Free Volume Approximation Research Articles

Validation study of using the free volume approximation to confined thermotropic and lyotropic liquid-crystalline fluids.

We examined the accuracy of the free volume approximation (FVA) to calculate the isotropic-nematic (IN) transition properties of thermotropic and lyotropic rods between two parallel hard walls. This approximation has been proposed to ease the calculation of the confined systems. It approximates the free energy of the confined particles with a bulk free energy. It predicts a special point for these two types of liquid crystals where the first-order IN transition changes to the second one by decreasing either the temperature, the density or the pore width. This prediction is in contradiction (in spite of some qualitative agreement) with those of the other publications where the authors note that the discontinuous transition terminates at the critical point when the walls are completely impenetrable.

Two-stage kinetic phase transition in a platelet–colloid mixture

The semi-grand canonical ensemble theory augmented by the free volume approximation and the scaled particle theory was applied to construct the Helmholtz free energy density function f of a mixture of uncharged colloidal hard spheres and a depletion agent of colloidal hard platelets. The episode of the phase-separation phenomenon is then described by a composite fm which is written as a sum of the coexisting free energy densities fi (i = gas, liquid or solid) and the latter is weighted by Vi/V, Vi and V being the ith spatial volume and total volume, respectively. In this work, we applied the free energy density minimization method to fm [16] and calculated the domains of phases in coexistence instead of delineating coexisting phase-diagram boundaries obtained by computing the pressure and chemical potential in the conventional thermodynamic equilibrium condition. The calculated coexistence domains thus have the same patterns as many colloidal laboratory experiments. We obtained in our calculated platelet–colloid phase diagram the well-known triangular area of triple-phases coexistence. This area, however, has to be realized as some kind of a kinetic process showing coalescence of two sets of coexisting biphases whenever any set of initial input number concentrations of platelets and colloids that falls inside this triangular area. In this kinetic phase-transition process, we find the triple phases of the platelet–colloid system always residing at the same three vertices of the triangle, but their spatial volumes are generally different depending on the initial number concentrations of platelets and colloids. It would be interesting and, a challenge as well, if laboratory experiments at the same quantitative level as that previously reported for the polymer–colloidal mixtures [7] can be carried out to confirm our theoretical findings.

Free Volume Approximation and Equation of State for the fcc Phase of Polydisperse Hard Spheres

Monte Carlo simulations of the fcc phase of polydisperse hard spheres near close packing are reported. An experimental equation of state (EoS) is determined numerically in the NpT ensemble with variable shape of the periodic box. The close packing volume extrapolated from the obtained data shows a good agreement with earlier experiments performed by other methods. A new theoretical EoS, based on the free volume approximation, is proposed. The modified EoS fits experimental data for polydisperse hard spheres better than the approximation used before.

Elasticity of two-dimensional crystals of polydisperse hard disks near close packing: Surprising behavior of the Poisson's ratio

The equation of state, elastic constants, and Poisson's ratio of a crystalline two-dimensional polydisperse hard disk system were determined in the close packing limit. Monte Carlo simulations in the NpT ensemble with variable shape of the periodic box reveal that the pressure and elastic constants grow with increasing polydispersity. The equation of state and the bulk modulus are well described by the free volume approximation. The latter approximation fails, however, for the shear modulus. The simulations also show that the introduction of any amount of size polydispersity in the hard disk systems causes a discontinuous "jump" of the Poisson's ratio in the close packing limit from the value ν(δ=0) = 0.1308(22), obtained for equidiameter hard disks, to ν(δ>0) ≈ 1, estimated for the polydisperse disks.

Elastic properties of the degenerate crystalline phase of two-dimensional hard dimers

Hard homonuclear dimers, each consisting of two identical and tangent hard disks, exhibit a thermodynamically stable, aperiodic solid phase, known as a degenerate crystal. The elastic properties of this phase are determined by the box fluctuation Monte Carlo method in the NpT ensemble. The computed elastic constants are well described by the free volume approximation and the coefficients of the leading singularities are estimated in the close packing limit. Three mechanisms decreasing Poisson’s ratio are indicated: one is based on increasing the pressure to temperature ratio, another comes from the constrains imposed on the disks forming a dimer, yet another, working in the close packing limit, arises from molecular motions. It is also shown that Poisson’s ratio increases with increasing concentration of vacancies.

Poisson’s ratio of the fcc hard sphere crystal at high densities

Elastic constants and the Poisson ratio of the fcc hard-sphere crystalline phases, free of defects and with vacancies, are determined by two Monte Carlo methods: (i) the analysis of the box fluctuations in the constant pressure ensemble with variable box shape (N-P-T) and (ii) by the free-energy differentiation with respect to deformation in the fixed box ensemble (N-V-T). Very good agreement is observed for the extrapolated to the infinitely large system limit results of both the methods. The coefficients of the leading singularities of the elastic constants near close packing are estimated; they are well described by the free volume approximation. Two mechanisms influencing the Poisson ratio are studied. (i) It is shown that at high densities particle motions decrease the Poisson ratio with respect to the static case which corresponds to zero temperature. Simulations performed for systems of soft spheres, interacting through n-inverse-power potentials, r-n, show that the elastic constants of the hard spheres can be obtained in the limit n-->infinity. When T-->0 the elastic constants of the soft spheres tend to those of the static model. (ii) It is also shown that vacancies decrease C11 and C44 and increase C12 and, hence, increase the Poisson ratio with respect to the defect-free state of the system.

Mixtures of charged colloid and neutral polymer: Influence of electrostatic interactions on demixing and interfacial tension

The equilibrium phase behavior of a binary mixture of charged colloids and neutral, nonadsorbing polymers is studied within free-volume theory. A model mixture of charged hard-sphere macroions and ideal, coarse-grained, effective-sphere polymers is mapped first onto a binary hard-sphere mixture with nonadditive diameters and then onto an effective Asakura-Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954)]. The effective model is defined by a single dimensionless parameter-the ratio of the polymer diameter to the effective colloid diameter. For high salt-to-counterion concentration ratios, a free-volume approximation for the free energy is used to compute the fluid phase diagram, which describes demixing into colloid-rich (liquid) and colloid-poor (vapor) phases. Increasing the range of electrostatic interactions shifts the demixing binodal toward higher polymer concentration, stabilizing the mixture. The enhanced stability is attributed to a weakening of polymer depletion-induced attraction between electrostatically repelling macroions. Comparison with predictions of density-functional theory reveals a corresponding increase in the liquid-vapor interfacial tension. The predicted trends in phase stability are consistent with observed behavior of protein-polysaccharide mixtures in food colloids.

Effects of colloid polydispersity on the phase behavior of colloid-polymer mixtures

We study theoretically the equilibrium phase behavior of a mixture of polydisperse hard-sphere colloids and monodisperse polymers, modeled using the Asakura-Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954)] within the free volume approximation of H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren [Europhys. Lett. 20, 559 (1992)]. We compute full phase diagrams in the plane of colloid and polymer volume fractions, using the moment free energy method. The intricate features of phase separation in pure polydisperse colloids combine with the appearance of polymer-induced gas-liquid coexistence to give a rich variety of phase diagram topologies as the polymer-colloid size ratio xi and the colloid polydispersity delta are varied. Quantitatively, we find that polydispersity disfavors fluid-solid against gas-liquid separation, causing a substantial lowering of the threshold value xi(c) above which stable two-phase gas-liquid coexistence appears. Phase splits involving two or more solids can occur already at low colloid concentration, where they may be kinetically accessible. We also analyze the strength of colloidal size fractionation. When a solid phase separates from a fluid, its polydispersity is reduced most strongly if the phase separation takes place at low colloid concentration and high polymer concentration, in agreement with experimental observations. For fractionation in gas-liquid coexistence we likewise find good agreement with experiment, as well as with perturbative theories for near-monodisperse systems.

Elastic properties of two-dimensional hard disks in the close-packing limit

Elastic constants and the Poisson ratio of defect-free hard-disk solid are determined by two independent methods: (1) analysis of the box side fluctuations in the N-p-T ensemble with variable box shape and (2) numerical differentiation (with respect to strain components) of the free energy computed in the N-V-T ensemble; N, p, V, and T denote the number of particles, the pressure, the volume, and the temperature, respectively. The efficiency of the applied methods is compared. It is shown that reasonable estimates of the elastic properties can be obtained by studying small systems in the N-p-T ensemble and that the singular behavior of the elastic constants near close packing is well described by the free volume approximation; the coefficients of the leading singularities are estimated.

Elastic properties of dense solid phases of hard cyclic pentamers and heptamers in two dimensions.

Systems of model planar, nonconvex, hard-body "molecules" of fivefold and sevenfold symmetry axes are studied by constant pressure Monte Carlo simulations with variable shape of the periodic box. The molecules, referred to as pentamers (heptamers), are composed of five (seven) identical hard disks "atoms" with centers forming regular pentagons (heptagons) of sides equal to the disk diameter. The elastic compliances of defect-free solid phases are computed by analysis of strain fluctuations and the reference (equilibrium) state is determined within the same run in which the elastic properties are computed. Results obtained by using pseudorandom number generators based on the idea proposed by Holian and co-workers [Holian et al., Phys. Rev. E 50, 1607 (1994)] are in good agreement with the results generated by DRAND48. It is shown that singular behavior of the elastic constants near close packing is in agreement with the free volume approximation; the coefficients of the leading singularities are estimated. The simulations prove that the highest density structures of heptamers (in which the molecules cannot rotate) are auxetic, i.e., show negative Poisson ratios.

Free-volume theory of the freezing of polydisperse hard-sphere mixtures: Initial preparation, fractionation, and terminal polydispersity

The freezing of hard-sphere mixtures of arbitrary polydispersity is studied within a van der Waals-type free-volume approximation. The present theory is simple enough to allow for a thorough numerical investigation of all the polydispersity effects on the order–disorder transition of hard spheres. Within this context we have studied the influence on the order–disorder transition of the initial preparation, the subsequent fractionation, and the possible terminal polydispersity. It is found that the order–disorder transition occupies a finite domain of the initial density–initial polydispersity plane and the frontier of this domain is determined. Considerable variation within this domain is found with respect to the initial density, while the influence of the specific form of the initial size-distribution is found to be only marginal.

Free-volume approximations for predicting the phase behaviour of asymmetrichard-sphere mixtures

Two different levels of the free-volume approximation for the calculation of thephase diagram of a asymmetric hard-sphere mixture are compared to directsimulations of the phase behaviour. The mean-field-level free-volume approach,where only the statistically pre-averaged free volume is accounted for, is alreadyin satisfactory agreement with the simulation results. Taking correlations intoaccount within the free-volume approach improves the agreement with the solidbranch of the phase diagram but, remarkably, this version performs somewhatworse for the fluid branch.

The importance of correlations in free-volume models for depletion phenomena

Two different levels of the free-volume approximation for describing one-dimensional mixtures of hard rods of two different lengths are analysed and tested against the exact solution. The mean-field level free-volume approach, where only the statistically pre-averaged free volume of the larger rods is accounted for, yields a spurious phase transition. Taking correlations into account not only gives better quantitative results, it also shows qualitative right physics, i.e. no phase transition.

An exactly solvable model for a colloid–polymer mixture in one-dimension

We consider the equilibrium properties of a simple one-dimensional model for a colloid–polymer mixture in which the colloids are modelled by hard rods and the polymers are treated as ideal particles excluded from the colloids by a certain distance. By mapping the binary mixture to an effective one-component Hamiltonian the model can be solved exactly for size ratios σ p / σ c ≡ q<1, where σ c and σ p are the lengths of colloid and polymer rods, respectively. We compare exact results for the osmotic pressure and free-volume fraction with those of the free-volume approximation, developed in the study of model colloid–polymer mixtures in three dimensions, in order to establish where that approximation is accurate and where it breaks down.

An exactly solvable model for depletion phenomena

The equilibrium properties of a one-dimensional mixture of hard rods of two different lengths are obtained. These are used to test the free-volume approximation, which has in the past been applied in the study of hard-sphere mixtures in three dimensions. Three different versions of the free-volume approximation are tested. In the most primitive of these the smaller spheres (or rods) are assumed not to interact with each other. In higher levels of the approximation the interactions between the smaller molecules are partly taken into account. The results show the ranges of compositions and densities over which the approximations are accurate and they identify one form of the approximation as the most reliable.

On an anomalous kinetic in irradiated polymers around the glass transition temperature

Anomalies occurring in irradiated polymers within the glass transition range, as reported by various authors are critically reviewed. A theoretical description for such processes, is developed within the free volume approximation. The discrepancies between the temperature at which such anomalies are noticed and the glass transition temperature have been ascribed to the fact that the volume of radiation-induced radicals is not equal with the segmental one. The agreement between experiment and theory is good.

The glass transition phenomenon in macromolecular systems

An extended Williams-Landel-Ferry equation is derived within the free volume approximation. This equation relates the jump frequency of an arbitrary molecule (around the glass transition temperature) to the segmental jump frequency at the glass transition temperature. The connection with the usual WLF equation is discussed.

Van der Waals theory of order - disorder transitions

In analogy with the well-known Van der Waals theory for disordered fluid phases we propose a simple analytic expression for the Helmholtz free energy of an ordered crystalline phase. It is based on a free-volume approximation to the entropic contribution due to the hard-core repulsions and a static lattice energy approximation to the contribution of the attractions. In this way we are able to describe in a simple way phase transitions between two disordered phases, between two ordered phases and between an ordered and a disordered phase in systems of any spatial dimensionality.

Spin probe investigation of molecular motions in polyepichlorohydrin: 1

Spin probe investigations on the molecular motions in the glass transition range are reported. Within the free volume approximation, a Fox—Flory dependence of the narrowing temperature on the molecular masses is derived. The experimental results support this theoretical prediction. The glass transition temperature is estimated from the spin probe data, within various approaches, and compared to the results obtained by differential scanning calorimetry.

Calculation of the Grüneisen parameter by the free-volume approximation method

A method for calculating the temperature dependence of the Grüneisen parameter γ of solids in the framework of the free volume approximation is presented. For illustration, the γ values of b.c.c. iron with the Morse potential and f.c.c. neon with the 6–12 Lennard-Jones potential have been computed with the aid of this method. The computed results are in good agreement with those previously evaluated by other methods. Moreover, it is found that the γ values of a lattice depend more strongly on the molecular separation as well as on the curvature of the intermolecular potential well rather than on the dissociation energy of the lattice.