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The melting temperaturepressure phase diagram [Tm\(P)P] for semiconductor, molecular crystals and oxide are predicted through the Clapeyron equation where the pressuredependent volume difference is modeled by introducing the effect of surface stress induced pressure. Semiconductor, molecular crystals and oxide have been employed to test the reliability of the model, because of its important role. For Si and Ge, the stable state under normal pressure is the diamond structure (SiI and GeI). Through pressure, this change in the diamond structure for beta Sn structure (SiII and GeII), and with the increase of temperature, phase I and II are going to be melting into a liquid (L). For the CO_{2} crystal (commonly known as dry ice), it is a molecular solid with a structure of Pa3 at low temperature and low pressure (CO_{2}I), and can be widely used for cooling. Al_{2}O_{3} has been extensively investigated because of its widely ranging industrial applications. This includes applications as a refractory material both of high hardness and stability up to high temperatures, as a support matrix in catalysis. MgO is a material of key importance to earth sciences and solidstate physics: it is one of the most abundant minerals in the Earth and a prototype material for a large group of ionic oxides.
Phase transitions, Pressuredependent, Temperature, Oxides.
High pressure research is a powerful tool to explore key and electronic states of nature in solids. Due to high pressure often produce a lot of new structure in the elements or compounds, so it has become a significant change in a way of atomic distance and coordination number. With the development of high pressure research unceasingly thorough, the temperature pressure phase diagram (T/P) caused by theoretical and experimental home interest. In recent years, the generation and measurement of simultaneous high pressures and high temperatures has undergone rapid development with the diamond anvil cell (DAC) technique [14]. We choose Si, Ge, CO_{2}, Al_{2}O_{3} and MgO to verify the reliability of the model, because, although the TP phase diagram of this substances have been obtained through the experiments is established, but the measurement accuracy under high temperature and pressure is not high, so most of the phase diagram are experimental, or is the schematic, the existing data has not been confirmed theoretically. Therefore, more indepth theoretical work is necessary.
For Si and Ge, as everyone knows, the steady state under normal pressure is the diamond structure (SiI and GeI). Through pressure, this change in the diamond structure for beta Sn structure (SiII and GeII), and with the increase of temperature, phase I and II will melt, into a liquid (L). For the CO_{2} crystal (commonly known as dry ice), it is a molecular solid with a structure of Pa3 at low temperature and low pressure (CO_{2}I), and can be widely used for cooling. As the pressure increases, several high pressure phases appeared, including the P4_{2}/mnm symmetric structure (CO_{2}II), Cmca orthorhombic structure (CO_{2}III), the structure of Pbcn (CO_{2}IV) and polymerization of the quartz structure (CO_{2}V) etc. However, for this high pressure phase is still uncertain, especially have yet to determine precisely their stable field. Al_{2}O_{3} and MgO have been employed to test the reliability of the model, because of its important role. Al_{2}O_{3 }has been extensively investigated because of its widely ranging industrial applications. This includes applications as a refractory material both of high hardness and stability up to high temperatures, as a support matrix in catalysis, as well as a variety of fundamental interests [58]. MgO is a material of key importance to earth sciences and solidstate physics: it is one of the most abundant minerals in the Earth (especially its lower mantle) and a prototype material for a large group of ionic oxides. The classic Clapeyron equation governing all firstorder phase transitions of pure substances may be useful to determine the Tm (P)P curve theoretically in the following form [9],
$d p=\frac{\Delta H\left(T_{m}, P\right)}{\Delta V\left(T_{m}, P\right) T_{m}} d T_{m}$ (1)
where H (T_{m},P) show the gramatom melting enthalpy and V(T_{m},P) is gramatom volume change during the melting with Δ denoting the change. Eq. (1) can describe the joint rate of change dP/dT_{m} along the phase equilibrium lines and estimate the derived properties of DH and DV. To utilize Eq. (1) for determination of phase diagram, a T_{m}(P) function or an integration of Eq. (1) is needed. Since both DH(T_{m},P) and DV(T_{m},P) are functions of temperature and pressure, and the necessary separation of variables cannot be accomplished in any direct and known manner, the integration of Eq. (1) has been carried out through approximate methods ever since the equation was first established in the 19th century [9]. Although when DP = P–P_{0} and DT = T_{m}–T_{m0} are small, DH(T_{m},P) » DH(T_{m0},P_{0}) and DV(T_{m},P) » DV(T_{m0},P_{0}) have minor error where the subscript 0 denotes the initial points and D denotes the difference [6], as DP and DT increase, exact functions of DH(T_{m},P) and DV(T_{m},P) must be known [9]. Thus, a successful application of Clapeyron equation for T_{m}(P)P phase diagram depends on establishing accurate ΔH(T_{m}, P) and ΔV(T_{m}, P) functions.
Recently, a general equation without any free parameter for surface stress f has been established as follows [10],
f (T_{m0}) = (h/2)[3DS_{vib}DH_{m0}/(k_{S}V_{S}R)]1/2 (2)
where h is atomic diameter, DS_{vib} is the vibrational part of the overall melting entropy DS_{m}, S_{m}=S_{el}+S_{pos}+S_{vib}, S_{el} is negligibly small, and S_{pos}=R[x_{A}ln(x_{A})+ x_{v}ln(x_{v})], where x_{A}= 1/(1+ΔV_{m}) and x_{v}=ΔV_{m} /(1+ΔV_{m}) are the molar fractions of the host material and vacancies, respectively [11], for oxides DS_{vib} » DS_{m}DS_{pos}, k = DV/(V_{P}) is compressibility of the crystal, V_{S }is gramatom volume of crystals, R shows the ideal gas constant and DH_{m0} is bulk melting enthalpy at T_{m0}.
The predicted f values of various materials in terms of Eq. (2) are in agreement with the known experimental and theoretical results obtained from the first principle and the classic mechanics calculations [10]. Since the measured thermodynamic amounts in Eq. (2) has reflected usually unknown surface states of materials [10], Eq. (2) supplies an easy way to establish a relationship between the surface stress induced internal pressure Pi for small particles and T_{m}, which brings out a possibility to determine DV (T_{m}, P) function.
DH_{m}(T_{m}) function can be determined by Helmholtz function, DH_{m}(T_{m}) = DG_{m}(T_{m})T_{m}dDG_{m}(T_{m})/dT_{m}, where DG_{m}(T_{m}) denotes the temperature dependent solidliquid Gibbs free energy difference. For oxides, DG_{m}(T_{m}) = DH_{m0}T_{m}(T_{m0}T_{m})/T_{m0}^{2} where DH_{m0} is the melting enthalpy at the melting temperature T_{m0} [12]. This function was modeled by treating DH_{m}(T_{m}) to be a linear function where the heat capacity difference ΔC_{p} between crystal and liquid to be a constant. Using a mathematic approximation that the quantity ln (T_{m0}/T_{m}) is approximately equal to 2(T_{m0}T_{m})/(T_{m0}+T_{m}) with neglecting of higher order terms. Thus,
DH_{m}(T_{m}) = DH_{m}(T_{m}/T_{m0})^{2}. (3)
In this contribution, through assuming that DV(T_{m},P) and DH(T_{m},P) functions may be determined by Eqs. (2) and (3), T_{m}(P) curves are obtained with an integration of Clapeyron equation when suitable original points for each integration are selected. It is found that the model prediction of the Tm(P)P phase diagram of corundum is consistent with the experimental results and other theoretical predictions [1618].
To find a solution of Clapeyron equation, as a first order approximation, DH_{m}(T_{m},P) » DH_{m}(T) and DV_{m}(T_{m},P) » DV_{m}(P) are assumed [9], which lead to a simplification of Eq. (1),
$d P=\frac{\Delta H_{m}\left(T_{m}\right)}{\Delta V_{m}(P) T_{m}} d T_{m}$ (4)
DV_{m}(P) = (V_{L}V_{S})+(DV_{L}DV_{S}) where the subscripts S and L denote solid and liquid, respectively. DV_{S} = V_{S}P_{S}k_{S} and DV_{L} = V_{L}P_{L}k_{L} where V_{L} and V_{S} are known data. To find a solution of the equation, a relationship between P_{L} and P_{S} must be found. To do that, a spherical particle with a diameter D is considered. In light of the LaplaceYoung equation, P_{S} = 4f/D and P_{L} = 4g/D where g is the surface energy of the liquid [10]. Thus, DV_{L} = V_{L}P_{S}(g/f)k_{L} because P_{S}/P_{L} = f/g. Substituting this relationship into DV_{m}(P) function,
DV_{m}(P) = V_{L}V_{S} +[V_{S}k_{S} V_{L}(g/f)k_{L}]P (5)
where P_{S} has been simplified as P. when the initial point of (P_{0}, T_{0}) is selected as (0,T_{m0}) where T_{m0} is the melting temperature under ambient pressure, integrating Eq. (5) with DH_{m}(T_{m}) and ΔV_{m}(P) functions in terms of Eqs. (3) and (5),
$\int_{0}^{P}\left\{V_{L}V_{S}+\left[V_{S} \kappa_{S}V_{L}\left(\gamma / f_{S}\right) \kappa_{L}\right] P\right\} d P =\left(\Delta H_{m 0} / T_{m 0}^{2}\right) \int_{T_{m 0}}^{T_{m}} T_{m} d T_{m}$,
or
$T_{m}(P) =T_{m 0} \sqrt{1+\left\{2\left(V_{L}V_{S}\right) P+\left[V_{S} \kappa_{S}V_{L}(\gamma / f) \kappa_{L}\right] P^{2}\right\} / \Delta H_{m 0}}$ (6)
Although the above discussion on P is related with the surface stress induced internal pressure Pi for a spherical particle, they may be extended to a general case for the pressure effect on T_{m}, which is illustrated as follows: Let P denote the sum of Pi and the external pressure Pe, namely [16],
P=Pi+Pe. (7)
When Pe » 0, P = Pi. This is the case of the sizedependent melting. When Pi » 0 with D®¥, P = Pe, which is the usual situation of the pressuredependent melting for bulk materials. Since any pressure source should have the same effect on materials properties, Pi can be substituted by Pe. Thus, although P denotes Pi in above T_{m}(P) equation, it has been considered as Pe and is simplified as P.
Figs.1 and Figs.2 describes the melting curve on the Si TP phase diagram and Si nano crystals and TP phase diagram of Ge according to equation (6) compared model predicted results and experimental results and other theoretical results, parameters used are listed in Table. 1. Figure. 1 present a comparison for TP phase diagram of bulk Si and the melting curve of Si nanocrystal among the model predictions and experimental and other theoretical results. The solid lines denote model predictions in terms of Eqs. (6) Where necessary parameters used are listed in Table .1. The symbols ○, □, ∆ show experimental results. The dash and dot lines denote other theoretical results. Other symbols denote III transition pressure at room temperature where n denotes the theoretical result, + denotes the experimental results under nonhydrostatic pressure, ♦, ◊ and ∇ denote the experimental results under hydrostatic pressure. For the melting of Si nanocrystal, the two dash lines show the predicted results where the corresponding T_{mI} (D) values denoted as × (1478 K) and ▲ (1371 K) are obtained from other theoretical result and Eq. (6), respectively. For comparison, the dot line gives the theoretical result for melting of Si nanocrystal. Figs.1 and Figs.2 in the PT relationship is through the small particles produced by internal pressure considered and applied in bulk crystals of external pressure equivalent to the idea of the establishment of the. However, equation (6) limit the size of nano crystal used in Pi must be taken into account; it is equal to 6h. For the IL transformation, the corresponding 6hI = 1.4112 nm, according to PI = 4 f more I/D, we obtain the limit pressure for the Pl = 10.5 GPa. while at Pl < P < Pt this pressure range, the error of PT curve and the experimental results of our model predicted only small, but this will still give the triple point (Pt determination of Tt), bring some inaccuracy. In order to improve the accuracy and the experimental results more in line with that, we take the approach in determining P > Pl curve is: the curve tangent direction line extend along at P = 10.5 GPa place, this is because the experimental results confirm the pressure is large enough melting curve is approximately a straight line. Similarly, for the transformation of the Ge IL, when 6hI = 1.47 nm, Pl = 6.12 GPa. We have taken and the Si class Like the way to deal with Pl < P < Pt the range of pressure curve. For the TP phase diagram of Si and Ge, III phase boundary shift is very fuzzy, change the pressure distribution in the reported a wide range, especially at low temperatures, and change due to the very slow lag. As shown in the figure, although we chose different experimental results at room temperature with a mean value of PIII to determine the transition curve of III, but the results of model predictions and the experimental results are in good agreement, this shows that the average results of the phase boundary of this transition and experiment is very close to the. In fact, our model predicted results and experimental results or have a certain error, this may be caused because we neglected the effect of compression of the pressure coefficient.
Figure 1. A comparison for TP phase diagram of bulk Si and the melting curve of Si nanocrystalamong the model predictions and experimental and other the oretical results
Figure 2. A comparison for TP phase diagram of bulk Ge among the model predictions and experimental and other theoretical results
The solid lines denote model predictions in terms of Eqs. (6) where necessary parameters used are listed in Table 1. The ymbols Δ , + and Ο show experimental results. The dash lines denote other theoretical results.
Table 1. Necessary parameters for calculating TP phase diagram of Si and Ge and the melting curve of Si nanocrystals in terms of Eqs. (6)

IL transition 
III transition 
IIL transition 


Si 
Ge 

Si 
Ge 

Si 
Ge 
T_{mI} 
1693 
1210.4 
T_{III}^{d} 
273 
273 
T_{t}^{f} 
960 
714 



P_{III}^{d} 
12 
10 
P_{t}^{f} 
11.6 
9.915 
V_{I}^{a} 
12.06 
13.64 
V_{I}^{a} 
11.00 
11.93 
V_{II}^{a} 
8.53 
9.66 
V_{L}^{a} 
10.93 
12.94 
V_{II}^{a} 
8.53 
9.66 
V_{L}^{a} 
10.93 
12.94 
κ_{I}^{b} 
1.02 
1.33 



κ_{II}^{b} 
0.885 
1.19 
κ_{L} 
10.00 
10.00 



κ_{L} 
10.00 
10.00 
f _{I}^{c} 
3.707 
2.252 



f _{II}^{c} 
2.797 
1.589 
γ 
0.765 
0.581 



γ 
0.765 
0.581 
ΔH_{IL} 
50.55 
36.94 
ΔH_{III}^{e} 
0.78 
0.2 
ΔH_{IIL}^{e} 
53.67 
37.74 
Fig.3 A comparison for TIL(P) curve of bulk CO_{2}I between the model prediction and experimental results. The solid line denotes the model prediction in terms of Eqs. (6) where necessary parameters used are: T_{mI} = 215.55 K, V_{L} = M/ρ_{L} = 12.43 cm^{3}⋅gatom^{1} with M =14.67 g⋅gatom^{1} and ρL = 1.18 g⋅cm^{3}, V_{I} = 10.65 cm^{3}⋅gatom^{1}, κ_{I} = 1/BI =8.06×10^{11} Pa^{1} with BI = 12.4 GPa, κ_{L} ≈ 100×10^{11} Pa^{1} as a first order approximation is equal to the κL value of CS_{2} since CS_{2} as a linear molecule has a similar behavior of CO_{2},ΔH_{IL} = 2.649 KJ⋅gatom^{1}, γ = 0.00913 J⋅m^{2}, and fI = 0.5613 J⋅m^{2} is calculated by Eq.(6) with hI = (2^{1/2}/2)a_{I} = 0.3575 nm where a_{I} = 0.5056 nm for Pa^{3} structure and ΔSvib^{IL} =8.856 J⋅gatom^{1}⋅K^{1}. The dot lines show the experimental phase diagram.
Figure 3. A comparison for TIL(P) curve of bulk CO_{2}I between the model prediction and experimental results
Figure.4 presents comparison between the model predictions of Eq. (6) and experimental result of pressure dependent melting of Al_{2}O_{3}. As shown in the figure, Tm increases as P increases. This is evidently induced by positive volume change DV during the melting. Thus, However, the low size limit of nanocrystals for the application of Pi in Eq. (6) must be considered, a crystal is characterized by its longrange order and the smallest nanocrystal should have at least a half of the atoms located within the particle [17]. Hence, the smallest Dmin is 2D_{0}. For Al_{2}O_{3}, Dmin =1.146 nm in term of D_{0}=3h for nanocrystals [7]. The curvatureinduced pressure P approximately equals to 20.10 GPa by LaplaceYoung equation Pi=4f/D associated with Eq. (2). The value of Tm on the melting curve Tm(P) must be determined. As shown in the figure, Clapeyron equation, without any adjustable parameter, is consistent with the experimental result and other theoretical prediction [1315]. The Tm(P)P relationship in Figs. 1 is made by a generalization where the internal pressure of small particles is considered to be equivalent to that of the bulk one.
Figure 4. Presents comparison between the model predictions of Eq. (6) and experimental result of pressure dependent melting of Al_{2}O_{3}
Figs.5 presents comparison between the model predictions of Eq. (6) and experimental result of pressure dependent melting of MgO. As shown in the figure, Tm increases as P increases. This is evidently induced by positive volume change DV during the melting.Thus, However, the low size limit of nanocrystals for the application of Pi in Eq. (6) must be considered, a crystal is characterized by its longrange order and the smallest nanocrystal should have at least a half of the atoms located within the particle [17]. Hence, the smallest Dmin is 2D_{0}. For MgO, Dmin =1.284nm in term of D_{0}=3h for nanocrystals . The curvatureinduced pressure P approximately equals to 25.39 GPa by LaplaceYoung equation Pi=4f/D associated with Eq. (2). The value of Tm on the melting curve Tm(P) must be determined. As shown in the figure, Clapeyron equation, without any adjustable parameter, is consistent with the experimental result and other theoretical prediction. The Tm(P)P relationship in Figs.2 is made by a generalization where the internal pressure of small particles is considered to be equivalent to that of the bulk one.
Figure 5. Presents comparison between the model predictions of Eq. (6) and experimental result of pressure dependent melting of MgO
Since Clapeyron equation may govern all firstorder phase transitions, the above consideration may be generalized for different phase transitions. Moreover, the success of Eq. (6) implies that the both assumptions of essentially ΔHm being a function of temperature and ΔV_{m} being a function of pressure are reasonable.
Fig.1 A comparison for TP phase diagram of bulk Si and the melting curve of Si nanocrystal among the model predictions and experimental and other theoretical results. The solid lines denote model predictions in terms of Eqs. (6) where necessary parameters used are listed in Table.1. The dash and dot lines denote other theoretical results. Other symbols denote III transition pressure at room temperature where n denotes the theoretical result, + denotes the experimental results under nonhydrostatic pressure, ♦ , ◊ and ∇ denote the experimental results under hydrostatic pressure. For the melting of Si nanocrystal, the two dash lines show the predicted results where the corresponding TmI(D) values denoted as × (1478 K) and ▲ (1371 K) are obtained from other theoretical result and Eq. (6), respectively. For comparison, the dot line gives the theoretical result for melting of Si nanocrystal.
Figure. 2 A comparison for TP phase diagram of bulk Ge among the model predictions and experimental and other theoretical results. The solid lines denote model predictions in terms of Eqs. (6) where necessary parameters used are listed in Table.1. The symbols Δ, + and Ο show experimental results. The dash lines denote other theoretical results.
Figure. 3 A comparison for TIL(P) curve of bulk CO_{2}I between the model prediction and experimental results. The solid line denotes the model prediction in terms of Eqs. (6) where necessary parameters used are: T_{mI} = 215.55 K, V_{L} = M/ρ_{L }= 12.43 cm^{3}⋅gatom^{1} with M =14.67 g⋅gatom^{1}and ρL = 1.18 g⋅cm^{3}, VI = 10.65 cm3⋅gatom^{1}, κI = 1/BI =8.06×10^{11} Pa^{1} with BI = 12.4 GPa, κL ≈ 100×10^{11} Pa^{1} as a first order approximation is equal to the κL value of CS_{2} since CS_{2} as a linear molecule has a similar behavior of CO_{2},ΔHIL = 2.649 KJ⋅gatom^{1}, γ = 0.00913 J⋅m^{2}, and fI = 0.5613 J⋅m^{2} is calculated by Eq.(6) with hI = (21/2/2)aI = 0.3575 nm where aI = 0.5056 nm for Pa3 structure and ΔS_{vib}IL =8.856 J⋅gatom^{1}⋅K^{1}. The dot lines show the experimental phase diagram.
Figure. 4 The pressuretemperature melting diagram of Al_{2}O_{3}, where the solid line shoes the model prediction of Eq. (6). The theoretical and experimental results are also plotted in the figure. The symbols ■ and ▲ denote the theoretical estimations and the experimental observations [13]. The necessary parameters in Eq. (6) are as follows: T_{m} = 2327 K [18], H_{m} = 21.76 KJ gatom^{1}[19],Sm=Hm/Tm=9.35Jgatom^{1}K^{}1, S_{m}=S_{el}+S_{pos}+S_{vib}, S_{el} is negligibly small, and S_{pos}=R[x_{A}ln(x_{A})+ x_{V}ln(x_{V})], where x_{A}= 1/(1+ΔV_{m}) and x_{V}=ΔV_{m} /(1+ΔV_{m}) are the molar fractions of the host material and vacancies, respectively [20], and ΔV_{m} is the volume difference between the crystal and corresponding fluid at Tm. As result, S_{vib}= S_{m}S_{pos} or S_{vib} » S_{m}+ R[x_{A}ln(x_{A})+ x_{V}ln(x_{V})] [21]. ΔV_{m}=(V_{L}V_{S})/V_{S}, S_{vib}=5.6 Jgatom^{1}K^{1}, h = 0.191 nm [18], g = 0.690 Jm^{2 }, k_{S} =3.86×1012 Pa^{1} is determined by k =1/B with B =289.55 GPa being the bulk modulus [21], VS = 5.14 cm3 gatom [21], V_{L} = 6.16 cm^{3}gatom^{1} [18], k_{L} =57.9×1012 Pa^{1} as a firstorder approximation under higher pressure, we assume kL » 15kS [13]. f is calculated through Eq. (2) and f = 4.5 Jm^{2}.
In summary, we have demonstrated the reliability of simple thermodynamic model in calculating the high pressure melting of solid by comparisons between obtained melting temperature and experimental melting data for Al2O3 and MgO. It is found that the model predictions are consistent with the present experimental and theoretical results. Since the Clapeyron equation may govern all firstorder phase transitions, the Clapeyron equation supplies a new way to determine the TP phase diagram of materials.
This work was supported by Science and technology project of Jilin province education department during the Twelfth Fiveyear Plan Period (No.2013232).
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