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Surface energy for brittle fracture of alkali halides from lattice dynamics
Surface Science 48 (1975) 5 6 1 - 5 7 6 © North-Holland Publishing Company
SURFACE ENERGY FOR BRITTLE FRACTURE OF ALKALI HALIDES FROM LATTICE DYNAMICS # G. B E N E D E K Gruppo Nazionale di Struttura della Materia del C.N.R., and Istituto di Fisica dell'Universitb, 20133 Milano, Italy S. B O F F I [stituto di Fisica dell'Universitb, 27100 Pavia, Italy, and Istituto Nazionale di Fisica Nucleare, Gruppo di Pavia, Italy G. C A G L I O T I lstituto di Ingegneria Nucleare, CESNEF, Politecnico di Milano, 20133 Miiano, Italy and J.C. B I L E L L O * Department of Materials Science, State University of New York at Stony Brook, Stony Brook, New York 11790, U.S.A. Received 6 August 1974
A lattice dynamics approach to the surface energy "r for brittle fracture of several ionic crystals is presented, based on recent work on surface dynamics and a reformulation of a model previously worked out for metals. The model requires the knowledge of the crystal structure, the eigenfrequencies and eigenvectors of the normal modes of vibration, as well as an analytical definition of the critical interplanar displacement appropriate to the cleavage mode. Some temperature dependent values of 7 are also computed for the (100) and (110) planes. Qualitatively the results indicate the correct cleavage systems for all cases and quantitively are reasonably consistent with the available experimental values.
1. I n t r o d u c t i o n In previous works [I-3]
a m e t h o d for calculating w i t h o u t a r b i t r a r y p a r a m e t e r s
"~Work supported under a contract with the Consiglio Nazionale deUe Ricerche (Italy) and the U.S. Atomic Energy Commission (U.S.A.) under the auspices of the CNR-NSF (Italy-U.S.A.) cooperative research agreement. * Currently on sabbatical leave at the: Istituto di Ingegneria Nucleare, CESNEF, Politecnico, 20133 Milano, Italy.
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G. Benedek et al./Surface energy for brittle fracture
the surface energy for cleavage of several monoatomic metallic crystals has been proposed, based on the knowledge of both the mean square atomic displacement at the melting and boiling temperatures, as well as on the lattice dynamics concept of interplanar force constants [4]. This earlier approach cannot be simply extended to ionic NaC1 structures, where the unit cell contains two atoms of different masses and opposite charges. A new concept is required, borrowed from surface dynamics [5], of force constants fp(1) between a single positive (p = +) or negative (p = - ) ion in the /th plane below the fracture plane and the whole, rigid half-crystal above the fracture plane. Furthermore, the present report introduces a new concept for defining analytically the "critical displacement", Uc, necessary to cause interplanar separation. Using u c the "work" to produce cleavage in cubic crystals can be calculated for the principal low index (hkl) planes with a more realistic treatment of the crystalline anisotropy. This is shown to remove the difficulties which arose from defining the critical displacement in terms of the mean square atomic amplitude (U2m)at melting. This earlier, first order approach assumed that higher index planes, of closer spacing, cleave when subjected to the same additional separation, as that invoked for (100) cleavage, because of the inherent isotropy of u m (hkl). The displacement u c is derived from the criterion for mechanical instability of a system of rigid planes coupled by a realistic interplanar potential function. For (100) planes it is shown that this new approach (i.e. using Uc) g~yes near!y the same results as those calculated from the earlier meth1 llZ.~ od ( 1' .e. us'ng m), but for high index planes the results differ significantly. In section 2 the model is presented, in section 3 the results are reported, and the conclusions, sections 4, contain a discussion which includes a comparison with the approach based on interplanar potentials [6] as well as considerations of the cleavage plane actually observed in real crystals, where plastic relaxation can occur [7].
2. The model The creation of two new surfaces by cleavage might be conceived as activated by either separation of the planes (hkl) of the crystal in a particular [hkl] direction until the system becomes mechanically unstable or by successive steps of local melting and boiling of the crystal along the fracture plane, as originally proposed [1 ]. The concepts involved are shown to be nearly equivalent for easy (100) cleavage, but diverge for higher index planes. With reference to the first process, the achievement of the critical separation requires a tensile stress acting against the interplanar potential. As this stress is applied, the shape of this potential is modified while a uniform stretch of the interplanar bonds occurs. As the amount of dilational strain increases, the perfect lattice configuration becomes unstable and other configurations of the system, exhibiting a pair of free surfaces, become possible. The transition between the stretched and the split configurations is assumed to occur when the second derivative of the interplanar potential with respect to the dilational strains is zero (Appendix A). The critical displacement depends, as the interplanar potential, on the tensile direction.
G. Benedek et al./Surface energy for brittle fracture
563
(I)
I
Ifp(3)
~~-~P I I
Fig. 1. Force constants fp(l) between a single ion in the lth plane below the fracture surface and the whole rigid half crystal above. In the case originally proposed, local melting, where a rearrangement of interatomic bonds is operative, is obtained by loading the system of spring constantsfp(l) (fig. 1) by a potential energy 7 m sufficient to separate the two to-be-split-half-crystals by an amount corresponding to the atomic square displacement: 27~ : f f ~ f p ( l ) $ [ u 2 ( T m ) - U 2 ( T ) ] 1
np,
( p = +,-),
(1)
where np is the number-density o f p atoms on the crystal plane, u_2(T) is the temperp ature dependent mean square atomic displacement. Local boiling, where the rupture of the bonds actually occurs, is roughly estimated here by adding to 7 m = 7 m + 7 m the thermal (potential) energy: ,yb =½kB(T b _ Tm ) (n+ + n _ ) ,
(2)
where T b and T m are the boiling and melting temperatures respectively and k B the Boltzmann constant . The relevant surface energy is obtamed as .
°
* The introduction of the factors 1/3 inserted in eq.(1) and 1/2 in eq.(2) is discussed in the last paragraph of section 4.1.
G. Benedek et al./Surface energy for brittle Jracture
564
*
m
7 = 3'+ +
,ym + ,)lb.
(3a)
For the first process the surface energy is given by: 7 = ")'c + ,),b,
(3b)
where 7c, as given in eq. (A.16), takes the same form as eq.(1), with the factor ~ replaced by an anisotropy factor B. A numerical treatment of eq.(1) requires the solution of the dynamic equations of the atomic motions in Fourier space, i.e. the knowledge of the frequencies w(q,j) and the polarization vectors, ep(q, j), of the vibrational modes of wave vector q and branch index j. The mean square displacement of the atom of mass Mp is then given, by: u2(r)=N
.
2Mp~(q,j) [eP (q'j)12 [ 2 n ( q , j ) + 11,
(4)
where N is the number of crystal cells and n(q, j) is the phonon population factor [exp (l~w(q, j)/kgT ) - 1]-1 (see e.g. ref. 8). Each of the interplanar force constants building the set {fp(1)} is a sum over the negative half of the crystal planes (the left hand side of fig. 1) defined as {-l'}, of the interatomic force constants q~c~a(in the present context the index c~, omitted in eq. (1), refers to the third cartesian component, orthogonal to the fracture plane):
where L and L' are the cell indices. In order to relate fpc~(l) to the eigenfrequencies and the eigenvectors, it is necessary to express ~ba,~in terms of its Fourier representation via the dynamical matrix and using the equations of motion in Fourier space. One thus obtains:
fpa(l)=N~
~ %Mp,)~e~p,(q,j)eo~p(q,J) l'p',q 3,]
× co2(q,/) exp [ia
q3(l +/')],
q = (0,0,q3),
(6)
where N L is the number of atomic layers, or crystal planes, along the direction 3, orthogonal to the fracture plane. It should be noticed that Elfp(l) reduces to the linear combination, K, of interplanar force constants, kr, NL
K =~
rkr,
f= 1
utilized in other papers on the subject.
(7)
565
G. Benedek et al./Surface energy for brittle fracture
r
i
X
/
[ oo]
[.o]
Fig. 2. Schematic representation of 1/8 of the first Brillouin zone used to calculate the eigenfrequencies and eigenvectors. 3. Results The above model has been applied to several ionic crystals with NaCI structure. The eigenfrequencies and eigenvectors entering eqs. (4) and (6) have b e e n c o m p u t e d on the basis o f the breathing shell model [9], which is k n o w n to give an excellent fit to the experimental dispersion relations [10]. Table 1 Temperature dependence of 3'* for (100) planes for some selected crystals Crystal
T (°K)
~+
LiF
0 300
78 40
KC1
0 300
NaC1
3'm
3,b
3'*
119 90
144 14-2
341 272
50 31
56 37
51 50
158 119
0 300
48 32
68 51
54 53
170 137
KBr
0 300
44 27
53 33
46 45
143 104
NaI
0 300
36 19
51 31
44 43
131 93
KI
0 300
46 20
51 26
37 36
134 82
G. Benedek et al./Surface energy for brittle fracture
566
Table 2 S u m m a r y of calculations of surface energy 3, (erg cm-2 ) according to eqs. (3) and (4), with comparison with other theoretical and experimental results Crystal
Plane
3m
,,/m _
3"b
3* [eq.(3)]
"rc
3'
Exp. (E) and other (T) values
LiF
(100)
78
119
144
341
209
353
340 a (E, 78°K), 374 b (T)~ 363 c (T), 414 ~ (T), 780 b (T)
(110)
84
110
102
297
515
617
LiC1
(100) (110)
19 59
69 62
75 52
163 173
67 195
142 247
LiBr
(100) (110)
19 50
57 50
66 46
142 147
54 161
120 207
Lil
(100) (110)
20 58
45 39
56 40
121 137
39 119
95 159
NaF
(100) (110)
78 82
94 92
91 65
263 239
147 259
238 324
NaCI
(100)
48
68
54
170
91
145
(110)
54
56
38
149
163
201
280 270 (E), 214 (T) 345
NaBr
(100) (110)
37 45
61 50
49 35
147 130
62 121
111 156
248 b (T) 288 b (T)
NaI
(100) (110)
36 38
51 41
44 31
131 110
65 107
109 138
KF
(100) (110)
64 60
60 66
64 45
188 171
94 158
159 203
KCI
(100)
50
56
51
158
80
131
(110)
45
47
36
128
116
152
(100)
44
53
46
143
69
116
(110)
40
43
32
115
102
134
306 b (T), 157 c (T), 94 d (T) 253 b (T)
KI
(100) (110)
46 37
51 38
37 26
134 101
66 91
103 128
233 b (T) 165 b (T)
RbF
(100) (110)
59 54
58 73
64 45
180 172
75 131
139 176
RbC1
(100) (110)
40 38
45 40
43 30
128 108
55 85
98 115
RbBr
(100) (110)
34 34
40 34
39 27
113 95
48 75
87 102
RbI
(100) (110)
29 26
34 27
34 24
96 77
41 61
75 86
KBr
110 318 (T), 271
e (E, 78°K), f (E), 3 7 0 g 310 b (T), c (T), 151 d b (T)
h (E, 298°K), b (T)j 171 c 108 u (T) b (T)
a Ref. 19. b Ref. 14. c Ref. 20. d Ref. 12. e Ref. 21. f Ref. 22. g Ref. 23. h Ref. 24.
G. Benedek et aL/Surface energy for brittle fracture
567
In the calculation of the mean square displacement a mesh of 4096 points in the Brillouin zone has been considered, while in the calculation offpa(l) the sum over q3 in eq. (6) runs over 49 equally spaced values on the FX segment of fig. 2 for the (100) cleavage plane, and over 49 values on the VX' segment for the (110) cleavage plane. The resulting surface energies have been numerically computed both at 0°K and 300°K, using consistently the appropriate input data. In practice the values l ~< 5 are found sufficient for the sum of eq. (1), since the series Zlfp(l) converges rapidly, in spite of the long range Coulomb potential, in ionic crystals [5]. Nevertheless, for the sake of completeness l values up to 20 were considered. The numerical values of ~,m and ,,/b and of the total surface energy ~,* are presented for 0°K and 300°K in tal~le 1. In table 2 the values of 7 and 7 are compared for 0°K with experimental values and other theoretical results.
4. Discussion and conclusion 4.1. Comparison with other models For alkali halides the classical calculations of surface energies are based on the Born model [ I 1]. It was originally proposed for the purpose of evaluating the (cohesive) lattice energy of crystals arising from interactions between the ions. In this model the surface energy of a finite crystal is computed as one half (of the negative) of the interaction energy of the finite crystal itself and an infinite crystal enveloping it. One then obtains for the specific surface energy in the NaC1 structure the following simple relation [12] ~/ = A Ze2 /a30,
(8)
where A is a constant depending on the selected surface [e.g., A = 0.1173 for the (100) face] and a 0 is the lattice parameter. More detailed calculations are reviewed in ref. 13. Many other approaches have been suggested to compute the surface energies of crystals. The ones which seem particularly relevant to the present discussion include Gilman's [14], where ionic crystals were specifically considered in the frame of the continuum mechanics, and the recent work of Linford and Mitchell [6] based on the concept of an interplanar potential. Present values of 7 obtained from using the critical displacement, Uc, systematically produce 7 (100) < 7 (110) as would be expected qualitatively from the observed cleavage modes of these crystals. The magnitude of 7 is also reasonably consistent with available experimental results. Earlier work [14] which had attempted to specify the correct cleavage plane of ionic crystals from calculated 7 values had predicted the wrong cleavage plane in half of the considered cases. The possibility exists that plastic relaxation could account for the apparent discrepancy of the results in ref. 14.
568
G. Benedek et al./SurJhce energy ]br brittle ]kacture
However, it will be shown below that, at least for ionic crystals, including plastic energy considerations is of no help and in fact leads to predicting the (110) cleavage plane for all NaCl-type ionic crystals. The (110) plane has never been observed as the primary cleavage plane for this type of crystalline solid. The interplanar potential model [6] and the present one are shown to lead to similar answers in Appendix A. This connection demonstrates that the harmonic approximation can be coherently framed within the interplanar potential approach, although in general it should be applied cautiously to problems involving large displacements. The present values are in general agreement with other theoretical models (see the last column of table 1), with the exception of the Shuttleworth and the Born-Stern models, where the (110) values are exceptionally high. Before concluding this subsection a few comments will be made linking the present work to an earlier one which used a similar physical model as a starting point to evaluate 7 for metals [ 1 ]. The definition ym and 7 b in eqs. (1) and (2) are different from the corresponding ones in eqs. (5) and (8) of ref. 1. In eq. (1) the surface energy for local melting is computed substracting the thermal energy already delivered to the crystal for the very reason that the crystal is kept at a finite temperature T. The advantage of the current procedure is that it allows one to directly compute the temperature dependence of 7 (neglecting plastic relaxation to be considered below). Furthermore, as indicated before, 7 m and 7 b are suggested here to be smaller by a factor ½ and ½ respectively. The introduction of these factors shows a limit, recently discussed in refs. 15 and 16, to the possibility of simply extending to ionic crystals the arguments of ref. 1, valid for metals. The factor three - expressing the ratio between the total mean square atomic displacement and its projection along the direction of separation - has been chosen here for two reasons: (i) The dependence of the interatomic potentials in metals as compared with ionic crystals shows a repulsive part at distances larger than the equilibrium one. This suggests that metals should be intrinsically more tenacious than ionic crystals. (ii) The generally lower Poisson's ratio found in ionic crystals evidences smaller transverse energy dissipation (i.e. within the plane of cleavage) than occurring in metals; thus the x3 factor corresponds more closely to the single degree of freedom process envisaged here. The factor of two difference between the present eq. (2) and eq. (8) of ref. 1, is made plausible by the fact that the actual process of boiling away of one surface with respect to the other, can be conceived as due to the constraint of purely potential energy to the single degree of freedom leading to separation. 4.2. Role o f plastic relaxation
The calculation for 7, made above, was based on the inherent assumption that the external energy supplied to the crystal to produce cleavage went solely into creating two perfect crystalline surfaces. Such an assumption would not be valid for a real
G. Benedek et al./Surface energy for brittle fracture
569
crystal where defect generation accompanies cleavage and creates an additional energy term which must be considered. The associated plastic energy for a particular combination of cleavage and slip systems and the role this plays in governing the cleavage plane for NaCl-type crystals has been evaluated on the basis of a model presented for bcc metals by Tyson et al. [7] (see Appendix B for formulation). Physically the stress field at the crack-tip can relax via plastic deformation to a degree which, for any slip system, is proportional to the corresponding resolved shear stress (RSS). The magnitude of RSS depends on the mutual orientation factor f(O) determined by the crack propagation direction within its cleavage plane and the slip systems (hkl) [h'k'f]. Table 3 presents maximum f(O) values for various cleavage-slip system combinations. Whenf(O) = 1, the entire stress field of the crack is transferred as shear stress to the slip system, and at the other extreme, when ](O) = 0, no component of the shear stress is effective. Thus, small f(O) values correspond to low driving force for plastic relaxation and hence minimum plastic energy. A particular cleavage system will have the lowest amount of plastic energy associated with it if f(O) is a minimum for all possible slip system combinations. For NaCl-type crystals table 3 shows that this occurs for (110) [001] cleavage. Within this frame, due to the probability of higher plastic energy to create (100) cleavage, invoking plastic energy requirements to rationalize theories which evaluate 3'110 slightly lower than or nearly equal to 3'100 appear inconsistent with the experimentally observed (100) cleavage. Thus 3'110 > 3'100 (for that matter, 3"hkl > 3'100 ) as derived in previous sections, is the most likely consistent anatytical result that can explain the (100) cleavage which occurs for all the NaCl-type ionic crystals. Table 3 Maximum values of the orientation factor f(o) for various cleavage systems and the most common slip systems for NaCl-type crystals, following ref. 7 Cleavage systems
(001) (001) (001) (001) (001) (110) (110) (110) (110) (110) (110)
[010] [1501 [1301 [120] [110] [0011 [113] [~12] [111 ] [2211 [71Ol
Slip systems, f(O) values (101) [1011 (701) [101]
(011) [Oil] (Oil) [0111
(110) [7101 (710) [1101
0.539 0.547 0.561 0.583 0.652 0.506 0.528 0.516 0.495 0.501 0.513
O.77O 0.761 0.746 0.722 0.652 0.506 0.528 0.516 0.495 0.501 0.513
0.252 0.233 0.202 0.151 0.0 0.0 0.426 0.577 0.816 0.943 1.0
G. Benedek et al./Surface energy for brittle fracture
570
4.3. Concluding remarks The main features of the model presented in this paper are the following. (i) The model can be applied to any crystal having more than one atom per unit cell (such as several ceramic materials), as well as to crystals whose point group might not contain the center of inversion (such as silicon). For these complex crystals the approach of ref. 1, ignoring the basic prescription eq. (6), would be inapplicable. (ii) The mean square atomic displacements are computed on the actual frequency distribution in the Brillouin zone, without referring to the Lindemann and Debye approximation. (iii) The effects of any possible ambiguity in the physical definition of interplanar force constants [17] are overcome by the above mentioned procedure. (iv) The calculation of the fracture energy using the critical displacement takes into account the anisotropy required to separate the crystal along directions other than that of large interplanar spacing, [100]. For this direction these results are nearly equivalent to those obtained considering local melting. This would be expected from the physical difference between a crystal undergoing (homogeneous) melting, as compared to one constrained under a state of stress leading to fracture. (v) These calculations are related to the thermodynamic surface energies one would obtain for zero surface entropy changes, namely at 0°K. Since the instantaneous cleavage process is conceived to be adiabatic, the temperature dependence of 7c derives only from that of the eigenfrequencies and critical displacements. Thus comparison of 7c with experimental values (table 2, last column) are appropriate at any temperature provided plastic relaxation does not actually occur.
Appendix A The lattice instabilities induced by an external uniaxial stress can be studied in terms of the interplanar potentials for the strain direction. For alkali halides, and for (100) and (11 O) directions, both repulsive and attractive interplanar potentials can be described by exponential forms; the total repulsive and attractive potential energies are written as
Urep =~- c ~
exp(--Irmn l/P),
mn
gat t =-12 b ~
exp(-Irmnl/~ ),
(A.1)
mn
where c and b are positive constants, and rmn is the distance between the ruth and the nth plane. The repulsive term is derived from the interionic potential in the classical Huggins-Mayer form, where the repulsive parameter p is the same for all alkali halides [13] : p = 0.3394 A for (100) direction and p = Vc2 x 0.3394 A for (110) direction. The exponential decay of interplanar attractive potentials is inferred from
G. Benedek et al./Surface energy for brittle fracture
571
2.
2.
(110)
(1oo) -1
lO
10 -2
2.
3
U
x
3
10-3
° t 10-4~
\ 1()-si-T
i
l
l
l
6
l
. . . . . . . . . .
I
1o\ i',,
,
. . . . .
Fig. 3. Decay of interplanar force constant fp(l) versus indicated neighbour order number for LiF the analogous behaviour of the interplanar force constants in the long range region, as shown in previous work [5] and in fig. 3; the values of a fitted on the calculated interplanar force constants are reported in table 4. The array of the 2Nplanes of the lattice, normal to the strain direction, is represented in fig. 4 by a linear chain. Since we are interested in the instability of this system produced by a uniform strain, which creates the two new surfaces at n = -½ and n = ½, we consider a constant interplanar distance X between all pairs of adjacent planes, except between -½ and ½ planes, whose distance is X + a; a works as a configurational coordinate. The sums of eqs. (A.1) are easily calculated c U r e p - eX / p -
Uatt -
b eX / a -
(2N 1
1 - e-a/° ~ - 1 ---~X/;f' e
[gN 1 ~-"
1 - e "-a/a 1 e --X~a]"
(A.2)
572
G. Benedek et al./Surface energy for brittle fracture
Table 4 Attractive range c~from interplanar force constants in terms of the interatomic first nearest neighbour a o Crystal
ao (A)
c~loo/ao
cq lo/ao
LiF LiC1 LiBr LiI NaF NaCI NaBr NaI KF KCI KBr KI RbF RbCI RbBr RbI
1.996 2.565 2.7506 3.0 2.31 2.789 2.9866 3.21 2.6735 3.116 3.262 3.489 2.82 3.2905 3.427 3.671
0.3076 0.3431 0.3268 0.33 0.3938 0.4314 0.4271 0.5179 0.4488 0.5067 0.5118 0.5212 0.4633 0.5059 0.5100 0.5288
0.5696 0.6244 0.6603 0.7120 0.4637 0.5763 0.6071 0.6546 0.4970 0.5620 0.5819 0.6183 0.5169 0.5487 0.5707 0.6004
The e q u i l i b r i u m i n t e r p l a n a r d i s t a n c e X 0 for the u n s t r a i n e d lattice (a = 0) is o b t a i n e d b y s e t t i n g d [Ure p + U a t t ] / d X = 0; X 0 is the s o l u t i o n o f t h e e q u a t i o n X0 X0 c a ( c o s h - - - 1) = bp ( c o s h - 1). p c~
(A.3)
U n d e r a n increasing e x t e r n a l tensile stress, t h e e q u i l i b r i u m d i s t a n c e X increases f r o m X 0 t o a critical value X c at w h i c h t h e p e r f e c t lattice c o n f i g u r a t i o n s (a = 0) b e c o m e s u n s t a b l e : this o c c u r s w h e n d 2 [Urep + U a t t ] / d a 2 = 0 at a = 0, n a m e l y w h e n
I I Ill----
- -
5
2
3 2
2
2
3 2
2
7 2
Fig. 4. Schematic drawing of the instability parameter a.
573
G. Benedek et al./Surface energy for brittle fracture
X X col2 (cosh --~c- 1)= bp 2 (cosh -5-c- 1).
p
(A.4)
c~
This situation corresponds to the "local melting" described in the text and requires a work per interplanar bond: 1
1
W=CleXC/Pl
1
1
eXofP_l]-b[eXc/-a_l
eXo/c~
1] .
(A.5)
For small critical displacement u c = X c - X 0' eqs. (A.5), (A.3) and (A.4) can be respectively approximated by:
c*a e -X°/p = b*p e-X°/~,
(A.6)
c*c~2 e -xc/° = b*o 2 e -Xc/°,
(A.7)
W ~ c*(e -xc/p - e -x°/p)
-
b*(e =xcla - e=X°/c~),
(A.8)
with
c* = c/(1 - e-X°/P) 2
and
b* = b/(1 - e-X°/a) 2.
The harmonic force constant ~bbetween the two half crystals is therefore the sum of all the force constants qSmn connecting planes on the opposite sites, namely:
~b= m<0 ~ ~bmn= m<0 ~ [~-2 exp(--lrmnl/P)--~ exp(-lrmn[/O0] , (A.9) n>0 n>0 ~X=X° * e_XO/O _ _b* ck = c___ _ e_XO/a" p2 a2
(A. 10)
Therefore, by setting r = alp and eliminating b* and c* in the above equations, one obtains: uc = p ~
r
log r,
(A.11)
27c = nW =-~ ckpucf(r) n,
(A.12)
where n is the number of surface unit cells per unit area and f(r)=
2 r- I logr
FLI
r+ l.l rr/(r-] ) ] "
(A.13)
The expression (A.12) for 27c can be transformed into the quasi-harmonic form, given by eq. (1) of the text by identifying the critical displacement for the easy direction, u c, with the thermal strain at the melting temperature. The thermal strain y is related to the atomic mean square displacement via cubic anharmonicity, ¢', by the well known relation [18]
G. Benedeket al./Surfaceenergyfor brittlefracture
574
~t
y --~ - ~
u2(T) = 1 ( 1 +1) u2(T)
(A.14)
and therefore pu c ~- (1 + 1 ) u2(Tm)
(for easy direction).
(A.15)
Thus, for the easy [100] direction eq. (1.12) becomes 27c 2 ½ ~buZ(Tm)B n,
(1.16)
where B = (1 + 1/r) f ( r ) is a slowly varying function o f t , approximately equal to 1.1 for 1 < r < 10, namely, for the usual values obtained for all alkali halides. For other directions, [hkl], eq. (A. 16) still holds for appropriate values of B [hkl] given by: p [hkl] u c [hkl] B[hkl] -
(A.17)
B, ,Ouc
where the quantities without indices refer to the easy direction. For the [110] direction uc[ll0] uc [ l l 0 ] B [ l l 0 ] =Vt2 ~ // B ~ 1.56 - - - -u C
(A.18)
C
The factor B[hkl] introduces in a simple way the anisotropy in the critical displacement necessary for uniform strain to produce cleavage. More sophisticated calculations can be envisaged but simple physical basis for introducing anisotropy awaits more detailed information on the microscopic nature of fracture.
Appendix B In this appendix the relevant formulae of the model proposed in ref. 7 calculating the stress at a crack tip are reviewed and applied to NaCl-structures. The force per unit length of the dislocation line acting in the slip direction is f=b.n.n,
(B.1)
where b is the Burgers vector, ~ is the stress tensor, and n the unit vector perpendicular to the slip plane. As in ref. 7, one frame of reference, S1, is selected in such a way that the x , y and z axes are directed normally to the fracture plane, in the direction of crack propagation and along the crack front, respectively. Assuming a plane strain condition, then
~= OXXoxY o~Oxy ozOz ,
(B.2)
G. Benedek et al./Surface energy for brittle fracture Ozz = p(a xx + Oyy),
575 (B.3)
and v is Poisson's ratio. Usually, b and n are given in the frame S 2 defined by the edges o f the elementary crystalline cube. To calculate the product j n eq. (B.1) in the frame S 2 one has to rotate 6 from S 1 into S 2 o -+ R" ~ " R - 1 .
(B.4)
The appropriate rotation matrix R for the (001) [ab0] crack system is (a2+ b 2 = 1):
R=
0 0
a b
-b a
1
0
0
,
(B.5)
and for the (110) [3cd] crack system it is (2c 2 + d 2 = 1):
- ~1 - c
_@2_2d
-L a 0
d
I
-V~ c
Assuming for oij the expression given by Cottrell, a computer program was set up to calculate the dependence of f on the cylindrical angle O between the x-axis o f S 1 and the normal to the relevant slip plane. Table 3 shows the results, as far as only the angular dependence f(O) is considered, obtained for the maximum value o f f ( O ) for the relevant slip systems o f NaCl-structures, and for u = 0.248.
References [ 1] [2] [3] [4] [5]
G. Caglioti, G. Rizzi and J.C. Bilello, J. Appl. Phys. 42 (1971) 4271. G. Caglioti, G. Rizzi and J.C. Bilello, J. Appl. Phys. 43 (1972) 3600. J.C. Bilello and G. Caglioti, Scripta Met. 6 (1972) 1041. A.J.E. Foreman and W.M. Lomer, Proc. Phys. Soc. (London) B 70 (1957) 1143. G. Benedek, Phys. Status Solidi 58 (1973) 661; G. Benedek, in: Proc. Intern. School "Enrico Fermi", Course 58, Varenna, 1973 (Academic Press, to be published). [6] R.G. Linford and L.A. Mitchell, Surface Sci. 27 (1971) 142. [7] W.R. Tyson, R. Ayres and D.F. Stein, Acta Met. 21 (1971) 621. [8] A.A. Maradudin, E.W. Montroll, G.H. Weiss and I.P. Ipatova, Solid State Phys. Suppl. 3 (2nd ed. 1971) 62. [9] U. Schr6der, Solid State Commun. 4 (1966) 347. [10] V. N/isslein and V. SchriSder, Phys. Status Solidi 21 (1967) 309.
576
G. Benedek et al./SurJace energy Jbr brittle fracture
[11] M. Born and K. ttuang, Dynamical Theory of Crystal Lattices (Oxford Univ. Press, London, 1954). [12] M. Born and O. Stern, Sitzber. Preuss. Akad. Wiss. Berlin 48 (1919) 901. [13] M.P. Tosi, Solid State Phys. 16 (1964) 1. [14] J.J. Gilman, in: Fracture, Eds. B.L. Averbach, D.K. Felbeck, G.T. Hahn, D.A. Thomas (Wiley, New York, (1959)) p. 193. [15] S. Purushothaman and J.J. Burton, J. Appl. Phys. 45 (1974) 1466. [16] G. Caglioti, G. Rizzi and J.C. Bilello, J. Appl. Phys. 45 (1974) 1468. [17] R.S. Leigh, B. Szigeti and V.K. Tewary, Proc. Roy. Soc. (London A230 (1971) 505. [18] E. Fermi, Molecules, Crystals and Quantum Statistics (Benjamin, New York, 1966). [19] J.J. Gilman, J. Appl. Phys. 31 (1960) 2208. [20] R. Shuttleworth, Proc. Phys. Soc. (London) A62 (1949) 167. [21] P.L. Gutshall and G.E. Gross, J. Appl. Phys. 36 (1965) 2459. [22] R.R. Weiler, J. Beeckmans and R. Mclntosh, Can. J. Chem. 39 (1961) 1360. [23] S.M. Weiderhorn, R.L. Moses and D.L. Bean, J. Am. Ceram. Soc. 53 (1970) 18. [24] J.J. Gihnan, J. Appl. Phys. 34 (1963) 3085.
SURFACE ENERGY FOR BRITTLE FRACTURE OF ALKALI HALIDES FROM LATTICE DYNAMICS # G. B E N E D E K Gruppo Nazionale di Struttura della Materia del C.N.R., and Istituto di Fisica dell'Universitb, 20133 Milano, Italy S. B O F F I [stituto di Fisica dell'Universitb, 27100 Pavia, Italy, and Istituto Nazionale di Fisica Nucleare, Gruppo di Pavia, Italy G. C A G L I O T I lstituto di Ingegneria Nucleare, CESNEF, Politecnico di Milano, 20133 Miiano, Italy and J.C. B I L E L L O * Department of Materials Science, State University of New York at Stony Brook, Stony Brook, New York 11790, U.S.A. Received 6 August 1974
A lattice dynamics approach to the surface energy "r for brittle fracture of several ionic crystals is presented, based on recent work on surface dynamics and a reformulation of a model previously worked out for metals. The model requires the knowledge of the crystal structure, the eigenfrequencies and eigenvectors of the normal modes of vibration, as well as an analytical definition of the critical interplanar displacement appropriate to the cleavage mode. Some temperature dependent values of 7 are also computed for the (100) and (110) planes. Qualitatively the results indicate the correct cleavage systems for all cases and quantitively are reasonably consistent with the available experimental values.
1. I n t r o d u c t i o n In previous works [I-3]
a m e t h o d for calculating w i t h o u t a r b i t r a r y p a r a m e t e r s
"~Work supported under a contract with the Consiglio Nazionale deUe Ricerche (Italy) and the U.S. Atomic Energy Commission (U.S.A.) under the auspices of the CNR-NSF (Italy-U.S.A.) cooperative research agreement. * Currently on sabbatical leave at the: Istituto di Ingegneria Nucleare, CESNEF, Politecnico, 20133 Milano, Italy.
562
G. Benedek et al./Surface energy for brittle fracture
the surface energy for cleavage of several monoatomic metallic crystals has been proposed, based on the knowledge of both the mean square atomic displacement at the melting and boiling temperatures, as well as on the lattice dynamics concept of interplanar force constants [4]. This earlier approach cannot be simply extended to ionic NaC1 structures, where the unit cell contains two atoms of different masses and opposite charges. A new concept is required, borrowed from surface dynamics [5], of force constants fp(1) between a single positive (p = +) or negative (p = - ) ion in the /th plane below the fracture plane and the whole, rigid half-crystal above the fracture plane. Furthermore, the present report introduces a new concept for defining analytically the "critical displacement", Uc, necessary to cause interplanar separation. Using u c the "work" to produce cleavage in cubic crystals can be calculated for the principal low index (hkl) planes with a more realistic treatment of the crystalline anisotropy. This is shown to remove the difficulties which arose from defining the critical displacement in terms of the mean square atomic amplitude (U2m)at melting. This earlier, first order approach assumed that higher index planes, of closer spacing, cleave when subjected to the same additional separation, as that invoked for (100) cleavage, because of the inherent isotropy of u m (hkl). The displacement u c is derived from the criterion for mechanical instability of a system of rigid planes coupled by a realistic interplanar potential function. For (100) planes it is shown that this new approach (i.e. using Uc) g~yes near!y the same results as those calculated from the earlier meth1 llZ.~ od ( 1' .e. us'ng m), but for high index planes the results differ significantly. In section 2 the model is presented, in section 3 the results are reported, and the conclusions, sections 4, contain a discussion which includes a comparison with the approach based on interplanar potentials [6] as well as considerations of the cleavage plane actually observed in real crystals, where plastic relaxation can occur [7].
2. The model The creation of two new surfaces by cleavage might be conceived as activated by either separation of the planes (hkl) of the crystal in a particular [hkl] direction until the system becomes mechanically unstable or by successive steps of local melting and boiling of the crystal along the fracture plane, as originally proposed [1 ]. The concepts involved are shown to be nearly equivalent for easy (100) cleavage, but diverge for higher index planes. With reference to the first process, the achievement of the critical separation requires a tensile stress acting against the interplanar potential. As this stress is applied, the shape of this potential is modified while a uniform stretch of the interplanar bonds occurs. As the amount of dilational strain increases, the perfect lattice configuration becomes unstable and other configurations of the system, exhibiting a pair of free surfaces, become possible. The transition between the stretched and the split configurations is assumed to occur when the second derivative of the interplanar potential with respect to the dilational strains is zero (Appendix A). The critical displacement depends, as the interplanar potential, on the tensile direction.
G. Benedek et al./Surface energy for brittle fracture
563
(I)
I
Ifp(3)
~~-~P I I
Fig. 1. Force constants fp(l) between a single ion in the lth plane below the fracture surface and the whole rigid half crystal above. In the case originally proposed, local melting, where a rearrangement of interatomic bonds is operative, is obtained by loading the system of spring constantsfp(l) (fig. 1) by a potential energy 7 m sufficient to separate the two to-be-split-half-crystals by an amount corresponding to the atomic square displacement: 27~ : f f ~ f p ( l ) $ [ u 2 ( T m ) - U 2 ( T ) ] 1
np,
( p = +,-),
(1)
where np is the number-density o f p atoms on the crystal plane, u_2(T) is the temperp ature dependent mean square atomic displacement. Local boiling, where the rupture of the bonds actually occurs, is roughly estimated here by adding to 7 m = 7 m + 7 m the thermal (potential) energy: ,yb =½kB(T b _ Tm ) (n+ + n _ ) ,
(2)
where T b and T m are the boiling and melting temperatures respectively and k B the Boltzmann constant . The relevant surface energy is obtamed as .
°
* The introduction of the factors 1/3 inserted in eq.(1) and 1/2 in eq.(2) is discussed in the last paragraph of section 4.1.
G. Benedek et al./Surface energy for brittle Jracture
564
*
m
7 = 3'+ +
,ym + ,)lb.
(3a)
For the first process the surface energy is given by: 7 = ")'c + ,),b,
(3b)
where 7c, as given in eq. (A.16), takes the same form as eq.(1), with the factor ~ replaced by an anisotropy factor B. A numerical treatment of eq.(1) requires the solution of the dynamic equations of the atomic motions in Fourier space, i.e. the knowledge of the frequencies w(q,j) and the polarization vectors, ep(q, j), of the vibrational modes of wave vector q and branch index j. The mean square displacement of the atom of mass Mp is then given, by: u2(r)=N
.
2Mp~(q,j) [eP (q'j)12 [ 2 n ( q , j ) + 11,
(4)
where N is the number of crystal cells and n(q, j) is the phonon population factor [exp (l~w(q, j)/kgT ) - 1]-1 (see e.g. ref. 8). Each of the interplanar force constants building the set {fp(1)} is a sum over the negative half of the crystal planes (the left hand side of fig. 1) defined as {-l'}, of the interatomic force constants q~c~a(in the present context the index c~, omitted in eq. (1), refers to the third cartesian component, orthogonal to the fracture plane):
where L and L' are the cell indices. In order to relate fpc~(l) to the eigenfrequencies and the eigenvectors, it is necessary to express ~ba,~in terms of its Fourier representation via the dynamical matrix and using the equations of motion in Fourier space. One thus obtains:
fpa(l)=N~
~ %Mp,)~e~p,(q,j)eo~p(q,J) l'p',q 3,]
× co2(q,/) exp [ia
q3(l +/')],
q = (0,0,q3),
(6)
where N L is the number of atomic layers, or crystal planes, along the direction 3, orthogonal to the fracture plane. It should be noticed that Elfp(l) reduces to the linear combination, K, of interplanar force constants, kr, NL
K =~
rkr,
f= 1
utilized in other papers on the subject.
(7)
565
G. Benedek et al./Surface energy for brittle fracture
r
i
X
/
[ oo]
[.o]
Fig. 2. Schematic representation of 1/8 of the first Brillouin zone used to calculate the eigenfrequencies and eigenvectors. 3. Results The above model has been applied to several ionic crystals with NaCI structure. The eigenfrequencies and eigenvectors entering eqs. (4) and (6) have b e e n c o m p u t e d on the basis o f the breathing shell model [9], which is k n o w n to give an excellent fit to the experimental dispersion relations [10]. Table 1 Temperature dependence of 3'* for (100) planes for some selected crystals Crystal
T (°K)
~+
LiF
0 300
78 40
KC1
0 300
NaC1
3'm
3,b
3'*
119 90
144 14-2
341 272
50 31
56 37
51 50
158 119
0 300
48 32
68 51
54 53
170 137
KBr
0 300
44 27
53 33
46 45
143 104
NaI
0 300
36 19
51 31
44 43
131 93
KI
0 300
46 20
51 26
37 36
134 82
G. Benedek et al./Surface energy for brittle fracture
566
Table 2 S u m m a r y of calculations of surface energy 3, (erg cm-2 ) according to eqs. (3) and (4), with comparison with other theoretical and experimental results Crystal
Plane
3m
,,/m _
3"b
3* [eq.(3)]
"rc
3'
Exp. (E) and other (T) values
LiF
(100)
78
119
144
341
209
353
340 a (E, 78°K), 374 b (T)~ 363 c (T), 414 ~ (T), 780 b (T)
(110)
84
110
102
297
515
617
LiC1
(100) (110)
19 59
69 62
75 52
163 173
67 195
142 247
LiBr
(100) (110)
19 50
57 50
66 46
142 147
54 161
120 207
Lil
(100) (110)
20 58
45 39
56 40
121 137
39 119
95 159
NaF
(100) (110)
78 82
94 92
91 65
263 239
147 259
238 324
NaCI
(100)
48
68
54
170
91
145
(110)
54
56
38
149
163
201
280 270 (E), 214 (T) 345
NaBr
(100) (110)
37 45
61 50
49 35
147 130
62 121
111 156
248 b (T) 288 b (T)
NaI
(100) (110)
36 38
51 41
44 31
131 110
65 107
109 138
KF
(100) (110)
64 60
60 66
64 45
188 171
94 158
159 203
KCI
(100)
50
56
51
158
80
131
(110)
45
47
36
128
116
152
(100)
44
53
46
143
69
116
(110)
40
43
32
115
102
134
306 b (T), 157 c (T), 94 d (T) 253 b (T)
KI
(100) (110)
46 37
51 38
37 26
134 101
66 91
103 128
233 b (T) 165 b (T)
RbF
(100) (110)
59 54
58 73
64 45
180 172
75 131
139 176
RbC1
(100) (110)
40 38
45 40
43 30
128 108
55 85
98 115
RbBr
(100) (110)
34 34
40 34
39 27
113 95
48 75
87 102
RbI
(100) (110)
29 26
34 27
34 24
96 77
41 61
75 86
KBr
110 318 (T), 271
e (E, 78°K), f (E), 3 7 0 g 310 b (T), c (T), 151 d b (T)
h (E, 298°K), b (T)j 171 c 108 u (T) b (T)
a Ref. 19. b Ref. 14. c Ref. 20. d Ref. 12. e Ref. 21. f Ref. 22. g Ref. 23. h Ref. 24.
G. Benedek et aL/Surface energy for brittle fracture
567
In the calculation of the mean square displacement a mesh of 4096 points in the Brillouin zone has been considered, while in the calculation offpa(l) the sum over q3 in eq. (6) runs over 49 equally spaced values on the FX segment of fig. 2 for the (100) cleavage plane, and over 49 values on the VX' segment for the (110) cleavage plane. The resulting surface energies have been numerically computed both at 0°K and 300°K, using consistently the appropriate input data. In practice the values l ~< 5 are found sufficient for the sum of eq. (1), since the series Zlfp(l) converges rapidly, in spite of the long range Coulomb potential, in ionic crystals [5]. Nevertheless, for the sake of completeness l values up to 20 were considered. The numerical values of ~,m and ,,/b and of the total surface energy ~,* are presented for 0°K and 300°K in tal~le 1. In table 2 the values of 7 and 7 are compared for 0°K with experimental values and other theoretical results.
4. Discussion and conclusion 4.1. Comparison with other models For alkali halides the classical calculations of surface energies are based on the Born model [ I 1]. It was originally proposed for the purpose of evaluating the (cohesive) lattice energy of crystals arising from interactions between the ions. In this model the surface energy of a finite crystal is computed as one half (of the negative) of the interaction energy of the finite crystal itself and an infinite crystal enveloping it. One then obtains for the specific surface energy in the NaC1 structure the following simple relation [12] ~/ = A Ze2 /a30,
(8)
where A is a constant depending on the selected surface [e.g., A = 0.1173 for the (100) face] and a 0 is the lattice parameter. More detailed calculations are reviewed in ref. 13. Many other approaches have been suggested to compute the surface energies of crystals. The ones which seem particularly relevant to the present discussion include Gilman's [14], where ionic crystals were specifically considered in the frame of the continuum mechanics, and the recent work of Linford and Mitchell [6] based on the concept of an interplanar potential. Present values of 7 obtained from using the critical displacement, Uc, systematically produce 7 (100) < 7 (110) as would be expected qualitatively from the observed cleavage modes of these crystals. The magnitude of 7 is also reasonably consistent with available experimental results. Earlier work [14] which had attempted to specify the correct cleavage plane of ionic crystals from calculated 7 values had predicted the wrong cleavage plane in half of the considered cases. The possibility exists that plastic relaxation could account for the apparent discrepancy of the results in ref. 14.
568
G. Benedek et al./SurJhce energy ]br brittle ]kacture
However, it will be shown below that, at least for ionic crystals, including plastic energy considerations is of no help and in fact leads to predicting the (110) cleavage plane for all NaCl-type ionic crystals. The (110) plane has never been observed as the primary cleavage plane for this type of crystalline solid. The interplanar potential model [6] and the present one are shown to lead to similar answers in Appendix A. This connection demonstrates that the harmonic approximation can be coherently framed within the interplanar potential approach, although in general it should be applied cautiously to problems involving large displacements. The present values are in general agreement with other theoretical models (see the last column of table 1), with the exception of the Shuttleworth and the Born-Stern models, where the (110) values are exceptionally high. Before concluding this subsection a few comments will be made linking the present work to an earlier one which used a similar physical model as a starting point to evaluate 7 for metals [ 1 ]. The definition ym and 7 b in eqs. (1) and (2) are different from the corresponding ones in eqs. (5) and (8) of ref. 1. In eq. (1) the surface energy for local melting is computed substracting the thermal energy already delivered to the crystal for the very reason that the crystal is kept at a finite temperature T. The advantage of the current procedure is that it allows one to directly compute the temperature dependence of 7 (neglecting plastic relaxation to be considered below). Furthermore, as indicated before, 7 m and 7 b are suggested here to be smaller by a factor ½ and ½ respectively. The introduction of these factors shows a limit, recently discussed in refs. 15 and 16, to the possibility of simply extending to ionic crystals the arguments of ref. 1, valid for metals. The factor three - expressing the ratio between the total mean square atomic displacement and its projection along the direction of separation - has been chosen here for two reasons: (i) The dependence of the interatomic potentials in metals as compared with ionic crystals shows a repulsive part at distances larger than the equilibrium one. This suggests that metals should be intrinsically more tenacious than ionic crystals. (ii) The generally lower Poisson's ratio found in ionic crystals evidences smaller transverse energy dissipation (i.e. within the plane of cleavage) than occurring in metals; thus the x3 factor corresponds more closely to the single degree of freedom process envisaged here. The factor of two difference between the present eq. (2) and eq. (8) of ref. 1, is made plausible by the fact that the actual process of boiling away of one surface with respect to the other, can be conceived as due to the constraint of purely potential energy to the single degree of freedom leading to separation. 4.2. Role o f plastic relaxation
The calculation for 7, made above, was based on the inherent assumption that the external energy supplied to the crystal to produce cleavage went solely into creating two perfect crystalline surfaces. Such an assumption would not be valid for a real
G. Benedek et al./Surface energy for brittle fracture
569
crystal where defect generation accompanies cleavage and creates an additional energy term which must be considered. The associated plastic energy for a particular combination of cleavage and slip systems and the role this plays in governing the cleavage plane for NaCl-type crystals has been evaluated on the basis of a model presented for bcc metals by Tyson et al. [7] (see Appendix B for formulation). Physically the stress field at the crack-tip can relax via plastic deformation to a degree which, for any slip system, is proportional to the corresponding resolved shear stress (RSS). The magnitude of RSS depends on the mutual orientation factor f(O) determined by the crack propagation direction within its cleavage plane and the slip systems (hkl) [h'k'f]. Table 3 presents maximum f(O) values for various cleavage-slip system combinations. Whenf(O) = 1, the entire stress field of the crack is transferred as shear stress to the slip system, and at the other extreme, when ](O) = 0, no component of the shear stress is effective. Thus, small f(O) values correspond to low driving force for plastic relaxation and hence minimum plastic energy. A particular cleavage system will have the lowest amount of plastic energy associated with it if f(O) is a minimum for all possible slip system combinations. For NaCl-type crystals table 3 shows that this occurs for (110) [001] cleavage. Within this frame, due to the probability of higher plastic energy to create (100) cleavage, invoking plastic energy requirements to rationalize theories which evaluate 3'110 slightly lower than or nearly equal to 3'100 appear inconsistent with the experimentally observed (100) cleavage. Thus 3'110 > 3'100 (for that matter, 3"hkl > 3'100 ) as derived in previous sections, is the most likely consistent anatytical result that can explain the (100) cleavage which occurs for all the NaCl-type ionic crystals. Table 3 Maximum values of the orientation factor f(o) for various cleavage systems and the most common slip systems for NaCl-type crystals, following ref. 7 Cleavage systems
(001) (001) (001) (001) (001) (110) (110) (110) (110) (110) (110)
[010] [1501 [1301 [120] [110] [0011 [113] [~12] [111 ] [2211 [71Ol
Slip systems, f(O) values (101) [1011 (701) [101]
(011) [Oil] (Oil) [0111
(110) [7101 (710) [1101
0.539 0.547 0.561 0.583 0.652 0.506 0.528 0.516 0.495 0.501 0.513
O.77O 0.761 0.746 0.722 0.652 0.506 0.528 0.516 0.495 0.501 0.513
0.252 0.233 0.202 0.151 0.0 0.0 0.426 0.577 0.816 0.943 1.0
G. Benedek et al./Surface energy for brittle fracture
570
4.3. Concluding remarks The main features of the model presented in this paper are the following. (i) The model can be applied to any crystal having more than one atom per unit cell (such as several ceramic materials), as well as to crystals whose point group might not contain the center of inversion (such as silicon). For these complex crystals the approach of ref. 1, ignoring the basic prescription eq. (6), would be inapplicable. (ii) The mean square atomic displacements are computed on the actual frequency distribution in the Brillouin zone, without referring to the Lindemann and Debye approximation. (iii) The effects of any possible ambiguity in the physical definition of interplanar force constants [17] are overcome by the above mentioned procedure. (iv) The calculation of the fracture energy using the critical displacement takes into account the anisotropy required to separate the crystal along directions other than that of large interplanar spacing, [100]. For this direction these results are nearly equivalent to those obtained considering local melting. This would be expected from the physical difference between a crystal undergoing (homogeneous) melting, as compared to one constrained under a state of stress leading to fracture. (v) These calculations are related to the thermodynamic surface energies one would obtain for zero surface entropy changes, namely at 0°K. Since the instantaneous cleavage process is conceived to be adiabatic, the temperature dependence of 7c derives only from that of the eigenfrequencies and critical displacements. Thus comparison of 7c with experimental values (table 2, last column) are appropriate at any temperature provided plastic relaxation does not actually occur.
Appendix A The lattice instabilities induced by an external uniaxial stress can be studied in terms of the interplanar potentials for the strain direction. For alkali halides, and for (100) and (11 O) directions, both repulsive and attractive interplanar potentials can be described by exponential forms; the total repulsive and attractive potential energies are written as
Urep =~- c ~
exp(--Irmn l/P),
mn
gat t =-12 b ~
exp(-Irmnl/~ ),
(A.1)
mn
where c and b are positive constants, and rmn is the distance between the ruth and the nth plane. The repulsive term is derived from the interionic potential in the classical Huggins-Mayer form, where the repulsive parameter p is the same for all alkali halides [13] : p = 0.3394 A for (100) direction and p = Vc2 x 0.3394 A for (110) direction. The exponential decay of interplanar attractive potentials is inferred from
G. Benedek et al./Surface energy for brittle fracture
571
2.
2.
(110)
(1oo) -1
lO
10 -2
2.
3
U
x
3
10-3
° t 10-4~
\ 1()-si-T
i
l
l
l
6
l
. . . . . . . . . .
I
1o\ i',,
,
. . . . .
Fig. 3. Decay of interplanar force constant fp(l) versus indicated neighbour order number for LiF the analogous behaviour of the interplanar force constants in the long range region, as shown in previous work [5] and in fig. 3; the values of a fitted on the calculated interplanar force constants are reported in table 4. The array of the 2Nplanes of the lattice, normal to the strain direction, is represented in fig. 4 by a linear chain. Since we are interested in the instability of this system produced by a uniform strain, which creates the two new surfaces at n = -½ and n = ½, we consider a constant interplanar distance X between all pairs of adjacent planes, except between -½ and ½ planes, whose distance is X + a; a works as a configurational coordinate. The sums of eqs. (A.1) are easily calculated c U r e p - eX / p -
Uatt -
b eX / a -
(2N 1
1 - e-a/° ~ - 1 ---~X/;f' e
[gN 1 ~-"
1 - e "-a/a 1 e --X~a]"
(A.2)
572
G. Benedek et al./Surface energy for brittle fracture
Table 4 Attractive range c~from interplanar force constants in terms of the interatomic first nearest neighbour a o Crystal
ao (A)
c~loo/ao
cq lo/ao
LiF LiC1 LiBr LiI NaF NaCI NaBr NaI KF KCI KBr KI RbF RbCI RbBr RbI
1.996 2.565 2.7506 3.0 2.31 2.789 2.9866 3.21 2.6735 3.116 3.262 3.489 2.82 3.2905 3.427 3.671
0.3076 0.3431 0.3268 0.33 0.3938 0.4314 0.4271 0.5179 0.4488 0.5067 0.5118 0.5212 0.4633 0.5059 0.5100 0.5288
0.5696 0.6244 0.6603 0.7120 0.4637 0.5763 0.6071 0.6546 0.4970 0.5620 0.5819 0.6183 0.5169 0.5487 0.5707 0.6004
The e q u i l i b r i u m i n t e r p l a n a r d i s t a n c e X 0 for the u n s t r a i n e d lattice (a = 0) is o b t a i n e d b y s e t t i n g d [Ure p + U a t t ] / d X = 0; X 0 is the s o l u t i o n o f t h e e q u a t i o n X0 X0 c a ( c o s h - - - 1) = bp ( c o s h - 1). p c~
(A.3)
U n d e r a n increasing e x t e r n a l tensile stress, t h e e q u i l i b r i u m d i s t a n c e X increases f r o m X 0 t o a critical value X c at w h i c h t h e p e r f e c t lattice c o n f i g u r a t i o n s (a = 0) b e c o m e s u n s t a b l e : this o c c u r s w h e n d 2 [Urep + U a t t ] / d a 2 = 0 at a = 0, n a m e l y w h e n
I I Ill----
- -
5
2
3 2
2
2
3 2
2
7 2
Fig. 4. Schematic drawing of the instability parameter a.
573
G. Benedek et al./Surface energy for brittle fracture
X X col2 (cosh --~c- 1)= bp 2 (cosh -5-c- 1).
p
(A.4)
c~
This situation corresponds to the "local melting" described in the text and requires a work per interplanar bond: 1
1
W=CleXC/Pl
1
1
eXofP_l]-b[eXc/-a_l
eXo/c~
1] .
(A.5)
For small critical displacement u c = X c - X 0' eqs. (A.5), (A.3) and (A.4) can be respectively approximated by:
c*a e -X°/p = b*p e-X°/~,
(A.6)
c*c~2 e -xc/° = b*o 2 e -Xc/°,
(A.7)
W ~ c*(e -xc/p - e -x°/p)
-
b*(e =xcla - e=X°/c~),
(A.8)
with
c* = c/(1 - e-X°/P) 2
and
b* = b/(1 - e-X°/a) 2.
The harmonic force constant ~bbetween the two half crystals is therefore the sum of all the force constants qSmn connecting planes on the opposite sites, namely:
~b= m<0 ~ ~bmn= m<0 ~ [~-2 exp(--lrmnl/P)--~ exp(-lrmn[/O0] , (A.9) n>0 n>0 ~X=X° * e_XO/O _ _b* ck = c___ _ e_XO/a" p2 a2
(A. 10)
Therefore, by setting r = alp and eliminating b* and c* in the above equations, one obtains: uc = p ~
r
log r,
(A.11)
27c = nW =-~ ckpucf(r) n,
(A.12)
where n is the number of surface unit cells per unit area and f(r)=
2 r- I logr
FLI
r+ l.l rr/(r-] ) ] "
(A.13)
The expression (A.12) for 27c can be transformed into the quasi-harmonic form, given by eq. (1) of the text by identifying the critical displacement for the easy direction, u c, with the thermal strain at the melting temperature. The thermal strain y is related to the atomic mean square displacement via cubic anharmonicity, ¢', by the well known relation [18]
G. Benedeket al./Surfaceenergyfor brittlefracture
574
~t
y --~ - ~
u2(T) = 1 ( 1 +1) u2(T)
(A.14)
and therefore pu c ~- (1 + 1 ) u2(Tm)
(for easy direction).
(A.15)
Thus, for the easy [100] direction eq. (1.12) becomes 27c 2 ½ ~buZ(Tm)B n,
(1.16)
where B = (1 + 1/r) f ( r ) is a slowly varying function o f t , approximately equal to 1.1 for 1 < r < 10, namely, for the usual values obtained for all alkali halides. For other directions, [hkl], eq. (A. 16) still holds for appropriate values of B [hkl] given by: p [hkl] u c [hkl] B[hkl] -
(A.17)
B, ,Ouc
where the quantities without indices refer to the easy direction. For the [110] direction uc[ll0] uc [ l l 0 ] B [ l l 0 ] =Vt2 ~ // B ~ 1.56 - - - -u C
(A.18)
C
The factor B[hkl] introduces in a simple way the anisotropy in the critical displacement necessary for uniform strain to produce cleavage. More sophisticated calculations can be envisaged but simple physical basis for introducing anisotropy awaits more detailed information on the microscopic nature of fracture.
Appendix B In this appendix the relevant formulae of the model proposed in ref. 7 calculating the stress at a crack tip are reviewed and applied to NaCl-structures. The force per unit length of the dislocation line acting in the slip direction is f=b.n.n,
(B.1)
where b is the Burgers vector, ~ is the stress tensor, and n the unit vector perpendicular to the slip plane. As in ref. 7, one frame of reference, S1, is selected in such a way that the x , y and z axes are directed normally to the fracture plane, in the direction of crack propagation and along the crack front, respectively. Assuming a plane strain condition, then
~= OXXoxY o~Oxy ozOz ,
(B.2)
G. Benedek et al./Surface energy for brittle fracture Ozz = p(a xx + Oyy),
575 (B.3)
and v is Poisson's ratio. Usually, b and n are given in the frame S 2 defined by the edges o f the elementary crystalline cube. To calculate the product j n eq. (B.1) in the frame S 2 one has to rotate 6 from S 1 into S 2 o -+ R" ~ " R - 1 .
(B.4)
The appropriate rotation matrix R for the (001) [ab0] crack system is (a2+ b 2 = 1):
R=
0 0
a b
-b a
1
0
0
,
(B.5)
and for the (110) [3cd] crack system it is (2c 2 + d 2 = 1):
- ~1 - c
_@2_2d
-L a 0
d
I
-V~ c
Assuming for oij the expression given by Cottrell, a computer program was set up to calculate the dependence of f on the cylindrical angle O between the x-axis o f S 1 and the normal to the relevant slip plane. Table 3 shows the results, as far as only the angular dependence f(O) is considered, obtained for the maximum value o f f ( O ) for the relevant slip systems o f NaCl-structures, and for u = 0.248.
References [ 1] [2] [3] [4] [5]
G. Caglioti, G. Rizzi and J.C. Bilello, J. Appl. Phys. 42 (1971) 4271. G. Caglioti, G. Rizzi and J.C. Bilello, J. Appl. Phys. 43 (1972) 3600. J.C. Bilello and G. Caglioti, Scripta Met. 6 (1972) 1041. A.J.E. Foreman and W.M. Lomer, Proc. Phys. Soc. (London) B 70 (1957) 1143. G. Benedek, Phys. Status Solidi 58 (1973) 661; G. Benedek, in: Proc. Intern. School "Enrico Fermi", Course 58, Varenna, 1973 (Academic Press, to be published). [6] R.G. Linford and L.A. Mitchell, Surface Sci. 27 (1971) 142. [7] W.R. Tyson, R. Ayres and D.F. Stein, Acta Met. 21 (1971) 621. [8] A.A. Maradudin, E.W. Montroll, G.H. Weiss and I.P. Ipatova, Solid State Phys. Suppl. 3 (2nd ed. 1971) 62. [9] U. Schr6der, Solid State Commun. 4 (1966) 347. [10] V. N/isslein and V. SchriSder, Phys. Status Solidi 21 (1967) 309.
576
G. Benedek et al./SurJace energy Jbr brittle fracture
[11] M. Born and K. ttuang, Dynamical Theory of Crystal Lattices (Oxford Univ. Press, London, 1954). [12] M. Born and O. Stern, Sitzber. Preuss. Akad. Wiss. Berlin 48 (1919) 901. [13] M.P. Tosi, Solid State Phys. 16 (1964) 1. [14] J.J. Gilman, in: Fracture, Eds. B.L. Averbach, D.K. Felbeck, G.T. Hahn, D.A. Thomas (Wiley, New York, (1959)) p. 193. [15] S. Purushothaman and J.J. Burton, J. Appl. Phys. 45 (1974) 1466. [16] G. Caglioti, G. Rizzi and J.C. Bilello, J. Appl. Phys. 45 (1974) 1468. [17] R.S. Leigh, B. Szigeti and V.K. Tewary, Proc. Roy. Soc. (London A230 (1971) 505. [18] E. Fermi, Molecules, Crystals and Quantum Statistics (Benjamin, New York, 1966). [19] J.J. Gilman, J. Appl. Phys. 31 (1960) 2208. [20] R. Shuttleworth, Proc. Phys. Soc. (London) A62 (1949) 167. [21] P.L. Gutshall and G.E. Gross, J. Appl. Phys. 36 (1965) 2459. [22] R.R. Weiler, J. Beeckmans and R. Mclntosh, Can. J. Chem. 39 (1961) 1360. [23] S.M. Weiderhorn, R.L. Moses and D.L. Bean, J. Am. Ceram. Soc. 53 (1970) 18. [24] J.J. Gihnan, J. Appl. Phys. 34 (1963) 3085.