Crystal nucleation, computer simulation

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

As a final remark it may be stated that the predictive simulation of crystal nucleation from a solvent, and of the consequent polymorph selectivity, is the grand challenge of computational crystallography in the next few decades. 

The temporal evolution of twins has been investigated with an aid of computer simulation by Wild et al, [84]. Figure 6.1 shows a result of a twin evolution on a (111) face of a cubo-octahedral crystal, assuming that a =1.75. Note that time t is in arbitrary units, and the crystal size is normalized. It is seen that a small twin on a (111) face at t = 0 laterally increases the area, reaches the adjacent (100) faces, and induces a secondary nucleation on the (100) faces. According to the computer simulation, there are three types of twin evolutions, as shown in Figure 6.2. The types and characteristics of the twins shown in Figure 6.2 are summarized below. Note that the definition Tjijk here is different from that of Section 5.5. 

J. S. van Duijneveldt and D. Prenkel (1992) Computer-simulation study of free-energy barriers in crystal nucleation. J. Chem. Phys. 96, pp. 4655-4668... 

Computer simulations were also used to show that the crystallization nucleus is more likely to form in the subsurface than in the bulk phase of the water slab. This result can have far reaching atmospheric implications. It has been suggested that formation of an ice nucleus at the interface would be hampered by contamination of the surface by organic surfactants. The effect of the adsorbed material will surely propagate towards the subsurface as well, however it will be smaller than in the topmost layer. Therefore, the anthropogenic emissions should have an effect on the radiative balance of the Earth atmosphere. This effect should, however, be smaller than predicted using the assumption of surface nucleation. 

The statement is frequently made that one zeolite synthesis is faster than another but the measurement criteria may be extremely loosely defined. Often, there is no distinction between the induction time and the growth period (section 6), so that it is impossible to tell whether a reaction is slow because it takes a long time to nucleate or because the crystals grow slowly in the given circumstances. A comparison of some hypothetical synthesis reactions is shown in Table 2. The data are derived from a computer simulation of zeolite growth based on a very simple kinetic crystal growth model [50,73], i.e. 

In simple liquids studied by computer simulation the problem of glass formation through structural arrest cannot be dissociated from the problem of nonequilibrium crystallization (homogeneous nucleation), since under many conditions the time scales for the two processes are comparable. A broad review emphasizing phase changes has recently been given by Frenkel and McTague, to which we refer the interested reader for a more... 

Figure 10.14 Effect of the number of seeds on primary nucleation rate in a controlled cooling batch crystallization. Computer simulation results for citric acid, as reported by Bohlin and Rasmuson (1992a).

A mathematical phase-field model for the kinetics of isothermal polymorphic crystallization has recently been proposed [47], according to which crystallization involves rapid relaxation of the metastable state followed by nucleation and growth of the polycrystalline phase. Computer simulations were used to obtain results which could be tested experimentally using X-ray scattering experiments. Growth rates of different polymorphic polymers have also been investigated [48]. Simultaneous development of spherulites of different polymorphs occurs at different rates under isothermal conditions. From observation of interspherulitic boundaries between the a- and y-forms of polypivalolactone. 

Sinee the experiments on bulk liquids including water cannot be performed below the homogeneous nucleation temperature (Th for bulk water = -38°C), where crystal formation is found to become inevitable, it is not possible to test whether the apparent divergences of the above three quantities at low temperature are indeed divergences or something else. However, experiments on nano-confined water and extensive computer simulation studies (which have been possible since the formation of crystals is difficult in such systems and we can study the liquid water well below its homogeneous nucleation temperature) find that these quantities do not diverge but rather have a maximum at low temperature. 

The experimental methods of dilfraetion and spectroscopy are uniquely applicable to the study of crystalhne microporous solids and their chemistry. Nevertheless, there are important aspects of zeolite science that are not readily accessible to these techniques the species involved in nucleation and crystal growth, the structure of sites (often present at low concentration) that are active for adsorption and catalysis or the reaction intermediates present in catalysis. In these cases computational atomistic simulation offers great possibilities for improved understanding. Furthermore, many experimental measurements, such as calorimetric studies of heats of adsorption, and NMR or neutron scattering studies of dynamics, may be very expensive and time-consuming. Computer simulation methods, which promise to predict the performance of materials as adsorbents and catalysts rapidly and at reasonable expense, are therefore highly attractive. Excellent recent texts and useful reviews are available that deal with the simulation of microporous materials. Here I summarise the most widely used methods and the information they give. 

The effect is similar with the prior phase separation for the acceleration of crystal nucleation in the concentrated phase, as described in Sect. 11.3. Figure 11.10 shows the snapshots of the single chains in the random coil state, the collapsed globule state and the crystalline folding state, respectively, obtained in computer simulations. 

We note that such a transition from a circular to a quadratic envelope of the crystals has also been reproduced by computer simulations [13,14]. There, this transition is due to the reduction of the growth front nucleation probability. Wliile the disk-like pattern consists of multiple crystals, the square-shaped pattern represents a single crystal. We thus assume that, for the given film thickness, we observed a transition from a poly crystalline structure to a single crystal within the temperature interval from 45 to 50°C. 

In addition to new insights gained from computer simulation, new experimental techniques have led to some reassessments of previously accepted wisdom. The suggestion that spinodal decomposition of the melt could exist as a precursor to polymer nucleation has already been mentioned. Several groups have suggested that crystallization occurs through a series of intermediate stages and that the final crystal structure is not a refiection of the structure that initially formed from the supercooled melt (137-141). 

Although secondary nucleation theory was, for a period, widely accepted, it is now coming under increasing pressure, from experimental data, from computer simulation, and from new approaches to the fundamental process of crystalhzation. It is not clear at this stage whether all that is required is a few adjustments to the theory, or whether the idea of a nucleation barrier is flawed, or even if the idea that the crystal thickness seen is the fastest growing is correct. With the development of new theoretical tools, and the increased integration of theory with computer simulation, it is hoped that a more complete model for polymer crystallization can be developed. 

The Avrami equationhas been extended to various crystallization models by computer simulation of the process and using a random probe to estimate the degree of overlap between adjacent crystallites. Essentially, the basic concept used was that of Evans in his use of Poisson s solution of the expansion of raindrops on the surface of a pond. Originally the model was limited to expansion of symmetrical entities, such as spheres in three dimensions, circles in two dimensions, and rods in one, for which n = 2,2, and 1, respectively. This has been verified by computer simulation of these systems. However, the method can be extended to consider other systems, more characteristic of crystallizing systems. The effect of (a) mixed nucleation, ib) volume shrinkage, (c) variable density of crystallinity without a crystallite, and (random nucleation were considered. AH these models approximated to the Avrami equation except for (c), which produced markedly fractional but different n values from 3, 2, or I. The value varied according to the time dependence chosen for the density. It was concluded that this was a powerful technique to assess viability of various models chosen to account for the observed value of the exponent, n. 

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