Thus, the activation energy for bubble nucleation, Δ F *, is reduced by a factor ϕ compared to the case of homogeneous nucleation (Hurwitz and Navon 1994 ): 15 $$ \Updelta F^* = \frac{{16\pi \sigma _{\rm LB} ^3 }}{{3\left( {\mathop P\nolimits_{\rm B}^* - P_{\rm L} } \right)^2 }}\phi, $$ where 0 ≤ ϕ ≤ 1 is a geometrical factor depending on the wetting relationships between the bubble and the crystal, as measured by the contact angle θ (Fig.  7 ): 16 $$ \phi = \frac{{\left( {2 - \cos \theta } \right)\left( {1 + \cos \theta } \right)^2 }}{4} $$ Angle θ is controlled by the relative values of the surface tensions associated with the bubble–liquid, bubble–crystal, and liquid–crystal interfaces, σ _LB , σ _SB , and σ _SL , respectively: 17 $$ \cos \theta = \frac{{\sigma _{\rm SB} - \sigma _{\rm SL} }}{{\sigma _{\rm LB} }} $$ Recalling that wetting refers to the tendency of a liquid to spread over a solid surface (like in the case of a droplet of liquid on a solid substrate; Eustathopoulos et al.

1999 ), the case θ > 90° (Fig.  7 a) corresponds to poor wetting conditions, and the case θ < 90° (Fig.  7 b) to good wetting conditions.

The geometrical factor ϕ in Eq.  15 depends on the shape of the nucleation substrate, the expression in Eq.  16 corresponding to the case of a planar substrate (Fig.  7 ).

( 2000 ) model and the experimental data of Mourtada-Bonnefoi and Laporte ( 2004 ); $$ \Updelta P_{\rm N}^{\rm liq} $$ is the supersaturation pressure required to produce homogeneous bubble nucleation in a hydrous rhyolitic liquid with 6 wt% H ₂ O, calculated using σ_LB  = 0.103 N/m; θ is an estimate of the contact angle of water bubbles on hematite crystals (Fig.  7 ), obtained using Eq.  18 ; $$ \sigma _{\rm LB}^{\rm E} $$ is the equivalent surface tension, i.e.

At some point, the degree of water supersaturation is everywhere reduced below Δ X_N and the range of water diffusion is sufficiently short to prevent Δ X from increasing with decreasing pressure: at this point, nucleation stops and the system enters a regime where N is stationary (see also Fig.  7 in Yamada et al.

The main parameter is the contact angle θ (Fig.  7 ), which in the case of magmas covers almost the whole range from 0° to 180°: (1)

In the case of hematite (this study), the contact angle is equal to ≈90° and ΔP_N is reduced by only 33% relatively to the homogeneous nucleation case: at 27.8 kPa/s, ΔP_N is equal to 131 MPa in the presence of hematite as compared to 195 MPa with no crystals present.