critical parameters are determined for macroion to counterion charge asymmetries of 2:1, 3:1, and 10:1. Our results suggest that binary electrolyte mixtures are type-I mixtures, where the two components mix continuously.

Yang-Yang anomalies and coexistence diameters: simulation of asymmetric fluids

A general method for estimating the Yang-Yang ratio, R(mu) , of a model fluid via Monte Carlo simulations is presented on the basis of data for a hard-core square-well (HCSW) fluid and the restricted primitive model (RPM) electrolyte. The isothermal minima of Q(L)(triple bond)<m(2)>2L/<m(4)>L scaling are evaluated at T(c) in an LxLxL box where m=rho-<rho>L is the density fluctuation. The "complete" finite-size scaling theory for the Q(+/-)(min) (T(c);L) incorporates pressure mixing in the scaling fields, thereby allowing for a Yang-Yang anomaly. It yields a dominant term in the asymmetry, Q(+)(min)-Q(-)(min) , varying as L(-beta/nu) with an amplitude proportional to the crucial pressure-mixing coefficient, j(2) . The reliably known critical order-parameter distribution for (d=3) Ising systems then enables one to estimate j(2) , thereby yielding R(mu) , from the Q minima together with information on the nonuniversal amplitudes for the order parameter and the susceptibility. The detailed analysis needed to estimate j(2) for an HCSW fluid and the RPM is presented. Furthermore, the Q-minima below T(c) can also provide the coexistence-curve diameters, rho(diam) (T) (triple bond)1/2 (rho(+) + rho(-)) , very close to T(c) for both models: here rho +/-(T) are the densities of the coexisting liquid and gas phases. The recently developed recursive scaling algorithm for Deltarho(infinity) (T) (triple bond)rho(+)-rho(-) is adapted to investigate the corresponding universal scaling functions. The two extremal forms of these scaling functions are computed with the aid of the exactly soluble decorated lattice-gas model. The critical densities for the RPM and HCSW fluid found via this route are consistent with previous estimates obtained from the data above T(c) ; the magnitudes of the |T- T(c)|(2beta) and |T- T(c)|(1-alpha) corrections to rho(diam)(T) are estimated.

Phase diagrams in the lattice restricted primitive model: from order-disorder to gas-liquid phase transition

The phase behavior of the lattice restricted primitive model (RPM) for ionic systems with additional short-range nearest neighbor (NN) repulsive interactions has been studied by grand canonical Monte Carlo simulations. We obtain a rich phase behavior as the NN strength is varied. In particular, the phase diagram is very similar to that of the continuum RPM for high NN strength. Specifically, we have found both gas-liquid phase separation, with associated Ising critical point, and a first-order liquid-solid transition. We discuss how the line of continuous order-disorder transitions present for the low NN strength changes into continuum-space behavior as one increases the NN strength and compare our findings with recent theoretical results by Ciach and Stell [Phys. Rev. Lett. 91, 060601 (2003)].

Molecular dynamics simulations of the surface tension of ionic liquids

We report molecular dynamics computer simulations of the surface tension and interfacial thickness of ionic liquid-vapor interfaces modeled with a soft core primitive model potential. We find that the surface tension shows an anomalous oscillatory behavior with interfacial area. This observation is discussed in terms of finite size effects introduced by the periodic boundary conditions employed in computer simulations. Otherwise we show that the thickness of the liquid-vapor interface increases with surface area as predicted by the capillary wave theory. Data on the surface tension of size-asymmetric ionic liquids are reported and compared with experimental data of molten salts. Our data suggest that the surface tensions of size-asymmetric ionic liquids do not follow a corresponding states law.

Convergence of Fine-Lattice Discretization for Near-Critical Fluids

In simulating continuum model fluids that undergo phase separation and criticality, significant gains in computational efficiency may be had by confining the particles to the sites of a lattice of sufficiently fine spacing, a(0) (relative to the particle size, say a). But a cardinal question, investigated here, then arises; namely, How does the choice of the lattice discretization parameter, zeta identical with a/a(0), affect the values of interesting parameters, specifically, critical temperature and density, T(c) and rho(c)? Indeed, for small zeta ( less, similar 4-8) the underlying lattice can strongly influence the thermodynamic properties. A heuristic argument, essentially exact in d = 1 and d = 2 dimensions, indicates that, for models with hard-core potentials, both T(c)(zeta) and rho(c)(zeta) should converge to their continuum limits as 1/zeta((d)(+1)/2) for d </= 3 when zeta --> infinity; but the behavior of the error is highly erratic for d >/= 2. For smoother interaction potentials, the convergence is faster. Exact results for d = 1 models of van der Waals character confirm this; however, an optimal choice of zeta can improve the rate of convergence by a factor 1/zeta. For d >/= 2 models, the convergence of the second virial coefficients to their continuum limits likewise exhibits erratic behavior, which is seen to transfer similarly to T(c) and rho(c); but this can be used in various ways to enhance convergence and improve extrapolation to zeta = infinity as is illustrated using data for the restricted primitive model electrolyte.