Appendix I: Derivation of the Toy Model

Transcription

1 SPEA ET AL.: DYNAMICS AND THEMODYNAMICS OF MAGMA HYBIDIZATION Thermdynamic Parameters Appendix I: Derivatin f the Ty Mdel The ty mdel is based upn the thermdynamics f an isbaric twcmpnent (A and B) phase diagram. The definitin f quantities is given in Table 1 in the text. Figure 1 gives the ty mdel phase diagram. X is the mass fractin f cmpnent B and Y is the mass fractin f cmpnent A such that X+Y=1 in any phase. There are three pssible phases in this system: crystals f, crystals f β, r liquid (L). Melt f eutectic cmpsitin is represented by Le fr which X=Xe. There are five pssible phase assemblages in this system: L, +L, β+l, Le++β r the crystalline assemblage +β. The phase diagram and thermchemistry are defined by specificatin f Xe, Te, the fusin enthalpies f ( Δh ) and β ( Δh β ) at their respective melting temperatures f T β and T and a single cnstant isbaric specific heat fr crystals f either r β (CS) and fr melt (CL). The liquidii f the ty mdel are linearized in T-X space. This apprximatin makes little difference t any f the basic insights gained by study f the ty mdel regarding magma hybridizatin. The characteristic cncave-dwn shape f liquidii culd easily be captured using fusin entrpies and taking accunt f the entrpy, vlume and enthalpy f mixing (i.e., nn ideality) as in a standard liquidus curve calculatin. Hwever, the algebra becmes mre cumbersme and nthing new is gained cnceptually. Hence the tw branches f the liquidii in T-X space are linearized such that fr X< Xe, T liquidus = T T e X e X + T (A1) whereas fr X > Xe, T liquidus = T T β e Y e Y + T β. (A2) 1

2 SPEA ET AL.: DYNAMICS AND THEMODYNAMICS OF MAGMA HYBIDIZATION Characterizatin f the initial state f M and magmas Once the phase diagram, magma thermchemical prperties and Φ are defined, five additinal parameters are required t initialize the magma hybridizatin. The initial state f M and are defined by specifying their bulk cmpsitins ( X M, X ), temperatures ( T M,T ) and the fractin f M magma in the magma mixture (f). Given bulk cmpsitins and initial temperatures f M and, phase assemblages in each can be determined frm the phase diagram (lever rule) and liquidii T-X relatins. Once the phase assemblage and liquid cmpsitins (if applicable) fr M and are knwn, the specific enthalpy f each can be calculated and, by apprpriate weighting, the specific enthalpy f the mixture cmputed. When magma hybridizatin is isenthalpic (-hybridizatin), the final enthalpy f the hybrid magma is identical t the sum f the mass weighted specific enthalpies f M and. When the prcess is diabatic (FC-hybridizatin), then the specific enthalpy f H magma is Φ times the initial enthalpy f M+, the remainder (1-Φ) being dissipated externally. The starting assemblage f M and depend n their bulk cmpsitin and initial temperature and hence expressins fr the specific enthalpy take int accunt phase state and prprtins. The relevant expressins are cllected in Table A1, which give the cntributins that M and make t the specific enthalpy f the mixture. As ne example f many initial pssibilities, cnsider initial M magma f bulk cmpsitin X M < X e is just at its liquidus (all melt) and that magma f cmpsitin X > X e lies at a temperature between the β-saturated liquidus and the eutectic. In this case, is a tw-phase assemblage f β + L whereas M is a crystal-free liquid dented by the subscript L in Table A1. In this case, the initial specific enthalpy f the mixture is given by h = h M L is: + h β+l which frm Table A1 2

3 SPEA ET AL.: DYNAMICS AND THEMODYNAMICS OF MAGMA HYBIDIZATION ( ) h = f # C S T M + Δh + X M (Δh β Δh )+ ΔC X M $ (T T β )+ (T M T ) % & # ' + (1 f ) C S T + - ) $ - ( * ',Δh β + Y (Δh Δh β )+ ΔC Y β ) (T + ( ' )+ ) ( (A3) * *%,(T β ),. + + &. Any cmbinatin f states f M and can be cnstructed using apprpriate pairs frm Table A1. The cmpsitin f the melt alng the liquidus in eq (A3) is fund frm eq (A2) by setting Tliquidus equal t T and slving fr Y, the cmpsitin f melt alng the β-saturated liquidus. As a secnd example, cnsider a β-saturated M magma that receives stped whlly crystalline blcks f f assemblage (+β) In this case, the initial specific enthalpy f the mixed magma is: )! f C S T M + + # * + " M M $! &Δh β + Y M (Δh Δh β )+ ΔC Y M β # (T % " + (1 f )) * C S T, - (A5)! )+ # " M M $ $, &(T M β )&. % %-. which accunts fr the β-phase saturatin f M and the crystalline nature f. T cmplete initializatin f the system, the bulk cmpsitin f the hybrid magma is simply fund as the mass-weighted average f M and bulk cmpsitins accrding t: X H = f X M + (1 f )X (A4) This cmpletes characterizatin f the initial state when magmas M and are mixed and hybridized (i.e., reach thermdynamic equilibrium). Characterizatin f the final phase assemblage f hybrid (H) magma The specific (per unit mass) enthalpy h f the H magma is given by h H = Φ(h M + h ) (A5) The weighted cntributin f M and t the mixture are given in Table A2. The parameter Φ defines the type f hybridizatin. If Φ =ne, the mixing is isenthalpic 3

4 SPEA ET AL.: DYNAMICS AND THEMODYNAMICS OF MAGMA HYBIDIZATION (adiabatic) als called -hybridizatin. If 0 < Φ <1, the mixing is diabatic and termed FC-hybridizatin. There are five pssible state assemblage utcmes when M and hybridize. The final hybrid magma can cnsist f either Liquid (L), crystals + liquid (+L), β crystals + liquid (β+l), eutectic liquid+ crystals+β crystals (Le++β), r crystals f and β (+β). Slid phase identities, liquid cmpsitin and temperature are fund by cmparing the specific enthalpy f H magma cmputed frm Eq. (A5) t enthalpy limits defined a priri fr the five pssible utcmes. These phase assemblage limits in h-t space are depicted schematically in Figure 3 f the text. Once X H is given, the h-t diagram fr that cmpsitin can be determined using the expressins given in Table A2. The five pssible final state assemblages ccupy distinct regins n the h-t diagram. There are three special enthalpies n this diagram dented hmax, hmid and hmin. These values separate phase assemblages. Fr example, when the specific enthalpy f hybrid magma h H f bulk cmpsitin X H exceeds hmax, then the final hybridized magma must lie in the L field n the phase diagram. Similarly, if X H > Xe and hmid < h H < hmax, then hybrid magma will cnsist f β+l r if X H < Xe, and hmid < h H <hmax, the H magma assemblage is +L. When the hybrid magma enthalpy lies in the range hmin < h H < hmid, then the assemblage is Le++β and the amunt f eutectic liquid is determined by enthalpy balance. In this case, the temperature is identically equal t Te, the eutectic temperature. Finally, if h H < hmin, the assemblage is a mixture f and β crystals in prprtins dictated by the lever rule and the temperature is less than Te. In summary, in rder t find the final state f the hybrid magma, the value f h H is cmpared t the ranges given in Table A2 t discver which f the five pssible assemblage utcmes is relevant. Characterizatin f the temperature and phase cmpsitin(s) f hybrid (H) magma Once the phase state r utcme is knwn by cmparing h H t the limits specified in Table A2 (see Figure 3), the final state f hybrid magma can be determined. The state depends first n cmparisn f X H with Xe and then n the value f h H. The cnditins and final state values are given in Table A3 when X H < Xe, 4

5 SPEA ET AL.: DYNAMICS AND THEMODYNAMICS OF MAGMA HYBIDIZATION Table A4 is valid when X H > Xe and Table A5 is valid when X H =Xe (exactly). Nte that in the latter case, the +L r β+l fields are nt pssible. As a summary example, cnsider the pssibilities when X H <Xe. Frm the phase diagram, the state f H magma can be ne f fur states (L, L+, Le++β, +β). If h H > hmax, then H is a single phase melt f cmpsitin equal t the bulk cmpsitin and its temperature is given frm the expressin in the first rw f Table A3. If instead, hmid <h H <hmax then the H magma cnsists f liquid plus crystals. Simultaneus slutin f the tw expressins in rw three f Table A3 gives T H and the cmpsitin f melt in H magma ( X H = X H ) in the L+ field, thereby defining the apprpriate tie line. If hmin < h H < hmid, the state is defined by the invariant pint assemblage f Le++β. In this case, T H = Te and X H = X e. The mass fractins f Le, and β crystals are given in rw 4 f Table A3. Finally, when h H <hmin, the assemblage is whlly crystalline (+β crystals) in prprtins given in the fifth rw f Table A3. Table A4 gives analgus slutins when X H >Xe and Table A5 is apprpriate when X H =Xe, exactly. Table A6 cllects thermdynamic parameters that apprximately mdel the system CaMgSi2O6-CaAl2Si2O8 at 10 5 Pa (1-bar). The ty cde can be fund at the fllwing UL Once dwnladed, a user is free t change any f the thermdynamic parameters and run cmputatins fr any binary eutectic system with knwn parameters. 5