Nucleation Rate I

The scientific goal of nucleation study is to obtain an experimental real image of nano-nucleation and to propose a correct nucleation theory that can explain and predict the nucleation. We have to approach the scientific goal from two different angles, one experimental and the other theoretical. 

Two approaches can explain and predict nucleation. The experimental approach tries to clarify the real image of nucleation how the size, shape, and number of nuclei evolve with the increase of crystallization time (t), that is, to make clear the size distribution f(N, t), where N is the number of particles within a nucleus that 

The theoretical approach constructs a basic equation to explain the observed f(N, t). CNT proposes a fundamental kinetic equation as a basic equation of nucleation by using/(N, t) [3]. However, the kinetic equation of CNT (as presented below) does not satisfy the fundamental mass conservation law, which means that the kinetic equation cannot be regarded as a basic equation. Any basic equation includes parameters (the so-called kinetic parameters determined experimentally) that give actual information about nucleus, nucleation, and so on. It remains an important problem to obtain correct kinetic parameters of nano-nucleation. 

We consider nucleation from the melt and assume a three-dimensional (3D) rectangular parallelepiped nucleus with sizes of /, m, and n, which are counted by the number of particles or repeating units. I is counted along the polymer chains. N is given by 

We assume m = 1 for the 2D nucleus. The free energy for formation of a nucleus (AG(N)) is defined as 

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The probability per site of forming a nucleus on an infinite substrate in the absence of other nuclei is taken to be equivalent to the nucleation rate, i. This obviously assumes that neighbouring patches do not collide during their formation, which is fully consistent with the nucleation model (see Sect. 3.4.4). 

Entanglement dependence of the nucleation rate I is qualitatively obtained for the first time by changing the number density of entanglement (ve) within the melt. An experimental formula of I as a function of ve was obtained on PE, I(ve) a exp(- yve) where y is a constant. 

The primary nucleation process is divided into two periods in CNT one is the so called induction period and the other is the steady (or stationary) nucleation period (Fig. 2) [16,17]. It has been proposed by CNT that small (nanometer scale) nuclei will be formed spontaneously by thermal fluctuation after quenching into the supercooled melt, some of the nuclei could grow into a critical nucleus , and some of the critical nuclei will finally survive into macroscopic crystals. The induction period is defined as the period where the nucleation rate (I) increases with time f, whereas the steady period is that where I nearly saturates to a constant rate (fst). It should be noted that I is a function of N and t,I = I(N, t). In Fig. 2, N and N mean the size of a nucleus and that of the critical nucleus, respectively. The size N is defined... 

The molecular weight (M) dependence of the steady (stationary) primary nucleation rate (I) of polymers has been an important unresolved problem. The purpose of this section is to present a power law of molecular weight of I of PE, I oc M-H, where H is a constant which depends on materials and phases [20,33,34]. It will be shown that the self-diffusion process of chain molecules controls the Mn dependence of I, while the critical nucleation process does not. It will be concluded that a topological process, such as chain sliding diffusion and entanglement, assumes the most important role in nucleation mechanisms of polymers, as was predicted in the chain sliding diffusion theory of Hikosaka [14,15]. 

In this regime, the lateral growth rate (g) is significantly greater than the secondary nucleation rate (i), so that the latter is rate-determining for the growth rate (G). For g i, the whole substrate is covered by stems as soon as the first stem is nucleated. Thus, monolayers are added one by one. If Lp is the substrate... 

If the lateral growth rate g is comparable or smaller than the nucleation rate i, then further layers are deposited before the first layer is fully formed [as sketched in Fig. 1.15(c)]. Under these circumstances (regime II), the hnear growth rate is given by... 

In aromatic solvents the step propagation rate v seems to be disproportionately more suppressed on both 100 and 110 faces compared to the nucleation rate i. This results in a-axis lenticular crystals (Fig. 28) and in nearly circular crystals with curvature on both 110 and 100 edges (Fig. 27). Also, as vis suppressed on 100 faces and Guo remains relatively high, fo-axis lenticular crystals with straight-faced pointed ends (type A crystals [33]) are often... 

Another parameter often used to characterise nucleation is the induction time or period, t. This is defined as the time taken for the formation of crystals after creating a supersaturated solution. Hence, the measured induction period does depend upon the sensitivity of the recording technique. It is generally assumed that t is inversely proportional to the nucleation rate, i.e. 

FIG. 39. Pseudobinary systems in a composition FIG. 40. The temperature dependence of tetrahedron. nucleation rate (i) and crystal growth... 

Figure 2.29 Variation of nucleation rate, I, with temperature, T.

The expressions derived above can be used to predict the shape of the curve for the nucleation rate, I, versus temperature. Since glasses are usually formed during cooling from a melt initially held at a temperature above T, discussion of the effect of temperature on the nucleation rate traditionally follows the same path, Le., from higher toward lower temperature. This process is illustrated schematically in Figure 2.1. 

It is obvious that the larger nucleus density, the thinner is the thickness of the metal film required to isolate the substrate from the solution. At the same time, a thinner surface film will be less coarse than a thicker one. This means that a smoother and thiimer surface film will be obtained at larger deposition overpotentials and nucleation rates, i.e., by electrodeposition processes characterized by high cathodic Tafel slopes and low exchange current densities. 

Fig. 19.1. Temperature dependencies of the primary nucleation rate (I) (A) and the linear crystal growth rate (G) (Q) for poly(ethylene succinate) (PEISU) [14] with a molecular weight (M) of 8,770. The solid and broken lines are results from the best fitting procedure for G based on Eq. (19.2) and for I based on Exj. (19.11) by the Arrhenius and the WLF expressions of the molecular transport term, respectively...

Fig. 19.13. Relationship between the nominal nucleation rate (I) and the rate constant (Jo) for i-PS crystallized at various temperatures from the melt at 230 C...

The role of epitaxy of nucleating agent in nucleation mechanism of polymers was studied to formulate the nucleation rate, I, as a function of the concentration of nucleation agent, Cna> the lateral size of a nucleation agent crystal, Th following relationship was considered ... 

The observable growth rate, G, is a combination of the secondary nucleation rate, I, and the size of crystal growth face and is described by basic relationships ... 

The nucleation rate I refers to the rate of formation of critical nuclei since only these can grow to produce liquid droplets. A pseudo-thermodynamic treatment of vapor-to-liquid transformation gives the result that I is proportional to exp (-AGJkT), where k is the Boltzmann constant and AGc is given by Eq. 

Figure 11.6 Schematic plots for the primary nucleation rate (I) and crystal growth rate (G) as a function of the isothermal crystallization (or nucleation) temperature. Adapted from Lorenzo and Muller [60].

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