Wetting transitions in polydisperse fluids

The properties of the coexisting bulk gas and liquid phases of a polydisperse fluid depend not only on the prevailing temperature, but also on the overall parent density. As a result, a polydisperse fluid near a wall will exhibit density-driven wetting transitions inside the coexistence region. We propose a likely topology for the wetting phase diagram, which we test using Monte Carlo simulations of a model polydisperse fluid at an attractive wall, tracing the wetting line inside the cloud curve and identifying the relationship to prewetting.Comment: 4 Pages, 4 figures. Accepted for publication in Physical Review Letter

Accurate simulation estimates of cloud points of polydisperse fluids

We describe two distinct approaches to obtaining cloud point densities and coexistence properties of polydisperse fluid mixtures by Monte Carlo simulation within the grand canonical ensemble. The first method determines the chemical potential distribution $\mu(\sigma)$ (with $\sigma$ the polydisperse attribute) under the constraint that the ensemble average of the particle density distribution $\rho(\sigma)$ matches a prescribed parent form. Within the region of phase coexistence (delineated by the cloud curve) this leads to a distribution of the fluctuating overall particle density n, p(n), that necessarily has unequal peak weights in order to satisfy a generalized lever rule. A theoretical analysis shows that as a consequence, finite-size corrections to estimates of coexistence properties are power laws in the system size. The second method assigns $\mu(\sigma)$ such that an equal peak weight criterion is satisfied for p(n)$for all points within the coexistence region. However, since equal volumes of the coexisting phases cannot satisfy the lever rule for the prescribed parent, their relative contributions must be weighted appropriately when determining$\mu(\sigma)$. We show how to ascertain the requisite weight factor operationally. A theoretical analysis of the second method suggests that it leads to finite-size corrections to estimates of coexistence properties which are {\em exponentially small} in the system size. The scaling predictions for both methods are tested via Monte Carlo simulations of a novel polydisperse lattice gas model near its cloud curve, the results showing excellent quantitative agreement with the theory.Comment: 8 pages, 6 figure

Phase behaviour and particle-size cutoff effects in polydisperse fluids

We report a joint simulation and theoretical study of the liquid-vapor phase behaviour of a fluid in which polydispersity in the particle size couples to the strength of the interparticle interactions. Attention is focussed on the case in which the particles diameters are distributed according to a fixed Schulz form with degree of polydispersity $\delta=14$. The coexistence properties of this model are studied using grand canonical ensemble Monte Carlo simulations and moment free energy calculations. We obtain the cloud and shadow curves as well as the daughter phase density distributions and fractional volumes along selected isothermal dilution lines. In contrast to the case of size-{\em independent} interaction strengths (N.B. Wilding, M. Fasolo and P. Sollich, J. Chem. Phys. {\bf 121}, 6887 (2004)), the cloud and shadow curves are found to be well separated, with the critical point lying significantly below the cloud curve maximum. For densities below the critical value, we observe that the phase behaviour is highly sensitive to the choice of upper cutoff on the particle size distribution. We elucidate the origins of this effect in terms of extremely pronounced fractionation effects and discuss the likely appearance of new phases in the limit of very large values of the cutoff.Comment: 12 pages, 15 figure

Liquid-vapor interface of a polydisperse fluid

We report a Grand Canonical Monte Carlo simulation study of the liquid-vapor interface of a model fluid exhibiting polydispersity in terms of the particle size $\sigma$. The bulk density distribution, $\rho^0(\sigma)$, of the system is controlled by the imposed chemical potential distribution $\mu(\sigma)$. We choose the latter such that $\rho^0(\sigma)$ assumes a Schulz form with associated degree of polydispersity $\approx 14$. By introducing a smooth attractive wall, a planar liquid-vapor interface is formed for bulk state points within the region of liquid-vapor coexistence. Owing to fractionation, the pure liquid phase is enriched in large particles, with respect to the coexisting vapor. We investigate how the spatial non-uniformity of the density near the liquid-vapor interface affects the evolution of the local distribution of particle sizes between the limiting pure phase forms. We find (as previously predicted by density functional theory, Bellier-Castella {\em et al}, Phys. Rev. {\bf E65}, 021503 (2002)) a segregation of smaller particles to the interface. The magnitude of this effect is quantified for various $\sigma$ via measurements of the relative adsorption. Additionally, we consider the utility of various estimators for the interfacial width and highlight the difficulties of isolating the intrinsic contribution of polydispersity to this width.Comment: 9 pages, 10 Fig

Polydisperse hard spheres at a hard wall

The structural properties of polydisperse hard spheres in the presence of a hard wall are investigated via Monte Carlo simulation and density functional theory (DFT). Attention is focussed on the local density distribution $\rho(\sigma,z)$, measuring the number density of particles of diameter $\sigma$ at a distance $z$ from the wall. The form of $\rho(\sigma,z)$ is obtained for bulk volume fractions $\eta_b=0.2$ and $\eta_b=0.4$ for two choices of the bulk parent distribution: a top-hat form, which we study for degrees of polydispersity $\delta=11.5$ and $\delta=40.4$, and a truncated Schulz form having $\delta=40.7$. Excellent overall agreement is found between the DFT and simulation results, particularly at $\eta_b=0.2$. A detailed analysis of $\rho(\sigma,z)$ confirms the presence of oscillatory size segregation effects observed in a previous DFT study (Pagonabarraga {\em et al.}, Phys. Rev. Lett. {\bf 84}, 911 (2000)). For large $\delta$, the character of these oscillation is observed to depend strongly on the shape of the parent distribution. In the vicinity of the wall, attractive $\sigma$-dependent depletion interactions are found to greatly enhance the density of the largest particles. The local degree of polydispersity $\delta(z)$ is suppressed in this region, while further from the wall it exhibits oscillations.Comment: 12 pages revte

Modelling credit grade migration in large portfolios using cumulative t-link transition models

For a credit portfolio, we are often interested in modelling the migration of accounts between credit grades over time. For a large retail portfolio, data on credit grade migration may be available only in the form of a series of (typically monthly) population transition matrices representing the gross flow of accounts between each pair of credit grades in the given time period. The challenge is to model the transition process on the basis of these aggregate flow matrices. Each row of an observed transition matrix represents a sample from an ordinal probability distribution. Following [Malik, M. and Thomas, L.C. (2012). Transition matrix models of consumer credit ratings. International Journal of Forecasting, 28, 261-272.], [Feng, D., Gourieroux, C. and Jasiak, J. (2008). The ordered qualitative model for credit rating transitions. Journal of Empirical Finance, 15, 111-130.] and [McNeil, A.J. and Wendin, J.P. (2006). Dependent credit migrations. Journal of Credit Risk, 2, 87-114.], we assume a cumulative link model for these ordinal distributions. Common choices of link function are based on the normal (probit link) or logistic distributions, but the fit to observed data can be poor. In this paper, we investigate the fit of alternative link specifications based on the t-distribution. Such distributions arise naturally when modelling data which arise through aggregating an inhomogeneous sample of obligors, by combining a simple structural-type model for credit migration at the obligor level, with a suitable mixing distribution to model the variability between obligors