(440d) Growth of Multiparticle Clusters in Sedimenting Suspensions

Sedimentation of multiparticle clusters (aggregates) in a quiescent fluid is relevant to differential phase separation, and to natural processes in marine systems. All computer simulations so far assumed that a rigid multiparticle aggregate is formed by a version of the Monte Carlo (MC) method  (typically, in the limits of diffusion-limited cluster aggregation (DLCA) or reaction-limited cluster aggregation (RLCA)), when already existing clusters are moved randomly to meet and form larger clusters (with some sticking probability). The library of clusters built in this manner is then used to simulate aggregate sedimentation by Stokesian Dynamics, Lattice-Boltzmann or multipole methods. However, these purely geometrical MC methods for cluster formation are entirely ad hoc and they do not correspond to the physical situation in sedimenting suspensions, at least for non-Brownian particles; there is obviously a disconnection between these methods and subsequent sedimentation simulations.

In contrast, the present work offers, for the first time, rigorous and highly-accurate, first-principle simulations of multiparticle aggregation as it naturally occurs in dilute sedimenting suspensions. The primary particles are monodisperse spheres; Brownian and inertial effects are neglected. For the cluster growth to start, some rigid vertically-aligned doublets are assumed to be already present (their concentration is much smaller than that for primary particles and does not affect the results). A doublet, in its downward motion, captures a new primary particle to form a rigid triplet. In general, once we have a rigid aggregate of N particles, a new primary particle is randomly generated far upstream from the cluster with uniform probability density (a suitable assumption for locally-homogeneous suspensions, prior to aggregation), and the particle-aggregate relative trajectory is computed. If no capture with the aggregate occurs, a new random initial particle position is tested, etc., until a colliding trajectory is found. The new rigid-body aggregate is then allowed to reach a steady sedimentation regime (which, for N>3, is not a simple downward translation, but a spiral motion about some axis of steady rotation) before a new primary particle is generated upstream, etc. Particle-aggregate collision requires singular van der Waals attraction (to overcome lubrication) which we assume to be present, but short-range, not appreciable until the particle-cluster surface clearance is much smaller than the particle radius. Technically, the algorithm consists of several parts:

1.  A semi-analytical solution to determine the regime of steady sedimentation for a random isolated cluster. Tensor representations for the aggregate translational and rotational velocities via the total applied force reduce the Stokes flow problem to only three necessary solutions at the initial cluster position, resulting in very efficient time integration. The steady-state rotational velocity (about the vertical axis) is found to be an eigenvector of the appropriate coupling tensor but, in general, there are up to three eigenvalues; the only stable solution is selected by time integration. The axis of steady rotation for the cluster is found to be typically quite away from the center of mass, at a distance much larger than the cluster hydrodynamical radius.

 2. A special strategy for generating random initial positions for a new primary particle, far upstream. This issue is non-trivial due to cluster rotation and fractal-like shapes. The upstream collision circle (where the particle emerges with uniform probability density) is found by backward integration of many relative trajectories from the cluster vicinity, using the far-field approximation to the particle-cluster hydrodynamical interaction; it is crucial here to include the cluster rotation. It is shown that the so-defined interception area is always an upper bound and, hence, includes all possible collisions when the full hydrodynamical problem is solved. Still, on the average, for moderate van der Waals attraction, 45 trajectories have to be generated until one collision is found to add a new particle to a large cluster. Simulating 80 growth realizations with N up to 100 spheres required a total of 300 000 particle-cluster relative trajectories.

 3. A highly efficient and precise multipole algorithm to solve the full Stokes flow problem for cluster-particle interaction, including the case of near-contact. This algorithm, vastly different from other, approximate schemes for multiparticle problems,  incorporates high-order multipoles (up to order 100) with rotational transformations of spherical harmonics by Wigner functions [1], economical truncation of  multipole expansions [1,2]  and the idea of geometry perturbation [3] to effectively include lubrication. The Stokes problem for an isolated cluster is a much simpler, particular case, and does not require high-order multipoles.

It is shown that the aggregate settling velocity, relative to the primary particle velocity, when ensemble-averaged for every N over many growth realizations, quickly attains a fractal scaling and is well approximated by 1.12 N^0.48 for all N>20 and “moderate” van der Waals attraction. The exponent 0.48 is found to be independent of the strength of the molecular force (provided that it remains short-range), while the prefactor shows only a few percent variation when the non-dimensional Hamaker constant is varied in a broad range.  For our aggregates, the average gyration radius, a usual measure to characterize fractal clusters, is extremely close to the average hydrodynamical radius for all N >4. The fractal dimension of rigid monodisperse aggregates formed in sedimentation is predicted to be 1.92 as the only possible value for the growth process with sequential particle addition and realistic hydrodynamics; in contrast, in DLCA and RLCA Monte Carlo ad hoc models, the fractal dimension is an arbitrary parameter. It is also found for our clusters that, on the average, the outer cluster radius is twice as large as the hydrodynamical radius. In a cluster of size N>>1, most particles are connected to one or two neighbors (33% and 43% of particles, respectively), although there are also particles with 3 or 4 neighbors (19% and 5%, respectively).

[1] Zinchenko A.Z. and Davis R.H. 2000 An efficient algorithm for hydrodynamical interaction of many deformable drops. J. Comput. Phys., vol. 157, pp.539-587.

 [2] Zinchenko A.Z., Rother M.A. and Davis R.H. 2011 Gravity-induced collisions of spherical drops covered with compressible surfactant. J. Fluid Mech., vol. 667, p.369-402.

 [3] Zinchenko A.Z. 1998 Effective conductivity of loaded granular materials by numerical simulation.  Phil. Trans. Roy. Soc.  Lond.  A, vol. 356, pp. 2953-2998.