Generalization of Clausius-Mossotti approximation in application to short-time transport properties of suspensions

In 1983 Felderhof, Ford and Cohen gave microscopic explanation of the famous Clausius-Mossotti formula for the dielectric constant of nonpolar dielectric. They based their considerations on the cluster expansion of the dielectric constant, which relates this macroscopic property with the microscopic characteristics of the system. In this article, we analyze the cluster expansion of Felderhof, Ford and Cohen by performing its resummation (renormalization). Our analysis leads to the ring expansion for the macroscopic characteristic of the system, which is an expression alternative to the cluster expansion. Using similarity of structures of the cluster expansion and the ring expansion, we generalize (renormalize) the Clausius-Mossotti approximation. We apply our renormalized Clausius-Mossotti approximation to the case of the short-time transport properties of suspensions, calculating the effective viscosity and the hydrodynamic function with the translational self-diffusion and the collective diffusion coefficient. We perform calculations for monodisperse hard-sphere suspensions in equilibrium with volume fraction up to 45%. To assess the renormalized Clausius-Mossotti approximation, it is compared with numerical simulations and the Beenakker-Mazur method. The results of our renormalized Clausius-Mossotti approximation lead to comparable or much less error (with respect to the numerical simulations), than the Beenakker-Mazur method for the volume fractions below $\phi \approx 30\%$ (apart from a small range of wave vectors in hydrodynamic function). For volume fractions above $\phi \approx 30 \%$, the Beenakker-Mazur method gives in most cases lower error, than the renormalized Clausius-Mossotti approximation

Multipole matrix elements of Green function of Laplace equation

Multipole matrix elements of Green function of Laplace equation are calculated. The multipole matrix elements of Green function in electrostatics describe potential on a sphere which is produced by a charge distributed on the surface of a different (possibly overlapping) sphere of the same radius. The matrix elements are defined by double convolution of two spherical harmonics with the Green function of Laplace equation. The method we use relies on the fact that in the Fourier space the double convolution has simple form. Therefore we calculate the multipole matrix from its Fourier transform. An important part of our considerations is simplification of the three dimensional Fourier transformation of general multipole matrix by its rotational symmetry to the one-dimensional Hankel transformation

Rotational self-diffusion in suspensions of charged particles: Revised Beenakker-Mazur and Pairwise Additivity methods versus numerical simulations

To the present day, the Beenakker-Mazur (BM) method is the most comprehensive statistical physics approach to the calculation of short-time transport properties of colloidal suspensions. A revised version of the BM method with an improved treatment of hydrodynamic interactions is presented and evaluated regarding the rotational short-time self-diffusion coefficient, $D^r$ , of suspensions of charged particles interacting by a hard-sphere plus screened Coulomb (Yukawa) pair potential. To assess the accuracy of the method, elaborate simulations of $D^r$ have been performed, covering a broad range of interaction parameters and particle concentrations. The revised BM method is compared in addition with results by a simplifying pairwise additivity (PA) method in which the hydrodynamic interactions are treated on a two-body level. The static pair correlation functions re- quired as input to both theoretical methods are calculated using the Rogers-Young integral equation scheme. While the revised BM method reproduces the general trends of the simulation results, it systematically and significantly underestimates the rotational diffusion coefficient. The PA method agrees well with the simulation data at lower volume fractions, but at higher concentrations $D^r$ is likewise underestimated. For a fixed value of the pair potential at mean particle distance comparable to the thermal energy, $D^r$ increases strongly with increasing Yukawa potential screening parameter.Comment: 24 pages, 13 figure

Thermodynamics of stationary states of the ideal gas in a heat flow

There is a long-standing question as to whether and to what extent it is possible to describe nonequilibrium systems in stationary states in terms of global thermodynamic functions. The positive answers have been obtained only for isothermal systems or systems with small temperature differences. We formulate thermodynamics of the stationary states of the ideal gas subjected to heat flow in the form of the zeroth, first, and second law. Surprisingly, the formal structure of steady state thermodynamics is the same as in equilibrium thermodynamics. We rigorously show that $U$ satisfies the following equation $dU=T^{*}dS^{*}-pdV$ for a constant number of particles, irrespective of the shape of the container, boundary conditions, size of the system, or mode of heat transfer into the system. We calculate $S^{*}$ and $T^{*}$ explicitly. The theory selects stable nonequilibrium steady states in a multistable system of ideal gas subjected to volumetric heating. It reduces to equilibrium thermodynamics when heat flux goes to zero

Rotational self-diffusion in suspensions of charged particles: simulations and revised Beenakker–Mazur and pairwise additivity methods

We present a comprehensive joint theory-simulation study of rotational self-diffusion in suspensions of charged particles whose interactions are modeled by the generic hard-sphere plus repulsive Yukawa (HSY) pair potential. Elaborate, high-precision simulation results for the short-time rotational self-diffusion coefficient, D^r, are discussed covering a broad range of fluid-phase state points in the HSY model phase diagram. The salient trends in the behavior of D^r as a function of reduced potential strength and range, and particle concentration, are systematically explored and physically explained. The simulation results are further used to assess the performance of two semi-analytic theoretical methods for calculating D^r. The first theoretical method is a revised version of the classical Beenakker–Mazur method (BM) adapted to rotational diffusion which includes a highly improved treatment of the salient many-particle hydrodynamic interactions. The second method is an easy-to-implement pairwise additivity (PA) method in which the hydrodynamic interactions are treated on a full two-body level with lubrication corrections included. The static pair correlation functions required as the only input to both theoretical methods are calculated using the accurate Rogers–Young integral equation scheme. While the revised BM method reproduces the general trends of the simulation results, it significantly underestimates D^r. In contrast, the PA method agrees well with the simulation results for D^r even for intermediately concentrated systems. A simple improvement of the PA method is presented which is applicable for large concentrations

Parameters of state in the global thermodynamics of binary ideal gas mixtures in a stationary heat flow

We formulate the first law of global thermodynamics for stationary states of the binary ideal gas mixture subjected to heat flow. We map the non-uniform system onto the uniform one and show that the internal energy $U(S^*,V,N_1,N_2,f_1^*,f_2^*)$ is the function of the following parameters of state: a non-equilibrium entropy $S^*$, volume $V$, number of particles of the first component, $N_1$, number of particles of the second component $N_2$ and the renormalized degrees of freedom. The parameters $f_1^*,f_2^*$, $N_1, N_2$ satisfy the relation $x_1f_1^*/f_1+x_2f_2^*/f_2=1$ ($f_1$, where $x_i$ is the fraction of $i$ component, and $f_2$ are the degrees of freedom for each component respectively). Thus only 5 parameters of state describe the non-equilibrium state of the binary mixture in the heat flow. We calculate the non-equilibrium entropy $S^{*}$ and new thermodynamic parameters of state $f_1^*, f_2^*$ explicitly. The latter are responsible for heat generation due to the concentration gradients. The theory reduces to equilibrium thermodynamics, when the heat flux goes to zero. As in equilibrium thermodynamics, the steady-state fundamental equation also leads to the thermodynamic Maxwell relations for measurable steady-state properties.Comment: 8 pages, 1 figur

Steady state thermodynamics of ideal gas in shear flow

Equilibrium thermodynamics describes the energy exchange of a body with its environment. Here, we describe the global energy exchange of an ideal gas in the Coutte flow in a thermodynamic-like manner. We derive a fundamental relation between internal energy as a function of parameters of state. We analyze a non-equilibrium transition in the system and postulate the extremum principle, which determines stable stationary states in the system. The steady-state thermodynamic framework resembles equilibrium thermodynamics

Droplet-based digital antibiotic susceptibility screen reveals single-cell clonal heteroresistance in an isogenic bacterial population

Since antibiotic resistance is a major threat to global health, recent observations that the traditional test of minimum inhibitory concentration (MIC) is not informative enough to guide effective antibiotic treatment are alarming. Bacterial heteroresistance, in which seemingly susceptible isogenic bacterial populations contain resistant sub-populations, underlies much of this challenge. To close this gap, here we developed a droplet-based digital MIC screen that constitutes a practical analytical platform for quantifying the single-cell distribution of phenotypic responses to antibiotics, as well as for measuring inoculum effect with high accuracy. We found that antibiotic efficacy is determined by the amount of antibiotic used per bacterial colony forming unit (CFU), not by the absolute antibiotic concentration, as shown by the treatment of beta-lactamase-carrying Escherichia coli with cefotaxime. We also noted that cells exhibited a pronounced clustering phenotype when exposed to near-inhibitory amounts of cefotaxime. Overall, our method facilitates research into the interplay between heteroresistance and antibiotic efficacy, as well as research into the origin and stimulation of heterogeneity by exposure to antibiotics. Due to the absolute bacteria quantification in this digital assay, our method provides a platform for developing reference MIC assays that are robust against inoculum-density variations

Thermodynamics of stationary states of the ideal gas in a heat flow

There is a long-standing question as to whether and to what extent it is possible to describe nonequilibrium systems in stationary states in terms of global thermodynamic functions. The positive answers have been obtained only for isothermal systems or systems with small temperature differences. We formulate thermodynamics of the stationary states of the ideal gas subjected to heat flow in the form of the zeroth, first, and second law. Surprisingly, the formal structure of steady state thermodynamics is the same as in equilibrium thermodynamics. We rigorously show that U satisfies the following equation dU= T* dS* -pdV for a constant number of particles, irrespective of the shape of the container, boundary conditions, size of the system, or mode of heat transfer into the system. We calculate S* and T* explicitly. The theory selects stable nonequilibrium steady states in a multistable system of ideal gas subjected to volumetric heating. It reduces to equilibrium thermodynamics when heat flux goes to zero