Anisotropic aggregation in a simple model of isotropically polymer-coated nanoparticles

Jurriaan A. Luiken and Peter G. Bolhuis

Phys. Rev. E 88, 012303 – Published 8 July 2013

Abstract

We report a numerical study of a simple, modified Asakura-Oosawa model for nanoparticles that are isotropically grafted with polymer chains. We perform canonical and grand-canonical Monte Carlo simulations to establish a qualitative morphology diagram, as well as quantitative phase diagrams. The morphology diagram qualitatively reproduces experimental observations and theoretical approaches employing more complex models. In addition, we establish the transition lines for a microphase separation and show that the phase behavior saturates for larger polymer sizes. An analytical treatment on the level of the second virial coefficient indicates that this saturation effect is caused by less effective shielding of nanoparticles by longer polymers. Our simple model enables large-scale particle-based simulations of self-assembly of polymer-coated particles.

Received 13 April 2013

DOI:https://doi.org/10.1103/PhysRevE.88.012303

Authors & Affiliations

Jurriaan A. Luiken and Peter G. Bolhuis

van 't Hoff Institute for Molecular Sciences, University of Amsterdam, P.O. Box 94157 1090 GD Amsterdam, The Netherlands

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Issue

Vol. 88, Iss. 1 — July 2013

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Images

Figure 1
In the top graph, the probability that a NP is not shielding another NP at distance $R$ with any of its $f$ polymers, ${[χ_{1} (R)]}^{f}$ , is plotted for various $f$ and $q$ . In the bottom graph we plotted the interaction strengths $ɛ_{c}$ for which the virial coefficient is zero. The observed trend is that the phase boundaries saturate for large polymer sizes.Reuse & Permissions
Figure 2
Morphological phase diagram of the output structures from the NVT simulations. We identified spherical (circles), cylindrical (diamonds), (interconnected) sheets or flattened (squares), stringlike/branched (up-pointing triangles), dispersions of short strings and/or small aggregates (empty down-pointing triangles), and fully isolated particles (filled down-pointing triangle). Morphological trends are visualized using dashed lines, and simulation snapshots are provided for all aggregate shapes (polymers are transparent for clarity). Shown at the bottom are contact order histograms for parameter coordinates $(f, q)$ that each assemble into different types of aggregates.Reuse & Permissions
Figure 3
Results (including error bars) from the canonical NVT simulations. The transition interaction strengths $ɛ_{c}$ for which half the NPs are present in aggregates are plotted on the left for all particle types. In most cases, the error bars are smaller than the symbols. NPs with $f = 12$ and $q \geq 0.35$ do not reach a phase transition below $ɛ = 10$ $k_{B} T$ and are, hence, not shown. Plotted on the right is the aggregation fraction $η$ as a function of $ɛ$ for $f = 6$ and various grafted polymer sizes.Reuse & Permissions
Figure 4
On the left, small polymers (dashed cyan circles) reside in the interstitial space between colloids (red circles). Increasing the polymer size induces interactions that lead to exclusion of polymers from this space and hence decrease their chain entropy. This entropy penalty is captured by the grafting potential and subsequently leads to reorganization of colloids into lower dimensional morphologies to reduce this penalty at the cost of stabilization energy, e.g., from a spherical ( $\sim 6 - 12 ɛ$ ) to hexagonal sheet ( $\sim 6 ɛ$ ) arrangement.Reuse & Permissions
Figure 5
Shown at the top are plots of the average number of clusters of size $n$ for NP type $(4, 0.40)$ for $μ = - 8.5$ $k_{B} T$ and increasing $ɛ$ (a), and the same plot but then for fixed $ɛ = 6.75$ $k_{B} T$ with increasing $μ$ (b). Here, we observe the formation of finite clusters due to the added long-range repulsion, thus preventing the formation of a bulk phase. Phase transition lines for colloids with $f = 6$ and $q \in {0.2, 0.3, . . ., 0.7}$ (c) and transition points for $μ = - 8.5$ $k_{B} T$ and $f \in {4, 6, 8}$ (d) are shown at the bottom.Reuse & Permissions
Figure 6
Snapshot of two nearby NPs with $f = 1$ . The bottom right polymer is bound to the left NP at a distance $r$ , with the other NP at distance $R$ . Here, the presence of the right NP is excluding a fraction $1 - p_{av}$ of the surface with radius $r$ around the left NP, in which the polymer can then not move.Reuse & Permissions

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