# colloids

Colloids are particles with a diameter in the range of 10 nanometer to a few micrometers, suspended in a liquid. Calculating the nonequilibrium properties of colloidal suspensions is highly nontrivial, because these depend both on the short-time thermal Brownian motion and the long time hydrodynamic behaviour of the solvent.

The fundamental difficulty of fully including the detailed solvent dynamics in computer simulations becomes apparent when considering the enormous time- and length-scale differences between mesoscopic colloidal and microscopic solvent particles. For example, a typical colloid of diameter 1 micrometer will displace on the order of 10^10 water molecules. Furthermore, a molecular dynamics scheme for the solvent would need to resolve time scales on the order of a femtosecond to describe the intermolecular forces, while a colloid of diameter 1 micrometer in water diffuses over its own diameter in about 1 second. Clearly, simulating even an extremely crude molecular model for the solvent on the time scales of interest is completely out of the question: some form of coarse-graining is necessary.

# Brownian dynamics of rods

At the simplest level, the effects of the solvent can be taken into account through Brownian dynamics, which assumes that collisions with the solvent molecules induce a random displacement of the colloidal particle positions, as well as a local friction proportional to their velocity.

Using this approach, we have followed the dynamics of rigid rods, with a length over diameter ratio of 60. The model accounts for excluded volume interactions between rods, but neglects hydrodynamic interactions. The self-rotational diffusion coefficients of the rods were calculated by standard methods and by a new, more efficient method based on calculating average restoring torques. The collective decay of orientational order was calculated by means of equilibrium and nonequilibrium simulations. Our results showed that the decay times in both cases are virtually identical. Moreover, the observed decay of diffusion coefficients with volume fraction is much quicker than predicted by the Doi-Edwards-Kuzuu theory, which is attributed to an oversimplification of dynamic correlations in the theory.

- Y. Tao, W.K. den Otter, J.T. Padding, J.K.G. Dhont, and W.J. Briels,
*Brownian dynamics simulations of the self and collective rotational diffusion coefficients of rigid long thin rods*, J. Chem. Phys.**122**, 244903 (2005).

# Modeling hydrodynamic interactions and Brownian motion

Although, due to its simplicity, Brownian dynamics is understandably very popular, it completely neglects momentum transport through the solvent as described by the Navier-Stokes equations which leads to long-range hydrodynamic interactions (HI's) between the suspended particles. These HI's may fall off as slowly as 1/distance and can qualitatively affect the dynamical behaviour of the suspension.

Ard Louis (currently at University of Oxford) and I have investigated in detail how to implement a coarse-grained hybrid molecular dynamics and (so-called)
stochastic rotation dynamics simulation technique that captures the combined effects of Brownian and
hydrodynamic forces in colloidal suspensions. The importance of carefully tuning the simulation parameters to
correctly resolve the
multiple time and length scales of this problem was emphasized. We systematically analysed how our coarsegraining
scheme resolves dimensionless hydrodynamic numbers such as the Reynolds number **Re**, which
indicates the importance of inertial effects, the Schmidt number **Sc**, which indicates whether momentum
transport is liquidlike or gaslike, the Mach number **Ma**, which measures compressibility effects, the Knudsen
number **Kn**, which describes the importance of noncontinuum molecular effects, and the Peclet number **Pe**, which
describes the relative effects of convective and diffusive transport. With these dimensionless numbers in the
correct regime the many Brownian and hydrodynamic time scales can be telescoped together (see figure below) to maximize
computational efficiency while still correctly resolving the physically relevant processes.

We also showed how to control a number of numerical artifacts, such as finite-size effects and solvent-induced attractive depletion interactions. When all these considerations are properly taken into account, the measured colloidal velocity autocorrelation functions and related self-diffusion and friction coefficients compare quantitatively with theoretical calculations.

- J.T. Padding, A. Wysocki, H. Lowen, and A.A. Louis,
*Stick boundary conditions and rotational velocity auto-correlation functions for colloidal particles in a coarse-grained representation of the solvent*, J. Phys.: Condens. Matter**17**S3393 (2005). A preprint can be found here. - J.T. Padding and A.A. Louis
*Hydrodynamic interactions and Brownian forces in colloidal suspensions: Coarse-graining over time and length-scales*, Phys. Rev. E**74**, 031402 (2006). A preprint can be found here. - J. Sane, J.T. Padding and A.A. Louis,
*Hydrodynamics of confined colloidal fluids in two dimensions*, Phys. Rev E.**79**, 051402 (2009). A preprint is available here. - J. Sane, J.T. Padding and A.A. Louis,
*The crossover from single file to Fickian diffusion*, Faraday Discussions**144**, 285 (2010). A preprint is available here. - A. Imperio, J.T. Padding and W.J. Briels,
*Force calculation on walls and embedded particles in Multi-Particle Collision Dynamics simulations*, Phys. Rev. E**83**, 046704 (2011). - J. Sane, J.T. Padding and A.A. Louis,
*Taylor dispersion of colloidal particles in narrow channels*, Mol. Phys.**113**, 2538 (2015). - G. Fiorucci, G.M. Coli, J.T. Padding and M. Dijkstra,
*The effect of hydrodynamics on the crystal nucleation of nearly hard spheres*, Journal of Chemical Physics**152**, 064903 (2020).

# Sedimentation of colloids

We applied the hybrid molecular dynamics and stochastic rotation dynamics technique to study the steady-state sedimentation of hard sphere particles (see picture to the right) for Peclet number Pe ranging from 0.08 to 12. Hydrodynamic backflow causes a reduction of the average sedimentation velocity relative to the Stokes velocity. We found that this effect is independent of Pe number. Velocity fluctuations show the expected effects of thermal fluctuations at short correlation times. At longer times, nonequilibrium hydrodynamic fluctuations are visible, and their character appears to be independent of the thermal fluctuations. The hydrodynamic fluctuations dominate the diffusive behaviour even for modest Pe number, while conversely the short-time fluctuations are dominated by thermal effects for surprisingly large Pe numbers.

Inspired by recent experiments, we also study finite sedimentation in a horizontal planar slit. We were the first to observe the emergence of distinct lateral patterns in simulations, in agreement with observations in the experiments. The figure below shows a top view and side view at two times: near the begining of the simulation (left) and when the colloidal particles have sedimented halfway the slit (right).

In a more recent project with the university of Granada (Spain) we have investigated the sedimentation dynamics of colloids which have mutual attractions. This has a profound effect on the sedimentation behaviour, including a surprising non-monotonic dependence of the sedimentation velocity on colloidal concentration.

- J.T. Padding and A.A. Louis,
*Hydrodynamic and Brownian Fluctuations in Sedimenting Suspensions*, Phys. Rev. Lett.**93**, 220601 (2004). Also available at cond-mat/0409133. - J.T. Padding and A.A. Louis,
*Interplay between hydrodynamic and Brownian fluctuations in sedimenting colloidal suspensions*, Phys. Rev. E**77**, 011402 (2008). A preprint is available here. - A. Moncho-Jordá, A.A. Louis, and J.T. Padding,
*The effects of inter-particle attractions on colloidal sedimentation*, Phys. Rev. Lett.**104**, 068301 (2010). A preprint is available here. - K. Milinković, J.T. Padding and M. Dijkstra,
*Hydrodynamic Rayleigh-Taylor-like instabilities in sedimenting colloidal mixtures*, Soft Matter**7**, 11177 (2011). - A. Moncho-Jorda, A.A. Louis and J.T. Padding,
*How Peclet number affects microstructure and transient cluster aggregation in sedimenting colloidal suspensions*, J. Chem. Phys.**136**, 064517 (2012).

# Diffusive dynamics of colloidal rods and spheres near walls

The hydrodynamic interaction functions for a single rod-like particle and a wall are not known to sufficient accuracy. This severely limits the possibility to develop analytical theory, for dilute suspensions and even more for concentrated suspensions of rods. In collaboration with scientists at the Juelich Forschungszentrum (Germany) we have started detailed measurements of the friction between a rod and a solid wall by means of constrained stochastic rotation dynamics simulations. Independently, we have corroborated our findings in collaboration with scientists at Monash University (Australia) by looking at translation-rotation coupling of carbon fibers near a solid substrate. Recently we have also applied the computational method to study the diffusivity of spherical colloids in a finite cylindrical microcavity.

- J.T. Padding and W.J. Briels,
*Translational and rotational friction on a colloidal rod near a wall*, J. Chem. Phys.**132**, 054511 (2010). A preprint is available here. - A. Neild, J.T. Padding, L. Yu, B. Bhaduri, W.J. Briels and T.W. Ng,
*Translational and Rotational coupling in Brownian rods near a solid surface*, Phys. Rev. E**82**, 041126 (2010). - A. Imperio, J.T. Padding and W.J. Briels,
*Diffusion of spherical particles in microcavities*, J. Chem. Phys.**134**, 154904 (2011). - G. Fiorucci, J.T. Padding, and M. Dijkstra,
*Small asymmetric Brownian objects self-align in nanofluidic channels*, Soft Matter**15**, 321 (2019).

# bionetworks

A large fraction of cellular materials such as the cytoskeleton (green filaments in the picture to the right) or the cell membrane have structural functions. The physical properties of these materials are crucial for, e.g., cell integrity, cell division, and the response of cells to mechanical stress and external mechanical stimuli. Because the interior of cells is highly non-homogeneous, it is necessary to measure viscoelastic properties of these soft materials on sub-micrometer scale. Several "microrheology" techniques have been developed in recent years that make this possible. The experiments usually involve active manipulation of particles by external fields or analysis of the Brownian motion of embedded particles. Bio-microrheology is still in its infancy, however, because of the difficulty to interpret the hydrodynamic coupling between fluid flow generated by a probe particle on the one hand, and the cytoskeleton and cell membrane surface fluctuations on the other hand.

We aim to advance bio-microrheology by means of state-of-the-art computer simulations of the interactions between a probe particle and a biopolymer network coupled to a cell membrane (see figure 1 below). The response of a cell membrane and cytoskeleton will be investigated for different probe sizes, distances to the cell membrane, driving amplitudes and frequencies. At relatively high driving frequencies, viscous effects will be taken into account, for the first time, occurring in complicated geometries near cell membranes. Further away from the membrane, the distribution in mesh sizes and mechanical properties of the cytoskeleton can be measured. All this requires quantitative modeling of the flow field and thermal fluctuations in both the membrane and the cytoplasm.

**Figure 1. Model of a bionetwork attached to a lipid-bilayer cell membrane, seen edge-on. Thermal fluctuations of the membrane cause spontaneous fluctuations, coupled to a flow field in the surrounding cytoplasm (blue arrows). The cytoskeloton consists of a network of actin filaments (black lines). It is attached to the cell membrane by association with integral membrane proteins (blue spheres). A probe sphere can be used to measure how the cytoskeleton changes the mechanical properties and dynamics of the membrane.**

- J.T. Padding and A.A. Louis
*Hydrodynamic interactions and Brownian forces in colloidal suspensions: Coarse-graining over time and length-scales*, Phys. Rev. E**74**, 031402 (2006). A preprint can be found here. - J.T. Padding and A.A. Louis,
*Interplay between hydrodynamic and Brownian fluctuations in sedimenting colloidal suspensions*, Phys. Rev. E**77**, 011402 (2008). A preprint is available here. - J.T. Padding,
*Efficient simulation of non-crossing fibers and chains in a hydrodynamic solvent*, J. Chem. Phys.**130**, 144903 (2009). A preprint is available here. - J.T. Padding and W.J. Briels,
*Translational and rotational friction on a colloidal rod near a wall*, J. Chem. Phys.**132**, 054511 (2010).

# Asphaltene deposition in rock pores

The aggregation and deposition of colloidal asphaltene in reservoir rock is a significant problem in the oil industry. To obtain a fundamental understanding of this phenomenon, in a project with Schlumberger Cambridge we have studied the deposition and aggregation of colloidal asphaltene in capillary flow by experiment and simulation. For the simulation, we have used the stochastic rotation dynamics method, in which the solvent hydrodynamic emerges from the collisions between the solvent particles, while the Brownian motion emerges naturally from the interactions between the colloidal asphaltene particles and the solvent. The asphaltene colloids interact through a screened Coulomb potential. In the simulations, we impose a pressure drop over the capillary length and measure the corresponding solvent flow rate. We observe that the transient solvent flow rate decreases when the asphaltene particles become more "sticky". For relatively small attractions, a monolayer deposits on the capillary wall. With increasing attraction, the capillary becomes totally blocked. The clogging is transient for intermediate attraction strengths, but appears to be permanent for higher attractions. We compared our simulation results with flow experiments in glass capillaries, where we used extracted asphaltenes in toluene, reprecipitated with n-heptane. In the experiments, the dynamics of asphaltene precipitation and deposition were monitored in a slot capillary using optical microscopy under flow conditions similar to those used in the simulation. Maintaining a constant flow rate of 5 micoliters per minute, we found that the pressure drop across the capillary first increased slowly, followed by a sharp increase, corresponding to a complete local blockage of the capillary. Doubling the flow rate to 10 microliters per minute, we observed that the initial deposition occurs faster but the deposits are subsequently entrained by the flow. We calculated the change in the dimensionless permeability as a function of time for both experiment and simulation. By matching the experimental and simulation results, we obtain information about (1) the interaction strength for the particular asphaltenes used in the experiments and (2) the flow conditions associated with the asphaltene deposition process.

- E.S. Boek, H.K. Ladva, J.P. Crawshaw and J.T. Padding,
*Deposition of colloidal asphaltene in capillary flow: experiments and mesoscopic simulation*, Energy and Fuels**22**, 805 (2008). - E.S. Boek, H.K. Ladva, J.P. Crawshaw and J.T. Padding,
Colloidal Asphaltene Deposition and Aggregation in Capillary Flow: Experiments and Mesoscopic Simulation,
AIP Conf. Proceedings
**1027**, 273 (2008). - E.S. Boek, T. Headen and J.T. Padding,
*Multi-scale simulation of asphaltene aggregation and deposition in capillary flow*, Faraday Discussions**144**, 271 (2010). - A.D. Wilson, E.S. Boek, H.K. Ladva, J. Crawshaw and J.T. Padding,
*Recent Developments in the Deposition of Colloidal Asphaltene in Capillary Flow. Experiments and Mesoscopic Simulation*, Society of Petroleum Engineers Journal (SPE) 122197-MS (2009). - E.S. Boek, A.D. Wilson, J.T. Padding, T. Headen and J. Crawshaw,
*Multi-scale simulation and experimental studies of asphaltene aggregation and deposition in capillary flow*, Energy and Fuels**24**, 2361 (2010). - Edo Boek, Ali Fadili, Michael John Williams and Johan Padding,
*Prediction of asphaltene deposition in porous media by systematic upscaling from a colloidal pore scale model to a Deep Bed Filtration model*, Society of Petroleum Engineers Journal (SPE) 147539-MS (2011). - E.S. Boek, S. Clark and J.T. Padding,
*Deposition of colloidal asphaltene in capillary flow from computer simulation and homogeneous deposition models*, submitted to Soft Matter (2014).

# Collaboration

Many people have collaborated with me on the various topics involving colloidal particles. The new way of coupling colloidal particles to a mesoscopic hydrodynamic solvent, i.e. stochastic rotation dynamics, was initiated during my stay in the group of Dr Ard Louis at the University of Cambridge from 2003 until 2006. Since then Dr Louis has started a new group in the Rudolf Peierls Centre for Theoretical Physics at the University of Oxford. We also collaborate with Dr Arturo Moncho Jorda, now at the University of Granada in Spain, on the subject of sedimentation of attractive colloids. I have collaborated with Prof. Jan Dhont and Dr Peter Lang at the Soft Condensed Matter group at the Juelich Forschungszentrum in Germany, to study hydrodynamic interactions between rods and solid walls. I also collaborated with Dr. Tuck Wang Ng and Dr. Adrian Neild at Monash University in Australia, to study hydrodynamic and tethering behaviour of carbon nanofibers near walls.

I received a VENI grant from the Netherlands Organisation for Scientific Research (NWO) to do the research on microrheology. Important preparatory work was done by investigating how to couple colloidal particles to a mesoscopic hydrodynamic solvent (stochastic rotation dynamics). Moreover, the results of the study of hydrodynamic interactions between rods and a (solid) wall can now be extended to the case of semiflexible chains and fluctuating walls.

We also received a grant from the Industrial Partnership Programme "Biorelated Materials" to look further into the relation between fiber-scale mechanical properties and the rheological properties of fibrillar networks when loaded by a colloidal particle. We collaborated directly with the biophysical engineering group at the University of Twente, who performed state-of-the-art Atomic Force Microscopy experiments on single fibers as well as 3d networks.

I also collaborate with Prof. Marjolein Dijkstra and her (former) PhD student Kristina Milinković and current PhD student Giulia Fiorucci at the University of Utrecht in The Netherlands, to study the interplay of hydrodynamic interactions and Brownian fluctuations on the sedimentation dynamics of binary colloidal mixtures and flow of asymmetric dumbbell particles through microfluidic channels. Another twist was given to the simulation code together with Dr Edo Boek at Schlumberger Cambridge Research, now at Queen Mary University of London, to apply the method to the case of asphaltene deposition, as described above.