Abstract
In this chapter, the basis for the tool to interpret crystal size distribution (CSD) will be explained. First, we will address the background of CSD introduction, the physical interpretation of the exponential CSD often observed in nature, and the examples of studies on CSD in nature. Then, we will introduce two theoretical methods of describing CSD’s temporal development; the Eulerian and Lagrangian descriptions, which were introduced in the chapters on vesiculation and crystallization. In this chapter on CSD, we will obtain general solutions of CSD under various conditions in closed and open systems. The closed system is a system in which CSD is only determined by the dynamics of nucleation and growth of crystals, and liquid and crystals do not move. An open system is that in which exchange with the external system of crystals essentially influences CSD. An advantage of the mathematical disscussion of CSD is that when CSD obtained by models is compared with that obtained by experiments, the relationship between observable CSD and model parameters can be clearly undrestood than when we use an empirical rule obatained using numerical calculations. Finally, we will derive the CSD corresponding to the Avrami plot, which is often conducted in phade change experiments.
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Notes
- 1.
When a constant nucleation rate (\(J=J_{0}\) = constant) is assumed,
$$\begin{aligned} J(t) =J_{0} =G_{0} F(0,t) = G_{0} F(R_{0},0) \exp \left[ c \left( t + \frac{R_{0}}{G_{0}} \right) \right] \end{aligned}$$(9.19)is obtained from the boundary condition. Therefore,
$$\begin{aligned} \exp \left[ c \left( t + \frac{R_{0}}{G_{0}} \right) \right] = \frac{J_{0}}{G_{0} F(R_{0},0)}, \, \, c= \frac{\ln J_{0} /( G_{0} F(R_{0}, 0))}{t + R_{0}/G_{0}} \end{aligned}$$(9.20)is obtained, which indicates that c depends on time t.
- 2.
 Note that this is different from the maximum particle size in the system, \(R_{\max }\)(described later).
- 3.
This is a condition by which increase in the degree of supersaturation due to cooling and decrease in the degree of supersaturation due to decrease in the concentration of components in liquid with progress of crystallization are brought into balance. Although a similar result can be obtained by a simple condition (\(\phi =\phi _\mathrm{n}^{*}\)) that nucleation terminates when the concentration of the crystallization component in liquid decreases by more than a certain critical value, whether or not a critical concentration depends on the cooling rate or other parameters at that time is unclear at this point in time. In other words, the dependence of \(\phi _\mathrm{n}^{*}\) on parameters will be clarified when the extremum condition is used (Eq. 9.93).
- 4.
That is, a particle size the population density of which exceeds a certain value.
- 5.
Moreover, the decrease of CSD on the smaller side of particle size occurs spontaneously with progress of crystallization. Its causes include (1) decrease in the nucleation rate associated with decrease in the degree of supersaturation and (2) decrease in the effective nucleation area due to increase in the amount of crystals.
- 6.
Note that to express unified time change of the degree of supercooling of the entire system, the part of et’ /tG in this equation should not be \(e^{(t^{\prime } - t^{\prime \prime })/t_{G}}\).
- 7.
Spohn et al. (1988) linearly approximated Eq. (7.103). In that case, \(G=G_{0} (1+ t/t_{G})\) is obtained and the size distribution is obtained as follows:
$$\begin{aligned} F(R)&= \frac{J_{0} \exp \left( -\frac{t_{G}}{t_{J}} \left( 1 + \sqrt{1 + \frac{2 }{G_{0} t_{G}}(R_{\max } -R)} \right) \right) }{G_{0} \sqrt{1 + \frac{2 }{G_{0}t_{G} }(R_{\max } -R)} } \end{aligned}$$(9.117)$$\begin{aligned} R_{\max }&= G_{0} t \left( 1 + \frac{1}{2} \frac{t}{t_{G}} \right) \end{aligned}$$(9.118)However, the integrated result of the volume fraction is complicated and it is analytically difficult to evaluate the termination time of nucleation \(t^{*}\).
- 8.$$\begin{aligned} \int _{0}^{\infty } x^{3} (1+x)^{\lambda -1} dx&{}={}&6 f_{\lambda } \end{aligned}$$(9.138)$$\begin{aligned} f_{\lambda }= & {} \frac{1}{\lambda (\lambda +1)(\lambda +2)(\lambda +3)} \end{aligned}$$(9.139)
are obtained.
- 9.
At this time, \(\Gamma _{1} /\Gamma _{2}\) monotonously changes from 1/24 to 1/1680.
- 10.
Assuming \(G(t) =dR/dt = G_{0} t^{\bar{n}}\), integrating it from \(t=t^{\prime }\) to t, and letting \(R(t^{\prime })=0\) yields \(t^{\prime } = (t -R/G_{0} )^{1/(\bar{n}+1)}\). Moreover, \(F(R(t^{\prime },t),t) = J_{4} / G(t^{\prime })\) holds, and substituting this into the integral in question and calculating it gives
$$\begin{aligned} \dot{\phi }&=3 \alpha _\mathrm{s} J_{4} t^{\bar{n}} \int _{0}^{R(0,t)} R(0,t^{\prime })^{2} \left[ t^{\bar{n}+1} - \frac{R(0,t^{\prime })}{G_{0}} \right] ^{- \frac{\bar{n}}{\bar{n}+1}} dR(0,t^{\prime }) \nonumber \\& = \frac{6 \alpha _\mathrm{s} J_{4} \left[ G_{0}(1+\bar{n}) t^{1+ \bar{n} } \right] ^{3}}{(\bar{n}+2)(2\bar{n}+3)} \end{aligned}$$(9.151)For this equation to be independent of time, the exponent of t, \(3(\bar{n}+1)\), should be 0, i.e., \(\bar{n} =-1\). However, at this time, the coefficient \((1+\bar{n})^{3}\) will be 0. Because when we integrated dR/dt to obtain the relationship among R, t, and \(t^{\prime }\), we assumed \(\bar{n} \ne -1\), we have to return to that point and confirm it.
- 11.
Note that we do not assume \(G=G_{0} t^{\bar{n}}\). In the case where \(G=G_{0} t^{\bar{n}}\), all crystals have uniform growth rate in the unified time flow regardless of nucleation time.
- 12.
The Beta function is then defined by
$$\begin{aligned} B(a,b)= \int _{0}^{1} x^{a-1} (1-x)^{b-1} dx \end{aligned}$$(9.215)and has the following relationship with the Gamma function.
$$\begin{aligned} B(a,b) = \frac{\Gamma (a) \Gamma (b)}{\Gamma (a+b)} \end{aligned}$$(9.216).
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Toramaru, A. (2022). CSD (Crystal Size Distribution). In: Vesiculation and Crystallization of Magma. Advances in Volcanology. Springer, Singapore. https://doi.org/10.1007/978-981-16-4209-8_9
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