Nucleation rate, definition

The maximum superheat which can be achieved with a nonboiling liquid is definitely pressure sensitive. This is evident in Fig. 18 and also in Fig. 28. Both plots show that the possible superheat (and therefore the possible values of ATc) decreases to zero as the critical pressure is approached. This is in agreement with the equation-of-state theory and also the nucleation-rate theory. 

By definition, therefore, the nucleation rate, which is the birth of crystals at size = 0 is shown in Eq. (5.3). As will be shown later, EP is also a function of other properties of the system and the crystallizer. 

Induction Period. By a common definition, the time elapsed between mixing two reacting solution and the appearance of first crystals of precipitate is termed an induetion period. In diluted or slowly reacting systems preeipitation starts slowly, then accelerates, and eventually slows down. This initial part is well characterized by an induction period. The induction period r/ jhas beeome an experimental basis to determine the nucleation rate J, using a simple relationship. 

The limit thexmodynamlc nonequilibrium corresponds to the attainment of liquid superheats at which intensive spontaneous boiling is observed on nucleus bubbles of fluctuation nature. The physical definiteness of this boundary is conditioned by a very shazi> dependence of the nucleation rate J(T. P ) on the Gibbs number G- = A T where is the work of formation of a critical, nucleus [4 5j. is the Boltzmaim constant. By the homogeneous nucleation theory the value of J is calculated making use of thermodynamic parameters. Thus, for water at atmospheric pressux T have J ... 

For short channels we suggest identifying the choking pressure (effective critical backpressure) with the pressure /5 (T) at which the homogeneous nucleation rate has a sufficiently high value. For definiteness we assume... 

The boundary condition at x = 0 is obtained by argument that the probability there is a particle (nucleus) between 0 and X(0) dt on the size coordinate during time t and t + dt can be obtained in two different ways. First, it is given by the left-hand side of the equation appearing below by definition of the first-order product density. Second, it is also given by the right-hand side by definition of the nucleation rate, which is the transition probability for the appearance of a nucleus in the time interval t, t + dt). Thus,... 

Whilst programmed cooling (i.e. operation at constant nucleation rate within the metastable zone) increases the mean product crystal size cf. natural cooling, is it the optimum in producing the largest possible crystals The problem is to find the maximum of the integral of crystal growth over the batch time. Thus because batch operation is by definition transient, a functional has to be maximized over time rather than just a function at some point in time. Jones (1972, 1974) addressed this problem by application of a particular result in... 

The crucial points in the derivation of an explicit ejq)ression for the stationary nucleation rate are (i) the definition of the frequencies of attachment and detachment of single atoms to and from the fth site of the electrode surface (equations (2.10) and 2.11)) and (ii) the elucidation of the correlation between the ratios and o>Jco+ . (equations (2.12) and... 

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

Jacobs and Tompkins6 consider only structure-sensitive nucleation, which occurs at definite sites in the lattice where the activation energy is least, such as lattice defects and dislocations. The rate of nucleus formation thus depends on both the defect density and on the activation energy. 

In the case of a perfectly oriented (111) surface, such as the sidewalls of an etched eavity on a (100) wafer, there may still be a definite etch rate due to the faet that steps may be generated from surface defeets such as vacancies and disloeations. The eteh rate of the perfectly oriented (111) surfaee can be viewed as being limited by the rate of step generation, i.e., a nucleation proeess. 

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