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Analysis of melting based on the diffusional force theory for alkali halides
Journal of Physics and Chemistry of Solids 63 (2002) 1935±1937
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Analysis of melting based on the diffusional force theory for alkali halides K.S. Singh* Department of Physics, R.B.S. College, Agra 282002, Uttar Pradesh, India Received 24 August 2001; accepted 9 January 2002
Abstract The theory of melting based on the concept of diffusional force given by Bosi is used for studying the melting of alkali halides. Values of thermal expansivity and the Anderson±Gruneisen parameter are used to predict the interionic separations in 16 alkali halides at melting temperatures with the help of the Anderson formula. A model for melting is developed by estimating the diffusional force from the knowledge of interionic potentials based on ultrasonic data for bulk modulus and its temperature and pressure derivatives. The model thus developed is found to yield satisfactory results in agreement with the experimental data on melting. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: D. Crystal structure; D. Diffusion; D. Thermal expansion
1. Introduction Melting of solids is a commonly observed phenomenon at high temperatures. Several attempts have been made to review various theories of melting of solids based on different criteria [1±6]. Bosi [7] has developed a model for predicting the melting temperatures of ionic solids using the concept of diffusional force. By considering the balance between the diffusional force and the coulombian force existing in ionic materials, Bosi found that much larger values of dielectric constants (1 ) were needed to obtain an agreement with the experimental values of melting temperatures of alkali halides. The values of 1 for alkali halides at higher temperatures close to melting temperatures (T ! Tm) thus required are about three to ®ve times larger than the corresponding values of 1 at room temperature [8]. The larger values of 1 at higher temperatures (T ! Tm) have been explained on the basis of the data for molar lattice energy and molar heat of fusion [7]. The direct experimental study of 1 for NaCl at 1000 K [9] also yields larger value for 1 in agreement with the value predicted by Bosi. In the present paper we develop a model for melting using the concept of diffusional force and Maxwell's thermodynamic relationship. The coulombian force considered by * Tel.: 191-562-520179; fax: 191-562-351288. E-mail address: [email protected] (K.S. Singh).
Bosi is replaced by the interionic force derived from the lattice potential energy. We use the Born±Mie inverse power form for the potential energy function using the parameters determined by ultrasonic experimental studies [10± 13]. We use the interionic distances for alkali halides at melting (rm) estimated with the help of Anderson's formula for thermal expansivity [6]. 2. Method of analysis In a liquid two microscopic spherical particles have a tendency to wander away from each other by means of three dimensional `random walk'. This is supposed to be due to the presence of a diffusional force which can be expressed as [7] dS Fr T 1 dr T where S is entropy and r is the distance between particles. The radial force Fr is responsible for an increase in r owing to the tendency towards maximum entropy. Using the Maxwell's thermodynamic relations we can write dS dP aKT 2 dV T dT V where a 1=V dV=dTP is the thermal expansivity and KT 2V dP=dVT is the isothermal bulk modulus. Eqs.
0022-3697/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(02)00180-4
1936
K.S. Singh / Journal of Physics and Chemistry of Solids 63 (2002) 1935±1937
Table 1 Values of thermal expansivity a (10 24 K 21), isothermal bulk modulus KT (kbar) from Ref. [13], Anderson±Gruneisen parameter d T derived from ultrasonic data [10±13], experimental values of melting temperature Tm in K [6], interionic separation rm (10 28 cm) calculated from Eq. (8), df=dr (10 24 dyn) at r rm calculated from Eq. (9) and melting temperature Tm (K) calculated from Eq. (5) Cyrstal
a
KT
dT
Tm Exp. [6]
rm
(df /dr) at r rm
Tm Eq. (5)
LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI
0.999 1.320 1.500 1.800 0.960 1.190 1.260 1.370 1.020 1.110 1.160 1.230 0.940 1.030 1.080 1.230
665.1 296.8 235.2 172.6 464.8 236.8 194.7 148.7 302.2 173.5 146.4 115.1 266.8 155.8 132.4 104.9
6.15 6.69 7.01 7.32 5.77 5.95 6.23 6.46 6.26 6.29 5.88 5.83 6.86 6.73 6.64 6.53
1115 887 820 723 1261 1074 1028 924 1119 1049 1003 959 1048 988 955 915
2.090 2.665 2.856 3.120 2.421 2.948 3.128 3.373 2.781 3.272 3.424 3.665 2.918 3.398 3.556 3.801
2.064 1.580 1.524 1.404 2.029 1.602 1.521 1.319 1.650 1.321 1.188 1.075 1.400 1.131 1.058 1.035
1185 946 882 774 1293 1090 1056 948 1153 1068 994 942 1093 1018 975 925
(1) and (2) yield 2
Fr 6r T aKT
3
We have used the relationship V 2r 3 for crystals with NaCl structure in deriving Eq. (3). At melting, Bosi has considered the equivalence between the diffusional force and the attractive coulombian force between the ions in a medium of dielectric constant 1 to obtain the formula for melting temperature. In the present study we consider that the melting of solids takes place when the diffusional force is balanced by the interionic force derived from the lattice potential energy f . Thus at melting we can write df 4 Fr dr rrm Eqs. (3) and (4) then lead to the following relationship for the melting temperature 1 df 5 Tm 2 6rm aKT dr rrm For determining the values of rm we use the thermal expansivity equation obtained by Anderson [6] which is expressed as follows
a a0 1 2 a0 dT T 2 T0 21
6
where a is thermal expansivity and d T is the Anderson± Gruneisen parameter, a 0 is the value of thermal expansivity a at T T0 ; the initial temperature. It should be mentioned that Eq. (6) is based on the assumptions that d T and the product aKT do not change with temperature along an isobar [6,14]. These assumptions hold approximately well at higher temperatures above the Debye temperature u D.
Therefore Eq. (6) is valid only at temperatures T $ uD : Now using the de®nition a 1=V dV=dTP in Eq. (6) and integrating it we get 1 2d
V V0 1 2 a0 dT T 2 T0
T
7
where V0 is the volume V at T T0 : Using the relationship V=V0 r=r0 3 in Eq. (6) we obtain r r0 1 2 a0 dT T 2 T0
1 2 3d
T
8
It has been found [15,16] that Eq. (8) yields satisfactory results for interionic separation r versus temperature in case of alkali halides starting from room temperature T0 300 K up to their melting temperatures. It is mainly because the values of u D for most of the alkali halides are close to the room temperature. We have therefore used Eq. (8) to calculate r rm ; the interionic separation at T Tm ; the melting temperature. For estimating the values of df=dr at r rm we use the following expression for the lattice potential energy f based on the Born±Mie potential function 2n Me2 r 1B f2 9 r r0 where M is the electrostatic Madelung constant, e is electronic charge, r is interionic separation, r0 is the value of r at T T0 : The ®rst term on the right of Eq. (9) represents the Madelung electrostatic energy and the last term is the Pauli's overlap repulsive energy expressed by the Born± Mie inverse power form. The parameters B and n are taken those derived from the ultrasonic experimental data [10±13].
K.S. Singh / Journal of Physics and Chemistry of Solids 63 (2002) 1935±1937
3. Results and discussions The results along with input data for various quantities used in calculations are given in Table 1. The values of rm calculated from the thermal expansivity Eq. (8) are about four percent larger than the corresponding values of interionic separation r0 at room temperature. The computer simulation studies on melting of alkali halides [4,17] also reveal an increase of interionic separations by the same amount. We have assessed the validity of Eq. (5) using the approximation that the product aKT remains constant with the variation in temperature. Thus a KT at T ! Tm is taken to be the same as that at T T0 ; the room temperature. The approximate validity of this assumption has been discussed by earlier investigators [6,18]. For estimating the values of df=dr at r rm we have used the potential energy expression given in the form of Eq. (9). Using the values of various quantities give in Table 1 we have obtained the values of Tm with the help of Eq. (5). The values of Tm thus obtained are found to present fairly good agreement with the experimental values of melting temperatures. It should be mentioned that the use of another potential form such as the Born±Mayer exponential form in place of the Born±Mie inverse power form yields the similar results for df=dr and Tm for all the alkali halides under study. Acknowledgements Author is indebted to Prof. Jai Shanker, Institute of Basic Sciences, Agra for valuable discussions. The Referees
1937
comments have been found useful in revising the manuscript.
References [1] A.R. Ubbelohde, The Molten State of Matter: Melting and Crystal Structure, Wiley, New York, 1978. [2] R.M.J. Cotterill, J. Cryst. Growth 48 (1980) 582. [3] L.L. Boyer, Phase Transitions 5 (1985) 1. [4] J. Shanker, M. Kumar, Phys. Status Solidi B 158 (1990) 11. [5] J.P. Poirier, Introduction to the Physics of the Earth's Interior, Cambridge University Press, New York, 1991. [6] O.L. Anderson, Equations of State of Solids for Geophysics and Ceramic Sceince, Oxford University Press, New York, 1995. [7] L. Bosi, Phys. Status Solidi A 101 (1987) K111. [8] R.P. Lowndes, D.H. Martin, J. Phys. C 2 (1969) 1595. [9] K.V. Rao, A. Smakula, J. Appl. Phys. 36 (1965) 3953. [10] R.W. Roberts, C.S. Smith, J. Phys. Chem. Solids 31 (1970) 619. [11] R.W. Roberts, C.S. Smith, J. Phys. Chem. Solids 31 (1970) 2397. [12] K.O. McLean, C.S. Smith, J. Phys. Chem. Solids 33 (1972) 279. [13] C.S. Smith, L.S. Cain, J. Phys. Chem. Solids 36 (1975) 205. [14] S.S. Kushwah, J. Shanker, Indian J. Pure Appl. Phys. 36 (1998) 46. [15] R.K. Pandey, Indian J. Phys. 72A (2) (1998) 125. [16] R.K. Pandey, J. Phys. Chem. Solids 59 (1998) 1157. [17] R.W. Hockney, Contemp. Phys. 20 (1979) 149. [18] J.L. Tallon, J. Phys. Chem. Solids 41 (1980) 837.
www.elsevier.com/locate/jpcs
Analysis of melting based on the diffusional force theory for alkali halides K.S. Singh* Department of Physics, R.B.S. College, Agra 282002, Uttar Pradesh, India Received 24 August 2001; accepted 9 January 2002
Abstract The theory of melting based on the concept of diffusional force given by Bosi is used for studying the melting of alkali halides. Values of thermal expansivity and the Anderson±Gruneisen parameter are used to predict the interionic separations in 16 alkali halides at melting temperatures with the help of the Anderson formula. A model for melting is developed by estimating the diffusional force from the knowledge of interionic potentials based on ultrasonic data for bulk modulus and its temperature and pressure derivatives. The model thus developed is found to yield satisfactory results in agreement with the experimental data on melting. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: D. Crystal structure; D. Diffusion; D. Thermal expansion
1. Introduction Melting of solids is a commonly observed phenomenon at high temperatures. Several attempts have been made to review various theories of melting of solids based on different criteria [1±6]. Bosi [7] has developed a model for predicting the melting temperatures of ionic solids using the concept of diffusional force. By considering the balance between the diffusional force and the coulombian force existing in ionic materials, Bosi found that much larger values of dielectric constants (1 ) were needed to obtain an agreement with the experimental values of melting temperatures of alkali halides. The values of 1 for alkali halides at higher temperatures close to melting temperatures (T ! Tm) thus required are about three to ®ve times larger than the corresponding values of 1 at room temperature [8]. The larger values of 1 at higher temperatures (T ! Tm) have been explained on the basis of the data for molar lattice energy and molar heat of fusion [7]. The direct experimental study of 1 for NaCl at 1000 K [9] also yields larger value for 1 in agreement with the value predicted by Bosi. In the present paper we develop a model for melting using the concept of diffusional force and Maxwell's thermodynamic relationship. The coulombian force considered by * Tel.: 191-562-520179; fax: 191-562-351288. E-mail address: [email protected] (K.S. Singh).
Bosi is replaced by the interionic force derived from the lattice potential energy. We use the Born±Mie inverse power form for the potential energy function using the parameters determined by ultrasonic experimental studies [10± 13]. We use the interionic distances for alkali halides at melting (rm) estimated with the help of Anderson's formula for thermal expansivity [6]. 2. Method of analysis In a liquid two microscopic spherical particles have a tendency to wander away from each other by means of three dimensional `random walk'. This is supposed to be due to the presence of a diffusional force which can be expressed as [7] dS Fr T 1 dr T where S is entropy and r is the distance between particles. The radial force Fr is responsible for an increase in r owing to the tendency towards maximum entropy. Using the Maxwell's thermodynamic relations we can write dS dP aKT 2 dV T dT V where a 1=V dV=dTP is the thermal expansivity and KT 2V dP=dVT is the isothermal bulk modulus. Eqs.
0022-3697/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(02)00180-4
1936
K.S. Singh / Journal of Physics and Chemistry of Solids 63 (2002) 1935±1937
Table 1 Values of thermal expansivity a (10 24 K 21), isothermal bulk modulus KT (kbar) from Ref. [13], Anderson±Gruneisen parameter d T derived from ultrasonic data [10±13], experimental values of melting temperature Tm in K [6], interionic separation rm (10 28 cm) calculated from Eq. (8), df=dr (10 24 dyn) at r rm calculated from Eq. (9) and melting temperature Tm (K) calculated from Eq. (5) Cyrstal
a
KT
dT
Tm Exp. [6]
rm
(df /dr) at r rm
Tm Eq. (5)
LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI RbF RbCl RbBr RbI
0.999 1.320 1.500 1.800 0.960 1.190 1.260 1.370 1.020 1.110 1.160 1.230 0.940 1.030 1.080 1.230
665.1 296.8 235.2 172.6 464.8 236.8 194.7 148.7 302.2 173.5 146.4 115.1 266.8 155.8 132.4 104.9
6.15 6.69 7.01 7.32 5.77 5.95 6.23 6.46 6.26 6.29 5.88 5.83 6.86 6.73 6.64 6.53
1115 887 820 723 1261 1074 1028 924 1119 1049 1003 959 1048 988 955 915
2.090 2.665 2.856 3.120 2.421 2.948 3.128 3.373 2.781 3.272 3.424 3.665 2.918 3.398 3.556 3.801
2.064 1.580 1.524 1.404 2.029 1.602 1.521 1.319 1.650 1.321 1.188 1.075 1.400 1.131 1.058 1.035
1185 946 882 774 1293 1090 1056 948 1153 1068 994 942 1093 1018 975 925
(1) and (2) yield 2
Fr 6r T aKT
3
We have used the relationship V 2r 3 for crystals with NaCl structure in deriving Eq. (3). At melting, Bosi has considered the equivalence between the diffusional force and the attractive coulombian force between the ions in a medium of dielectric constant 1 to obtain the formula for melting temperature. In the present study we consider that the melting of solids takes place when the diffusional force is balanced by the interionic force derived from the lattice potential energy f . Thus at melting we can write df 4 Fr dr rrm Eqs. (3) and (4) then lead to the following relationship for the melting temperature 1 df 5 Tm 2 6rm aKT dr rrm For determining the values of rm we use the thermal expansivity equation obtained by Anderson [6] which is expressed as follows
a a0 1 2 a0 dT T 2 T0 21
6
where a is thermal expansivity and d T is the Anderson± Gruneisen parameter, a 0 is the value of thermal expansivity a at T T0 ; the initial temperature. It should be mentioned that Eq. (6) is based on the assumptions that d T and the product aKT do not change with temperature along an isobar [6,14]. These assumptions hold approximately well at higher temperatures above the Debye temperature u D.
Therefore Eq. (6) is valid only at temperatures T $ uD : Now using the de®nition a 1=V dV=dTP in Eq. (6) and integrating it we get 1 2d
V V0 1 2 a0 dT T 2 T0
T
7
where V0 is the volume V at T T0 : Using the relationship V=V0 r=r0 3 in Eq. (6) we obtain r r0 1 2 a0 dT T 2 T0
1 2 3d
T
8
It has been found [15,16] that Eq. (8) yields satisfactory results for interionic separation r versus temperature in case of alkali halides starting from room temperature T0 300 K up to their melting temperatures. It is mainly because the values of u D for most of the alkali halides are close to the room temperature. We have therefore used Eq. (8) to calculate r rm ; the interionic separation at T Tm ; the melting temperature. For estimating the values of df=dr at r rm we use the following expression for the lattice potential energy f based on the Born±Mie potential function 2n Me2 r 1B f2 9 r r0 where M is the electrostatic Madelung constant, e is electronic charge, r is interionic separation, r0 is the value of r at T T0 : The ®rst term on the right of Eq. (9) represents the Madelung electrostatic energy and the last term is the Pauli's overlap repulsive energy expressed by the Born± Mie inverse power form. The parameters B and n are taken those derived from the ultrasonic experimental data [10±13].
K.S. Singh / Journal of Physics and Chemistry of Solids 63 (2002) 1935±1937
3. Results and discussions The results along with input data for various quantities used in calculations are given in Table 1. The values of rm calculated from the thermal expansivity Eq. (8) are about four percent larger than the corresponding values of interionic separation r0 at room temperature. The computer simulation studies on melting of alkali halides [4,17] also reveal an increase of interionic separations by the same amount. We have assessed the validity of Eq. (5) using the approximation that the product aKT remains constant with the variation in temperature. Thus a KT at T ! Tm is taken to be the same as that at T T0 ; the room temperature. The approximate validity of this assumption has been discussed by earlier investigators [6,18]. For estimating the values of df=dr at r rm we have used the potential energy expression given in the form of Eq. (9). Using the values of various quantities give in Table 1 we have obtained the values of Tm with the help of Eq. (5). The values of Tm thus obtained are found to present fairly good agreement with the experimental values of melting temperatures. It should be mentioned that the use of another potential form such as the Born±Mayer exponential form in place of the Born±Mie inverse power form yields the similar results for df=dr and Tm for all the alkali halides under study. Acknowledgements Author is indebted to Prof. Jai Shanker, Institute of Basic Sciences, Agra for valuable discussions. The Referees
1937
comments have been found useful in revising the manuscript.
References [1] A.R. Ubbelohde, The Molten State of Matter: Melting and Crystal Structure, Wiley, New York, 1978. [2] R.M.J. Cotterill, J. Cryst. Growth 48 (1980) 582. [3] L.L. Boyer, Phase Transitions 5 (1985) 1. [4] J. Shanker, M. Kumar, Phys. Status Solidi B 158 (1990) 11. [5] J.P. Poirier, Introduction to the Physics of the Earth's Interior, Cambridge University Press, New York, 1991. [6] O.L. Anderson, Equations of State of Solids for Geophysics and Ceramic Sceince, Oxford University Press, New York, 1995. [7] L. Bosi, Phys. Status Solidi A 101 (1987) K111. [8] R.P. Lowndes, D.H. Martin, J. Phys. C 2 (1969) 1595. [9] K.V. Rao, A. Smakula, J. Appl. Phys. 36 (1965) 3953. [10] R.W. Roberts, C.S. Smith, J. Phys. Chem. Solids 31 (1970) 619. [11] R.W. Roberts, C.S. Smith, J. Phys. Chem. Solids 31 (1970) 2397. [12] K.O. McLean, C.S. Smith, J. Phys. Chem. Solids 33 (1972) 279. [13] C.S. Smith, L.S. Cain, J. Phys. Chem. Solids 36 (1975) 205. [14] S.S. Kushwah, J. Shanker, Indian J. Pure Appl. Phys. 36 (1998) 46. [15] R.K. Pandey, Indian J. Phys. 72A (2) (1998) 125. [16] R.K. Pandey, J. Phys. Chem. Solids 59 (1998) 1157. [17] R.W. Hockney, Contemp. Phys. 20 (1979) 149. [18] J.L. Tallon, J. Phys. Chem. Solids 41 (1980) 837.