arXiv subjects

Karol Makuch

Publications and source records attributed to Karol Makuch.

15 recordsLinked to original sources

arXiv2025

Diffusion of rod-like particles in complex fluids

Diffusion of particles in complex fluids and gels is difficult to describe and often lies beyond the scope of the classical Stokes-Einstein relation. One of the main lines of research over the past few decades has sought to relate diffusivity to a fundamental dissipative property of the fluid: the wave-vector-dependent shear viscosity function. Here, we use linear response theory to extend this viscosity function framework to rod-like particles. Using a dimer (two-bead particle) as a minimal rod-like probe, we derive explicit expressions for its diffusion coefficients parallel and perpendicular to its axis in terms of the viscosity function. We show that this description captures the full range of behaviors, from nearly isotropic diffusion of the rod-like probe to highly anisotropic, reptation-like motion. The method is based on a microscopic statistical-mechanical treatment of the Smoluchowski dynamics, yet leads to simple final formulas, providing a practical tool for interpreting diffusion experiments on rod-like tracers in complex fluids. We also clarify the limitations of this approach, emphasizing that the present formulation is primarily suited to complex liquids like polymer solutions and only indirectly applicable to gels.

cond-mat.soft↗

arXiv2025

Brownian motion at various length scales with hydrodynamic and direct interactions

Brownian motion is essential for describing diffusion in systems ranging from simple to complex liquids. Unlike simple liquids, which consist of only a solvent, complex liquids, such as colloidal suspensions or the cytoplasm of a cell, are mixtures of various constituents with different shapes and sizes. Describing Brownian motion in such multiscale systems is extremely challenging because direct and many-body hydrodynamic interactions (and their interplay) play a pivotal role. Diffusion of small particles is mainly governed by a low viscous character of the solution, whereas large particles experience a highly viscous flow of the complex liquid on the macro scale. A quantity that encodes hydrodynamics on both length scales is the wave-vector-dependent viscosity. Assuming this quantity to be known -- in contrast to most studies in which the solvent shear viscosity is given -- provides a new perspective on studying the diffusivity of a tracer, especially in situations where the tracer size can vary by several orders of magnitude. Here, we start systematic studies of exact formal microscopic expressions for the short- and long-time self-diffusion coefficients of a single probe particle in a complex liquid in terms of short-ranged hydrodynamic response kernels. We study Brownian motion as a function of the probe size, contrasting most theories that focus on self-diffusion as a function of the crowder volume fraction. We discuss the limits of small and large probe sizes for various levels of approximations in our theory, and discuss the current successes and shortcomings of our approach.

cond-mat.soft↗

arXiv2025

Continuous non-equilibrium transition driven by the heat flow

We discovered an out-of-equilibrium transition in the ideal gas between two walls, divided by an inner, adiabatic, movable wall. The system is driven out-of-equilibrium by supplying energy directly into the volume of the gas. At critical heat flux, we have found a continuous transition to the state with a low-density, hot gas on one side of the movable wall and a dense, cold gas on the other side. Molecular dynamic simulations of the soft-sphere fluid confirm the existence of the transition in the interacting system. We introduce a stationary state Helmholtz-like function whose minimum determines the stable positions of the internal wall. This transition can be used as a paradigm of transitions in stationary states and the Helmholtz-like function as a paradigm of the thermodynamic description of these states.

cond-mat.stat-mech↗

arXiv2024

Tailoring the first law of thermodynamics for convective flows

Equilibrium thermodynamics is grounded in the law of energy conservation, with a specific focus on how systems exchange energy with their environment during transitions between equilibrium states. These transitions are typically characterized by quantities such as heat absorption and the work needed to alter the system's volume. This study is inspired by the potential to develop an analogous, straightforward thermodynamic description for systems that are out of equilibrium. Here, we explore the global energy exchanges that occur during transitions between these nonequilibrium states. We study a system with heat flow and an external (gravity) field that exhibits macroscopic motion, such as Rayleigh-Bénard convection. We show that the formula for system's energy exchange has the same form as in equilibrium. It opens the possibility of describing out-of-equilibrium systems using a few simple laws similar to equilibrium thermodynamics.

cond-mat.stat-mech↗

arXiv2024

Direction of spontaneous processes in non-equilibrium systems with movable/permeable internal walls

The second law of equilibrium thermodynamics explains the direction of spontaneous processes in a system after removing internal constraints. When the system only exchanges energy with the environment as heat, the second law states that spontaneous processes at constant temperature satisfy: $\textrm{d} U - δQ \leq 0$. Here, $\textrm{d} U$ is the infinitesimal change of the internal energy, and $δQ$ is the infinitesimal heat exchanged in the process. We will consider three different systems in a heat flow: ideal gas, van der Waals gas, and a binary mixture of ideal gases. We will also study ideal gas and van der Waals gas in the heat flow and gravitational field. We will divide each system internally into two subsystems by a movable wall. We will show that the direction of the motion of the wall, after release, at constant boundary conditions is determined by the same inequality as in equilibrium thermodynamics. The only difference between equilibrium and non-equilibrium law is the dependence of the net heat change, $δQ$, on the state parameters of the system. We will also consider a wall thick and permeable to gas particles and derive Archimedes' principle in the heat flow. Finally, we will study the ideal gas's Couette flow, where the direction of the motion of the internal wall follows from the inequality $\textrm{d} E - δQ - δW_s \leq 0$, with $\textrm{d} E$ being the infinitesimal change of the total energy (internal and kinetic) and $δW_s$ the infinitesimal work exchanged with the environment due to shear force. Ultimately, we will synthesize all these cases in a framework of the second law of non-equilibrium thermodynamics.

cond-mat.stat-mech↗

arXiv2024

The first law of thermodynamics in hydrodynamic steady and unsteady flows

We studied planar compressible flows of ideal gas as models of a non-equilibrium thermodynamic system. We demonstrate that internal energy $U(S^{*},V,N)$ of such systems in stationary and non-stationary states is the function of only three parameters of state, i.e. non-equilibrium entropy $S^{*}$, volume $V$ and number of particles $N$ in the system. Upon transition between different states, the system obeys the first thermodynamic law, i.e. $dU=T^{*}dS^{*}-p^{*}dV+μ^{*}dN$, where $U=3/2 NRT^{*}$ and $p^{*}V=NRT^{*}$. Placing a cylinder inside the channel, we find that U depends on the location of the cylinder $y_{c}$ only via the parameters of state, i.e. $U(S^{*}(y_{c}),V,N(y_{c}))$ at V=const. Moreover, when the flow around the cylinder becomes unstable, and velocity, pressure, and density start to oscillate as a function of time, t, U depends on t only via the parameters of state, i.e. $U(S^{*}(t),V,N(t))$ for V=const. These examples show that such a form of internal energy is robust and does not depend on the particular boundary conditions even in the unsteady flow.

cond-mat.stat-mech↗

arXiv2023

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.

cond-mat.stat-mech↗

arXiv2023

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.

cond-mat.stat-mech↗

arXiv2023

Steady thermodynamic fundamental relation for the interacting system in a heat flow

There is a long-standing question of whether it is possible to extend the formalism of equilibrium thermodynamics to the case of non-equilibrium systems in steady states. We have made such an extension for an ideal gas in a heat flow [Hołyst \emph{et al.}, J. Chem. Phys. 157, 194108 (2022)]. Here we investigate whether such a description exists for the system with interactions: the Van der Waals gas in a heat flow. We introduce the parameters of state, each associated with a single way of changing energy. The first law of non-equilibrium thermodynamics follows from these parameters. The internal energy $U$ for the non-equilibrium states has the same form as in equilibrium thermodynamics. For the Van der Waals gas, $U(S^*, V, N, a^*,b^* )$ is a function of only 5 parameters of state (irrespective of the number of parameters characterizing the boundary conditions): the entropy $S^*$, volume $V$, number of particles $N$, and the rescaled Van der Waals parameters $a^*$, $b^*$. The state parameters, $a^*$, $b^*$, together with $S^*$, determine the net heat exchange with the environment.

cond-mat.stat-mech↗

arXiv2022

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.

cond-mat.stat-mech↗

arXiv2021

Internal energy in compressible Poiseuille flow

We analyse a compressible Poiseuille flow of ideal gas in a plane channel. We provide the form of internal energy U for a non-equilibrium stationary state (NESS) that includes viscous dissipation and pressure work. We demonstrate that U depends strongly on the ratio Δp/p_0, where Δp is the pressure difference between inlet and outlet and p_0 is the outlet's pressure. In addition, U depends on two other variables: the channel aspect ratio and the parameter equivalent to Reynolds number. The stored internal energy, ΔU=U-U0, is small compared to the internal energy U0 of the equilibrium state (ES) for a moderate range of values of Δp/p_0. However, ΔU can become large for big Δp or close to vacuum conditions at the outlet (p_0~0 Pa).

physics.flu-dyn↗

arXiv2015

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 $ ϕ\approx 30\% $ (apart from a small range of wave vectors in hydrodynamic function). For volume fractions above $ϕ\approx 30 \%$, the Beenakker-Mazur method gives in most cases lower error, than the renormalized Clausius-Mossotti approximation.

cond-mat.soft↗

arXiv2015

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.

cond-mat.soft↗

arXiv2015

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.

math-ph↗

arXiv2012

Scattering series in mobility problem for suspensions

The mobility problem for suspension of spherical particles immersed in an arbitrary flow of a viscous, incompressible fluid is considered in the regime of low Reynolds numbers. The scattering series which appears in the mobility problem is simplified. The simplification relies on the reduction of the number of types of single-particle scattering operators appearing in the scattering series. In our formulation there is only one type of single-particle scattering operator.

cond-mat.stat-mech↗