Abstract
In the previous chapter, we learned that, for bubble generation, a supersaturated state is required. So, how do bubbles actually form in a supersaturated liquid? How do these bubbles grow to bubbles that we can see? To answer these questions and understand bubble formation, it is absolutely necessary for us to consider the meaning of the size and number of bubbles present in highly vesiculated volcanic rocks such as pumice and scoria, described in Chap. 1. These concepts are necessary for us to understand the dominant factors controlling eruption styles and intensity from the bubble texture in highly vesiculated volcanic rocks. In this chapter, we will understand the mechanisms of such bubble formation and behavior of bubbles in liquids based on both theories and experiments.
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Notes
- 1.
The word stable used in this sentence actually means metastable. Water can be stable as liquid water even if it is cooled down to 0 \(^{\circ }\!\)C, i.e., the freezing point. This state is a metastable state and such water is sometimes called supercooled water. Let us consider a metastable state from a point of view of energy under gravity as an analogy. When you place a cuboid on a horizontal plane, you can do it in two ways: where the longest sides are perpendicular to the horizontal plane and where the longest sides are parallel to the horizontal plane. Although the latter has lower potential energy and is stable, the former is stable unless pressed by an external force that is larger than a certain degree of force.
- 2.
This bubble is a very small one comprising 100–1000 water molecules.
- 3.
The number of possible states indicates the number of possible combinations, under a given total energy of a designated system, of the arrangement of different energy states that microscopic parts constituting the system contain. The maximum value of it corresponds to the arrangement that has the most ways of combination, i.e., the case that has the maximum arrangement of microscopic energy states is an equilibrium state, and entropy at the state reaches a maximum, given by Eq. (3.1).
- 4.
Conventionally, the probability of nucleation has been evaluated by translating internal energy into work (reversible work) under constant entropy and using it based on Landau and Lifshitz (1958). This book uses free energy to determine direct connection to calculations that are handled later.
- 5.
The mark \(*\) is used as a mark representing saturation states.
- 6.
A critical nucleus is a bubble nucleus or a crystal nucleus having a critical radius, which is called the critical nucleus radius or simply the critical radius, i.e., the radius of a critical nucleus.
- 7.
Here it is not definitely stated whether the number of bubbles at the critical size is a value in the equilibrium size distribution \(N_\mathrm{E}^{*}\) or a value in the stationary size distribution \(N_\mathrm{S}^{*}\) discussed later. They are in proportion to each other, as seen later.
- 8.
Note that the migration velocity used here is different from the growth rate. The migration velocity used here includes the effect of fluctuation. \(F^{*}\) can be represented by the equilibrium size distribution \(F_\mathrm{E}^{*}\). Even if the stationary distribution \(F_\mathrm{S}\) is assumed, the flux J of clusters with fluctuation in the size space cannot be simply expressed by the product of size distribution \(F_\mathrm{S}\) and the growth rate A (actual rate at which clusters at the critical size grow, i.e., zero), \(A \cdot F_\mathrm{S}\). J should have the fluctuation term (B), i.e., \(J=A \cdot F - B \cdot \partial F / \partial R\) (Eq. 3.34). It is rather preferable that \(F^{*}\) is symbolically represented by a value of the equilibrium size distribution \(F_\mathrm{E}^{*}\) at the critical size.
- 9.
This is an expedient approach to hold change in size associated with coalescence and fragmentation of bubbles and conservation of mass and to simplify an expression of the addition and subtraction of the size. It will be converted to the radius of bubble later. Assuming that the molecular volume of gas is \(v_\mathrm{G}, l v_\mathrm{G} = 4 \pi R^{3}/3\).
- 10.
Therefore, this equation can be universally applied to other aspects such as change in populations of human beings or insects by replacing the size in Eq. (3.27) with a human being’s income or the size of the insects. However, it is possible only when the transition probability can be completely described as a function of size.
- 11.
See Sect. 11.4 in the Appendix (Chap. 11). As the ambiguity concerning \(F_\mathrm{E}^{0}\) remains to the end, it should be accurately described from the microscopic point of view. Regarding homogeneous reaction of solution and gas, a partition function of the transition state is statistically defined and the transition rate is calculated by the theory of the transition state.
- 12.
This is the approach of Lifshitz and Pitaevskii (1981) and is convenient because the result is determined only by well-known macroscopic physical properties. Another method is to calculate a reaction rate in accordance with the theory of the transition state in terms of statistical mechanics.
- 13.
Toramaru (1995) introduces \(y=C_\mathrm{eq}/C\) to obtain \(R_{0} = 2 \gamma /P \cdot P/P_\mathrm{G}^{*}= 2 \gamma y^{2} /P \) and defines \(v_\mathrm{G}\) by \(v_\mathrm{G}= k_\mathrm{B} T/P\). In this case,
$$\begin{aligned} B(R) =\frac{v_\mathrm{G}^{2} D Cy^{2} }{8 \pi R_\mathrm{C} R^{2}} \end{aligned}$$(3.74)is obtained, which contains \(y^2\) in its numerator. Note that \(v_\mathrm{G}\) is the molecular volume of gas under the liquid pressure P.
- 14.
If the coefficient of the equilibrium size distribution \(F_\mathrm{E}^{0}\) is left as it is,
$$\begin{aligned} J_\mathrm{S} = Z \frac{v_\mathrm{G}^{2} D C F_\mathrm{E}^{0}}{8 \pi R_\mathrm{C}^{3}} \exp \left( - \displaystyle \frac{4 \pi \gamma R^{2}_\mathrm{C} }{3 k_\mathrm{B} T } \right) \end{aligned}$$(3.81)is.
- 15.$$\begin{aligned} J_\mathrm{S} = \frac{D C^{2} ( 1-y^{2} ) }{8 \pi } \sqrt{\displaystyle \frac{k_\mathrm{B} T}{\gamma }} \exp \left( - \displaystyle \frac{4 \pi \gamma R^{2}_\mathrm{C} }{3 k_\mathrm{B} T } \right) = Z \frac{v_\mathrm{G} D C^{2} y^{2}}{8 \pi R_\mathrm{C}} \exp \left( - \displaystyle \frac{4 \pi \gamma R^{2}_\mathrm{C} }{3 k_\mathrm{B} T } \right) \end{aligned}$$(3.83)
where \(v_\mathrm{G}=k_\mathrm{B} T / P\text {D}\) .
- 16.
Force balance in a direction perpendicular to the interface between the crystal and liquid is compensated by the strength of the crystal.
- 17.
\(c_{n}\) is used rather than c because there can be multiple constants.
- 18.
This equation is a typical second-order ordinary differential equation
$$\begin{aligned} \frac{d^{2} y(x)}{dx^{2}} + p(x) \frac{dy(x)}{dx} + q(x) y(x) =0 \end{aligned}$$(3.108)in which the coefficient is not a constant. Many equations appearing in mathematical physics, such as the Legendre equation and Bessel’s equation, result in this form. Because the case involving constant coefficients is familiar, the case in which coefficients are not constants seems to be easily solved. However, the normal method of solving it is very complicated.
- 19.
This solution is expressed by the result in which solutions to an infinite number of eigenvalues are superimposed.
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Toramaru, A. (2022). Mechanism of Bubble Formation. In: Vesiculation and Crystallization of Magma. Advances in Volcanology. Springer, Singapore. https://doi.org/10.1007/978-981-16-4209-8_3
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