# Drying-mediated self-assembly of nanoparticles

Original entry: Tony Orth, APPHY 226, Spring 2009

Eran Rabani, David R. Reichman, Phillip L. Geissler & Louis E. Brus., Nature vol. 426 p.271 [1]

### Abstract

"...Here we present a coarse-grained model of nanoparticle self-assembly that explicitly includes the dynamics of the evaporating solvent. Simulations using this model not only account for all observed spatial and temporal patterns, but also predict network structures that have yet to be explored..."

### Surface Tension Phenomena

While surface tension is not mentioned specifically in this paper is included implicitly in the author's analysis of a drying suspension of nanoparticles. In this study, the authors simulate the drying of a particle-laden fluid via a Monte Carlo method. The model used in the simulation is depicted in figure 1. A drying planar substrate is modelled as a 2D grid with each cell assigned to one of three "phases". The nanoparticles are assigned to a conglomeration of 3x3 cells, while the 2D planar substrate may occupied by either liquid or a bare patch (solid) by as fine as one cell resolution. Because the Monte Carlo method is inherently an energy-minimization algorithm, the heart of this simulation of the energy definition. In this paper, the authors us an energy function <math>H = -\epsilon_l \Sigma l_i l_j - \epsilon_{nl}\Sigma n_i l_j - \mu\Sigma l_i</math> where the doubly-indexed sums are run over all nearest neighbours. The <math>\epsilon</math> factors are energy couplings between the appropriate phases (labelled by their subscripts) and <math>\mu</math> is the chemical potential of the liquid. Of particular interest are the <math>\epsilon_l</math> and <math>\epsilon_{nl}</math> terms: buried in these weighting factors are the surface tensions of the liquid-liquid and liquid solid interfaces. At each time step, a random evaporation/condensation on the whole lattice is attemped. The change in configuration is accepted with the familiar Boltzmann (Metropolis) probability. Additionally, the nanoparticles are allowed a random displacement every <math>n</math> steps, where <math>n</math> determines the relative time scale of the diffusive dynamics to the evaporative dynamics. As the time index is stepped forward, the system evolves into final configuration which displays fractal-like properties.

Figure 2 shows a comparison between experimental and simulation results for a system of PbSe nanoparticles in octane: the top row of images are experimental micrographs of dried nanoparticle assemblies at different times during the drying process. The corresponding simulation snapshots are placed in the bottom row. The domain size is also plotted as a function of time along with its distribution in graphs d-e. Some more experimental and simulation snapshots are shown in figure 3 where the nanoparticle cell concentration rises from 5-60% from left to right. The change in the structure of drying patterns throughout this range of nanoparticle number density is striking. The authors discuss the thermodynamics of the system in some length. In particular, a wetting layer is seen to persist if the "binding energies" (ie. <math>\mu</math> and <math>\epsilon</math> factors, the latter of which contains the surface tension energy) are relatively high compared to the thermal energy. When the chemical potential lies within <math>-2\epsilon_1 < \mu < \mu_{sp}(T)</math> (where <math>\mu_{sp}</math> corresponds to the spinodal limit of stability), the liquid is metastable and so the drying pattern is evolved through a nucleative process.

It is somewhat surprising at first that such rich fractal patterns can be observed by using such a simple model. The authors mention that the model can be refined to include roughness and hydrodynamic interactions. However, as shown in the paper, these likely do not change the qualitative results of the simulation; one may adequately model the system by writing it in terms of only a few energy coupling terms.