SUPPLEMENTARY INFORMATION
Onset of the Crystalline Phase in Small Assemblies of Colloidal Particles
Ray M. Sehgal†, Joshua G. Cogan‡, David M. Ford†, and Dimitrios Maroudas†
†Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003
‡ SLAC National Accelerator Laboratory, Stanford, CA 94305
In this supplemental information section, we provide a detailed description of the
interparticle potential used in this work, of the Monte Carlo umbrella sampling scheme
implemented for the construction of the free-energy landscapes (FELs), and of the calculation of
the equilibrium constant K used in the computation of the phase behavior metric D.
Interparticle Potential
The interparticle interaction potential is given by the sum of two terms, a screened
electrostatic repulsion term, UE, and an attractive depletion term, UD ,
𝑟
𝑟 3
3( )
( )
𝑈(𝑟) 𝑈𝐸 (𝑟) 𝑈𝐷 (𝑟)
𝐵
𝑟
Π 4
𝑎 +
𝑎
=
+
=
exp (−𝜅𝑎 ( − 2)) −
[ 𝜋(𝑎 + 𝑎𝐿)3 (1 −
)].
𝑘𝐵 𝑇
𝑘𝐵 𝑇
𝑘𝐵 𝑇
𝑘𝐵 𝑇
𝑎
𝑘𝐵 𝑇 3
4(1 + 𝐿) 16(1 + 𝐿)3
(1)
Values for the parameters in Eq. (1) are chosen to model an aqueous system of colloidal silica
particles in the presence of a hydrogel depletant as reported in Ref. [1]. The particle radius, a, is
set to 1100 nm. The potential parameters in the electrostatic contribution are set to B/kBT =
2497.0 and κa ≈ 140.5, indicating a repulsion of high strength but very short range. The
dimensionless parameter L represents the relative size of the depletant particles and, thus,
controls the range of the attractive potential; its value is set to 0.1. The factor Π/kBT is related to
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the depletants concentration and controls the depth of the attractive potential well. This factor is
treated as a control parameter in our study and ranges from 1.05×10-7 nm-3 to 1.15×10-7 nm-3.
Monte Carlo Umbrella Sampling (MC-US) Method
Our MC-US technique consists of three steps.
First, the coarse-variable space is
subdivided into overlapping target regions, and a biasing scheme based on a genetic algorithm is
used to create initial configurations within the bounds of each region. The genetic algorithm
attempts to find one system in each target region; unlike previously published algorithms [2], the
one employed here searches for all targets simultaneously. The algorithm performs unbiased
Monte Carlo (MC) sampling on 48 independent systems. After each of these systems completes
a short sampling run (<105 MC moves), the final system configurations are saved for future use
and those lying within an undiscovered target are noted. Regardless of the generation that they
belong to, the four configurations that are closest to any still-undiscovered target region are
cloned 12 times and the next generation of 48 systems begins. To prevent the algorithm from
stalling in areas of steep free energy, target regions are assigned a weight that decreases each
time they are attempted. A system close to only low-weight target regions is less likely to be
chosen and, if it is, fewer MC moves are performed on this system. With the first step complete,
we have a configuration for each target region. Starting from this configuration, we allow the
system to thermally sample the states available within that region via an MC simulation, carried
out with periodic boundary conditions at a low volume fraction (φ = 0.0695). We apply infinite
hard walls at a given target's boundaries in coarse-variable space to keep the system within the
target region, instead of using a harmonic biasing function [3, 4]. This guarantees that each
system stays within the appropriate sampling region without biasing the observed system
behavior within that region. With an appropriate restriction scheme in place, we apply the MCUS method to generate a FEL for each target region of phase space [4]. Finally, once the first
two steps are completed for all accessible regions of coarse-variable space, the individual FELs
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are stitched together to create a continuous FEL for the entire range of coarse-variable space
examined [3]. The resulting FEL provides all the required information for the phase behavior of
the system at equilibrium, with the free-energy minima corresponding to locally stable
configurations.
Calculation of Equilibrium Constant K
Computing the ratio of the probability for the particle cluster to be contained in each of
the two free-energy basins provides us with an estimate of the relative probability, K, of
observing the system in the crystalline phase instead of the fluid phase, which is expressed by
𝐾=
ℱ
ℱ
𝒫[C]
− x
− x
= ∫ e k B T ⁄ ∫ e kB T .
𝒫[F]
xϵC
(2)
xϵF
In Eq. (2), P[C] and P[F] are the probabilities of finding the assembly in the crystalline and
fluid basins, respectively, Fx/kBT is the free energy evaluated at a point x in the FEL, and the
crystalline, C, or fluid, F, regions are defined as being the corresponding free-energy basins on
either side of the saddle point on the minimum-F path. The ratio K ≡ P[C]/P[F] is, equivalently,
an equilibrium constant for the transitions between the two phases [5]. The definition of Eq. (2)
implies that 0 ≤ K < ∞.
References
[1]
G. E. Fernandes, D. J. Beltran-Villegas, and M. A. Bevan, Langmuir 24, 10776 (2008).
[2]
D. M. Deaven and K. M. Ho, Phys. Rev. Lett. 75, 288 (1995).
[3]
D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press,
New York, NY, 1987).
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[4]
D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to
Applications (Academic Press, San Diego, CA, 2002), 2nd edn.
[5]
R. S. Berry, C. R. Phys. 3, 319 (2002).
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