Differentiation of silicates from H2O ice in an icy body induced by ripening
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013
Received: 23 September 2012
Accepted: 8 July 2013
Published: 6 December 2013
One of the probable scenarios of differentiation between silicate-ice in an icy object is the settling of a silicate particle in water after the melting of the object. In order for settling to proceed or occur, the size of the particle should be sufficiently large such that the settling velocity of the particle exceeds the background flow velocity induced by thermal convection. The sizes of the particles change because of dissolution and precipitation. This process is called ripening. In this study, the critical particle sizes required for settling, and the timescales for the growth of the particles to these sizes through ripening, are analytically derived. It is observed that settling is possible if the silicate particles coagulate with each other to form a network in water. If the particles do not coagulate, the probability of the occurrence of settling is low, because the time duration required for the particle growth to the critical size is large. The coagulation of silicate particles strongly depends on the pH of the water.
The outer regions of the solar system are populated by objects such as icy planets, satellites, and cometary nuclei, along with Kuiper belt objects. The internal structures of several such icy satellites have been estimated by space missions. The process of differentiation is crucial to the internal structure of such objects. Ganymede can be differentiated as an iron core, rock mantle, and icy mantle (Showman and Malhotra, 1999). On the other hand, only a partial differentiation between ice and rock can be made for Callisto (Kuskov and Kronrod, 2005). The most probable scenario for ice–rock differentiation is the gravitational settling of silicate particles after the melting of the silicate-ice mixture in the object. For example, Barr and Canup (2010) investigated the differentiation of Ganymede due to impact melting. They showed the size of a silicate particle settling downward to be larger than 30μm.
However, it should be noted that the building blocks of the above-mentioned icy objects are interstellar grains which are sub-micron sized icy grains containing silicate cores (Li and Greenberg, 1997). Let us suppose that a sub-micron-sized silicate particle is placed in water. The question is whether the particle can settle down or not. The Stokes settling velocity for water is quite small; it is on the order of 10−7 cm s−1 inside a 1000-km-radius object. This velocity is obtained from the Stokes’s drag law (Eq. (2) shown later with parameters below Eq. (5)). Moreover, this small settling velocity cannot be achieved due to thermal convection inside the object. Therefore, differentiation between silicates and ice is improbable for sub-micron particle sizes.
In this paper, the critical size of a particle for settling is determined, taking into account the coarsening due to Ost-wald ripening. Correspondingly, this study has two aspects. One is the determination of the background flow velocity against which a particle settles down. This is achieved using the formulas of convective velocity and of the permeable flow velocity. The other aspect is devoted to calculating the timescale for the particle growth through ripening. Ostwald ripening has been mainly studied in the field of metallurgy. The theoretical framework of Ostwald ripening has been well developed (Ratke and Voorhees, 2002). Moreover, an analytical solution for the evolution of the size distribution, appropriate to the conditions in this study, has been obtained (Kumaran, 1998).
In the next section, the scenario of this model is described, because this study involves several mechanisms which are unfamiliar in the field of geophysics. In Section 3, the critical size for differentiation is analytically derived. In Section 4, the timescale for particle growth is estimated, and the requirement for the composition of ice is discussed in Section 5. Section 6 presents the conclusions.
2. Scenario of This Model
The scenario of this study is schematically displayed in Fig. 1. Here, we assume that a molten layer of an icy object is formed by a heating event. There are several possible origins of the heating event causing the melting. The decay of a radionuclide, such as 40K or 26Al, is a promising mechanism which leads to melting (Prialnik and Merk, 2008). Tidal heating is also a powerful source of heating as seen in Europa’s underground ocean (Ross and Schubert, 1987). Impact melting is another reasonable mechanism (Barr and Canup, 2010) because impact is a common event in the planetary formation process. Here, I do not address the origin of the heat source and simply assume the existence of a molten layer of thickness D inside an icy object.
If melting occurs, silicate particles are immersed in water. The constituent molecules dissolve in the water according to their solubility. It should be noted that the solubility depends on the surface curvature (or size) of a given particle. The solubility of a small particle is larger than that of a large particle because of the Gibbs-Thomson effect (Ratke and Voorhees, 2002). As a result, a molecule originates from a small particle and precipitates on a large particle (Fig. 1(a)), leading to a coarsening of the particles. If the settling velocity of the large particle becomes larger than the background flow velocity of water, the settling and differentiation of silicate proceeds.
The discussion above implicitly assumes that each particle is kept isolated without coagulating with other particles. This is true if there is a repulsive interaction between the silicate particles in water. In many cases, the surface of a sub-micron sized particle is charged. The origin of charging is the ionization of the surface material, or ions contained in the water. The former depends on the pH of water, and the latter depends on the amount of salt in the water. Repulsion of particles is caused by the osmotic pressure induced between two approaching particles. This mechanism is summarized as DVLO theory (Safran, 1994). In addition to this mechanism, it is possible that organic molecules formed in the interstellar space are dissolved in the water. This molecule, when absorbed by a particle, may induce steric stabilization of silica particles (Iler, 1979).
On the other hand, there is a particular pH value where the surface charge on the particle is zero. This pH is known as the point of zero charge (PZC). The coagulation of particles efficiently proceeds around PZC. If the amount of salt in the water is large, the osmotic pressure is not induced because the contrast of ion concentration in the gap between the particles disappears and coagulation proceeds.
All of these mechanisms depend on the composition of the water. The composition is determined through the chemical evolution in the molecular cloud and in the pro-toplanetary nebula, and is highly uncertain. Therefore, only two extreme cases are considered in this paper. One is the case without coagulation of particles. In this case, the particles do not coagulate and remain isolated throughout the evolution. In the other case, coagulation proceeds efficiently. The particles coagulate with each other to form a spanning network throughout the molten layer inside the icy object.
3. Critical Sizes for Settling
3.1 Without coagulation
The formation of a molten layer inside an icy object has been discussed in many studies (Multhaup and Spohn, 2007; Schubert et al., 2007; Castillo-Rogez et al., 2012; Czechowski, 2012). Heating by the decay of 26Al efficiently melts the interior of Rhea, a Saturnian satellite with a radius of 764km (Czechowski, 2012). The temperature uniformly exceeds 260 K up to 0.8×satellite’s radius from the center, and the molten region extends to 0.5×satellite’s radius. The temperature difference between the center and the surface induces an intensive convection of water-rock mixture which flattens the temperature distribution. The radius of the molten region critically depends on the amount of 26Al present at the beginning. The decay of 26Al leads to the melting of Enceladus with a radius of 250 km (Schubert et al., 2007) and the small satellite Phoebe, 106 km in radius (Castillo-Rogez et al., 2012). Even if 26Al is depleted upon the formation of an icy object, the decay of a long-lived radionuclide can induce the melting, provided that the ice contains ~ 10 weight % ammonia (Nagel et al., 2004; Multhaup and Spohn, 2007).
If Vf is larger than the upward velocity of the flow, the particle settles down and differentiation proceeds.
3.2 Setting under coagulation
The fractal dimension d is ≃ 2.5 for a diffusion-limited aggregate (Meakin, 1983). Because the mass ratio between silicate and water is roughly 1:1, based on the solar composition (Greenberg, 1998), the volume fraction of silicate is ϕ ≃ 0.3, taking into account the density ratio of 3 between silicate and water. In this case, the size of the aggregate becomes ra ≃ 10r.
The mean collision time of a silicate particle is given by 1/nσv, where n = ϕ/(4πr3/3) is the number density of the silicate particles, σ = π(2r)2 is the collisional cross-section between two spherical particles of radius r, and is the collisional velocity induced by the Brownian motion (ρs is the density of a silicate particle). The timescale is 1/nσv = 5.5 × 10−6 s with r = 0.1 μ m, T = 300 K, ϕ = 0.33, and ρs = 3.0 g cm−3. The timescale changes as the aggregates grow. If the aggregates have the same size of ra, the number density becomes n/(ra/r) d , the cross-section is σ(ra/r)2, and the velocity is v/(ra/r)d /2, where (ra/r) d is the number of particles contained in an aggregate. The collision timescale for the fractal aggregate of dimension d is then given by (ra/r)2−3d/2/nσv. The timescale decreases as the aggregate grows, if d = 2.5. As a result, the timescale of growth of silicate particles is still roughly given by 1/nσv = 5.5 × 10−6s. Therefore, the formation of a spanning network is completed shortly after the formation of the molten layer. The typical size of the pore (filled with water) is almost the same as the size of a silicate particle, because the volume fraction of silicate particles is ϕ ≃ 0.3. When the thermal convection of water takes place, water flows inside these pore spaces.
If the mechanical strength of the network of particles is sufficiently large, the network can sustain its weight against self gravity. Because the network of particles is immersed in water which is in hydrostatic equilibrium, the effective density of the network is ϕΔρ, where Δρ is the density difference between silicate and water. Then, the typical pressure applied to the network in a molten layer inside an object of size R is ~ G(ϕΔρ)2R2 = 2.9 × 108(R/1000km)2 dyncm−2, where G is the gravitational constant, and Δρ = 2 g cm−3 is the density difference between silicate (density ρs = 3gcm−3) and water (density ρw = 1 gcm−3). The volume fraction of silicate ϕ = 0.33 is adopted in this formula.
The growth of silicate particles via Ostwald ripening also proceeds in this coagulated case (see Fig. 1(b)). Coagulation of particles forms a neck between them. The neck has a concave surface and low solubility, thereby leading to the precipitation of solutes and to the growth of the neck. As a result, a series of particles loses necks and the local thickness reflects the initial size of the particles (Fig. 1(c)). The strength of the network of grains increases by this growth. Without neck growth, the volume fraction of the aggregates composed of 0.76-μm-radius SiO2 particles in air increases to ϕ ≃ 0.33, when a pressure of ~ 107 dyn cm− is applied (Blum and Schrapler, 2004). Therefore, the strength of the network of grains with growing necks should be larger than this value. A typical compressive strength of oxide ceramics, which is an analogous material to the network of sub-micron-sized grains with necks grown by sintering, is on the order of 109dyncm−2 (Carter and Norton, 2007). In this case, neither a compaction of the network, nor the differentiation of silicate grains from water, proceeds.
Because the solubility at the relatively thinner section is larger than that at thicker sections (the curvature perpendicular to the network axis is large at a thin section), the thinner part dissolves and shrinks. Eventually, the network is broken at the thinner sections. This fragmentation is a well-known phenomena in the synthesis of monolithic silica from sol-gel reactions (Nakanishi, 2011).
The successive disconnection in the above-mentioned manner produces a fragment that is completely separated from the remaining network of connected particles. A fragment produced by this process settles down provided that the background flow velocity is small. Consequently, the fragment again sticks to the network, and the density, or the packing fraction occupied by the silicate particles, increases.
4. Timescale for Settling
In the settling process, the molecules constituting a silicate particle dissolve into, and precipitate from, water. As described before, the difference in solubility depending on the surface curvature promotes dissolution from a small particle and precipitation on a large particle. Eventually, the small particle disappears and a relatively thin section in a network of particles disconnects, while a large particle, or a thick section in a network, grows. As a result, the size distribution (or the thickness distribution of a network) of particles evolves.
The last approximation is valid because rs/r ≪1. The concentration of the solvent SE is determined through the total balance of the dissolution and deposition. SE is not determined here, because it is irrelevant to the timescale of particle ripening.
These timescales are small compared to the cooling timescale of a molten layer. If the thickness D is larger than , the cooling timescale τc is longer than 100 yr and settling is possible. Here, the thermal diffusiv-ity of ice is adopted as κice = 6.9 × 102/T2 cm2 s−1 with T = 270 K (Haruyamaet al., 1993).
In the absence of coagulation, the critical size for settling is ~ 1 cm (Eq. (5)). Because τg ∝ r2, the timescale is more than 1010 yr. Therefore, if the silicate particles do not coagulate, the probability of differentiation occurring is low.
Two extreme cases, coagulation and without coagulation, have been discussed. Actual internal evolution of an icy body might deviate from these two cases. If the volume fraction of silicate grains ϕ is smaller than the canonical value ϕ = 0.33, the permeability K(Eq. (8)) becomes large because the low packing fraction enables a high-velocity permeable flow. This effect lowers the critical radius of grains (Eq. (9)). The critical size is larger than the typical grain size 10−5 cm if ϕ < 0.22. Therefore, differentiation is only possible with a high volume fraction of silicate grains.
Another possibility is the fragmentation of the silicate network due to stress generated by the convective flow. A fragment larger than the critical size (Eq. (5)) settles down against the convective flow. However, the porosity inside the fragment does not change by the settlement. To reduce the internal porosity, fragmentation by ripening followed by settling of silicate particles is required.
As shown in the previous sections, one critical quantity that determines the settling is the pH of the water. Two factors depend on this pH. First, the coagulation of silicate particles is affected by the pH value. The surface of a particle is charged positively or negatively depending on the pH. There is a particular pH called “point of zero charge” (PZC) at which the surface of a particle is neutral. The surface charge increases as the pH deviates from this value. The PZCs for representative silicates are as follows: amorphous silica/quartz 1.8–3.5, feldspar 2.0–2.4, olivine 4.1, kaolinite 3.3–6.0, and serpentine 9.6–11.8 (Parks, 1967). Coagulation proceeds efficiently around the PZC. An electrostatic repulsive force prevents coagulation as the deviation from the PZC increases. This is critical because the probability of differentiation reduces if coagulation does not take place.
The second mechanism involving pH is the dependence of the dissolution rate as seen in Fig. 3. This effect influences the timescale for breaking the network by two orders of magnitude.
The pH of water is determined by water–rock reaction. A calculation of an oceanic composition of Enceladus (Zolotov, 2007) starting from a mixture of water and CI chon-drite composition rock showed that the pH of the ocean is alkaline, ranging from 8 to 11. This range of pH would be buffered by secondary minerals (saponite, serpentine) and an abundance of cations formed by aqueous/hydrothermal alterations. The range of pH includes the PZC of serpentine. If this is the case, coagulation of serpentine grains proceeds and differentiation of serpentine from water occurs as a result of ripening. The combination of the secondary minerals and the pH determined from water–rock reaction is critical in the differentiation of an icy body.
The differentiation of ice–silicate in icy objects is accompanied by the melting of the object. In order for differentiation to proceed, the size of a silicate particle should be larger than a critical size for which the settling velocity is larger than the background convective velocity of water. In this study, the critical sizes for settling and the timescale for growth of silicate particles to these sizes have been derived. If the silicate particles coagulate with each other in the water-silicate mixture, differentiation is possible because the flow velocity of water is slow. On the other hand, if the silicate particles cannot coagulate because of repulsive forces between them, differentiation does not proceed because the time duration for growth to the critical size for settling is large.
The author greatly thanks the constructive comments by two anonymous reviewers. The author is grateful to Dr. Maria Teresa Capria who provided a comfortable stay in Instituto di Astrofisica Spaziale e Fisica Cosmica (Roma), where this work was carried out.
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