Isotropic-nematic spinodals of rigid long thin rodlike colloids by event-driven Brownian dynamics simulations

Naturally derived colloidal rods in microfluidic flows

Naturally derived colloidal rods (CR) are promising building blocks for developing sustainable soft materials. Engineering new materials based on naturally derived CR requires an in-depth understanding of the structural dynamics and self-assembly of CR in dispersion under processing conditions. With the advancement of microfabrication techniques, many microfluidic platforms have been employed to study the structural dynamics of CR under flow. However, each microfluidic design has its pros and cons which need careful evaluation in order to fully meet the experimental goal and correctly interpret the data. We analyze recent results obtained from naturally derived CR and relevant rod-like macromolecules under microfluidic flows, with emphasis on the dynamical behavior in shear- and extensional-dominated flows. We highlight the key concepts required in order to assess and evaluate the results obtained from different CR and microfluidic platforms as a whole and to aid interconnections with neighboring fields. Finally, we identify and discuss areas of interest for future research directions.

Geometric percolation of hard-sphere dispersions in shear flow

We combine a heuristic theory of geometric percolation and the Smoluchowski theory of colloid dynamics to predict the impact of shear flow on the percolation threshold of hard spherical colloidal particles, and verify our findings by means of molecular dynamics simulations. It appears that the impact of shear flow is subtle and highly non-trivial, even in the absence of hydrodynamic interactions between the particles. The presence of shear flow can both increase and decrease the percolation threshold, depending on the criterion used for determining whether or not two particles are connected and on the Péclet number. Our approach opens up a route to quantitatively predict the percolation threshold in nanocomposite materials that, as a rule, are produced under non-equilibrium conditions, making comparison with equilibrium percolation theory tenuous. Our theory can be adapted straightforwardly for application in other types of flow field, and particles of different shape or interacting via other than hard-core potentials.

Scaling law of Brownian rotation in dense hard-rod suspensions

Self-diffusion in dense rod suspensions are subject to strong geometric constraints because of steric interactions. This topological effect is essentially anisotropic when rods are nematically aligned with their neighbors, raising considerable challenges in understanding and analyzing their impacts on the bulk physical properties. Via a classical Doi-Onsager kinetic model with the Maier-Saupe potential, we characterize the long-time rotational Brownian diffusivity for dense suspensions of hard rods of finite aspect ratios, based on quadratic orientation autocorrelation functions. Furthermore, we show that the computed nonmonotonic scalings of the diffusivity as a function of volume fraction can be accurately predicted by an alternative tube model in the nematic phase.

Single-file transport in periodic potentials: The Brownian asymmetric simple exclusion process

Single-file Brownian motion in periodic structures is an important process in nature and technology, which becomes increasingly amenable for experimental investigation under controlled conditions. To explore and understand generic features of this motion, the Brownian asymmetric simple exclusion process (BASEP) was recently introduced. The BASEP refers to diffusion models where hard spheres are driven by a constant drag force through a periodic potential. Here we derive general properties of the rich collective dynamics in the BASEP. Average currents in the steady state change dramatically with the particle size and density. For an open system coupled to particle reservoirs, extremal current principles predict various nonequilibrium phases, which we verify by Brownian dynamics simulations. For general pair interactions we discuss connections to single-file transport by traveling-wave potentials and prove the impossibility of current reversals in systems driven by a constant drag and by traveling waves.

Persistent Anti-Correlations in Brownian Dynamics Simulations of Dense Colloidal Suspensions Revealed by Noise Suppression

Transport properties of a hard-sphere colloidal fluid are investigated by Brownian dynamics simulations. We implement a novel algorithm for the time-dependent velocity-autocorrelation function (VACF) essentially eliminating the noise of the bare random motion. The measured VACF reveals persistent anti-correlations manifested by a negative algebraic power-law tail t^{-5/2} at all densities. At small packing fractions the simulations fully agree with the analytic low-density prediction, yet the amplitude of the tail becomes dramatically suppressed as the packing fraction is increased. The mode-coupling theory of the glass transition provides a qualitative explanation for the strong variation in terms of the static compressibility as well as the slowing down of the structural relaxation.

Translational and rotational diffusion of rod shaped molecules by molecular dynamics simulations

The results of molecular dynamics simulations of the dynamical evolution of assemblies of linear rigid rods of variable aspect ratio, a, and number density, ρ, in the isotropic phase are reported. The rods consist of m equally spaced sites interacting with the Weeks-Chandler-Andersen repulsive pair potential, where 2 < m < 16. With increasing m, features specific to long rods, such as anisotropic self-diffusion, become apparent. There is also an increasing separation between the characteristic relaxation times of the torque, angular velocity, and reorientational time correlation functions with increasing density. The latter is exponential at high densities even for dimers. The isotropic translational diffusion coefficient, D_i, and rotational diffusion coefficient, D_r, are reported as a function of m and ρ or volume fraction, ξ. The mD_i data scale with ξ throughout much of the simulated range, while the rotational diffusion coefficients scale approximately as m³D_r against ρ at low densities but as ∼m⁶D_r at high ρ, consistent with theories of colloidal and noncolloidal rod-containing liquids. The crossover density between the two regimes is parameterized in analytic form. The probability distribution functions for displacements and angular jumps in a given time show evidence of non-Gaussian behavior with increasing density. The shear viscosity and D_i scale approximately as m and m^-1, respectively, in the semidilute regime, which is consistent with a Stokes-Einstein-like relationship. At high concentrations, a frustrated or glassy structure formed in which the rods were randomly oriented.

Dynamically crowded solutions of infinitely thin Brownian needles

We study the dynamics of solutions of infinitely thin needles up to densities deep in the semidilute regime by Brownian dynamics simulations. For high densities, these solutions become strongly entangled and the motion of a needle is essentially restricted to a one-dimensional sliding in a confining tube composed of neighboring needles. From the density-dependent behavior of the orientational and translational diffusion, we extract the long-time transport coefficients and the geometry of the confining tube. The sliding motion within the tube becomes visible in the non-Gaussian parameter of the translational motion as an extended plateau at intermediate times and in the intermediate scattering function as an algebraic decay. This transient dynamic arrest is also corroborated by the local exponent of the mean-square displacements perpendicular to the needle axis. Moreover, the probability distribution of the displacements perpendicular to the needle becomes strongly non-Gaussian; rather, it displays an exponential distribution for large displacements. On the other hand, based on the analysis of higher-order correlations of the orientation we find that the rotational motion becomes diffusive again for strong confinement. At coarse-grained time and length scales, the spatiotemporal dynamics of the needle for the high entanglement is captured by a single freely diffusing phantom needle with long-time transport coefficients obtained from the needle in solution. The time-dependent dynamics of the phantom needle is also assessed analytically in terms of spheroidal wave functions. The dynamic behavior of the needle in solution is found to be identical to needle Lorentz systems, where a tracer needle explores a quenched disordered array of other needles.

Tube Concept for Entangled Stiff Fibers Predicts Their Dynamics in Space and Time

We study dynamically crowded solutions of stiff fibers deep in the semidilute regime, where the motion of a single constituent becomes increasingly confined to a narrow tube. The spatiotemporal dynamics for wave numbers resolving the motion in the confining tube becomes accessible in Brownian dynamics simulations upon employing a geometry-adapted neighbor list. We demonstrate that in such crowded environments the intermediate scattering function, characterizing the motion in space and time, can be predicted quantitatively by simulating a single freely diffusing phantom needle only, yet with very unusual diffusion coefficients.

The Connection between Biaxial Orientation and Shear Thinning for Quasi-Ideal Rods

The complete orientational ordering tensor of quasi-ideal colloidal rods is obtained as a function of shear rate by performing rheo-SANS (rheology with small angle neutron scattering) measurements on isotropic fd-virus suspensions in the two relevant scattering planes, the flow-gradient (1-2) and the flow-vorticity (1-3) plane. Microscopic ordering can be identified as the origin of the observed shear thinning. A qualitative description of the rheological response by Smoluchowski, as well as Doi⁻Edwards⁻Kuzuu theory is possible, as we obtain a master curve for different concentrations, scaling the shear rate with the apparent collective rotational diffusion coefficient. However, the observation suggests that the interdependence of ordering and shear thinning at small shear rates is stronger than predicted. The extracted zero-shear viscosity matches the concentration dependence of the self-diffusion of rods in semi-dilute solutions, while the director tilts close towards the flow direction already at very low shear rates. In contrast, we observe a smaller dependence on the shear rate in the overall ordering at high shear rates, as well as an ever-increasing biaxiality.

Hydrodynamics and Brownian motions of a spheroid near a rigid wall

In this work, we study in detail the hydrodynamics and the Brownian motions of a spheroidal particle suspended in a Newtonian fluid near a flat rigid wall. We employ 3D Finite Element Method (FEM) simulations to compute how the mobility tensor of the spheroid varies with both the particle-wall separation distance and the particle orientation. We then study the Brownian motion of the spheroid by means of a discretized Langevin equation. We specifically focus on the additional drift terms arising from the position and orientational dependence of the mobility matrix. In this respect, we also propose a numerically convenient approximation of the orientational divergence of the mobility matrix that is required in the solution of the Langevin equation. Our results illustrate that both hydrodynamics and Brownian motions of a spheroidal particle near a confining wall display novel features from those of a sphere in the same type of confinement.

DNA origami nanoneedles on freestanding lipid membranes as a tool to observe isotropic-nematic transition in two dimensions

We introduce a simple experimental system to study dynamics of needle-like nanoobjects in two dimensions (2D) as a function of their surface density close to the isotropic-nematic transition. Using fluorescence correlation spectroscopy, we find that translational and rotational diffusion of rigid DNA origami nanoneedles bound to freestanding lipid membranes is strongly suppressed upon an increase in the surface particle density. Our experimental observations show a good agreement with results of Monte Carlo simulations of Brownian hard needles in 2D.

Two-dimensional system of hard ellipses: a molecular dynamics study

We have simulated the dynamics of a two-dimensional system of hard ellipses by event-oriented molecular dynamics in microcanonical NVE ensemble. Various quantities, namely longitudinal and transverse velocity auto-correlation functions, translational and rotational diffusion mean-squared displacements, pressure, intermediate self-scattering function, radial distribution function, and angular spatial correlation, have been obtained and their dependence on packing fraction is characterized. Despite absence of prominent positional ordering, the orientational degree of freedom behaves nontrivially and exhibits interesting features. Slowing down is observed in the angular part of the motion near isotropic-nematic phase transition. It is shown that above a certain packing fraction the rotational mean-squared displacement exhibits a three-stage temporal regime including a plateau. Comparison to 2D system of hard needles is made and it is shown that from positional viewpoint, the ellipse system is more ordered.

Brownian dynamics simulations with hard-body interactions: spherical particles

A novel approach to account for hard-body interactions in (overdamped) Brownian dynamics simulations is proposed for systems with non-vanishing force fields. The scheme exploits the analytically known transition probability for a Brownian particle on a one-dimensional half-line. The motion of a Brownian particle is decomposed into a component that is affected by hard-body interactions and into components that are unaffected. The hard-body interactions are incorporated by replacing the "affected" component of motion by the evolution on a half-line. It is discussed under which circumstances this approach is justified. In particular, the algorithm is developed and formulated for systems with space-fixed obstacles and for systems comprising spherical particles. The validity and justification of the algorithm is investigated numerically by looking at exemplary model systems of soft matter, namely at colloids in flow fields and at protein interactions. Furthermore, a thorough discussion of properties of other heuristic algorithms is carried out.

Event-driven Langevin simulations of hard spheres

The blossoming of interest in colloids and nanoparticles has given renewed impulse to the study of hard-body systems. In particular, hard spheres have become a real test system for theories and experiments. It is therefore necessary to study the complex dynamics of such systems in presence of a solvent; disregarding hydrodynamic interactions, the simplest model is the Langevin equation. Unfortunately, standard algorithms for the numerical integration of the Langevin equation require that interactions are slowly varying during an integration time step. This is not the case for hard-body systems, where there is no clear-cut distinction between the correlation time of the noise and the time scale of the interactions. Starting first from a splitting of the Fokker-Plank operator associated with the Langevin dynamics, and then from an approximation of the two-body Green's function, we introduce and test two algorithms for the simulation of the Langevin dynamics of hard spheres.

Hard-wall interactions in soft matter systems: exact numerical treatment

An algorithm for handling hard-wall interactions in simulations of driven diffusive particle motion is proposed. It exploits an exact expression for the one-dimensional transition probability in the presence of a hard (reflecting) wall and therefore is numerically exact in the sense that it does not introduce any additional approximation beyond the usual discretization procedures. Studying two standard situations from soft matter systems, its performance is compared to the heuristic approaches used in the literature.

Temporal and structural characteristics of a two-dimensional gas of hard needles

We have simulated the dynamics of a 2D gas of hard needles by event-oriented molecular dynamics. Various quantities namely translational and rotational diffusion constants and intermediate self-scattering function have been explored and their dependence on density is obtained. Despite absence of positional ordering, the rotational degree of freedom behaves nontrivially. Slowing down is observed in the angular part of the motion. It is shown that above a certain density the rotational mean-square displacement exhibits a three-stage regime including a plateau.

Nonlinear rheology of active particle suspensions: insights from an analytical approach

We consider active suspensions in the isotropic phase subjected to a shear flow. Using a set of extended hydrodynamic equations we derive a variety of analytical expressions for rheological quantities such as shear viscosity and normal stress differences. In agreement to full-blown numerical calculations and experiments we find a shear-thickening or -thinning behavior depending on whether the particles are contractile or extensile. Moreover, our analytical approach predicts that the normal stress differences can change their sign in contrast to passive suspensions.

Shear-stress-controlled dynamics of nematic complex fluids

Based on a mesoscopic theory we investigate the nonequilibrium dynamics of a sheared nematic liquid, with the control parameter being the shear stress σ xy (rather than the usual shear rate, γ). To this end we supplement the equations of motion for the orientational order parameters by an equation for γ, which then becomes time dependent. Shearing the system from an isotropic state, the stress-controlled flow properties turn out to be essentially identical to those at fixed γ. Pronounced differences occur when the equilibrium state is nematic. Here, shearing at controlled γ yields several nonequilibrium transitions between different dynamic states, including chaotic regimes. The corresponding stress-controlled system has only one transition from a regular periodic into a stationary (shear-aligned) state. The position of this transition in the σ xy-γ plane turns out to be tunable by the delay time entering our control scheme for σ xy. Moreover, a sudden change in the control method can stabilize the chaotic states appearing at fixed γ.

The role played by self-orientational properties in nematics of colloids with molecules axially symmetric

The self-orientational structure factor as well as the short-time self-orientational diffusion coefficient is computed for colloids composed by nonspherical molecules. To compute the short-time dynamics the hydrodynamic interactions are not taken into account. The hard molecules with at least one symmetry axis considered are: rods, spherocylinders, and tetragonal parallelepipeds. Because both orientational properties in study are written in terms of the second and fourth order parameters, these automatically hold the features of the order parameters. That is, they present a discontinuity for first order transitions, determining in this way the spinodal line. In order to analyze the nematic phase only, we choose the appropriate values for the representative quantities that characterize the molecules. Different formalisms are used to compute the structural properties: de Gennes-Landau approach, Smoluchowski equation and computer simulations. Some of the necessary inputs are taken from literature. Our results show that the self-orientational properties play an important role in the characterization and the localization of axially symmetric phases. While the self-structure decreases throughout the nematics, the short-time self-diffusion does not decrease but rather increases. We study the evolution of the second and fourth order parameters; we find different responses for axial and biaxial nematics, predicting the possibility of a biaxial nematics in tetragonal parallelepiped molecules. By considering the second order in the axial-biaxial phase transition, with the support of the self-orientational structure factor, we are able to propose the density at which this occurs. The short-time dynamics is able to predict a different value in the axial and the biaxial phases. Because the different behavior of the fourth order parameter, the diffusion coefficient is lower for a biaxial phase than for an axial one. Therefore the self-structure factor is able to localize continuous phase transitions involving axially symmetric phases and the short-time self-orientational diffusion is able to distinguish the ordered phase by considering the degree of alignment, that is, axial or biaxial.

Self-assembly of three-legged patchy particles into polyhedral cages

The self-assembly of rigid three-legged building blocks into polyhedral cages is investigated by patchy particle simulations. A four-site anisotropic interaction potential is introduced to make pairs of overlapping legs bind in an anti-parallel fashion, thereby forming the edges of a polyhedron of pentagons and hexagons. A torsional potential, reflecting an asymmetry or polarity in the legs' binding potential, proves crucial for the successful formation of closed fullerene-like cages. Self-assembly proceeds by a nucleation-and-growth mechanism, with a high success rate of cage closure. The size distribution of the self-assembled buckyballs is largely determined by the pucker angle of the particle. Nature explores a similar building block, the clathrin triskelion, to regulate vesicle formation at the cell membrane during endocytosis.

The force acting on a polymer partially confined in a tube

We consider the force acting on a polymer part of whose length is configurationally confined in a tube and the rest of which is free. This situation arises in many different physical contexts, including a flexible synthetic polymer partially confined in a nanopore and a stiff viral genome partially ejected from its capsid. In both cases the force acting to pull the chain molecule out of its confinement is argued to be constant once a few persistence lengths are "free"/"outside". We present Brownian dynamics simulations that confirm the constancy of the force for different chain lengths and illustrate the dependence of the force on the strength of tube confinement. Experimental results are reported for genome ejection from viral capsids, from which we estimate the pulling force to be a few tenths of a piconewton.

Isotropic-to-nematic nucleation in suspensions of colloidal rods

Using computer simulations, we study the isotropic-to-nematic nucleation in a fluid of colloidal hard rods as well as in a mixture of colloidal rods and non-adsorbing polymer. In order to follow the transformation of the system from the isotropic to the nematic phase, we use a new cluster criterion that enables us to distinguish the nematic clusters from the isotropic fluid phase. Applying this criterion in Monte Carlo simulations, we find two different regimes depending on the supersaturation. At low supersaturation we find nucleation and growth, while at higher supersaturation spinodal decomposition is observed. We determine the height of the nucleation barrier, and we study the structure as well as the shape of the nematic clusters. We discuss our simulation results in the light of classical nucleation theory.

Polar nano-rods under shear: from equilibrium to chaos

The orientational dynamics of rod-like particles with permanent (electric or magnetic) dipole moments in a plane Couette shear flow is investigated using mesoscopic relaxation equations combined with a generalized Landau free energy. The free energy contribution due to the coupling between average alignment and dipole orientation is derived on a microscopic basis. Numerical results of the resulting eight-dimensional dynamical system are presented for the case of longitudinal dipoles and thermodynamic conditions where the equilibrium state is a (polar or non-polar) nematic. Solution diagrams reveal presence of a large variety of periodic, transient chaotic, and chaotic dynamic states of the average alignment and dipole moment, respectively, appearing as a function of Deborah number and tumbling parameter. Compared to rods without dipoles we observe a significant preference of out-of-plane kayaking-tumbling states and, generally, a higher sensitivity to the initial conditions including bistability. We also demonstrate that the average (electric) dipole moment characterizing most of the observed states yields electrodynamic (magnetic) fields of measurable strength.

Dynamical density functional theory for anisotropic colloidal particles

We generalize the formalism of dynamical density functional theory for translational Brownian dynamics toward that of anisotropic colloidal particles which perform both translational and rotational Brownian motion. Using a mean-field approximation for the density functional and a Gaussian-segment model for the rod interaction, the dynamical density functional theory is then applied to a concentrated rod suspension in a confined slab geometry made by two parallel soft walls. The walls are either expanded or compressed and the relaxation behavior is investigated for an equilibrated starting configuration. We find distinctly different orientational ordering during expansion and compression. During expansion we observe preferential parallel ordering of the rods relative to the wall while during compression there is homeotropic ordering perpendicular to the wall. We find a nonexponential relaxation behavior in time. Furthermore, an external field which aligns the rods perpendicular to the walls is turned on or switched off and similar differences in the relaxational dynamics are found. Comparing the theoretical predictions to Brownian dynamics computer simulation data, we find good agreement.

Event-driven Brownian dynamics for hard spheres

Brownian dynamics algorithms integrate Langevin equations numerically and allow to probe long time scales in simulations. A common requirement for such algorithms is that interactions in the system should vary little during an integration time step; therefore, computational efficiency worsens as the interactions become steeper. In the extreme case of hard-body interactions, standard numerical integrators become ill defined. Several approximate schemes have been invented to handle such cases, but little emphasis has been placed on testing the correctness of the integration scheme. Starting from the two-body Smoluchowski equation, the authors discuss a general method for the overdamped Brownian dynamics of hard spheres, recently developed by one of the authors. They test the accuracy of the algorithm and demonstrate its convergence for a number of analytically tractable test cases.

A computer simulation study of the formation of liquid crystal nanodroplets from a homogeneous solution

The aggregation of liquid crystal nanodroplets from a homogeneous solution is an important but not well understood step in the preparation of various advanced photonic materials. Here, the authors performed molecular dynamics computer simulations of the formation of liquid crystalline nanodroplets, starting from an isotropic and uniform binary solution of spherical Lennard-Jones (solvent) and elongated ellipsoidal Gay-Berne (solute) rigid particles in low (<10%) concentration. They studied the dynamics of demixing and the mesogen ordering process and characterized the resulting nanodroplets assessing the effect of temperature, composition, and specific solute-solvent interaction on the morphology, structure, and anisotropy. They find that the specific solute-solvent interaction, composition, and temperature can be adjusted to tune the nanodroplet growth and size.

Periodic orientational motions of rigid liquid-crystalline polymers in shear flow

The collective periodic motions of liquid-crystalline polymers in a nematic phase in shear flow have, for the first time, been simulated at the particle level by Brownian dynamics simulations. A wide range of parameter space has been scanned by varying the aspect ratio L/D between 10 and 60 at three different scaled volume fractions Lphi/D and an extensive series of shear rates. The influence of the start configuration of the box on the final motion has also been studied. Depending on these parameters, the motion of the director is either characterized as tumbling, kayaking, log-rolling, wagging, or flow-aligning. The periods of kayaking and wagging motions are given by T=4.2(Lphi/D)gamma(-1) for high aspect ratios. Our simulation results are in agreement with theoretical predictions and recent shear experiments on fd viruses in solution. These calculations of elongated rigid rods have become feasible with a newly developed event-driven Brownian dynamics algorithm.