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Optimum lattice parameters for cation transport
Volume 56A, number 3
PHYSICS LETTERS
OPTIMUM
LATFICE
E.
PARAMETERS
22 March 1976
FOR CATION
TRANSPORT
RUCKENSTEIN and D.B. DADYBURJOR
Faculty of Engineering and Applied Sciences, State University of New York at Buffalo, Buffalo, NY 14214, USA Received 26 January 1976 The distance parameter of the cs-silver-like lattices is computed to minimize the calculated activation energy, maximizing the transport of Ag’.
The existence of abnormally high coefficients of conduction and diffusion in certain ionic crystals was related [1] to the ability of the mobile species (usually cations) to move as through a liquid sublattice. Flygare and Huggins [2] calculated the effect of the ionic radius of the mobile species on the activation energy of the transport process. They showed that, over a small range of values of the cationic radius, the activation energy for transport in a lattice of the a Ag! structure changed considerably and even went through a minimum. The value of the cationic radius at this minimum activation energy (corresponding to a maximum in the rate of the transport process) was that of Ag~.The inverse problem, perhaps of equal interest, is to design a lattice that will maximize the diffusion rate of a given cation, say Agt This will help in selecting and designing materials to be used as solid electrolytes in batteries and fuel cells. At the present moment, investigation of such compounds is done on a case by case basis. The structure of double salts and organometallic halides of silver are well known [3—5]. However, for simplicity in this work we consider only compounds similar in structure to a-Ag!.
io~ I
o
91
ac
w
Ix
O~
490
498
5.00
505
Fig. 1. Interaction activation energy as a function of lattice distance for mono-divalent cs-silver compounds. Other lattice parameters are linearly interpolated and extrapolated from sulfide and selenide values. ~E for cs-AgI is an order of magnitude greater because of the monovalence of the anion and the
larger anionic radius. Table 1 Anionic parameters for cs-silver compounds Anionic radius
r
Edge of bcc Anionic polarizability unit cell
1 [6]
c1 [7]
~ 3[81
A
A
A
1.84 1.98 2.20
4.89 4.99 5.044
3.06 4.29 5.58
The structure of a-Ag! has been given in detail [1] Briefly, the lattice is bcc in the anions, all of which are assumed fixed. The Ag+ ions, much smaller in size are free to move in the tunnels of the anionic lattice. The total interaction cation and an anionic lattice energy doublybetween infinite ainmobile all three directions is given by the anion-cation shell overlap repul-
Ag 2S Ag2Se AgI
________
________________________
sion energy and the interaction between the cation and the dipole induced by it on an anion, both terms summed over all anions. Coulomb attractions between 233
Volume 56A, number 3
PHYSICS LETTERS
anion and cation are negated by corresponding cationcation repulsions; hence, they are not considered. Also ignored are the anion-induced dipoles on the cation and the multipole interactions. If the cation is assumed to be moving in, say, the 1 direction, for any given value of lit will be found at a position of minimum interaction energy. Plotting these values of minimuminteraction energy as a function of l gives the energy path of the cation. Flygare and Huggins [2] have obtained a series of these for various values of the cationic radius in the compound a-Ag!. The activation energy is defined as the difference between the peak and the valley of this path. In table 1 are given the anionic parameters of vanous silver compounds similar in structure to a-Ag!. It is assumed that the sulfide and the selenide combine m all proportions and that the mixture is homogeneous with parameters varying smoothly from those of one pure compound to those of the other. Energy paths were obtained for a series of these hypothetical compounds. The corresponding activation energies are shown in fig. 1 as a function of the lattice parameter c1. There is a maximum as well as a minimum in the
234
22 March 1976
curve. The maximum activation energy occurs in a mixture of approximately 20% selenide. The minimum activation energy, or maximum diffusion rate, occurs in a lattice slightly larger than that of the selenide. The drop in activation energy from the sulfide to the selenide is consistent with values reported in the literature for tracer diffusion and conductivity experiments. An expanded version of our results will appear later.
References [1] L.W. Strock, Z. Phys. Chem. B25 (1934) 441. [2] W.H. Flygare and R.A. Huggins, J. Phys. Chem. Solids 34 (1973) 1199. [3] S. Hoshino, J. Phys. Soc. Japan 10 (1955) 197. [4] S. Geller, Nature 176 (1972) 1016. [5] 5. Geller and M.D. Lind, J. Chem. Phys. 52 (1970) 5854. [6] R.D. Shannon and C.T. Prewitt, Acta Cryst. B25 (1969) 925. [71 J.D.H. Donney, ed., Crystal Data Determinative Tables, 2nd Ed. ACA Monograph No. 5, American Crystallographic Association (1963).
[81 R.
Rich, Periodic correlations (W.A. Benjamin Inc., New
York, 1965) pp. 60-61.
PHYSICS LETTERS
OPTIMUM
LATFICE
E.
PARAMETERS
22 March 1976
FOR CATION
TRANSPORT
RUCKENSTEIN and D.B. DADYBURJOR
Faculty of Engineering and Applied Sciences, State University of New York at Buffalo, Buffalo, NY 14214, USA Received 26 January 1976 The distance parameter of the cs-silver-like lattices is computed to minimize the calculated activation energy, maximizing the transport of Ag’.
The existence of abnormally high coefficients of conduction and diffusion in certain ionic crystals was related [1] to the ability of the mobile species (usually cations) to move as through a liquid sublattice. Flygare and Huggins [2] calculated the effect of the ionic radius of the mobile species on the activation energy of the transport process. They showed that, over a small range of values of the cationic radius, the activation energy for transport in a lattice of the a Ag! structure changed considerably and even went through a minimum. The value of the cationic radius at this minimum activation energy (corresponding to a maximum in the rate of the transport process) was that of Ag~.The inverse problem, perhaps of equal interest, is to design a lattice that will maximize the diffusion rate of a given cation, say Agt This will help in selecting and designing materials to be used as solid electrolytes in batteries and fuel cells. At the present moment, investigation of such compounds is done on a case by case basis. The structure of double salts and organometallic halides of silver are well known [3—5]. However, for simplicity in this work we consider only compounds similar in structure to a-Ag!.
io~ I
o
91
ac
w
Ix
O~
490
498
5.00
505
Fig. 1. Interaction activation energy as a function of lattice distance for mono-divalent cs-silver compounds. Other lattice parameters are linearly interpolated and extrapolated from sulfide and selenide values. ~E for cs-AgI is an order of magnitude greater because of the monovalence of the anion and the
larger anionic radius. Table 1 Anionic parameters for cs-silver compounds Anionic radius
r
Edge of bcc Anionic polarizability unit cell
1 [6]
c1 [7]
~ 3[81
A
A
A
1.84 1.98 2.20
4.89 4.99 5.044
3.06 4.29 5.58
The structure of a-Ag! has been given in detail [1] Briefly, the lattice is bcc in the anions, all of which are assumed fixed. The Ag+ ions, much smaller in size are free to move in the tunnels of the anionic lattice. The total interaction cation and an anionic lattice energy doublybetween infinite ainmobile all three directions is given by the anion-cation shell overlap repul-
Ag 2S Ag2Se AgI
________
________________________
sion energy and the interaction between the cation and the dipole induced by it on an anion, both terms summed over all anions. Coulomb attractions between 233
Volume 56A, number 3
PHYSICS LETTERS
anion and cation are negated by corresponding cationcation repulsions; hence, they are not considered. Also ignored are the anion-induced dipoles on the cation and the multipole interactions. If the cation is assumed to be moving in, say, the 1 direction, for any given value of lit will be found at a position of minimum interaction energy. Plotting these values of minimuminteraction energy as a function of l gives the energy path of the cation. Flygare and Huggins [2] have obtained a series of these for various values of the cationic radius in the compound a-Ag!. The activation energy is defined as the difference between the peak and the valley of this path. In table 1 are given the anionic parameters of vanous silver compounds similar in structure to a-Ag!. It is assumed that the sulfide and the selenide combine m all proportions and that the mixture is homogeneous with parameters varying smoothly from those of one pure compound to those of the other. Energy paths were obtained for a series of these hypothetical compounds. The corresponding activation energies are shown in fig. 1 as a function of the lattice parameter c1. There is a maximum as well as a minimum in the
234
22 March 1976
curve. The maximum activation energy occurs in a mixture of approximately 20% selenide. The minimum activation energy, or maximum diffusion rate, occurs in a lattice slightly larger than that of the selenide. The drop in activation energy from the sulfide to the selenide is consistent with values reported in the literature for tracer diffusion and conductivity experiments. An expanded version of our results will appear later.
References [1] L.W. Strock, Z. Phys. Chem. B25 (1934) 441. [2] W.H. Flygare and R.A. Huggins, J. Phys. Chem. Solids 34 (1973) 1199. [3] S. Hoshino, J. Phys. Soc. Japan 10 (1955) 197. [4] S. Geller, Nature 176 (1972) 1016. [5] 5. Geller and M.D. Lind, J. Chem. Phys. 52 (1970) 5854. [6] R.D. Shannon and C.T. Prewitt, Acta Cryst. B25 (1969) 925. [71 J.D.H. Donney, ed., Crystal Data Determinative Tables, 2nd Ed. ACA Monograph No. 5, American Crystallographic Association (1963).
[81 R.
Rich, Periodic correlations (W.A. Benjamin Inc., New
York, 1965) pp. 60-61.