U. M. Titulaer

The kinetic boundary layer around an absorbing sphere and the growth of small droplets

Deviations from the classical Smoluchowski expression for the growth rate of a droplet in a supersaturated vapor can be expected when the droplet radius is not large compared to the mean free path of a vapor molecule. The growth rate then depends significantly on the structure of the kinetic boundary layer around a sphere. We consider this kinetic boundary layer for a dilute system of Brownian particles. For this system a large class of boundary layer problems for a planar wall have been solved. We show how the spherical boundary layer can be treated by a perturbation expansion in the reciprocal droplet radius. In each order one has to solve a finite number ofplanar boundary layer problems. The first two corrections to the planar problem are calculated explicitly. For radii down to about two velocity persistence lengths (the analog of the mean free path for a Brownian particle) the successive approximations for the growth rate agree to within a few percent. A reasonable estimate of the growth rate for all radii can be obtained by extrapolating toward the exactly known value at zero radius. Kinetic boundary layer effects increase the time needed for growth from 0 to 10 (or 2 1/2) velocity persistence lengths by roughly 35% (or 175%).

The moment method for boundary layer problems in Brownian motion theory

We apply Grads moment method, with Hermite moments and Marshak-type boundary conditions, to several boundary layer problems for the Klein-Kramers equation, the kinetic equation for noninteracting Brownian particles, and study its convergence properties as the number of moments is increased. The errors in various quantities of physical interest decrease asymptotically as inverse powers of this number; the exponent is roughly three times as large as in an earlier variational method, based on an expansion in the exact boundary layer eigenfunctions. For the case of a fully absorbing wall (the Milne problem) we obtain full agreement with the recent exact solution of Marshall and Watson; the relevant slip coefficient, the Milne length, is reproduced with an accuracy better than 10−6. We also consider partially absorbing walls, with specular or diffuse reflection of nonabsorbed particles. In the latter case we allow for a temperature difference between the wall and the medium in which the particles move. There is noa priori reason why our method should work only for Brownian dynamics; one may hope to extend it to a broad class of linear transport equations. As a first test, we looked at the Milne problem for the BGK equation. In spite of the completely different analytic structure of the boundary layer eigenfunctions, the agreement with the exact solution is almost as good as for the Klein-Kramers equation.

Onsager-Casimir symmetry properties of the Burnett equations

We demonstrate the approximate nature of the Onsager-Casimir relations for the example of the linearized Burnett equations for a dilute gas. For any discussion of Onsager relations the choice of a correct set of thermodynamic forces and fluxes is of course crucial. By retracing the Chapman-Enskog procedure, we show that the usual expressions for the thermodynamic forces require modifications at the Burnett level. However, inclusion of these terms does not remedy the violation of Onsager symmetry first noticed by McLennan. A modified version of the Onsager symmetry that involves thermodynamic forces derived from an entropy Lagrangian rather than from the entropy itself does remain valid on the Burnett level. Throughout, we allow for the presence of an external potential; the Burnett equations including potential terms are derived in an appendix for a set of variables particularly suited for our discussion. We stress that in discussing Onsager relations one should use the full thermodynamic fluxes rather than their dissipative parts only, in spite of the fact that only the latter contribute to the entropy production.

The kinetic boundary layer for the linearized Boltzmann equation around an absorbing sphere

We construct approximate solutions of the linearized Boltzmann equation for a gas outside of a completely absorbing sphere, a simple model for a liquid droplet growing in a supersaturated vapor. The solutions are linear combinations of two Chapman-Enskog-type solutions, which carry heat and particle currents, and boundary layer eigenfunctions that decay with increasing distance from the sphere on a distance of the order of a mean free path. To construct the boundary layer eigenfunctions and the linear combination that satisfies the boundary condition at the sphere, we expand the solution in Burnett functions and truncate the resulting system of equations for the expansion coefficients. For one particular truncation prescription, which generalizes Grads 13-moment scheme, good initial convergence with increasing order of truncation is obtained for both moderately small and large radii of the sphere; the results for small radii extrapolate smoothly toward the known limit of zero radius. We present results for the reaction rate (the particle current arriving at the sphere divided by the density at infinity) and for the density and temperature profiles in the boundary layer. The explicit calculations are carried out for Maxwell molecules, but the method appears to be suitable for more general intermolecular potentials.

Force-Induced Lysozyme—HyHEL5 Antibody Dissociation and Its Analysis by Means of a Cooperative Binding Model

Dynamic force spectroscopy probes the kinetic properties of molecules interacting with each other such as antibody-antigen, receptor-ligand, etc. In this article, a statistical model for the dissociation of such cooperative systems is presented. The partner molecules are assumed to be linked by a number of relatively weak bonds that can be grouped together into cooperative units. Single bonds are assumed to open and close statistically. Our model was used to analyze molecular recognition experiments of single receptor-ligand pairs in which the two molecules are brought into contact using an atomic force microscope, which leads to the formation of a strong and specific bond. Then a prescribed time-dependent force is applied to the complex and the statistical distribution of forces needed to pull the molecules completely apart is measured. This quantity is also calculated from our model. Furthermore, its dependence on the model parameters, such as binding free energy, number of bonds and groups, number of cooperative elementary bonds and degree of cooperativity within a group, influence of the force on the binding free energy, and the rate of change of the pulling force, is determined.

Kinetic Boundary Layers in Gas Mixtures: Systems Described by Nonlinearly Coupled Kinetic and Hydrodynamic Equations and Applications to Droplet Condensation and Evaporation

We consider a mixture of heavy vapor molecules and a light carrier gas surrounding a liquid droplet. The vapor is described by a variant of the Klein-Kramers equation, a kinetic equation for Brownian particles moving in a spatially inhomogeneous background; the gas is described by the Navier-Stokes equations; the droplet acts as a heat source due to the released heat of condensation. The exchange of momentum and energy between the constituents of the mixture is taken into account by force terms in the kinetic equation and source terms in the Navier-Stokes equations. These are chosen to obtain maximal agreement with the irreversible thermodynamics of a gas mixture. The structure of the kinetic boundary layer around the sphere is then determined from the self-consistent solution of this set of coupled equations with appropriate boundary conditions at the surface of the sphere. For this purpose the kinetic equation is rewritten as a set of coupled moment equations. A complete set of solutions of these moment equations is constructed by numerical integration inward from the region far away from the droplet, where the background inhomogeneities are small. A technique developed in an earlier paper is used to deal with the severe numerical instability of the moment equations. The solutions so obtained for given temperature and pressure profiles in the gas are then combined linearly in such a way that they obey the boundary conditions at the droplet surface; from this solution source terms for the Navier-Stokes equation of the gas are constructed and used to determine improved temperature and pressure profiles for the background gas. For not too large temperature differences between the droplet and the gas at infinity, self-consistency is reached after a few iterations. The method is applied to the condensation of droplets from a supersaturated vapor, where small but significant corrections to an earlier, not fully consistent version of the theory are found, as well as to strong evaporation of droplets under the influence of an external heat source, where corrections of up to 40 % are obtained.

A kinetic model for droplet growth in the transition regime

A variant of the moment expansion method, used in an earlier paper to describe the flow of a gas toward an absorbing sphere, is applied to a more realistic model of a droplet condensing from a supersaturated vapor. In the simplest version a spherical droplet absorbs all incoming vapor molecules, but spontaneously emits molecules with a Maxwellian distribution at the droplet temperature and with the corresponding saturated vapor density. From a solution of the stationary linearized Boltzmann equation with these boundary conditions we obtain expressions for the heat and mass currents toward the sphere as a function of the supersaturation and the temperature difference between the droplet and the vapor at infinity. For small droplet radii the known free flow limit is obtained in a natural way. From the calculated expressions for the heat and mass current we derive evolution equations for the radius and temperature of the droplet. The temperature evolves more rapidly and can thus be eliminated adiabatically; the resulting growth curve for the radius shows a sharp transition from a kinetically controlled regime for small radii to a regime dominated by heat conduction for large radii. The effect of incomplete absorption at the surface is also studied. The actual calculations are carried out for Maxwell molecules, with parameters corresponding to argon at 0.65Tcand 100% supersaturation.

The Onsager-Casimir relations revisited

We study the fate of the Onsager-Casimir reciprocity relations for a continuous system when some of its variables are eliminated adiabatically. Just as for discrete systems, deviations appear in correction terms to the reduced evolution equation that are of higher order in the time scale ratio. The deviations are not removed by including correction terms to the coarse-grained thermodynamic potential. However, via a reformulation of the theory, in which the central role of the thermodynamic potential is taken over by an associated Lagrangian-type expression, we arrive at a modified form of the Onsager-Casimir relations that survives the adiabatic elimination procedure. There is a simple relation between the time evolution of the redefined thermodynamic forces and that of the basic thermodynamic variables; this relation also survives the adiabatic elimination. The formalism is illustrated by explicit calculations for the Klein-Kramers equation, which describes the phase space distribution of Brownian particles, and for the corrected Smoluchowski equation derived from it by adiabatic elimination of the velocity variable. The symmetry relation for the latter leads to a simple proof that the reality of the eigenvalues of the simple Smoluchowski equation is not destroyed by the addition of higher order corrections, at least not within the framework of a formal perturbation expansion in the time scale ratio.

The effect of a magnetic field on the recombination of a radical pair

Abstract We analyze the exact solution of the recombination probability of a radical pair with the Δg mechanism for the spin dynamics and diffusion for translational motion. This reproduces the B 12 dependence of the recombination probability in a small magnetic field and the saturation effect in a strong magnetic field.

The Influence of Temperature Gradients on the Kinetic Boundary Layer Problem for a Condensing Droplet

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state; the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.