The short--time self diffusion coefficient of a sphere in a suspension of rigid rods is calculated in first order in the rod volume fraction. For low rod concentrations the correction to the Einstein diffusion constant of the sphere is a linear function of the rod volume fraction with the slope proportional to the equilibrium averaged mobility diminution trace of the sphere interacting with a single freely translating and rotating rod. The two--body hydrodynamic interactions are calculated using the so--called bead model in which the rod is replaced by a stiff linear chain of touching spheres. The interactions between spheres are calculated numerically using the multipole method. Also an analytical expression for the diffusion coefficient as a function of the rod aspect ratio is derived in the limit of very long rods. We show that in this limit the correction to the Einstein diffusion constant does not depend on the size of the tracer sphere. The higher order corrections depending on the applied model are computed numerically. An approximate expression is provided, valid for a wide range of aspect ratios.
We investigate confined shear thickening suspensions for which the sample thickness is comparable to the particle dimensions. Rheometry measurements are presented for densely packed suspensions of spheres and rods with aspect ratios 6 and 9. By varying the suspension thickness in the direction of the shear gradient at constant shear rate, we find pronounced oscillations in the stress. These oscillations become stronger as the gap size is decreased, and the stress is minimized when the sample thickness becomes commensurate with an integer number of particle layers. Despite this confinement-indu…
Aging in a two dimensional suspension containing rods was studied by means of kinetic Monte Carlo simulations. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The initial state was produced using a random sequential adsorption model. During the aging, the rods underwent translational and rotational Brownian motions. The simulations were run at different values of number density (the number of rods per unit area), $ρ$, and of the initial orientation order parameter, $S_i$. The rods were assumed to have core---shell structures. The evolutions …
We present a study of the isotropic-nematic interface in a system of hard spherocylinders. First we compare results from Monte Carlo simulations and Onsager density functional theory for the interfacial profiles of the orientational order parameter and the density. Those interfacial properties that are not affected by capillary waves are in good agreement, despite the fact that Onsager theory overestimates the coexistence densities. Then we show results of a Monte Carlo study of the capillary waves of the interface. In agreement with recent theoretical investigations (Eur.Phys.J. E {\bf 18} 40…
In order to investigate the coupling between the degrees of alignment of elongated particles in binary nematic dispersions, surfactant stabilized single-wall carbon nanotubes (CNTs) have been added to nematic suspensions of colloidal rodlike viruses in aqueous solution.We have independently measured the orientational order parameter of both components of the guest-host system by means of polarized Raman spectroscopy and by optical birefringence, respectively. Our system allows us therefore to probe the regime where the guest particles (CNTs) are shorter and thinner than the fd virus host parti…
We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the length, the diameter and the connectedness criterion, and invoke bimodal, Gaussian and Weibull distributions for these. The main finding from our simulations is that the percolation threshold shows quasi universal behaviour, i.e., to a good approximation it depends only on certain cumulants of the full size and connectivity distribution. Our connectedness percolation theory hinges on a Lee-Pars…
The sedimentation of a two dimensional suspension containing rods was studied by means of kinetic Monte Carlo (MC) simulations. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The initial state before sedimentation was produced using a model of random sequential adsorption. During such sedimentation, the rods undergo translational and rotational Brownian motions. The MC simulations were run at different initial number densities (the numbers of rods per unit area), $ρ_i$, and sedimentation rates, $u$. For sediment films, the spatial distribut…
There is now convincing evidence that inter-particle frictional contacts are essential for observing shear-thickening in concentrated suspensions of compact particles. While this has inspired many strategies to tailor the rheology in these systems, in the more general case, viz-a-viz suspensions of anisotropic particles, the mechanism of shear-thickening remains unclear. Here through simultaneous measurements of the bulk viscosity and the first Normal stress difference, we show that strong shear thickening in suspensions of colloidal rods is accompanied by large positive normal stresses, indic…
We compute the entropic interactions between two colloidal spheres immersed in a dilute suspension of semi-flexible rods. Our model treats the semi-flexible rod as a bent rod at fixed angle, set by the rod contour and persistence lengths. The entropic forces arising from this additional rotational degree of freedom are captured quantitatively by the model, and account for observations at short range in a recent experiment. Global fits to the interaction potential data suggest the persistence length of fd-virus is about two to three times smaller than the commonly used value of $2.2 μ{m}$.
In a mixed suspension of rods and small polymer coils, the rods adsorb onto a hard wall in contact with the suspension. This adsorption is studied in the low density of rods limit. It is driven by depletion forces and is much stronger for long rods than for spheres. This is shown by means of exact, numerical, calculations and an approximate theory.
Isotropic-Nematic and Nematic-Nematic transitions from a homogeneous base state of a suspension of high aspect ratio, rod-like magnetic particles are studied for both Maier-Saupe and the Onsager excluded volume potentials. A combination of classical linear stability and asymptotic analyses provides insight into possible nematic states emanating from both the isotropic and nematic non-polarized equilibrium states. Local analytical results close to critical points in conjunction with global numerical results (Bhandar, 2002) yields a unified picture of the bifurcation diagram and provides a conve…
We investigate geometric percolation and scaling relations in suspensions of nanorods, covering the entire range of aspect ratios from spheres to extremely slender needles. A new version of connectedness percolation theory is introduced and tested against specialized Monte Carlo simulations. The theory accurately predicts percolation thresholds for aspect ratios as low as 10. The percolation threshold for rod-like particles of aspect ratios below 1000 deviates significantly from the inverse aspect ratio scaling prediction, thought to be valid in the limit of infinitely slender rods and often u…
Above a certain density threshold, suspensions of rod-like colloidal particles form system-spanning networks. Using Monte Carlo simulations, we investigate how the depletion forces caused by spherical particles affect these networks in isotropic suspensions of rods. Although the depletion forces are strongly anisotropic and favor alignment of the rods, the percolation threshold of the rods decreases significantly. The relative size of the effect increases with the aspect ratio of the rods. The structural changes induced in the suspension by the depletant are characterized in detail and the sys…
When non-adsorbing polymers are added to an isotropic suspension of rod-like colloids, the colloids effectively attract each other via depletion forces. We performed Monte Carlo simulations to study the phase diagram of such rod-polymer mixture. The colloidal rods were modelled as hard spherocylinders; the polymers were described as spheres of the same diameter as the rods. The polymers may overlap with no energy cost, while overlap of polymers and rods is forbidden. Large amounts of depletant cause phase separation of the mixture. We estimated the phase boundaries of isotropic-isotropic coexi…
We present experimental measurements of the cholesteric-smectic phase transition of suspensions of charged semiflexible rods as a function of rod flexibility and surface charge. The rod particles consist of the bacteriophage M13 and closely related mutants, which are structurally identical to this virus, but vary either in contour length and therefore ratio of persistence length to contour length, or vary in surface charge. Surface charge is altered in two ways; by changing solution pH and by comparing M13 with {\it fd} virus, a mutant which differs from M13 only by the substitution of a singl…
Understanding the relaxation dynamics of colloidal suspensions is crucial to identify the elements that influence the mobility of their constituents, assess their macroscopic response across the relevant time and length scales, and thus disclose the fundamentals underpinning their exploitation in formulation engineering. In this work, we specifically assess the impact of long-ranged ordering on the relaxation dynamics of suspensions of soft-repulsive rod-like particles, which are able to self-organise into nematic and smectic liquid-crystalline phases. By performing Dynamic Monte Carlo simulat…
Orientational ordering driven by mechanical distortion of soft substrates plays a major role in material transformation processes such as elastocapillarity and surface anchoring. We present a theoretical model of the orientational response of anisotropic rods deposited onto a surface of a soft, elastic substrate of finite thickness. We show that anisotropic rods exhibit a continuous isotropic-nematic phase transition, driven by orientational interactions between surface deposited rods. This interaction is mediated by the deformation of the underlying elastic substrate, and is quantified by the…
We study theoretically the effect of a gravitational field on the equilibrium behaviour of a colloidal suspension of rods with different length-to-width aspect ratios. The bulk phases of the system are described with analytical equations of state. The gravitational field is then incorporated via sedimentation path theory, which assumes a local equilibrium condition at each altitude of the sample. The bulk phenomenology is significantly enriched by the presence of the gravitational field. In a suspension of elongated rods with five stable phases in bulk, the gravitational field stabilizes up to…
We study by simulation and theory how the addition of insulating spherical particles affects the conductivity of fluids of conducting rods, modeled by spherocylinders. The electrical connections are implemented as tunneling processes, leading to a more detailed and realistic description than a discontinuous percolation approach. We find that the spheres enhance the tunneling conductivity for a given concentration of rods and that the enhancement increases with rod concentration into the regime where the conducting network is well established. By reformulating the network of rods using a critic…
Suspensions of non-Brownian rods: droplet pinch-off, onset of heterogeneity and effective extensional viscosity
PMMH, CNRS, ESPCI Paris, Sorbonne Université, Université Paris-Cité（PMHH，法国国家科学研究中心，巴黎高等物理化工学院，索邦大学，巴黎 Cité 大学）；Department of Mechanical Engineering, University of California, Santa Barbara（加州大学圣塔芭芭拉分校机械工程系）；Institute of Physics, University of Amsterdam（阿姆斯特丹大学物理研究所）；Department of Mechanical Engineering, University of Maryland（马里兰大学机械工程系）；Department of Chemical and Biomolecular Engineering, University of Maryland（马里兰大学化学与生物分子工程系）
The rheology of nanoparticle suspensions for nanoparticles of various shapes with equal mass is studied using molecular dynamics simulations. The equilibrium structure and the response to imposed shear are analyzed for suspensions of spheres, rods, plates, and jacks in an explicit solvent for both charged and uncharged nanoparticles. For the volume fraction studied, ?$ϕ_{vf}=0.075$, the uncharged systems are all in their isotropic phase and the viscosity is only weakly dependent on shape for spheres, rods, and plate whereas for the jacks the viscosity is an order of magnitude larger than for t…
We study suspensions of semi-flexible colloidal rods and biopolymers using an Onsager-type second-virial functional for a segmented-chain model. For suspensions of thin and thick fd virus particles we calculate phase diagrams in quantitative agreement with experimental observations, and we find their effective state-point dependent shape to be much shorter and thicker than the actual shape. We also calculate the stretching of worm-like micelles in a host fd virus solution, again finding agreement with experiments. For both systems, our results show that the fd virus stiffness can play a key ro…
We present an experimental study of the isotropic-nematic phase transition in an aqueous mixture of charged semi-flexible rods (fd virus) and neutral polymer (Dextran). A complete phase diagram is measured as a function of ionic strength and polymer molecular weight. At high ionic strength we find that adding polymer widens the isotropic-nematic coexistence region with polymers preferentially partitioning into the isotropic phase, while at low ionic strength the added polymer has no effect on the phase transition. The nematic order parameter is determined from birefringence measurements and is…
Coexistence between the isotropic and the nematic phase in suspensions of rods is studied using grand canonical Monte Carlo simulations with a bias on the nematic order parameter. The biasing scheme makes it possible to estimate the interfacial tension gamma in systems of hard and soft rods. For hard rods with L/D=15, we obtain gamma ~ 1.4 kB T/L^2, with L the rod length, D the rod diameter, T the temperature, and kB the Boltzmann constant. This estimate is in good agreement with theoretical predictions, and the order of magnitude is consistent with experiments.
The criticality of self-assembled rigid rods on triangular lattices is investigated using Monte Carlo simulation. We find a continuous transition between an ordered phase, where the rods are oriented along one of the three (equivalent) lattice directions, and a disordered one. We conclude that equilibrium polydispersity of the rod lengths does not affect the critical behavior, as we found that the criticality is the same as that of monodisperse rods on the same lattice, in contrast with the results of recently published work on similar models.
We elucidate the molecular forces leading to assembly of two dimensional membrane-like structures composed of a one rod-length thick monolayer of aligned rods from an immiscible suspension of hard rods and depleting polymers. We perform simulations which predict that monolayer membranes are thermodynamically stable above a critical rod aspect ratio and below a critical depletion interaction length scale. Outside of these conditions alternative structures such as stacked smectic columns or nematic droplets are thermodynamically stable. These predictions are confirmed using an experimental model…
The phase behavior of rodlike molecules with polydisperse length and solvent-mediated attraction and repulsion is described by an extension of the Onsager theory for rigid rods. A phenomenological square-well potential is used to model these long-range interactions, and the model is used to compute phase separation and length fractionation as a function of well depth and rod concentration. The model closely captures experimental data points for isotropic/liquid crystalline phase coexistence of single-walled carbon nanotubes (SWNTs) in superacids. The model also predicts that the isotropic-biph…
We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally-resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a st…
Penetration of Rigid Rods, Flexible Rods, and Granular Jets into Low-Density Granular Media
Instituto de Física, Benemérita Universidad Autónoma de Puebla（普埃布拉自治大学物理研究所）；Gulliver, CNRS, ESPCI Paris, Université PSL（高勒维尔实验室，法国国家科学研究中心，巴黎高等物理化工学院，巴黎文理研究大学）
We analyze the behavior of a suspension of rigid rod-like particles in shear flow using a modified version of the Doi model, and construct diagrams for phase coexistence under conditions of constant imposed stress and constant imposed strain rate, among paranematic, flow-aligning nematic, and log-rolling nematic states. We calculate the effective constitutive relations that would be measured through the regime of phase separation into shear bands. We calculate phase coexistence by examining the stability of interfacial steady states and find a wide range of possible ``phase'' behaviors.
The vertical drying of a colloidal film containing rod-like particles was studied by means of kinetic Monte Carlo (MC) simulation. The problem was approached using a two-dimensional square lattice and the rods were represented as linear $k$-mers (i.e., particles occupying $k$ adjacent sites). The initial state before drying was produced using a model of random sequential adsorption (RSA) with isotropic orientations of the $k$-mers (orientation of the $k$-mers along horizontal $x$ and vertical $y$ directions are equiprobable). In the RSA model, overlapping of the $k$-mers is forbidden. During t…
We study the elastic response of concentrated suspensions of rigid wire frame particles to a step strain. These particles are constructed from infinitely thin, rigid rods of length $L$. We specifically compare straight rod-like particles to bent and branched wire frames. In dense suspensions the wire frames are frozen in a disordered state by the topological entanglements between their arms. We present a simple, geometric method to find the scaling of the elastic stress with concentration in these glassy systems. We apply this method to a simple 2D model system where a test particle is placed …
In the presence of a non-adsorbing polymer, monodisperse rod-like colloids assemble into one-rod-length thick liquid-like monolayers, called colloidal membranes. The density of the rods within a colloidal membrane is determined by a balance between the osmotic pressure exerted by the enveloping polymer suspension and the repulsion between the colloidal rods. We developed a microfluidic device for continuously observing an isolated membrane while dynamically controlling the osmotic pressure of the polymer suspension. Using this technology we measured the membrane rod density over a range of osm…
We study the elastic response of rigid, wire frame particles in concentrated, glassy suspensions to a step strain by applying the simple, geometric methods developed in part I. The wire frame particles are comprised of thin, rigid rods of length $L$ and their number density, $ρ$, is such that $ρL^3 \gg 1$. We specifically compare rigid rods to L-shapes made of two equal length rods joined at right angles. The behaviour of wire frames is found to be strikingly different from that of rods. The linear elasticity scales like $ρ^3 L^6$ for L-shaped particles, whereas it scales proportional to $ρ$ f…
We study the elastic response of composites of rods embedded in elastic media. We calculate the micro-mechanical response functions, and bulk elastic constants as functions of rod density. We find two fixed points for Poisson's ratio with respect to the addition of rods in 3D composites: there is an unstable fixed point for Poisson's ratio=1/2 (an incompressible system) and a stable fixed point for Poisson's ratio=1/4 (a compressible system). We also derive an approximate expression for the elastic constants for arbitrary rod density that yields exact results for both low and high density. The…
Phototaxis is the movement of microorganisms towards or away from light. In presence of dim (intense) light, microorganisms move towards (away from) the light sources, which is known as positive (negative) phototaxis. In this study, we present a model in which isotropic scattering algae suspension is illuminated by both diffuse and collimated irradiation, and the top surface is assumed to be rigid. Bimodal steady states result from the scattering Effect (for almost purely scattering suspension), which transits into a unimodal steady state as diffuse light intensity increases. The linear stabil…
We develop a novel biased Monte-Carlo simulation technique to measure the force-extension curves and the distribution function of the extension of fluctuating filaments stretched by external force. The method is applicable for arbitrary ratio of the persistence length to the contour length and for arbitrary forces. The simulation results agree with analytic expressions for the force-extension curves and for the renormalized length-scale-dependent elastic moduli, derived in the rigid rod and in the strong force limits. We find that orientational fluctuations and wall effects produce non-Gaussia…
We report on the dynamics in colloidal suspensions of stiff viral rods, called fd-Y21M. This mutant filamentous virus exhibits a persistence length 3.5 times larger than the wild-type fd-wt. Such a virus system can be used as a model system of rodlike particles for studying their self-diffusion. In this paper, the physical features, such as rod contour length and polydispersity have been determined for both viruses. The effect of viral rod flexibility on the location of the nematic-smectic phase transition has been investigated, with a focus on the underlying dynamics studied more specifically…
Self-diffusion in isotropic and liquid crystalline phases of fd virus colloidal rods: a combined single particle tracking and differential dynamic microscopy study
We consider an ideal gas of infinitely rigid rods near a perfectly repulsive wall, and show that the interfacial tension of a surface with rods on one side is lower when the surface bends towards the rods. Surprisingly we find that rods on both sides of surfaces also lower the energy when the surface bends. We compute the partition functions of rods confined to spherical and cylindrical open shells, and conclude that spherical shells repel rods, whereas cylindrical shells (for thickness of the shell on the order of the rod-length) attract them. The role of flexibility is investigated by consid…
Purely entropic systems such as suspensions of hard rods, platelets and spheres show rich phase behavior. Rods and platelets have successfully been used as models to predict the equilibrium properties of liquid crystals for several decades. Over the past years hard particle models have also been studied in the context of non-equilibrium statistical mechanics, in particular regarding the glass transition, jamming, sedimentation and crystallization. Recently suspensions of hard anisotropic particles also moved into the focus of materials scientists who work on conducting soft matter composites. …
The isotropic-to-nematic transition in an athermal solution of long rigid rods subject to a gravitational (or centrifugal) field is theoretically considered in the Onsager approximation. The new feature emerging in the presence of gravity is a concentration gradient which coupled with the nematic ordering. For rodlike molecules this effect becomes noticeable at centrifugal acceleration g ~ 10^3--10^4 m/s^2, while for biological rodlike objects, such as tobacco mosaic virus, TMV, the effect is important even for normal gravitational acceleration conditions. Rods are concentrated near the bottom…
We construct colloidal ``sticky'' rods from the semi-flexible filamentous fd virus and temperature-sensitive polymers poly(N-isopropylacrylamide) (PNIPAM). The phase diagram of fd-PNIPAM system becomes independent of ionic strength at high salt concentration and low temperature, i.e. the rods are sterically stabilized by the polymer. However, the network of sticky rods undergoes a sol-gel transition as the temperature is raised. The viscoelastic moduli of fd and fd-PNIPAM suspensions are compared as a function of temperature, and the effect of ionic strength on the gelling behavior of fd-PNIPA…
Random sequential adsorption of straight rigid rods of length $k$ ($k$-mers) on a simple cubic lattice has been studied by numerical simulations and finite-size scaling analysis. The calculations were performed by using a new theoretical scheme, whose accuracy was verified by comparison with rigorous analytical data. The results, obtained for \textit{k} ranging from 2 to 64, revealed that (i) in the case of dimers ($k=2$), the jamming coverage is $θ_j=0.918388(16)$. Our estimate of $θ_j$ differs significantly from the previously reported value of $θ_j=0.799(2)$ [Y. Y. Tarasevich and V. A. Cher…
Dense suspensions of self-propelled rod-like particles exhibit a fascinating variety of non-equilibrium phenomena. By means of computer simulations of a minimal model for rigid self-propelled colloidal rods with variable shape we explore the generic diagram of emerging states over a large range of rod densities and aspect ratios. The dynamics is studied using a simple numerical scheme for the overdamped noiseless frictional dynamics of a many-body system in which steric forces are dominant over hydrodynamic ones. The different emergent states are identified by various characteristic correlatio…
Using a combination of theory and experiments we study the interface between two immiscible domains in a colloidal membrane composed of rigid rods of different lengths. Geometric considerations of rigid rod packing imply that a domain of sufficiently short rods in a background membrane of long rods is more susceptible to twist than the inverse structure, a long-rod domain in a short-rod membrane background. The tilt at the inter-domain edge forces splay, which in turn manifests as a spontaneous edge curvature whose energetics are controlled by the length asymmetry of constituent rods. A thermo…
The viruses studied are genetically engineered, charged, semiflexible filamentous bacteriophages that are structurally identical to M13 virus, but differ either in contour length or surface charge. While varying contour length (L) we assume the persistence length (P) remains constant, and thus we alter the rod flexibility (L/P). Surface charge is altered both by changing solution pH and by comparing two viruses, fd and M13, which differ only by the substitution of one charged for one neutral amino acid per virus coat protein. We measure both the isotropic and cholesteric coexistence concentrat…
The yet virtually unexplored class of soft colloidal rods with small aspect ratio is investigated and shown to exhibit a very rich phase and dynamic behavior, spanning from liquid to nearly melt state. Instead of nematic order, these short and soft nanocylinders alter their organization with increasing concentration from isotropic liquid with random orientation to one with preferred local orientation and eventually a multi-domain arrangement with local orientational order. The latter gives rise to a kinetically suppressed state akin to structural glass with detectable terminal relaxation, whic…
We introduce a Rigid-Body Fluctuating Immersed Boundary (RB-FIB) method to perform large-scale Brownian dynamics simulations of suspensions of rigid particles in fully confined domains, without any need to explicitly construct Green's functions or mobility operators. In the RB-FIB approach, discretized fluctuating Stokes equations are solved with prescribed boundary conditions in conjunction with a rigid-body immersed boundary method to discretize arbitrarily-shaped colloidal particles with no-slip or active-slip prescribed on their surface. We design a specialized Split--Euler--Maruyama tempo…
We introduce methods for large scale Brownian Dynamics (BD) simulation of many rigid particles of arbitrary shape suspended in a fluctuating fluid. Our method adds Brownian motion to the rigid multiblob method at a cost comparable to the cost of deterministic simulations. We demonstrate that we can efficiently generate deterministic and random displacements for many particles using preconditioned Krylov iterative methods, if kernel methods to efficiently compute the action of the Rotne-Prager-Yamakawa (RPY) mobility matrix and it "square" root are available for the given boundary conditions. W…
A framework for performing event-driven, adaptive time step simulations of systems of rigid bodies interacting under stepped or terraced potentials in which the potential energy is only allowed to have discrete values is outlined. The scheme is based on a discretization of an underlying continuous potential that effectively determines the times at which interaction energies change. As in most event-driven approaches, the method consists of specifying a means of computing the free motion, evaluating the times at which interactions occur, and determining the consequences of interactions on subse…
