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We discuss crystal formation in supersaturated suspensions of monodisperse hard spheres with a concentration of hard spheres randomly pinned in space and time. The pinning procedure introduces an external length scale and an external time scale that restrict the accessible number of configureurations and ultimately the number of pathways leading to crystallization. We observe a significant drop in the nucleation rate density at a characteristic pinning concentration that can be directly related to the structure of the critical nucleus and the dynamics of its formation in the unpinned system.

Nucleation, Random Pinning, Confinement, Hard Spheres

Homogeneous as well as heterogeneous crystallization are of importance in materials design and production. But even for one of the most simple models for liquids, the suspension of monodisperse hard spheres, the crystallization process is not fully understood [

For the hard sphere system, the transition from the supersaturated fluid to the crystal is purely entropic. It is a first order transition, hence in the case of packing fractions slightly higher than the coexistence packing fraction, the system prevails in its meta-stable fluid state for a characteristic induction time before it is transformed irreversibly into a crystal.

The idea of the present work is to modify and restrict the possible number of pathways to crystallization in a controlled manner to understand how sensitive the crystallization process and, in particular, the induction time are with respect to changes in configureuration space.

The method we employ is to take a configureuration of hard spheres and to pin a randomly chosen fraction of them to their current positions.

This approach is called the random pinning model (RP) in the literature. Even though the dynamics change, the configureurations correspond to typical equilibrated fluid configureurations [

In the present work, the pinning concentrations are kept sufficiently low and do not reach the glass transition, because we are interested in crystallizing trajectories. Our study is based on suspensions of hard spheres and is split in two parts: In the first part we discuss crystallization for static random pinning, i.e. once the pinned hard spheres are chosen, they stay pinned for the rest of the simulation. The concentration of pinned hard spheres can directly be translated into a length scale that interferes with the typical size of a critical cluster. In the second part we alter the selection of pinned hard spheres in time intervals

We investigate

The time evolution of the system is calculated using an event driven molecular dynamics algorithm (EDMD), see [

We first discuss static pinning. Here, a set of hard spheres of average concentration

is the natural time unit of the simulation algorithm. The pinning time intervals discussed are

During the molecular dynamics simulation the local

For a hard sphere

where

Two neighbors

We start out with the dynamic properties of the supersaturated fluid in the presence of pinned hard spheres. As it has been reported already in [

The sum is restricted to the mobile particles

In addition to the mean squared displacement, we discuss the properties of the self part of the dynamic structure factor

In

For large concentrations,

As already mentioned in the introduction, slow relaxation becomes important at high concentrations, but it is still insignificant for concentrations

Pinning a given concentration of hard spheres introduces a characteristic length scale

The diameter of the critical nucleus in the case without pinning at a packing fraction

Crystal nucleation rate densities are presented in

where

We observe a sharp decrease in the nucleation rate density around

indicates internal stresses inside the nucleus, leading to a more irregular structure. This interpretation is supported by the analysis of the radius of gyration

with

The radius of gyration of the recorded nuclei for

We further ask whether pinned hard spheres are part the growing nuclei or whether the nuclei grow such that they avoid them. In

For the two concentrations

percentage for clusters

From

Our findings motivate the next section of this work, where pinned hard spheres are only held immobile for a given pinning time interval

Static pinning induces defects inside the growing nuclei. We can release the defects on long time scales if we apply periodic pinning, i.e. if a new set of pinned hard spheres is chosen after given time intervals

The mean percentage of pinned hard spheres inside the growing crystal is presented in

The nucleation rate densities that we obtain for different

The time it takes to develop a critical nucleus in the unpinned case is

pared to

For

We have presented a simulation study of crystallization in suspensions of hard spheres under the constraint of random static and periodic pinning. This approach allows us to directly restrict the number of accessible configureurations and the number of possible paths leading to crystallization.

We have shown that already a small pinning concentration is sufficient to suppress crystallization completely. We observe a sudden drop in the nucleation rate densities when the length scale introduced by the pinned hard spheres becomes smaller than the diameter of the critical nucleus of the unpinned system (i.e. at a concentration of

In a second step we extended the pinning procedure to periodic pinning at a fixed concentration of

The procedure of pinning a low concentration of hard spheres in the overcompressed fluid allows one to obtain information of the static and dynamic properties of the critical nucleus through observing the drop in the nucleation rate densities without exploring the details on the microscopic scale. The results presented here could be experimentally verified, for example in colloidal suspensions using laser trapping to pin hard spheres [

This project has been financially supported by the DFG (SPP1296) and by the National Research Fund, Luxembourg co-funded under the Marie Curie Actions of the European Commission (FP7-COFUND) and under the project FRPTECD. Computer simulations presented in this paper were carried out using the HPC facility of the University of Luxembourg.