Simple models for disequilibrium fractional melting and batch melting with application to REE fractionation in abyssal peridotites

doi:10.1016/j.gca.2015.10.020

Geochimica et Cosmochimica Acta

Volume 173, 15 January 2016, Pages 181-197

https://doi.org/10.1016/j.gca.2015.10.020 Get rights and content

Abstract

Disequilibrium melting arises when the kinetics of chemical exchange between a residual mineral and partial melt is sluggish compare to the rate of melting. To better understand the role of a finite crystal–melt exchange rate on trace element fractionation during mantle melting, we have developed a disequilibrium melting model for partial melting in an upwelling steady-state column. We use linear kinetics to approximate crystal–melt mass exchange rate and obtain simple analytical solutions for cases of perfect fractional melting and batch melting. A key parameter determining the extent of chemical disequilibrium during partial melting is an element specific dimensionless ratio (ε) defined as the melting rate relative to the solid–melt chemical exchange rate for the trace element of interest. In the case of diffusion in mineral limited chemical exchange, ε is inversely proportional to diffusivity of the element of interest. Disequilibrium melting is important for the trace element when ε is comparable to or greater than the bulk solid–melt partition coefficient for the trace element (k). The disequilibrium fractional melting model is reduced to the equilibrium perfect fractional melting model when ε is much smaller than k. Hence highly incompatible trace elements with smaller mobilities in minerals are more susceptible to disequilibrium melting than moderately incompatible and compatible trace elements. Effect of chemical disequilibrium is to hinder the extent of fractionation between residual solid and partial melt, making the residual solid less depleted and the accumulated melt more depleted in incompatible trace element abundances relative the case of equilibrium melting.

Application of the disequilibrium fractional melting model to REE and Y abundances in clinopyroxene in abyssal peridotites from the Central Indian Ridge and the Vema Lithospheric Section, Mid-Atlantic Ridge revealed a positive correlation between the disequilibrium parameter ε and the degree of melting, which can be explained by an increase in melting rate and a decrease in REE diffusion rate in the upper part of the melting column. Small extent of disequilibrium melting for LREE and equilibrium melting for HREE in the upper part of the melting column can explain the elevated LREE abundances or spoon-shaped REE patterns in clinopyroxene in more refractory abyssal peridotites. The latter has often been attributed to melt refertilization.

Introduction

The distribution and fractionation of trace element between minerals and coexisting melt are important to the interpretation of mantle melting and melt migration processes. Interpretation of the trace elements in mantle rocks is based largely on simple melting models such as batch melting, fractional melting, dynamic or continuous melting, and flux melting models (e.g., Shaw, 2006 and references therein). These simple models are also referred to as equilibrium melting models because residual solid and coexisting melt are in instantaneous chemical equilibrium during melting and melt migration. When the rate of chemical exchange between a residual mineral and partial melt is smaller than the rate of melting, chemical disequilibrium arises. Incomplete chemical exchange between interiors of mineral grains and their surrounding melt results in lesser extent of fractionation between residual solid and partial melt when compared to equivalent cases of equilibrium melting.

Models for trace element fractionation during disequilibrium mantle melting were presented in a number of studies (e.g., Allègre and Minster, 1978, Prinzhofer and Allègre, 1985, Iwamori, 1992, Iwamori, 1993, Qin, 1992, Van Orman et al., 2002, Liang, 2003a, Rudge et al., 2011). The earlier model of Allègre and Minster (1978) and Prinzhofer and Allègre (1985) assumes no chemical exchange between interiors of residual minerals and their surrounding melt. This is a case of complete disequilibrium melting that may be more relevant to melting under crustal (and hydrous) conditions where temperature is low and kinetics is sluggish. In a more general case of disequilibrium melting, melt and residual minerals exchange at finite rates compared to the rate of melting. A common treatment of disequilibrium melting is to consider grain-scale diffusive exchange between residual minerals and their surrounding melt during melting (Iwamori, 1992, Iwamori, 1993, Qin, 1992, Van Orman et al., 2002). This is a moving boundary problem that has no analytical solution. Although numerical solutions to the grain-scale moving boundary problem are straightforward, they become a formidable computational task at larger length scales and higher spatial dimensions. For large-scale geochemical mass transfer problems, such as melt generation beneath mid-ocean ridge spreading center, it is convenient and practical to reformulate the grain-scale mass transfer problem on continuum length scale where mineral and melt compositions are defined in a representative elementary volume (REV) that is much larger than mineral grain size through proper averaging and upscaling (e.g., Bear and Bachmat, 1990, Quintard and Whitaker, 1996, Whitaker, 1999). A simple treatment of grain-scale kinetics is to approximate diffusive flux at the mineral–melt interface by linear kinetics and uses an average rate constant to characterize mineral–melt diffusive exchange within the REV. This boundary layer approximation for solid–melt mass transfer in porous media has been widely used in studies of chemical chromatography and mantle metasomatism (e.g., Vermeulen, 1953, Glueckauf, 1955, Navon and Stolper, 1987, Bodinier et al., 1990, Cussler, 1997, Liang, 2003b). Using linear kinetics for mineral–melt diffusive exchange, Liang (2003a) generalized the dynamic melting model of McKenzie (1985, see also Albarède, 1995, Zou, 1998, Shaw, 2000) by assuming constant porosity in the solid and melt mass conservation equations and solved the system of coupled ordinary differential equations numerically following the temperature–pressure path of a subducting slab. Rudge et al. (2011) discussed continuum scale formulations for disequilibrium melting of a multicomponent mantle from perspectives of non-equilibrium thermodynamics. Their expressions for mineral–melt mass transfer are essentially the same as those discussed in Liang, 2003a, Liang, 2003b based on grain-scale averaging.

With the exception of the complete disequilibrium melting models of Allègre and Minster (1978) and Prinzhofer and Allègre (1985), there is no analytical solution for disequilibrium melting. The main purpose of the present study is to develop simple models for trace element fractionation during disequilibrium fractional melting and batch melting. Fractional melting and batch melting are two limiting styles of mantle melting. Melting in a one-dimensional steady-state upwelling column without instantaneous melt extraction is equivalent to batch melting (Ribe, 1985, Asimow and Stolper, 1999). The very depleted nature of incompatible trace elements in abyssal peridotites, olivine-hosted melt inclusions, and cumulates crystallized from basalts suggests that melting in the oceanic mantle is fractional or near fractional (e.g., Johnson et al., 1990, Johnson and Dick, 1992, Ross and Elthon, 1993, Sobolev and Shimizu, 1993, Shimizu, 1998). Fractional or near fractional melting requires efficient isolation of partial melt from residual solid. This can be achieved physically by channelized melt migration and chemically by disequilibrium melting and melt transport (Spiegelman and Kenyon, 1992, Hart, 1993, Kelemen et al., 1997). In the next section, we first outline a steady-state model for non-modal disequilibrium melting in an upwelling melting column. We then focus on a simplified problem of disequilibrium perfect fractional melting and present analytical solutions for a trace element in the partial melt and residual solid (Section 2). (Approximate solutions for near equilibrium batch melting are summarized in Appendices B and C.) We discuss key features of disequilibrium fractional melting and show that small extent of chemical disequilibrium can have a significant effect on the abundance and distribution of highly incompatible trace elements in residual solid (Section 3). As an example of geological application, we invert for degree of melting and a disequilibrium parameter that measures the extent of chemical disequilibrium from abundances of REE and Y in clinopyroxene in abyssal peridotites using the disequilibrium fractional melting model (Section 4). We assess effects of disequilibrium batch melting in the lower part of the melting column, melt refertilization in the upper most part of the melting column, and mantle metasomatism on REE patterns in clinopyroxene and discuss physical meanings of the inverted parameters (Section 5). Small extent of chemical disequilibrium may be present during adiabatic melting of the oceanic mantle.

Section snippets

Model setup

We consider melting and melt migration in a one-dimensional steady-state upwelling column in which part of the melt generated is extracted to nearby channels or conduits. We are interested in steady-state distributions of a trace element in the interstitial melt (concentration $C_{f}$ ), residual bulk solid (C_s) and minerals ( $C_{s}^{j}$ , j = 1, 2, … , N) in the upwelling column when the system is locally out of chemical equilibrium, where N is the number of minerals in the system. For simplicity, we neglect

General features of disequilibrium fractional melting

To obtain simple analytical solutions, we assumed that the mineral–melt exchange rate constant R_j is the same for all the minerals in the melting column. In terms of diffusive mass transfer through mineral grains, this assumption implies that zoning profiles for the trace element of interest in minerals are the same for all the minerals in the system when normalized to grain sizes and rim concentrations. Consequently, (average) concentrations of the trace element in the minerals are

REE and Y depletion in abyssal peridotites

The abundance of REE in cpx in abyssal peridotites has often been used to infer the degree of melting and style of melt extraction in the oceanic mantle (e.g., Johnson et al., 1990, Johnson and Dick, 1992, Dick and Natland, 1996, Niu and Hèkinian, 1997, Hellebrand et al., 2002, Hellebrand and Snow, 2003, Niu, 2004, Brunelli et al., 2006, Brunelli et al., 2014, Seyler et al., 2007, Liang and Peng, 2010). Interpretations of the trace element data in previous studies are based on equilibrium

The assumption of R₁ = R₂ = ⋯ = R_N

The simple analytical solutions (Eqs. (5a), (5b), (5c)) were obtained under the assumption that the mineral–melt exchange rate constants are the same for all the minerals in the residue. According to Cherniak and Liang (2007), REE diffusion in enstatite is not sensitive to ionic radius and their diffusivities are comparable to those of Gd and Dy in diopside (Van Orman et al., 2001). To assess the effect of REE diffusion in orthopyroxene (opx) on the inverted melting parameters, we solved the

Acknowledgments

YL wishes to thank Frank Richter for many simulating discussion regarding disequilibrium melting during the 1991–1992 academic year at the University of Chicago where solutions similar to the near equilibrium batch melting model outlined in Appendix B were first obtained. We thank Soumen Mallick for pointing out the existing isotope data for the VLS samples and for useful discussion. This paper benefited from thoughtful review comments from Daniele Brunelli, Eric Hellebrand, and Richard Katz.

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Citation Excerpt :
Inverse modeling of REE depletion in clinopyroxenes in abyssal peridotites demonstrated the need of a small extent of chemical disequilibrium to account for the less fractionated LREE patterns in a large fraction of residual peridotite samples from Mid-Atlantic Ridge, Central Indian Ridge, Southwest Indian Ridge, Lena Trough, and American–Antarctic Ridge (101 out of 135 samples, Liang and Liu, 2016; Liu and Liang, 2017a). A finite solid–melt exchange rate due to slow cation diffusion in clinopyroxene reduces the extent of depletion of incompatible trace elements in residual solid, increasing the effective solid–melt partition coefficient (Qin, 1992; Iwamori, 1994; Van Orman et al., 2002; Liang and Liu, 2016). Hence for slow diffusing cations, such as LREE and Hf, their stretching factors will be less than those predicted by the equilibrium melting models.
The source region of basalt in the upper mantle is heterogeneous and may consist of depleted background mantle and blobs of enriched mantle. The size, shape, and distribution of the enriched blobs in the upper mantle are unknown but may play an important role in controlling variations in isotope ratios and trace element abundances in basalts and residual peridotites. During decompression melting, the mass flux of interstitial melt increases while the mass flux of residual solid decreases upward from the solidus, resulting in an acceleration of the effective transport velocity for an incompatible trace element in the residue. Consequently, a blob of chemical heterogeneity is stretched during its transit through the melting column. Here we quantify the melt migration induced size change by allowing trace element abundances and isotope ratios in the mantle source to vary as a function of time and space. We use simple analytical solutions for the time-dependent batch melting and fractional melting models to illustrate how a trace element or an isotope ratio varies spatially and temporally in an upwelling and chemically heterogeneous melting column. We show that an enriched blob as marked by isotope or incompatible trace element anomaly is variably stretched along the direction of melt flow during its transit through the melting column. The amount of stretching depends on the extent of melting, style of melt extraction (batch vs. fractional), porosity of the melting column, and partition coefficient, and can be quantified by a dimensionless parameter called the stretching factor. For radiogenic isotopes U, Th, Pb, Sr, Nd, and Hf, a factor of $2 \sim 8$ stretching is expected for the residue and a factor of at least 30 is found for the channel melt. For near fractional melting beneath mid-ocean ridge, an enriched Nd isotope signal takes approximately 10 times more time to transit through the low-porosity matrix than through the high-porosity channel. Hence chemical heterogeneities observed in residual peridotites and extracted melts are decoupled both spatially and temporally, which has important implications for the interpretation of isotope and trace element characteristics of the basalts and residual peridotites.

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Simple models for disequilibrium fractional melting and batch melting with application to REE fractionation in abyssal peridotites

Abstract

Introduction

Section snippets

Model setup

General features of disequilibrium fractional melting

REE and Y depletion in abyssal peridotites

The assumption of R1 = R2 = ⋯ = RN

Acknowledgments

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Chem. Geol.

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An analysis of variations in isentropic melt productivity

Phil. Trans. R. Soc. Lond.

Calculation of peridotite partial melting from thermodynamic models of minerals and melts, IV. Adiabatic decompression and the composition and mean properties of mid-ocean ridge basalts

J. Petrol.

Introduction to Modeling of Transport Phenomena in Porous Media

Mechanisms of mantle metasomatism: geochemical evidence from the Lherz orogenic peridotite

J. Petrol.

Discontinuous melt extraction and weak refertilization of mantle peridotites at the Verma lithospheric section (Mid-Atlantic Ridge)

J. Petrol.

Oceanic crust generated by elusive parents: Sr and Nd isotopes in basalt–peridotite pairs from Mid-Atlantic Ridge

Geology

Diffusion Mass Transfer in Fluid Systems

Abyssal peridotites, very slow spreading ridges and ocean ridge magmatism

Magmatism in the Ocean Basins

Geological Society Special Publication

Late-stage melt evolution and transport in the shallow mantle beneath the East Pacific Rise

Proc. Ocean Drilling Program Sci. Results

Chemical trends in abyssal peridotites: refertilization of depleted suboceanic mantle

J. Geophys. Res.

The assumption of R₁ = R₂ = ⋯ = R_N