1. On the understanding of self-assembly of anisotropic colloidal particles using computer simulation methods
Nikoletta Pakalidou1✣ and Carlos Avendaño1
1. School of Chemical Engineering and Analytical Science, The University of Manchester ✣
Introduction and background
Self-assembly is the process in which building blocks, such as colloidal and nanoparticles, self-assembly to form organised structures under specific conditions. This process takes place under equilibrium conditions and is driven by non-covalent interactions such as van der Waals and electrostatic forces. For self-assembled colloidal materials, it is known that the shape of the particles influences the behaviour of the system. To understand the stability of the new materials formed, it is important to understand both thermodynamic and kinetics of self-assembly. While thermodynamic allow us to determine if a system can be formed, kinetics provide information about the time scale required for the process. In this work, we present preliminary work to understand the thermodynamics and kinetics of self-assembly of colloidal suspensions comprised of particles with different shapes.
Methodology
Particle models
The phase diagram for a hard-sphere system is mapped out using Monte Carlo simulation in the NPT ensemble. TI method is applied in order to calculate the free energy of hard spheres: A reversible path connects the system of interest and a reference system (the free energy is known). Two methods are used in performance of the TI method: Widom particle insertion method for a fluid (reference systems: ideal gas, extremely dilute fluid) and Einstein Crystal method for a solid (reference system: non-interacting einstein crystal). Phase diagram and several order parameters are used to detect the position of phase transition for platelets for two degrees of roundness.
Rounded square Spherical caps1, Hard spheres
colloidal platelets2,3
Simulation results
Simulation details and theory
Phase diagram for hard
spheres
The question is about the
liquid-solid coexistence
point. The solution is the
determination of hysteresis
phenomenon range
(from green dotted line to
blue one). To this end, the
free energy calculations
using Widom method is
performed.
(a) Reference system: very low density (b) Reference system: ideal gas
Widom method is applied
for hard spheres. Free
energy can be calculated
using that method for a
hard-sphere system.
We can use as a reference system either an ideal gas, or a dilute liquid, to create a reversible path between the reference system and system of interest.
Phase diagram for rounded square colloidal platelets (roundness 0.25, 0.50)
Potential energy of a hard sphere:
Simulation details: A Monte Carlo simulation and a Metropolis method is applied for a constant-NPT and a constant-NVT ensemble, for 1372 hard spheres particles. The simulation runs for 250,000 total cycles. For each cycle the average of packing fraction η for each value of dimensionless pressure P* is calculated, and the phase diagrams for each particle shape (hard spheres and platelets) are mapped out.
Widom method5: A “ghost” particle is added into a configuration at liquid state. Helmholtz free energy in reduced units F* of the system is calculated to compute the chemical potential for a liquid.
(a) Reference system: very low density6
, where
(b) Reference system: ideal gas6
Einstein Crystal method7: Each particle of a solid is connected with an harmonic spring λ: for very high density, switch on the springs and switch off the interactions. Helmholtz free energy is computed.
ri = center-of-mass position for i
ro,i = ideal lattice position for i
σ = diameter of the particle
Bond order parameters:
Global orientational order:
Solid
Liquid A B
Gas
Conclusions and future work
References
Using MC simulation and Metropolis method, the phase diagrams for hard spheres and rounded squares are mapped out. The basic aim is to study liquid-solid phase transition, using both TI (Widom and Einstein Crystal method) and order and global parameters. Simulation for hard spheres agrees well with theory for both liquid and solid. The hysteresis phenomenon is observed and the free energy is calculated using Widom method for liquid phase.
Degree of roundness (larger roundness [0.50] is connected with more squared shape than smaller one [0.25]) plays significant role for phase transition, as the parameters indicate. It has been observed that as the density is increased, isotropic phase is transformed into a hexagonal rotator phase (RHX) as a result of the first-order phase transition.
In the future, Einstein Crystal code will be structured in order to be applied for TI method for a solid. Also, the range of hysteresis phenomenon an dthe position of interface will be determined.
C. Avendaño, C.M. Liddell-Watson, and F.A. Escobedo, Soft Matter, 9:9153 (2013).
C. Avendaño and F.A. Escobedo, Soft Matter, 8:4675 (2012).
K. Zhao, R. Bruinsma, and T.G. Mason, PNAS, 108:2684 (2010).
S. Sacanna and D. Pine, Curr. Opin. Colloid Interf. Sci., 16:96 (2011).
B. Widom, J. Chem. Phys., 39:2802 (1963).
M. Dijkstra, Physics of Complex Colloids, 184:229 (2013).
D. Frenkel and A. Ladd, J. Chem. Phys., 81:3188 (1984).
Acknowledge
I gratefully acknowledge the funding received towards my Ph.D. from the Faculty of Engineering, The University of Manchester.