Analysis of thermal expansivity of solids at extreme compression

Thermodynamics of solids in the limit of infinite pressure formulated by Stacey reveals that the thermal expansivity (alpha) of solids tends to zero at infinite pressure. The earlier models for the volume dependence of thermal expansivity do not satisfy the infinite pressure behaviour of thermal expansivity. The expressions for the volume dependence of the isothermal AndersonGrüneisen parameter (delta T) considered in the derivation of earlier formulations for alpha (V) have been found to be inadequate. A formulation for the volume dependence of delta T is presented here which is similar to the model due to Burakovsky and Preston for the volume dependence of the Grüneisen parameter. The new formulation for alpha (V) reveals that delta T infinity must be greater than zero for satisfying the thermodynamic result according to which alpha tends to zero at infinite pressure. It is found that our model fits well the experimental data on thermal expansivity alpha (V) for hcp iron corresponding to a wide range of pressures (0–360 GPa).


Introduction
The Anderson-Grüneisen parameter is an important physical quantity for understanding the thermoelastic properties of solids at high pressures and high temperatures [1].In the present paper we study the volume dependence of the Anderson-Grüneisen parameter and derive an improved formula for thermal expansivity of solids which is found to be consistent with the thermodynamic constraint at extreme compression.The isothermal Anderson-Grüneisen δ T is defined as [1] where α is the thermal expansivity or volume thermal expansion coefficient and K T , the isothermal bulk modulus Using the thermodynamic identity in equation (1) we get Equation ( 5) can be integrated to obtain α as a function of volume V , provided we know the dependence of the Anderson-Grüneisen parameter δ T on V .It has been found by Anderson and Isaak [2] that δ T depends on V in the following manner where δ T = δ 0 T at V = V 0 , the reference state (P = 0).k is a dimensionless thermoelastic parameter defined as [2] An alternative form for δ T (V ) has been considered by Chopelas and Boehler [3] as follows Equation ( 8) is the basis for an equation of state (EOS) formulated by Kumar [4,5] which turns out to be the same as the usual Tait EOS [6].Equation ( 6) with k = 1 is also used for developing an EOS and for investigating the thermoelastic properties of solids [7][8][9][10].
The Chopelas-Boehler formulation (equation 8) when used in equation ( 5) gives the following relationship [7,8] where α 0 is the thermal expansivity at P = 0. On the other hand, when the Anderson-Isaak formulation (equation 6) is used in equation ( 5), we get Anderson et al. [8] have made a comparative study of equations ( 9) and (10).It should be emphasized here that the Chopelas-Boehler relationship (equation 9) and the Anderson-Isaak formulation (equation 10) are not consistent with the infinite pressure behaviour based on thermodynamics.The thermal expansivity α should tend to zero at extreme compression, V approaching zero [11,12].But equation ( 9) predicts α → ∞, and equation (10) gives a finite value of α at extreme compression (V → 0, P → ∞).We therefore present a revised formulation for α(V).

Formulation based on the Burakovsky-Preston model
Burakovsky and Preston [13] have recently formulated a model for the volume dependence of the Grüneisen parameter γ based on the following expression where γ ∞ is the value of γ at extreme compression V → 0. The value of γ ∞ = 1/2 or 2/3 based on the Thomas-Fermi model [13,14].γ ∞ is treated as a universal constant, i.e., the same for all materials, whereas a, b and n(> 1) are material-dependent parameters.It was found [15] that the Burakovsky-Preston model equation ( 11) satisfies the thermodynamic constraints γ → γ ∞ , q = (d ln γ/d ln V ) T → 0, and λ = (d ln q/d ln V ) T → λ ∞ , where γ ∞ and λ ∞ are finite positive values.
It is appropriate to consider that the Anderson-Grüneisen parameter δ T follows a volume dependence similar to equation (11).A similarity for the volume dependence of γ and δ T was pointed out earlier by Tallon [16].We can thus write where δ T∞ represents the value of δ T at V → 0. In analogy with equation ( 11), δ T∞ can be considered as a universal constant and c 1 , c 2 and m as constants for a given material.When we use equation (12) in equation ( 5), and then on integrating we find the following expression for the thermal expansivity α, On comparing equation ( 13) with equations ( 9) and ( 10) we note that δ T∞ = −1 in the Chopelas-Boehler formulation (equation 9), and δ T∞ = 0 in the Anderson-Isaak formulation.The infinite pressure condition for α based on thermodynamics (α → 0 at V → 0) is satisfied only when δ T∞ is greater than zero.This is a result similar to that (δ S∞ > 0) obtained by Stacey and Davis [11].Here δ S is the adiabatic Anderson-Grüneisen parameter related to the temperature derivative of adiabatic bulk modulus K S [1] It should be mentioned that K S and K T are related by the thermodynamic identity Stacey and Davis [11] emphasized that isothermal and adiabatic properties become identical in the limit of infinite pressure.So, when δ S∞ > 0, we should also have δ T∞ > 0.
An independent expression for δ T∞ can be obtained from the thermodynamic identity [1] δ which gives at P → ∞ To know more about δ T∞ , we use the following thermodynamic identities [1,12] ∂ ln(αK T ) and

Results and discussions
The Grüneisen parameter is related to the thermal and elastic properties of the material by the formula where C V and C P are the specific heats at constant volume and constant pressure, respectively.It follows from equation (20) that α should decrease with a decreasing volume or an increasing pressure since gamma decreases and bulk modulus increases faster than 1/V .It is desirable to judge the suitability of equation ( 13) for α (V) which is based on a model for δ T (V) (equation 12) similar to that (equation 11) formulated by Burakovsky and Preston [13] for γ(V).
We have in all five parameters viz.α 0 , δ T∞ , c 1 , c 2 and m in equation (13).Equation ( 12) at V = V 0 gives We take δ T∞ = 2/3 based on equation ( 17) using K ∞ = 5/3 derived from the Thomas-Fermi model [13,14], and neglecting the last term in equation (17) for the volume derivative of C V since it is very small at high pressure and high temperature [1,11].The value 2/3 for δ T∞ can also be supported from the identities ( 18) and (19).Equation ( 18) can be integrated along an isotherm whereas equation ( 19) can be integrated along an adiabat.We integrate equation ( 19) between the limits Since q = (d ln γ/d ln V ) T becomes zero at V → 0 [11], it is found from equation ( 22) that the product αK T tends to infinity at P → ∞ or V → 0. Equation ( 18) was used by Anderson [17] and others [18,19] to discuss the nature of variation of αK T with volume.According to equation ( 18), αK T → ∞ only when δ T − K T is negative at P → ∞.This reveals that δ T∞ must be less than K ∞ .Thus the value of δ T∞ should be constrained as follows: The value of 2/3 for δ T∞ taken in the present study satisfies the above constraint.δ T∞ should be considered as a universal constant in the same sense as γ ∞ and K ∞ .The other parameters δ 0 T , α 0 , c 1 , c 2 and m depend on the material chosen for the study.To judge the suitability of equation ( 13) for α(V ) we use the experimental data for hcp iron which was well studied for a wide range of pressures [20,21].For hcp iron we take δ 0 T = 5.32 and α 0 = 7.83 • 10 −5 K −1 from Isaak and Anderson [20].Using δ 0 T = 5.32 and δ T∞ = 2/3 in equation ( 21) we have The parameters c 1 , c 2 and m are now fitted to experimental data [20,21] for hcp iron in the pressure range 0-360 GPa given in table 1.The fitted parameters are found to have the values c 1 = 3.60, c 2 = 1.05, and m = 1.5.With the help of these parameters, values of α(V ) are determined using equation ( 13) and then compared with the experimental values reported by Isaak and Anderson [20] in figure 1.We find that our model fits the experimental data well particularly in view of the fact that the experimental data become increasingly imprecise as pressure increases due to non-hydrostaticity, comparably low quality of pressure standards, recrystallization etc. Table 1.Experimental data for the thermal expansivity α (P, V) in 10 −5 K −1 for hcp iron [20,21].

Conclusions
It was emphasized by Tallon [16] that the volume dependence of δ T should be similar to that of γ.The volume dependence of δ T is required for investigating the variation of thermal expansivity α with volume V .We have presented a formulation for α(V ) (equation 13) using a model for δ T (V ) (equation 12) which is similar to the model for γ(V ) (equation 11) originally due to Burakovsky and Preston [13].In both the models (equation (11) and equation ( 12)) the first term on the right is a universal constant (γ ∞ or δ T∞ ), and the remaining two terms depend on the volume representing the concave up and concave down behaviour [13].The experimental data for thermal expansivity of hcp iron [20,21] for a wide pressure range up to 360 GPa have been fitted well with the help of equation ( 13) using the reasonable values of parameters, m > 1 and c 1 /c 2 = 3.4, in agreement with the original model due to Burakovsky and Preston [13].