Isotopic Gradients Across Fluid – Mineral Boundaries

The distribution of stable and radiogenic isotopes within and among phases provides a critical means of quantifying the origin, residence and cycling of materials through terrestrial reservoirs (Wahl and Urey 1935; Epstein and Mayeda 1953; Johnson et al. 2004; Eiler 2007; Porcelli and Baskaran 2011; Wiederhold 2015). While isotopic variability is globally observable, the mechanisms that govern both their range and distribution occur largely at atomic (e.g., radioactive decay), molecular (e.g., the infl uence of mass on the free energy of atomic bonds) and pore (e.g., diffusive transport to reactive surface) scales. In contrast, the vast majority of isotope ratio measurements are based on sample sizes that aggregate multiple pathways, species and compositions. Inferring process from such macroscale observations therefore requires unraveling the relative contribution of a variety of potential mechanisms. In effect, the use of isotopes as proxies to infer a specifi c parameter, such as temperature (Urey 1947) or residence time (Kaufman and Libby 1954), carries the implicit requirement that one mechanism is the primary infl uence on the measured isotopic composition of the composite sample.


INTRODUCTION
The distribution of stable and radiogenic isotopes within and among phases provides a critical means of quantifying the origin, residence and cycling of materials through terrestrial reservoirs (Wahl and Urey 1935;Epstein and Mayeda 1953;Johnson et al. 2004;Eiler 2007;Porcelli and Baskaran 2011;Wiederhold 2015).While isotopic variability is globally observable, the mechanisms that govern both their range and distribution occur largely at atomic (e.g., radioactive decay), molecular (e.g., the infl uence of mass on the free energy of atomic bonds) and pore (e.g., diffusive transport to reactive surface) scales.In contrast, the vast majority of isotope ratio measurements are based on sample sizes that aggregate multiple pathways, species and compositions.Inferring process from such macroscale observations therefore requires unraveling the relative contribution of a variety of potential mechanisms.In effect, the use of isotopes as proxies to infer a specifi c parameter, such as temperature (Urey 1947) or residence time (Kaufman and Libby 1954), carries the implicit requirement that one mechanism is the primary infl uence on the measured isotopic composition of the composite sample.

Druhan, Brown & Huber
In the present chapter, we consider a wide variety of macro-scale observations of isotope partitioning across fl uid-solid phase boundaries.For this purpose we defi ne the continuum scale as a representation in which interfaces are averaged over elementary volumes, as opposed to the pore scale in which these interfaces are explicitly resolved.Throughout this review it will be demonstrated that observations of isotope partitioning across fl uid-solid boundaries require some representation of the isotopic composition of the solid surface and surrounding fl uid distinct from 'bulk' or 'well mixed' reservoirs.For example, this distinction is necessary in order to (1) quantify the partitioning of radioactive and radiogenic species, (2) describe transport limitations that may impact the macroscopic partitioning of isotope ratios, (3) explain observations of transient fractionation due to dissolution through preferential release at the solid surface, and (4) parameterize apparently variable fractionation factors during precipitation.The ability to describe isotopic partitioning specifi c to the phase boundary then infl uences the accuracy of simulations from highly variable, fi eld scale systems (e.g., Druhan et al. 2013) to highly controlled laboratory experiments (e.g., Tang et al. (2008)).These observations imply that quantifying the composition of solids and fl uids as averages across a given representative volume carries some inherent loss of information that isotopes appear to be sensitive to.This conclusion leads us to consider mechanistic descriptions of isotope partitioning that could be signifi cantly improved by modeling approaches that are capable of resolving spatial zoning within individual solids (e.g., Li et al. 2006;Tartakovsky et al. 2008;Molins et al. 2012;Molins 2015, this volume;Yoon et al. 2015, this volume).
The suite of experimental data and quantitative approaches described herein support a conceptual framework in which macroscopic observables, such as an apparent fractionation factor, are the emergent result of multiple interacting processes that are strongly infl uenced by the physical and chemical characteristics of the fl uid-solid boundary.In this sense we consider the pore scale as a characteristic length over which the unique physical and chemical properties of the phase interface may be described as distinct from bulk or aggregate values.This defi nition of the pore scale provides a critical reference frame over which molecular scale mechanisms combine to yield the macroscopic observables (Steefel et al. 2013).

A conceptual model of isotope partitioning at the pore scale
The fl ux of solutes in porous media is infl uenced by the development of spatial and temporal gradients resulting from the combined effects of transport and reactivity.How the pore-scale nature of these processes emerges into continuum scale observations of fl uid-solid interaction is yet unresolved and requires multiscale (e.g., fractal) upscaling methods that remain a challenge.For example, Darcy scale fl uid fl ow may occur along a fi xed gradient, but the size, shape and distribution of individual solid grains is such that at the pore scale velocity vectors vary in both magnitude and direction.As a result, the principle mechanism of solute transport in some areas is advective, while in others it is diffusive.Mixing between these distinct regimes is approximated at the continuum scale by either a heterogeneous conductivity fi eld (Li et al. 2010;Sudicky et al. 2010), a hydrodynamic dispersion coeffi cient (Gelhar and Axness 1983;Steefel and Maher 2009) or a non-uniform fl uid travel time distribution (Maloszewski and Zuber 1982;Bellin and Tonina 2007).
Reactions that occur between fl uid and solid phases are infl uenced by these local porescale transport regimes.For example, where transport of solutes between a solid surface and a surrounding fl uid is accomplished primarily by diffusion, the rate of reaction across that interface may be limited by either the delivery of solute to the reactive surface or the approach to equilibrium.When advection is dominant, the same process may be governed by the relative rates of reactivity and fl ow (Rolle et al. 2009;Maher 2010;Hochstetler et al. 2013).The factors infl uencing isotopic partitioning are then equally variable across the pore scale (Fig. 1).In areas of the domain governed by diffusion, the partitioning of isotopes may refl ect a transport limitation or diffusive fractionation, whereas in areas where fl ow is relatively fast, the isotopic composition is likely governed by the partitioning associated with a reaction (Lemarchand et al. 2004;Fantle and DePaolo 2007;DePaolo 2011).The accumulation of newly formed solids is also anticipated to vary with the local transport regime, which in turn exerts an infl uence on the isotopic composition of the fl uid-solid interface.
This relationship between the dominant mechanisms of transport and fractionation suggests that over a representative volume of porous media a variety of fractionation factors may be observed.For example a fractionation factor characteristic of diffusive transport between the solid-fl uid surface and the well-mixed fl uid, or a fractionation factor associated with the difference in isotope composition between the fl uid and solid at each reactive surface.The continuum-scale or observable fractionation factor over the timescale of the reaction (e.g., seconds to years) is then subject to the location and volume of material sampled.Over much longer periods of time, processes like recrystallization and solid-state diffusion homogenize spatial zoning within the solid.This implies that transient isotopic partitioning should be observable over signifi cant periods of time, and that the observed macroscopic fractionation factor is potentially (1) a combination of multiple, distinct mechanisms and (2) variable as a result of parameters such as saturation state and fl ow rate.
From this perspective, variability in the magnitude of an observed fractionation factor is in some ways analogous to the discrepancy in rate constants observed across natural systems (Malmstrom et al. 2000;White and Brantley 2003;Maher et al. 2006b).This wide range in what should in principle be a fi xed value has been attributed to the relative infl uence of a variety of processes, such as changes in reactive surface area (VanCappellen 1996), transport limitation (Steefel and Lichtner 1998;Maher 2011;Li et al. 2014); and even climate variations (Kump et al. 2000;Maher and Chamberlain 2014).Similarly, the apparent isotopic partitioning observed in fl ux-weighted or homogenized natural samples is often distinct from that obtained under controlled experimental conditions.These effects have been noted in a Figure 1.Conceptual model of isotopic partitioning at the pore scale during fl uid-rock interaction.Darcyscale fl ow occurs from left to right, whereas at the pore scale local variations in both magnitude and direction lead to areas of the domain that are either advection or diffusion dominated.The bulk fl uid isotopic composition also varies from left to right as a result of reaction progress, but the isotopic composition of fl uid and the solid surface is subject to multiple fractionating processes and thus variable.The precipitation of new solid or dissolution of existing solid is then also isotopically variable and distinct from the bulk.variety of contexts, including oxygen and nitrogen isotope fractionation in marine sediments (Brandes and Devol 1997), selenium isotope fractionation in wetlands (Clark and Johnson 2008), compound-specifi c stable isotope analysis (CSIA) of organics (Van Breukelen 2007) and chromium isotope fractionation in a contaminated aquifer (Berna et al. 2010).
To the extent that pore structure infl uences the distribution of isotope ratios in reactive systems, using volume-averaged sample measurements to quantify related parameters, such as reaction progress or mixing, can result in signifi cant uncertainty.For example, both analytical and numerical solutions have demonstrated that neglecting the effects of hydrodynamic dispersion on observed stable isotope ratios can lead to an underestimation of reactivity in through-fl owing systems (Abe and Hunkeler 2006;Van Breukelen and Prommer 2008).As noted above, dispersion is one method of parameterizing mixing at the continuum scale, and thus approximating the contribution of distinct fl uid isotopic compositions at the pore scale (Fig. 1).Improved accuracy should then be achievable by describing distributed porescale isotopic compositions as the result of a variety of mechanisms, e.g., multiple fractionating reactions, a difference in the diffusion coeffi cient of the isotopologues of a compound or the dampening of a kinetic fractionation as a result of transport limitation.In the current article, we consider observations of isotopic partitioning across fl uid-solid boundaries associated with a variety of fractionating mechanisms.In reviewing these examples we emphasize the aspects of each process that lead to pore-scale heterogeneity and thus a disconnect in behavior between the scale of mechanism and the scale of observation.These categories are by no means an exhaustive list of all processes that result in isotopic partitioning across reactive interfaces, but serve as examples in which interpreting the response of continuum-scale isotopic values to variations in external parameters is improved by consideration of pore-scale isotopic distributions.

Organization of article
The structure of the article is broadly divided into three sections.First, we provide a brief explanation of the notation used to describe isotope partitioning, with an accompanying discussion of primary mechanisms and models for fractionation.Second, we describe experimental observations for four examples of isotopic partitioning across fl uid-mineral interfaces: D-recoil, diffusion, dissolution, and precipitation.Across this wide range of processes, a common observation is that the interface between fl uid and solid phases (1) governs material transfer between reservoirs and (2) displays isotopic compositions distinct from bulk values.Associated models for the mass balance at phase boundaries are described with particular emphasis on the mechanisms necessary in order to quantify macroscale observations.This leads to the second section in which current modeling techniques for the description of transient isotopic fi nal and the development of zoned mineral grains are discussed, concluding with a novel application of pore-scale modeling techniques to describe the distribution of stable isotopes across a fl uid-mineral boundary as an initially oversaturated system establishes equilibrium.Throughout this review the principal intent is to describe experimental and modeling studies of isotopic partitioning with reference to the ways in which the composition and gradients across fl uid-mineral boundaries is distinct from bulk-averaged values and uniquely infl uences continuum-scale observations.

NOTATION
The exchange of material across fl uid-mineral surfaces infl uences both radiogenic and stable isotope distributions.Unlike (most) stable isotope fractionation, variations in radiogenic isotopes are mass-independent and arise due to the radioactive decay of a parent nuclide to an intermediate radioactive nuclide or a stable, radiogenic nuclide.The uranium decay series (Fig. 2) illustrates the relationship amongst parent and daughter isotopes, and the terminology is applicable to other radioactive-radiogenic systems.
The 238 U isotope is radioactive and for N nuclei at time t in a closed system the number of decaying nuclei dN in the time interval dt follows a homogeneous Poisson statistical law and is proportional to N: where lambda (O) is the decay constant.The expression on the right side is a probability per unit time of a radioactive decay event.Integration of Equation (1) yields an exponential equation for radioactive decay: where the integration constant (N 0 ) is equal to the initial number of nuclei at t = 0.The decay constant is related to the nuclide half-life For intermediate daughter products such as 234 U and 226 Ra, isotopic abundance is commonly discussed in terms of activity (A) where: and thus the initial activity at t = 0 is A 0 = ON 0 .In a closed system that starts with no daughter isotopes, for example the 238 U-234 U system, the initial activity of 234 U is zero and it will increase until Finally, for stable radiogenic daughter isotopes (e.g., 206 Pb, in the 238 U series), accumulation of isotope N 2 is related to the initial abundance of the parent isotope (N 0 ) and time: such that as t goes to infi nity N 2 goes to N 0 .Krane (1988) presents an exhaustive discussion of radioactive decay including considerations for branched decay, and very short-lived intermediate daughter isotopes.The subsequent discussion will use the 238 U-234 U system to consider the infl uence of pore-scale effects in quantifying D-recoil damage and associated alterations to solid surfaces and solution chemistry.
As noted above, there can be ambiguity in the notation used to describe the partitioning of stable isotopes.In the subsequent text we primarily adhere to the guidelines put forth by Coplen (2011).The isotope ratio (R) of a particular reservoir is defi ned as where N is the number of atoms of i E and j E, the isotopes i and j of the element E in substance P.This value is commonly reported relative that of a standard ratio (std), referred to as the delta value (G), and defi ned as This delta value is nondimensional, but is commonly multiplied by 1000 to report values as parts per thousand or per mil (‰).The difference between the isotope ratios (Δ) of two compounds or phases (P and Q) is then The apparent or net isotopic fractionation factor does not make use of delta notation, but is defi ned as Conversion between Δ i E P/Q and D i E P/Q may be approximated as Δ i E P/Q ≈ ln D i E P/Q .From these relationships it is noted that both the difference between the isotope ratios (Eqn.8) and the apparent fractionation factor (Eqn. 9) between two reservoirs are obtained from direct measurement regardless of the variety of reaction pathways necessary to produce them, and are therefore a function of the representative volume of the sample in all but the most simplifi ed systems.In contrast, a fractionation factor associated with a specifi c reaction pathway may be derived in reference to the rate law for that reaction.These types of fractionating processes are discussed in the subsequent section.Finally, an isotopic mole fraction (X) may be defi ned as the ratio of the amount of a particular isotope in a given species, compound, or reservoir divided by the total amount of that element in the same group This value is used in the derivation of isotope-specifi c reversible rate expressions that involve a solid phase.

A note on fractionation
In this chapter, stable isotope fractionation will be discussed in terms of equilibrium and kinetic processes.For this purpose chemical equilibrium is described as a dynamic state that occurs when two elementary reactions, the forward reaction from reactants to products and the backward reaction from products to reactants, are in balance.For example, the hydroxylation of dissolved CO 2 : which is at equilibrium when in a closed system the distribution of species is invariant in time.
The ratio of the elementary forward and backward rates is then termed the equilibrium constant (K eq ).Isotopic equilibrium may be described similarly, for example considering carbon isotope equilibrium between the carbon-bearing species in Equation ( 11): leading to a separate equilibrium constant to account for this isotopic partitioning.A comparable relationship could be written for the partitioning of oxygen isotopes between CO 2 and HCO 3 -, or between HCO 3 -and OH -.While all of these equilibria are described using the same law of mass action, it is important to note that they are not predicated on one another.In other words, these descriptions allow for the establishment of chemical equilibrium without necessarily requiring isotopic equilibrium.For reactions involving simple stoichiometric relationships, where one atom of the element of interest occurs in all reactant and product species, the isotopic equilibrium constant is equivalent to the fractionation factor (Eqn. 9); however, the relationship is often more complex (Schauble 2004).In general, the equilibrium fractionation factor is temperature dependent and larger for low mass elements and for isotopes of the same element that have large differences in mass.Typically, the partitioning of stable isotopes between two phases at equilibrium preferentially incorporates the heavier isotope in the phase with lower bond energy.
An imbalance between the forward and backward rates leads to a net accumulation of either the product or reactant, and the rate of this mass transfer is described by kinetics.Many reactions take place under conditions in which the reverse reaction is in some way prohibited, or the system is very far from equilibrium, such that isotopic partitioning is entirely kinetic.A kinetic fractionation factor is expressed as the ratio of the isotopic composition of the instantaneously generated product species (P inst ) and the residual reactant (Q) through a single reaction pathway: This D kin can be related to the kinetic rate constant of the reaction (k), depending on the order of the reaction.For example, a fi rst order dependence on concentration (e.g., dP/dt = kQ), leads to an expression for D kin = i k/ j k (Mariotti et al. 1981).From an observational perspective it is often diffi cult to categorically identify equilibrium vs. kinetic effects on isotope partitioning.In low-temperature systems equilibration can be extremely slow, and, in addition, open system conditions may support the establishment of a steady state that appears balanced but is not necessarily equilibrated.In this sense the distinction between a specifi c state that is dynamic equilibrium, and an observable net rate of precipitation or dissolution does not imply an exclusive infl uence of equilibrium vs. kinetic fractionation.The approach towards an equilibrated system implies instead a continuum between pure kinetic fractionation and equilibrium fractionation.Later sections of this chapter explore the consequences of such a model and the extent to which pore scale treatments are capable of improving upon current approaches.
Under certain conditions, it is possible to develop theoretical models to predict the evolution of the isotopic composition of products and reactants.These models offer the advantage that they are simple to use, but are generally limited by particular assumptions and do not refl ect the complexity of the reaction pathway or the relationship between transport and reaction in the isotopic mass balance description.Rayleigh fractionation, for instance, assumes an open system distillation process, where the reactant is progressively consumed, such that the isotope ratio of the reactant follows (Rayleigh 1902): where the isotope ratio (Eqn.6) is equal to the product of the initial isotope ratio (R 0 ) and the fraction of reactant remaining relative to the initial concentration (f) raised to the power (D1).This relationship requires a constant fractionation factor D, and produces an exponential relationship between reaction progress and isotopic partitioning.This model is only strictly intended for systems in which the reactant is continually supplied (f cannot go to zero) and the product of the reaction is instantaneously removed or segregated from the reactive system (Criss 1999).In practice the Rayleigh model is used in a wide variety of systems because it offers a simple relationship between reaction progress and fractionation without requiring knowledge of the reaction pathway or transport mechanisms.As a result, several studies have pointed out limitations to the Rayleigh model in application to hydrogeochemical systems (Brandes and Devol 1997;Abe and Hunkeler 2006;Van Breukelen and Prommer 2008).A goal of the current chapter is to describe a variety of common fractionating mechanisms that often undermine the assumptions of such simplifi ed models and promote new quantitative methods for explicit treatment of reactivity and transport in the description of isotope partitioning.

Alpha recoil
Prior to the development of modern radioactive decay counting and mass spectrometry techniques it was widely assumed that the 234 U/ 238 U AR could not deviate from secular equilibrium.This was assumed to be the case because the small mass difference (~1.7%) of the two isotopes would not produce stable isotope effects like those observed for hydrogen and oxygen.Careful study of natural rocks and minerals in the 1950's, however, revealed variations from secular equilibrium (Chalov 1959).Since the chemical behavior of 234 U and 238 U should be nearly identical, and yet variations in 234 U/ 238 U AR can exceed 500%, researchers suggested that the energy associated with 238 U decay could directly eject the 234 Th daughter isotope from the mineral surface (10's of nm) or damage the mineral lattice allowing preferential leaching of the daughter isotope (Rosholt et al. 1963;Kigoshi 1971).Kigoshi (1971) carried out a pioneering study where fi ne-grained zircon sand was suspended in dilute aqueous solutions and the addition of 234 Th and 234 U to the solution was quantifi ed.The 234 Th activities of the solutions were consistent with predicted D-recoil injection to the solution based upon a spherical grain model and an D-recoil distance of 55 nm.Earlier inferences (Turkowsky 1969) and later laboratory studies (Fleischer 1988) suggest that the recoil distance is closer to 30-40 nm.Both the 234 Th and 234 U in the fl uid increased with time at a rate greater than the increase in 238 U, demonstrating that 234 Th has a recoil distance of tens of nm and can be ejected from the mineral structure or preferentially leached from lattice defects (Kigoshi 1971;Fleischer and Raabe 1978).Kigoshi (1971) calculated the expected rate of addition of 234 Th to a fl uid (Q) due to D-recoil ejection from a mineral grain based on: where L is the D-recoil range, S is the surface area, U is the solid density, and N 238 and O 238 are the number of238 U atoms per gram of solid and the decay constant, respectively.The activity of 234 Th in solution is a production-decay equation (combining Eqns. 4 and 5) of the form: All of the terms on the right side of the equation are known or can be measured independently, however, in natural systems the grain surface area (S) and its infl uence in directing D-recoil to the fl uid phase is arguably the most important variable.For example, Kigoshi (1971) assumed spherical grain geometry for their 1-10 μm diameter zircon sand and then fi t the measured 234 Th activities to calculate a characteristic recoil distance of 55 nm.Once L is known from similar experiments or determined by other methods, the probability of an D-particle (f D ) being ejected from a mineral grain assuming a spherical geometry can be approximated: Subsequent papers have explored the evolution of the solid phase and the effects of nonideal grain geometry on the production of daughter products to the fl uid phase (Kigoshi 1971;Fleischer and Raabe 1978;Fleischer 1980;1988;Maher et al. 2004;DePaolo et al. 2006DePaolo et al. , 2012;;Lee et al. 2010;Handley et al. 2013).A general conclusion from these studies, particularly those of Lee et al. (2010) and Handley et al. (2013), is that the chemical treatment of sediments and the model of grain surface structure need to be carefully considered and normalized across multiple laboratories in order for results to be broadly useful.Similar observations are noted, with important caveats for the differing chemical behavior, in the elements Th, Ra and Rn (Torgersen 1980;Semkow 1990;Sun and Semkow 1998;Porcelli and Swarzenski 2003).For example Ra is strongly adsorbed to mineral surfaces at low ionic strength but soluble at high ionic strength, thus the solution chemistry is critical for interpreting Ra activities (Moore 1976).
The general equations describing the evolution of uranium series isotopic ratios in pore fl uids and minerals are presented by Ku et al. 1992;Porcelli et al. 1997;Henderson et al. 2001;Tricca et al. 2001;Porcelli and Swarzenski 2003;Maher et al. 2004.The evolution of pore fl uid composition is related to primary mineral dissolution, secondary mineral precipitation reactions, D-recoil and preferential leaching of daughter isotopes to the pore fl uid, sorptiondesorption reactions and diffusion-advection of fl uid in and out of a pore.Alpha-recoil loss, preferential leaching near the mineral surface, and solid-state diffusion, will all affect the solid composition with respect to time.However, these processes operate at different timescales, allowing mineral grains to evolve distinct 234 U/ 238 U AR domains.There is considerable discussion in the uranium series literature about the relative roles of preferential leaching and direct D-recoil leading to daughter isotope accumulation in the pore fl uid (Rosholt et al. 1963;Vigier et al. 2005;DePaolo et al. 2006;Dosseto et al. 2006).Preferential leaching may occur due to mineral lattice damage associated with 234 Th recoil (Rosholt et al. 1963) or due to preferential oxidation of the 234 U in damaged mineral lattice by aqueous fl uids (Kolodny and Kaplan 1970).Regil et al. (1989), Roessler (1983Roessler ( , 1989) ) and Adloff and Roessler (1991) present detailed models of 234 U oxidation due to D-recoil.
It is diffi cult to ascertain the exact mechanism that transfers 234 U preferentially from the solid phase to the pore fl uid, but this process has important implications for the interpretation of uranium series isotopes in mineral-fl uid systems.DePaolo et al. (2006) suggest that based on fi ne-grained alluvial sediments, the leaching depth into grains is not appreciably greater than the recoil distance.For the purposes of this discussion we proceed under the assumption that direct D-recoil is the primary mechanism of 234 U transfer, but acknowledge this may be unwarranted, particularly in coarser-grained or uranium rich minerals.
The mass conservation equations presented below are applied to the 234 U and 238 U isotopes and illustrated schematically for distinct pore fl uid, solid surface, and solid interior compositions in Figure 3.This formulation can be applied to other intermediate daughter products with additional consideration for differing chemical behavior (e.g., strong sorption/ secondary mineral partitioning of thorium and radium under certain conditions).At the scale of a single pore surrounded by mineral grain surfaces, the 234 U/ 238 U AR of the fl uid will evolve based on the dissolution-precipitation reactions and D-recoil ejection to the fl uid.
The 234 U/ 238 U AR evolution of mineral grains with time is: It is apparent from Equation ( 18) that probability of D-particle ejection (f D ) and time are the critical variables that affect the activity ratio of the solid grains (DePaolo et al. 2006).
Estimating f D requires knowledge of the mineral volume surface area and recoil distance.DePaolo et al. (2006DePaolo et al. ( , 2012) ) calculated f D as a function of grain diameter and shape (surface area), demonstrating, for example, that mineral grains of 10-Pm diameter could have greater than a factor of 10 variability in f D (Fig. 4).A subsequent study by Bourdon et al. 2009 suggests employing a combined BET sorption surface area measurements and a fractal dimension model to overcome the uncertainty of mineral surface areas in natural sediments.
where S BET is the measured surface area, a is the molecule diameter for the adsorbate and D is the fractal dimension of the grain surface, which is measured independently.Non-ideal (spherical) surface area is particularly important at the pore scale, where roughness (Lee et al. 2010) can increase the surface area of a mineral grain by a factor of 17.The sorption data of Bourdon et al. (2009) have been successfully applied to studies of soils (Oster et al. 2012) and to date ice using trapped fi ne-grained sediments (Aciego et al. 2011).
The evolution of 234 U/ 238 U AR of pore fl uids is described by Equation (20): where C is the concentration and A is the 234 U/ 238 U AR of the solid (s) and fl uid (f), R d is the rate of dissolution in inverse time, R f is the retardation factor accounting for sorbed uranium and M s is the solid/fl uid volume ratio (Tricca et al. 2001;Maher et al. 2004).
The conservation equation can be rearranged to yield the following approximation for the pore fl uid at steady state (A f(ss) ): This relationship can then be manipulated to use the activity ratios to calculate the solid dissolution rate: Maher et al. (2004) used this relationship to calculate the dissolution rates of deep-sea sediments over timescales longer than those accessible by laboratory experiments.
The models described above are a small sample of the work on uranium series isotopes in hydrogeologic systems.At the pore scale additional complexities may be considered, particularly when the steady-state assumption is not valid.For example, rapid changes in the dissolution rate could dissolve the 234 U depleted layer of mineral grains, lowering the 234 U/ 238 U AR in the pore fl uid.Additionally, where mineral grains are in direct contact with one another the ejected daughter isotope may be implanted into an adjacent mineral grain and not accumulated in the pore fl uid.The net effect of signifi cant implantation would be to underestimate the sediment dissolution rate (Eqn.22) or the age of fi ne-grained sediments (DePaolo et al. 2006).In general the aforementioned studies demonstrate that uranium series disequilibria is largely controlled by the surface structure of individual grains that affect the isotopic composition of fl uids considered at the continuum scale.Uncertainty in the relative contributions of D-recoil and preferential leaching to isotopic fractionation in the uranium series and precise measurements of mineral surface area and recoil tracks currently limit the fi delity of existing models.Future directions in this area may include incorporating daughter isotopes with different recoil distances such as 230 Th and 226 Ra to better constrain the f D parameter and more sophisticated imaging techniques to quantify mineral surface structure.Additionally, precise measurements of the D-recoil distance (L) and the relationship between surface area and f D will be necessary to improve the application of pore scale 234 U/ 238 U AR to diverse problems such as sediment dating, mineral dissolution rates and vadose zone transport.

Diffusive fractionation
The change in concentration (C) of a solute due to a net divergence of the fl ux (J) in an arbitrary volume is often approximated with the linear transport theory that relates the diffusive fl ux to the gradient in chemical potential or concentration, i.e., Fick's fi rst law: Here the transport coeffi cient D is the molecular diffusivity of the solute and has units of length squared per time.as the product of the mean free path (l) between collisions, and the square root of the product of the gas constant (R) and temperature (T) divided by the mass (m) of the molecule or atom.This relationship suggests that the mass of the particle infl uences the transport coeffi cient and by extension the fl ux to the reactive surface.Therefore the lighter the particle the larger the diffusion coeffi cient, which leads to the expectation that if two particles differ in their masses (m 1 and m 2 ), then for the same conditions the difference in their diffusion coeffi cients would be However, this relationship is specifi c to an ideal gas, and in liquids the factors that contribute to the value of an individual species diffusion coeffi cient become much more complex (Moller et al. 2005;Yamaguchi et al. 2005).For instance, in electrolyte solutions, the infl uence of electrochemical forces on the interaction of diffusing species cannot be neglected, and thus requires accounting for cross-diagonal terms representing the infl uence of one species' or isotopic activity gradient or electrochemical potential gradient on another.This effect may be quantifi ed in the absence of charged surfaces by including treatment of electrochemical migration using the Nernst-Plank equation (Steefel and Maher 2009;Steefel et al. 2015;Rasouli et al. 2015), while treatment of surface charge requires further inclusion of a complete electrical double layer model (Tournassat and Steefel 2015, this volume).
In the context of stable isotope fractionation in aqueous solutions, this complexity has only recently been considered.Richter et al. (2006) conducted a series of experiments to determine the fractionation of isotopes associated with diffusion in comparison with previously estimated values.Their experimental design allowed a small volume of aqueous solution containing a dissolved salt (V 1 ) to be suspended inside of a much larger volume of dilute fl uid (V 2 ).The length and aperture of the tube connecting the two reservoirs was selected such that the solution inside of V 1 would remain well mixed over the timescale necessary for dissolved solutes to diffuse into V 2 .Richter et al. (2006) considered departure from Equation.(26) by recasting the power as a parameter (E), which they then fi t to their experimental data.
The results of their study determined E values less than 0.5, but greater than 0.0, thus providing evidence that kinetic fractionation due to diffusion could contribute to observed variations in stable isotope ratios in aqueous solutions.However, their method required the implicit assumption that an inverse power law (Eqn.27) is an accurate description of the dependence of a diffusion coeffi cient on the mass of the solute.Bourg and Sposito (2007) sought to directly test this assumption by utilizing molecular dynamics (MD) simulations.The use of a numerical method allowed them to obtain the diffusion coeffi cient for more than two isotopes of the same species, and thus test the generality of the E values reported by Richter et al. (2006).Their results showed good agreement with the Richter et al. (2006) values, demonstrating that the value of E for a given mass pair in aqueous solutions is typically less the value of 0.5 predicted by kinetic theory for an ideal gas (Eqn.26).Following on the agreement of these results, Bourg et al. (2010) extended both the experimental and numerical results to include additional ions.The results of these studies are summarized in Table 1.
Two key aspects of these observations are that the monovalent solutes show a larger contrast in the diffusivity of their isotopes than divalent solutes, and this difference does not follow a direct correlation with the ratio of the masses.Bourg et al. (2010) demonstrated that this relationship occurs as a result of (1) the size of the solute radius, and (2) the strength of attractive interactions between the solute and solvating water (effective mass of the diffusing species), both of which infl uence coupling of motion between the solute and solvent.Using the residence time of water (W) in the fi rst solvation shell of the ion obtained from the MD simulations as a proxy for this coupling, Bourg et al. (2010) showed a clear correlation between the inverse of W and E (Fig. 5).
The aforementioned experimental observations and MD simulations indicate that pore scale gradients in both concentration and isotopic abundance between a reactive surface and the surrounding fl uid (Fig. 1) are infl uenced by diffusion coeffi cients that differ as a result of the size and charge of ions and the difference in masses of their isotopes.The effects of ion size and charge are explicitly treated at the resolution of MD simulations, but must be approximated at larger scales.Therefore accurate representation of these diffusive effects on isotopic partitioning at the pore and continuum scale requires independent knowledge of the magnitude of fractionation associated with the isotopes of each ion of interest (Table 1).Furthermore, the extent to which observed differences in diffusivity relate to fractionation associated with dispersion at larger scales is unknown.
From the results of Richter et al. (2006), Bourg and Sposito (2007) and Bourg et al. (2010), the value of E relating the ratio of the masses to the ratio of the diffusion coeffi cients is highly variable among ions and solution compositions.In principle, the ratio of the diffusion coeffi cients could be directly obtained for two isotopologues, specifi c to a given solution composition, through a simple diffusion experiment.For example, implementing a generic tracer in a discretized domain using the CrunchFlow (Steefel et al. 2015) reactive transport code such that at the start of the simulation an elevated concentration exists in the center of a closed system, one may obtain the spatial partitioning of the tracer as it diffuses through the system (Fig. 6A).If this total tracer is broken into two components, where one 'isotope' composes the majority of the tracer concentration and the other is a trace amount, then a slight difference in the diffusion coeffi cients of the two species results in a spatial profi le of their ratio through time (Fig. 6B).We emphasize that this simple modeling example demonstrates a proof of concept wherein an experimental system could provide a time series of the tracer concentration (Fig. 6C) and isotope ratio (Fig. 6D) at a fi xed observation point.For any system in which variation in the isotope ratios are large enough to be detected beyond measurement error, a time series dataset of this nature could be used to obtain unique values for the diffusion coeffi cients.If the difference in the masses of the isotopologues is known, these values could then be related to obtain direct estimates of E for a given system.
The sensitivity of isotope ratio measurements are such that diffusive fractionation may infl uence the interpretation of observed fractionation, but these effects have only recently begun to be considered in hydrologic studies.Analytical and numerical models have demonstrated that even in the absence of true diffusive fractionation, neglecting the effects of diffusion and dispersion leads to a lower overall estimate of reaction progress based on stable isotope ratios than when these factors are considered (Abe and Hunkeler 2006;Van Breukelen and Prommer 2008).These results have lead to a recent focus on the effects of dispersive mixing on stable isotope ratios, with particular emphasis on compound-specifi c stable isotope labels used to track contaminant degradation.Hydrodynamic dispersion is a fundamentally distinct process from molecular diffusion, arising from the fl uctuations in velocity within and among connected pores during fl ow.Under certain assumptions, dispersion has been shown to display a diffusive behavior.In practice, most efforts to directly simulate fractionation due to dispersion in through-fl owing systems have made use of estimated relationships between the ratio of the masses and the ratio of the dispersion coeffi cients.Dispersive isotope fractionation has been simulated at the continuum scale for both hydrogen and carbon isotopologues of solutes during fl ow through heterogeneous porous media using a variety of approximations for the difference in isotope-specifi c dispersivities.LaBolle et al. ( 2008) calculated isotope-specifi c values of dispersivity by substituting the reduced masses of the solutes into Equation (26).Rolle et al. (2009), Eckert et al. (2012) and Van Breukelen and Rolle (2012) used an empirical correlation from Worch (1993) close to Equation (26) but with a power of 0.53.Thus far it is not evident what, if any relationship may exist between diffusive and dispersive fractionation, or even if the power law relationship between the ratio of the masses and ratio of the diffusion coeffi cients validated by Bourg and Sposito (2007) holds true for the ratio of two isotopic dispersion coeffi cients.

Dissolution
Fractionation directly associated with the dissolution of material from the solid phase is often considered negligible over the reactive timescales represented by many natural samples.This assumption is justifi ed in that many of the mechanisms thought to contribute to isotopic partitioning, such as transport (discussed above), and changes in solvation (Hofmann et al. 2012), are restricted within a solid phase.Furthermore, where apparent shifts in isotopic ratio are observed during net dissolution of natural samples, it is diffi cult to distinguish preferential mobilization of individual, isotopically distinct mineral phases from true fractionation (Ryu et al. 2011).However, a wide variety of studies now appear to demonstrate observable fractionation during dissolution, usually during the initial stages.These observations are signifi cant in that they implicitly require that the solid become isotopically differentiated as material is removed from the reactive surface.This implies the development of a surface that is compositionally distinct from the interior of the grain, a process that is not readily compatible with bulk or volume fraction representations of solids.The earliest observations of fractionation during dissolution involved dissimilatory reduction of ferric iron by anaerobes (Beard et al. 1999(Beard et al. , 2003;;Brantley et al. 2001Brantley et al. , 2004)), which naturally led to the issue of whether observed iron isotope partitioning was a vital effect or if it could be reproduced in abiotic systems.From this starting point, a wide range of experimental conditions, including both pure mineral phases and whole rocks, as well as a variety of stable isotope ratios, have shown evidence for partitioning of isotopes during net dissolution (Table 2).The compilation in Table 2 is limited to low temperature and ambient pressure experimental conditions in which efforts have been made to minimize fractionation of fl uid phase isotopes due to additional effects, such as secondary mineral precipitation.
Despite wide variability in experimental conditions, a common observation is that during dissolution the isotopic composition of the fl uid phase tends to be lighter than that of the initial solid, i.e. the light isotope tends to be preferentially dissolved.Furthermore, in studies where the isotopic value of the fl uid is monitored as a function of time, the magnitude of this fractionation appears to vary.When a solid is introduced to an aqueous solution such that the system is undersaturated, the fractionation between the fl uid and solid phase tends to exhibit a maximum value relatively soon after the experiment is initiated (Fig. 7).For example, the dissolution of goethite at pH 3 (Wiederhold et al. 2006) promoted by ligand complexation leads to a maximum fractionation between dissolved Fe and the bulk solid of -1.83‰, but this difference lessens as dissolution continues (Fig. 7A).A similar temporal trend is noted in iron isotopes during proton-promoted dissolution of granite (Kiczka et al. 2010)( Fig. 7b), in zinc isotopes during both proton and ligand promoted dissolution of biotite (Weiss et al. 2014) and in silicon isotopes during proton promoted dissolution of Hawaiian basalt (Ziegler et al. 2005).At higher temperatures and pressures, the same trend is noted in lithium isotopes during the dissolution of basalt (Verney-Carron et al. 2011) and in magnesium isotopes during the dissolution of basalt and forsterite (Wimpenny et al. 2010), but magnesium isotopes show the opposite behavior during magnesite dissolution (Pearce et al. 2012).A low-temperature exception to this general trend may be the fractionation of copper isotopes associated with oxidative dissolution of sulfi de minerals.Mathur et al. (2005) reported Δ 65 Cu values of +2.74‰ and +1.32‰ for the abiotic dissolution of chalcocite and chalcopyrite, respectively.They suggested this isotopic enrichment of the solute was associated with the accumulation of secondary phases, however, a subsequent study demonstrating +2.0‰ values for dissolution of whole rock containing chalcopyrite argued that secondary mineral formation was negligible (Fernandez and Borrok 2009).
The general observation of an initially negative Δ fl uid-bulksolid value that trends towards zero with further reaction progress is common to a variety of reactive pathways, minerals and isotope systems, and is observable even during dissolution of natural samples.For example, the biotite and chlorite enriched component of a granite sample dissolved in the presence of hydrochloric and oxalic acids demonstrated negative Δ fl uid-solid values that were most pronounced early in the reaction (Kiczka et al. 2010) (Fig. 7B).This study explored a range of dissolution rates by varying the potassium concentration, which served to slow down the rate of iron release in the HCl experiments, and observed a greater enrichment in the light isotopes of the fl uid phase with decreasing dissolution rate.In contrast, addition of potassium to the oxalic acid experiments only made a small difference in the Fe release rates, and little difference in the Δ fl uid-bulksolid values were observed.Reaction of a biotite granite with the same acids yielded similar temporal trends, though the maximum Δ fl uid-bulksolid values showed lighter isotope composition for the fl uid phase with more acidic solutions (Chapman et al. 2009).These observations suggest that the partitioning of stable isotopes may potentially be utilized as an indicator of the mechanisms of mineral weathering, leading to development of a variety of conceptual and quantitative models for fractionation during congruent dissolution.The preferential removal of light isotopes during the initial stages of a dissolution reaction is commonly considered a kinetic fractionation associated with enrichment of the solid surface in the heavy isotope.Some studies have considered whether or not the initial stage in which this surface layer is established could be described as a distillation or Rayleigh process.Wiederhold et al. (2006) discussed this conceptual model in the context of iron isotopes and Weiss et al. (2014) in the context of zinc isotopes.Both studies point out that a Rayleigh model would carry the implicit assumption that the isotopes of the solid phase are well mixed, and thus homogeneously released.However, the development of either a depth into the solid that is isotopically zoned or a non-uniform distribution of reactive surface sites along a monoatomic surface layer both contradict this requirement.The consistent observation of a maximum Δ fl uid-bulksolid early in the reaction progress that becomes less depleted with time (Fig. 7) then requires consideration of an additional mechanism.
The development of an isotopically distinct solid surface has been modeled in terms of the propagation of a reacted surface layer into the unreacted solid.This is based on the idea of a preferentially reacted or 'leached' layer as commonly used to describe non-stoichiometric dissolution of silicates, e.g., Casey et al. (1993); Casey and Ludwig (1996); Hellmann et al. (1990);Oelkers (2001).Brantley et al. (2004) derived a model for isotopic fractionation  during dissolution based on a transfer of material from the bulk solid through a layer altered by dissolution to the fl uid.The depth of the altered surface layer (x), the fl ux of material from the bulk solid to this layer (F) and the rate of dissolution from the surface layer to the fl uid (R = kC(t)) all contribute to the transient value of Δ fl uid-bulksolid (Fig. 8A).The model is given as a ratio of the rates of 54 where 54 k and 56 k are the rate constants for the dissolution of material from the surface layer to the fl uid for 54 Fe and 56 Fe, respectively.Similarly 54 C o and 56 C o are the respective concentrations of the isotopes of iron in the bulk solid and t is the time since dissolution started.As discussed in Brantley et al. (2004), this equation effectively describes the period of time from the start of dissolution to the establishment of a steady state value for a given surface layer depth (x) (Fig. 8B and C).At very early time, relative rates of dissolution of the two isotopes are governed by the ratio of the rate constants, but as time progresses, the fl ux from the unreacted solid to the surface layer becomes more infl uential and eventually controls the composition of material transferred to the fl uid.Conceptually, this transition occurs because at t = 0, the surface layer has not been formed yet and the fl uid is in contact with the bulk solid, allowing for preferential release of one isotope vs. another as a function of the ratio of the rate constants.
As the surface layer develops and eventually reaches steady state, the transfer of material to this layer from the underlying, unreacted solid becomes the governing rate.In the Brantley et al. (2004) model, this steady state value is achieved quickly if the thickness of the surface layer This model approach clearly yields a temporal trend comparable to those observed during many of the dissolution experiments discussed in the observations section (Table 2).Furthermore, it suggests that the establishment of a steady state value is independent of the fraction of solid that is mobilized, but is rather a function of the establishment of a steady state leach layer.This implies that the trend towards a Δ fl uid-bulksolid of approximately 0.0‰ as dissolution progresses is not the result of a depletion in the abundance of the preferentially released isotope (i.e., a mass balance restriction) but rather a shift from the infl uence of dissolution (into the fl uid) to the infl uence of material transfer (into the leach layer).There are some aspects of the assumptions built into this model that may be further developed.For example, the thickness of the leach layer is fi xed and allowed to propagate inward throughout the simulation, but one may conceptualize a period at the very beginning of the model in which the distinct surface layer has yet to establish and thus x = 0.This would then require consideration of a period over which the surface layer grows into the solid and establishes a fi xed depth.Furthermore, in Equation ( 28) an eventual Δ fl uid-bulksolid value of 0.0‰ results from a fl ux of Fe from the bulk solid to the surface layer (F) that is non-fractionating.This distinction between the mechanism through which iron is supplied to the leach layer and the mechanism by which it is dissolved into solution are potentially diffi cult to reconcile, and may benefi t from consideration of solid state diffusion through the reacted zone (Verney-Carron et al. 2011).
Recent analytical developments have supported increasingly high-resolution observations of the structure and chemical composition of solid surfaces and a corresponding refi nement in the conceptual model of reacted zones.Nano-scale resolution of reactive fronts are becoming more commonly described as a dissolution-reprecipitation process (Hellmann et al. 2012(Hellmann et al. , 2015) ) leading to formation of a distinct, often amorphous phase at the solid surface.The extent to which the Brantley et al. (2004) model might be reworked to consider fractionating dissolution in systems where a reacted surface layer does not form has only recently begun to be addressed.Wiederhold et al. (2006) quantifi ed stable isotope partitioning during dissolution of a solid phase that does not form an observable reacted surface layer using a model they describe as conceptually similar to the Brantley et al. (2004) approach.In their derivation for iron isotope partitioning during goethite dissolution, the reacted surface depth was replaced by a percentage of the total iron contained in the solid phase that was considered to exist within a reactive, monoatomic surface layer.The isotopes of iron could be preferentially removed from the reactive surface sites at the step-edge of this receding monoatomic layer, but are replenished by the isotopic composition of the bulk solid.Their derivation results in two free parameters, the fractionation associated with dissolution of Fe from the reactive surface layer and the percentage of the total solid phase Fe composing the reactive surface site pool.Importantly, they fi t these two parameters and obtained a best estimate for the reactive surface site pool (2.4%) that matched quite closely with the value they independently quantifi ed based on the surface site density, the measured surface area, and the molar mass of goethite (2.7%).This model was successfully extended by Kiczka et al. (2010) to consider the contribution of two independent types of reactive surface site in a more complex granite consisting of enriched biotite and chlorite fractions.
Macroscopic or Darcy-scale reactive transport models are now becoming a common means of quantifying stable isotope partitioning by separating the isotopes of interest into individual 'species' that follow parallel reaction networks.The fractionating reaction in these simulations is commonly a homogeneous aqueous process (Dale et al. 2009;Gibson et al. 2011;Druhan et al. 2012Druhan et al. , 2014;;Jamieson-Hanes et al. 2012;Wanner and Sonnenthal 2013).To date, models designed to simulate the infl uence of dissolution on the isotope ratio of fl uids Unauthenticated Download Date | 6/4/19 7:10 AM rely on the assumption that the solid-phase isotopic composition remains fi xed over time.In doing so, dissolution of the solid acts as a source to the fl uid phase that is isotopically invariant with time (Steefel et al. 2014a;Wanner et al. 2014).This approach is necessary given that at the continuum scale mineralogy is described as a volume fraction of the total solid in each grid cell, and thus does not readily support the development of an isotopically zoned mineral surface.Some approximation to this zoning might be achievable through the use of a nested-grid or multiple interacting continua approach (e.g., Xu and Pruess 2001), but this would severely restrict the size of the spatial domain and would require independent, a-priori knowledge of the length scales of zoning.Specifying a fi xed isotopic composition of a solid phase also restricts the range of saturation states that can be accurately simulated.If a system were to become supersaturated with respect to a solid phase where the isotopic composition of that solid was pre-specifi ed, accumulation of new precipitate would induce fractionation of the surrounding fl uid.Stated differently, specifying a fi xed solid phase isotopic composition effectively means that the rate law describing the dissolution of that solid is no longer inherently reversible.
The observed temporal trend in fractionation associated with dissolution indicates that these effects are likely negligible to long term processes, but potentially infl uential over shorter time periods.This implies that models for isotope fractionation applied to contaminant remobilization, microbially catalyzed oxidative dissolution and system response to short term environmental perturbations may benefi t from an improved capability to simulate the isotopic partitioning characteristic of the initial stages of dissolution.Based on current modeling approaches, it appears that accounting for fractionation due to dissolution of a solid within a reactive transport framework requires a means of describing the development of a distinct solid surface that infl uences isotopic partitioning.This implies the need to resolve the geometry and surface of individual grains, i.e., a pore-scale model.Precipitation is less problematic, and has been more extensively described by Darcy-scale simulations, but as will be discussed in the next section this process is also infl uenced by the unique composition of the reactive interface.

Precipitation
In the previous section concerning studies of net dissolution, experimental conditions largely prohibited the accumulation of secondary minerals.Where secondary solids composed of the fractionating element were able to form, precipitation likely infl uenced the observed stable isotope ratio.For example Mathur et al. (2005) observed partitioning of copper stable isotopes during abiotic oxidative dissolution of chalcocite and chalcopyrite where the Δ 56 Cu fl uid-bulksolid was positive, in conjunction with mass balance calculations that suggested removal of copper from solution.Similarly, Wimpenny et al. (2010) report positive Δ fl uid-bulksolid values for lithium when the pH range allowed for concurrent secondary mineral precipitation.
The formation of a precipitate from solution is broadly recognized as a fractionating mechanism in a variety of stable isotope systems.For example, partitioning of the lighter 6 Li isotope into the solid phase was predicted by ab initio calculations (Yamaji et al. 2001) and observed during carbonate precipitation (Marriott et al. 2004), clay formation (Williams and Hervig 2005;Vigier et al. 2008) and accumulation of secondary silicates (Wimpenny et al. 2010).These observations suggest that precipitation of secondary minerals contributes to enrichment in heavy lithium isotope in rivers relative to primary mineralogy in a wide variety of weathering environments (Vigier et al. 2009;Lemarchand et al. 2010;Millot et al. 2010;Dellinger et al. 2014).Similar partitioning of silicon isotopes associated with precipitation was predicted (Meheut et al. 2007(Meheut et al. , 2009) ) and inferred from observations of progressive basalt weathering (Ziegler et al. 2005;Georg et al. 2007) and riverine composition (Georg et al. 2006;Opfergelt et al. 2013).Recently, Geilert et al. (2014) demonstrated a temperature dependent preferential incorporation of the lighter isotope of silicon into amorphous silica, while Oelze et al. (2015) demonstrated a rate dependence of silicon fractionation during sequential precipitation of amorphous silica in the presence of dissolved aluminum.Johnson (2011) noted that in general the isotopic partitioning of redox sensitive elements are governed by pathways that alter the redox state of reacting species, and thus precipitation in the absence of electron transfer tends to impart comparatively minor fractionation in these systems.However, Skulan et al. (2002) demonstrated that under extremely rapid precipitation rates kinetic effects were observable in iron isotopes.
The Skulan et al. (2002) study offers a closed-system kinetic end-member example of the isotopic partitioning associated with precipitation.They utilized experimental conditions in which dissolved ferric iron was rapidly precipitated from solution to form hematite, resulting in approximately 80% removal of Fe(III) aq in one hour.These conditions lead to a large kinetic fractionation (Fig. 9A), which was observable in both the residual aqueous phase and the bulk hematite stable isotope ratios of iron.This observation was considered a kinetic effect due to the rapid rate of accumulation prohibiting isotopic equilibration between phases and because longterm exposure of hematite to solution resulted in small Δ 56 Fe values.Furthermore, enrichment in the heavy isotope composition of the fl uid as the reaction progressed followed a Rayleigh model (Eqn.14), adhering to an apparently constant fractionation factor of D = 0.99868 ± 12 and suggesting the infl uence of a single fractionating process.This distillation effect was pronounced enough that the shift in isotopic ratio as a function of reaction progress was also observable in the ' 56 Fe of the bulk hematite.Although this cumulative value only increased by approximately 0.8 ‰ compared with a fl uid enrichment of roughly 4.4 ‰, mass balance dictates that the isotope ratio of the hematite surface forming from the surrounding fl uid varied as much as the fl uid phase, leading to formation of a solid that was isotopically heterogeneous, or zoned.This zoning was evident when hematite crystals synthesized by this rapid process were subsequently dissolved in HCl (Fig. 9B).Despite a bulk hematite G 56 Fe composition of -0.29 ±0.05 ‰, dissolution of the fi rst 6.8 % of the solid yielded a fl uid value of +1.16 ± 0.11 ‰, which subsequently decreased with continued dissolution to reach the value of the bulk solid.Two aspects of the Skulan et al. (2002) results are relevant to the current discussion.First, a solid phase can preserve an isotopic signature associated with the kinetic effects of formation, and second in a closed system the kinetic signal imparted during formation of the solid phase varies as a function of the progress of the reaction, which in turn produces an isotopically heterogeneous or zoned precipitate.Dissolution of such a solid could then lead to the appearance of fractionation as an artifact of zoning.This concern was noted in several  of the dissolution studies discussed in the previous section, for example Crosby et al. (2007) conducted separate partial dissolution measurements to ensure the homogeneity of their hematite and goethite samples.Based on the Skulan et al. (2002) results, if the light isotope is preferentially incorporated into the solid during precipitation, and this solid is subsequently dissolved, zoning would seem to counteract the general trend of light isotopes preferentially mobilizing in the fl uid phase during dissolution (Table 2).The extent to which preferential mobilization of the light isotope infl uenced the values shown in Fig. 9B cannot be disentangled from the effect of the zoned hematite composition.
If the Skulan et al. (2002) results represent one extreme, in which the isotopic composition of a precipitate is entirely governed by unidirectional, kinetic fractionation, then the other end member is the formation of a solid at slow enough rates that isotopic equilibrium is maintained between phases.Many natural systems exist in a range of saturations close to chemical equilibrium (Helgeson et al. 1984) and solids are thus often anticipated to form over timescales suffi cient to refl ect isotopic equilibrium.For example, the use of stable isotope ratios as paleoproxies commonly requires that the timescales associated with formation of solids are suffi cient to maintain isotopic equilibrium.This has been a particular issue in the interpretation of carbon and oxygen isotope compositions in carbonates, where multiple studies have demonstrated a combination of equilibrium and kinetic effects (Romanek et al. 1992;Kim and Oneil 1997;Chacko et al. 2001;Mickler et al. 2006;Dietzel et al. 2009;Gabitov et al. 2012;Gabitov 2013;Watkins et al. 2013).More recently, the same issues have been considered in the mid-mass stable isotope ratios of cations in carbonates (Fantle and Tipper 2014).For example both magnesium and calcium isotopes have been shown to partition during carbonate mineral formation, typically such that the fl uid is isotopically heavy relative to the solid (Galy et al. 2002;Lemarchand et al. 2004;Gussone et al. 2005;Tang et al. 2008;Kisakuerek et al. 2009;Immenhauser et al. 2010;Rustad et al. 2010;Reynard et al. 2011;Wombacher et al. 2011;Mavromatis et al. 2013).
Observations of both light-and mid-mass stable isotope partitioning during carbonate precipitation seem to indicate a combination of kinetic and equilibrium effects.These observations imply that the mechanisms of fractionation should not be viewed as a binary system, but rather as a continuum between the two processes.As a result, a range of reaction rates should exist near equilibrium where the apparent or observed fractionation factor associated with formation of a solid phase is rate dependent (Fig. 10).Within a closed system, this effect leads to the accumulation of an isotopically heterogeneous solid as a function of both changes in reaction rate and distillation of the reactant reservoir.Within open systems, in the absence of any supply or transport limitations, the isotopic composition of the solid surface may still vary as a result of changes in the saturation state of the system.Thus experiments designed to quantify the magnitude of isotopic partitioning between a fl uid and precipitating solid often utilize open systems in which the chemical composition of the fl uid is continually titrated to maintain a fi xed saturation state (Tang et al. 2008;Watkins et al. 2013), thereby ensuring that the accumulated solid phase is isotopically homogeneous.In natural systems, lack of such controls on saturation state or reaction rate lead to the formation of solids such that the observed fractionation factor appears to vary both spatially (e.g., Tipper et al. 2012) and temporally (e.g., Druhan et al. 2013).Therefore models that describe the exchange of material between a coevolving fl uid and solid surface should be capable of describing this variability.
Several models have been proposed for the partitioning of stable isotopes during net precipitation under controlled experimental conditions.Tang et al. (2008) utilized the surface entrapment model (SEMO) developed by Watson and Liang (1995) and Watson (2004) to describe their variable fractionation factor as a function of precipitation rate (Fig. 10).The SEMO model was developed to describe zoning of trace elements and isotopes in solids that were thought to form under such slow growth rates that the fl uid and solid surface must refl ect equilibrium partitioning and an absence of aqueous transport limitations.In order to explain such observations, the SEMO model assumes that although the isotope ratios of the interior of the solid and the bulk fl uid refl ect the equilibrium fractionation factor, there exists a surface layer of the solid in which the rare isotope is depleted, with a maximum partitioning defi ned by a surface entrapment factor (F).Under growth conditions, this depleted surface layer may become 'entrapped' in the crystal beneath newly formed solid, leading to a disequilibrium fractionation.This effect is offset by re-equilibration due to diffusion within the crystal lattice.A key aspect of the model is the requirement that ion mobility, in this case cast as the solid-state diffusion coeffi cient governing re-equilibration (Watson 2004), increases from the crystal interior to a higher value at the solid surface.In application this approach was able to accurately reproduce the trend in fractionation as a function of precipitation rate for the Tang et al. (2008) dataset.However, the diffusion coeffi cient at the solid surface required to achieve this result was roughly 16 orders of magnitude higher than that estimated for trace element diffusion in calcite.
An alternate approach to the SEMO model was proposed by DePaolo (2011), which considers a kinetic effect in the fl uid boundary layer surrounding the precipitating solid (Fantle and DePaolo 2007).This model represents a fundamental shift from both the SEMO approach and the models used to describe dissolution discussed in the previous section, in that there is no description of a depth or volume percentage considered to represent a reactive surface.Instead, the only aspect of the solid considered is the surface in contact with the surrounding fl uid (Fig. 11A), which dictates the isotopic composition of the backward reaction.As discussed in the introduction it is the cumulative effect of an irreversible forward addition of material to the mineral surface and an irreversible backward removal of material from the mineral surface that infl uences the observable fractionation factor, despite the fact that the overall system is supersaturated and thus precipitate is accumulating.In principle, this creates a coupled parameter set, in that the isotope ratio of the fl uid and the solid surface are both evolving  2011) simplifi es this relationship by assuming that the isotopic composition of the solid surface is at steady state, as in the case of the buffered titration experiments in which a fi xed supersaturation is maintained (Tang et al. 2008;Watkins et al. 2013).In doing so, the formulation leads to a closed form solution for the observed fractionation factor (D obs ) at steady state where R f and R b are the forward and backward reaction rates (note R f may be recast as R net + R b to utilize the net observed reaction rate), D f is the fractionation factor associated with the forward addition of material to the precipitating mineral and D eq is the fractionation factor associated with dynamic equilibrium between the solid surface and surrounding fl uid.In order to apply this model, the value of the backward rate of material removed from the solid surface, as well as the two fractionation factors D f and D eq must be supplied.DePaolo (2011) demonstrated that for an R b of 2.16x10 3 μmol/m 2 /hr roughly corresponding to the values reported by Chou et al. (1989) and fractionation factors D f = 0.9984 and D eq = 0.9995 the model captures the trend observed by the Tang et al. ( 2008) study (Fig. 11B).
This model is based on the same concept of a dynamic system discussed in the introduction section.An implicit assumption of this affi nity-based approach is that reactivity and fractionation are describable based on the composition of the fl uid phase relative to equilibrium with the solid.In practice, the parameters required to apply this type of model often vary between systems, for example requiring a higher order dependence on the departure from equilibrium, or a distinct kinetic rate constant (Stack 2014).DePaolo (2011) points out that the value of D eq used to reproduce the Tang et al. ( 2008) dataset (0.9995) is inconsistent with independent estimates (Fantle and DePaolo 2007;Jacobson and Holmden 2008).An alternate parameter set is also able to fi t the Tang et al. (2008) data using an D eq of 1.000 and a variable backward reaction rate (R b ), which DePaolo (2011) calculates as a function of the fl uid saturation state.
Recent efforts have focused on replacement of the TST derivation with crystal growth models that include a description of the type and prevalence of reactive sites at the mineral surface in an effort to provide a more mechanistic and predictive capability (Wolthers et al. 2012;Bracco et al. 2013).This type of crystal growth model has been applied in combination with the model of DePaolo (2011) to derive solutions for calcium isotope partitioning during calcite precipitation as a function of solution stoichiometry (Nielsen et al. 2012) and for oxygen isotope partitioning during calcite precipitation as a function of pH (Watkins et al. 2013).The key development of the recent modeling approaches is to allow the infl uence of factors in addition to reaction affi nity to infl uence the reaction rate and thus the isotopic compositions of the fl uid and solid.However, these derivations still largely incorporate a steady state solution as demonstrated in Figure 11.Recently, Steefel et al. (2014a) discuss the need to explicitly describe the time-variant evolution of both the fl uid and solid surface in order to model isotopic re-equilibration of a kinetically fractionated system.In the following section, we describe current modeling approaches to resolve the time dependence of isotopic exchange between fl uids and solid surfaces in reactive systems during net precipitation, and show that aspects of this problem are uniquely resolvable at the pore scale.

TREATMENT OF TRANSIENT ISOTOPIC PARTITIONING AND ZONING
Extending the concept of a variable apparent fractionation factor to through-fl owing, fi eld scale systems has thus far been accomplished primarily through the use of continuum-scale reactive transport models.As discussed in the dissolution section, these models have commonly been applied to simulate fractionation during homogeneous (aqueous) reactions.Accumulation of a solid phase that is composed of two isotopes of the same element is problematic in that any change in the isotope ratio of the fl uid through time needs to be incorporated into the solid phase without imposing an erroneous fractionation factor.Druhan et al. (2013) accommodated a variable solid phase isotopic composition in the context of calcium isotope partitioning during carbonate mineral precipitation using a solid solution.In common practice, the overall rate of carbonate mineral precipitation: where k is the kinetic rate constant, K eq is the equilibrium constant and a represents the activity of the individual species, is simplifi ed by the fact that the activity of a pure solid phase is equal to unity ( 3(s) CaCO a = 1).In the Druhan et al. (2013) model, the isotopes of calcium are simulated as separate 'species', which are coupled through their precipitation rates as: The key to this approach is the recognition that if  accuracy of simulations from highly variable, fi eld scale systems (e.g., Druhan et al. 2013) to highly controlled laboratory experiments (e.g., Tang et al. 2008).This observation implies that .mechanisticdescriptions of isotope partitioning could be signifi cantly improved by modeling approaches that are capable of resolving spatial zoning both within a single phase and within individual crystals or grains (e.g., Li et al. 2006;Tartakovsky et al. 2008;Molins et al. 2012;Molins 2015, this volume;Yoon et al. 2015, this volume).
One such method is the use of a lattice Boltzmann approach (Yoon et al. 2015, this volume), which is capable of explicitly describing the location and shape of a boundary between independently discretized fl uid and solid phases.This approach has previously been applied to describe heterogeneous reactivity (Kang et al. 2002(Kang et al. , 2003(Kang et al. , 2004) ) including multicomponent systems (Kang et al. 2006(Kang et al. , 2007(Kang et al. , 2010;;Yoon et al. 2012). Recently, Huber et al. (2014) developed an approach to describe the transfer of mass across a phase boundary that is independent of the shape of the solid surface.This approach is well suited to the description of fractionating reactions that are coupled through isotope partitioning across a phase boundary.As a proof of concept, Figure 13 demonstrates application of the Huber et al. (2014) model to a closed system in which the fl uid phase surrounding a grain of calcite is initially supersaturated with respect to the solid.Two isotopes of calcium are explicitly tracked using the formulations given in Equation (31a,b) over time as the system reaches chemical and isotopic equilibrium.Figure 13 depicts a particular snapshot in time, during which the fl uid phase has not yet reached equilibrium with the solid.This disequilibrium is evident in the gradients of reactive species as well as the fl uid isotope ratio with distance from the solid phase boundary.Importantly, the surface of the solid phase, which is composed of pure CaCO 3(s) is also isotopically variable, and supplies a more accurate estimate of the values of the mole fraction for each isotope X (Eqn.32) than previous bulk averaged estimates.This is not readily observed in the spatial distribution of isotope ratios because only a very small portion of the solid phase surface is considered in contact with the surrounding fl uid.However, a time series plot of the isotope composition of the calcite surface clearly demonstrates this variability, which feeds back into the rate at which the fl uid phase established isotopic equilibrium with the reactive solid surface.This proof of concept exercise suggests that signifi cant improvements could be gained from pore scale descriptions of isotopic partitioning during precipitation, dissolution, transport and decay.For example, application of the Druhan et al. (2013) derivation in a system where X (Eqn.32) is only describable as the bulk solid phase value would fail to reproduce observed isotope partitioning during dissolution.The same is true for the closed form solutions of DePaolo (2011) and Brantley et al. (2004).The result is a situation in which reactions that generate isotope partitioning, which are formulated as a reversible departure from equilibrium, cannot be described across the range of saturation states from net dissolution to net precipitation with a single model.The use of pore scale discretization may provide a means of circumventing this discontinuity.2014) methodology, and including two isotopes of calcium using the modifi ed TST rate laws of Druhan et al. (2013).A solid phase calcite grain (textured) is surrounded by a fl uid phase that is initially supersaturated with respect to calcite.The domain is defi ned by no-fl ux boundary conditions.Spatial concentration and isotope ratios are shown for a given point in time during the transient approach to equilibrium.The isotopic composition of the solid surface in contact with the fl uid is tracked, used to defi ne X in Equation ( 32), and varies in tandem with the isotope ratio of the fl uid through time.

Figure 3 .
Figure 3. Mineral grains in contact with pore fl uids lose 234 U by alpha recoil, preferential leaching along recoil tracks, dissolution-precipitation and diffusion.The recoil distance of Th (L) defi nes the depleted surface area depth.Adapted from Maher et al. (2006a).

Figure 4 .
Figure 4. Estimates of f D as a function of grain size modifi ed from DePaolo et al. (2006), Maher et al. (2006a) and DePaolo et al. (2012).Values of O r represent a roughness factor accounting for deviations in grain shape and surface area from an ideal sphere.Grey shaded zone indicates the region of measured values from the King's River Fan (Lee et al. 2010), dust from ice cores (Aciego et al. 2011), marine sea core sediment (DePaolo et al. 2006) and sediment from Hanford, WA (Maher et al. 2006a).

Figure 5 .
Figure 5. E values as a function of inverse residence time of water in the fi rst solvation shell for ions where both experimental and MD simulation data are available.Figure modifi ed from Bourg et al. (2010).

Figure 6 .
Figure6.Numerical simulation of a tracer diffusing through time in a discretized domain demonstrating how experimental diffusion data could be used to constrain relative diffusion coeffi cients.The system is closed, and initially an elevated concentration exists in the center of the domain.(A) the concentration of the tracer across the domain through arbitrary time points 0.25, 1 and 5; (B) the isotope ratio relative to an arbitrary standard in delta notation (‰) for the same time points.In (A) and (B) the vertical dashed line corresponds to the location over which time series of concentration (C) and isotope ratio (D) are monitored.

Figure 7 .
Figure 7. Fractionation between dissolved and bulk solid iron isotopes as a function of time since the start of dissolution.A Δ-value of 0.0 represents no fractionation.(A) Goethite dissolution promoted by oxalate in the absence of light, which removes the effects of oxidation, modifi ed from Wiederhold et al. (2006).(B) Granite (biotite and chlorite enriched fraction) dissolution promoted by hydrochloric acid, modifi ed from Kiczka et al. (2010).

Figure 8 .
Figure 8. (A) Diagram of modeled reservoirs and material transfer describing parameters of the Brantley et al. (2004) derivation.See main text for defi nition of variables.(B) Ratio of release rates of iron isotopes to the fl uid, relative to ratio of isotopes in bulk solid as a function of time.This value is infl uenced by the thickness of the leach layer, the rate of transfer of material from the bulk solid to this leach layer, and the rate of dissolution into the fl uid.(C) 54 C(t) and 56 C(t) obtained by numerical integration of eq. 12 presented as Δ fl uid-bulksolid vs. time.Results are presented for arbitrarily chosen values 56 k = 0.5, 54 k/ 56 k = 1.001, 56 Co = 4, 54 Co/ 56 Co = 0.001, F = 1, x = 0.01 and 20.

Figure 9 .
Figure 9. (A) measured values of fl uid-phase G 56 Fe (open circles) and cumulative solid-phase G 56 Fe of hematite (fi lled circles) during rapid precipitation.Modeled fl uid-phase isotope fractionation (black line, Eqn.14) and cumulative solid-phase fractionation (grey line) agree with measured values.Modeled solid surface isotopic composition (dashed line) is inferred by mass balance.(B) corresponding values during subsequent dissolution.Data from Skulan et al. (2002).

Figure 10 .
Figure10.Calcium isotope fractionation between bulk fl uid and calcite precipitated at steady state for a range of net precipitation rates(Tang et al. 2008).Each of these values are generated by precipitating calcite at a fi xed oversaturation for a period of time suffi cient to accumulate enough isotopically homogeneous solid for measurement.
then the activities of the individual isotopologues of calcite must sum up to unity.Assuming an ideal solid solution,

Figure 11 .
Figure 11.(A) schematic of the DePaolo (2011) model for calcium isotope fractionation during precipitation of calcite for a reversible reaction.(B) Application of the DePaolo (2011) model (heavy dashed line) to the Tang et al. (2008) dataset.The model is bounded between two values defi ned by D f and D eq .

Figure 12 .
Figure 12.Fluid calcium isotope ratios and calcium concentrations measured as a function of time during unbuffered carbonate mineral precipitation.(A) Measured values (open circles) in comparison with Rayleigh models (solid lines) for three values of G, showing that the data do not conform to a constant fractionation factor, modifi ed from Druhan et al. (2013).(B) Reactive transport model of calcium isotope data as a function of time using Equations (32a,b), demonstrating good agreement with measured fl uid values by approximating the mole fractions X with the bulk carbonate isotope composition.[Used with permission from Elsevier Limited, from Druhan JL, Steefel CI, Williams KH, DePaolo DJ (2013) Calcium isotope fractionation in groundwater: Molecular scale processes infl uencing fi eld scale behavior.Geochimica et Cosmoschimica Acta, Vol.119, Fig. 7A, p. 107.]

Figure 13 .
Figure 13.Lattice Boltzmann model of calcite precipitation based on the Huber et al. (2014) methodology, and including two isotopes of calcium using the modifi ed TST rate laws ofDruhan et al. (2013).A solid phase calcite grain (textured) is surrounded by a fl uid phase that is initially supersaturated with respect to calcite.The domain is defi ned by no-fl ux boundary conditions.Spatial concentration and isotope ratios are shown for a given point in time during the transient approach to equilibrium.The isotopic composition of the solid surface in contact with the fl uid is tracked, used to defi ne X in Equation (32), and varies in tandem with the isotope ratio of the fl uid through time.
At this point the supply of 234 U from 238 U decay is balanced by the decay of 234 U to230Th, representing a steady state where the ratio of O limits the maximum possible intermediate daughter activity.When A 1 = A 2 this is a special condition

Table 2 .
Compilation of measured fractionation between dissolved and initial solid phases during net dissolution at ambient temperature and pressure.Where fractionation is measured as a function of time, the maximum value is reported.Values are divided into abiotic and microbially mediated experimental conditions.Values reported in italics are considered within error of 0.0‰.