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Appendix

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Vesiculation and Crystallization of Magma

Part of the book series: Advances in Volcanology ((VOLCAN))

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Abstract

In this chapter as appendix, some fundamental knowledge and concepts, derivation of equations, material properties related to magmas, and practical methods of textural observation are briefly summarized for reader’s convenience. For more comprehensive understanding, readers will find the helpful illustration in cited articles.

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Notes

  1. 1.

    For chemical potential \(\mu \), see Box in Sect. 2.5.1.

  2. 2.

    A special combination between an intensive variable and an extensive variable whose product (the intensive variable) \(\times \) (the extensive variable) makes one with the unit of energy. In particular, \(T \times S\), \(P \times V\), and \(\mu \times N\). Memorizing these combinations helps you manipulate thermodynamic equations.

  3. 3.

    Specifically, the total sum of the number of molecules contained in bubbles of all size is equal to the concentration; however, in the case where the degree of supersaturation is not so large, the approximation described in the text can be used.

  4. 4.

    Furthermore, the integrand other than the exponential part, \(R^{2}\), can be moved outside the integral by representing the value at \(R_\mathrm{C}\), which gives the local maximum value. The integral In can then be written as follows:

    $$\begin{aligned} In= & \frac{8 \pi R_\mathrm{C} \exp \left( \displaystyle \frac{ 4 \pi \gamma \left( R_\mathrm{C} \right) ^{2} }{3 k_\mathrm{B} T} \right) }{ v_\mathrm{G} D C^{2}} \int _{0}^{\infty } \exp \left( \frac{ \displaystyle \frac{1}{2} \displaystyle \frac{d^{2} { \Delta \mathcal{F}}}{dR^{2}} (R -R_\mathrm{C} )^{2} }{k_\mathrm{B} T} \right) dR \end{aligned}$$
    (11.36)

    Next, using the change of variables \(R_\mathrm{C} x = R -R_\mathrm{C}\) yields \(R: 0 \rightarrow \infty \), \(x: -1 \rightarrow \infty \). Moreover, because \(dR = R_\mathrm{C} dx\) holds, the integral part of Eq. (11.36) becomes

    $$\begin{aligned} R_\mathrm{C} \int _{-1}^{\infty } \exp \left( -a x^{2} \right) dx \approx R_\mathrm{C} \int _{- \infty }^{\infty } \exp \left( -a x^{2} \right) dx \end{aligned}$$
    (11.37)

    where

    $$\begin{aligned} a= \frac{1}{2} \left| \frac{d^{2} \mathcal{W}}{ d R^{2}} \right| \frac{(R_\mathrm{C})^{2} }{k_\mathrm{B} T} = \frac{4 \pi \gamma (R_\mathrm{C})^{2}}{k_\mathrm{B} T} = \pi (Z R_\mathrm{C})^{2} \end{aligned}$$
    (11.38)

    The integral (11.37) then becomes \(R_\mathrm{C} \sqrt{\pi / a}= \Gamma (1/2) R_\mathrm{C} a^{-1/2}\) = \(Z^{-1}\) using the Gaussian integral.   Here, \(\Gamma \) is the Gamma function.

  5. 5.

    The integral whose integrand is the Gaussian function \(e^{-x^2}\) is known as the Gaussian integral, and the integral from \(- \infty \) to \(\infty \) becomes \(\sqrt{\pi }\). That is,

    $$\begin{aligned} \int _{- \infty }^{\infty } \exp \left( - x^{2} \right) dx = \sqrt{\pi } \end{aligned}$$
    (11.39)

    The integral (11.37) then becomes \(\sqrt{\pi /a}\) by change of variables \(y=a^{1/2}x\).

  6. 6.

    Strictly, the modulus of elasticity is distinguished between the bulk (volumetric) modulus \(M^{*}_\mathrm{v}\) and the shear modulus \(M^{*}_\mathrm{s}\) and the viscosity coefficient is then distinguished between the volumetric viscosity \(\eta ^{*}_\mathrm{v}\) and the shear viscosity \(\eta ^{*}_\mathrm{s}\), which are expressed as follows:       

    $$\begin{aligned} M^{*}&{}={}&M^{*}_\mathrm{v} + \frac{4}{3} M^{*}_\mathrm{s} \end{aligned}$$
    (11.48)
    $$\begin{aligned} \eta ^{*}= & {} \eta ^{*}_\mathrm{v} + \frac{4}{3} \eta ^{*}_\mathrm{s} \end{aligned}$$
    (11.49)

    However, they are not distinguished here.

  7. 7.

    Although this expression is used often, the direction of stress is positive in the compression direction. If this expression is incorporated in the equation of motion, the expression should follow the direction of the common coordinate system and should be \(\sigma \) = \(- k_\mathrm{E} \epsilon _{1} \) = \(- \eta \dot{\epsilon }_{2}\).

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  1. Department of Earth and Planetary Sciences, Kyushu University, Fukuoka, Japan

    Atsushi Toramaru

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Toramaru, A. (2022). Appendix. In: Vesiculation and Crystallization of Magma. Advances in Volcanology. Springer, Singapore. https://doi.org/10.1007/978-981-16-4209-8_11

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