Clapeyron Equation in Condensed State Equilibria

G(ii). Along the phase line, we also have dGi = dGii. Inserting the condition of equilibrium dG =−SdT + V dP into the line, we obtain -SⁱdT + VidP =

−SⁱⁱdT + ViidP . Rearranging the equation, dP /dT = (Si − Sⁱⁱ)/(Vi− Vⁱⁱ).

From the relation between heat and change of entropy in a reversible process, there is ΔHi→ii(T, P ) = T ΔSi→ii(T, P ). Combining the last two equations where all amounts are in molar units, it reads

dP

dT = ΔHi→ii(T, P )

T ΔVi→ii(T, P ). (4.2)

Equation (4.2) is the Clapeyron equation, which, geometrically speaking, not only expresses the slope of the equilibrium line, but also gives the rate, where the P must change with T for two phases to remain in equilibrium. It applies to all changes of phases where there is a discontinuity in S and V at the transition that is the first order phase change.

4.2 Clapeyron Equation in Condensed State Equilibria [3, 4]

For the most substances, the slope of the solid-liquid line is positive. The greater the P on a given substance, the closer the molecules of the substance are brought to each other, which increases the effect of the substance’s inter-molecular forces. Thus, the substance requires a higher T for its molecules to have enough energy to break out the fixed pattern of the solid phase and enter the liquid phase. The reason is also interpreted qualitatively from Eq. (4.2), which shows that this is associated with the fact that the most substances expand on melting and therefore have both ΔV and ΔH positive. Water is an exception in that it expands on freezing so that the solid-liquid boundary has a negative slope. Thus, in the case of water, it is possible by increasing P isothermally to pass from vapor to solid to liquid (e.g. at T1 in Fig. 4.1(b)) whereas for most substances, the solid is the high-pressure phase. As stated in Chapter 3, when chemical bonding is changed, such as semiconductors, ΔV could also be negative where coherent bonding with CN ≈ 4 − 8 in solids changes to metallic bonding with CN ≈ 10 − 11 in liquids.

To utilize Eq. (4.2) for determination of the phase diagram, one is fre-quently interested in knowing the relation between the equilibrium values of P and T instead of their mutual rate of change. This leads to trying and integrating Eq. (4.2) where experimental ΔHi→ii(T, P ) and ΔVi→ii(T, P ) functions for a certain material are needed. Since both ΔHi→ii(T, P ) and ΔVi→ii(T, P ) are functions of T and P , and the necessary separation of vari-ables cannot be accomplished in any direct and known manner, the integra-tion of Eq. (4.2) has been carried out through approximate methods ever since the equation was first established in the 19th century.

The crudest approximation is to take ΔHi→ii and ΔVi→ii as constants.

122 Chapter 4 Phase Diagrams

When ΔP = P − P⁰ and ΔT = T − T⁰ are small where the subscript “0”

denotes the initial point, ΔHi→ii(T, P )≈ ΔHⁱ→ii(T0, P0) and ΔVi→ii(T, P )≈ ΔVi→ii(T0, P0) have a minor error. Equation (4.2) then integrates to give P = (ΔH_0,i→iilnT /ΔV_0,i→ii) + a^ with a^ being a constant.

Instead of assuming that ΔHi→iiwas constant, we may make the better approximation that ΔHi→ii is a function of T while ΔVi→ii is a function of P, ΔHi→ii(T, P )≈ ΔHi→ii(T ) and ΔVi→ii(T, P )≈ ΔVi→ii(P ), which brings out a simplification of Eq. (4.2) [5],

dP

dT = ΔHi→ii(T )

T ΔVi→ii(P ). (4.3)

In fact, the P and T dependent molar volumes of the both phases for wide ranges of P and T are rarely known. Simple analytical forms of ΔV (T, P ) have to be used.

In order to assess the P effect on T , the surface stress-induced internal pressure Pinfor a spherical particle is extended to the bulk case. Let P denote the sum of Pin and the external pressure Pe, P = Pin+ Pe. When Pe ≈ 0, P = Pin. This is the case for the size-dependent transition of low-dimensional materials. When Pin≈ 0 with r → ∞, P = Pe, which is the usual situation of pressure-dependent transition for bulk materials. Since any pressure source should have the same effect on materials properties, Pincan be substituted by Pe. In our case, we will use the expression of Pinas that of P in the following [6, 7].

As an example, the T -P phase diagram of Ge determined by Clapeyron equation is introduced [8]. In the T -P phase diagram of Ge, there are to-gether three transitions with the corresponding T (P ) functions, which are the melting of Ge-I, the solid transition between Ge-I and Ge-II where I and II denote different solid states, and the melting of Ge-II. These tran-sitions are identified by subscripts of I-L, I-II and II-L, respectively. Ac-cording to Eq. (4.3), three T (P ) curves are obtained by integrating Eq.

(4.3) after suitable considerations for initial points and ΔH(T ) and ΔV (P ) functions, which are described here. The first one in consideration is the I-L transition. For this transition, ΔHm(T ) function can be determined by Helmholtz function, ΔHm(T ) = ΔGm(T )−T dΔGm(T )/dT . For semiconduc-tors, ΔGm(T ) = ΔH_m,0T (T_m,0−T )/(Tm,0)²where the subscript “0” denotes the reference state [9] (for details see Sec. 4.6). Thus,

ΔHm(T ) = ΔH_m,0(T /T_m,0)². (4.4) When T < TKwith TK= T_m,0/2 being the Kauzmann temperature or isen-tropic temperature where ∂ΔGm(T )/∂T = 0 [10], ΔHm(T ) = ΔHm(TK). As a result, at T < TK,

ΔHm(T ) = ΔH_m,0/4. (4.5)

Another function of Eq. (4.3) is ΔVI-^L(PI) = (V_L,0− VI,0) + (ΔVL− ΔV^I) where V_L,0 and V_I,0 as molar volumes of liquid and solid are known data

4.2 Clapeyron Equation in Condensed State Equilibria 123

while ΔVL = −V^LPLβL and ΔVI = −V^IPIβI with the available compre-ssibility β =−ΔV/(V P ).P^L and PIin the above equations are considered by assuming a spherical particle with a radius r where there exists a curvature-induced pressure. In light of the Laplace-Young equation, PI = 2fI/r and PL = 2γsv/r [11]. Essentially γsv describes a reversible work per unit area to form a new surface while f denotes a reversible work per unit area due to the elastic deformation, which equals the derivative of γsv with respect to the strain tangential to the surface. For solid, f = γ while for liquid f = γsv. Thus, ΔVL = −VL,0PI(γ/fI)βL because PI/PL = fI/γsv. Substituting this relationship into ΔVI-^L(PI) function, it reads

ΔVI-^L(PI) = V_L,0− VI,0+ [V_I,0βI− VL,0(γsv/fI)βL]PI (4.6) where f has been expressed as

f = (h/2)[3ΔSvibΔHm(T )/(βV R)]^1/2. (4.7) When the initial point of (P0, T0) is selected as (0, T_mI,0) where T_mI,0 is the melting temperature of Ge-I under ambient pressure, integrating Eq. (4.3) with ΔHI-^L(TI-^L) and ΔVI-^L(PI) functions in terms of Eqs. (4.4) and (4.6),

 _PI

0

{VL,0− VI,0+ [V_I,0βI− VL,0(γsv/fI)βL]PI}dPI

= [ΔHI-L,0/(T_mI,0)²]

 _T

TmI,0

T dT , or,

T (PI) = T_mI,0

1+{2(VL,0−VI,0)PI+[V_I,0βI−VL,0(γsv/fI)βL]P_I²}/ΔHI-L,0. (4.8) Note that Eq. (4.8) is also applicable to nanosized material if T_mI,0 in Eq.

(4.8) is substituted by Tm(r) at P = 0, T_mI,0(r). Thus, T_mI,0(r) is a function of r and P . The value of T_mI,0(r) has been deduced as Eq. (3.80). In Sec.

4.5.2, we will discuss the size-dependent phase diagram in detail.

For I-II transition, the subscript “I^” is used to substitute I for distinguish-ing this P -induced transition from meltdistinguish-ing transition of the Ge-I phase since the initial point as the boundary condition is selected as (PI-II,0, 273), where P > 0 and related parameters have been affected by P . Since PI varies a little in the full transition temperature range, as a first order approximation, ΔVII≈ ΔVI is assumed and thus V_II,0fIIβII≈ VI,0fIβI. As a result,

ΔVI-^II(PI)≈ VII,0− VI,0, (4.9) and

fII≈ (V^I,0/V_II,0)(βI/βII)fI. (4.10)

124 Chapter 4 Phase Diagrams

The corresponding thermodynamic parameters of Ge-II differ from Ge-I due to their distinct structures. Since the specific heat difference between different polymorphous solid phases ΔC_P is small, it is assumed that

ΔSvibII-^L≈ ΔSvibI-^L+ ΔSvibI-^II (4.11) where ΔSI-^II= ΔHI-II,0/TI-II,0, which may be determined by

ΔSvibI-^II= (ΔHI-II,0/TI-II,0)(ΔSvibI-^L/ΔSmI) (4.12) where ΔSvibI-II/ΔSI-II is supposed to be equal to ΔSvibI-L/ΔSmI as a first order approximation.

With the neglect of ΔC_P, ΔHI-II,0≈ ΔHII-L− ΔHI-L. As the transition occurs at T < TK, in terms of Eq. (4.5),

ΔHI-II,0= (ΔHII-^L− ΔHI-L,0)/4. (4.13) Note that Eqs. (4.4) and (4.5) are also applicable to ΔHII-^L(T ). In terms of Eqs. (4.5), (4.7) and (4.10), (VI,0/V_II,0)(βI/βII)hI[ΔSvibI-^LΔH_mI,0/ (βIVI,0)]^1/2= hII[ΔSvibII-^L× ΔHmII/(βIIV_II,0)]^1/2, it yields

ΔHII-^L = (VI^,0/V_II,0)(βI^/βII)(h²_I/h²_II)(ΔS_I^vib-^L/ΔS_II^vib-^L)ΔHI-L,0. (4.14) In terms of Eqs. (4.11) – (4.14), ΔHI-II,0is obtained as

ΔHI-II,0={[(4T^I-II,0ΔSmI− ΔH^I-L,0)²

+16ΔHI-L,0TI-II,0ΔSmI(VI^,0/V_II,0)(βI^/βII)(h²_I/h²_II)]^1/2

−ΔHI-L,0− 4TI-II,0ΔSmI}/8. (4.15) Integration of PI from PI-II,0to PI and T from TI-II,0to T in terms of Eqs.

(4.9) and (4.15) gives

 _PI

PI-II,0

(V_II,0− V^I,0)dPI = ΔHI-II,0

 _T

TI-II,0

1

TdT , which brings out

T (PI) = TI-II,0exp[(V_II,0− VI,0)(PI− PI-II,0)/ΔHI-II,0]. (4.16) Letting Eq. (4.8) = Eq. (4.16), the Ge-I/Ge-II/liquid triple point (Pt, Tt) is obtained, which is considered as the known threshold point for the melting curve of Ge-II. Since all the three phases (Π = 3) coexist there, there is no freedom in the system and the condition for thermodynamic equilibrium leads to unique T and P defining the triple point, or there is only one combination of T and P where three phases coexist in a single-component system.

If it is assumed that Eqs. (4.4) and (4.6) are applicable to the ΔHII-^L(T ) and ΔVII-^L(P ) functions through substituting the initial point of (0, T_mI,0) by (Pt, Tt), Eq. (4.3) is integrated from Ptto PIIfor PIIand from Ttto T for T to

4.2 Clapeyron Equation in Condensed State Equilibria 125

give

 _PII

Pt

{V^L−V^II+[VIIβII−V^L(γ/fII)βL]PII}dP^II= (ΔHII-^L/T_t²)

 _T

Tt

T dT , which in turn results in

T (PII)

= Tt

1+{2(VL−VII)(PII−Pt)+[VIIβII−VL(γ/fII)βL](PII−Pt)²}/ΔHII-L. (4.17) Figure 4.2 presents functions of Eqs. (4.8), (4.16) and (4.17) and experimental results of T -P phase diagram of Ge where the necessary parameters are listed in Table 4.1. Note that Pinin tiny droplets of a few molecules may reach GPa range studied here where the pressure has a real effect on V of solid and liquid.

Fig. 4.2 A comparison of T -P phase diagrams of Ge between Eqs. (4.8), (4.16) and (4.17) (solid line) and experimental results shown with different symbols come from distinct sources.(Reproduced from Ref. [8] with permission of Elsevier)

The P -T relationship in Fig. 4.2 is established by a generalization from the consideration of internal pressure on small particles to the bulk case.

Thus, the low size limit of nanoparticles for Pin at 6h in Eqs. (4.8) and (4.17) must be considered. For I-L transition, at 6hI = 1.47 nm, P_l = 6.12 GPa in terms of Eq. (4.7). As shown in the figure, Pt= 9.915 GPa. In order to predict the P -T relationship from 6.12 GPa to 9.915 GPa, the value of T (P = 6.12 GPa) and the corresponding slope are linearly extended up to Ptpoint. This consideration is based on the experimental results shown in the figure where when P is large enough, the melting curve approximately changes linearly.

With a similar consideration, for II-L transition, at 6hII = 1.62 nm, P2 = 3.92 GPa in terms of Eq. (4.10) and the largest applicable pressure is 13.835 GPa, which is the sum of Pt and P2.

126 Chapter 4 Phase Diagrams

Table 4.1 Necessary parameters for calculating T (P ) phase diagram of Ge in terms of Eqs. (4.8), (4.16) and (4.17). T is in K, P in GPa, V in cm³·mol⁻¹, β in 10⁻¹¹Pa⁻¹, γ and f in J·m⁻²and ΔH is in kJ·mol⁻¹

I-L transition I-II transition II-L transition

TmI,0 1210.4 TI-II,0 273^d Tt 714^f

PI-II,0 10^d Pt 9.915^f

VI,0 13.64^a V_I,0 11.93^a VII 9.66^a

VL,0 12.94^a VII,0 9.66^a VL 12.94^a

βI 1.33^b βII 1.19^b

βL 10.00 βL 10.00

fI 2.252^c fII 1.589^c

γ 0.581 γ 0.581

ΔHI-L,0 36.94 ΔHI-II,0 0.2^e ΔHII-L 37.74^e

aVI,0 = Mw/ρI and VL,0 = Mw/ρL with Mw = 72.59 g·mol⁻¹ being the molar weight and ρ being the density, ρI = 5.32 g·cm⁻³ and ρL = 5.61 g·cm⁻³·VII,0= NAvIIand V_I,0= NAvI where NAdenotes the Avogadro’s constant and v is the mean atom volume within the corresponding crystalline structures. vII= a²_IIcII/4 = 0.016,053 nm³with the lattice constants aII= 0.4884 nm and cII= 0.2692 nm for β-Sn structure and vI = (aI)³/8 = 0.019,815 nm³ where the lattice constant aI= 0.5412 nm for diamond structure.

bβI= 1/Bm,Iand βII= 1/Bm,IIwith Bmbeing the bulk modulus, Bm,I= 75.0 GPa and Bm,II= 84.0 GPa. Note that as the first-order approximation, βI≈ βIis assumed.

cfIis calculated by Eq. (4.7) with hI= (3^1/2/4)aI= 0.2450 nm due to its diamond struc-ture where aI= 0.5658 nm denoting the lattice constant and ΔS^vibI-L = 4.6 J·mol⁻¹·K⁻¹.fIIis calculated through Eq. (4.10) where fI = 1.151 J·m⁻²determined by Eqs. (4.5) and (4.7) with βI≈ βI= 1.33 × 10⁻¹¹Pa⁻¹being a weak function of pressure and hI= (3^1/2/4)aI= 0.2343 nm for diamond structure.

dThis value is the more recent value and approximately the mean value among the experi-mental results.

eΔHI-II,0is calculated by Eq. (4.15) where ΔSmI= ΔHI-L,0/TmI,0= 30.52 J·mol⁻¹·K⁻¹ and hII= cII= 0.2692 nm due to itsβ-Sn structure. The value of ΔHII-Lis determined by Eq.

(4.13).

fThe triple point (Pt, Tt) is determined by letting Eq. (4.8) = Eq. (4.16).

High pressure work on Ge is plagued with the problems of sluggish phase transitions and possible metastable phases. Thus, the I-II phase boundary in the T -P diagram is very indefinite with a wide reported transition pres-sure where the transition with hysteresis is very slow, especially at the lower temperature range. As shown in the figure, although a mean value of PI-II,0

among different experiments has been selected to ascertain the I-II transition curve, the predicted I-II phase boundary in terms of Eq. (4.16) is indeed approximately equal to the mean value of the experimental results.

4.3 Solution, Partial Molar Properties and Chemical

In document Qing Jiang Zi (pagina 140-145)