Widom model

As early as 1969, Wlieeler and Widom [73] fomuilated a simple lattice model to describe ternary mixtures. The bonds between lattice sites are conceived as particles. A bond between two positive spins corresponds to water, a bond between two negative spins corresponds to oil and a bond coimecting opposite spins is identified with an amphiphile. The contact between hydrophilic and hydrophobic units is made infinitely repulsive hence each lattice site is occupied by eitlier hydrophilic or hydrophobic units. These two states of a site are described by a spin variable s., which can take the values +1 and -1. Obviously, oil/water interfaces are always completely covered by amphiphilic molecules. The Hamiltonian of this Widom model takes the form... 

Even though the basic idea of the Widom model is certainly very appealing, the fact that it ignores the possibihty that oil/water interfaces are not saturated with amphiphiles is a disadvantage in some respect. The influence of the amphiphiles on interfacial properties cannot be studied in principle in particular, the reduction of the interfacial tension cannot be calculated. In a sense, the Widom model is not only the first microscopic lattice model, but also the first random interface model configurations are described entirely by the conformations of their amphiphilic sheets. 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

Ginzburg-Landau models can be derived in a straightforward way from all microscopic lattice models of microemulsions. This has been done explicitly for the Widom model [43], for the three-component model [44], for vector models [45], and for the charge-frustrated Ising model [37]. In the case of the three-component model of Eqs. (2) and (3), the derivation shows, for example, that... 

The gradient model has been combined with two equations of state to successfully model the temperature dependence of the surface tension of polar and nonpolar fluids [54]. Widom and Tavan have modeled the surface tension of liquid He near the X transition with a modified van der Waals theory [55]. 

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

Stillinger F 1973 Structure in aqueous solutions from the standpoint of scaled particle theory J. Solution Chem. 2 141 Widom B 1967 Intermolecular forces and the nature of the liquid state Sc/e/ ce 375 157 Longuet-Higgins H C and Widom B 1964 A rigid sphere model for the melting of argon Mol. Phys. 8 549... 

Phase transitions in two-dimensional layers often have very interesting and surprising features. The phase diagram of the multicomponent Widom-Rowhnson model with purely repulsive interactions contains a nontrivial phase where only one of the sublattices is preferentially occupied. Fluids and molecules adsorbed on substrate surfaces often have phase transitions at low temperatures where quantum effects have to be considered. Examples are molecular layers of H2, D2, N2 and CO molecules on graphite substrates. We review the path integral Monte Carlo (PIMC) approach to such phenomena, clarify certain experimentally observed anomalies in H2 and D2 layers, and give predictions for the order of the N2 herringbone transition. Dynamical quantum phenomena in fluids are analyzed via PIMC as well. Comparisons with the results of approximate analytical theories demonstrate the importance of the PIMC approach to phase transitions where quantum effects play a role. 

In 1970 Widom and Rowlinson (WR) introduced an ingeniously simple model for the study of phase transitions in fluids [185]. It consists of two species of particles, A and B, in which the only interaction is a hard core between particles of unlike species i.e., the pair potential v jsir) is inflnite if a P and r < and is zero otherwise. WR assumed an A-B demixing phase transition to occur in dimensions D >2 when the fugacity... 

B. Widom. Lattice model of microemulsions. J Chem Phys 54 6943-6954, 1986. 

Beck, T. L., Quantum path integral extension of Widom s test particle method for chemical potentials with application to isotope effects on hydrogen solubilities in model solids, J. Chem. Phys. 1992, 96, 7175-7177... 

Polach, K.J. and Widom, J. (1995) Mechanism of protein access to specific DNA sequences in chromatin a dynamic equilibrium model for gene regulation. J. Mol. Biol. 254, 130-149. 

Widom, J. (1989) Toward a unified model of chromatin folding. Atmu. Rev. Biophys. Biophys. 

Barkema, G. T., Marko.J. F., and Widom, B. (1994). Electrophoresis of charged polymers Simulation and scaling in a lattice model of reptation. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 49(6), 5303—5309. 

Widom [9] realized the importance of this problem for statistical mechanics and showed that the centers of the particles of a hard disk gas, in an equilibrium position, are not uniformly random distributed. The available area for a nevt particle power series in particle density 6 = Nnr2/A, where N is the number of adsorbed panicles, r their radius and A the total area of the surface. The coefficients of the series terms are identical up to the second power of 9 for the equilibrium and the RSA models. The differences in the higher powers coefficients lead for RSA to jamming for Op = 0.76, 0.547 and 0.38 for the ID (segments on a line), 2D (disks on a surface) and 3D (spheres on a volume), respectively, while for the equilibrium configurations the close-packing occurs at 9 = 1, 0.91 and 0.74, respectively. 

Schaaf and Talbot [15] continued Widom s analytical approach by calculating the available area for the RSA of disks and obtained the coefficient of fZ3, which is different in RSA and equilibrium models. The next coefficient was obtained independently by Dickman et al. [16] and Given [17]. The first five terms of the series are not, however, very useful for the calculation of the jamming point Op. Indeed, using the five known terms, there is no jamming, because analytical continuations based on Pade approximants P[i,j] pro-... 

Widom has recently formulated a lattice model that takes into account the amphiphilic nature of surfactant and introduces molecular interactions that explicitly affect curvature of surfactant sheetlike structures [25]. 

The PDT that is a central feature of this book dates from this period (Widom, 1963 Jackson and Klein, 1964), as does the related but separately developed scaled-particle theory (Reiss et al, 1959). Both the PDT and scaled-particle approaches have been somewhat bypassed as features of molecular theory, in contrast to their evident utility in simulation and engineering applications. Scaled-particle theories have been helpful in the development of sophisticated solution models (Ashbaugh and Pratt, 2004). Yet the scaled-particle results have been almost orthogonal to pedagogical presentations of the theory of liquids. This may be due to the specialization of the presentations of scaled-particle theory (Barrat and Hansen, 2003). 

A conceptually complementary approach to describe hydrophobic effects has been introduced by Pratt and colleagues (78, 96). Their iifformation theory (IT) model is based on an application of Widom s potential distribution theorem (97) combined with the perception that the solvation free energy of a small hard sphere, which is essentially governed by the probability to find an empty sphere, can be expressed as a limit of the distribution of water molecules in a cavity of the size... 

Kasaian MT, Donaldson DD, Tchistiakova L, Marquette K, Tan XY, Ahmed A, Jacobson BA, Widom A, Cook TA, Xu X, Barry AB, Goldman SJ, Abraham WM.. Efficacy of IL-13 neutralization in a sheep model of experimental asthma. Am. J. Respir. Cell. Mol. Biol. 2007 36 368-376. 

Certainly the most important models for the development of modem scaling theory of critical phenomena have been the discrete Ising model of ferromagnetism and its antipode - the continuum van der Waals model of fluid. The widespread belief is that real fluids and the lattice-gas 3D-model belong to the same universality class but the absence of any particle-hole-type symmetry in fluids requires the revised scaling EOS. The mixed variables were introduced to modify the original Widom EOS and account the possible singularity of the rectilinear diameter. 

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