Abstract
In this chapter, we will discuss the kinetics of the cooling crystallization of magma, i.e., nucleation and growth, to use it as a foundation for making tools for observing unknown information from crystallization texture similar to the vesiculation processes (equilibrium conditions Chap. 2, nucleation process Chap. 3, growth process Chap. 4, temporal development of vesiculation Chap. 5, and complex processes related to vesiculation Chap. 6) dealt with in previous chapters. In this chapter, for crystallization-related problems, we will explain the similar contents in one go because both vesiculation and crystallization are the formation of a new phase from solution or a phase transition process and have similarity. First, using a binary eutectic system as an example of a multi-component system, we will explain crystallization-related conditions from a liquid and define a solid—liquid equilibrium as well as supersaturation in terms of thermodynamics. Next, we will conduct the formulation of crystal nucleation and crystal growth, evaluate experiments using these results, and examine the classical theory. As the crystal growth-related problems, the compositional zoning that is often observed in crystals is discussed for simplified cases, involving the movement of the interface, which has been extensively studied in metallurgy.
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Notes
- 1.$$\begin{aligned} \Delta s_\mathrm{A}&{}={}&\Delta s_\mathrm{A} (T_{0}, X_{0}) = s_\mathrm{Liquid}^\mathrm{A}(T_{0},X_{0}) -s_\mathrm{Solid}^\mathrm{A} (T_{0},1) \end{aligned}$$(7.4)$$\begin{aligned}= & {} s_\mathrm{Liquid}^\mathrm{A}(T_{0},X_{0}) \!-s_\mathrm{Liquid}^\mathrm{A} (T_{0},1) \!+ s_\mathrm{Liquid}^\mathrm{A} (T_{0},1) \!- s_\mathrm{Solid}^\mathrm{A} (T_{0},1)\ \ \end{aligned}$$(7.5)$$\begin{aligned}= & {} \frac{\partial s_\mathrm{Liquid}^\mathrm{A}}{\partial X} (X_{0} - 1) + \Delta s_\mathrm{A} (T_{0},1) \end{aligned}$$(7.6)$$\begin{aligned}= & {} \Delta s_\mathrm{A} (T_\mathrm{A},1) + \frac{\partial s_\mathrm{Liquid}^\mathrm{A}}{\partial X} (X_{0} - 1) + \frac{\partial \Delta s_\mathrm{A}}{\partial T} (T_{0} - T_\mathrm{A}) \end{aligned}$$(7.7)
Because \(\Delta s_\mathrm{A} (T_\mathrm{A},1) = \Delta s_\mathrm{A}^{0}\) holds, \(\Delta s_\mathrm{A} = \Delta s_\mathrm{A}^{0}\) can be considered to hold when the correction term for composition in the second term of the right-hand side and the correction term for temperature in the third term are small.
- 2.
In the case of crystals in liquid, mechanical equilibrium can be considered to always hold, unlike in the case of bubbles.
- 3.
When temperature \(T_{0}\) is assumed to be undercooling temperature, \(T_\mathrm{L}\) higher than \(T_{0}\) has to be assumed. From Eq. (7.8), \(T_\mathrm{L} -T_{0} = \Delta \tilde{h}^{-1} T_\mathrm{L} \ln (C_\mathrm{L}/C_{0})\) holds. On the other hand, from Eq. (7.17), \(1- T_{0} /T_\mathrm{L}= R_{T}^{*}/R\) holds. From these two equations,
$$\begin{aligned} \frac{C_\mathrm{L}}{C_{0}} = \exp \left[ \Delta \tilde{h} \frac{R_{T}^{*}}{R} \right] = \exp \left( \frac{R_{X}^{*}}{R} \right) \qquad \qquad \qquad {(22)} \end{aligned}$$where \(\Delta \tilde{h} = \Delta h/ (k_\mathrm{B} T_{0})\). \(R_{X}^*\) at this time is \(2\gamma v_\mathrm{S}/k_\mathrm{B}T_0\) and \(T_0\) is used as temperature.
- 4.
For this equation, when supersaturated concentration C is considered as a reference concentration \(C_{0}\), the ratio of equilibrium concentration at the degree of undercooling \(T_{0} -T\) (or difference in concentration, i.e., the degree of supersaturation \(1 -C_{0}/C_\mathrm{eq}\)) as first order approximation of logarithm) is defined using the critical radius. In other words, this equation becomes Eq. (7.19) when Eq. (7.3), which defines liquidus, is used.
- 5.
The ratio \(R_{X}^{*} / R_{T}^{*}\) is equal to the dimensionless number \(\alpha _{2}\) (Eq. 7.10), which defines the slope of liquidus:
$$\begin{aligned} \frac{R_{X}^{*}}{R_{T}^{*}} = \frac{\Delta s}{k_\mathrm{B}} = \alpha _{2}. \end{aligned}$$(7.31) - 6.
Note that it differs from \(R_{X}^{*}\) in the case of bubbles only by a factor \(y^{2}\).
- 7.
When \(\Delta g^{*} = (C_{\infty } -C_{R})/C_{\infty }\)= \(R_{X}^{*}/R_\mathrm{C} -R_{X}^{*}/R\) is considered as the driving energy for growth, Eq. (7.67) then becomes
$$\begin{aligned} \frac{dR}{dt} = \frac{v_\mathrm{S} \cdot D \cdot C_{\infty }}{R} \left\{ 1- \exp \left( - \frac{\Delta g^{*}}{k_\mathrm{B} T} \right) \right\} \qquad \qquad \qquad {(69)} \end{aligned}$$which is equivalent in the form to the expression of the reaction-limited growth explained in the next section, although the coefficient differs. However, note that the difference in concentration used for the driving energy for growth is the difference in concentration due to the difference in distance between the interface and a distant point. .
- 8.
The case in which \(R_\mathrm{C}\) (or \(\Delta \)) is assumed to change with time and used as time in transforming the equation will be the Lifshitz-Slyozov theory, which is discussed in Sect. 6.1.2.
- 9.
- 10.
Refer to the Footnote 7.
- 11.
Because \(\Delta g^{*} + \Delta g_{+} = \Delta g_{-}\) is assumed, if \(\Delta g_{-}\) is assumed to be considered as the activation energy of diffusion in the solid phase here, the free energy of solid-liquid phase transition would be equal to the difference in the activation energy of diffusion between solid and liquid. However, it is unclear whether this relation holds.
- 12.
Although the right-hand side is an approximate equation when the Frenkel-Wilson law is used, because \(\Delta \tilde{h}/\tilde{q}\) cannot become much smaller than 1, it cannot be considered as an approximation from the middle part of the equation.
- 13.
Lasaga used the inverse of viscosity as the index.
- 14.
The case following the G-T law in dilute solution was dealt with in Sect. 7.4.
- 15.
The Av number and \(\alpha _{3}\) are mutually related. Assuming that the coefficient of the nucleation rate is \(J_{0}\),
$$\begin{aligned} \alpha _{3} = \frac{(G_{0}/J_{0})^{1/4}}{R_{X}^{*}} \mathrm{Av}^{-1/4} \end{aligned}$$(7.122)holds.
- 16.
A programmed cooling system in which temperature change occurs because of release of the latent heat is regulated by electrical circuits or computer programs to ensure cooling at a constant rate.
- 17.
In the case of vesiculation, if a system is finite and mechanically interacts with its surroundings, the system receives pressure-related feedback.
- 18.
Although this equation approximately holds, it does not hold with respect to the exponent denoting the parameter dependence, to be accurate. This is because sudden change in the nucleation rate is represented only by \(J_\mathrm{MAX}\) and \(\delta t_\mathrm{n}\).
- 19.
The discussion here assumes the case where the pre-exponential factor of the nucleation rate is sufficiently large and nucleation does not continue for a long time.
- 20.
For narrow dikes, magma movement due to forced convection caused by cooling from the wall of the dike is negligible compared with the cooling time. Similarly, for thin sills, convection does not occur because the Rayleigh number is small and forced convection caused by cooling from the side boundary is negligible.
- 21.
A method in which a realistic crystallization sequence is assumed and by which the ratio of each mineral contained in rocks after complete crystallization of magma is calculated. Using this method, chemical composition of magma can be expressed by the ratios of each mineral component and the chemical composition of magma can be plotted on an equilibrium phase diagram.
- 22.
In typical binary solid solution such as plagioclase and olivine, chemical composition is sequentially changed by reaction between the liquid phase and solid phase on crystallization. If chemical equilibrium in the whole system (chemical potential of any chemical component in the system is equal everywhere in the space) holds perfectly, crystal composition should be homogeneous in grains at every step of crystallization. Moreover, chemical equilibrium in the whole system holds in the case where reaction in the solid-liquid interface and diffusion in the solid phase and the liquid phase are sufficiently fast or the case in which physical conditions such as temperature and pressure change sufficiently slowly. As per the situation, because diffusion in solid is slow, the composition at the vicinity of the interface often differs from the composition at the interior of grains. This is observed as growth zoning.
- 23.
- 24.
Because chemical potential
$$\begin{aligned} \mu ^\mathrm{A}_\mathrm{L} = \left( \frac{\partial \mathcal{G}_\mathrm{L}}{\partial N_\mathrm{A}} \right) _{T,P} \end{aligned}$$(7.167)has a characteristic of partial molar quantity, chemical potentials of individual components are mutually related under Eq. (7.167). In this case, because temperature and pressure are constant, the same constraint as the Gibbs-Duhem equation holds.
- 25.
Hereafter, the composition (molar fraction) X indicates that of component B unless otherwise stated.
- 26.
Any units such as the molar fraction X and the number of molecules per unit volume C can be used for concentration but wt% is used here to maintain consistency with the discussion on decompression-induced crystallization (Chap. 8).
- 27.
Note that the space coordinate x independent of the movement of the interface is used here.
- 28.
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Toramaru, A. (2022). Cooling Crystallization of Magma. In: Vesiculation and Crystallization of Magma. Advances in Volcanology. Springer, Singapore. https://doi.org/10.1007/978-981-16-4209-8_7
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