Pure appl. geophys. 151 (1998) 605–6180033–4553/98/040605–14 $ 1.50+0.20/0

Free Energy Minimization by Simulated Annealing withApplications to Lithospheric Slabs and Mantle Plumes

CRAIG R. BINA1

Abstract—An optimization algorithm based upon the method of simulated annealing is of utility incalculating equilibrium phase assemblages as functions of pressure, temperature, and chemical composi-tion. Operating by analogy to the statistical mechanics of the chemical system, it is applicable both toproblems of strict chemical equilibrium and to problems involving metastability. The method reproducesknown phase diagrams and illustrates the expected thermal deflection of phase transitions in thermalmodels of subducting lithospheric slabs and buoyant mantle plumes. It reveals temperature-inducedchanges in phase transition sharpness and the stability of Fe-rich � phase within an �+� field in coldslab thermal models, and it suggests that transitions such as the possible breakdown of silicate perovskiteto mixed oxides can amplify velocity anomalies.

Key words: Phase relations, simulated annealing, subduction zones, mantle plumes.

Introduction

Determining stable mineral phase assemblages as functions of pressure, temper-ature, and composition is an important part of modeling the structure andcomposition of earth’s interior. For systems with few chemical components, suchproblems are typically approached (e.g., HELGESON et al., 1970; BINA and WOOD,1987; POWELL and HOLLAND, 1988) by numerically solving sets of equilibriumequations, ��

i =��i , for all phases � and � for each component i, where the

chemical potential of component i in phase � (��i ) is a function of pressure (P),

temperature (T), and composition (as expressed by the mole fractions X�i ). For

more complex systems with larger numbers of chemical components, such problemsare commonly approached (e.g., VAN ZEGGEREN and STOREY, 1970; SMITH andMISSEN, 1982) by minimizing the Gibbs free energy G=�i, � X�

i ��i , so as to find

the equilibrium X�i as functions of P and T. This latter approach amounts to the

minimization of a highly nonlinear objective function (G) of many variables (P, Tand the X�

i ) subject to several linear equality constraints (mass balance conditions)and linear inequality constraints (mole fractions between zero and one).

Many methods have been employed to solve such free energy minimizationproblems, including linear (e.g., ‘‘steepest-descent’’) algorithms (STOREY and VAN

1 Department of Geological Sciences, Northwestern University, 1847 Sheridan Road, Evanston, IL60208-2150, U.S.A. E-mail: craig@earth.nwu.edu

Craig R. Bina606 Pure appl. geophys.,

ZEGGEREN, 1964; ERIKSSON, 1975; SAXENA, 1982; WOOD and HOLLOWAY, 1984;BINA and WOOD, 1987; WOOD, 1987; SAXENA, 1996), quadratic (e.g., ‘‘quasi-New-ton,’’ ‘‘conjugate-gradient’’) algorithms (SUNDMAN et al., 1985; GHIORSO, 1985; DE

CAPITANI and BROWN, 1987; HARVIE et al., 1987; GHIORSO and SACK, 1995), orhybrid linear-quadratic methods (BINA, 1987). Such techniques, which explore thefree energy hypersurface by accumulating information on local slope and curvature,work well for the study of gradual phase transitions, in which equilibrium phaseproportions and compositions vary slowly and smoothly. However, they oftenencounter difficulties when applied to more abrupt phase transitions in non-idealmulticomponent systems, in which elemental partitioning may shift dramatically orindividual phases may appear or disappear over small ranges of pressure, tempera-ture, or composition. Such problems can be partially alleviated by the use of activeset methods (GILL et al., 1981), but this does not change the fact that the propertiesof the free energy hypersurface can become unstable for small values of X�

i (giventhe logarithmic dependence of ��

i upon X�i ).

In response to such issues, as well as to the more general problem of avoidingentrapment in local minima (ERIKSSON and HACK, 1990), stochastic optimizationmethods such as Simulated Annealing (KIRKPATRICK et al., 1983; KOREN et al., 1991),Genetic Algorithm (GOLDBERG, 1989; SAMBRIDGE and DRIJKONINGEN, 1992), andTaboo Search (GLOVER, 1989, 1990; CVIJOVIC and KLINOWSKI, 1995) have beenapplied to certain classes of optimization problems. Here I present a free energyminimization technique based upon the method of simulated annealing (KIRK-

PATRICK et al., 1983). Because it is a stochastic method which does not rely uponestimation of local hypersurface slope and curvature, it works well for both gradualand abrupt phase transitions, and avoids entrapment in local minima. Because itoperates by analogy with the statistical mechanics of the chemical system, it is alsoideal for the study of metastability. I illustrate the method using simple compositionalsystems subject to various thermal structures which represent subduction zones andmantle plumes. Upon solving for the stable phase assemblages by free energyminimization, I also calculate corresponding geophysical properties, such as density,buoyancy, seismic wave speed structures, and potential elastic anisotropy contrasts.

Method

KIRKPATRICK et al. (1983) identify four key elements of a simulated annealingalgorithm: a description of system configuration, a random generator of rearrange-ments, a quantitative objective function, and an annealing schedule. The systemconfiguration here is described in terms discrete units (thus allowing use of efficientinteger arithmetic) of chemical components, in this case corresponding to moles ofconstituent oxides (e.g., MgO). These components ‘‘occupy’’ various phases, ac-cording to the stoichiometry appropriate to their structural formulae (e.g., n molesof Mg2SiO4 forsterite will consist of n units of SiO2 and 2n units of MgO).

Mantle Phase Relations by Simulated Annealing 607Vol. 151, 1998

Random rearrangements of the system configuration are generated by a proce-dure analogous to reaction mechanisms (Fig. 1). The components occupying arandom number of phases are transferred to an (amorphous) ‘‘excited state.’’ Allcomponents are then allowed to ‘‘precipitate’’ from the excited state into randomlyselected phases. (Note that, in order to ensure full evacuation of the excited statefor each rearrangement, the constituent oxides must be part of the set of allowablephases.)

For each such rearrangement, the objective function G is evaluated, and theMetropolis algorithm (METROPOLIS et al., 1953) is employed to determine whetheror not the rearrangement should be adopted as the new system configuration. TheMetropolis algorithm mimics a Boltzmann distribution function, always acceptingthe rearrangement if it results in an energy decrease (�G�0), otherwise (�G�0)randomly accepting the rearrangement with probability exp(−�G/�), where � isthe annealing ‘‘temperature’’ characterizing this stage of the minimization. Thisprocedure allows rearrangements resulting in an energy increase to occur with highprobability early in a minimization but with falling probability as � decreases laterin the minimization, thus permitting the algorithm to escape from local minima. Arandom number of rearrangements are tested at each �.

Figure 1Cartoon illustrating method by which random rearrangements of system configuration are generated. Ina procedure analogous to reaction mechanisms, components occupying random phases are transferred to

an ‘‘excited state’’ and then allowed to ‘‘precipitate’’ into random phases.

Craig R. Bina608 Pure appl. geophys.,

Figure 2Test phase diagram for olivine polymorphs. Temperature (T) is fixed at 1700 K. Bulk composition(XMg ) varies from 0 (Fe2SiO4) to 1 (Mg2SiO4). Pressure varies from 0 to 30 GPa. Equilibrium phase

distributions show ���+��� and (at high XMg ) ���+��� transitions.

The annealing temperature � is gradually lowered according to an annealingschedule. Convergence is assumed when many random rearrangements yield nosuccessful reductions in free energy at a given � or at several successive �s. In thiscase, I employ the simple empirical expedient of reducing � by a constant fraction(1/2) at each iteration. While this does result in convergence, adoption of moresophisticated schedules which evolve dynamically in response to the nature ofenergy changes in the problem, such as those of ‘‘constant thermodynamic speed’’(MOSEGAARD and VESTERGAARD, 1991; KOREN et al., 1991), ‘‘adaptive’’ (INGBER,1989, 1996), or ‘‘critical temperature’’ (BASU and FRAZER, 1990) annealing, wouldresult in increased efficiency and more rapid convergence of minimization. Furtherinsurance against entrapment in local minima is obtained by simultaneouslyminimizing several systems in parallel (a process ideally suited to multiprocessorcomputing) starting from the same initial configuration.

Some Examples

In the following examples, I consider systems which can be fully characterizedby the three components MgO, FeO, and SiO2. As hosts for these components, Iconsider only the following eleven phases: the �, �, and � phases of Mg2SiO4 andFe2SiO4, the perovskite phase of MgSiO3 and FeSiO3, the magnesiowustite phaseof MgO and FeO, and the stishovite phase of SiO2. For calculating the free energyG, I employ the thermodynamic data set of FEI et al. (1991) for these phases. Forsimplicity of illustration, only two-dimensional geometries are shown.

I first test the algorithm by computing a simple phase diagram for the olivinepolymorphs. Fixing temperature at 1700 K, a P-X section is constructed byallowing bulk composition to vary from Fe2SiO4 to Mg2SiO4 along the x-axis andpressure to increase from 0 to 30 GPa along the y-axis. The resulting equilibrium

Mantle Phase Relations by Simulated Annealing 609Vol. 151, 1998

distributions of the phases � and � (Fig. 2), for example, clearly depict the���+��� and ���+��� transitions at high pressures (KATSURA and ITO,1989). Henceforth, I illustrate cases in which the bulk composition is fixed globallyat (Mg0.9Fe0.1)2SiO4 (i.e., a pure forsterite-90 olivine mantle model). Furthercomplexity can be introduced by including such components as CaO and Al2O3 inpyroxene and garnet phases and by allowing bulk composition to vary with spatialposition, thus allowing, for example, investigation of compositional layering in thelithosphere (HELFFRICH et al., 1989).

I next investigate a subduction zone thermal model, computed on a 120×90grid using N. H. Sleep’s finite difference algorithm (TOKSOZ et al., 1973). Themodel (Fig. 3a) is constructed in a 890×686.63 km box for 140 Ma lithospherewith an initial GDH1 (STEIN and STEIN, 1992) thermal structure, subducting at adip angle of 60 degrees with a velocity of 8 cm/yr (KIRBY et al., 1996; LEFFLER,pers. comm.), Pressures are determined by vertical integration of a reference densityprofile (DZIEWONSKI et al., 1975). Upon computing the equilibrium phase distribu-

Figure 3aTop: Temperature (T) and pressure (P) for subduction zone thermal model from finite differencealgorithm (KIRBY et al., 1996; LEFFLER, pers. comm.). Bottom: Density (�) and bulk sound velocity

(V� ) structures attending consequent equilibrium phase relations in (Mg0.9Fe0.1)2SiO4 composition.

Craig R. Bina610 Pure appl. geophys.,

Figure 3bTop: Maximum potentially observable elastic anisotropy, at room P and T, in VS (kS ) and VP (kP ) forequilibrium phase relations in subduction zone thermal model of Figure 3a. Bottom: Same quantities for

case in which �-olivine persists metastably at temperatures below 1000 K.

tions and compositions for an (Mg0.9Fe0.1)2SiO4 bulk composition (Plate 1), theexpected thermal deflection of phase boundaries (TURCOTTE and SCHUBERT, 1971,1972; SCHUBERT et al., 1975) is evident—the ���+��� and ���+���

transitions deflect upwards while the ���+p�+mw�p�+mw transition deflectsdownwards—as is a temperature-dependent change in transition sharpness withinthe slab (HELFFRICH and BINA, 1994). In the coldest core of the slab, the uplifted�+� region gives way to a shallower region of �+� stability (GREEN andHOUSTON, 1995). Furthermore, Fe-rich � phase is stable within this �+� zone. Thedensity and velocity structures associated with these equilibrium phase assemblages(Fig. 3a) give rise to corresponding buoyancy and velocity anomalies (Plate 1), theformer of which contribute significantly to both convective dynamics (RICHTER,1973; SCHUBERT et al., 1975; CHRISTENSEN and YUEN, 1984, 1985; BINA and LIU,1995) and seismicity-related stresses (ISACKS and MOLNAR, 1971; TURCOTTE andSCHUBERT, 1971, 1972; ITO and SATO, 1992; BINA, 1996). The velocity anomaliesmay be further examined by comparing these results for the equilibrium case tothose obtained for the case of metastably persisting �-olivine, in which the higherpressure olivine polymorphs are not permitted to precipitate at temperatures below

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Plate 1Top: Equilibrium distribution of phases (X�, X�, X�, Xp�, Xmw ) for (Mg0.9Fe0.1)2SiO4 bulk compositionin slab thermal model of Figure 3. Corresponding equilibrium composition (X�

Mg ) of phase �. Buoyancy(F� ) and bulk sound velocity (V� ) anomalies corresponding to equilibrium density and velocitystructures of Figure 3a. Residual bulk sound velocity anomalies (�V� ) for case of metastable �-olivine,after subtraction of anomalies for equilibrium case. Bottom: Equilibrium distribution (X� ) and compo-

sition (X�Mg ) of phase � for (Mg0.9Fe0.1)2SiO4 composition in slab thermal model of Figure 4.

Craig R. Bina612 Pure appl. geophys.,

Figure 4Temperature (T) and pressure (P) for subduction zone thermal model from two-dimensional numerical

convection simulation (KINCAID and SACKS, 1997).

1000 K (RUBIE and ROSS, 1994). The resulting metastable olivine wedge (SUNG andBURNS, 1976; KIRBY et al., 1996; BINA, 1996) yields a narrow low-velocity zonewithin the slab (Plate 1). Finally, using the elastic moduli of the relevant minerals(BASS, 1995) at room pressure and temperature, I compute the maximum VS andVP anisotropy for each phase (MAINPRICE and SILVER, 1993). Neglecting the(possibly substantial) pressure- and temperature-dependence (KARATO, 1997), Ithen construct an approximate measure of the potential maximum elastic an-isotropy (kS, kP ) of each assemblage, applying a simple mole-weighted average ofthe individual phases. The results, for both the equilibrium and the metastable �

cases (Fig. 3b), illustrate how a metastable olivine wedge may be expected to exhibitlarger potential anisotropy to greater depths than an equilibrium slab.

I also investigate a different subduction zone thermal model, this one computedwithin a two-dimensional numerical convection simulation employing temperature-

Figure 5Temperature (T) and pressure (P) for quasi-steady state, axisymmetric, mantle plume thermal model,

with dimensionless basal temperature maximum (KELLOGG and KING, 1997) scaled to 2500°C.

Mantle Phase Relations by Simulated Annealing 613Vol. 151, 1998

dependent viscosity (Rayleigh number �8×105). The model (Fig. 4) is constructedin a 700×1400 km box in which temperature varies over 0–1800°C (KINCAID

and SACKS, 1997). Again, pressures are determined by vertical integration of areference density profile (DZIEWONSKI et al., 1975). The computed equilibriumphase assemblages for an (Mg0.9Fe0.1)2SiO4 bulk composition are broadly similar tothe previous case but with a more marked asymmetry to the isotherms. Again (Plate1), the expected thermal deflection of phase boundaries is evident, althoughdeflection of the ���+p�+mw�p�+mw transition is not observed since therelatively young slab has not yet penetrated to so great a depth. The temperature-dependence of transition sharpness is again detected, along with the stability ofFe-rich � phase in a cold �+� field. While the oceanic lithosphere thins dramati-cally at the edge of this dynamical model, the major features within the slab itselfremain robust.

Finally, I investigate a thermal model for a quasi-steady state, axisymmetric,mantle plume. The model (Fig. 5) is constructed within the framework of a stronglytemperature dependent viscosity (reference viscosity Rayleigh number �107) byheating a patch at the base of the mantle, without accounting for any additionaleffects of internal heating or latent heat of phase change (KELLOGG and KING,1997). I have scaled the dimensionless temperature of the heated patch to 2500°C.Pressures are determined by radial integration of a reference density profile(DZIEWONSKI et al., 1975). Again, the computed equilibrium phase assemblages foran (Mg0.9Fe0.1)2SiO4 bulk composition reveal (Plate 2) thermal deflection of phaseboundaries, although in the opposite sense to that observed for a cold slab. Theassociated density anomalies generate buoyancy forces (Plate 2) important toconvective dynamics, forces which are reversed relative to those associated withslabs. It is important to note that this scaled dynamical model, representing a largethermal upwelling and its counterflow, features unrealistically low temperatures atthe edges of the model which are unlikely to be found in earth’s mantle. Nonethe-less, it is interesting to note (Plate 2) that the thermodynamic parameters of FEI etal. (1991) predict the breakdown of silicate perovskite to mixed oxides in the coldregions at the edge of this plume model. While the thermodynamic description ofSiO2 stishovite in the FEI et al. (1991) data set, derived from the silica polymorphparameterizations of FEI et al. (1990), yields a narrow region of perovskitebreakdown, more recent analyses of the coesite-stishovite transition (AKAOGI et al.,1995; LIU et al., 1996) yield a greater thermodynamic stability for stishovite and

Plate 2Left: Equilibrium distribution of phases (X�, Xst ) for (Mg0.9Fe0.1)2SiO4 bulk composition in plumethermal model of Figure 5. Stishovite parameters from F90 (FEI et al., 1990) and from L96 (LIU et al.,1996). Right: Buoyancy (F� ) and bulk sound velocity (V� ) anomalies in upper half of model mantlearising from corresponding density and velocity structures for L96. Total (left) velocity anomalies (V� )contain component (right) due only to p��mw+st reaction, obtained by removing thermal signature

from total.

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thus a broader region of perovskite breakdown (Plate 2). Because the possibility ofsuch perovskite breakdown has been proposed at higher temperatures in the mantle(HEMLEY and COHEN, 1992; MEADE et al., 1995; SAXENA et al., 1996), it isinstructive to note that this reaction (p��mw+st) is accompanied by a corre-sponding fast velocity anomaly. Thus, in this plume model, the disproportionationof perovskite to mixed oxides serves to amplify the velocity anomaly arising simplyfrom the low temperatures (Plate 2). Despite the low model temperatures, thecontinuing evolution of the thermodynamic parameterization of silicate perovskite(FABRICHNAYA, 1995), and ambiguity about post-stishovite silica phases (BE-

LONOSHKO et al., 1996), these calculations illustrate the principle that interpretationof seismic velocity anomalies solely in terms of thermal structure can be compli-cated by superposition of phase stability boundaries.

Concluding Remarks

An optimization algorithm based upon the method of simulated annealing canbe of utility in calculating equilibrium phase assemblages as functions of pressure,temperature, and chemical composition in complex multicomponent systems. Themethod reproduces known phase diagrams, and it illustrates the expected thermaldeflection of phase transitions in thermal models of subducting lithospheric slabsand buoyant mantle plumes. Application of the method to two different slabthermal models demonstrates temperature-induced changes in phase transitionsharpness and reveals the stability of Fe-rich � phase in an �+� field within a coldslab. Application of the method to a plume thermal model illustrates that reactionssuch as the breakdown of silicate perovskite to mixed oxides can amplify thermalvelocity anomalies. Finally, it is important to note that this method operates byanalogy to the statistical mechanics of the chemical system. Thus, it is not restrictedto problems of strict chemical equilibrium. It is also ideal for the investigation ofmetastability effects (BINA, 1996).

Acknowledgements

I thank L. Kellogg, C. Kincaid, S. Stein, and L. Leffler for providing usefulthermal models. H.-C. Nataf and S. Saxena for helpful reviews, and J. Linton forproducing Figure 1. All other figures were produced using GMT software (WESSEL

and SMITH, 1995). I acknowledge the support of the National Science Foundation(EAR-9158594).

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(Received October 13, 1996, revised April 30, 1997, accepted May 22, 1997)

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