Aggregation of Finite-Size Particles with Variable Mobility

Abstract

New model equations are derived for dynamics of self-aggregation of finite-size particles. Differences from standard Debye-Hückel [1] and models are: the mobility of particles depends on the configuration of their neighbors and linear diffusion acts on locally-averaged particle density. The evolution of collapsed states in these models reduces exactly to finite-dimensional dynamics of interacting particle clumps. Simulations show these collapsed (clumped) states emerge from smooth initial conditions, even in one spatial dimension. Keywords: gradient flows, blow-up, chemotaxis, parabolic-elliptic system, singular solutions PACS numbers:Modeling finite size effects in the aggregation of interacting particles requires modifications of the class of Debye-Hückel equations. This problem is motivated by recent experiments using self-assembly of nano-particles in the construction of nano-scale devices [3]. Fundamental principles underlying the self-assembly at nano-scales are non-local particle interaction and nonlinear motion due to variations of mobility at these scales. The model should account for the change of mobility due to the finite size of particles and the nonlocal interaction among the particles. The local density (concentration) of particles is denoted by ρ. For particles interacting pairwise via the potential −G(|r|), the total potential at a point r is Φ(r) = − ρ(r ′ )G(|r − r ′ )dr ′ = G * ρ where * denotes convolution and G > 0 for attracting particles. The velocity of the particle is assumed to be proportional to the gradient of the potential times the mobility of a particle, µ. The mobility can be computed explicitly for a single particle moving in an infinite fluid. However, when several particles are present, especially in highly dense states, the mobility of a particle may be hampered by interactions with its neighbors. These considerations are confirmed, for example, by the observation that the viscosity of a dense suspension of hard spheres in water diverges, when the density of spheres tends to its maximum value. Many authors have tried to incorporate the dependence of mobility on local density by putting µ = µ(ρ) and assuming µ(ρ) → 0 as ρ → ρ max = 1 [4]. Vanishing mobility leads to the appearance of weak solutions in the equations, to singularities and, in general, to massive complications and difficulties in both theoretical analysis and computational simulations of the equations.Alternatively, we suggest that the mobility µ should depend on an averaged density ρ over some sampling volume, rather than on either the potential, or the exact value of the density at a point. This assumption makes sense from the viewpoints of both physics and mathematics. From the physical point of view, the mobility of a finite-size particle must depend on the configuration of particles in its vicinity. While attempts have been made to approximate this dependence by using derivatives of the local density, this approach may lead to unphysical negative diffusivity [5]. Hence, we assume instead that the local mobi...

A Nonlocal Continuum Model for Biological Aggregation

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Emergent singular solutions of nonlocal density-magnetization equations in one dimension

“…We investigate these equations numerically and study their evolution and aggregation properties. One aspect of non-local problems, already mentioned in [11], is the effect of competition between the length scales of non-locality on the system evolution. We shall highlight this effect with a linear stability analysis of the full density-magnetization equations.…”

“…In a series of papers, Holm, Putkaradze and Tronci [6,7,8,9,10,11] have focused on the derivation of aggregation equations that possess emergent singular solutions. Continuum aggregation equations have been used to model gravitational collapse and the subsequent emergence of stars [12], the localization of biological populations [13,14,15], and the self-assembly of nanoparticles [16].…”

Stability and clustering of self-similar solutions of aggregation equations

“…As we can see from Figure 5, for γ < 4, the clusters solution for any m is unstable and eventually expands to a circle, but for γ > 4, m clusters always deform to 3 clusters, except for some cases when 4 < γ < 6 with m ∈ {4, 5} which agrees precisely with Table III. Figure 5 and Table III give the whole picture of cluster stability in R 2 for f defined in (13). However, the argument does not extend naturally to higher dimensions.…”

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