Growth rate as a function of temperature

Prepare a plot of the predicted growth rates as a function of temperature, including the results from all six sets of simulations, as well as the experiment. 

Figure 11 Silicon film growth rate as a function of temperature for different pressures.

Table I. Maximum Growth Rates as a Function of Temperature...

The constants kg and kg are temperature-dependent and are usually fit to the Arrhenius equation to obtain a general expression for growth rate as a function of temperature. The Arrhenius equation can be written as... 

For a condensed-matter transformation that occurs on cooling below an equilibrium temperature Tg, the overall growth rate as a function of temperature is shown in Figure 6.23. The overall growth rate is a result of two factors the driving force for transformation, which increases linearly with decreasing temperature below Tg, and the mass transfer term v = which decreases exponentially with... 

To determine the parameters Go and Kg, one needs to measure the growth rate G(7). For materials with slow crystallization kinetics, one can easily measure the spherulite growth rate as a function of temperature from micrographs (Fig. 4.2). Then Gq and Kg are determined by plotting In G + W/Rg T - T ) against T + T ) jlT AT. 

FIGURE 5.2 Plot of nucleation and crystal growth rates as a function of temperature. 

Figure 98. (a) Crack growth rate as a function of temperature for Types 304 SS, Type 316L SS, Alloy 690, and Alloy 625 (in deaerated water at 500°C) and at a pressure of 25.5 MPa. (b) Arrhenius plots for crack growth rate. Reprinted from Ref 166, Copyright (1993) with permission from NACE International. 

Van Putte and Bakker [33] proposed a first numerical approach to model crystallization kinetics of palm oil. They took the approach of solvent crystallization (see Section II.F) and observed the crystallization of the saturated part of TAGS in palm oil. For their system, they determined nucleation and growth rates as a function of temperature from microscopic observations at very low percentages of solids (fs around 0.1%), as shown in Figmes 3 and 4, respectively. 

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. 

Fig. 3 Ising model calculations of the normalized growth rate as a function of the driving force. The surface temperatures are 0.40T , closed circles, 0.54 T/j, open circles, and 1.08 T/j, squares.

Figure 4-6 Interface reaction rate as a function of temperature, pressure, and composition. The vertical dashed line indicates the equilibrium condition (growth rate is zero), (a) Diopside growth and melting in its own melt as a function of temperature with the following parameters Te= 1664K at 0.1 MPa, A5m-c = 82.76J mol K , E/R —30000 K, 4 = 12.8 ms K, and AV c l. l x 10 m /mol. The dots are experimental data on diopside melting (Kuo and Kirkpatrick, 1985). (b) Diopside growth and melting in its own melt as a function of pressure at 1810 K (Tg = 1810 K at 1 GPa from the equilibrium temperature at 0.1 MPa and the Clapeyron slope for diopside). (c) Calcite growth and dissolution rate in water at 25 °C as a function of Ca " and CO concentrations.

Other computer simulations were made to test the classical theory. Recently, Ford and Vehkamaki, through a Monte-Carlo simulation, have identified fhe critical clusters (clusters of such a size that growth and decay probabilities become equal) [66]. The size and internal energy of the critical cluster, for different values of temperature and chemical potential, were used, together with nucleation theorems [66,67], to predict the behaviour of the nucleation rate as a function of these parameters. The plots for (i) the critical size as a function of chemical potential, (ii) the nucleation rate as a function of chemical potential and (iii) the nucleation rate as a function of temperature, suitably fit the predictions of classical theory [66]. 

Fig. 17.3 Temperature and flow-rate dependence of silicon thin-film growth for a silane CVD process. The left-hand panel shows the temperature dependence for a fixed inlet flow rate. The right-hand panel shows normalized growth rate as a function of inlet velocity for three different surface temperatures. The actual growth rate at U = 10 cm/s is stated parenthetically under the temperature call out.

For a disk temperature of 800 K, calculate the Si growth rate as a function of disk rotation rate, over the range 500 to 1300 rpm. At 500 rpm and at 1300 rpm, what fraction of the Si growth is attributable to the reaction of SiH4 itself with the surface (i.e.,reaction 1 in silicon, surf) ... 

Fig. 6.36. Growth rate as a function of the doping level, i.e., the B2H6/DEZ ratio for LP-CVD ZnO B films deposited at a substrate temperature of 155°C and a process pressure of 0.5mbar...

CSVT allows short experiments and easy determination of the growth rate. In this technique, a source and a substrate are placed at very short distance ( 1mm). The experiments were achieved under a hydrogen flow of 1 l/min at two source temperatures, 550 and 600°C. In Fig. 1 is pictured the instantaneous growth rate as a function of the inverse substrate temperature for these two source temperatures. The solid and dashed lines correspond to a fit using a theoretical model. ... 

In dry oxidation the first reaction dominates whereas the second reaction dominates in wet oxidation. The growth rate is a function of temperature, oxide thickness, and substrate orientation. As an example, the average growth rate on the (100) surface in wet oxygen at 1000 °C after Ih of oxidation is about 1 A/s. The structure of thermally grown oxide is amorphous and typically has exact stoichiometric composition. The oxide layer formed on a silicon substrate is about 2.27 times the thickness of the consumed silicon and contains about 2.2 x 10 molecules/cm of Si02. 

Figure 2.31 Variation of the crystal growth rate, U and nucleation rate, /, as a function of temperature, T (schematic).

Figure 5-8. The growth rate as a function of deposition temperature for Si02 films deposited using TEOS as the silicon source (after [39]).

Equation 6.102 can be modified to describe the relative growth rate as a function of supersaturation (j at a given temperature under the influence of a given impurity concentration... 

Table 3.4 refers to a number of crystaUizable miscible polymer blends for which the sphemhte growth rate as a function of the crystallization temperature has been investigated. For most blends, only a part of the bell-shaped curve could be measured. In Fig. 3.8, the complete bell-shaped spherulitic growth rate curve of iPS in iPS/PS blends containing 0,15, and 30 wt% PS is shown. Due to the addition of impurity (e.g., the amorphous PS), a suppression of the growth rate is observed, which is greater than the concentration of the impurity added. Important parameters of the impurity added to the crystaUizable component are the type, concentration, and molecular weight (Keith and Padden 1964). 

Fig. 19.29. Conunon logarithm of the growth rate as a function of common logarithm of molecular weight for PEO [77] at three different temperatures of 54.5°C circle), 56.9 C triangle) and 59.1 C (square)...

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