Linear growth rate, crystals

First, on a purely physical basis, we determine when departures from a linear growth rate may be expected, that is when the crystal dimensions do not increase proportionally with time, but also depend on the size of the crystal (and maybe even other factors). Then we show how these limits relate to the possible values of i and g. 

Figure 13 Top, plot of linear growth rates of polyfethylene adipate) spherulites as a function of crystallization temperature for indicated molecular weight fractions. Spherulites shown correspond to the indicated range of temperatures. (A) Crystallization at the lower temperature range (B) at intermediate temperatures (C) crystallization at high temperatures. Reproduced with permission from Ref. [216]. Copyright 1956,...

We define the linear growth rate Vg as the linear velocity of displacement of a crystal face relative to some fixed point in the crystal. vg may be known as a function of c and c , derived from the theory of transport control, and as a function of c and cs as well, derived from the theory of surface control. Then c may be eliminated by equating the two mathematical expressions... 

Methods used for the measurement of crystal growth rates are either a) direct measurement of the linear growth rate of a chosen crystal face or b) indirect estimation of an overall linear growth rate from mass deposition rates measured on individual crystals or on groups of freely suspended crystals 35,41,47,48). 

Because the rate of growth depends, in a complex way, on temperature, supersaturation, size, habit, system turbulence and so on, there is no simple was of expressing the rate of crystal growth, although, under carefully defined conditions, growth may be expressed as an overall mass deposition rate, RG (kg/m2 s), an overall linear growth rate, Gd(= Ad./At) (m/s) or as a mean linear velocity, // (= Ar/At) (m/s). Here d is some characteristic size of the crystal such as the equivalent aperture size, and r is the radius corresponding to the... 

In this case, the linear growth rate is usually a function of the mass and heat transfer conditions at the surface of the crystal. The linear growth rate may... 

Concentration at crystal surface in equilibrium with T, kg/kg. Slope of solution freezing curve at the freezing point, kg/(kg-h). linear growth rate, nun/h. 

NUCLEATION IN POLYMER CRYSTALLIZATION The nucleation part of the linear growth rate is... 

The unit of the growth rate u in Equation 4-31 is mole per unit area per unit time. If growth rate in m/s (referred to as linear growth rate) is needed. Equation 4-31 must be multiplied by the molar volume of the crystal (VJ ... 

The total volume growth of crystals in a melt is the sum of the volume growth of all individual crystals. For a crystal nucleated at time x with linear growth rate u (which may depend on time), the radius and volume of crystal i at time t can be written as... 

Figure 16.5. Supersaturation behavior, (a) Schematic plot of the Gibbs energy of a solid solute and solvent mixture at a fixed temperature. The true equilibrium compositions are given by points b and e, the limits of metastability by the inflection points c and d. For a salt-water system, point d virtually coincides with the 100% salt point e, with water contents of the order of 10-6 mol fraction with common salts, (b) Effects of supersaturation and temperature on the linear growth rate of sucrose crystals [data of Smythe (1967) analyzed by Ohara and Reid, 1973],...

Vc = volume of holdup in the tank n = number of crystals per unit volume L = length of the crystal G = linear growth rate of the crystal t = time... 

The last column is of the summation Wj/Lj at corresponding values of crystal length L. The volumetric shape factor is av =0.866, the density is 1.5g/mL, and the mean residence time was 2.0 hr. The linear growth rate G and the mideation rate B° will be found. 

Adamski and Klimczyk analyzed cholesteryl pelargonate36) and caproate 37) liquid crystal to fully-ordered-crystal transitions over a temperature range of about 25 K. Again, the appearance of the fully ordered crystals was that of a spherulitic superstructure. The nucleation was time dependent, and the linear growth rate of the spherulites decreased with decreasing temperature by a factor 1/2 to 1/3, in contrast to the nonanoate and acetate. The Avrami exponent was close to 4 as judged from the measurement of the crystallized volume in the field of view under the microscope. 

Supersaturation has been observed to affect contact nucleation, but the mechanism by which this occurs is not clear. There are data that infer a direct relationship between contact nucleation and crystal growth these data showed that the number of nuclei produced by an impact was proportional to the linear growth rate of the impacted face. This could indicate that the effect of supersaturation is to alter growth rates and, concomitantly, the characteristics of the impacted crystal faces alternatively, what appears to be a mechanistic relationship actually could be a result of both nucleation and growth depending upon supersaturation. 

Fig. 29 Linear growth rates from melt of extended-chain crystals of PEO fractions 3000, 4000 and 6000 molecular weight as a function of crystallization temperature (re-plotted from [18]). Dashed lines are maximum slope tangents (after [25])...

Here, Cp is the concentration of the dissolved solute in the bulk of the liquid, Cp is the concentration of the solute at the liquid-crystal interface, and Cp is the solubility. Note that the nucleation rate (Jn) and the linear growth rate (G) have been transformed into molar units by using appropriate multiplying factors. It should be emphasized that, while these equations capture the phenomena under consideration, to be correct, they should be expressed in terms of activities in stead of concentrations. 

Na = rtf ia the number of particles-II (formed by the growth of the particles of quasicrystalline phase released from the gel during the crystallization process), both contained in a unit mass of zeolite formed at the end of the crystallization process, Kg is the constant of the linear growth rate of zeolite particles and 0 = 6/(q+1)(q+2) (q+3). The numerical values of the constants K0 = G N0Kg and Ka = G BNaK, calculated by the procedure described earlier (25), as well as the ratios Na/ N0 = Ka/0 K0, are listed in Table IV. as functions of time ta of the gel ageing. 

M the mass of one mole of the crystal, p the density of the crystal, and c the concentration of the solute. Equation (31) assumes that the change in concentration of the solution can be solely attributed to the growth of a single characterised solid. The mean linear growth rate of the suspension, R, may then be defined as... 

A more general treatment of growth in suspensions accounts for the differences in the growth rate on different crystal faces. If each crystal face of type hkl has a linear growth rate R, which is only dependent on the growth affinity, then... 

When mass transfer limits crystal growth, the interfacial concentration is often taken to be the saturation concentration, Cj, at the growth temperature and a linear growth rate G may be defined as... 

Although it is accepted that within a given crystallizer the linear growth rate is independent of size per se, the growth rates may differ due to effects such as the dependence of the free settling velocity on size. If all crystals, regardless of their size, were treated in exactly the same manner, the equations presented in this section could be used to predict product size for any seed size distribution. 

Feigelson (1988) found that their surface linear growth rates are comparable. When the number of molecules added per unit time is considered, protein crystals do grow more slowly. However, because protein molecules are much larger, fewer are needed to achieve the linear growth rates observed for small molecules. 

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