- Email: [email protected]
The relative surface energy of hexagonal close-packed crystals
Materials Chemistry and Physics 60 (1999) 70±78
The relative surface energy of hexagonal close-packed crystals Z.A. Matysina State University, Dnepropetrovsk 320000, Ukraine Received 19 January 1999; accepted 12 February 1999
Abstract Surface energies of hexagonal close-packed (hcp) crystals for three families of lattice site planes have been computed and their angular correlations have been determined. The broken bond method and the model of stepped microprojections of crystal faces were employed. The atomic interactions were taken into account in two coordinate spheres. Relative surface energies have been computed for Be, Cd, Co, Hf, Mg, Ti, Zn, Zr. The theoretical results are compared with experimental data. # 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Surface energy; Hexagonal close-packed crystal; Broken-bond model
1. Introduction Atoms on the crystal surface have a smaller coordination than atoms in bulk and their energy states differ. It can simulate a redistribution of surface atoms as well as reconstruction of a solid surface. The surface energy of crystals is determined by surface-producing work and it depends upon a character of distribution and redistribution of the atoms on the surface [1±10]. The evaluation of surface energy and its connection with processes in solids can contribute to more understanding of the nature of many surface effects and helps to prede®ne their in¯uence on bulk properties of materials. The many works [1±30] have been devoted to investigations of surface processes in solids. Many theoretical evaluations and experiments have been carried out to determine crystal surface energies [31±44]. There are some positive results in experimental and theoretical approach to this question. In this paper we describe some results of surface energy evaluation of various lattice sites planes in crystals. We also describe some results of theoretical determination on relative con®gurational surface energy for three families p p of lattice site planes in perfect crystals (c/a 2 2= 3) with hcp lattice of A3 (Mg type) and for one of lattice site planes in real crystal (c/a 6 p family p 2 2= 3). The angular dependence of surface energy has been determined.
The results have ben obtained by making use of the atoms pairwise interaction and by broken-bond model of crystal surfaces and compared with experimental data.
2. The surface energy of a perfect crystal The speci®c surface energy of a material in a condensed state can be determined as a work of producing a surface unit area which, according to Gibbs±Helmholtz's equation, consist of con®gurative and of entropy components. The second component has been experimentally proved to be comparatively small which account for its omission from the present treatment [4]. The speci®c con®gurational surface energy hkil of crystal face (hkil) can be calculated as the sum of twin interaction energy of atoms on a unit surface area and of those which are out off by a free face (really absent). The works [27,28] have proved that the value hkil is proportional to the number of broken bonds on the surface unit area hkil Zhkil :
(1)
We shall be especially interested in the values of the relative speci®c surface energy hkil
hkil Zhkil ; 0 Z0
0254-0584/99/$ ± see front matter # 1999 Published by Elsevier Science S.A. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 9 9 ) 0 0 0 5 0 - 4
(2)
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
71
and radii. Z1 12;
Z2 6;
r1 a;
p r 2 a 2;
(4)
The number of broken bonds Zhkil will be as follows: n Zhkil nmg ; (5) st where n is the effective number of broken bonds for one step of a lattice site plane of one atomic width, m the number of steps per unit of length, s equal to 1/m is the length of a step, t equal to 1/g is the width of a step which is equal to an interatomic space across the step and g the number of atoms per unit of length across the step. The effective number of broken bonds n is as follows: n n1 k1 n2 k2 k1 n1 n2 ;
are double lined. Fig. 1. The hcp lattice. (a) The planes (0001), (1010) The plane (1210) which is normal to (0001)and (10 10) planes is shown with dotted lines. (b) The atoms surrounded by lattice sites of the first (solid arrows) and the second (dotted arrows) coordinate spheres projected on the plane (1210).
where hkil are Miller's indexes corresponding to axes x, y, u, z, and 0, Z0 are the surface energy and the numbers of broken bonds per unit surface area of some site plane under consideration. Consequently, the task of the relative surface energy evaluation comes to computing the procedure of calculating broken bonds on the (hkil) site plane surface, it is convenient to consider its projection upon the perpendicular site plane. The atoms located at one side of the plane (for example, under the plane) are assumed as belonging to the crystal. The atoms located at the other side of the plane (above the plane) are considered as being really absent and possessing only broken bonds with the atoms of the surface. Meanwhile, some nearest neighbor bonds might be broken. Therefore, the lattice site plane (hkil) will really be a stepped surface with its projection upon a normal plane presenting a regularly stepped line. Let us try to evaluate the surface energy for various crystal faces of hcp structures and its angular correlation. Fig. 1 shows a coordinate hcp lattice with p c 2 2 p ; (3) a 3 Taking into consideration the interaction of atoms only in two coordinate spheres we receive coordinational numbers
(6)
where n1, n2 are the number of broken bonds for a step in the ®rst and the second coordinate spheres, respectively; k1, k2 are the coef®cients of effectiveness of the broken bonds for the ®rst and the second coordinate spheres, respectively; and
k2 =k1 . To evaluate , we must keep in mind that when interatomic space r grows, the interatomic bonding force decreases as 1/r7 [1,39], which when combined with Eq. (4) becomes 7 r1 1 p 0:088; (7)
r2 8 2 The relative surface energy, using Eqs. (5) and (6) is as follows: hkil
n1 n2 =sthkil n1 n2 =st0
(8)
If we try to evaluate the energy of (hkil) plane with reference to a zero site plane, provided both are of the same type, we assume that their steps are of the same width, thkil t0 , and Eq. (8) can be expressed as follows: hkil
n1 n2 =shkil n1 n2 =s0
(9)
Thus, the task of speci®c surface energy evaluation comes to the calculation of n1, n2 and s values. The speci®c surface energy will be evaluated for site planes of the next three families. 1. The family of site planes (hkil) (h0hl), normal to the plane (1210). 2. The family of site planes (hkil) (hhil), normal to the plane (1100). 3. The family of site planes (hkil) (hki0), normal to the plane (0001). 3. The surface energy of the crystal site planes (h0hl) This family of site planes is selected as follows. Suppose we have an arbitrary plane which at the beginning is in
72
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
Fig. 2. The family of lattice site planes (ho hl) projected on the plane (1210) (solid lines). One step of each plane is shown. Full circles mark the lattice sites which put on the corresponding lattice site plane and the step.
coincidence with a plane (0001). Further on, we rotated this plane about y-axis at 908. In the process of the rotation, this arbitrary plane will coincide with various lattice site planes of (h0hl) family. When 908 it will be (10 10). All these site planes are normal to ( 12 10) plane. Fig. 1 shows the above mentioned planes (0001), (1010) by double lines, and the plane ( 12 10) by broken lines. Fig. 1 also shows projected on (1210) plane the crystal lattice sites surrounded by the neighbour lattice sites of the ®rst (solid arrows) and the second (dot arrows) coordinate spheres. The double arrows mean the double bonds for corresponding distances. Fig. 2 shows projected on ( 12 10) plane and normal to it the planes above mentioned, which make angle with the (0001) plane. One step, consisting as a rule of several substeps, is shown by the broken line for each to the lattice site planes. Full circles mark the surface lattice sites which are located on the site plane as well as the lattice sites of a step. All the surface sites and broken-bonded lattice sites located near the surface are included in the step. All the planes in Fig. 2 pass through some central lattice site 0. These site planes at some magnitudes of cannot provide optimal conditions for calculation of surface energy. The dotted lines in Fig. 2 indicate the site plane which are parallel to the ®rst ones and spaced from them at distances much smaller then the lattice parameters. The calculation of the surface energy for them gives more real value because for this planes, as for (10 11) plane, the number of broken bonds decreases. In the case of (10 13) plane, such parallel
Fig. 3. (a) The relative surface energy of the family of lattice site planes (h0 hl) plotted against angle . (b) The graphic presentation of the function h0hl plotted in polar coordinates.
dislocation of a site plane doesn't change the number of broken bonds. Comparing the lattice sites of the steps in Fig. 2 with the corresponding bonds above the site planes in Fig. 1(b), one can easily calculate the number n1 and n2 of the broken bonds of steps for each of the lattice site planes. The geometry in Fig. 2 helps in calculating the length s of the step and the tangent of angle . Fig. 3(a) shows a plot of relative surface energy as a function of angle, i.e. h0hl . The function ( ) in the interval 0 < < 90 possesses two maxima and three minima. From Fig. 3, it can be seen that the surface energy processes the least value for the lattice site lane (0001) at 0 ; further on, as angle grows, the value of ( ) increases reaching the maximum for (1013) plane and later on it drops again to minimum for (1011) plane. Further increasing of angle is of less effect on the value of h0hl . Fig. 3(b) provides the graphic presentation of h0hl ( ) plotted in polar coordinates. The parts of the magnitudes of h0hl , which exceed a unit, are scaled up for better perception. Presented in this ®gure are axes z and projections on x and u axes. As it can be seen from Fig. 3(b), the function h0hl ( ) occurred to be symmetric about the lattice site planes (0001) ( 08) and (1010) ( 908). It corresponds to the symmetry of crystal lattice.
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
73
Fig. 5. The lattice site planes of (hhil) family, projected on (1 100) plane. Designations are identical to those in Fig. 2. Fig. 4. (a) The planes (0001), (11 20) (are double-lined) and normal to them (1100) plane (dot-lined). (b) Surrounding of atoms by lattice sites of the first and of the second coordination spheres, projected on (1 100) plane.
Graphic presentation of the function h0hl , both in rectangular and in polar coordinates, looked smooth for the selected lattice site planes. The character of the function h0hl enables to state that the energetically advantageous free face of a crystal may be plane of (0001) kind and, almost of the same probability, the planes of (10 11) and of (10 10) kind. Taking into account the interactions of atoms only in the ®rst coordinate sphere, when 0, results in increasing the values of ( ), and the general character of the function ( ) remains unchanged (dotted lines on the Fig. 3). The same is valid for bcc crystals [36]. 4. The surface energy of the lattice sites planes for (hhil) family The family of site planes (hhil) is received by means of rotation of some arbitrary plane about an axes which is normal to u-axis and parallel with the plane (0001). All these planes are normal to the (1 100) plane. Fig. 4 shows the cell of hcp lattice. The planes (0001), (1120) are double-lined, and the plane (1 100) is dot-lined. Fig. 4 also shows projected on (1 100) plane the lattice site, surrounded by neighbor atoms of the ®rst and of the second coordinate spheres. Fig. 5 shows the family of site planes
(hhil) under consideration, projected on (1100) plane. Each plane of the family has a step which enables, by means of the geometry in Figs. 5 and 4(b), to calculate the number of broken bonds as well as the tangent the angle and the length of the step s. The calculated structural parameters and the relative surface energies for the family of lattice site planes (hhil) are graphic presented in Fig. 6(a and b) in the rectangular and in the polar coordinates, respectively. Comparing the results of calculations for the family of lattice site planes (h0hl) and (hhil), we can see that the surface energies hhil for all 6 08, are larger than those of h0hl . So the maximum value of hhil has occurred to reach 1.25 for (1124) plane at 398, but h0hl has the maximum value of 1.19 for (1013) plane at 328. For the meaning of > 508, the hhil shows insigni®cant change in the range of 1.2±2.25 unlike the corresponding values of 0001. The function hhil has occurred to be smooth, as well as for the family of planes (h0hl). Discounting the interaction of atoms on the second coordinate sphere gives the rise in calculated values of hhil (the dot-lined curve in Fig. 6(a)). The graphic presentation of hhil in polar coordinates shows symmetry of this function about lattice site planes (0001) and (1120) which agrees with crystal lattice symmetry. Based upon external appearances of Fig. 6(a and b) it may be argued that the formation of crystals, with their free faces being the planes of (hhil) family with > 08, is of a small probability.
74
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
Fig. 7. (a) The planes (10 10), ( 12 10) (double lined) and the plane (0001) which is normal to them (dot-lined). (b) Surrounding of atoms by the lattice sites of the first two coordinate spheres in projection on the plane (0001). Fig. 6. (a) The relative surface energy of (hhil) family plotted against angle. (b) The graphic presentation of the function hhil in polar coordinates.
5. The surface energy of lattice site planes of the (hki0) family
The calculated values of relative surface energy for the family of lattice site planes under consideration were found to differ in character from those of (h0hl) and (hhil) family.
All lattice site planes of the family are parallel to the axis z, i.e. they are normal to site plane (0001). They are selected by means of rotating an arbitrary plane about the axis z. When 08, we receive (1 210) plane; at 908 we receive (1010) plane (see Figs. 7 and 8). In Fig. 8 we see the solid-lined projections of site planes (hki0), passing thought the central lattice site 0, and the dot-lined projections of some lattice site planes of this family for which there may be provided an optimal calculations of broken bonds number. A less number of broken bonds for (1010), (3120), (2110) planes is calculated by means of comparing data from Figs. 8 and 7(b) for these planes, shown in dotted lines. As for (5410), its less number of broken bonds was calculated for solid-lined planes. The results of the calculations are presented in Fig. 9(a and b). The relative surface energy hki0 hki0 1010 was plotted with respect to (10 10) plane of the same family, so the calculations where performed with Eq. (9). Calcula0 hki0 =0001 with respect to (0001) plane, tions of hki0 using Eq. (8), yields some larger values.
Fig. 8. Family of the lattice site planes (hki0) projected on (0001) plane. Designations are the same as those in Fig. 2.
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
75
type from the family of (hkil) planes seems to be the most probable. Nevertheless, the face (0001), which is of another lattice site family, possesses the least energy. Therefore, it may be argued that the texture of monolith polycrystals or ®lms, produced by precipitation of metals on a substrate, e.c. by electrolytic precipitation or by the method of thermal spraying, will show predominance of the crystals with the faces of (0001) ad (1100) kind. These faces must be formed as free ones while growing single crystals of hcp structures. 6. The surface energy of the lattice site planes (h0hl) for real crystals Let us calculate the relative surface energy for metals of hcp lattice with the parameters a and c not corresponding to idealization, i.e. p a 2 2 6 p 1:63: c 3
Fig. 9. relative surface energies of (hki0) family plotted against angle. (b) Graphic presentation of the function hki0 in the polar coordinates.
As can be seen from Fig. 9(a and b), the relationship hki0 is not found to be smooth but has revealed an abrupt change when passing from one lattice site plane to another. This leads to conclusion, that the formation of a free face, e.c. of (1100) or (10 10) type, will take place strictly on these planes, because even a small deviation of angle yields a signi®cant energetic disadvantage, i.e. an abrupt rise in surface energy. The function hki0 was found to be symmetric about lattice site planes (1 2 10) ( 08), (1 100) ( 308); (2110) ( 608); and (1010) ( 908). This corresponds to symmetry of crystal lattice about planes parallel to the axis z. According to the symmetry of the lattice, its sections by planes (1100) or (10 10) are adequate, i.e. these planes as surface planes, possess an equal number of broken bonds per unit of area. The same is valid for (1 2 10) and (2 1 10) planes. That is why the lattice site planes, symmetric about both mentioned, are of identical character. Such are planes (2310) (3210), (3120) or (3 410), (4 3 10), (4 1 30) or (4510), (5410), (5140) or (5 610), (6 5 10), (6 1 50) etc. For each of these triplets the number of broken bonds n1, n2, the step width s and the relative surface energies hki0 are of the same value. The planes (1100), (10 10), (01 10) produced by parallel to z sections across the axes x, y, u, possess minimal surface energies. Consequently, the formation of free face of (1010)
We shall take into account the atomic interaction with the same approximation as for the perfect crystals. But in the case under consideration, each atom has neighbors spading at r1, r2, r3 (Fig. 10), i.e. atomic interaction p pis accounted in three coordinate spheres. If c=a 2 2= 3 , then p r 3 a 2; (10) r1 r2 a; i.e. we receive the above-mentioned variant (Eq. (4)). If c=a > 1:63 (the lattice is extended along z direction), parameters r1, r2, r3 are determinated with the next equations: r r a2 c 2 4a2 c2 ; r3 : (11) r2 r1 a; 3 4 3 4 If c=a < 1:63 (the lattice is contracted along z direction), r1 and r2 permute, i.e.: r r a2 c 2 4a2 c2 ; r2 a; : (12) r3 r1 3 4 3 4
Fig. 10. p The p hcp structure with a, b, c, r1, r2, r3 parameters at > 2 2= 3.
76
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
Fig. 11. The bonds of an atom projected on (1 210) plane with all their neighbors spaced at r1, r2, r3 depicted p by psolid, long-dashed pand pshortdashed arrowed lines for > 2 2= 3(a) and < 2 2= 3 (b), respectively.
The coordination numbers of three coordination spheres for hcp lattice are correspondingly equal to Z1 6;
Z2 6;
Z3 6:
(13)
Let us perform the calculations of the energy for a lattice site plane with Miller's indexes (hkil) for axes x, y, u, z with reference to the energy of (0001), which has been proved to be minimal by experiments and calculations [32,38,40,42] hkil
hkil Zhkil ; 0001 Z0001
(14)
where Zhkil, Z0001 are the numbers of broken bonds per unit of area for the surface site plane mentioned above. Further calculations will be performed for the family of lattice site planes of (hkil) kind, for which k 0, i h and l 0,1,2,. . .. Fig. 11 shows the bonds of an atom projected on (1210) plane with all its neighbors spaced at r1, r2, r3 for c=a > 1:63 (a) and c/a<1.63 (b). Comparing this ®gure with Fig. 2 for each of the surfaced and near-surfaced atoms of each of lattice site planes from (h0 hl) family, we can easily determinate the number of broken bond n1, n2, n3 per step for three coordination spheres. The number of broken bonds per unit of surface area for a site plane can be calculated as follows: Zh0hl n1 k1 n2 k2 n3 k3 mg:
(15)
The bond breaking ef®ciencies k1, k2, k3 are evaluated here on the assumption that interatomic forces are directly proportional to 1=r 7 [1,39]. These ef®ciencies or the ratios 7 7 k2 r1 k3 r1 ;
(16)
0 k1 r2 k1 r3 depend upon r1, r2, r3 values Eqs. (11) and (12), and they are de®ned by the ratio of c/a. The number of steps m per unit length of a site plane, as well as the length of a step s 1=m, depend upon angle and, consequently, upon c/a value. The width t 1=g of an atomic row across a step for the family of site planes under consideration is equal to the lattice parameter a. The numerical values of the relative surface energy will be different for the same site plane of metals provided the rations s/a are different.
family of the lattice site Fig. 12. The relative surface energies of (h0hl) planes for Be and Cd metals (solid lines) and for a perfect crystal (dotted lines), plotted against angle in the rectangular (a) and in the polar (b).
7. The numerical evaluation of the surface energy for hcp metals: discussion 7.1. Comparison of the theoretically predicted values with the experimental data Calculations of surface energies have been performed for the Be, Hf, Ti, Zr, Mg, Co, Zn, Cd metals. The results are arranged in ascending order of the ratios c/a. Fig. 12 shows such plots Be, Cd and for a perfect crystal p pfor ( 2 2= 3). Plots for other metals are placed between those pfor pBe and Cd below the dotted line when p< 2p2 = 3 and above the dotted line when 2 2= 3. All plots display common regularity and possess two maxima and three minima. The latter fall at the lattice site planes (0001), (1011), (1010) for all elements. With the increasing of value, the maxima and the middle minimum, falling at (1011), shift in the direction of increasing angle. Meanwhile, the values of 1011 and 1010 energies grow, being approximately equal in magnitude. The points to equal probability of formation free faces of (1011) and (1010) kinds for all metals. For small numbers of (Be, Hf, Ti metals) the formations of free faces of (0001), (1010),
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
(1011) kinds are equiprobable. As for Be, the surface energy of (1010) was found to be some fewer than that of (0001) face. Consequently, the probability of forming free faces of (1010) kind must increase as compared to that of (0001), but the difference, being negligible, has occurred to be beyond the accuracy of the experiments [45]. For Zn, Mg and Co, the probability of formation free faces (10 11) or (1010) decreases as compared to (0001). For Zn and Cd with p p > 2 2= 3, the energy of (10 11), (10 10) faces is 1.5 times more than the face energy of (0001) face, therefore, the most probable for such crystals will be formation of free faces of (0001) kind. The increase of stipulates the rise in the surface energy of lattice site planes which points to a decrease in the packing density of crystal atoms. The features mentioned in changing of hkil, while passing from one site plane to another are in a qualitative correlation with known experimental data and with the results that have been obtained on the basis of theory by some other authors for Ti, Mg, Zn, Cd crystals [22,32,38]. Our values of surface energies for Ti, Mg, Cd are smaller than those obtained by other authors. As for our values of h0hl for Zn, they are within the range of those given in other works [32,38,43]. Our results showed satisfactory correlations with theory regarding the changing of the surface energy when passing from one lattice site plane to another as well as to occurrences of faces with theoretically predicted lattice site planes [43]. The authors of the work [22] succeeded in receiving a free face of (1012) kind for Ti with a signi®cantly high surface energy as was predicted on the basis of calculations. 8. Conclusions Let us note that the general features of the relationship hkil for the mentioned metals have been found to be the same as obtained by us for perfect crystals. Consequently, it is probably possible to con®ne oneself to observing a perfect crystal when dealing with many physical problems on the basis of molecular-kinetic theory. Thus, knowing the value of surface energy and its anisotropy with respect to the kind of site planes, we can predetermine the kind of free faces of growing crystals, the character of polycrystal textures during recrystallization, slip and cleavage planes of materials under deformation. We hope that juxtaposition of our theoretically predicted values of relative surface energy will be veri®ed by as yet unknown to us experimental data other metals. We think the theory may be extended as to originating a more perfect model, to accounting relaxation of the surface layer of a lattice and effect of electron subsystem of a crystal upon formation of a surface etc.
77
References [1] Ya.I. Frenkel, Elektricheskaya teoria tvyordykh tel. Leningrad. Krasnaya November 1924. [2] V.D. Kuznetsov, Poverkhnostnaya energia tverdykh tel. Moscow. GITTL, 1954. [3] K. Hauffe Reaktionen in and on festen stoffen. Springer, Berlin. 1955. [4] B.Ya. Pines, Ocherki po metallofizike. Kharkov. Univers Press, 1961. [5] B.V. Tsarevsky, S.I. Popel, Poverkhnostnye yavleniya v metallakh i splavakh i ikh rol v protsessakh poroshkovoi metallurgii. Kiev. Akad. Nauk., 1961. [6] V.N. Eryomenko (Ed.), Poverkhnostnye yavlenia v metallakh i splavakh i ikh rol v protsessakh poroshkovoi metallurgii. Kiev. Akad. Nauk., 1961. [7] A.N. Frumkin (Ed.), Poverkhnostnye svoistva poluprovodnikov. Akad. Nauk., Moscow, 1962. [8] B.A. Presnov (Ed.), Poverkhnostnye i kontaktnye yavleniya v poluprovodnikakh. Tomsk. University Press, 1965. [9] A.A. Appen, Ya.E. Geguzin (Eds.), Poverkhnostnye yavleniya v rasplavakh i voznikayuschikh iz nikh tverdykh fazakh. Nalchik. Univers. Press, 1965. [10] D.V. Skobeltsyn (Ed.), Poverkhnostnye svoistva poluprovodnikov i dinamika ionnykh kristallov. Nauka, Moscow, 1969. [11] E.A. Flad (Ed.), The Solid-Gas Interface, Marcel Dekker, New York, 1967. [12] M.A. Green (Ed.), Solid State Surface Science. Marcel Dekker, New York, 1969. [13] M.A. Kerdivarenko (Ed.), Poverkhnostnye yavleniya i adsorbtsia. Kishinev. Shtiintsa, 1972. [14] F.F. Volkenstein, Fiziko-khimiya poverkhnosti poluprovolnikov. Moscow. Nauka, 1973. [15] D.M. Skorov, A.I. Dashkovsky, Maskalets, V.N., Khizhny V.K. Poverkhnostnaya energia tverdykh metallicheskikh fas. Moscow Atomizdat., 1973. [16] G.P. Cherepanov, Mekhanika khrupkogo razrusheniya. Nauka, Moscow, 1974. [17] V.N. Eremenko (Ed.), Poverkhnostnye yavlenia v rasplavakh i voznikayuschikh iz nikh tverdykh fazakh. Kishinev. Shtiintsa., 1974. [18] W. Missol, Energia powierzchni rozdzialu faz w metalach. Katowice Wydawnictwo `Slask', 1974. [19] S.S. Gorelik (Ed.), Poverkhnostnye yavlenia v poluprovodnikakh. Metallurgia, Moscow, 1976. [20] S.R. Morrison, The Chemical Physiks of Surfaces. Plenum Press, New York, 1977. [21] Yu.V. Naidich (Ed.), Poverkhnostnye svoistva rasplavov. Naukova Dumka, Kiev, 1982. [22] B.A., Kolachev, Malkov A.V. Fizicheskie osnovy razrusheniya titana. Metallurgia, Moscow, 1983. [23] D. Bakli, Poverkhnostnye yavlenia pri adgezii i friktsionnom vzaimodeistvii. Moscow. Mashinostroenie, 1986. [24] S.F. Borisov, Fizika poverkhnosti. Sverdlovsk. University Press, 1987. [25] G.A. Abduragimov, Poverkhnost kristalla i epitaksia. University Press, Rostov na Donu, 1987. [26] N.B. Narozhny, Poverkhnost i effekty neodnorodnosti tvyordogo tela. Moscow. Energoatomizdat, 1987. [27] S.Yu. Zaginaichenko, Z.A. Matysina, V.I. Ryzhkov, Atomy na poverkhnosti i v ob'yome kristalla. No 6202-B87. Moscow. VINITI, 1987. [28] M.A. Vasiliev, Struktura i dinamika poverkhnosti perekhodnykh metallov. Naukova Dumka, Kiev, 1988. [29] V.P. Zhdanov, Elementarnye fiziko-khimicheskie protsessy na poverkhnosti. Nauka, Novosibirsk, 1988.
78
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
[30] R. Hoffman, Solid and Surfaces: A Cemist's View of Bonding in Extended Structures. Angewandte Chemie, Reviews of Modern Phisics, VCH Publisher, 1988. [31] Ya.E. Geguzin, N.N. Ovcharenko, Uspekhi Fis Nauk. 76(2) (1962) 283. [32] S.N. Zadumkin, I.G. Shebzuhova, Fiz. Met. I Metalloved. 22(1) (1962) 121. [33] V.Yu. Novikov, Ibid. 16(1) (1963) 129. [34] B.G. Livshits, V.Yu. Novikov, T.Yu. Tolstova, Ibid. 18(4) (1964) 580. [35] Y.S. Avraamov, A.G. Gvozdev, B.G. Lifshyts, Ibid. 21(1) (1966) 136. [36] Yu.S. Avraamov, A.G. Gvozdev, Ibid. 23(3) (1967) 405. [37] S.N. Zadumkin, V.G. Egiyev, Ibid. 28(3) (1969) 434. [38] W.A. Miller, G.J.C. Carpentner, G.A. Chadwik, Phil. Mag. 19(158) (1969) 305.
[39] G.M. Vorobiov, V.M. Demura, Fiz. Met. I Metalloved. 30(4) (1970) 807. [40] J.M. Poate, K.N. Tu, J.W. Mayer (Eds.), Thin Films-Interdiffusion and Reactions. Wiley-Interscience, New York, 1985. [41] W.R. Tyson, Canad. Met. Quart. 14 (1985) 307. [42] O.B. Girin, R.V. Ermakova, S.Yu. Zaginaichenko, Z.A. Matysina, in: Sb. Vsesoyuzn. sovesch. `Rentgen'. University Press, ChernovtsyErevan, 1987, 193 pp.. [43] Z.A. Matysina, L.M. Chuprina, S.Yu. Zaginaichenko, J. Phys. Chem. Solid. 53(1) (1992) 167. [44] Z.A. Matysina, L.Ye. Miroshnichenko, O.S. Pogorelova, Otnositelnaya poverkhnostnaya energiya kristallov GPU struktur. No 1718B92. Moscow. VINITI, 1992. [45] D. Webster, G.J. London, D.R. Floyd, J.N. Lowe (Eds.), Beryllium Science and Technology. Plenum Press, New York, 1973.
The relative surface energy of hexagonal close-packed crystals Z.A. Matysina State University, Dnepropetrovsk 320000, Ukraine Received 19 January 1999; accepted 12 February 1999
Abstract Surface energies of hexagonal close-packed (hcp) crystals for three families of lattice site planes have been computed and their angular correlations have been determined. The broken bond method and the model of stepped microprojections of crystal faces were employed. The atomic interactions were taken into account in two coordinate spheres. Relative surface energies have been computed for Be, Cd, Co, Hf, Mg, Ti, Zn, Zr. The theoretical results are compared with experimental data. # 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Surface energy; Hexagonal close-packed crystal; Broken-bond model
1. Introduction Atoms on the crystal surface have a smaller coordination than atoms in bulk and their energy states differ. It can simulate a redistribution of surface atoms as well as reconstruction of a solid surface. The surface energy of crystals is determined by surface-producing work and it depends upon a character of distribution and redistribution of the atoms on the surface [1±10]. The evaluation of surface energy and its connection with processes in solids can contribute to more understanding of the nature of many surface effects and helps to prede®ne their in¯uence on bulk properties of materials. The many works [1±30] have been devoted to investigations of surface processes in solids. Many theoretical evaluations and experiments have been carried out to determine crystal surface energies [31±44]. There are some positive results in experimental and theoretical approach to this question. In this paper we describe some results of surface energy evaluation of various lattice sites planes in crystals. We also describe some results of theoretical determination on relative con®gurational surface energy for three families p p of lattice site planes in perfect crystals (c/a 2 2= 3) with hcp lattice of A3 (Mg type) and for one of lattice site planes in real crystal (c/a 6 p family p 2 2= 3). The angular dependence of surface energy has been determined.
The results have ben obtained by making use of the atoms pairwise interaction and by broken-bond model of crystal surfaces and compared with experimental data.
2. The surface energy of a perfect crystal The speci®c surface energy of a material in a condensed state can be determined as a work of producing a surface unit area which, according to Gibbs±Helmholtz's equation, consist of con®gurative and of entropy components. The second component has been experimentally proved to be comparatively small which account for its omission from the present treatment [4]. The speci®c con®gurational surface energy hkil of crystal face (hkil) can be calculated as the sum of twin interaction energy of atoms on a unit surface area and of those which are out off by a free face (really absent). The works [27,28] have proved that the value hkil is proportional to the number of broken bonds on the surface unit area hkil Zhkil :
(1)
We shall be especially interested in the values of the relative speci®c surface energy hkil
hkil Zhkil ; 0 Z0
0254-0584/99/$ ± see front matter # 1999 Published by Elsevier Science S.A. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 9 9 ) 0 0 0 5 0 - 4
(2)
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
71
and radii. Z1 12;
Z2 6;
r1 a;
p r 2 a 2;
(4)
The number of broken bonds Zhkil will be as follows: n Zhkil nmg ; (5) st where n is the effective number of broken bonds for one step of a lattice site plane of one atomic width, m the number of steps per unit of length, s equal to 1/m is the length of a step, t equal to 1/g is the width of a step which is equal to an interatomic space across the step and g the number of atoms per unit of length across the step. The effective number of broken bonds n is as follows: n n1 k1 n2 k2 k1 n1 n2 ;
are double lined. Fig. 1. The hcp lattice. (a) The planes (0001), (1010) The plane (1210) which is normal to (0001)and (10 10) planes is shown with dotted lines. (b) The atoms surrounded by lattice sites of the first (solid arrows) and the second (dotted arrows) coordinate spheres projected on the plane (1210).
where hkil are Miller's indexes corresponding to axes x, y, u, z, and 0, Z0 are the surface energy and the numbers of broken bonds per unit surface area of some site plane under consideration. Consequently, the task of the relative surface energy evaluation comes to computing the procedure of calculating broken bonds on the (hkil) site plane surface, it is convenient to consider its projection upon the perpendicular site plane. The atoms located at one side of the plane (for example, under the plane) are assumed as belonging to the crystal. The atoms located at the other side of the plane (above the plane) are considered as being really absent and possessing only broken bonds with the atoms of the surface. Meanwhile, some nearest neighbor bonds might be broken. Therefore, the lattice site plane (hkil) will really be a stepped surface with its projection upon a normal plane presenting a regularly stepped line. Let us try to evaluate the surface energy for various crystal faces of hcp structures and its angular correlation. Fig. 1 shows a coordinate hcp lattice with p c 2 2 p ; (3) a 3 Taking into consideration the interaction of atoms only in two coordinate spheres we receive coordinational numbers
(6)
where n1, n2 are the number of broken bonds for a step in the ®rst and the second coordinate spheres, respectively; k1, k2 are the coef®cients of effectiveness of the broken bonds for the ®rst and the second coordinate spheres, respectively; and
k2 =k1 . To evaluate , we must keep in mind that when interatomic space r grows, the interatomic bonding force decreases as 1/r7 [1,39], which when combined with Eq. (4) becomes 7 r1 1 p 0:088; (7)
r2 8 2 The relative surface energy, using Eqs. (5) and (6) is as follows: hkil
n1 n2 =sthkil n1 n2 =st0
(8)
If we try to evaluate the energy of (hkil) plane with reference to a zero site plane, provided both are of the same type, we assume that their steps are of the same width, thkil t0 , and Eq. (8) can be expressed as follows: hkil
n1 n2 =shkil n1 n2 =s0
(9)
Thus, the task of speci®c surface energy evaluation comes to the calculation of n1, n2 and s values. The speci®c surface energy will be evaluated for site planes of the next three families. 1. The family of site planes (hkil) (h0hl), normal to the plane (1210). 2. The family of site planes (hkil) (hhil), normal to the plane (1100). 3. The family of site planes (hkil) (hki0), normal to the plane (0001). 3. The surface energy of the crystal site planes (h0hl) This family of site planes is selected as follows. Suppose we have an arbitrary plane which at the beginning is in
72
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
Fig. 2. The family of lattice site planes (ho hl) projected on the plane (1210) (solid lines). One step of each plane is shown. Full circles mark the lattice sites which put on the corresponding lattice site plane and the step.
coincidence with a plane (0001). Further on, we rotated this plane about y-axis at 908. In the process of the rotation, this arbitrary plane will coincide with various lattice site planes of (h0hl) family. When 908 it will be (10 10). All these site planes are normal to ( 12 10) plane. Fig. 1 shows the above mentioned planes (0001), (1010) by double lines, and the plane ( 12 10) by broken lines. Fig. 1 also shows projected on (1210) plane the crystal lattice sites surrounded by the neighbour lattice sites of the ®rst (solid arrows) and the second (dot arrows) coordinate spheres. The double arrows mean the double bonds for corresponding distances. Fig. 2 shows projected on ( 12 10) plane and normal to it the planes above mentioned, which make angle with the (0001) plane. One step, consisting as a rule of several substeps, is shown by the broken line for each to the lattice site planes. Full circles mark the surface lattice sites which are located on the site plane as well as the lattice sites of a step. All the surface sites and broken-bonded lattice sites located near the surface are included in the step. All the planes in Fig. 2 pass through some central lattice site 0. These site planes at some magnitudes of cannot provide optimal conditions for calculation of surface energy. The dotted lines in Fig. 2 indicate the site plane which are parallel to the ®rst ones and spaced from them at distances much smaller then the lattice parameters. The calculation of the surface energy for them gives more real value because for this planes, as for (10 11) plane, the number of broken bonds decreases. In the case of (10 13) plane, such parallel
Fig. 3. (a) The relative surface energy of the family of lattice site planes (h0 hl) plotted against angle . (b) The graphic presentation of the function h0hl plotted in polar coordinates.
dislocation of a site plane doesn't change the number of broken bonds. Comparing the lattice sites of the steps in Fig. 2 with the corresponding bonds above the site planes in Fig. 1(b), one can easily calculate the number n1 and n2 of the broken bonds of steps for each of the lattice site planes. The geometry in Fig. 2 helps in calculating the length s of the step and the tangent of angle . Fig. 3(a) shows a plot of relative surface energy as a function of angle, i.e. h0hl . The function ( ) in the interval 0 < < 90 possesses two maxima and three minima. From Fig. 3, it can be seen that the surface energy processes the least value for the lattice site lane (0001) at 0 ; further on, as angle grows, the value of ( ) increases reaching the maximum for (1013) plane and later on it drops again to minimum for (1011) plane. Further increasing of angle is of less effect on the value of h0hl . Fig. 3(b) provides the graphic presentation of h0hl ( ) plotted in polar coordinates. The parts of the magnitudes of h0hl , which exceed a unit, are scaled up for better perception. Presented in this ®gure are axes z and projections on x and u axes. As it can be seen from Fig. 3(b), the function h0hl ( ) occurred to be symmetric about the lattice site planes (0001) ( 08) and (1010) ( 908). It corresponds to the symmetry of crystal lattice.
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
73
Fig. 5. The lattice site planes of (hhil) family, projected on (1 100) plane. Designations are identical to those in Fig. 2. Fig. 4. (a) The planes (0001), (11 20) (are double-lined) and normal to them (1100) plane (dot-lined). (b) Surrounding of atoms by lattice sites of the first and of the second coordination spheres, projected on (1 100) plane.
Graphic presentation of the function h0hl , both in rectangular and in polar coordinates, looked smooth for the selected lattice site planes. The character of the function h0hl enables to state that the energetically advantageous free face of a crystal may be plane of (0001) kind and, almost of the same probability, the planes of (10 11) and of (10 10) kind. Taking into account the interactions of atoms only in the ®rst coordinate sphere, when 0, results in increasing the values of ( ), and the general character of the function ( ) remains unchanged (dotted lines on the Fig. 3). The same is valid for bcc crystals [36]. 4. The surface energy of the lattice sites planes for (hhil) family The family of site planes (hhil) is received by means of rotation of some arbitrary plane about an axes which is normal to u-axis and parallel with the plane (0001). All these planes are normal to the (1 100) plane. Fig. 4 shows the cell of hcp lattice. The planes (0001), (1120) are double-lined, and the plane (1 100) is dot-lined. Fig. 4 also shows projected on (1 100) plane the lattice site, surrounded by neighbor atoms of the ®rst and of the second coordinate spheres. Fig. 5 shows the family of site planes
(hhil) under consideration, projected on (1100) plane. Each plane of the family has a step which enables, by means of the geometry in Figs. 5 and 4(b), to calculate the number of broken bonds as well as the tangent the angle and the length of the step s. The calculated structural parameters and the relative surface energies for the family of lattice site planes (hhil) are graphic presented in Fig. 6(a and b) in the rectangular and in the polar coordinates, respectively. Comparing the results of calculations for the family of lattice site planes (h0hl) and (hhil), we can see that the surface energies hhil for all 6 08, are larger than those of h0hl . So the maximum value of hhil has occurred to reach 1.25 for (1124) plane at 398, but h0hl has the maximum value of 1.19 for (1013) plane at 328. For the meaning of > 508, the hhil shows insigni®cant change in the range of 1.2±2.25 unlike the corresponding values of 0001. The function hhil has occurred to be smooth, as well as for the family of planes (h0hl). Discounting the interaction of atoms on the second coordinate sphere gives the rise in calculated values of hhil (the dot-lined curve in Fig. 6(a)). The graphic presentation of hhil in polar coordinates shows symmetry of this function about lattice site planes (0001) and (1120) which agrees with crystal lattice symmetry. Based upon external appearances of Fig. 6(a and b) it may be argued that the formation of crystals, with their free faces being the planes of (hhil) family with > 08, is of a small probability.
74
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
Fig. 7. (a) The planes (10 10), ( 12 10) (double lined) and the plane (0001) which is normal to them (dot-lined). (b) Surrounding of atoms by the lattice sites of the first two coordinate spheres in projection on the plane (0001). Fig. 6. (a) The relative surface energy of (hhil) family plotted against angle. (b) The graphic presentation of the function hhil in polar coordinates.
5. The surface energy of lattice site planes of the (hki0) family
The calculated values of relative surface energy for the family of lattice site planes under consideration were found to differ in character from those of (h0hl) and (hhil) family.
All lattice site planes of the family are parallel to the axis z, i.e. they are normal to site plane (0001). They are selected by means of rotating an arbitrary plane about the axis z. When 08, we receive (1 210) plane; at 908 we receive (1010) plane (see Figs. 7 and 8). In Fig. 8 we see the solid-lined projections of site planes (hki0), passing thought the central lattice site 0, and the dot-lined projections of some lattice site planes of this family for which there may be provided an optimal calculations of broken bonds number. A less number of broken bonds for (1010), (3120), (2110) planes is calculated by means of comparing data from Figs. 8 and 7(b) for these planes, shown in dotted lines. As for (5410), its less number of broken bonds was calculated for solid-lined planes. The results of the calculations are presented in Fig. 9(a and b). The relative surface energy hki0 hki0 1010 was plotted with respect to (10 10) plane of the same family, so the calculations where performed with Eq. (9). Calcula0 hki0 =0001 with respect to (0001) plane, tions of hki0 using Eq. (8), yields some larger values.
Fig. 8. Family of the lattice site planes (hki0) projected on (0001) plane. Designations are the same as those in Fig. 2.
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
75
type from the family of (hkil) planes seems to be the most probable. Nevertheless, the face (0001), which is of another lattice site family, possesses the least energy. Therefore, it may be argued that the texture of monolith polycrystals or ®lms, produced by precipitation of metals on a substrate, e.c. by electrolytic precipitation or by the method of thermal spraying, will show predominance of the crystals with the faces of (0001) ad (1100) kind. These faces must be formed as free ones while growing single crystals of hcp structures. 6. The surface energy of the lattice site planes (h0hl) for real crystals Let us calculate the relative surface energy for metals of hcp lattice with the parameters a and c not corresponding to idealization, i.e. p a 2 2 6 p 1:63: c 3
Fig. 9. relative surface energies of (hki0) family plotted against angle. (b) Graphic presentation of the function hki0 in the polar coordinates.
As can be seen from Fig. 9(a and b), the relationship hki0 is not found to be smooth but has revealed an abrupt change when passing from one lattice site plane to another. This leads to conclusion, that the formation of a free face, e.c. of (1100) or (10 10) type, will take place strictly on these planes, because even a small deviation of angle yields a signi®cant energetic disadvantage, i.e. an abrupt rise in surface energy. The function hki0 was found to be symmetric about lattice site planes (1 2 10) ( 08), (1 100) ( 308); (2110) ( 608); and (1010) ( 908). This corresponds to symmetry of crystal lattice about planes parallel to the axis z. According to the symmetry of the lattice, its sections by planes (1100) or (10 10) are adequate, i.e. these planes as surface planes, possess an equal number of broken bonds per unit of area. The same is valid for (1 2 10) and (2 1 10) planes. That is why the lattice site planes, symmetric about both mentioned, are of identical character. Such are planes (2310) (3210), (3120) or (3 410), (4 3 10), (4 1 30) or (4510), (5410), (5140) or (5 610), (6 5 10), (6 1 50) etc. For each of these triplets the number of broken bonds n1, n2, the step width s and the relative surface energies hki0 are of the same value. The planes (1100), (10 10), (01 10) produced by parallel to z sections across the axes x, y, u, possess minimal surface energies. Consequently, the formation of free face of (1010)
We shall take into account the atomic interaction with the same approximation as for the perfect crystals. But in the case under consideration, each atom has neighbors spading at r1, r2, r3 (Fig. 10), i.e. atomic interaction p pis accounted in three coordinate spheres. If c=a 2 2= 3 , then p r 3 a 2; (10) r1 r2 a; i.e. we receive the above-mentioned variant (Eq. (4)). If c=a > 1:63 (the lattice is extended along z direction), parameters r1, r2, r3 are determinated with the next equations: r r a2 c 2 4a2 c2 ; r3 : (11) r2 r1 a; 3 4 3 4 If c=a < 1:63 (the lattice is contracted along z direction), r1 and r2 permute, i.e.: r r a2 c 2 4a2 c2 ; r2 a; : (12) r3 r1 3 4 3 4
Fig. 10. p The p hcp structure with a, b, c, r1, r2, r3 parameters at > 2 2= 3.
76
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
Fig. 11. The bonds of an atom projected on (1 210) plane with all their neighbors spaced at r1, r2, r3 depicted p by psolid, long-dashed pand pshortdashed arrowed lines for > 2 2= 3(a) and < 2 2= 3 (b), respectively.
The coordination numbers of three coordination spheres for hcp lattice are correspondingly equal to Z1 6;
Z2 6;
Z3 6:
(13)
Let us perform the calculations of the energy for a lattice site plane with Miller's indexes (hkil) for axes x, y, u, z with reference to the energy of (0001), which has been proved to be minimal by experiments and calculations [32,38,40,42] hkil
hkil Zhkil ; 0001 Z0001
(14)
where Zhkil, Z0001 are the numbers of broken bonds per unit of area for the surface site plane mentioned above. Further calculations will be performed for the family of lattice site planes of (hkil) kind, for which k 0, i h and l 0,1,2,. . .. Fig. 11 shows the bonds of an atom projected on (1210) plane with all its neighbors spaced at r1, r2, r3 for c=a > 1:63 (a) and c/a<1.63 (b). Comparing this ®gure with Fig. 2 for each of the surfaced and near-surfaced atoms of each of lattice site planes from (h0 hl) family, we can easily determinate the number of broken bond n1, n2, n3 per step for three coordination spheres. The number of broken bonds per unit of surface area for a site plane can be calculated as follows: Zh0hl n1 k1 n2 k2 n3 k3 mg:
(15)
The bond breaking ef®ciencies k1, k2, k3 are evaluated here on the assumption that interatomic forces are directly proportional to 1=r 7 [1,39]. These ef®ciencies or the ratios 7 7 k2 r1 k3 r1 ;
(16)
0 k1 r2 k1 r3 depend upon r1, r2, r3 values Eqs. (11) and (12), and they are de®ned by the ratio of c/a. The number of steps m per unit length of a site plane, as well as the length of a step s 1=m, depend upon angle and, consequently, upon c/a value. The width t 1=g of an atomic row across a step for the family of site planes under consideration is equal to the lattice parameter a. The numerical values of the relative surface energy will be different for the same site plane of metals provided the rations s/a are different.
family of the lattice site Fig. 12. The relative surface energies of (h0hl) planes for Be and Cd metals (solid lines) and for a perfect crystal (dotted lines), plotted against angle in the rectangular (a) and in the polar (b).
7. The numerical evaluation of the surface energy for hcp metals: discussion 7.1. Comparison of the theoretically predicted values with the experimental data Calculations of surface energies have been performed for the Be, Hf, Ti, Zr, Mg, Co, Zn, Cd metals. The results are arranged in ascending order of the ratios c/a. Fig. 12 shows such plots Be, Cd and for a perfect crystal p pfor ( 2 2= 3). Plots for other metals are placed between those pfor pBe and Cd below the dotted line when p< 2p2 = 3 and above the dotted line when 2 2= 3. All plots display common regularity and possess two maxima and three minima. The latter fall at the lattice site planes (0001), (1011), (1010) for all elements. With the increasing of value, the maxima and the middle minimum, falling at (1011), shift in the direction of increasing angle. Meanwhile, the values of 1011 and 1010 energies grow, being approximately equal in magnitude. The points to equal probability of formation free faces of (1011) and (1010) kinds for all metals. For small numbers of (Be, Hf, Ti metals) the formations of free faces of (0001), (1010),
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
(1011) kinds are equiprobable. As for Be, the surface energy of (1010) was found to be some fewer than that of (0001) face. Consequently, the probability of forming free faces of (1010) kind must increase as compared to that of (0001), but the difference, being negligible, has occurred to be beyond the accuracy of the experiments [45]. For Zn, Mg and Co, the probability of formation free faces (10 11) or (1010) decreases as compared to (0001). For Zn and Cd with p p > 2 2= 3, the energy of (10 11), (10 10) faces is 1.5 times more than the face energy of (0001) face, therefore, the most probable for such crystals will be formation of free faces of (0001) kind. The increase of stipulates the rise in the surface energy of lattice site planes which points to a decrease in the packing density of crystal atoms. The features mentioned in changing of hkil, while passing from one site plane to another are in a qualitative correlation with known experimental data and with the results that have been obtained on the basis of theory by some other authors for Ti, Mg, Zn, Cd crystals [22,32,38]. Our values of surface energies for Ti, Mg, Cd are smaller than those obtained by other authors. As for our values of h0hl for Zn, they are within the range of those given in other works [32,38,43]. Our results showed satisfactory correlations with theory regarding the changing of the surface energy when passing from one lattice site plane to another as well as to occurrences of faces with theoretically predicted lattice site planes [43]. The authors of the work [22] succeeded in receiving a free face of (1012) kind for Ti with a signi®cantly high surface energy as was predicted on the basis of calculations. 8. Conclusions Let us note that the general features of the relationship hkil for the mentioned metals have been found to be the same as obtained by us for perfect crystals. Consequently, it is probably possible to con®ne oneself to observing a perfect crystal when dealing with many physical problems on the basis of molecular-kinetic theory. Thus, knowing the value of surface energy and its anisotropy with respect to the kind of site planes, we can predetermine the kind of free faces of growing crystals, the character of polycrystal textures during recrystallization, slip and cleavage planes of materials under deformation. We hope that juxtaposition of our theoretically predicted values of relative surface energy will be veri®ed by as yet unknown to us experimental data other metals. We think the theory may be extended as to originating a more perfect model, to accounting relaxation of the surface layer of a lattice and effect of electron subsystem of a crystal upon formation of a surface etc.
77
References [1] Ya.I. Frenkel, Elektricheskaya teoria tvyordykh tel. Leningrad. Krasnaya November 1924. [2] V.D. Kuznetsov, Poverkhnostnaya energia tverdykh tel. Moscow. GITTL, 1954. [3] K. Hauffe Reaktionen in and on festen stoffen. Springer, Berlin. 1955. [4] B.Ya. Pines, Ocherki po metallofizike. Kharkov. Univers Press, 1961. [5] B.V. Tsarevsky, S.I. Popel, Poverkhnostnye yavleniya v metallakh i splavakh i ikh rol v protsessakh poroshkovoi metallurgii. Kiev. Akad. Nauk., 1961. [6] V.N. Eryomenko (Ed.), Poverkhnostnye yavlenia v metallakh i splavakh i ikh rol v protsessakh poroshkovoi metallurgii. Kiev. Akad. Nauk., 1961. [7] A.N. Frumkin (Ed.), Poverkhnostnye svoistva poluprovodnikov. Akad. Nauk., Moscow, 1962. [8] B.A. Presnov (Ed.), Poverkhnostnye i kontaktnye yavleniya v poluprovodnikakh. Tomsk. University Press, 1965. [9] A.A. Appen, Ya.E. Geguzin (Eds.), Poverkhnostnye yavleniya v rasplavakh i voznikayuschikh iz nikh tverdykh fazakh. Nalchik. Univers. Press, 1965. [10] D.V. Skobeltsyn (Ed.), Poverkhnostnye svoistva poluprovodnikov i dinamika ionnykh kristallov. Nauka, Moscow, 1969. [11] E.A. Flad (Ed.), The Solid-Gas Interface, Marcel Dekker, New York, 1967. [12] M.A. Green (Ed.), Solid State Surface Science. Marcel Dekker, New York, 1969. [13] M.A. Kerdivarenko (Ed.), Poverkhnostnye yavleniya i adsorbtsia. Kishinev. Shtiintsa, 1972. [14] F.F. Volkenstein, Fiziko-khimiya poverkhnosti poluprovolnikov. Moscow. Nauka, 1973. [15] D.M. Skorov, A.I. Dashkovsky, Maskalets, V.N., Khizhny V.K. Poverkhnostnaya energia tverdykh metallicheskikh fas. Moscow Atomizdat., 1973. [16] G.P. Cherepanov, Mekhanika khrupkogo razrusheniya. Nauka, Moscow, 1974. [17] V.N. Eremenko (Ed.), Poverkhnostnye yavlenia v rasplavakh i voznikayuschikh iz nikh tverdykh fazakh. Kishinev. Shtiintsa., 1974. [18] W. Missol, Energia powierzchni rozdzialu faz w metalach. Katowice Wydawnictwo `Slask', 1974. [19] S.S. Gorelik (Ed.), Poverkhnostnye yavlenia v poluprovodnikakh. Metallurgia, Moscow, 1976. [20] S.R. Morrison, The Chemical Physiks of Surfaces. Plenum Press, New York, 1977. [21] Yu.V. Naidich (Ed.), Poverkhnostnye svoistva rasplavov. Naukova Dumka, Kiev, 1982. [22] B.A., Kolachev, Malkov A.V. Fizicheskie osnovy razrusheniya titana. Metallurgia, Moscow, 1983. [23] D. Bakli, Poverkhnostnye yavlenia pri adgezii i friktsionnom vzaimodeistvii. Moscow. Mashinostroenie, 1986. [24] S.F. Borisov, Fizika poverkhnosti. Sverdlovsk. University Press, 1987. [25] G.A. Abduragimov, Poverkhnost kristalla i epitaksia. University Press, Rostov na Donu, 1987. [26] N.B. Narozhny, Poverkhnost i effekty neodnorodnosti tvyordogo tela. Moscow. Energoatomizdat, 1987. [27] S.Yu. Zaginaichenko, Z.A. Matysina, V.I. Ryzhkov, Atomy na poverkhnosti i v ob'yome kristalla. No 6202-B87. Moscow. VINITI, 1987. [28] M.A. Vasiliev, Struktura i dinamika poverkhnosti perekhodnykh metallov. Naukova Dumka, Kiev, 1988. [29] V.P. Zhdanov, Elementarnye fiziko-khimicheskie protsessy na poverkhnosti. Nauka, Novosibirsk, 1988.
78
Z.A. Matysina / Materials Chemistry and Physics 60 (1999) 70±78
[30] R. Hoffman, Solid and Surfaces: A Cemist's View of Bonding in Extended Structures. Angewandte Chemie, Reviews of Modern Phisics, VCH Publisher, 1988. [31] Ya.E. Geguzin, N.N. Ovcharenko, Uspekhi Fis Nauk. 76(2) (1962) 283. [32] S.N. Zadumkin, I.G. Shebzuhova, Fiz. Met. I Metalloved. 22(1) (1962) 121. [33] V.Yu. Novikov, Ibid. 16(1) (1963) 129. [34] B.G. Livshits, V.Yu. Novikov, T.Yu. Tolstova, Ibid. 18(4) (1964) 580. [35] Y.S. Avraamov, A.G. Gvozdev, B.G. Lifshyts, Ibid. 21(1) (1966) 136. [36] Yu.S. Avraamov, A.G. Gvozdev, Ibid. 23(3) (1967) 405. [37] S.N. Zadumkin, V.G. Egiyev, Ibid. 28(3) (1969) 434. [38] W.A. Miller, G.J.C. Carpentner, G.A. Chadwik, Phil. Mag. 19(158) (1969) 305.
[39] G.M. Vorobiov, V.M. Demura, Fiz. Met. I Metalloved. 30(4) (1970) 807. [40] J.M. Poate, K.N. Tu, J.W. Mayer (Eds.), Thin Films-Interdiffusion and Reactions. Wiley-Interscience, New York, 1985. [41] W.R. Tyson, Canad. Met. Quart. 14 (1985) 307. [42] O.B. Girin, R.V. Ermakova, S.Yu. Zaginaichenko, Z.A. Matysina, in: Sb. Vsesoyuzn. sovesch. `Rentgen'. University Press, ChernovtsyErevan, 1987, 193 pp.. [43] Z.A. Matysina, L.M. Chuprina, S.Yu. Zaginaichenko, J. Phys. Chem. Solid. 53(1) (1992) 167. [44] Z.A. Matysina, L.Ye. Miroshnichenko, O.S. Pogorelova, Otnositelnaya poverkhnostnaya energiya kristallov GPU struktur. No 1718B92. Moscow. VINITI, 1992. [45] D. Webster, G.J. London, D.R. Floyd, J.N. Lowe (Eds.), Beryllium Science and Technology. Plenum Press, New York, 1973.