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Thermodynamic Properties of Magmatic Liquid: Review and Future Directions
Toru SUGAWARA῍
Advances in calorimetric measurements for magmatic liquids in the recent two decades are reviewed. This
paper summarized previously reported values of heat capacities of silicate glasses and liquids and heats of fusionof minerals. Some of the values for heats of fusion are recalculated using the recently reported heats of
vitriﬁcation and more reliable heat capacities of solids compared with those previously used. Heats of mixing ofsilicate liquids are also re-examined using excess enthalpies of glasses, heat capacities of end-members, excess heat
capacities of liquids and recalculated values of heats of fusion. Excess enthalpies of pseudobinary silicate liquids
are generally within -* kJ/mol and mostly /+* kJ/mol. Although the excess enthalpies reach only -῍+*ῌ of enthalpies of fusion, those cannot be ignored, because liquidus temperatures and compositions of minerals
are greatly a#ected by small excess enthalpies. Based on the compilation of the excess enthalpies by direct andindirect measurements, we found that interactions among network-forming oxides (SiO,, NaAlO,, KAlO,),
network-modifying oxides (CaO, MgO) and intermediate oxide (CaAl, O.) control the excess enthalpy of silicate
liquid. As a preliminary test for generalized prediction of enthalpy of magmatic liquids, regular solution
parameters for K, O-Na, O-CaO-MgO-Al, O--SiO, liquids are determined using compiled calorimetric enthalpies
using a least square method. Finally, Adam-Gibbs theory for viscosity and conﬁgurational entropy of silicate
liquid are reviewed. In order to express Gibbs free energy of magmatic liquids, the following studies will be
required in future : (+) measurements of excess heat capacity by drop calorimetry for liquids including interactions
between NaAlO,, KAlO, and CaO, MgO, (,) measurements of excess enthalpy by solution calorimetry andrelative enthalpy by drop calorimetry for Fe-bearing multicomponent glasses and liquids, and (-) determinationsof entropy of mixing and conﬁgurational entropy by systematic viscosity measurements for multicomponent
silicate liquids.
Key words : magma, calorimetry, thermodynamics, silicate melt, enthalpy, entropy+ῌ ῐ ῎ ῑ ῏

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Institute for Study of the Earth’s Interior, Okayama
University at Misasa, 2,1 Yamada, Misasa, Tottori
02,῍*+3-, Japan.

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Fig. +. Schematic relationships between enthalpies
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f$R7 Pa

CpLEx, aL Cp Ex, adT(+/a)

XiHi,LTKxm % i aa

+,-L
XiH i,LT TS Mix,a(+/b)
H LEx, a, Tf+ aL Cp Ex, adT

ῌ 
T῍



῎



Cp GEx, a
dT
T






Cp LEx, a ῏
dTῐ
T
ῑ
(+/c)

]. ^

108
-ῌ. 

ῌ(G a,L T῏
m i,LT5῍ῐ῎ (ni ῑT a  i m  G  i   ni 
L
i, T

L
a, T

L
ῌ
ῌ(S Mix,
῏a῏
5H i,LT'T῍S i,LTk῍ῐῐkRTlng iL῎
῎ (ni ῑῑ   mLi,T !"#$ %&L
L
, CpLEx, a, CpGEx, a () SMix,a, H Ex, a, Tf
+'a

"#$



{|

*+

L
SMix,
 ./a  ,-L
ῌ(S Mix,a῏῍ῐ 5'RlnXi
῎ (ni ῑT

0#1* ,-23./0#1Mix  4 RTlng iL5+'Xi,WHkT'Tf+'a

L
S Mix,
a5klnWMix/W*

(+0)'TlnT/Tf+'aWCp467  k 89: #+ ,;.

(+2)

-<=> ?@3A*B

  R  gi ()};#{| i ~L
C% DE Temkin, +3./ ;7 SMix,a *467

S# ~+S#  
:J:<<$  L@Toop and Samis, +30, ; Weill et al., +32* ; Bhattacharya

/23g,-i1f4jB*=>

and Viswanath, +32/ ; Blander and Pelton, +321 ; Hillert,+331 ; Chartrand and Pelton, +333 FGH 3;+*

+II xLI{| Cp>@> HLEx, a, Tf +'a,

LA  CI 

I:J:$ C@AS >@>

@?@ ()A%A

L
Mix, aL
Ex, aHEx =>  K L !M

CBC+ D ()LLNO"#PQR$"STF$A%"#abI>E=> FNc U&*'VW C%%& (X@YZ[\])    7=>  

*R$^+_`ab7c,$L@G*M7 C%I >EH HEx d-.

# A*/0$ C% DE Thompson, +301 ; Berman and Brown, +32. ;Fei et al., +320 ; Hel#rich and Wood, +323 ; Jackson, +323 ;Mukhopadhayay et al., +33- ; Cheng and Ganguly, +33.

. 

IJ$K

DEeO1f@ab$  , = /2(+33,) {|LD  (+332) *+N M 2

3g,-I,-hD (*i1

- 9/ -N >?@¡¢£

f4j (Symmetric simple solution, Thompson, +301) +`¤O=> J*@7 > PY

L
L
()
BC SMix,a  HEx, a, Tf+'atuvw\$ x£¥7 C%

L
S Mix,
a5'RXilnXik+'Xiln+'Xi

LEx, a, Tf+'aH

5Xi+'XiWH¤O=> ¦1x&
(+1a)
(+1b)

Ytuvw\Q .7
§



.7¨L©ª7> (%& Qy! HLT «&%& *C WH  , <"lmS¬­vw\L|¬­

#+  HLEx, a, Tf+'an* I= CpLEx, a5CpGEx, a5* 

HGT'HST, HLT'HST 7vw\()7

564j (Regular solution) oE > CpGT CpLT * £`,@pq7=>



(Cp

L
Ex, a

Or (+1b) 8ab$

5Xi (+'Xi) WCp) s&Ytuvw\=>xH9y@pq7*:z7 >@>%& CpGEx, a5*  C i ®

.ῌ+  (Drop calorimetry)R¯O SET}U°$%!T x%vw\0±VS¯E²!

a>0 (,12ῒ,32 K) 4R¯³W7c
LN ! T  ,12ῒ,32 K 0 ® 4X²YZ <#X´µ*¶#[> >·\¸¹ºx%Xº»¼

· ·^j1~5 : ¸¹º »¼
 

(Southard, +3.+ ; Proks et al., +311a) fgh i109
H eH  , 
S
,32

L
T

#PQF# oK + cmL - cm p =Q  !"# $Tb, U#VWJ=*X% &'()

Y:/d SCQ Z*U=

(Ginnings andCorruccini, +3.1 ; Furukawa et al., +3/0 ; Bacon, +311 ;Carmichael et al., +311 ; Richet et al., +32, ; Cordfunkeand Westrum, +322)'()

 
(Lee and Itagaki, +320 ;Blachnik et al., +330) *+, #-./+0:bC":?Q F#[\]
SJ^'()0/23
 (Tamura et al., +31/ ; Gaune-Escard and Bros, +31. ;Pool et al., +313 ; Ziegler and Navrotsky, +320), j_`

jajQbc=



ON/+

*1, 2#*345

Setaram  MHTC ON Fig. , , 

67891:;< =>?@AB

KL::Ad d +21* K "/!C"D#EF#$ 67GHIJ

eb m¡S¢ g *MN &Mf

, F#$#KL*&M +ῌ, cm %glnN£*eb¤:1 ohR# Ar ip,NOP/+Q RSTUF#&V +ῌ3 g * + W

67#jkF#/+ Al, O-  Pt ¥¦§\l67X',

F# (-*ῌ+** mg) =%0/;,Y Z'1=  ?1= /#m67/VW¨lF#j: [[\()+F#*=]**^+

kF#F#A=%78*+J

(_/ K/s) /`,4-.abc 0d+QF#S©*nkbQ o¤:= +WZ//# HS,1-eHLT /+Q 4\:ª*p1«Z*+,  4\fgh i*I",  05j12Rfgh i e/+0*qZ/

STUF##k&V=]**^+/#lmn+, ;¬fgh'J 0&V#3fghi#+323 ; Tarina et al., +33. ; Kojitani and Akaogi, +33/, +331 ;

+ei p
HGoTf
eHLT b, 4()*lmn/+Q Tf,ei q,1-

Sugawara and Akaogi, ,**.) B CaMgSi, O0 (Ziegler and(Navrotsky et al.,Ni, SiO.  Co, SiO. (Sugawara and=/5r678bst9uF#v1

Navrotsky, +320):=  CQ3fgh i#

Akaogi, ,**-a) Fe, O-  NaFeSi, O0 $ (Sugawara

,ei p
,ei p
+ei p
HGoTf
eHGoTfeHLT wx/+, HS,1-eHLT, HGoTf,1T,1-

and Akaogi, ,**.) *= CQ :1

,ei p
,ei p
HGoTf
eHGoTf/ T '%:  &,1T,/:;0CQ CpLT ?Cx CpGT *I"

.ῌ-

,

r$

#rsF#r­4tr$

05j1F#2y<z'V==  

/+, RSTUF#=r$

67# +3.* >? +30* >?@:@{ABC| :# R/ HF  HNO- r\rs}~?1:DE16
+3/.)

CaTiSiO/ (King et al.,Mn, SiO. (Mah, +30*), Fe, SiO. (Orr, +3/-),FeTiO- (Naylor and Cook, +3.0) Cb

F1b=]*+/#lmn'b1|$



:'1®

(Solution calotimetry)(Proks et al., +301 ; Robie and Hemingway,+31,)  L (31*ῌ32/ K) / ,PbOB, O- rs

:'1®

oKleppa, +31, ; Navrotsky, +311 ; Akaogi,+33* ; uvw¯ +33.p *+, ]/# x°$

*1G H, & k5|MNy'()0/

2lmn*=  CQr$]fgh i3[/+Q -.

  oIc

Proks et al., +311b ; Carmichael etal., +311 ; Adamkovicova et al., +32* ; Stebbins et al., +32,,*'1:1,  oT'r
$Oxide melt solution calotimetryp /# 1

+32-, +32. ; Richet and Bottinga, +32.a,b ; Richet, +321 ;=zbC{X , v§*+Q  F#

Richet et al., +33+p,rsx°$ | #rsx°$(MN(Transposed-temperature-

, JK}±/t²J^³´-=  /# *JKE

XN S/":?Q F#ONPQCQR (,32 K) GS

r$]i oJ^p '()0/  T JKAQ=%:$% &

r$3, + W67/r$%F##rs.ῌ,
drop calorimetry)LMN:?QµJ, v§#¶~*bC:

º» ¼

110

Fig. ,. The transposed-temperature drop calorimeter (a) and its detector (b). Samples are dropped from outsideof the calorimeter at room temperature into the sample chamber at high temperature (Y+2** K). In thecalorimetric detector, the thermopile consists of ,2 pairs of thermocouples connected in series. Junctions ofthermocouples are arranged symmetrically and are placed alternately between the sample chamber andreference.
++ g  /ῌ+/ mg 





 Setaram   !"
/$01234 56789:;,<,.-=>?89:
C DH

S
i, T

@AB

GFTf +Di G
i, T

G 'H?,EFH DH GDFH DH
Sol
i, T

Sol
i, T

et al., +330) '+,-.
.ῌ. ῎ῌ῏῍ῒῐῑ (Di#erential scanning calorimetry)#$%&'()*+,-.

Tf+Di
Vit, i, T

RS@AB C (Kosa et al., +33, ; Adamkovicova

ssq2 J ,  
70 0P7;,<,j-$'J[ +ῌ+*

 IJ KL4MN89:7;,0ONPQ

K ῌmin $s'?$=0N89:=>? 89:RS@ABP7 GFTf +Di G
Ex, a

GFTf +Di G
a, T

GFTf +Di G
i, T0[7'

G',0= F DSCG 7  TU$'8

 ,   B' +,9:VW$X (Tf+DiYTf,Di) Z.J[ 89

$0¡) J[s@¢ £=

:\]^_0`ab+cJ[ de>?¤=0& ¥t ,¦§ DSC

f (+./ῌ,./ h) f89:Ogh 'ij kl$7¨,-. F©ª#« +320 ; ©¬ª% +33,GC H

EFH DHH?,

' Tf,Di 

Sol
a, T

P

GD῍FH DH
ῌ

Sol
i, T

mn- H?,o;,

Tf,Di
GFTf ,Di G
7s
<, DHVit,
i, T pqr H Ex, aqtKL4MN>?NuNvw

 (/*ῌ+/* mg) 7_ (Au, Pt) 
$

0;,<,-=0[ +? (Al, O-) 0.JTU5­®0P

C (Navrotsky et al.,7qt'H?, syz ¯89:8+32*, +32-, +33* ; Hovis, +32. ; Hervig and Navrotsky,

9:VW$qt¥ /*ῌ+** K °$ (Y+*** K) I

xyz89:RS@AB+32. ; Hervig et al., +32/) ' IJ7 { ±²s DSC '³´µ+,-pt 89:|B0M}4~c7 K

¶89:VW$Z!

789:

(Proks et al., +311b ; Adamkovicova et al.,+32* ; Weill·¸J[*+,-. F"¹¨ Stebbins et al., +32. ;et al., +32* ; Kosa et al., +32+ ; Zigo et al., +321) pqr

Lange and Navrotsky, +33- ; Toplis et al., ,**+ ; Tangeman

0cdc&DpK : ¸¹º8"»¶&¼d
and Lange, +332, ,**+ ; Roskosz et al., ,**.


  

111

/ 



/ῌ+ 



/ῌ+ῌ+ 

!"#  DSC "

@ABC,lk& D - #$ !"#% +2** K #&'#!`&F"G" & ^:; %

 !&() *+, - Fe, O- &.

X0]y Anderson, +3.0 ; Egorov et al., +31, ; Hirao and

+ (Lange and Navrotsky, +33,) CaMgSi, O0 &/

Soga, +32, :D&)# +32* ;< P. Richet01/ 23*" J.F. Stebbins W;;=>w+,&456+789 (Lange et al., +33. ; DeYoreo

 e#?@&A2$2&*+,"lk&

et al., +33/ ; Sugawara and Akaogi, ,**-b) :; ^ (Stebbins et al., +32,, +32-, +32. ; Stebbins

and Carmichael, +32. ; Richet et al., +32,, +320 ; Richet (Lange et al., +33+).ῌ/ 

and Bottinga, +32*, +32.a, b, +32/). Richet (&<=>?&#@ABC,*+,&D
HI8J5J+KL(

E&F"G"

^BC:; Stebbins &~86

D/L"g}w

Z)  Stebbins (M !;*+,"N2&O*+,&~86@&EF&h;

PQ PRST&NUV&/L*+&! Richet (qrEFlkG, -W&2&M XM YE#HIJ#wx; Kl) Na " K Z[\ , 23 (Fraser and Bottinga, +32/ ;k&LY  " , G&MN

Fraser et al., +32/)(1W; SiO, ]\ - 2

3 (Rammensee and Fraser, +321 ; Chastel et al., +321)() O#e Richet (v

&

^tu

P

G & ^ _HI8J5J+KL

Table + ?@$2&lk"*+,&&# !`"2&OPQab"#o-p&V"L  lk"*+,&) cdc$2&@ABC,*+,&e2f%g

W;;m() ¡:; :]"Na " K Z[\ , 23h:;

&E#
i ' j ( &  ) #  k l d Xe  CaO-MgO-Al,
O--SiO, 3 

-i'&*+Dm9

CpGi,T¢ai£biT£ciT¤,£diT¤*./(+3a)

Cpi,LT¢ai£biT(+3b)

,# i'R@ABC,*+,&NUV&!!# 5Q Cv "5Q Cp &T&UV$2 n%Qo-pqrs W;tu

kld.&n9i'2 (FeO, NiO, SiO,) &M
m9

Ev/wx;

 X0]y Belton

et al., +31- ; Taniguchi et al., +331 ; Matsuzaki et al., +332 ;Cv¢Cp¤XTVa,/bT!!# a 5QRST(,*)
bT 5Q¥TU  Fig.

Pagador z1 +332 ; Morita et al., ,*** 23E- V¦§¨ (gram-atom) )&lk"*+,&

4:; *

5Q&9 @ABC,lk#+, - FeO, Fe, O-, NiO, CoO 5&.+{|Cv " Cp & ©   W X +ῌ >  #   (Richet andcdc$2&@ABC,*+,!

4 5 6 + 7 8 (RTlngi) & $ 2 o - p  g } : ; 

Bottinga, +32*) lk& -R u"lk(Doyle and Naldrett, +320, +321 ; Doyle, +322 ; Holzheid etGª {K«*+,") ¬&5Qbal., +331 ; O’Neill and Eggins, ,**, ; Gaillard et al., ,**-)

# -R u Dulong-Petit YZ­[\®}_!&E#:; (Haggerty et al., +302 ; Richet and Bottinga, +320 ;L
6; gi 7 S Mix,a .~+9

&o- 6;7!`" 

Martens et al., +321)

2&.+{|456+78&#&# 

lk&&]#¯°] &

89&{|456+78 (HLEx) U 7! *+,&(±²³v#!"6; ^´§¨&µN_ !"! ¶´e§¨&` ·a#s!"( lkG
L
L
G -`  (Cp Conf,
i, Tf+¤i¢Cp i, Tf+¤i¤Cp i, Tf+¤i) b, " v/ &c$2 X0]y Di, AnTable +. Abbreviations, chemical formulae and heat capacities of glass and liquid.

112

῍ῌ ῎

¢£¢T¤G~¥¦ : §¨©%&ª«¬ 

 Qz, Ab  Cp

S,CDET6uva&B :L (Fig. -) Cp Conf,
i, Tf+i

L
Conf, i, Tf+i

CpGa, Tw῍Xi aixbiTxciT,xdiT*./

!"#$%&'()* +#,-./01234
567 /ῌ,ῌ, 89 !"#$%:;!,<$%6=>

ῌ

(,+)

AM Richet et al. (+33-b)  * K yzCDE12!,<$% {| !,<$% 6lV}~B /b/ῌ+ῌ, 
Richet113

(+321) ?@ABCDEFGHIBJK-** K y Jd, En, Gr CDET6

 ?@ABU%"& ^_` * K & ,32 K 

L@MNOPQRS,CDETU%"6A!,<$% (SGi,,32 SGi,*) jHk#6JK

&V WXYZ[M\] p (,+) mOy ; SGi,,32SGi,* jHk#6 Table

CDETU%"&^_`a&B ,1*b+*** K cABdefgh
N HiPQRS,CDEjHk#T6lVM j
Hk#TmO&nopqr stPQR, N

HiPQRS,+#,T +31* A

B 3q[M Carmichael et al. (+311), Stebbins
et al. (+32,, +32-, +32.) oT *

y ; An-Ab-Di i+#,T6M a@r
U%"\]+#,T& ^_`

Z_@ N HiPQRS,+#,jHk#TVJKL@M (Stebbins et al., +32.) ? Lange
and Navrotsky (+33,) T DSC  * Na, O-FeOFe, O--SiO, i CaO-FeO-Fe, O--SiO, iy;+#,T6Z JU%"& ^_`
+#,jHk#T6M  Richet
and Bottinga (+32/) oT U%"&

J SiO,-Na, O-Li, O-CaO-MgO-SrO i+#,T6A&V Al, O- 6Ah+#,*deBjHk#T6JKMAM Courtial andRichet (+33-)  MgO-Al, O--SiO, iBoT6

Z aiBdeB Al, O- jHk#T6JK
Fig. -. Heat capacity of CaSiO- (Wo), CaMgSi, O0(Di), CaAl, Si, O2 (An), NaAlSiO. (Ne),NaAlSi, O0 (Jd), NaAlSi- O2 (Ab), KAlSi- O2(Sa) and SiO, (Qz) glasses and liquids. Data
from Richet and Bottinga (+32.a, b) and Richet
et al. (+32,, +33*, +33+).

Table ,.

 a@r

Table , A&VM

jHk#T6eN HiPQRS,+#,T¡p *uvL@ :L
L
Cpa,
Tw῍Xi Cpi

ῌPartial molar quantities of S,32-S* of glass and heat capacities of glass and liquid.

(,,)

¦ §

114
Fig. .  An-Ab  An-Di  Ab-Di  21- K            

L
L
L
CpAbqDirXAb CpAb
sXDi CpDi
q32..XDi XAb,s--,.0XAb XDi,

(Richet and Bottinga, +32.a, b ; Stebbins et al., +32-, +32.)  (Richet, +321 ; Lange and Navrotsky, +33,) !"#$%&'()*+,-(  ./01 234(5$ 36  ./01 78349: %&,-'#5;<+="#$ > Table + "#?
 Richet / Al, O- @ABC%&
EFG#5H

D

 I J (   5 K Lange andNavrotsky (+33,)  DE,-LM(=$ NO Lange and Navrotsky (+33,) KIJ(
  P&QRST 

U015V+WXYWZ5[!\]^_`a]^Kb?[ Lc/d5$
Fig. . eW[?fgV#hij Ab-Di 

Kklm"#H WZ5$ H

(,-)

lm"#tu ( Ht SiO,-K, O Kvlm"#H

Kw/d5(Richet and Bottinga, +32/)$ lm x (+-) (+2) "#? {|}~ HLEx, ay (RTlngi) DE,-

lmyz
=5

1WZ5K Hd>W H An-Ab-Di _
BC%&W%&,-Kh/d=$ klmGk HLEx, a /ῌ-ῌ-KZ U

BC%&lm}  #5$/ῌ, ῎῏ῌῑ῍ῐ} (HTm) }

'015`W= ¡WZ5¢£ P&

lm} ¤¥#5n?opWi5 :Fig. .. Compositional variations of heat capacities of An-Di, An-Ab and Ab-Di liquids at +11- K and glasses at
21- K.

0¸G¹ : ;<= º»¼

115

Table -. Enthapy of fusion of diopside (kJ/mol) at
melting point, +00/ K.

(Stebbins et al., +32- ; Richet and Bottinga, +32.b ; Langeet al. +33+)'
Weill et al. (+32*)  32/ K  "-QR /0K!
Fig. /. Experimental data of relative enthalpies (HjT
HS,32, jhL, S or G) of CaMgSi, O0 liquid, supercooled liquid, glass and diopisde. Enthalpies ofglasses were calculated using enthalpy of
vitriﬁcation reported by Weill et al. (+32*).
Solid line represents relative enthalpy of diopside
calculated from formulation of heat capacity byRichet and Fiquet (+33+). Dotted lines areenthalpies of liquid and glass calculated fromheat capacity by Lange and Navrotsky (+33,)
and partial molar heat capacity by Richet(+321), respectively.

>$ Diopside NOP"

'

Tf,Di(DHVit,Di, 32/h2/.0i+.1 kJ/mol)'  Kelley (+30*)

\CpGDi  CpSDi, Ferrier (+302)  Carmichael et al.,(+311) \

L
CpDi
0

K!\

NOP_` (Tf,Dih+*,0, Briggs, +31/)  $  (.b) \dWHDi'+00/h+.,.1 kJῌmol j Weill et al. (+32*) f /0K!#U NOPm* 
+*aX 1D9%$^

VDhiTf+i Tf ,i   9A 

1J ' Stebbins et al. (+32-) _`ef\d ,3232/ K NOP

"

ENOP:$% , lK!4 V+ ler z 

 !"#$%% &' () *+,-./012

3456789%$

Diopside :$%;<=

>$ ? 9(@89%$ A7%,-/0(  BCDEFGHIJA '/ῌ,ῌ+Diopside ῍῔ΐ῎ῒ῏ῑῐῌ

9(K!D9%C

NOP , l 32/ K 4 NOPX 
Tf +iTf,i
h..1i+./ kJ ῌ mol(Tf+Di h+*,0 K,
g^8 DhDi

Tf,Dih32/ K) j ' ]81 ef0

K!\dL
CpDih-/- kJῌmol  $ \d WH h+-2.+i,.+ kJῌmol Z[4 ' E RichetDi+00/

and Bottinga (+32.b) _`"_` ¡¢$% Tf,Dih32/ K £!4 ' ]8

Diopside LM NOPQR Fig. / S4 T +00/K #U /VWH X  Table - (Di+00/

ef0K!\d
Tf+Di h+**/ K LCpDih --. kJ ῌ mol $  (.b)  (+*a) ^8 WHDi
+00/ h+-1.1i,.* kJῌmol 4 '+32* ¤¥.¦( _`ef0K! "-'Diopside WHDi+00/ 2YZ[ Ferrier (+302) \

QR0K!

' ] -/- K ^8 +22/ K _`abcdef0+*** K ª`(vw«89%$ ' 4^4K!>$ QRLM
 WH h+,2./i-.- kJῌmol jDi+00/\dNOP"A + l
q

D9r

g

VEFeDiopside LM"Dk1Cdem1n5 0op
LM1s4

tu9 X' 4^

$

§¨©¥

4%$



\d¬_­®K!§¨©1¯°D9 Ziegler
and Navrotsky (+320) ±² ³m´A ¬__`ef0K!Y%>' ]8 +/1. K^8 +100 K ^U% Diopside µ¶·¶
'g­®K!4 WHDi
+00/ h+-2./ kJ ῌ mol j4 Ferrier (+302) vwex-1 Di LMN Stebbins et al. (+32-)  Richet and BottingaOP(  Wo yz- d WHDi
+00/ {|}~(+32.b) \

d4%$ 11 D9%$

9(\5z%#d NOP

Tf+i Tf,i¢$%QR  DhDi

¼ d

116

 

94CDHTm "=7$

 Ni, SiO. DHTm @TABC !7$D , E94 Lange et al. (+33+)  Ziegler and Navrotsky(+320) 

+10, K " #$%& "'+,- Diopside !

 +.*- K

() *DSC./

(Table .) * TAF-@GV{0!7$ 0 H Ni, I"94Crs~0
<7

."')$.J k-.$(Sugawara and Akaogi, ,**-a) 

0"

1 $2 3 * -4 Diopside TAWX/:"94C Ni CH]

+0*0 K -CH];K Z|} Takahashi, +312 ;.567 +0/* K "#$,*ῌ  .7 89 2*ῌ  +/ K -:".

* ;< !"

=7$ >?@."ABC (DHDi+00/ E+-1.1 kJ/

Hart and Davis, +312 ; Beattie et al., +33+\  Ni, SiO. 

94CDHTm "')$BL"MN-4
Fig. 0 " Table + "!





-

.



K$

L7MNC

O P QJRDHTm  .

"' STU V H  "' STU W
 Richet and Bottinga (+32.a, b)  "]^_`aR"#7$ bc Di -()

7Rf'z$WX!  HO!

ZSTU VG

LL- Cpconf,
TECpTd-R\

DSTm = Cp P&"'9P&7 Cp
LT

1 ** XY Z[! \  $/"K$

3

K, Si, O/ (Richet and Bottinga, +32/) "1

Na, Si, O/

Diopside '"FGHI DSC CJ

(Sugawara and

Akaogi, ,**-a)

mol) !
/ῌ,ῌ, 

Ni-Mg

L
Conf, T DSTm27

.X C .

HO¡QRS¢ =T7$
 £

*



rsF2

C"1 $ .

eKX!

* - 943I()$f9 [ Richet and Bottinga (+320) -!

WXF¤;* -4  |}¥A2

$C- DhiTf +didTf,di %
ghi

.

"j&'"[ k$ (l-DSTm  En, Wo, Di "1 $ 0.-῍2.- JῌK-g atom - +32* )*+mn",o3 Zp"q- "#7 Jd!  STU0\  $rs .

J ῌ K-g atom:"¦

"1 $etu() Zvw./ A+ "c \ *<! WXF2 £! ",oxk Ryz^_ NaFeSi, O0 WX"f`a0BDHTm

DSTm (EDHTm/Tm)  Table. "[ k*. HVk$Ac DSTm  /.- yz /.+

*

 Jd

Ac PU7$ Fe-  Al-

WX



$"

 YA2 DSTm

 An  0., JῌK-g atom - "#7 Ab, Sa, Ne 

{01 Z|} Na, K, Cu, Ag, Mg, Zn\ ~B

DSTm '9¦C ,.3῍-.3 J ῌK-g atom -

D STm  1.+  +*./ J ῌ K-mol -9 (Kubaschewski and§$RZ- SiO, ¦C DSTm 

Alcock, +313) 2"'! R " Pj(DSCristobalite
E+./ J/K-g atom) mb"'9 DSTm W
+333 *,3DHTm 45{0I`0"1 $DSTm

6 ! 

k"7}7}

!$f9 Richards 3 7$5!$X¨B

 SiO, 

=7$

*



©



 *

**-[\9 SiO, ªC]tu«7$¬"#7 ^_`aR 0BDSTm ZJ/K-g atom, 1' ­ 1 §$CWX"f $^_7

]9 .X\ j7" +./`0 (SiO,, TiO,, Al, O-, MgO, CaO, Na, O · · · )!

+0 894 7 {0'9:"94C;

0<

 *k^_`aR- Richards 3"' DHTm = >-4C  *!DSTm J'C3-K[  <*[-"5!$`¯7ªC]-

7$(model-+) ­

1 Al  Ca,, Na, K "')$a°%±²#$. H C9 CaAl, O., NaAlO,, KAlO, C¯7ªC]-

94CDSTm ®

(model-,) ³ªC]^_ ´µyz´µb2{0xL´¶·¸µ`aR Ni, SiO. -,SiO. E+0.. J/K-g atom)
(DSNi
+3,-

^_XWXc M/(MAl)E*./ -9"C9

Ni, SiO.  .

k" 6 (+3,- K) -?

;

0!

! KAlO, ¹NaAlO, ¹CaAl, O. ¹

= !  (Hirschmann and Ghiorso, +33.) -MgAl, O. º-¯W0»7 Bottinga

9 94CDSTm FGH  -!

and Weill (+31,) ª3">-  Fig. 1a 

Table ..
Melting temperatures and enthalpies and entropies of fusion of minerals and silicates.


:
117

¨w ©

118

Fig. 0. Relationships between entropy of fusion ofsilicate minerals (An, Wo, Di, Py, Co, Qz, Ab,
Jd, Ne, Sa, En and Fo) and crystals (Na, Si, O/
and K, Si, O/) and heat capacity or conﬁgurational heat capacity of those liquids at ﬁctive
temperature. Dotted line represent a + : + corL
relation between STm and CpConf,
Tf.

  SiO,

 

SiO,

  model-+ model-,  STm

STm

SiO, 

!"#$% &# ' ()* CaAl, O., NaAlO,, KAlO, + ,./Table .

9

STm

01

234567model-,

b ).0 234567

234587) Fig. 1

STm : XSiO,

!"

SiO,

" ;9< =>87

STm :?

@AB#

CDE&F G H 3IJKLM23458
7

STm

Cation ﬁeld strength (Z/(+..Nr),, ZOP

Q rO3GRST (U)V0  WX

XSiO, Y0Z[C 3GR \]E ^_)9`
STm "DE;& (Fig. 1c) a b  2345Z[

STm : ' ( cd (XSiO,)

e3GR \]

E f , 9 ghiR(Mj E`? ".
/E` klmR(Mnj : Z[op"q`'(rs)&tuv wxyz ={|d}0 f~  #cd"DE; 3 DE&
 GR<oE`' (

STm "\];&? :  `/ῌ- 

+32* Sa:  d   +*** K V )#' ( mR nj
0? " ¡¢tuub) £)!¤¥ ¦§

Fig. 1. Compositional variations of entropy offusion of silicate minerals and crystals listed inTable .. (a) Plot of STm for silicate minerals
against mole fraction of SiO, by model-+ (An,Jd, Ac, Ne) and model-, (all minerals) ; (b)plot of STm for silicate minerals and Al-free K,
Na, Li-silicate crystals against mole fraction of
SiO, by model-, ; (c) plot of STm for ortho-, inoand phyllosilicates against cation ﬁeld strength.
The cation ﬁeld strength is deﬁned as : Z/(+..Nr),,with Z being the nominal cation charge and r
the Pauling-Ahrens radius in Angstroms.

´µ´ῌ6¶ : ·¸¹Q#º=»¼ῌ

H 
L
Ex

  

 L 

+0X
L
S Mix
ZcRXAn
ln
L
L .
e,cX
fe+dX
f
An
An 


Di-An !"῍#$%&( HLEx )"*+ ,'
SiO,-MO !-./L
.e+cX An
f
L
dXDi
ln
L ,
L ,
e+dX Anf e,cX Anf 

%&'0121( H 345678 9+LEx

:;( <= >? @ABCDEFGD6H
HLEx ?I"J$ K, O-Na, O-CaO-MgO-Al, O--SiO,!-./KLMNO.PQ3"R# (/ῌ-ῌ+ 



Di-An !%&'ST"ῌ6 U(#"VW( Fig. .

X$ Y  !-./ CpLEx, a

Z* #[(  -./\ F]%^_N`a. (+2)

 <b

119
L ,
An

(,0)

#H( e $B Weill et al., +32* "¡¢f Table . £ Table + -./¤EF Richet and Fiquet(+33+)( Di # An ¤"+  (,/) 01( Di-An !¥¦§¨# Osborn (+3.,)(%&'i©ª«"¤¬$ (Fig. 2) XAn Z.* wt
eDi # An ¥¦§¨­®f\}(¥¦§¨¯°"¤¬U(# WHZ* "[U(# kq.
 Di # An ¥¦§¨ijy<b .2 \F 21 K ±B Two-lattice q.<b(S m i,LTZH i,LTcTS i,LTd L de+cXiLf, WH
 (n i L
Mix+/ \ F 3* K ±B( #i©¥¦§¨(,.a)

4²U( YWH "³ (# kq.

 WHZc,+ kJῌmol,, Two-lattice q. WHZc+/
m Si, TZH Si, TcTS Si, TdRTlna Si(,.b)

kJ ῌ mol, 21 op  (+1b)

L
XAn
Z/*

#H(  i  Di R An "gU Di-An !
-./#h$+(]%ij k%HB Di
l> CaAl, SiO0 EF Mg, Si, O0
CaMgSi- O2m An l>m"]nU( (Murphy, +311) op ]%q.mr XSi  +st0 u+ $0$ 

1]nvwKLMNO.PQ (Sack and Ghiorso,+33.b ; Sugawara, ,**+) "x$ gLi  +yst0z+ {  |0}~ a ZX g Z+ #(#U
S
i

S S
i i

 &']%#%^_N`a.
 $+ emSi, TcmLi,Tf #L(S Mix
*ZeH i,LTcH Si, TfcTeS i,LTcS Si, Tfd L 
 (n i 

de+cXiLf, WH
H i,LTcH Si, TZDHTmid ῏ῐῌ

eCp i,LTcCp Si, TfdTῌ῎ ῍

S i,LTcS Si, TZDSTmid ῏
ῐ

ῌ

ῌ῎ ῍

eCp i,LTcCp Si, Tf
dT
T

(,/)

"2(
L
SMix

9+ , 9q."Y #9 -./ CaAl, Si, O2 # CaMgSi, O0 #+Ym01(+1a)(#[U(q. 

( ekq.f yY#

9 -./ .  (Tetrahedral sublattice) #G (Interstitial sublattice) "9" $ 
 Si # Al  = Ca # Mg U(#[  U ( q  .   ( e,   q  . Two-latticeLmodelf  SMix (+0) "+

Fig. 2. Comparisons between observed and calculated
liquids temperatures of An-Di system. (a)Simple oxide model ; (b) two-lattice model.

¼4 N

120
molῒ 

 HLEx/.- kJῌmol  HLEx-.2 kJῌmol  HLEx

 Di  An 

(Table .) -

Table /. Excess enthalpies at XSiO,*./ and SiO, contents of cristobalite liquidus at +1.-K in the MXOSiO, system and ionic potential (Z/r) of cation M.

ΐ.ῒ !"#$%& ' 3* K ()*+,-./ 0123456 % 76  
%  >)?@

89:;<; =AB A& 3C"



D3

E (F#G
Table /  SiO,-MX O H (MFe, Co, Mn, Zn, Mg,
Ca, Na, K) 3 =I JKL $M9:"B A XLi*./ 

JReyes and Gaskell, +32-  Havrotsky, +33. NO@ P$M1Q6R
L +1.- K S),ATU@A)*+,V
J@X\]

W@XYZ>[++3 pmF^]R@X_ J`a@X

_L #b6R L

cR SiO,-MX O H =I

 H d# (+) H & S ef g3
L
Ex

L
Ex

L
Mix

Gaskell (+32-) NO@
HLEx "#EV:%%( K= SiO,-MX O HB

A HLEx  J>{  (+1b) # ¡3E#!
L ?@¢Vgx
%L XSiO,

3(D{M (,) 9:% %hijklmHTm &d/ῌ-ῌ, 

 #G (-)SiO, n MO V n3 6   =IAB HLEx & n oEnp lD"£3E&¤C#G DE  (+) B%% %q - rs!&G 76  "M (,) ¦U, HGEx B

A3F¥

H &#t%$u%%& !"vwj

G/E

x#y& 3z#'(%%(D{ME (-) ¦U, HGEx f

L
Ex

SiO,-MX O HB

3r

A# @XYZ>[ &|}HLEx &|} S),ATU@A)*+|}"R S),ATU@A+,-GK SiO, p-&G (a g
L
SiO,

L
SiO,

L
$rR XSiO,.") E& S),ATU@A

L
SiO,

X

)*+,V l " %
L
*./ %EMH HLEx & XSiO,

%

f

A

1§¨¥#D
%¨¥#DE - rDE&G(+) F¥#,V~ SiO, )*>A3 HLEx |}
LgSiO,  ©-G/E DSC DE&G HVWI#
Vn %  HV!ªz-#©-
G/E
Sa, ,32

H«qE~ HLi,THSi, ,32  Ha,L T&"M EM¬ QE~ HLEx, a,T &

c :
(/

L
*./ # 6  
#GE XSiO,LSLS
H LEx, a, TJH a,
TH a, ,32LῒXiJH i, TH i, ,32L

῍

(,1)

 #~Rl8VM0
1&0 %E 3  `x&C"@X%r$­! (Table .)  ®©-G/`x!V#Ghj2 l3&!

E J1 -z¯ JK +2** KL ~(% 76 ~l801 Q4&& ((HV CpSi, CpLi, mHi, Tm &°±#56" E&@XYZ>[  H 

c6 %M z'²³ ML´ " %D{M%  9:3E&#! 3§µLEx =P  SiO,-MX O (MFe, Co, Mn, Zn,
Mg, Ca) H%  SiO, )V#7, 0

& SiO,-Na, O W K, O H#"% E
E SiO,-Na, O W K, O H&~8% HLEx
E9,##G %H( X

L
SiO,

3

*./ # H

L
Ex

L
S
H LEx, a, TJH a,
TH a, ,32LῌῒXi῍ΐ
῔
῍

῎

ῌ῎ ῍

ῌ

῏ῑῐ

ῌ῎ ῍

Cp SidT¶DHi, Tmi¶ ΐ῔῏ῑ

Cp iLdTῐ
(,2)

L
*./ V:&/#G& 7, XSiO,

EDE ¦U, HGEx ·¸DE J -,L  # HLEx & t  ; }%¹º»]#F¥# HLEx &"MM!&GE

 < 3  Reyes and±²±#ΐµ8¶ : ·¸¹{Cº»¼ 
 HLEx  

5T Cp

῍!"#ῒ$%&'

N-; (Tf+Sa STf+Si [Tf,Sa STf,Si) C(()+/ kJ/mol) *+,-./01

234-

L
G2Tf+Sa =,Sa =H LEx, a, T[HG2Tf=
Ex, a, TSC \2H a, TSH a, ,32/C; DE@/$% FGHILM TS= NFGOῑ

2KPLM TL=[*  opM %C U (-,) 

HLEx 56῔78-95:;<= %ῒ8->?@AB
DSC

121

G
EX, a




+Si =
S  Xi 2H i,LT S HG2Tf=S 
Cp LEx, a dT
i, ,32


2JK



QR

 

(--)

L
S
#ῒ H a,TLSH a, TS @TN U (,2)  T C ,32 K 5

VW TL C TSCe4 opMCVopM

X4YZ

TSC G#$%5;VW ;<LS
H LEx, a, TL[2H a,
TLSH a, TS =S Xi


 

 



L
Cp Si dT\DHi, Tmi \  Cp i dT



Sol
,Sa =
,Si =
SHG2Tf=
HG2Tf
Ex, a, TSC [Xi2H i, TSC
i, TSC



 



,Sa =
S2H Sol
SHG2Tf=a, TSCa, TSC(,3)

* H LEx, a, TL @TN ]^"



῍!"#

ῒ$%C_`<&'@a-; (()/ kJ/mol) C;
῎EC TL C TS  '@34b-C DSC $% c2#῏ῒ de cfR#gh;<JKLMHIE= @>? b;01

iFEj0Vk;l7mn@AB/C; DE@/(,) opM HGEx C῍!"#ῒ$%qrs ^"C t -+/ opM
uvwxyz{tU%| :
(-*)



+Sa=
+Sa=
+Si=
SHG2Tf\ XiHG2Tf
U (++) C *[HG2Tf
a, ,32a, ,32
i, ,32

G2Tf +Si =i, ,32

SXiH


5~Z H

L
Ex, a, T;<GbCH

G2Tf +Si =i, ,32C H SH
L
i, T

c

Cp

L
EX, a

  Ha,L T S

5U (--)

]C HLEx, a, T @TN
+Sa=
+Si =
U (--)  Ha,L T SHG2Tf
C HLi,T SHG2Tf
5Ca, ,32
i, ,32

opM_#5;<C,Sa =H LEx, a, T[HG2Tf
Ex, a, TSC

\ 


 

 

\




 

Xi2Cp iLSCp Gi=dT
Cp LEx, adTS 
Cp LEx, adT


 

(-/)

5T ]@ (-) opM HGEx C_#qrs]^"U (--)

^"C_`< 2++ C_#  @j<;-N= *a-;mn HLEx, a, T @T

G2Tf+Si =i, ,32

SXi2H SH

G2Tf +Sa=
a, ,32^"/

L
G2Tf+Sa =,Sa =H LEx, a, T[HG2Tf=
Ex, a, TSC \2H a, TSH a, ,32
L
i, T

(-.)

,Sa =
]  . (HG2Tf
Ex, a, TSC ) C ῍ ! " # ῒ $ %

G2Tf,Sa =
,Sa =
,Sa =
HG2Tf
SXiHG2Tf
Ex, a, T [H a, T
i, T

}U

V

N'NopMuvwxyz{@TN :

=



L
L
NC; ῎E@/@  CpEx, a, T   HEx, a, T

,Sa =
+Sa =
SHG2Tf=S2HG2Tfa, TSCa, ,32,Si =+Si =\Xi2HG2Tf
SHG2Tf=i, TSCi, ,32(-+)

C- U (-+)  , 5_#5;<C+Si =SXi2H i,LTSHG2Tf=S i, ,32





\X
i



 

 

S


 

 

\


 

 

@/ t¤ 

,Sa =
HG2Tf
Ex, a, TSC $%&vwxyz{5), kJῌmol § ¨ ©1]C@

Cp GEx, a dT

401 ++ CpLEX, a[* C%4-N ªK«¬@­,- HLEx, a, T % ­®-¯°
/C¦ZN

2Cp iLSCp Gi=dT/ῌ-ῌ- 



±²±wxyz{5%Ue]C2Cp iLSCp Gi=dT
2Cp LEx, aSCp GEx, a=dT

C]/j*

'C Tf+Sa ¥%&'5¦ῐ<+uL
G2Tf+Sa =,Sa =H LEx, a, T[HG2Tf=
Ex, a, TSC \2H a, TSH a, ,32 @01

N<;-; CpLEX, a .@¡¢ £ῌ -C; DE

5³8CC -´e
CaO-Al,O--SiO, 9=  HLEx *+(-,)

9 2῔Z

 2῔Z SiO,-CaAl,Si,O2-CaSiO-, CaAl,Si,O2-Ca,AlSi,O1-

U¼ V

122
CaSiO-  H 



L
Ex

 

  (+)  
  !"#$*'

%&'"#()

+, (,)  ! -./ 0123456.78
'012345 9:?@AB+C D"E ;<=>

/@A>

F

GHI'" (-) F' HLEx (Table /)/JKLM3N&"OPQ&
HLEx /

 &: ;:/ HLEx @&"L/,,M3Nij012345;A&"W ?-,F]   ij012345

/"A"'"/'
L
*./ /B"An/, kJῌmol
An-Di  HLEx  XAn

 (Fig. 3a) ,=0B:CW+3+,-
9$`ab3b3

HLEx n/.- kJ ῌ mol, Two-lattice a

HLExn-.2 kJῌmol DE 

Ab-Di $%M3Nij012345"CR'S!"!TLUV+, &W,@PL (Fig. 3c) HGEx ' b52/

&XIYZ##'+[ HLEx :) 9_ -/ /B"A CpLEx, a*  L XAb /B\])'  H ^ S_/B"A"A HLEx  0.- kJ ῌ mol 6? &W&_ (,-) /  $`'ab3 cdUV+,W+ 

CpLEx, a U>L +-1- K+31- K /&A

L
Ex

L
Mix



%e  ,? //7f"A H ?-LEx

HLEx+/-- kJῌmol 3+, ,+ (k)HaL &g],A"?./ h&$

"-:%ij012345';F&A" @A Ab-Di @F'?-(k)b52"A_ (-/) * (--) /

HLEx l-/>&- 9_ -/ /Bm Tf+na o )/p"A+,By(k)/;:HLEx @ij'!T/JK&A"

W GH/UV SiO,-K, O  HLEx F'A, /q&- 'B HLEx 6./r&A

I (Table /)-sb52.*'t/ LuA/v w

ij' JL (Richet and Bottinga, +32/) / M3NI

b52/x&A Table + '  Table . : B

&A"UV+,

yzl{0' b52 91V[ Courtial et al., ,*** ;M3N$%'&Aij012345@Bouhifd et al., ,**, "A012345

K/N&A" (Fig. 3a, b, c, d, f, g, i, j) ,|> & ,78/&A HLEx |}~@-

HLEx  HGEx x _ 9_ -/ /B"A CpLinCpGi ¡/@2

M3N HaL By HLEx 3+,A"w Table 0 /?.- ,+b52/7f) ,  (Fig. -) ?-$%L

(Tf+na, Tf+ni ¢Tf+na ¢Tf+nj) LB£

 HLEx  Fig. 3 /PL 'B Fig. 3(XiTf+ni ¤XjTf+nj)  HLEx  94 DSC

JK&A" 9+, A, ¦§

+11- K /"A "C,  & CpLEX, a* F'+[ _ (+-) /H /S!&'"

6 HLEx 

/&A_ (+1b) 7&A3+

L
Ex

5 

%9:

/ HGEx, HGEx (k)2"A_ (--)  }~],- H¨©NM3Nij012345F]> LK/p"AUªLF«¬ Table 1

 $ij012345?.- -«& HLEx $`

b5/'L¯C °¬'|}~O 

9:

G
Ex

 H

 ?C /@A± ²'-. 91b52"A"A_ (-/) /

}~],V[ SiO,  Si.O2, CaSiO-  Ca,Si,O0 HLEx  gram for-

L
Ex

'

, M3N HLEx F]'L /,,0­®  WH  Xi*./ /Bm,ij0123458L
An-Di  An-Ab  Ab-Di  (Fig. 3a, b, c)

M/N"¥/- HLEx 9: y(k)/ HLEx 9;:

mula weight 78/³~&- °/ /@A´³Pµ  . <b52@AB  ,+'/xQ&A"a3²'  ,R@

L F ,+/Bm:;: HLEx 'L = b52I

L M3NOL =¶ 9/῎,῎, ·¸:012345y01N¹45

ºSGH/

YZ/ 6.- /xY+C ,+

SiO,, NaAlO,, KAlO, AIYZ&U>&A;:AUV F Xi*./ /B"A 1V[ Ab-Di '" (, kJ/mol) (k)/;: An-Ab-Di 

῍ *.,/CaO¤*.,/MgO ´ *.,/SiO, ¤*.,/NaAlO,ῌ

? An-Fo  (Fig. 3d) By An-Wo  (Fig. 3m) /B³PµT»A"UV F  Pµ"AI+,AB  "C, HLEx /+/ kJῌmol/xQ&A" *./ a3   Ab-Sa 
Table 0.

:

Data list of excess enthalpy of silicate liquids.

123

124

῍ῌ ῎

Fig. 3. Excess enthalpies of pseudobinary silicate liquids and glasses. (a) An-Di ; (b) An-Ab ; (c) Ab-Di ; (d)An-Fo ; (e) An-.Qz ; (f) Ab-.Qz ; (g) Sa-.Qz ; (h) Di-Qz ; (i) Ab-Sa ; (j) En-Wo ; (k) Wo-Ak ; (l)Wo-Ge ; (m) An-Wo ; (n) An-Ak ; (o) Di-Ak ; (p) An-Ge. Dotted lines in each ﬁgure represent variationsof HLEx approximated with a symmetric simple solution.



Fig. 3.

:

Continued

125

»¼ >

126

< NF-NM )239 HLEx f g/ 

Table 1. Regular solution parameters (Wij) andexcess enthalpies at Xi¹*./ of pseudobinary
silicate liquids.

L
h* ij HLEx 9f
 SiO,-Mx O 6 XSiO,//Y
T/ kDI lm*/9*noX pq@rstui=>

 (+)NF / NM "DEv^ NF 9w!

Yx

{" #d01n9Tv

nAy^?z(,) tu*)

^$|9}~ tu
LM 1 R IM 9l$( D

nxnAy^9?, 6s i=>T%

9 ipq>=>$%&'(Y   NM q%-uL/ D
In NF-NM-IM !)23&u D  Y/GHT/ῌ-ῌ.



K, O-Na, O-CaO-MgO-

Al, O--SiO,
῍ *.,/KAlO,   *.,/NaAlO,ῌ *.,/   gram formula weight  HLEx   +    ))A '(n)*y+ tu "# Ay,

% _ DI! "#$%&'( )* )*y+/tu,L - !*Ay+ ,-./, 

% .$( M

4* 5+ ¡ / )+0 DT

01 )23  Table 2

Wo-Ak 6/ An-Ge 6 HLEx 789:

;

 ¤/

<
=>  NF)$%&'(¢T£ D

BC@A

!@A

?@A

(network former,(network modiﬁer, NM)(Intermediate, IM)

F

"DE

GHDI

M+ 5i<¥¦*'(9  Table 0_/: HLEx £(&K, O-Na, O-CaO-MgO-

Al, O--SiO, 6=> +11- K"#$%&'(f12£A*/§dD

SiO, NaAlO,, KAlO,  NF    CaO / MgO 

 HLEx /*  3¨n©49ª5

NM /9 6« Ghiorso and Sack (+33/)

JI Bottinga and Weill (+31,) 9K
CaAl, O.  NF LM NaAlO, N KAlO,

DI
/O

PQ R CaOSAl, O- / NM /JI/GHT : UV 1
NF / NM WXT

Y/IZ

CaAl, O.  IM /[*Table 2 

, 6

"

1 ©4¯   HLEx /)+0 D HLEx O§d
Ghiorso and Sack (+33/) 86pq>=>"# Si, Al, Mg, Ca, Na "DE K
H\ Ab-Di 6

DY/$

7T986)9¬l­T®

*  ° SiO,, Al, O-, Mg, SiO., CaSiO-, Na, SiO- "D

SiO, / NaAlO, 9 MgO ]E CaO /*

E KAlSiO. /  , 6 NF / NM )23901 ¤:³  6«Dx 0 ´T

±uf12²
Mg,

 R An-Fo 6 SiO, / CaAl, O. 9 MgO /

SiO. =>$%&'(µ*: NF / NM "DE IM / NM )23;<¶·< : 7T9¸ /6«I DI
01)23^, 6

_/:/

NF-IM, NM-NM, NM-IM )23`a , 6b

W HLEx 9cQ NF-NM  NF-NF )23d`a , 6 H 9/9 5Table 2 eLEx 5Table .

 +11- K  ,/.+ kJ ῌ mol  8 9     Ghiorso and Sack (+33/) £ D HLEx kDI=f  :
H LEx5recalculated<¹H LEx5original<º,/.+X LFo kJ/mol(-0)

Table 2.
Enthalpy change of exchange reaction in silicate liquids and a classiﬁcation of interactions.


:
127

º» ¼

128

 H (original)  Ghiorso and Sack (+33/) L
Ex HLEx 

ῌ
ῑ

 HLEx  



 HS,32, Berman and Brown (+32/)

Berman (+322)

Table 3. Regular solution parameters, Wij (kJ/mol,)of silicate liquids determned based on thecalorimetric enthalpies and of MELTS (Ghiorso
and Sack, +33/).

Richet and Fiquet (+33+) Courtial et al. (,***) Bouhifd

 CpSi, Table .  DHTmi ! " Table +et al. (,**,)

 Cp ῍#$ % (, a)Li!& An,+11- KAb, Di, Fo, En, Qz, Ak, Ge, Sa '()*+,(-.   /012*+,(-.3 Table 045)6780129: HLEx ;!

"% (++)

Ha,L +11- < =3> An-Ab-Di ?'() @#$ Navrotsky et al., (+323)# ῌ

;ABCD>

EF.,G/H#I

J#$ Ha,L +11- G/ Ghiorso and Sack (+33/) 012*+,(-. =K6 Fo '()F.,LM
B6NO3 H

L
Ex, a, +11- PQ

I
6$

Ghiorso and Sack (+33/) C0129:36$῍#$#

R#78012 (An, Ab, Di, En, Ak, Ge, Jd, Ne, Sa, Py,
Cd, Mw) *+,(-. S6$T HLEx, a, +11- 

UV</ HLEx, a, +11- ῍#$

H LEx, a, +11-WXiXjWij

(-1)

 

Xῌ^_`#$ %(Wij) E6 YZ[>\]

DHa,TmiWXiH i,L+11-XiXjWij E K,O-Na,O-CaO-MgO-Al,O--SiO,?'() Wij 3 Ghiorso and Sack (+33/) Cab6῎

3 WNacK defgC Ab-Sa ?F., h/

 

 

ῌ
c῍H Sa, ,32 3 R! LM WAlcK; Table 3

78~3C

> :

 


Cp aLdT῏ῑCp SadTῐ(-2)

i/j῏ E>R# WAlcKWWNacK k

An, Ab, Di, En, Ak, Ge, Jd, Ne, Sa, Py, Cd ! " Mw

3l6

6$ Hi, Tm 3i/; (Table .) ῑ

 HLEx 3ῌ

 HLEx  Fig.

mn6 Fig. ++  An-Ab, Ab-Di, An-Di ! "+*mn Fig. +,q ῑ

 Wij ῍#3

 - kJῌgfw (r+/ kJ/mol) AB~C

Wo-An ? !& +11- K '()3 ,32 K op*

$# Wij ῑ

+,(-.Bq Ghiorso and Sack (+33/) 

78012~  ῍> ΐ HLEx  c-*r* kJ ῌ mol D>kῌ

(Fig. +*a) ῑ

E$#R#

 CaMgSiO. (Monticellite) ~¡+¢+£ῒ' HLEx st3uvw6()¤ Ca 2¥~ ¦῎Rῒ§ῑkC O x/Cyz R{12?p|}~T+/K$#R#  Table 3  Wij, Richet andkJῌmol AB*+,(-.C;C</$#

Bottinga (+32/)

3q (Fig. ++) ῑ 9O Wij ῍#(+322)

3 HLEx ῑ> 6G6 1=An-Di, An-Wo, Wo-Gh ?I

S6$c+* kJῌmol U

'()2¨(m Berman

opῒ§F.,῍#$% (-2)

3 +11- K!&:©ª῔~36$

2..+ kJῌmol C</ (Table .)

 HLEx 3'()*+,(-.C>$#R#/ῌ. 

(Fig. +*b, Fig. ++b, d) 3 601/ῌ.ῌ+ 

2

6

 CaO 3 CaAl, O.

'




L
'()*+)«-. ¬­*+)«-. SVib,i, T() 6$ῌlCR#3q

L=% 1aI 3¥®*+)«-. SConf,i, T =% 1bI

C H ABTD># >R# 6G6 p|}±²³2¥ῒ§¦῎Rῑ

Table 3  Wij 3012'()*+,(-.῍i/῔k +32* ´µ¶·ῐ¸C¹

L
Ex

¯°



:

129

Fig. +*. Comparisons between observed and calculated excess enthalpies of silicate liquids. (a) MELTS (Ghiorso
and Sack, +33/) ; (b) Calorimetric regular solution parameters.

L
Fig. ++. Relative enthalpies (H+11HS,32) of An-Ab, An-Di, Ab-Di An-Wo systems by calorimetric measurements
and those calculated by regular solution parameters in Table 3.

°± ²

130





@AB

 !"#$% & '%()*+, Richet (+32.) -./01-23

@CD

E& Ae>i)IEJ K :;<LM-NO

Be>i

FGH

Adam and Gibbs(+30/) P.6 Bottinga and Richet (+330) @QRS SLConf, i, T

 4 & An, Di, Qz, Ab, Ne, Sa 56 SiO,-Na, O

TUV WX YZ[\-P]

7'%8& Adam and Gibbs (+30/) -.'L !" (S Conf,
i, Tf+>i) @^O& < (1b) ./_`

%9  !":;< :L< (3) -./a < (2) :;./& S Conf,
i, Tf+>i

log hi,T=Ae>i?

Be>iL
TS Conf,i, T

bcEJ Berman and Brown (+32/)(-3)

Courtial et al.

(,***) -.deWX& Table + TUVWX& Table ,  SGi,,32>SGi,* fghi& Table . jk !"@^O_` An, Ab, Di, En, Wo T
Ll
UV S Conf,
i, Tf+>i@ Table +* -b

Fig. +-

Scarfe et al. (+32-)

Taniguchi (+33,) -.

/() Di P.6 An '%9@mB '%9 &n[o TUVpqrst& u[o vwxyz{t./()cEJ +*,|+*3 poise }~ GH
 K()- 8 S l-"8]P/& '%9 [\I:; & n[o u[oEy@mB (Fig. +-a) & Table +* 
LS Conf,i, Tf+>i

L
Table + WX-./ SConf,i, T @al& Adam-

L
Gibbs :;< K< -3S - SConf,&i, T @-

'%9 H- B (Fig. +-b) (Richet, +32., Tauberand Arndt, +321 ; Taniguchi, +33,) :; & En,
Wo, Py 56 Gr  (Neuville and Richet, +33+)& 

56 (Neuville et al., +33-) E O Adam-Gibbs :;< 
'%9-X¡¢H£¤@¥¦l& §¨§'%
©ªH«¬-PO­O-O K®¦
Fig. +,. Comparison between measured and predictedenthalpies of fusion of silicate minerals (An, Ab,
Di, En, Ak, Ge, Jd, Ne, Sa, Py, Co and Mw).

¯& Davis, +333, Bouhifd et al. +333, Toplis et al., ,***,
Romano et al., ,**+ ; Toplis, ,**+S< (1b) YZ[\-P]!"@ITable +*. Conﬁgurational entropies at ﬁctive temperatures of glasses by calorimetry and viscometry andcomparisons between observed and predicted melting temperatures.

³´³῏yz µ : ¶·¸G¹º»¼

131

8 `dlmnompqrs tuCvrG
H2wE?!"4x0 ῏yz H  S 
1 z?`d {|}_ῒ`a!" Metastable
incongruent melting point~ $@2456
Richet and Bottinga (+32.b)  Clausius-Clapeyron 

NO&' Jd `d$ ++**+** K, Sa `d$+/**+** K e8'&6 STm HTm $`d4

@0K {( -~ `d HCDEFG5? H$5A@ {( ,b~6 ῐij Jd  Sa +** K `d H +/1- K |}`aCDEFG (SLi,+/1- #SLi,+/1-) h8'+,ῒ 
H$P6( (3) &' DSi, Tmi $x!" T+ `aCDFG (DHi,Ti)  Cp iL$%&' (
$ :
῔
L
-#i ~
S Conf,
SG{Tfi, Tf-#i
i, *
῕

ῌ῏ ῎

ῐ

ῌ
῍DHi,Ti ῔
῕
῎

῔
῕

ῌ῍ῒΐ῎

ῌ῏ ῎

'

Fig. +-. Viscosity of liquid CaAl, Si, O2 and
CaMgSi, O0 as a function of reciprocal temperaL
ture (a) and as a function of +/TSConf,i, T (b).
Data from Scarfe et al. (+32-) and Taniguchi(+33,).

Cp SidT

῏
ῑ

ῌ῏ ῎

ῌῑ

Cp iLdTῐ Tmi

Cp iLdT ῔῕ῐCp GidT

ῌ῍ῒΐ῎

(.*)

L
 S Conf,
i, Tf-#i , ῎

9: DHi,Ti  lm$%&' ((.*) $)*`d (Tmi) $3@2456 7*8'K An, Ab, Di, En, Wo `d$L
Table +* ]6 S Conf,
i, Tf-#i Y&'+*ῒ Tf-#i Y&',* K  H$ gh`d H41QR (Table +*)6 8' Jd
 Sa |}_ῒ`a!"$K 7 ++3- 

ῌ

!"

(Tf-#i) $%&' ( (-3) $)*+,-./

,- K : +/+.,/ K 6 @`d

1$%&@4 0JfCDE
FG$gh@2456L0 Ae#i, Be#i, S Conf,
i, Tf-#i123456 7*/ῌ.ῌ, 

L
8' An, Ab, Di, En, Wo  S Conf,
i, Tf-#ibc ¡¢ bc£¤¥¡Y&'

1 9: Tf-#i 2;<=ῑ

 !" (Tf+#i)  

ῌ¦G Adam-Gibbs ($§%@47
 !" (Tf-#a) £¤¥¡LMC>?@ABCDEFGH$( (+*b) 

0IJ81$ Table +* K6

L
DEFG {S Conf,
a, Tf-#a ~ 26 Q¨ £¤¥¡0¡¢CDEFG$%&'(LMCDEFG῎NOP1PQ

LMCDEFG

R8'&6 @@ STUV

(+.b) 06 ' ( (+.b) 4 Tf+#i 2῎W/XY8'Z%45[\ $]Tf-#i 9: TTf-#a Mj

8'&6

L
L
LMCDEFG (S Conf,
a, Tf-#a , S Conf, a, Tf-#i ) ῎

038LMCDEFG ^_ῒ`1 7£¤CDEFG

a$bc`de 0Jf?CD

L(SMix,
a) $K@2456 @W/ ©

EFGghKZ%456 ῐij Jd  Saª$M« ῎¦Gm¬­$¥®

^_ῒ`a$K`d2k'&?&6 8L¤¯°e/?±²῍ SMix,a ῎z?12

º» ¼

132

  

Neuville and Richet (+33+)  En-Wo  Gr-Py !"#$%
)*+,-./ Neuville and&'(Richet (+33+) 01 !"#$%23

45678 Table + 2349-
L
SMix,
nA
a -:;/<=- Fig. +.a >? @B C/D-EF<G HI)*JKL
( )*!"#$%M N+1aO P1
SMix  n

1 Fig. +.a Q

L
En-Wo (MgSiO--CaSiO-)  SMix,a  RR (XEn ln

XEnSXWo ln XWo) TU?VJ( Gr-Py (Ca- Al,
L
Si- O+,-Mg- Al, Si- O+,)  SMix,a R-R (XGr ln XGr

SXPy ln XPy) TW->? VV

<GX ]])*JK^VYZ[\J 

 En-Wo  + _B

Gr-Py  - _B `abcDCa  Mg  CdJK^ V- ?

X

/An-Di      F 

Taniguchi(+33,)  Richet (+32.) e!"#$%W-L
SMix,a -:;/<=+,  : ; N, f   H I ) * Two-lattice + , 
MELTSO g- Fig. +.b >? h 
L&'i SMix,a  XAn j?klhc

mnop? qr 9V
st!"u$% (Fig. 3a) -

vwxy

L
-? NeF9mn-q/ O SMix,
a

-z{

Fig. +.b |}Wh

An  DifHI)* NM +1aO ~ Two-lattice +, NM,0O )*!"#$%@? VW
L
e SMix,a 

 )*f  Adam-Gibbs xy H ,LEx9~h , fHI)*5

Fig. +.. Entropies of mixing obtained from viscosity
measurements (symbols) and a combination ofphase equilibrium and by calorimetry (shadedarea) and by statistical models (solid anddotted lines). (a) MgSiO--CaSiO- (En-Wo) and
Ca- Al,Si-O+,-Mg-Al,Si-O+, (Gr-Py) liquids ; (b)CaAl,Si,O2-CaMgSi,O0 (An-Di) liquid.

->/  @ MELTS (Ghiorso and Sack,+33/) 51  SiO,, Al, O-, Mg, SiO., CaSiO- ¡1hst!"u$% NSiO,-K, O O  ¢-z{ g/st9£op?qr ¤£V

Wh +,Jhi ¥ ¦)*fst9J§¨1 HI)* S

LMix, a(hV- /  Ab-Di © J NaAlO, ~ KAlO,  CaOª MgO «Cd NNF-NM v¬­O -!

0ῌ ῐ῏῍῎ῑ st9@£?   ®(f !"u$%!"hqr

r¯ Sa-Di, Ab-Fo, Jd-Di ª¢

#$%-Mc? h--°± -². fj/]{´

st9st!"u$%78K^1 >/    (hst!"u$% NAb-Di O ]

"9-³V

st9fc-µ¶V

J 
st!"u$%·-* kJῌmol (1  ¸iS/²R+* kJῌmol ¹qrq£¸ : ¹º»6lm¼¨ (Fig. 3)



Fig. 2 !"! #$% &'()*+"!, (-.+/ kJ/mol) /01234

56() 7 8 DSC

ABlm H4I12¯ ?@

I¬56)V i.)£()cd°± Fig. +.b)£* ,kM12I¬56²()y63ª@³)kM12I¬56VX

]*+

>/´$Vµ#G \ t=i

CDEFG H4I J()%(H4IkM12I¬56y63Vz{ ,)9:; <= > ?@


&'(

133

)8K=<LMNO P'() QR

 T@¦H4I¶y4 -/S  TU*+VW B XY,.kM12I¬56VXZ (-., kJ/mol) H4I/0123456Vy40·V uv   qrq<= ! [\ /0123456()

IH4I #'!¶y4kM12I¬56

@P'()

%R @b-- DSC

&'()V] X^  @_`a!b8cd () /0123456
)^eTfg h\`iB CaO-MgO-Al, O--SiO,
\

9:

@=j kMi hg

lm Fe Vnopqrq<= s

t=j kMi Quv
An-Ab-Sa iwS

Di-Fa-An-Ab, Fo-Fa-Di-x/0123456()y63z{ |`@ 8 ,x j K=
J123456 /0123456 } 8T~"!*+V (/ kJ/mol, Table .)  J j K=\B ()V  p
V) , lmB   K=123456kM<=/0
12345624 A \

!" IH4I123456V#$K=<= %G 
b T)H4I123456

 &)RGL@"' >Vx uv t=i gy4 (Sack and Ghiorso, +33.a, b ; Sugawara,,***, ,**+) H 4 I  (   4 g { ¡¢£(Gaetani et al., +332) t=iH4I¤¥y4G(Kress and Ghiorso, +33/) w

8 MELTS y

4V' N!¦§¨() 

,123456 \ p$ y4G©bAdam-Gibbs ªRV' IH

4I)£*y63 @H4I&«12I¬56
kM12I¬56 x)V ! [\ IH4I)£() H4I
()­x +,!; £  8t,cd )£*$¡¢£
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Anderson, S. (+3.0) Measurements of mean speciﬁc heat ofplate glass at high temperatures. J. Am. Ceram. Soc., ,3,-02ῌ-1*.
Ariskin, A. A. (+333) Phase equilibria modeling in igneous
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Earth Sci., /, +ῌ,3.
Ariskin, A. A., Frenkel, M. Y., Barmina, G. and Nielsen, R.
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Asimow, P. D. and Stolper, E. M. (+333) Steady-state
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Barkhatov, L. S., Kagan, D. N., Tsytsarkin, A. F.,
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Beattie, P. (+33-) Olivine-melt and orthopyroxene-melt
equilibria. Contrib. Mineral. Petrol., ++/, +*-ῌ+++.
Beattie, P., Ford, C. and Russell, D. (+33+) Partition
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Belton, G. R., Suito, H. and Gaskell, D. R. (+31-) Free
energies of mixing in the liquid iron-cobalt orthosilicatesat +./*C. Metallurgical Transactions, ., ,/.+ῌ,/.1.
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Forsterite  Richet et al. (+33-a)  DSC  ,+1. K  +.,+.
kJῌmol   Richet et al. (+33-a)  Fo    ! +** K "#$%&'()*+,-./01234

premelting 56

78 9:0;< premelting
8=>?@ABCDEF07GH 48I& DSC

J Al, O- (Barkhatov, et al. +31-) KLMNOP Q

R>?#$STUVW08= ;<I&

 X&9:Y XZ

Tangeman et al. (,**+)  containerless technique 
Fo [\  Fo ]^_`a 7b=cd8 487]^_Pe#$fgh?  (,32 Ki+*1/ K) j7  ,32 K  Fo
]^_kl'()*+, 0+...+.,/ kJῌmol 
m8 noJ7p Richet (+321) 
qAr*s Richet et al. (+33-a) $ Richet and Bottinga (+32/) ]^_]^_kl#
t*uqAr*s vw Richet and Fiquet (+33+) Fo s

P x (.b)   Fo '()*+,yBC
8z{ +-2.3,.0 kJῌmol m8 ;<]^_kl'()*+,;<]^_kl#$Y;< (,*K)  |=8pJ}  yBCp Richet et al.
(+33-a) p;<~}  noJa HFo, ,+1. +-2.3,.0 kJῌmol P8Richet and Bottinga (+32.a, b)  En, An, Ab, Sa, Jd,
Ne, Cd c]^_t*us  

]^_`'()*+,spP ZZ'()*+,p|=
8 I&s Robie et al. (+312) P
8 Robie et al. (+312) p"#Jz{ 4J#a '()*+,p@a Richet and Bottinga (+32.a, b) 4=8'()*+,p /@XZ

 noJ An  Richet and Fiquet (+33+)

#4J ¡ p Z¢

s#XZ

Berman and Brown (+32/)

 w Holland and Powell (+33*) sP

x (.b)  '()*+,yBC8

Table A+.

Enthalpies of vitriﬁcation (HVit) and heat capacities used to calculate enthalpies of fusion (HTm).


:
141

¯° ±

142

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' (# )*+
0Hi, Tm

1234A ,  

5 (-/) 6+789 :;<9=>?@

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ABCD (Tf+Ea) FG+ Tf+Ea =>HIJK6+LM%NKO,+) PQ:;<RS$% TUCD
VW9

JKX)Y

Z[\ Tf+Ea]^/,%_` )a b4bABCD ;<cde)ABCD,fgcd4__/h 5 (-/) i% Tf+EijTf,Ei i6k Tf+EajTf,Ea
Tf,Ea AK9l+

mnJKopeJKiq+rst

uvw9 ABCD9*+ Wo-En, Gr-Py, An-Di i6
k An-Ab x
Tf,EamnJKykpeJKz/./,

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Tf,Ea 

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Fig. A+.

Plot of 0Tf (j῍XiTf,EiETf,Ea) againstῌliquid compositions. The 0Tf can be apploximated as [+-*/*] Xi (+EXi) (solid curve).
This relationship was used to estimate Tf,Ea andto calculate excess enthalpy of the liquid of
which the Tf,Ea was not measured.

126b0Tf

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Toplis (+332) LM% Tf,Ea  J4 78 !"# Tf,Ea + 5$%%+ Tf,Ea

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V4 Tf,Ea )JKO,%_x$%`

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