Temperature Dependence of Crystallization

For practical purposes, one usually need not know the entire temperature dependence of crystal growth rate, the data on the maximum rate of growth often being satisfactory. Littleton (1931) pointed out that for glasses of the system Na20 — CaO -— Si02, a satisfactory agreement is provided by the relation... 

The crystalline phase typically grows as spherical aggregates called spherulites. However, other geometries such as disks or rods may be found with, as shown below, a consequent modification of the rate equation. M. Avrami [26] first derived these rate equations in the form used for polymer kinetics for the solidification of metals. The weight of the crystalline phase is calculated as a function of time at constant temperature. As will be described below, the temperature dependence of crystallization can be derived from classical nucleation theory. 

Besides an explanation of chain-folding lengths, the phenomenon of regime transitimis can also be explained by the LH theory. If crystal growth is controlled by secondary nucleatitHi, the temperature dependence of crystal growth rates can be dominated by... 

Fig. 5.28. sPP, crystallized at 25 °C and heated up to the melt Temperature dependencies of crystal thickness dc (circles) and long spacing dac (triangles) [53]... 

Chlorine, a member of the halogen family, is a greenish yellow gas having a pungent odor at ambient temperatures and pressures and a density 2.5 times that of air. In Hquid form it is clear amber SoHd chlorine forms pale yellow crystals. The principal properties of chlorine are presented in Table 15 additional details are available (77—79). The temperature dependence of the density of gaseous (Fig. 31) and Hquid (Fig. 32) chlorine, and vapor pressure (Fig. 33) are illustrated. Enthalpy pressure data can be found in ref. 78. The vapor pressure P can be calculated in the temperature (T) range of 172—417 K from the Martin-Shin-Kapoor equation (80) ... 

Results from magnetic susceptibiHty studies have been reported (50—53). Measurements (50) obtained by the Gouy method are shown in Figure 3. These are lower than those of other investigators. However, the temperature dependences of the magnetic susceptibiHties, for the various plutonium allotropes were similar. a-Plutonium single crystals show a slight anisotropy of (54). 

Fig. 1. Examples of temperature dependence of the rate constant for the reactions in which the low-temperature rate-constant limit has been observed 1. hydrogen transfer in the excited singlet state of the molecule represented by (6.16) 2. molecular reorientation in methane crystal 3. internal rotation of CHj group in radical (6.25) 4. inversion of radical (6.40) 5. hydrogen transfer in halved molecule (6.16) 6. isomerization of molecule (6.17) in excited triplet state 7. tautomerization in the ground state of 7-azoindole dimer (6.1) 8. polymerization of formaldehyde in reaction (6.44) 9. limiting stage (6.45) of (a) chain hydrobromination, (b) chlorination and (c) bromination of ethylene 10. isomerization of radical (6.18) 11. abstraction of H atom by methyl radical from methanol matrix [reaction (6.19)] 12. radical pair isomerization in dimethylglyoxime crystals [Toriyama et al. 1977].

The temperature dependences of k, calculated by Hancock et al. [1989], are given in fig. 48. The crossover temperature equals 25-30 K. The weak increase of k T) with decreasing temperature below is an artefact caused by extending the gas-phase theory prefactor to low temperatures without taking into account the zero-point vibrations of the H atom in the crystal. For the same reason the values of the constants differ by 1-2 orders of magnitude from the experimental ones. 

It is noteworthy that the above rule connects two quite different values, because the temperature dependence of is governed by the rate constant of incoherent processes, while A characterizes coherent tunneling. In actual fact, A is not measured directly, but it is calculated from the barrier height, extracted from the Arrhenius dependence k T). This dependence should level off to a low-temperature plateau at 7 < This non-Arrhenius behavior of has actually been observed by Punnkinen [1980] in methane crystals (see fig. 1). A similar dependence, also depicted in fig. 1, has been observed by Geoffroy et al. [1979] for the radical... 

The shape of the equilibrium line, or solubility curve, is important in determining the mode of crystallization to be employed in order to crystallize a particular substance. If the curve is steep, i.e. the substance exhibits a strong temperature dependence of solubility (e.g. many salts and organic substances), then a cooling crystallization might be suitable. But if the metastable zone is wide (e.g. sucrose solutions), addition of seed crystal might be necessary. This can be desirable, particularly if a uniformly sized product is required. If on the other hand, the equilibrium line is relatively flat (e.g. for aqueous common salt... 

Next we consider a molecular crystal composed of N2 molecules, (Vp = 0). Molecular N2 solids at low temperatures and low pressures are in the a structure (Pa3). Using PIMC simulations we studied the low temperature properties of N2 sohds [260] (B = 2.88 K, = 500). In Fig. 6 the temperature dependence of the molar volume is shown for our simulational as well as for experimental [289] data. We note that the classical simulations (corresponding to P = 1) lead to a nonzero slope of the volume at very low temperatures, which is in sharp contrast to the experimental behavior [289]. 

Figure 3.8 Anomalous temperature dependence of relative dielectric constant of ferroelectric crystals at the transition temperature (Curie point).

At each temperature one can determine the equilibrium lattice constant aQ for the minimum of F. This leads to the thermal expansion of the alloy lattice. At equilibrium the probability f(.p,6=0) of finding an atom away from the reference lattice point is of a Gaussian shape, as shown in Fig. 1. In Fig.2, we present the temperature dependence of lattice constants of pure 2D square and FCC crystals, calculated by the present continuous displacement treatment of CVM. One can see in Fig.2 that the lattice expansion coefficient of 2D lattice is much larger than that of FCC lattice, with the use of the identical Lennard-Lones (LJ) potential. It is understood that the close packing makes thermal expansion smaller. 

Figure 4. Temperature dependence of CRSS for the six different deformation modes observed in Ti-56 at.%Al single crystals.

Figure 5. Temperature dependence of CRSS for <101] superlattice slip, <110] ordinary slip and twinning observed for Ti-54 at.%Al and Ti-56 at. SjAl single crystals.

Figure 3. The temperature dependence of the crystal distortion for VjSi, (0), derived from the data in Figure 2 [curves (a) and (c)], as discussed in reference 5. The dot-dashed curve shows the same distortion parameter for In-26.5 at%Tl derived from the data in Figure 1. The inset shows in detail the data below the super-c-nducting critical temperature, Tc. (From reference 5)...

It seems that structural irregularities that cause spontaneous polarization are a relatively common property of niobium and tantalum oxyfluoride crystals. Fig. 100 shows the temperature dependence of SHG signals for several compounds that form island-type and chain-type structures. 

Fig. 106. Temperature dependence of vs(NbO) and vs(NbF) wave numbers for a single crystal of RbsNbsOF/s- Reproduced from [442], A. I. Agulyansky, J. Ravez, R. Cavagnat, M. Couzi, Ferroelectrics 152 (1993) 373, Copyright 1993, with permission of Taylor Francis, Inc., http //www.routledge-ny.com.

In the case of single crystals of K5Nb3OF 8, a maximum in the dielectric permittivity 33 was observed at about 400K. Fig. 108 shows the temperature dependence of b at different frequencies. 

Thermocurrent measurements were performed on crystals of RbsNbjOFig and K5Nb3OFi8 along the c direction [440, 443]. Fig. 112 shows the temperature dependences of the pyroelectric coefficients close to room temperature. 

In retrospect, by inspecting the literature, we find a confirmation of this variance (see for instance Ref. [67]). Peak intensities of bands originally assigned to Franck-Condon components of the excilonic emission have random relative intensities. This would not be possible if the bands were intrinsically vibronic. Since we know that the excilonic emission, as it is observed in single crystals, is rather sharp at low temperatures, we were forced to reconsider the assignment of the fluorescence of thin films. From the temperature dependence of the fluorescence effi-... 

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