Wilbur and Ague 4 WILBUR AND AGUE; APPENDIX DR1. Two-dimensional chemical maps as well as chemical profiles were done at 15 kv using

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4 Wlbur and Ague 4 WILBUR AND AGUE; APPENDIX DR1 MINERAL ANALYSES Two-dmensonal chemcal maps as well as chemcal profles were done at 15 kv usng the JEOL JXA-8600 electron mcroprobe at Yale Unversty usng wavelength-dspersve spectrometers, natural and synthetc standards, mean atomc number (MANB) and off-peak background correctons, and φ(ρz) matrx correctons. Peak T was nsuffcent to homogenze compostons by dffuson, so the chemcal zonaton patterns provde a valuable record of reacton hstores (cf. Kohn and Spear, 2000; Spear and Danel, 2001; Carlson, 2002). Chemcal maps were done wth MANB and ether 100 na beam current and 1.5 second dwell tmes, or 200 na beam current and 0.25 second dwell tmes. Hgh spatal resoluton profles ndcate smlar concentratons for T and Y across core-rm boundares n WepS garnets (not shown). NOTES ON MONTE CARLO CRYSTAL GROWTH SIMULATIONS We use the Monte Carlo (MC) method of Xao et al. (1988) to model crystal growth at constant P. The treatment ncludes dffuson of nutrent through the flud phase to the growng crystal, surface attachment knetcs, and dffuson along crystal faces (a form of surface relaxaton). We extend the method to allow for varable amounts of equlbrum oversteppng durng growth. The goal of the modelng s to nvestgate whether crystal morphology transtons n metamorphc rocks can be related to the degree of reacton oversteppng, not to reproduce crystal shapes and szes exactly. The smulatons are done on a two-dmensonal, trgonal lattce (1000 x 1000 grd sze). Nutrent aqueous speces dssolved n the flud are generated n a crcular source regon at some dstance from the growng crystal. Dffuson through the flud takes place accordng to a random-walk process through the grd. Intervenng mneral grans

5 Wlbur and Ague 5 could modfy dffuson pathways, but we note that f nutrent supples were strongly ansotropc, then the natural crystals would have grown wth marked asymmetres (cf. Xao et al., 1990). Although dffuson rates are not quantfed, lattce unts n the crystal have the same length as the mean free path for dffuson, approprate for growth from a lqud (Xao et al., 1990). The probablty that nutrent dffusng through the flud wll bond to the crystal surface on contact at ste (P ) s gven by the rate of mpngement of nutrent (K ) relatve to the sum of the rates of mpngement and detachment (K - ): P K K K = = Keq K eq exp( Δμ / kt ) exp( Δμ / kt ) ν exp( E / kt ) (1) where k s the Boltzmann constant, T s absolute temperature, E s the nteracton energy for the growth unt and ts nearest and next-nearest neghbors n the sold, and ν s a vbraton factor. Δμ / kt s the thermodynamc drvng force for crystallzaton that acts to bond growth unts to the crystal. The chemcal potental dfference Δ μ reflects the average Gbbs free energy dfference between the compostons of the supersaturated flud and the flud n equlbrum wth the sold drectly at the soluton-crystal nterface. Crystallzaton at low equlbrum favors growth of compact, facetted, euhedral crystals whereas hgh Δμ / kt Δ μ / kt near chemcal corresponds to strong dsequlbrum and results n anhedral or branched forms (Xao et al., 1988). E Φ1n Φ2 = m ; Φ 1 and Φ 2 are the nteracton energes per molecule between a gven ste n the lattce and ts n nearest neghbors and m next-nearest neghbors, respectvely. The normalzed bond strength Φ / kt ncreases as bond strength ncreases or T decreases. 1 K eq s the equlbrum value of K and s evaluated assumng local equlbrum drectly at the crystal surface (rates of attachment and removal are equal). The nutrent may bond to the crystal,

6 Wlbur and Ague 6 dffuse along the surface, or return to the flud phase. If the nutrent does not bond, the jump rate from ste on the crystal to ste j s: K j = ν s exp( ΔEj / kt ) (2) where ν s s a vbraton factor. Φ ( n n j ) Φ2( m m 1 j ΔEj s the actvaton energy approxmated by ) ; n and m denote numbers of nearest and next-nearest neghbors, respectvely, for ste and the potental jump ste j. The probablty of a jump from ste to a partcular neghbor ste j on the crystal or n the flud s gven by the rato of the K value for j the jump relatve to the sum of the K j values for all possble nearest-neghbor jumps. Computer soluton follows the flow chart gven n Fgure 2 of Xao et al. (1988), wth the excepton that we account for varable Δ μ / kt durng model crystal growth. We found that Φ / Φ 2 1 values less than ~ ±0.2 were best for replcatng the observed garnet morphologes. There are scale-dependent aspects to crystal growth (Xao et al., 1988), but the general conclusons regardng oversteppng and morphology wll be approprate for macroscopc crystals. For example, the transton from branched to compact, euhedral crystal forms wll shft to slghtly smaller Δ μ / kt and larger Φ / kt as model crystal sze ncreases (Xao et al., ), but growth n the 3 rd dmenson, whch we neglect, tends to have the opposte effect and stablzes compact, euhedral forms at larger two effects wll tend to cancel. Δ μ / kt (Xao et al., 1991). Thus we nfer that these Early growth at large Δ μ / kt followed by growth at lower Δ μ / kt s successful at qualtatvely reproducng the observed garnet morphologes. We examne two possbltes for varatons n Δμ / kt durng crystal growth. In the frst, growth occurs ntally at large Δ μ / kt and then drops to a low value at a prescrbed stage of the growth hstory. In the second, the

7 Wlbur and Ague 7 log 10 ( Δ μ / kt ) decreases lnearly from an ntal large value to a fnal small one. Many other possble Δ μ / kt paths are possble, but the two examned here both capture the basc morphologcal characterstcs of the crystals. Future refnement of the crystal growth models may allow Δμ / kt -tme paths to be accurately determned based on crystal morphologes. In all examples shown, the model crystals comprse 100,000 molecular growth unts. REFERENCES CITED IN APPENDIX AND FIGURES Ague, J.J., 2002, Gradents n flud composton across metacarbonate layers of the Wepawaug Schst, Connectcut, USA: Contrbutons to Mneralogy and Petrology, v. 143, p Carlson, W.D., 2002, Scales of dsequlbrum and rates of equlbraton durng metamorphsm: Amercan Mneralogst, v. 87, p Kohn, M.J., and Spear, F.S., 2000, Retrograde net transfer reacton nsurance for pressuretemperature estmates: Geology, v. 28, p Spear, F.S., and Danel, C.G., 2001, Dffuson control of garnet growth, Harpswell Neck, Mane, USA: Journal of Metamorphc Geology, v. 19, p Xao, R.F., Alexander, J.I.D., and Rosenberger, F., 1988, Morphologcal Evoluton of Growng Crystals: a Monte Carlo Smulaton: Physcal Revew A, v. 38, p Xao, R.F., Alexander, J.I.D., and Rosenberger, F., 1990, Growth morphology wth ansotropc surface knetcs: Journal of Crystal Growth, v. 100, p Xao, R.F., Alexander, J.I.D., and Rosenberger, F., 1991, Growth Morphologes of Crystal Surfaces: Physcal Revew A, v. 43, p

4.2 Chemical Driving Force

4.2 Chemical Driving Force 4.2. CHEMICL DRIVING FORCE 103 4.2 Chemcal Drvng Force second effect of a chemcal concentraton gradent on dffuson s to change the nature of the drvng force. Ths s because dffuson changes the bondng n a