Metastable growth

Metastable growth of diamond takes place from gases rich in carbon and hydrogen at low pressures where diamond would appear to be thermodynamically unstable. The subject has a long history, beginning with work in the United States and Russia as early as 1962 (30—32) but not achieving widespread interest and acceptance until about 1986 after successful work in Japan. 

Angus, J.C. and C.C. Hayman (1988). Low-pressure, metastable growth of diamond and diamondlike" phases. Science 241, 913. 

The position marked B at 4 GPa and temperatures between 1775 and 2075 K, gives the condition under which well faceted diamonds of about 200 pm diameter have been crystallized using specially treated phenolic resins as the source of carbon and molten cobalt as solvent [30]. The notable point of this particular crystallization is that it occurred well into the graphite stable region of Fig. 7 and is thus an example of the metastable growth of diamond . An explanation for this may be found in consideration of the relative solubilities of phenolic resins and diamond in molten cobalt allowing dissolution of one metastable form and precipitation of another, namely diamond. This is an example where rules governing the transitions between metastable states, such as the Ostwald and Ostwald-Volmer rules, can be applied [31]. 

Figure B3.3.10. Contour plots of the free energy landscape associated with crystal niicleation for spherical particles with short-range attractions. The axes represent the number of atoms identifiable as belonging to a high-density cluster, and as being in a crystalline environment, respectively, (a) State point significantly below the metastable critical temperature. The niicleation pathway involves simple growth of a crystalline nucleus, (b) State point at the metastable critical temperature. The niicleation pathway is significantly curved, and the initial nucleus is liqiiidlike rather than crystalline. Thanks are due to D Frenkel and P R ten Wolde for this figure. For fiirther details see [189].

Unlike melting and the solid-solid phase transitions discussed in the next section, these phase changes are not reversible processes they occur because the crystal stmcture of the nanocrystal is metastable. For example, titania made in the nanophase always adopts the anatase stmcture. At higher temperatures the material spontaneously transfonns to the mtile bulk stable phase [211, 212 and 213]. The role of grain size in these metastable-stable transitions is not well established the issue is complicated by the fact that the transition is accompanied by grain growth which clouds the inteiyDretation of size-dependent data [214, 215 and 216]. In situ TEM studies, however, indicate that the surface chemistry of the nanocrystals play a cmcial role in the transition temperatures [217, 218]. 

Under equiUbrium vapor pressure of water, the crystalline tfihydroxides, Al(OH)2 convert to oxide—hydroxides at above 100°C (9,10). Below 280°—300°C, boehmite is the prevailing phase, unless diaspore seed is present. Although spontaneous nucleation of diaspore requires temperatures in excess of 300 °C and 20 MPa (200 bar) pressure, growth on seed crystals occurs at temperatures as low as 180 °C. For this reason it has been suggested that boehmite is the metastable phase although its formation is kinetically favored at lower temperatures and pressures. The ultimate conversion of the hydroxides to comndum [1302-74-5] AI2O2, the final oxide form, occurs above 360°C and 20 MPa. 

Fig. 11. Effects of pH in the colloidal siUca-water system (1), where A represents the point of zero charge regions B, C, and D correspond to metastable gels, rapid aggregation, and particle growth, respectively. Positive and negative correspond to the charges on the surface of the siUca particle.

Whilst programmed eooling (i.e. operation at eonstant nueleation rate within the metastable zone) inereases the mean produet erystal size ef. natural eooling, is it the optimum in produeing the largest possible erystals The problem is to find the maximum of the integral of erystal growth over the bateh time. Thus beeause bateh operation is by definition transient, a funetional has to be maximized over time rather than just a funetion at some point in time. Jones (1972, 1974) addressed this problem by applieation of a partieular result in... 

With a finite value of A(i 0, the interface starts to move. In the mean-field approximation of a similar model, one can obtain the growth rate u as a function of the driving force Afi [49]. For Afi smaller than the critical value Afi the growth rate remains zero the system is metastable. Only above the critical threshold, the velocity increases a.s v and finally... 

FIG. 2 Growth rates as a function of the driving force A//. Comparison of theory and computer simulation for different values of the diffusion length and at temperatures above and below the roughening temperature. The spinodal value corresponds to the metastability limit A//, of the mean-field theory [49]. The Wilson-Frenkel rate WF is the upper limit of the growth rate. 

Unfortunately, even for low molecular weight material it is difficult to obtain clear experimental evidence for a roughening transition [71]. This is mainly due to the fact that during growth the interface generally assumes a metastable shape and relaxation times are long and increase with crystal size. Therefore we certainly cannot expect a definitive answer for macromolecules. We shall therefore just make several comments which hopefully will be of use when reading the literature. 

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