Hydrodynamic resistance matrices of colloidal particles with various shapes

The hydrodynamic resistance matrix is an important quantity for describing the dynamics of colloidal particles. This matrix encodes the shape- and size-dependent hydrodynamic properties of a particle suspended in a simple liquid at low Reynolds number and determines the particle's diffusion tensor. For this reason, the hydrodynamic resistance matrix is typically needed when modeling the motion of free purely Brownian, externally driven, or self-propelled colloidal particles or the behavior of dilute suspensions of such particles on the basis of Langevin equations, Smoluchowski equations, classical dynamical density functional theory, or other appropriate methods. So far, however, the hydrodynamic resistance matrix was available only for a few particle shapes. In this article, we therefore present the hydrodynamic resistance matrices for various particle shapes that are relevant for current research, including apolar and polar as well as convex and partially concave shapes. The elements of the hydrodynamic resistance matrices are given as functions of shape parameters like the aspect ratio of the corresponding particle so that the results apply not only to discrete but instead to continuous sets of particle shapes. This work shall stimulate and support future studies on colloidal particles with anisometric shapes.Comment: 12 pages, 4 figures, 10 table

The self-propelled Brownian spinning top: dynamics of a biaxial swimmer at low Reynolds numbers

Recently, the Brownian dynamics of self-propelled (active) rod-like particles was explored to model the motion of colloidal microswimmers, catalytically-driven nanorods, and bacteria. Here, we generalize this description to biaxial particles with arbitrary shape and derive the corresponding Langevin equation for a self-propelled Brownian spinning top. The biaxial swimmer is exposed to a hydrodynamic Stokes friction force at low Reynolds numbers, to fluctuating random forces and torques as well as to an external and an internal (effective) force and torque. The latter quantities control its self-propulsion. Due to biaxiality and hydrodynamic translational-rotational coupling, the Langevin equation can only be solved numerically. In the special case of an orthotropic particle in the absence of external forces and torques, the noise-free (zero-temperature) trajectory is analytically found to be a circular helix. This trajectory is confirmed numerically to be more complex in the general case involving a transient irregular motion before ending up in a simple periodic motion. By contrast, if the external force vanishes, no transient regime is found and the particle moves on a superhelical trajectory. For orthotropic particles, the noise-averaged trajectory is a generalized concho-spiral. We furthermore study the reduction of the model to two spatial dimensions and classify the noise-free trajectories completely finding circles, straight lines with and without transients, as well as cycloids and arbitrary periodic trajectories.Comment: 13 pages, 4 figures, 2 table

Symmetry-breaking in clogging for oppositely driven particles

The clogging behavior of a symmetric binary mixture of particles that are driven in opposite directions through constrictions is explored by Brownian dynamics simulations and theory. A dynamical state with a spontaneously broken symmetry occurs where one species is flowing and the other is blocked for a long time which can be tailored by the size of the constrictions. Moreover, we find self-organized oscillations in clogging and unclogging of the two species. Apart from statistical physics, our results are of relevance for fields like biology, chemistry, and crowd management, where ions, microparticles, pedestrians, or other particles are driven in opposite directions through constrictions.Comment: 5 pages, 5 figure

Microscopic and macroscopic theories for the dynamics of polar liquid crystals

We derive and analyze the dynamic equations for polar liquid crystals in two spatial dimensions in the framework of classical dynamical density functional theory (DDFT). Translational density variations, polarization, and quadrupolar order are used as order-parameter fields. The results are critically compared with those obtained using the macroscopic approach of time-dependent Ginzburg-Landau (GL) equations for the analogous order-parameter fields. We demonstrate that for both the microscopic DDFT and the macroscopic GL approach the resulting dissipative dynamics can be derived from a dissipation function. We obtain microscopic expressions for all diagonal contributions and for many of the cross-coupling terms emerging from a GL approach. Thus we establish a bridge between molecular correlations and macroscopic modeling for the dissipative dynamics of polar liquid crystals.Comment: 10 page

Stability of liquid crystalline phases in the phase-field-crystal model

The phase-field-crystal model for liquid crystals is solved numerically in two spatial dimensions. This model is formulated with three position-dependent order parameters, namely the reduced translational density, the local nematic order parameter, and the mean local direction of the orientations. The equilibrium free-energy functional involves local powers of the order parameters up to fourth order, gradients of the order parameters up to fourth order, and different couplings between the order parameters. The stable phases of the equilibrium free-energy functional are calculated for various coupling parameters. Among the stable liquid-crystalline states are the isotropic, nematic, columnar, smectic A, and plastic crystalline phases. The plastic crystals can have triangular, square, and honeycomb lattices and exhibit orientational patterns with a complex topology involving a sublattice with topological defects. Phase diagrams were obtained by numerical minimization of the free-energy functional. Their main features are qualitatively in line with much simpler one-mode approximations for the order parameters.Comment: Submitted to Physical Review

Brownian dynamics of a self-propelled particle in shear flow

Brownian dynamics of a self-propelled particle in linear shear flow is studied analytically by solving the Langevin equation and in simulation. The particle has a constant propagation speed along a fluctuating orientation and is additionally subjected to a constant torque. In two spatial dimensions, the mean trajectory and the mean square displacement (MSD) are calculated as functions of time t analytically. In general, the mean trajectories are cycloids that are modified by finite temperature effects. With regard to the MSD different regimes are identified where the MSD scales with t^a with a = 0,1,2,3,4. In particular, an accelerated (a = 4) motion emerges if the particle is self-propelled along the gradient direction of the shear flow.Comment: 6 pages, 4 figure

Extended dynamical density functional theory for colloidal mixtures with temperature gradients

In the past decade, classical dynamical density functional theory (DDFT) has been developed and widely applied to the Brownian dynamics of interacting colloidal particles. One of the possible derivation routes of DDFT from the microscopic dynamics is via the Mori-Zwanzig-Forster projection operator technique with slowly varying variables such as the one-particle density. Here, we use the projection operator approach to extend DDFT into various directions: first, we generalize DDFT toward mixtures of $n$ different species of spherical colloidal particles. We show that there are in general nontrivial cross-coupling terms between the concentration fields and specify them explicitly for colloidal mixtures with pairwise hydrodynamic interactions. Secondly, we treat the energy density as an additional slow variable and derive formal expressions for an extended DDFT containing also the energy density. The latter approach can in principle be applied to colloidal dynamics in a nonzero temperature gradient. For the case without hydrodynamic interactions the diffusion tensor is diagonal, while thermodiffusion -- the dissipative cross-coupling term between energy density and concentration -- is nonzero in this limit. With finite hydrodynamic interactions also cross-diffusion coefficients assume a finite value. We demonstrate that our results for the extended DDFT contain the transport coefficients in the hydrodynamic limit (long wavelengths, low frequencies) as a special case.Comment: 15 pages, 1 figur

Polar liquid crystals in two spatial dimensions: the bridge from microscopic to macroscopic modeling

Two-dimensional polar liquid crystals have been discovered recently in monolayers of anisotropic molecules. Here, we provide a systematic theoretical description of liquid-crystalline phases for polar particles in two spatial dimensions. Starting from microscopic density functional theory, we derive a phase-field-crystal expression for the free-energy density which involves three local order-parameter fields, namely the translational density, the polarization, and the nematic order parameter. Various coupling terms between the order-parameter fields are obtained which are in line with macroscopic considerations. Since the coupling constants are brought into connection with the molecular correlations, we establish a bridge from microscopic to macroscopic modeling. Our theory provides a starting point for further numerical calculations of the stability of polar liquid-crystalline phases and is also relevant for modeling of microswimmers which are intrinsically polar.Comment: 11 page

Derivation of a three-dimensional phase-field-crystal model for liquid crystals from density functional theory

Using a generalized order parameter gradient expansion within density functional theory, we derive a phase-field-crystal model for liquid crystals composed by apolar particles in three spatial dimensions. Both the translational density and the orientational direction and ordering are included as order parameters. Different terms involving gradients in the order parameters in the resulting free energy functional are compared to the macroscopic Ginzburg-Landau approach as well as to the hydrodynamic description for liquid crystals. Our approach provides microscopic expressions for all prefactors in terms of the particle interactions. Our phase-field-crystal model generalizes the conventional phase-field-crystal model of spherical particles to orientational degrees of freedom and can be used as a starting point to explore phase transitions and interfaces for various liquid-crystalline phases.Comment: 7 page

Microscopic approach to entropy production

It is a great challenge of nonequilibrium statistical mechanics to calculate entropy production within a microscopic theory. In the framework of linear irreversible thermodynamics, we combine the Mori-Zwanzig-Forster projection operator technique with the first and second law of thermodynamics to obtain microscopic expressions for the entropy production as well as for the transport equations of the entropy density and its time correlation function. We further present a microscopic derivation of a dissipation functional from which the dissipative dynamics of an extended dynamical density functional theory can be obtained in a formally elegant way.Comment: 10 page