Pressure derivatives of bulk modulus for materials at extreme compression

Transcription

1 Indian Journal o ure & Applied hysics Vol. 5, October, pp ressure derivatives o bulk modulus or materials at extreme compression Singh* & A Dwivedi Department o hysics, Institute o Basic Sciences, Dr B R Ambedkar University, hari Campus, Agra, India * pramod@yahoo.com Received December ; revised 4 March ; accepted 7 June The method based on the calculus o indeterminates or demonstrating that all the physically acceptable equations o state satisy the identities or the pressure derivatives o bulk modulus o materials at extreme compression, has been developed. The speciic examples o the Birch-Murnaghan inite strain equation, the oirier-tarantola logarithmic equation, the Rydberg-Vinet potential energy equation, the eane -primed equation the Stacey reciprocal -primed equation, have been considered. Expressions or the bulk modulus its pressure derivatives have been derived reduced to the limit o ininite pressure. The expressions thus obtained are useul or urther analysis o higher derivative thermoelastic properties. eywords: ressure derivatives, Bulk modulus, Equations o state, Extreme compression behaviour Introduction For investigating high-pressure properties o materials, we need equations o state representing the relationships between pressure volume V at a given temperature T,. Some important equations o state such as the Birch-Murnaghan inite strain equation 3, the oirier-tarantola logarithmic equation 4, the Rydberg-Vinet potential energy equation 5,6, the eane -primed equation 7 the Stacey reciprocal -primed equation 8 have been widely used or various materials. Theory All o these equations reveal that pressure bulk modulus both increase rapidly with the decreasing volume. both become ininite in the limit o extreme compression (V), but their ratio remains inite such that : = () where is the value o = d d, the pressure derivative o bulk modulus at ininite pressure. It should be mentioned that represents the rate o increase o bulk modulus with the increase in pressure. Value o has been ound to decrease continuously with the increase in pressure attains a constant positive value equal to in the limit o extreme compression. The basic condition or the physical acceptability o an equation o state (EOS) is that in the limit o extreme compression (V), the pressure must approach ininity, must remain inite positive. In the limit V tends to zero, changes with volume as V so that the bulk modulus = V d dv changes as V. This is consistent with Eq. (). All equations o state with greater than zero satisy Eq. () as demonstrated by Stacey. In the present study, we demonstrate that various equations o state satisying Eq. (), which has the status o an algebraic identity, also satisy the identities or the second order third order pressure derivatives o bulk modulus at ininitely large pressures. It should be emphasized that the boundary conditions at extreme compression or ininite pressure are equally important as they are at zero pressure. These boundary conditions are to be satisied by all physically acceptable equations o state. The Brich-Murnaghan EOS, the oirier-tarantola logarithmic EOS, the generalized Rydberg-Vinet EOS, all can be represented by the ollowing common ormula: ( x ) = + () where ( is a unction o x=v/v, V is the value o volume V at =. Values o ( are dierent

2 SINGH & DWIVEDI: RESSURE DERIVATIVES OF BUL MODULUS 735 or dierent EOS. Thus or the Birch-Murnaghan ourth order EOS, =/3, 3 4 ( 4 3) Bx + ( 3) 3 Ax Cx = 4 3 Ax Bx + Cx 3 D where the constants A,B,C,D are given below: (3) 5 A = + ( 4)( 5) + (4) B = 3 + ( 4)( 3 3) + (5) 5 C = 3 + ( 4)( 3 ) + (6) 35 D = + ( 4)( 3) + (7) where, are the values o, = d d, all at =. In case o the third-order Birch-Murnaghan EOS, we have = 3 D= (Eq.7), so that: = ( 4 3) Ax ( 3) Bx Ax Bx + C (8) For the oirier-tarantola logarithmic ourth order EOS we have =, where A ln x + 3 A (ln = () ln x + A (ln + A (ln 3 A = ( ) () A = ( ) () 6 In case o the third order oirier-tarantola logarithmic EOS, we have = A =, so that: In case o the generalized Rydberg-Vinet EOS, is a material-dependent parameter related to the zeropressure parameters as ollows : 5 = (3) For this EOS, we have : x x = +η ( x ) 3 (4) η = ( 3 ) 3 + (5) It is interesting to note that in the limit o extreme compression, V or x, ( becomes zero, Eq. () reduces to Eq. (). This is true or all the equations o state considered above, demonstrating the universality o Eq. (). Expressions or pressure derivatives o bulk modulus up to third order are obtained here by dierentiating Eq. () successively with respect to. Thus, we ind: = x ( (6) x x + = + (7) = x + 3 x + x (8) where = d dx, = d dx 3 3 =. In the derivation o Eqs (6-8), d dx we have used the ollowing relationship: d d = V = x dv dx Eqs (6-8) reveal that in the limit x: () A ln = () ln x + A (ln = ()

3 736 INDIAN J URE & AL HYS, VOL 5, OCTOBER = () = () provided the ollowing conditions are satisied in the limit o extreme compression (. x = (3) x 3 x = (4) = (5) In addition to the condition ( at x. The unction ( expressed by Eqs (3), (8), (), () (4) based on dierent EOS satisy these conditions. The quantities in Eqs () to () become individually zero, but their ratios remain inite at extreme compression 3,4. To demonstrate this we divide Eq. (7) by Eq. (6) to get : x x + x x + = (6) x Now taking the limit x, using Eq. () evaluating x x with the help o Eqs (3), (8), (), () (4), we ind: = 3 = = 3 (7) (8) () These equations are the results based on dierent EOS. The Brich-Murnaghan third order as well as ourth order EOS (Eqs 3 8) yield the same result in the orm o Eq. (7). The oirier-tarantola logarithmic third order as well as ourth order EOS (Eqs ) both give the same result as given by Eq. (8). The generalized Rydberg-Vinet EOS based on Eq. (4) or ( yields Eq. (). The ratio o third order pressure derivative second pressure derivative o bulk modulus can be obtained by dividing Eq. (8) by Eq. (7), rearranging the terms in numerator (N) denominator (D) in the ollowing manner : 3 N x + 3 x + x = D x + x where N = D = + (3) (3) (3) In the limit o extreme compression or ininite pressure, we have: N D 3 = = + (33) (34) Eqs (33) (34) have been obtained by using = taking ( ) to be inite at extreme compression. In view o Eqs (33) (34), Eq. (3) becomes : N D 3 x ( + 3x ( + x = x ( + x ( x (35) Values o ( its derivatives are evaluated using the expressions or ( based on dierent EOS. The ollowing results are obtained using Eq. (35) with the help o Eqs (7-), (33) (34). = + 3 (36)

4 SINGH & DWIVEDI: RESSURE DERIVATIVES OF BUL MODULUS 737 = (37) = + (43) = + 3 (38) The Birch-Murnaghan third order ourth order EOS yield Eq. (36), the oirier-tarantola logarithmic third order ourth order EOS give Eq. (37), the generalized Rydberg-Vinet EOS yields Eq.(38). It should be mentioned that have the same units o Ga, so / is dimensionless. is also dimensionless. is multiplied by, by, so that both are dimensionless. In the extreme compression limit, Eq. (6) reduces to the ollowing: x + x + = x x (3) Using the calculus o indeterminates, we have the ollowing relationship in the limit x, 3 x + x x + 3 x + x = x x + x (4) Eq. (4) is valid under the conditions given by Eqs (3) to (5) in addition to the condition that ( tends to zero in the limit x. Eq. (4) gives the ollowing relationship with the help o Eqs (35) (3), N = D (4) Substituting Eq. (33) or N, Eq.(34) D in Eq.(4), we get: = (4) Eq. (4) is satisied not only by all such EOS which are represented by Eq. (), but also some EOS based on the expressions or. The eane EOS is written as 7 : which gives: = = (44) (45) Eqs (44) (45) satisy Eq. (4). The reciprocal -primed EOS is given below 8 : = + Eq.(46) gives: ( ) = ( + ) = (46) (47) (48) Eqs (47) (48) satisy Eq. (4). In act, all equations o state which satisy Eq. () also satisy Eq. (4). 3 Results Discussion It should be emphasized that the pressure derivatives o bulk modulus are o central importance or determining thermoelastic properties o materials at high pressures high temperatures. The ormulations presented here are related to dierent equations o state which have been used recently 5- or investigating properties o materials at high pressures. We may thus conclude that Eq. (4) has the status o an identity which can be used or determining the third-order Grüneisen parameter. Acknowledgement The authors are thankul to Dr Jai Shanker Dr B S Sharma or helpul discussion.

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