.J. Chem. SOC., Faraday Trans. 1, 1988, 84(4), 899-916 The Interaction between Ions and the Activation Barrier of Elementary Events of Crystal Growth and Evaporation Vincent K. W. Cheng*'f and Bruce A. W. Coller Department of Chemistry, Monash University, Clayton 3168, Victoria, Australia Edgar R. Smith$ Department of Mathematics, University of Melbourne, Parkville 3052, Victoria, Australia The activation barrier of the elementary processes of the growth and evaporation of an NaC1-type ionic crystal in contact with its vapour such as surface diffusion, incorporation and detachment, are examined in terms of the sum of the pair potential energy between adions and lattice ions. Lattice- sum studies of the Coulombic part of the potential energy was evaluated exactly using integral transform which is equivalent to the solution of the Poisson equation derived earlier.The Coulombic potential-energy function experienced by adions decays rapidly to zero within a distance less than one interionic separation above the surface. Three forms of non-Coulombic potential energy were evaluated by direct lattice summation. Both the pair potential-energy function derived from ab initio quantum mechanics and the effective Born-Mayer potential-energy function are short-ranged. More- over, at a binding site, the ab initio potential energy experienced by an adion is attractive and slightly larger than that of the Coulombic part, whilst the Born-Mayer potential function gives rise to a small but repulsive contribution to the total potential energy. The total potential energy of an adion was shown not to be proportional to the number of nearest neighbours, despite being effectively short-ranged.The non-Coulombic potential energy derived from a combination of the Mott-Littleton polarisation energy and overlap repulsion is long-ranged, and at an equilibrium adsorption site it is attractive and is very large compared with the Coulombic potential-energy function. All non-Coulombic potential energies are dependent on the type of charge of the adion. The activation barrier to incorporation, either directly from the vapour phase or after surface diffusion, is zero, whilst the barrier to detachment is determined by the total potential energy of the surface ion. The activation barrier for surface diffusion, as calculated from the variation of adion potential energy along a diffusion path, is shown to be contributed almost exclusively from the Coulombic part.Of the two possible diffusion paths, the lateral path along the unit cell (path 1) has a larger activation barrier than the preferred diffusion path diagonal to the unit cell (path 2) which has a barrier of almost half the size. The corresponding diffusion will be more restricted on the surface of bivalent metal sulphates because of the larger charges of lattice ions. For both mathematical and computational simplicity, the dependence of the rate of growth and dissolution on concentration derived from the one-component Kossel crystal model is considered to be adequate for the analysis of experimental rate law for ionic crystals.Such analysis has been reported with apparent success in identifying the growth mechanism, such as nucleation at a large distance from eq~ilibrium.l-~ The rate- ? Present address: Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong. 1 Present address : Department of Mathematics, La Trobe University, Bundoora 3083, Victoria, Australia. 899900 Crystal Growth and Evaporation determining step, however, has shown to be inconclusively established on the basis of the rate law a10ne.~ It has been shown that crystal growth and dissolution dominated by surface diffusion or by direct transfer have the same form of dependence of rate on c~ncentration.~.~ Thus recourse to other data is essential in the establishment of the rate- determining step.The rate-determining step in a complex reaction sequence may sometimes be identified in terms of the relative magnitudes of the activation energies of the elementary steps. In a sequence of reactions it is supposed to have the largest activation energy, and in parallel, competitive reactions it is supposed to have the smallest. In crystal growth and dissolution the Arrhenius activation energy has been a useful indicator to discern volume (solvent) diffusion-controlled kinetics from those of surface control.6 Volume diffusion- controlled kinetics would have an activation energy of ca. 20 kJ mol-1 and that for surface-controlled process would be considerably higher. The same idea is expected to be a useful guide to the establishment of the rate-determining step among the various interfacial processes. We can justify our assumption from the following relationships’ [eqn (1) and (2)] for the choice of rate-determining step for crystal growth in solution in terms of the activation free energies for incorporation at kink, AGhink, deadsorption, AGieads, and surface diffusion, dGHdiff.If the rate-determining step is surface diffusion, then If the integration or detachment at kink is the rate-determining step, then The comparison of activation free energies can be reduced approximately to a comparison of activation energies of these elementary processes by neglecting or cancelling the various T A S and RTcontributions to AGI and A#. The numerical values of the activation energies of the elementary events, such as surface diffusion, incorporation and detachment at kinks, can be calculated from the interaction between ions, and they would provide some insight into the rate-determining step for the surface-controlled kinetics. When the BCF theory for crystal growth* is extended to ionic crystals there will be a barrier for surface diffusion arising from the repulsion between like charges along the lateral direction (path 1) of the surface unit cells or along the path diagonal to these unit cells (path 2) as shown in fig.1. An assessment of the activation energies of these interfacial elementary events can similarly be made in terms of the interactions between solid and fluid units in the simulation model. The fluid units, which may either be the molten liquid or the solvated lattice unit, are assumed to occupy the same volume as a solid cell.’ Activation is necessary in surface diffusion and detachment.* For example, the activation energy for a surface diffusion jump along (100) surface is simply the binding energy mq5, where m is the total number of neighbours at the departure site and g5 is the bond energy between a solid-fluid pair.’ In terms of the pair potential energy between the solid and fluid units, (3) # is given as The maximum in the barrier for the diffusion jump along either path 1 or path 2 is located midway between a kink site and its neighbour.If the solid/vapour interface is considered, g5 is a function of only the bond energy between a solid-solid pair. The potential energy of an ion adsorbed on a KC1 (100) crystal/vapour interface and its relation to the activation barriers for the elementary growth and dissolution events are studied in this paper using simple formulae.The total ion-ion potential is the sum of both Coulombic and non-Coulombic contributions. Furthermore the usefulness of the one-component crystal as a model for a multicomponent ionic crystal will be clarified with detailed knowledge of potential energies of ions at the crystal surface. The results will be generalised to the case of a bivalent metal sulphate. Although the vapour phase is known to contain a large number of ion pairs, whose interionic separation is less than $ = A f - + ( A s + Af).2T 7' @ad-ion 90 1 Fig. 1. Surface diffusion paths of an adion. that in the crystal lattice, the activation barrier of the abovementioned elementary events involving ion pairs and the influence of ion pairs on the kinetics of crystal growth and evaporation are not considered in this study.Hovel3 showed that at room temperature the lifetime of an ion pair adsorbed on the (100) KCl crystal surface is extremely short. Potential Energy of an Ion near an Ionic Crystal Surface The potential energy of an ion inside and outside an ionic crystal can be calculated as the sum of all the pair potential energies between the ion in question and the rest. Similarly the potential energy of the crystal can be given as the sum of all possible pair potential energies. This pair potential energy as a function of distance can be determined experimentally from gas-phase data or calculated from ab initio quantum mechanics.lO*l' It has long been recognised12v13 that these pair potentials can be split into a number of contributions such as Coulombic, polarisation, repulsion etc.However, it has been found that this pair potential-energy function needs to be replaced by a slightly different effective pair potential-energy function in order to account adequately for bulk cry st a1 proper ties. lo The Coulombic energy of a finite crystal can be accurately evaluated by adding all Coulombic interactions between (point) ion pairs of charge qi and qj and separation rij within the lattice: where a is the unit-cell parameter. Similarly the Coulombic potential energy V(r) for an ion near the surface of an ionic crystal can be calculated by the direct summation involving individual pair Coulombic interactions between the given ion of charge q and the ion of charge qi in the lattice at separation r - ri, where ri is the displacement of the ion relative to the same origin as r : Both r and ri are scaled by a.The alternating addition and subtraction of terms dictated by the arrangement of ions in unit cells results in a slowly and conditionally convergent series.14 Rapid convergence of the sum can be achieved by the integral transform method.15 One of us has developed analytical methods for such a calculation on the basis902 Crystal Growth and Evaporation of the integral transform. ''-la Alternatively, the Coulombic potential energy can be calculated from the Poisson equation approach.12 The polarisation potential energy is the result of the polarisation of the adsorbed ion by the electric field of the crystal and that due to the polarised crystal in the presence of the adsorbed ion.van der Waals forces between neighbouring ions give rise to the repulsive potential energy. Contributions due to polarisation and repulsion have been studied by Hove.13 Only crystal polarisation (by the adsorbed ion) and overlap repulsion are considered to be important. The lattice sum potential energy (in eV) for polarisation and repulsion for the KCl lattice were stated as, respectively, eqn (6) and (7): Vr = (81.20) (1 + 9.122) exp (-9.122) + (75) (1 + 10.15~) (cos nx) (cos ny) exp (10.152) + (90.1) (1 + 1 1 1.072) (cos 27rx + cos 2ny) exp (- 1 1.072) +(119)(1+ 12.73z)(cos2nx)(cos27ry)exp(- 12.732).(7) In eqn (6) a,( = a/2) is the interionic distance within the crystal and the lattice sum of the pair potential energy due to polarisation over all ions in the semi-infinite crystal. S , = M , c (a0/r,)4, is spilt (according to the sign of the ionic charge given) into contributions due to cations and anions in the lattice. The constants M , and M- (in short M + ) are derived in terms of the in-crystal ion polarisabilities a+ and dielectric constant-& according to the theory of Mott and Littleton." Here the positive subscript refers to a cation and negative subscript refers to an anion in the lattice. M+ - is given M , = - 47r(a,+aT) and the dielectric constant of the crystal E is related to a+ - by 8-1 4n a,+a- ~ + 2 - 3 a i 2 ' (9) We note that in our calculation, M , and E are dimensionless.Bulakh and Chernov2' and Hovel3 gives M , and M- as, respechely, 0.01 56 and 0.0690 for a rigid KCl lattice. These values can be calculated from the in-crystal ion polarisabilities given later in table 4. For an elastic lattice, M , and M- are, respectively, 0.0488 and 0.0771, which implies a small change in a+. The polarisation contribution to the binding energy of the adsorbed ion was considered to be significant in previous calculation^.^^^^^ However, the factor of 1/2 in eqn (6) was not present in its original derivation but appeared in the calculation of the potential energy experienced by a vacancy in the bulk of a crystal." More often the energy and static properties of an ionic solid, such as interionic separation and isothermal bulk modulus, are successfully calculated using an effective pair potential-energy function2' which contains the Coulombic part, qyl(r), in which the ions are assumed as point charges, and the short-ranged non-Coulombic part, and C .. D . . I/sf(r) = B . . exp ( - a i i r ) - 3 - - 2 i,j = +, - , r6 r8 The coefficients B,,, aii, Ci, and D , for various ionic crystals are given in tables 1 and 2. The abovementioned contributions to the pair potential energy due to Coulombic, repulsion and polarisation forces are incorporated into eqn (1 2), but the dipoledipole polarisation potential energy now has a 1 / r 6 dependence.2' Thus it is not as long-rangedV. K. W. Cheng, B. A . W. Coller and E. R. Smith 903 Table 1. Coefficients (in atomic units) for ab initio non-Coulombic potential-energy functionsll pair interactions A B C D E F Na-Na 140.383 99 -4.611 12 -0.257 01 1.181 66 13.190 09 5.571 62 Na-CI" 100.788 12 -3.042 14 -0.116 15 0.684 38 0.022 70 3.889 75 Na-C112 76.822 55 -3.333 44 -0.254 08 0.663 91 16.758 16 3.994 29" c1-Cl 2.141 67 - 1.034 75 -0.061 19 0.586 59 627 519.578 40 11.349 35 "The inclusion of a negative sign in ref.(12) gives an unphysical potential-energy curve. Table 2. Mayer coefficients for Born-Mayer non-Coulombic potential-energy functions'' Na-Na 3.155 25 16.08 1.68 0.8 Na-C1 11.2 233.0 Cl-Cl 116.0 13.9 K-K 2.967 3575.47 24.3 24.0 K-Cl 48.0 73.0 Cl-c1 124.5 250.0 as assumed by Mott and Litt1et0n.l~ Various sets of van der Waal coefficients Cii and D , have been proposed,21 but those reported by Born-Mayer are regarded to be most reliable.Subsequently Catlow et a1.22 proposed an improvement of C , and suggested that it would be suitable for simulation The effective pair potential-energy function in fact takes into account of the many-body interactions, e.g. polarisation of the ion pair in question by surrounding ions2* The calculations of ionic crystal static properties using accurately determined (from ah initio quantum mechanics) pair potential-energy function for NaCl has shown discrepancies with crystal properties obtained from experiment." This pair potential- energy function has a form similar to that of the effective pair potential-energy function of eqn (lo)-( 12), which contains both Coulombic and non-Coulombic parts. The non- Coulombic part is given as y;(r) = (Aij r-Bii + Cij r) exp (- D , r ) + Eij r-Fij i, j = + , - .The coefficients Aij to Fij, given in table 1 are fitting parameters and are not related to the various types of interaction mentioned above. This pair potential energy accurately reproduces experimental data for ion pairs in the gas phase.l0.l1 However, it predicts a smaller interionic separation in the crystal and hence the existence of covalent inter- action between ions. A more negative lattice energy but a dynamically less stable 1a.ttice was also predicted using this pair potential-energy function. lo Ions of opposite charges near the surface will relax from their bulk equilibrium positions in directions opposite to each other but perpendicular to the surface. It has been shown both e~perirnentally~~ and theoretically26 that in the few layers near the surface of NaCl, the anions are displaced upwards and cations downwards.The corresponding shift of equilibrium adsorption positions is expected. Lattice summations involving relaxed surfaces have been extensively reviewed,27 but no agreeable method has been identified. It has been suggested that in the NaCI-type surface lattice the change in position alone is expected to lower the surface energy by 2040 %. Relaxation from bulk lattice structure at surfaces may cause some degree of polarisation. We would expect, as a result, that higher total binding energies at equilibrium adsorption sites should arise904 Crystal Growth and Evaporation from the relaxation of the surface lattice structure.Any polarisation of surface ions as a result of the relaxation was ~uggested~~ to be mostly cancelled out and thus not to contribute to the surface energy. For simplicity of calculation, these additional contributions to the adion potential energy due to surface relaxation are neglected. Calculations of Coulombic Potential Energy of Ions near Ionic Crystal Surfaces In the present study the Coulombic potential energy V(r) of a univalent ion near the unrelaxed (100) KCl surface [eqn (5)] was calculated using a number of general half- space lattice Recently we derived a potential-energy expression by transforming l / x by the identity (hereafter defined as method 1) - 1 - $lom t-f exp (- t x 2 ) dt 1 " = Jq, t-f exp ( - t x 2 ) dt + t-f exp ( - tx2) dt.The integral in eqn (14) was actually split into two parts at the arbitrarily chosen point of q2, which allows us to control the convergence rate of the resulting lattice sums. The sum involving the first integral Vl(r) can be evaluated in terms of the complementary error function (16) to give a simple sum over the lattice A The remaining part of the potential V2(r) as a result of summing terms containing the second integral in eqn (15) is numerically cumbersome.'7v18 It depends on the unit-cell dipole moment and the shape of the crystal, and if the unit cell dipole moment is not zero, it diverges.l' More recently, we developed a different way of evaluating a half-space lattice sum for the special case of the surface containing (100) NaC1-type unit cells,lS which has a zero unit-cell dipole moment (hereafter defined as method 2).Instead of splitting the transformation of l / r [eqn (14)] into two terms, the integrand in eqn (14) was transformed using the identity 1 f a exp (- tyz) = 1 J exp [ - (uz/t + 2iuy)l du. d n -a The resulting integral over t in eqn (1 8) effectively transformed all space variables to its reciprocal lattice space and could be evaluated directly. The resulting lattice sum was then split into one containing only the reference reciprocal lattice and the other containing rest. The first sum needed to be evaluated with recourse to the (100) NaC1- type unit-cell structure and the second sum could be evaluated readily to give the Coulombic potential at a distance z above the surface as'' x cos (2n[k,(x - X i ) + k,Cy -&)I} (1 9) where N = 8 is the number of ions in a unit cell.In all the lattice sums, the origin of the position vector Y = (x, y , z ) and the two-dimensional reciprocal lattice vector k = (k,,k,) is taken at the centre of a unit cell in reciprocal space. It is sufficient to take the x and y components (along the surface) to within + 5 during the actual computation.V. K. W. Cheng, B. A . W. Coller and E. R. Smith 905 'The positions of the univalent cation are (a/4, a/4, a/4), (a/4, - a / 4 , - a/4), ( - a / 4 , - a/4, a/4), ( - a/4, a/4, - a / 4 ) and those for the anion are ( - a/4, a/4, a/4), (a/4, - a / 4 , a/4), ( - a / 4 , - a / 4 , - a / 4 ) . As a comparison, the Coulombic potential of an ion above the (100) KC1 surface was also calculated from the solution to the Poisson equation12.When the adsorption position is taken as a variable, these methods also provide the barriers to surface diffusion. The Coulombic activation barriers for surface diffusion .were calculated at a fixed distance of a / 2 (a,) above the surface for paths both parallel to a row of ions (path 1) and parallel to the diagonal of the top of the unit cell (path 2). 'The Coulombic potential above this surface was also calculated at various distances measured from the centre of a surface unit cell. The unit-cell parameter for the KC1 lattice was taken as 630 pm13 and that for NaCl as 564 pm.l0 For the purpose of comparison, the Coulombic energy of a monovalent ion at a lattice site at the end of a semi-infinite one-dimensional (non-polar and non-relaxed) KC1-type lattice array with interionic distance a, was also calculated using the direct summation formula Calculations of Non-Coulombic Potential Energy of Ions near Ionic Crystal Surfaces Non-Coulombic attraction, such as polarisation, increases the binding energy of an adion at its equilibrium adsorption site, and this attraction is expected to vary as the adion moves from, say, its initial equilibrium ad-site to another along a given surface diffusion path.We can assume, as in previous work,lg that the pair potential energy (in e.s.u.) experienced by an adion in the presence of another ion in the neighbourhood at a distance ri from it-takes the form - Two sets of in-crystal polarisabilities (table 4, see later) are used in this calculation: (1) those reported by Tessman et aLZ8 and used by Hovel3 and (2) recently improved data proposed by Fowler.29 In the latter set the polarisabilities of group I and I1 ions are similar to that of ref.(28), but the polarisability of an anion and a do ion such as Ag+ depends on its neighbouring ionic environment. We also assumed that the values of the polarisabilities for these ions are not significantly changed near the surface, although the surface ions may experience a different electric field than that in the bulk crystal. The lattice sum for the pair polarisation interaction given above is not a conditional sum because terms due to the positive and negative charges are not cancelling each other (all interactions are attractive), unlike the case of the Coulombic potential.Therefore if the pair potential is a long-range interaction, the nett potential function will remain long-range. This half-space sum was evaluated directly. The surface repulsion (for KCl only) was evaluated using eqn (7), and the total potential energy of an adion will be given as the difference between the total attraction and the repulsion. The direct lattice sum is also used to evaluate the non-Coulombic potential energy of adions on crystal surfaces, assuming that the ion-pair interaction V:i(r) follows that of the Born-Mayer theory eqn (12) or, for the NaCl surface, that proposed from ab initio quantum mechanics [eqn (1 3)]. Results Coulombic Potential Energy Among the methods adopted to calculate the Coulombic potential of an ion near the (100) KCl/vacuum interface, the summation formula developed by Hoskins et a1.16 was906 Crystal Growth and Evaporation Table 3.Contributions to the binding energy (eV) of univalent ions at the equilibrium adsorption site on the (100) KCI surface this work adion method 1 eqn (17) ref. (12) ref. (19) ref. (1 1) ref. (30)" electrostatic -0.47 -0.30 -0.30 -0.30 -0.30 - 0.60 repulsion 0.1 6b 0.1 6b 0.16b 0.16b 0.18 0.32 polarisation K+ -1.07 -1.07 -0.53 -0.53 - - c1- -0.97 -0.97 -0.48 -0.48 - - total K' -1.38 -1.21 -0.67 -0.67 -0.12 - 0.28 c1- -1.28 -1.10 -0.62 -0.62 -0.12 - 0.28 " Nearest-neighbour interactions only. Ref. (1 3). found to be divergent. The failure of the summation formula was later found to be caused by the inadequate treatment of the contour integration and the asymptotic expansion.Absolutely convergent numerical results for the Coulombic potentials near the surface were obtained from methods 1 and 2.17~ la However, the Coulombic potential calculated from method 1 is still very time consuming. The Coulombic potential from method 2 showed significant improvements in this aspect but it was still found to be less efficient than the solution to the Poisson equation." The Coulombic potential above the (100) KCl surface calculated from methods 1 and 217,18 is short-ranged (decaying to zero within one layer distance). The potential from method 2 is identical to that from the Poisson equation.12 Along a line immediately above the adsorption site, the potential calculated from the first method was found to oscillate slightly." A small activation barrier of ca.0.02 eV (2 kJ mol-l) would be associated with the deposition of monovalent ions from the neighbouring vacuum. Method 2 gives potentials above the surface only, and oscillation of potentials is not found. Furthermore, from both methods 1 and 2, the surface potential at a position equidistant from two neighbouring ions (of opposite charge) is zero. The Coulombic potential energies for a univalent ion in the vacuum space at a presumed equilibrium adsorption site (distance a,) above a lattice site of the (100) KCl crystal surface is given in table 3, together with other major contributions to the total potential energy such as polarisation [eqn (6)] and repulsive interactions [eqn (7)]. The Coulombic binding energy calculated by method 1 for a univalent adion at a supposed equilibrium adsorption site above the plane surface of a semi-infinite lattice array is found to be 0.47 eV (45 kJ mol-'), which is higher than those found in a number of previous works but lower than the nearest-neighbour value of Stranski3' (table 3).The smaller Coulombic potential of 0.30 eV (28.8 kJ mol-l) was found in method 2 [eqn (1 9)] and in the Poisson-equation approach.12 The results reported by Hovel3 indicate that it is reasonable to take just the leading term of the fast convergent expression derived from the Poisson-equation approach.12 On the other hand the Coulombic potential energy of a univalent ion at the lattice site next to a one-dimensional semi- infinite KC1 array [eqn (I 8)] is 0.22 eV (20 kJ mol-').The potential energy of the same ion inside an infinite linear KCl array would be 0.44 eV (40 kJ mol-l). The surface Coulombic potential calculated from method 1 gradually converts into the bulk potential (2.6 eV at a lattice site), which is many times larger than the surface potential (obtained from both methods 1 and 2), over a distance of two ionic layers beneath the surface, as predicted.17 The Coulombic potential of an ion at the surface layer (2.3 eV) is slightly smaller than that of the bulk ion. Thus a proportionality between the Coulombic potential and the number of neighbours cannot be established on the basis of the value of the surface and bulk Coulombic potential. The bulk potentialV. K . W. Cheng, B.A . W. Coller and E. R. Smith 907 1 .o >, 2 3 - 0.5 5 \ x .r( Y * a - C -0.5 Fig. 2. Variation of non-Coulombic potential energies of an Na-Cl ion pair with distance. The distance z is scaled by the crystal interionic separation, a, = 282 pm (-) Born-Mayer, (----) tzb initio (-.--.-), Mott-Littleton anion next to cation (-. .-- .-), Mott-Littleton cation next to anion. (scaled by 1/4mo) calculated with method 1 at (a/6, a/6, a/6) was found to obey the exact result of 2/3 for unit cells with a = 2.31 Also, at the centre of a unit cell, where complete cancellation of charges occurs, the Coulombic potential is zero. The magnitude of surface and bulk Coulombic potential energy for other NaC1-type (100) surface are determined by the value of the unit-cell parameter chosen.Thus the Coulombic binding energy of an adion above the NaCl (100) surface is larger than that for KC1 because of its smaller value of a. Non-Coulombic Energy 'The variation of the non-Coulombic potential energy of an NaCl pair with interionic separation as given by Mott and Littleton eqn (21). Born-Mayer eqn (12) and Clementi et al. eqn (1 3 ) are presented in fig. 2. All these potential functions decay to zero or a small value at large interionic separations. However, the Mott-Littleton potential-energy function calculation with both sets of ion polarisabilities cannot be regarded as being as short-ranged as those at large interionic separations ( r = 4 . 5 4 , the potential function remains at a small but almost constant value. The choice of a different set of ion polarisabilities only cause a small difference in the potential energy. At a given interionic separation, the attractive part of the Born-Mayer potential energy is the smallest and the Mott-Littleton potential energy is the largest.The magnitude of the effective pair potential with larger van der Waals coefficients given by Catlow (not shown in fig. 2) is only slightly larger than that with Mayer coefficients. The position of the potential- energy minimum occurs at r = 1 .35a0, whereas for the ab initio potential-energy function the minimum occurs at r = 1. la,. At a small interionic separation as shown in fig. 2, the Mott-Littleton polarisation pair potential-energy function for a cation in the neighbourhood of an anion is smaller than that experienced by an anion next to a cation.When the interionic separation gets larger, both potential functions converge to the same value. On the other hand, the potential energy of an ion next to another of opposite908 Crystal Growth and Evaporation charge as determined by the effective pair potential-energy function or the ab initio pair potential-energy function is the same regardless of the type of charge it carries. The attractive part of all these potentials are, however, much smaller than the Coulombic pair potential energy at a given interionic separation. The potential energy of an ion as a function of the distance above the crystal surface, as given in fig. 3, have largely the same shape as the pair potential-energy curve. The Mott-Littleton potential energy of an adion on a NaC1-type (100) surface using eqn (21) and polarisabilities given in table 4 proves to be very time-consuming.The summation of a considerably large number of terms is needed for good convergence. The potential energy as a function of the distance above the (100) NaCl surface is long-ranged, as shown in fig. 3. This form of the polarisation potential energy at an adsorption site, as given in table 3, appears to provide a major contribution to the binding energy of an adion. The polarisation energy of an adion above (1 00) NaCl was found to be larger than that above (100) KCI. Although the polarisability of Na+ is smaller than that of K+, the smaller interionic distance for crystalline NaCl seems to have more influence on the magnitude of the polarisation energy.Furthermore, this contribution to the adsorption energy is larger than that obtained by HoveL3 and Bulakh and Chernov20 using eqn (6) by a factor of 2. Fig. 4 and 5 show the variation of the van der Waals repulsion potential [eqn (7)], which is the same for both cation and anion, for an adion on a (100) KCI surface as it moves along diffusion paths 1 and 2, respectively. This repulsion contribution to the binding energy of the adion at an equilibrium adsorption site is given in table 3. Such a large repulsion, at the given ionic separation, suggest that there is an appreciable overlap of electron clouds between neighbouring ions in the crystal. On the other hand, the potential energy of adions as calculated by the Born-Mayer potential-energy function and the ab initio pair potential-energy function are short- ranged and thus only require the sum of a few terms for good convergence.The magnitudes of these potential energies are larger than that given by their respective pair potential-energy functions. They are even larger than the Coulomb interaction [eqn (19)], which was significantly reduced because of the cancellation of contributions due to opposite charges (table 3). The equilibrium binding energy of a negatively charged adion was found to be larger than that of a positively charged adion if it is determined by the effective pair potential-energy function (with either Mayer or Catlow coefficients) or the ab initio pair potential-energy function. On the contrary, the Mott-Littleton potential energy of a positively charged adion is larger.The potential-energy minimum of an adion due to both the effective and the ab initio potential-energy function again occurs, respectively, at r = 1 .35a0 and 1. la,. Activation Barriers to Surface Diffusion The overall barriers for both diffusion paths were calculated from the difference between the total binding energies at the equilibrium adsorption site and midway between two equilibrium adsorption sites. At these midway locations the adion of binding energies were found to be a minimum. The Coulombic contribution to the activation barriers to surface diffusion for the paths across the plane (100) KCl surface under consideration (paths 1 and 2) calculated from method 117 has a number of maxima and minima (fig. 4 and 5).On the other hand, the barrier obtained from method 218 appear to be physically more reasonable. The barriers found by method 2 for paths 1 and 2 are 0.52 and 0.26 eV (50 and 25.4 kJ mol-l), respectively, for the (100) KC1 surface. The barriers are comparable to Hove’s values of 0.60 and 0.19 eV,I3 without consideration of the polarisation contribution. The variation of all non-Coulombic potential energy along the two different paths are smooth and very small, as shown in tables 5 and 6. The barrier to surface diffusion for the Mott-Littleton potential energy appears to be larger, but it is only a small fraction of the corresponding equilibrium binding energy.V. K. W. Cheng, B. A . W. Coller and E. R. Smith 909 0.5 - 0 0.5 1.0 1.5 I Fig. 3. Variation of potential energy of an adion (Na+ or C1-) with distance above a surface lattice ion of opposite charge.The distance z is scaled by the crystal interionic separation, a, = 282 pm. A, Born-Mayer, anion; A, Born-Mayer, cation; I, ab initio, anion; 0, ab initio, cation; e, Mott-Littleton, anion ; 0, Mott-Littleton, cation. Table 4. Contributions to the potential energy (eV) of an ion at an equilibrium adsorption site a above a (100) NaC1-type surface surface adion ion polarisability Coulombic Born-Mayerb ab initio polarisation NaCl Na' 0.148 C1- 3.13 KCl K+ 0.79 (0.75)' C1- 3.39 (3.66)" -0.34 0.22 -0.63 -1.22 -0.34 0.12 - 1.03 -1.00 -0.30 0.16 - -1.07(-1.11)" -0.30 0.12 - -0.97 (- 1 .OO)' ~ a At a distance a, above the surface lattice ions. value stands for repulsion.Use polarisabilities given by Hove.13 Positive Table 5. Contributions to the activation energy (eV) for surface diffusion of an ion along the lateral path on a (100) NaC1-type surfacea surface adion Coulombic Born-Mayer ab initio polarisationb NaCl Na+ 0.68 - 0.006 -0.18 0.40 c1- 0.68 -0.016 -0.18 0.52 KCI K+ 0.60 -0.012 - 0.35 (0.13)' c1- 0.60 - 0.045 - 0.44 (0.17)c a At a distance a, above the surface lattice ions. " Use polarisabilities given by Hove.13 Positive value stands for positive barrier.910 Crystal Growth and Evaporation 0.4 I I *-*, 0.2 p-' 7A +/--* 0.5 1 1 1 1 1 1 Y I I 1 I 1 1 \ c t + r c,- Fig. 4. Various contributions to the potential energy of an ion above the KCl surface moving along the lateral diffusion path (path 1). The distance r is scaled by the crystal interionic separation, u, = 315 pm.A, Coulombic, method 1 ; A, Coulombic, method 2; 0, overlap repulsion, eqn (7). 0.2 + -0' \ '-. . . I Y [ -0.4 Fig. 5. Various contributions to the potential energy of an ion above the KC1 surface moving along the diagonal diffusion path (path 2). The distance Y is scaled by the crystal interionic separation, a, = 315 pm. A, Coulombic, method 1; A, Coulombic, method 2; 0, overlap repulsion, eqn (7).V. K. W. Cheng, B. A . W. Coller and E. R. Smith 91 1 Table 6. Contributions to the activation energy (eV) for surface diffusion of an ion along the diagonal path on a (100) NaC1-type surfacea surface adion Coulombic Born-Mayer ab initio polarisationb NaCl Na' 0.34 - 0.034 -0.176 0.34 c1- 0.34 - 0.073 -0.10 0.40 KCl K+ 0.30 -0.10 - 0.29 (0.22)' c1- 0.30 -0.12 - 0.32 (0.40)" a At a distance a, above the surface lattice ions.polarisabilities given by Hove.13 Positive value stands for positive barrier. " Use Furthermore, the barrier derived from both the effective potential-energy function and the ab initio pair potential-energy function is negative (i.e. the non-Coulombic potential energy midway between two equilibrium adsites along a given diffusion path is a minimum) and that from the Mott-Littleton potential-energy function is positive. Discussion The short-ranged nature of the Coulombic potential experienced by an adsorbed ion on an unrelaxed (100) NaC1-type surface is the result of the charge neutrality of the crystal lattice. Its strength is thus much less than the pair Coulomb interaction assumed by Stranski3' (fig.2 and table 3). The zero potential found at positions equidistant from two neighbouring ions arises because of cancellation of almost equal contributions due to ions of opposite charges. The small potential barrier near the crystal surface found using method 117 is likely to be of unphysical origin and not due to incomplete charge cancellation. Since the surface Coulombic potential calculated from method 1 differs from that of method 2 and from the Poisson equation, the surface potential derived from method 1 is likely to be in error. The accuracy of the sum cannot be checked experimentally. However, the Coulombic potential of an ion at a bulk lattice site as calculated by method 1 is expected to be reliable because it was shown to be capable of reproducing agreeable results obtained from exact calculations.31 The identical results obtained from method 2 and from the Poisson equation, which is a physically equivalent approach, validates method 2. The integral transform method has the advantage over the direct lattice sums in that it does not require a large number of terms for good convergence, although its rate of convergence is still lower than the expression derived from the Poisson equation.12 The ions between a lattice ion deep in the bulk and the surface ion would become a dielectric medium in our calculation of the surface Coulombic potential energy using eqn (19). However, the dielectric constant of the crystal was not taken into account. Because of the almost effective cancellation of the Coulombic potential energy due to opposite charges, the Coulombic potential calculated from eqn (19) is effectively contributed by the first layer of ions.This deduction has been validated'' using the Poisson equation. l2 For an NaCl ion pair, the total attractive potential energy at large separations is determined by the Coulombic contribution. On the other hand, the exact form of the non-Coulombic part determines the repulsive end and the shape of the potential well and hence ion pair properties such as vibrational force constant and dissociation energy. lo At a site on the (100) NaCl surface and within the crystal the cancellation of both the surface and bulk Coulombic potential due to opposite charges provides the possibility for the non-Coulombic potential energy to play a dominant role in determining the binding energy of an ion either at the crystal surface or inside the crystal.Whilst the ab initio potential-energy function can provide a large attractive influence on the912 Crystal Growth and Evaporation adsorption site binding energy, the corresponding influence due to the effective potential energy is repulsive. The dependence of the Mott-Littleton potential energy of an ion, either in an ion pair or on the crystal surface, on the charges of its neighbour(s) (and more precisely their polarisability) is consistent with that reported for the bulk crysta1.l' In the latter case, the Mott-Littleton potential energy of an electron at the cation site and a hole at an anion site were considered.Although such charge dependence is not found for the pair potential-energy calculation using either the effective potential function or the ab initio function, a neighbouring charge dependence of adion potential energy using these two potential functions in the lattice sum was found (fig. 3) because of the additional involvement of cation-cation or anion-anion pair interaction. Furthermore, the binding energy of a positively charged adion relative to that of a negatively charged adion as calculated using these two potential functions is opposite to that predicted from the Mott-Littleton potential function. This comparison is not expected to alter even if the binding energy of an adion include, in addition to the polarisation energy, the charge- independent repulsive potential energy [eqn (7)].We can therefore consider the Mott- Littleton potential together with the repulsive part adopted by Hovel3 and Bulakh and Chernov2' as an inappropriate representation of the non-Coulombic potential between ions in both the gas and crystal phase. One way to improve this potential function is to modify M , and one such modification has been shown to be necessary in the application of the M6tt-Littleton method to lattice-dynamical problems.21 We have shown that simply a small change to a different set of a+, and hence M , , values does not significantly change the potential energy calculated. Our polarisation-energy expression and results differ from those previously p~blishedl~*~' because we did not include the factor of 1/2 in eqn (6).This factor is absent in the original derivation by Mott and Littleton and it was only included in their calculation of the energy at a vacant site in the bulk lattice. On the contrary, with appropriately chosen coefficients, the Born-Mayer potential function is known to be able to predict agreeable lattice energy and other static lattice properties.1° Therefore we would expect it to provide a more accurate measure for the potential energy of an ion on the surface or in the lattice. As a result of this small potential-energy minimum at z > a,, we would expect a repulsive contribution to the binding energy of an adion from non-Coulombic interaction. The large adion binding energy calculated using the ab initio pair potential-energy function (Coulombic + non- Coulombic) relative to that obtained from the Born-Mayer effective pair potential- energy function is consistent with the larger lattice energy calculated with the ab initio pair potential-energy function.lo It has been suggested that a many-body interaction correction is needed to describe the crystal properties accurately. The lattice sum expressions for calculating the Coulombic and non-Coulombic potential energy of an ion in a bulk lattice site, in the surface layer or at an adsorption site can be separated into mutually exclusive but unequal components of half space one- dimensional and two-dimensional sums in the same way as that shown in eqn (19) and (20). At kink and step sites at large two-dimensional clusters or along infinitely long steps or spirals, we can similarly treat additional contributions to the lattice sums arising from those ions above a complete lattice plane separately as one- and two-dimensional sums.Again the Coulombic potential resulting from the contribution of these one- and/ or two-dimensional sums involving partial cancellation of charges of a large number of ions is expected to be short-ranged and similar in magnitude to those for an ion next to an infinitely long array of ions [eqn (20)] or above the surface of a lattice [eqn (19)]. Thus, for the crystal growth/dissolution kinetic models for non-polar surfaces, such as the (100) face of NaCl and KC1, we can still justifiably approximate the overall potential energy of ions at both bulk and surface lattice sites into contributions from nearest neighbours, but we cannot exactly establish the proportionality dependence of the short-V.K . W. Cheng, B. A . W. Coller and E. R. Smith 913 Table 7. Binding energy (eV) at a lattice and surface site for NaCl site lattice ion Coulombic Born-Mayer" ab initio polarisation lattice Na+ -2.70 1.25 -2.39 - 3.25 c1- -2.70 0.96 - 2.75 -2.21 surface Na+ - 2.36 1.03 - 1.76 - 2.03 c1- -2.36 0.84 - 1.72 - 1.21 " Positive value stands for repulsion. Table 8. Lattice parameters (in pm) for a number of sparingly soluble ionic crystals at room temperat~re,~ crystal a, a!l a, geometric mean CaCO, (calcite) 499.0 499.0 1706.4 751.8 BaSO, (barite) 887.8 545.0 715.2 702.1 SrSO, (celestite) 835.9 535.2 686.6 674.7 PbSO, (anglesite) 848.0 539.8 695.8 682.9 CaSO, (anhydrite) 699.1 699.6 623.8 673.2 530.0 720.0 840.0 680.0" 2(H,O) a Ref.(36). ranged potential energy on the number of them. On the other hand, this approximate picture may be invalid for ions on the (1 11) polar surfaces. They may experience a long- range contribution to the Coulombic potential energy depending on crystal 33 The activation barriers to surface diffusion along the two paths chosen for an adion on the unrelaxed (100) KCI surface are calculated from the difference in its maximum and minimum binding energy along the path. The preferred surface diffusion path is shown to be the one along path 2 (the diagonal path). The adion is assumed to move at a distance of a/2 (a,) above the surface lattice ions. It is not expected to be any closer to the surface lattice because of the strong overlap repulsion which determines the ionic radii.Furthermore, because of the short-range characteristic potential energy experienced by the adion, any upward displacement of the adion from its adsorption site would result in a reduction in its binding energy. The adion can be regarded as completely desorbed when the distance between it and its nearest neighbour is ca. a,,. The variations of the Coulombic and non-Coulombic potential energy, as determined by the Born-Mayer effective pair potential-energy function and the ab initio pair potential-energy function, with distance along a diffusion path indicate that, although the binding energy of an adion is determined to some extent by the non-Coulombic contribution, the activation barrier it experienced originates from the Coulombic part.Thus both types of non-Coulombic interactions show another consistency with each other : they both predict a negative but negligible surface diffusion barrier, because adion at the supposed site of minimum binding energy has in fact a slightly higher non- Coulombic binding energy. On the other hand the Mott-Littleton potential-energy function gives a positive and large barrier, which however, is still small compared with the binding energy. The small contribution from the non-Coulombic potential energy to the surface diffusion activation energy is a consequence of its attractive or repulsive nature throughout the surface. When the charge factors are considered for the BaS0,-type lattice, the Coulombic energy should be increased by a factor of 4.Unfortunately, the extension of the present lattice sum to calculate Coulombic energy for the barium sulphate crystal is complicated914 Crystal Growth and Evaporation because of the latter's orthorhombic lattice Because of the differences between lattice parameters at various faces, the electrostatic binding energies of ions on these crystal faces will be different. However, as shown from the small difference in the Coulombic potential energy of ions on a KC1 and an NaCl lattice (tables 4 and 7), we can assume, from a comparison of the geometric mean lattice parameter for a number of sparingly soluble ionic crystals (table 8) with that of KC1 (630 pm), that the mean surface Coulombic binding energy of ions on these crystal faces are not affected significantly by the difference in lattice parameters. Thus the larger Coulombic energy and activation barrier for surface diffusion expected for these crystals will be largely contributed by the charge factor.Similarly, the perturbation to the short-ranged Coulombic potential energy of surface ions as a result of the relaxation from the uniform lattice structure near the surface is expected to be small. Contributions to the Activation Energies of Elementary Events and the Rate- determining Step The Coulombic interaction between ions at the crystal/vapour interface has been shown to be the determining factor to the activation barrier for surface diffusion and, to some extent, detachment. Because of the presence of lateral neighbours at steps and kinks, the activation energy for detachment would be higher than that for surface diffusion and would be influenced by the presence of, for example, dislocation at the surface site under consideration.However, when an ion is adsorbed onto an equilibrium adsorption site of a plane ionic crystal surface from the vapour, the short-ranged net ion-ion attraction it experienced will not give rise to an activation barrier. Similarly, there is no activation barrier due to ion-ion interactions for the integration at a step or kink after the adion overcomes its last surface diffusion barrier or when it is incorporated directly from the vapour. The large activation barriers for surface diffusion are expected to reduce the average surface diffusion distance along either path.Additional contributions to the surface- diffusion activation barrier along a step from ions at steps make the overall barrier even larger than the corresponding barrier to movement along the plane ionic surface. Thus diffusion of ions along and away from steps are even less frequent, and adions would prefer to remain attached to a step or kink site, once they reached it. Kink generation would be more favoured than the one-component Kossel crystal model would indicate. It can be considered that ions at steps have been successfully incorporated into the lattice. Nevertheless, this does not suggest any dependence of the kink density along a step on the distance from equilibrium. The less frequent diffusion of growth units along steps into and out of kinks would slow down the growth and dissolution rate.According to the BCF theory,' however, only the rate constant is reduced to an extent determined by retardation factors. A comparison of the magnitude of the activation energy calculated for surface diffusion and incorporation at kink would support the former as the rate-determining step in crystal growth involving these two in sequence. On the other hand, surface diffusion is not the preferred rate-determining step compared with detachment at kinks during evaporation because the latter process have a higher activation energy. This comparison is consistent with the criteria given in eqn (1) and (2). When the surface diffusion-controlled growth process is competing with direct incorporation at kinks from the vapour, the high activation energy of surface diffusion would make it an unfavourable choice as the rate-determining step.However, such a choice of the rate- determining step between these two competing processes according to their activation energies is inconsistent with that stated in eqn (1) and (2), which is derived from a full consideration of the rate theory.lVi Eqn (1) and (2) would suggest that surface diffusion, by virtue of its large activation energy, can still be an important process when it isV. K. W. Cheng, B. A . W. Coller and E. R. Smith 915 competing with direct incorporation. For these competing processes there are other factors, such as the difference in the driving forces (distance from equilibrium) near steps and kinks and at the rest of the interface, which determine the mean lifetime of surface units and the other components in the activation free energy of surface diffusion and incorporation.Therefore this approximation of comparing the activation free energies by the activation energies is apparently only valid in the sequential process in which the continuity of the diffusion flux is implied in the derivation of eqn (1) and (2).4 Conclusion In this study we provide an alternative way of understanding the relative significance of various elementary events in the growth and evaporation of ionic crystals by considering the nature of their activation barriers in relation to the potential energy of ions at the surface. We have also shown the importance of the form of the non-Coulombic potential energy, and hence their influence, in determining the potential energy of ions at the unrelaxed (100) NaC1-type crystal surface and, perhaps, inside the crystal lattice.The non-Coulombic pair potential-energy functions considered in this work are different from each other in terms of both their origin and their actual value at a given interionic separation, although they have common features, particularly those between the Born- Mayer function and the ab initio function. Because the Born-Mayer function takes into account the complexity of the many-body interaction, we would regard the lattice sum of the Born-Mayer function with appropriately chosen coefficients to provide the most suitable description of the potential energy of ions at the crystal surface, and these coefficients are not expected to differ from those for the bulk crystal.Lattice-sum studies have indicated that the barrier to detachment reflects the total binding energy of an adion, and that there is no barrier to attachment. The ionic nature of the crystal surface gives rise to activation barriers to surface diffusion of adions by virtue of the repulsion between adions and lattice ions of like charges. The surface- diffusion activation energy was shown to depend on the migration path and is determined by the electrostatic potential energy of the adion. It is appreciably larger for crystals such as barium sulphate because of the charge factor in the Coulombic potential expression [eqn (1 9)]. The rate-determining step needs to be established from the theory of crystal growth.However, as a result of the approximation of more general a rite ria,^ it can be successfully determined in terms of a comparison of the activation energy only in the case of the sequential process of surface diffusion, incorporation or detachment. Other factors, such as the difference in the driving force factor for the competing processes of surface diffusion and direct transfer, must be considered before such a comparison is made in this case. The large Arrhenius activation energy measured for the growth of a number of ionic crystals in ~ o l u t i o n ~ ~ ~ ~ ~ appears to support the proposition of the importance of surface diffusion in the growth of ionic crystals. The influence of the detachment activation during evaporation has been illustrated in our previous work on the dependence of lattice stress (reduction in binding energy of adions) on the Arrhenius activation energy for dissol~tion.~~ In that case detachment at the defect sites is essential to promote dissolution.The influence of stress-field parameters on the binding energy (and hence activation energy of detachment) for KC1-type crystals has been studied.37 We thank the referees for valuable suggestions on the revision of the manuscript. 31 FAR 1916 Crystal Growth and Evaporation References 1 P. Bennema, J. Cryst. Growth, 1967, 1, 278; 287. 2 J. Christoffersen, J. Cryst. Growth, 1980, 49, 29. 3 G. M. van Rosmalen, M. C. van der Leeden and J. Gorman, KristaII Techn., 1980, 15, 4 P. Bennema, J. Cryst. Growth, 1969, 5, 331. 5 G. H. Gilmer, R. Ghez and N. Cabrera, J. Cryst. Growth, 1971, 8, 79. 6 L. L. Bircumshaw and A. C. Riddiford, Q. Rev. Chem. Soc., 1952, 7, 157. 7 P. Bennema, J. 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Smith, unpublished results (1982). 35 CRC Handbook (Chemical Rubber Company, Cleveland, 60th edn, 1979), pp. B186-B205. 36 V. K. Cheng, B. A. W. Coller and J. L. Powell, Faraday Discuss. Chem. SOC., 1984, 77, 243. 37 H. B. Hungtington, J. E. Dickie and R. Thomson, Phys. Rev., 1961, 100, 1117. 38 G. L. Gardner and G. H. Nancollas, J . Phys. Chem., 1983, 87,4699. 39 G. E. Cassford, W. A. House and A. D. Pethybridge, J. Chem. SOC., Faraday Trans. 1, 1983, 79, 1954), p. 388. Verlag, Berlin, 1982), p. 288. Verlag, Berlin, 1982), p. 115. 1967), vol. 1, p. 203. unpublished. 1617. Paper 6/ 1688 ; Received 19th August, 1986