Theory and experiment for one-dimensional directed self-assembly of nanoparticles

Mertens, Keith; Putkaradze, Vakhtang; Xia, Deying; Brueck, S. R. J.

doi:10.1063/1.1999029

Cited by 13 publications

(21 citation statements)

References 30 publications

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“…If the range of G (denoted by l) is sufficiently small, one may approximate (at least formally) the integral operator in Φ = G * ρ as a differential operator acting on the density ρ, namely, Φ = ρ + l∇ · (l∇ρ). As was demonstrated in [6], equation (1) then becomes a generalization of the viscous Cahn-Hilliard equation, describing aggregation of domains of different alloys. In the case that µ is constant,…”

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confidence: 95%

Aggregation of Finite-Size Particles with Variable Mobility

2005

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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...

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Aggregation of Finite-Size Particles with Variable Mobility

2005

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“…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]. These are complexes of atoms or molecules that form mesoscale structures with particle-like behavior.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed in this framework, it is possible to prescribe the dynamics of the particle-like structures after collapse. Thus, the model provides a description of directed self-assembly in nanophysics [16,17], in which the detailed physics is less important than the effective medium properties of the dynamics.…”

Section: Introductionmentioning

confidence: 99%

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

2008

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We investigate the emergence of singular solutions in a non-local model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the nonlocal effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of non-locality on the system's stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.

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“…Similar elliptic-parabolic equations (with hyper-diffusion, instead of diffusion) also arose recently as a leading order description of molecular beam epitaxy [9]. A variant of the system (1) re-appeared even more recently as a model of self-assembly of particles at nano-scales [11].…”

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“…A colloidal solution of 50nm-size particles is deposited on a grooved substrate; evaporation and receding contact line drags the particles into the channels. It has been shown [11] that the most important phenomenon effecting the distribution of particles in the channels is the capillary interaction between the particles at last stage of water evaporation, when all water and particles are confined to the channels. Examples of particle self-assembly in nano-channels is shown inFig.…”

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Formation of clumps and patches in self-aggregation of finite-size particles

Holm

Putkaradze

2006

Physica D: Nonlinear Phenomena

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New model equations are derived for dynamics of self-aggregation of finite-size particles. Differences from standard Debye-Hückel [1] and Keller-Segel [2] models are: a) the mobility µ of particles depends on the locally-averaged particle density and b) linear diffusion acts on that locally-averaged particle density. The cases both with and without diffusion are considered here. Surprisingly, these simple modifications of standard models allow progress in the analytical description of evolution as well as the complete analysis of stationary states. When 1 µ remains positive, the evolution of collapsed states in our model reduces exactly to finite-dimensional dynamics of interacting particle clumps. Simulations show these collapsed (clumped) states emerging from smooth initial conditions, even in one spatial dimension. If µ vanishes for some averaged density, the evolution leads to spontaneous formation of jammed patches (weak solution with density having compact support). Simulations confirm that a combination of these patches forms the final state for the system.

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Theory and experiment for one-dimensional directed self-assembly of nanoparticles

Cited by 13 publications

References 30 publications

Aggregation of Finite-Size Particles with Variable Mobility

Aggregation of Finite-Size Particles with Variable Mobility

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

Formation of clumps and patches in self-aggregation of finite-size particles

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