Geometric view of the thermodynamics of adsorption at a line of three-phase contact

We consider three fluid phases meeting at a line of common contact and study the linear excesses per unit length of the contact line (the linear adsorptions Lambda_i) of the fluid's components. In any plane perpendicular to the contact line, the locus of choices for the otherwise arbitrary location of that line that makes one of the linear adsorptions, say Lambda_2, vanish, is a rectangular hyperbola. Two of the adsorptions, Lambda_2 and Lambda_3, then both vanish when the contact line is chosen to pass through any of the intersections of the two corresponding hyperbolas Lambda_2 = 0 and Lambda_3 = 0. There may be two or four such real intersections. It is required, and is confirmed by numerical examples, that a certain expression containing \Lambda_{1(2,3)}, the adsorption of component 1 in a frame of reference in which the adsorptions Lambda_2 and Lambda_3 are both 0, is independent of which of the two or four intersections of Lambda_2 = 0 and Lambda_3 = 0 is chosen for the location of the contact line. That is not true of Lambda_{1(2,3)} by itself; while the adsorptions and the line tension together satisfy a linear analog of the Gibbs adsorption equation, there are additional, not previously anticipated terms in the relation that are required by the line tension's invariance to the arbitrary choice of location of the contact line. The presence of the additional terms is confirmed and their origin clarified in a mean-field density-functional model. The additional terms vanish at a wetting transition, where one of the contact angles goes to 0

Liquid drop in a cone - line tension effects

The shape of a liquid drop placed in a cone is analyzed macroscopically. Depending on the values of the cone opening angle, the Young angle and the line tension four different interfacial configurations may be realized. The phase diagram in these variables is constructed and discussed; it contains both the first- and the second-order transition lines. In particular, the tricritical point is found and the value of the critical exponent characterizing the behaviour of the system along the line of the first-order transitions in the neighbourhood of this point is determined.Comment: 11 pages, 4 figure

Model of Hydrophobic Attraction in Two and Three Dimensions

An earlier one-dimensional lattice model of hydrophobic attraction is extended to two and three dimensions and studied by Monte Carlo simulation. The solvent-mediated contribution to the potential of mean force between hydrophobic solute molecules and the solubility of the solute are determined. As in the earlier model, an inverse relation is observed between the strength and range of the hydrophobic attraction. The mean force no longer varies monotonically with distance, as it does in one dimension, but has some oscillations, reflecting the greater geometrical complexity of the lattice in the higher dimensions. In addition to the strong attraction at short distances, there is now also a local minimum in the potential of depth about $kT$ at a distance of three lattice spacings in two dimensions and one of depth about $2kT$ at a distance of two lattice spacings in three dimensions. The solubility of the solute is found to decrease with increasing temperature at low temperatures, which is another signature of the hydrophobic effect and also agrees with what had been found in the one-dimensional model.Comment: 4 pages, 4 figures, submitted to J. Chem. Phy

Molecular correlations and solvation in simple fluids

We study the molecular correlations in a lattice model of a solution of a low-solubility solute, with emphasis on how the thermodynamics is reflected in the correlation functions. The model is treated in Bethe-Guggenheim approximation, which is exact on a Bethe lattice (Cayley tree). The solution properties are obtained in the limit of infinite dilution of the solute. With $h_{11}(r)$, $h_{12}(r)$, and $h_{22}(r)$ the three pair correlation functions as functions of the separation $r$ (subscripts 1 and 2 referring to solvent and solute, respectively), we find for $r \geq 2$ lattice steps that $h_{22}(r)/h_{12}(r) \equiv h_{12}(r)/h_{11}(r)$. This illustrates a general theorem that holds in the asymptotic limit of infinite $r$. The three correlation functions share a common exponential decay length (correlation length), but when the solubility of the solute is low the amplitude of the decay of $h_{22}(r)$ is much greater than that of $h_{12}(r)$, which in turn is much greater than that of $h_{11}(r)$. As a consequence the amplitude of the decay of $h_{22}(r)$ is enormously greater than that of $h_{11}(r)$. The effective solute-solute attraction then remains discernible at distances at which the solvent molecules are essentially no longer correlated, as found in similar circumstances in an earlier model. The second osmotic virial coefficient is large and negative, as expected. We find that the solvent-mediated part $W(r)$ of the potential of mean force between solutes, evaluated at contact, $r=1$, is related in this model to the Gibbs free energy of solvation at fixed pressure, $\Delta G_p^*$, by $(Z/2) W(1) + \Delta G_p^* \equiv p v_0$, where $Z$ is the coordination number of the lattice, $p$ the pressure, and $v_0$ the volume of the cell associated with each lattice site. A large, positive $\Delta G_p^*$ associated with the low solubility is thus reflected in a strong attraction (large negative $W$ at contact), which is the major contributor to the second osmotic virial coefficient. In this model, the low solubility (large positive $\Delta G_p^*$) is due partly to an unfavorable enthalpy of solvation and partly to an unfavorable solvation entropy, unlike in the hydrophobic effect, where the enthalpy of solvation itself favors high solubility, but is overweighed by the unfavorable solvation entropy.Comment: 9 pages, 2 figure

Line adsorption in a mean-field density functional model

Recent ideas about the analog for a three-phase contact line of the Gibbs adsorption equation for interfaces are illustrated in a mean-field density-functional model. With $d¥tau$ the infinitesimal change in the line tension $¥tau$ that accompanies the infinitesimal changes $d¥mu_i$ in the thermodynamic field variables $¥mu_i$ and with $¥Lambda_i$ the line adsorptions, the sum $d¥tau + ¥Sigma ¥Lambda_i d¥mu_i$, unlike its surface analog, is not 0. An equivalent of this sum in the model system is evaluated numerically and analytically. A general line adsorption equation, which the model results illustrate, is derived.</p

Symmetry effects and equivalences in lattice models of hydrophobic interaction

We establish the equivalence of a recently introduced discrete model of the hydrophobic interaction, as well as its extension to continuous state variables, with the Ising model in a magnetic field with temperature-dependent strength. In order to capture the effect of symmetries of the solvent particles we introduce a generalized multi-state model. We solve this model - which is not of the Ising type - exactly in one dimension. Our findings suggest that a small increase in symmetry decreases the amplitude of the solvent-mediated part of the potential of mean force between solute particles and enhances the solubility in a very simple fashion. High symmetry decreases also the range of the attractive potential. This weakening of the hydrophobic effect observed in the model is in agreement with the notion that the effect is entropic in origin.Comment: 19 pages, 2 figure

Fluctuations in Mass-Action Equilibrium of Protein Binding Networks

We consider two types of fluctuations in the mass-action equilibrium in protein binding networks. The first type is driven by relatively slow changes in total concentrations (copy numbers) of interacting proteins. The second type, to which we refer to as spontaneous, is caused by quickly decaying thermodynamic deviations away from the equilibrium of the system. As such they are amenable to methods of equilibrium statistical mechanics used in our study. We investigate the effects of network connectivity on these fluctuations and compare them to their upper and lower bounds. The collective effects are shown to sometimes lead to large power-law distributed amplification of spontaneous fluctuations as compared to the expectation for isolated dimers. As a consequence of this, the strength of both types of fluctuations is positively correlated with the overall network connectivity of proteins forming the complex. On the other hand, the relative amplitude of fluctuations is negatively correlated with the abundance of the complex. Our general findings are illustrated using a real network of protein-protein interactions in baker's yeast with experimentally determined protein concentrations.Comment: 4 pages, 3 figure

The Dynamics of the One-Dimensional Delta-Function Bose Gas

We give a method to solve the time-dependent Schroedinger equation for a system of one-dimensional bosons interacting via a repulsive delta function potential. The method uses the ideas of Bethe Ansatz but does not use the spectral theory of the associated Hamiltonian