Adv. Polym. Sci. (2005) 173:149-207 DOI:10.1007/1399429 C Springer-Verlag Berlin Heidelberg 2005 Numerical Simulation of Crystal Nucleation in Colloids Stefan Auer1 and Daan Frenkel2 I Department of Chemistry, Cambridge University, Lensfield Road, Cambridge, CB2 lEW, United Kingdom, sa372@ cant ac.uk 2 FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands, frenkel@ aniolf. n1 Abstract This article reviews the recent progress that has been made in the application of computer simulations to study crystal nucleation in colloidal systems. We discuss the concept and the numerical methods that allow for a quantitative prediction of crystal nucleation rates. The computed nucleation rates are predicted from first principles and can be directly compared to experiments. These techniques have been applied to study crystal nucleation in hard-sphere colloids, polydisperse hard-sphere colloids, weakly charged or slightly soft colloids and hard- sphere colloids that are confined between two plane hard walls. Keywords Crystal nucleation, Colloids, Monte Carlo simulation, Nucleation barrier, Kinetic prefactor, Crystal nucleation rate 1 Introduction 149 2 Method 153 2.1 Calculation of the Cluster Size Distribution 154 2.2 Kinetic Prefactor 161 3 Hard-Sphere Colloids 163 4 Effect of Polydispersity 169 5 Weakly Charged Colloids 175 6 Poly-12-hydroxystearic Coated Polymethylmethacrylate Particles 186 7 Wall Induced Crystallization 190 8 Concluding Remarks 197 A Distribution of Cluster Sizes in Equilibrium 198 B Calculation of the Chemical Potential 201 C Surface Free Energies of Critical Nuclei 202 References 205 150 Stefan Auer and Daan Frenkel 1 Introduction Heating a block of ice will result in melting. Cooling the resultingwater will freeze it again. At a given pressure, water and icecan coexist at only one temperature. The water-ice coexistence temperature at ambientpressure is of such importance for every-day life, that it has been chosen as the zero-point of the widely usedtempera- ture scale invented by the Swedish physicist Celsius. Closer inspection of the melting and freezing transition shows that this transition is not quite symmetric. Ice heated above 0° C always melts, whereas cooling it below 0° C does not always result in immediate freezing. In fact water, and most other liquids,can be cooled significantly below their freezing temperature and kept there without crystallizing (1, 2]. This phenomena is known as supercooling. A supercooled liquid can be triggered into freezing by adding a little bit of the corresponding solid. A single snowflake ina glass of supercooled water will induce freezing of water that touches it andgrow rapidly into a big chunk of ice. Other disturbances, such as dust oreven shocks, can trigger the freezing of supercooled liquids as well. It thus seems that the freezing process has great difficulty to start spontaneously, but becomes very easy once it is initiated. The spontaneous formation of a piece of solid is an example of nucleation. The fact that a liquid can be supercooled is best understood qualitatively in the framework of classical nucleation theory (CNT) (see e.g. Ref. [3]). According to CNT the free energy of a spherical nucleus that forms in a supersaturated solution contains two terms. The first term accounts for the fact that the solid phase ismore stable than the liquid. This term is negative and proportional to the volume of the nucleus. The second term is a surface term. It describes the free energy needed to create a solid/liquid interface. This term is positive and proportional to the surface area of the nucleus. The (Gibbs) free energy of a spherical nucleus of radius R has the following form: 4 ,2 AG= 37r ps +471 R2y, (1) where ps is the number density of the bulk solid, A tt the difference in chemical po- tential between the solid and the liquid, and y is the solid/liquid surface free energy density. The function AG has a maximum at R 2y Aps I All I) and the correspond- ing height of the nucleation barrier is given by 16r y3 AG*= (2) 3(osliiii1)2. For small nuclei the surface term dominates and the free energy increases. Only if this nucleus exceeds a critical size does its free energy decrease and the crystallite can grow spontaneously (see Fig. 1). The probability for the formation of a critical nucleus depends exponentially on its free energy of formation: Pc or exp(AG7icsT). (3) Numerical Simulation of Crystal Nucleation In Colloids 151 30 20 10 % 0 10 20 0 1 2 3 4 5 Radius R Fig. 1. Free energy barrier of a spherical nucleus described by classical nucleation theory Eq. (1). For small radii the surface term dominates and the free energy increases. When the radius exceeds a critical size the bulk term dominates and the free energy decreases The crystal nucleation rate is given by the product of Pc and a kinetic factor r that describes the rate with which a critical nucleus grows. The CNT expression for the nucleation rate per unit volume is: 16r y3 /=rexpr (4) 3kBT(pslAy1)2]' with r fc+. Here p1 is the number density of the liquid, Z = y 1/67rkB7-n, is the Zeldovich factor and L.+ is the attachment rate of particles to the critical cluster. The Zeldovich factor arises from the fact that not all particles that are at the top of the nucleation barrier crystallize: some will recross the barrier and melt again. The attachment rate of particles to the critical nucleus can be estimated by multiplying the number of monomers available at the surface of the nucleus which is proportional to nc213-with a typical transition rate of these particles to become part of the nucleus. This transition rate is proportional to Ds/A2, where Ds is a self diffusion coefficient and A is a typical distance over which diffusion takes place: 24Dsne2/3 fc+ = Az (5) The above expression for the nucleation rate is the one most commonly used to analyze crystal nucleation rate experiments. The problem with the CNT approach is however that, in most cases, neither A nor y are accurately known. More often than not, both parameters are obtained by fitting the CNT expression to experimental nucleation data. To illustrate the problems associated with the fitting of CNT to experimental data, we give two examples. Let us start with Turnbull's first quantitative measurement of 152 Stefan Auer and Daan Frenkel a nucleation rate in liquid mercury [2] (see Fig. 2). For the interpretation of his data he used Eq. (4), where he estimated the difference in chemicalpotential between the two phases bypnaAh(Tn, T)/ Tm. Here Ah is the enthalpy changeper particle on freezing at coexistence, Tm is the coexistence temperature and T is thetempera- ture of the liquid mercury. A plot of log(/) vs. I/ TA T2 should givea straight line with the slope proportional to y 3 and the intercept equalto log(x). From this two 7.4 7.2 7 6 8 cr, --I6.6 6.4 6.2 6 1.01 1.02 1.03 lerro-rf Flg. 2. The steady state nucleation rate, I in units of l/(in3s), as a function of undercooling in Kelvin for liquid mercury from Ref. [2]. The open and the filled circles correspond to two different samples. The solid lines result from a two parameter fit of Eq. (4) to the experimental data parameter fit we see that the functional form given by CNT for the nucleation rate reproduces the experimental data. However, the resulting value oftc is a factor 107 larger than predicted from CNT. The corresponding estimate for the typical diffusion distance A is many orders of magnitude too small. To explain this Turnbull noted is his paper: "...suppose that y depends upon temperature according to the equation: y= yo + bT, [where yo is the value at coexistence and b is a constant], ... a value of b=0.0008/K is sufficient to change the apparent value of the kinetic factor by sixor- ders of magnitude." A remarkable statement which might be correct, but at the time direct corroboration was not possible because of the absence of a priori knowledge of both fit parameters. The major problem of experimental investigations of crystalliza- tion kinetics in atomic systems is the high speed of nucleus formation and subsequent crystal growth, as well as the difficulty of preventing heterogeneous nucleation. The second example we take from more recent experiments on the crystallization kinetics in a suspension of hard-sphere colloids. Crystallization in colloidal suspensions is in- teresting because it can be studied in considerable detail, since colloidal particlesare much larger than atoms. Colloids therefore crystallize on a timescale which is about ten orders of magnitude longer than that for an atomic liquid. Moreover because of Numerical Simulation of Crystal Nucleation In Colloids 153 their size, colloids can be probed by powerful optical methods such as time-resolved static laser light scattering and confocal microscopy. In these systems it is also some- what easier to control heterogeneous nucleation. In Fig. 3 we show the results from crystallization rate measurements in hard-sphere colloids, performed by two differ- ent groups [5, 6]. For this system the difference in chemical potential between the two phases can be calculated accurately from existing analytical expressions for the equation of state. The curves in the figure result from a two parameter fit of Eq. (4) to the experimental data. Palberg [4] fitted the data from Harland and van Megen [5] 1013 -0 °-© 1012 , 10" 4' 0 10lo cr, -2 E -e -4 oP In J' -8 -10 -12 10° -14 -16 I -18 107 :1 -20 M; 0 10 20 3040 1/(n, n142 1050.52 0.53 0.54 0.55 0.56 0.57 0.58 a> Fig.