Abstract
Bubbles nucleating in supersaturated liquids grow by \({\text {H}_{2}}\text {O}\) diffusion in melts into bubbles (the number of \({\text {H}_{2}}\text {O}\) molecules in the bubbles increases) and expand by decompression (the number of \({\text {H}_{2}}\text {O}\) molecules in the bubbles does not change but the molecular volume increases). In this manner, “growth” (“grow”) and “expansion” (“expand”) are used differently for different phenomena in this chapter. Although growth and expansion simultaneously progress, their behavior will be separately understood by ignoring the other at first. Then, the two phenomena will be combined to understand the experimental results. In this chapter, important factors including time constants of various elemental processes and their dimensionless number ratios, such as the Pélclet number, which is the ratio of the diffusion to the viscosity-limited expansion rate, are discussed.
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Notes
- 1.
It slightly differs from crystal growth. Reaction-controlled crystal growth rate is driven by difference between concentration in liquid and equilibrium concentration at interface (i.e., the degree of supersaturation).
- 2.
This equation slightly differs from the equation for crystallization (Eq. 7.38). This results from (1) difference in the equation of state between bubbles and crystals (This is attributed to the equation of state of gas, whereas crystals are assumed to be linear elastic bodies) and (2) difference in activity relation (water is proportional to the square of concentration, whereas crystals are assumed to be an ideal solution). If an ideal gas \(P_\mathrm{G}^{*} = k_\mathrm{B} T / v_\mathrm{G} \) is applied to gas, the equation agrees with the corresponding definitional equation of crystallization in the following form: \(R_{P}^{*} = 2 \gamma v_\mathrm{G} / k_\mathrm{B} T \).
- 3.
Assuming \(y=C_\mathrm{eq}/C, y^{-2} -1\) gives an estimate of the dimensionless degree of supersaturation, \(\Delta \), and the dimensionless degree of supersaturation \(\Delta \) and the critical nucleus radius \(R_\mathrm{C}\) generally have a relationship
then
can be written.
- 4.
In actuality, \(v_\mathrm{G}\) changes in accordance with the equation of state, e.g., \(v_\mathrm{G} = k_\mathrm{B} T/ P_\mathrm{G}\).
- 5.
Difference from the case of crystallization lies at 2 in the denominator.
- 6.
For the initial delay in bubble growth, when Proussevitch et al. (1993) first indicated it in calculation of bubble growth by a cell model, Sparks (1994) argued that the delay was attributed to viscosity. Sparks (1978) had reviewed the problem of the initial delay in bubble growth and had recognized the problem of the critical size at which the surface tension was effective, as well as the influence of viscosity. Against this, Proussevitch et al. (1993) showed additional simulation results and were adamant that it was the effect of interfacial tension. After that, Navon et al. (1998) revealed that the delay in bubbles’ viscosity-limited growth has exponential characteristics.
- 7.
- 8.
Taking \(\ddot{R}=d \dot{R} / dt = d \dot{R} /dR \cdot d R/ dt = \dot{R} \cdot d \dot{R} /dR\) into consideration, Eq. (4.76) can be rewritten as
Assuming \(y= \dot{R(t)}^{2} /2\),
$$ \frac{dy}{dR} + \frac{3}{R} y = \frac{1}{R} \left( \frac{\Delta P_\mathrm{G}}{\rho } - \frac{2 \gamma }{\rho } \frac{1}{R} \right) $$is obtained, which is an inhomogeneous first-order linear differential equation with respect to y as a function of R and can be solved by the variation of constants method. Assuming \(y=0\) when \(R=R(0)=R_\mathrm{B}\), the constants of integration (0) can be determined.
- 9.
Scaling of the above equation by this timescale (\(\tilde{R}=R/R(0)\), \(\tilde{t} = t/t_\mathrm{c}\)) gives
- 10.
The Péclet number defined here does not represent the ratio of advection to diffusion conventionally used in matter transport. The conventionally used ratio is the ratio of the advection term of the diffusion equation (the second term on the left-hand side of Eq. (4.1)) to the diffusion term (the right-hand side of Eq. (4.1)), such as the Péclet number described in Sect. 4.3.2. However, note that the advection term is not included in the discussion here.
- 11.
Between diffusivity and viscosity, the classical Stokes-Einstein relation holds
in some cases. In this equation, \(a_{\mathrm {M}}\) is the effective radius of a molecule; however, this relation does not hold between the viscosity and the molecular diffusion of silicate melt (Liang et al. 1997).
- 12.
You can convert the unit of concentration from wt% to (number of molecules/m\(^{3}\)) by replacing \(\rho _\mathrm{G} / \rho \)
\(P_\mathrm{G} / k_\mathrm{B} T = 1/ v_\mathrm{G}(P_\mathrm{G})\), where \(v_\mathrm{G}(P_\mathrm{G})\) is the volume of one water molecule in gas at the pressure \(P_\mathrm{G}\).
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Toramaru, A. (2022). Growth and Expansion of Bubbles. In: Vesiculation and Crystallization of Magma. Advances in Volcanology. Springer, Singapore. https://doi.org/10.1007/978-981-16-4209-8_4
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