J. Chem. Soc., Faraday Trans. I , 1987’83, 1381-1394 Potential-energy Calculations of the Mechanisms of Self-diffusion in Molecular Crystals Part 2 .-Naphthalene D. Harold Smith? Department of Chemistry, Northwestern University, Evanston, Illinois, U.S.A. Self-diffusion in crystalline naphthalene has been investigated by calculation of the activation energies for various postulated mechanisms. A first-order ‘rigid-lattice’ calculation is first performed in which only the diffusing molecule is allowed to move. This yields several conclusions: (1) self- diffusion normal to the ab plane occurs only by vacancy exchange between sixth-nearest-neighbour sites; (2) diffusion along the a-direction occurs by the same mechanism and also by vacancy exchange between nearest- neighbour sites; (3) diffusion along the b-direction occurs by the nearest- neighbour mechanism; and (4) the molecular trajectories in both mechanisms apparently do not cross saddle points on their respective potential surfaces.It is then postulated that diffusion occurs without rotation of the diffusing molecule when its initial and final lattice sites belong to the same sublattice (the sixth-nearest neighbour mechanism) and with a single, direct rotational flip of the diffusing molecule while it is translating between lattice sites when the sites belong to different sublattices (the nearest- neighbour mechanism). The theoretical activation energies for these trajectories in the rigid lattice model are twice as large as the experimental values, but most of the barrier comes from just four (sixth-nearest neighbour mechanism) or two (nearest-neighbouring mechanism) nearby molecules in the lattice.A second-order ‘ relaxed-lattice ’ calculation is then performed in which these ‘blocking’ molecules are allowed to rotate, and good agreement with experiment is found for both mechanisms when the blocking molecules rotate ca. 10-15” to their ‘equilibrium’ orientations in the ‘transition state’. The tentative conclusion that the operative trajectories in naphthalene do not cross saddle points is of considerable theoretical interest. Since the introduction of the radiotracer technique,’ self-diffusion coefficients for atomic and ionic crystals have been extensively measured.2 The development of interatomic potentials coupled with high-speed computing machinery has made it possible to simulate lattice defects in these solid^,^ and self-diffusion in crystalline rare gases has been studied in this Compared to molecular crystals the theoretical interpretation of diffusion in atomic systems has several major advantages.(1) The lattice symmetry is very high, often cubic; this greatly reduces the number of possible diffusion pathways as well as the number of calculations required for any one pathway. (2) The number of atomic species present is small, usually only one; in this case only one atomic pair potential is required. ( 3 ) The atoms can be treated as point masses; the three angular coordinates which would be needed to describe the orientation of a rigid body are not required, and the motion of each species in the crystal can be completely described by just three spatial coordinates.Present address : Morgantown Energy Technology Center, Morgantown, West Virginia 26507-880, U.S.A. 13811382 Self-difusion in Naphthalene In spite of the relative complexities, however, there exist many more molecular crystals than atomic ones. And molecular crystals possess many of the most interesting and useful properties found in the crystalline state. In recent years, primarily due to the careful radiotracer measurements of Sherwood and coworkers,8 reliable data for the temperature dependence of the intrinsic self- diffusion have become available for several aromatic compounds. Isotopic substitution experiments have shown that self-diffusion in benzene9 (and presumably other) crystals occurs by a vacancy mechanism.However, no experimental methods for obtaining detailed mechanical descriptions of diffusion mechanisms have been reported. For example, for no aromatic crystal has it been shown which pair(s) of lattice sites participate in the vacancy interchange or what role in the diffusion is played by molecular rotation. The lack of this type of information makes it impossible to apply modern dynamic theories of diffusion to real crystals and invariably necessitates the use of transition-state theory for the interpretation of experimental data. Detailed inter- pretations of the diffusion mechanism in molecular crystals have been considered impractical,8 however, owing to the difficulties in calculating the intermolecular forces.With the availability of accurate semi-empirical atomic pair potentials for carbon and hydrogen,l* it is now possible to obtain the intermolecular potential for any specified distance and relative orientation between two hydrocarbon m o l e c ~ l e s ; ~ ~ - ~ ~ this makes it feasible to attempt calculations of the self-diffusion pathways in these molecular cry~ta1s.l~ For all the compounds so far examined (adamantane, naphthalene, anthra- cene), the value of AU, calculated in the rigid-lattice approximation is almost exactly twice the experimental value, presumably owing to neglect of lattice relaxations. In eqn (3) of ref. (14) the factor 0.5 was introduced to correct for this effect; in the present paper the 0.5 factor is omitted, and we attempt to calculate the effect of lattice relaxation on A Uf directly.Naphthalene is one of the most studied molecular crystals; its monoclinic (P2,la) structure15~ l6 represents a typical molecular packing for aromatic c o m p o ~ n d s . ~ ~ ~ l8 Furthermore, this system has three unique axes hence, in principle, three different rates of self-diffusion. Radiotracer measurements of the self-diffusion normal to ab basal plane19 and of diffusion along sub-grain boundaries20 have been reported, as well as the results of creep flow studies.21 Sherwood has also made unpublished studies of the self-diffusion parallel to the a and b axes.22 The purposes of the present study were to (I) confirm the assumption that self-diffusion in naphthalene occurs by a vacancy mechanism or mechanisms, (2) discover which pair(s) of lattice sites participate in vacancy exchange, (3) determine the role of molecular rotation in the mechanism(s) and (4) investigate activation volume effects.A treatment similar to the present one has previously been reported by this author of the mechanisms of self-diffusion in plastic-crystalline adamantane.14 Transition-state Theory of Naphthalene Self-diffusion The experimental data8 indicate that self-diffusion in aromatic crystals takes place by activated processes in which a neighbouring molecule jumps into a single-vacancy hole in an otherwise ‘perfect’ region of the crystal. In the usual ‘transition-state’ theory the diffusion along direction q due to each type of neighbour (nearest, next-nearest etc.) may be represented2 by an equation of the form 0% = r,(a%)2 V, exp (ASJR) exp (- PAV,/RT) exp (- AU,/RT) in which vm is a vibrational frequency of the neighbour rn; a% is a component of its jump distance; y m depends on the number of rn neighbours; and ASm, AVm and AU, are theD.H. Smith I383 net entropy, volume and potential-energy changes associated with formation of the vacancy and the transition state. In contrast to the common assumption for atomic self-diffusion in cubic crystals, there is no a priori reason to restrict the jumps to nearest neighbours; nor is it reasonable to expect that -(AU,+PAV,)/RT should be equal for all m. In general, the measured diffusion coefficient for a given crystalline direction should be DQ = CDS, (2) in which each Dq, has a different temperature dependence and a different weighting factor v,(aS,)2vmexp (ASm/R).Empirically, however, it is found8 for each of the crystalline directions so far examined that the intrinsic or lattice self-diffusion can be expressed by the simple form D = Do exp(-AH,,,/RT). (3) Moreover, separate values of Do and AHact for non-equivalent lattice directions have not yet been experimentally resolved for any aromatic molecular crystal. Hence, in addition to obtaining an accurate value for AH,,,, the calculated potentials for each compound must show either (1) that AHm is prohibitively large for the diffusion of all but one type of neighbour, (2) that within experimental resolution AH, is the same for all important mechanisms and/or (3) conclude that Do is exceedingly small for any mechanism whose theoretical AH, is significantly smaller than the experimental AHact.Assuming, for the moment, that only one mechanism contributes significantly and dropping the subscript m, we may write the coefficient of self-diffusion as Dq = y ( ~ q ) ~ v exp [(AS,+ASf)/R] exp[-P(AVf+AV$)/Ra exp[-(AU,+AU)/RTJ where the subscript f and the superscript $ refer to the vacancy and the transition state, respectively.2 In this case the temperature dependence of the diffusion would be identical for all three directions. The absolute rates would be in the ratios of the ( ~ q ) ~ , but in general these rations are 5 2 and thus too small for accurate measurement. (4) From eqn (3) and (4) it follows that for the variable X (= S, U or V) AXact = AXf + AH.( 5 ) The enthalpy required to form a vacancy, AHf, is (neglecting small corrections) just the heat of sublimation,11’23 so that an experimental value for AH$ can be obtained by subtracting the heat of sublimation from the activation enthalpy obtained from the temperature dependence of D according to eqn (3).t Further experimental deconvolution of eqn (4) is more difficult. In principle, AV,,, (but not A& or AYI separately) can be obtained from the isothermal pressure dependence of D if the pressure dependence of v can be estimated:24 No pressure studies of the self-diffusion in these compounds have yet been reported. The value of AS,,, can be calculated with fair accuracy from Do = ya2v exp (AS,,,/R) (7) and reasonable estimates of 7, a2 and v. This has been done by S h e r ~ o o d , ~ ~ and the results were taken as further evidence for diffusion by the single-vacancy mechanism.It was t More exactly, AU, = AHsubl+AU0- RT, where W , is the zero-point energy, the usual approximations involving ideality of the gas and neglect of the volume of the solid are made, and temperature corrections involving the difference between the heat capacities of the solid and gas are neglected. Not only are these terms small, but they partially cancel.1384 Self-difksion in Naphthalene Table 1. Values of the parameters used in the atomic pair potentialsloU atomic pair A,/kJ nm6 mo1-I B,,/kJ mo1-I C,/ nm- * carbon-carbon 2238 x lop6 carbon-hydrogen 581.6 x hydrogen4 ydrogen 150.6 x lop6 31 1540 0.360 39 376 0.367 16 736 0.374 suggested that ASf N AS1 e ;AS act, but the bases for an accurate analysis are not available. The value of v was calculated from the Debye temperature, although it has been objected that the lattice vibrations of a molecule close to a vacancy will be much different from those of a molecule far from any defect.26 A more serious problem than the experimental determination of the transition-state parameters is the question of their physical meanings.Some of the objections to eqn (1) and (4) have been discussed elsewhere in connection with the formulation of alternate and much of the apparent success of the theory may be due to the lack of sufficiently stringent tests. The present treatment provides opportunities for a more critical evaluation of the theory. Diffusion Model and Calculation of Potentials for Vacancy Formation and Diffusion The potential energy between two molecules in the crystal is assumed to be the sum of all the atomic pair potentials between their constituent atoms.The atomic pair potentials are assumed to depend only on the distance between the two atoms and on their atomic species, i.e. to be unaffected by the presence of third atoms on either other molecules or on the molecules of which the two given atoms form a part. The exponential-6 function Vij = A i j P + Bij exp (- C,r) is used for the potentials between atoms i andj, with Williams’ values for the parameters A,, Bij and Cij.l0 The empirical justification for these assumptions and evidence for the accuracy of the potentials chosen may be found el~ewhere.~O-~~ The values of the parameters used are listed in table 1.The potential of a molecule M surrounded by any given configuration of other molecules can be obtained by calculation of each intermolecular potential and summation of the molecular pair potentials. In particular, if the positions of M and the surrounding molecules correspond to the equilibrium crystal structure (as determined by X-ray or neutron diffraction measurements), the calculated potential corresponds to - AUf, where AU, is the energy required to create a vacancy. Once AUf has been calculated, M may be moved successively through six-dimensional translational-rotational space to other points and orientations in the lattice and the potential calculated for each point. When the potential surface for a molecular trajectory between two lattice sites has been obtained in this way, AUt is equated to the maximum in the potential encountered by the molecule along the jump path actually taken, minus its initial potential.No attempt is made here to link the position of maximum potential with a thermodynamically definable transition state. The initial potential is just -AUf, less the molecular pair potential term within -AUf which exists between the diffusing molecule at its initial site and a molecule at the vacancy site when the latter site is actually filled. The origin is chosen to be at M, and the potential barriers for diffusion of molecule M into single vacancies at various other lattice sites are calculated and compared. In theD. H . Smith 1385 0 1.0 nm I Fig.1. The unit cell and molecular packing of naphthalene (hydrogens are omitted for clarity). Among the nearest neighbours to molecule M are : A(first), B(second), C(third), D(fourth), E(fifth), F(sixth). Table 2. Unit-cell dimensions of naphthalene (space group P2 1 /a)'6 a/nm b/nm c/nm a/O D/" Y/" 0.8266 0.5968 0.8669 90 122.92 90 initial calculation, all molecules except M are held fixed in the positions and orientations indicated by X-ray and neutron-diffraction studies. Hence, at this level of approximation, A vf and A Vx are zero and corresponding energy and enthalpy terms are equal. If desired, these conditions (i.e. the lattice) can be relaxed later, after the initial 'screening' calculation to determine which of the neighbouring molecules can actually jump into a vacancy.The unit cell and molecular packing of naphthalenel59 l6 are illustrated in Fig. 1, and the cell dimensions are listed in table 2. At ambient pressure the compound crystallizes in the P2Ja space group with two molecules per unit cell. It is convenient to think of the lattice as being composed of two equivalent sublattices, here designated as s and t , respectively. (The coordinates of the s and t molecule in the unit cell are interchangeable by a screw diad pera at ion.^^) A molecule can, in principle, jump into a vacancy in its own sublattice by pure translation. However, the exchange of molecules between sublattices must be accompanied by molecular rotation, or else leave the diffused molecule in a non-equilibrium orientation.In the computation of the potentials from eqn (8) carbon positions were taken from the published cell dimensions and fractional coordinate~,~~~ l6 and hydrogen positions were either taken from the neutron-diffraction study16 or were calculated from the carbon coordinates and an assumed C-H bond length of 1.08 A. In his calculations to obtain the parameters in table 1, Williams used C-H bond lengths of 1.027 A.loa However, the difference in length is small, and empirical justification has been found for using these potentials with the true C-H intemuclear distance.11-14 For the enthalpy of sublimation of naphthalene there is little difference among the value used by Williams (72.4 kJ mol-l), .1386 Self-diflusion in Naphthalene Table 3. Calculated potential barriers for molecular jumps into neighbouring vacancies: AUX, barrier of operative mechanism; A U*, ‘ Saddle-point ’ Barriera relaxed rigid lattice lattice distance no.of neighbour /nm neighbours sublattice AU/kJ mol-1 AU*/kJ rno1-I Ui/kJ mol - - - (0 - 0th 0.0 1 st 0.51 4 S 230 50 84 2nd 0.60 2 t 470 30 3rd 0.79 4 3 480 480 4th 0.81 2 t > 1000 > 1000 5th 0.83 2 t > 1000 > 1000 6th 0.87 2 t 210 40 95 - S - - - a Experimental barrier height AH$ z 106 kJ mol-l. Table 4 Comparison of relaxed-lattice model and experimental values (in kJ mol-I) of self-diffusion parameters for the three crystalline directions theoretical experimental direction mechanism AUf AUt AUact AHsubla AHJb AHact 11 to b axis nearest 71 84 155 73 ca.106 ca. 179c 1) to a axis nearest 71 84 155 73 ca.106 ca. 179c neighbour neighbour neighbour neighbour sixth-nearest 71 95 166 I to ab plane sixth-nearest 71 95 166 73 106 1 79d a Average of 5 experimental values.32* 36 directly. Ref. (22). Ref. (19). Difference between AHac+, and AH,,,,, not measurable the value calculated in the present work (71 .O kJ mol-l) and the average experimental value (72.8 kJ mol-1)33-37 (see below). The diffusing ‘molecule was assumed in all cases to move along the shortest transla- tional path between lattice sites. This assumption is justified for naphthalene by the molecular and crystal symmetries. 14-16 Early in the calculations evidence arose that the molecular trajectories might not cross saddle points in the six-dimensional potential surface. Hence, procedures for starting at the origin to seek the path of minimum potential between the initial and final sites were rejected in favour of the following method, which covers all of the reasonable possibilities in a computationally efficient manner.The complete potential surface in rotational space was calculated with the diffusing molecule midway between its initial and final translational coordinates. In some cases the potentials were sufficiently large at all molecular orientations to eliminate the possibility of diffusion between the given pair of neighbouring sites. When areas of reasonably low potential were found, the surfaces were refined by calculation of the potentials for additional orientations. Only after this winnowing procedure was it efficient to calculate the molecular potentials of M for other spatial coordinates and confirm that the barrier maximum did indeed occur at the jump mid-point.After it had been found between which pairs of lattice sites vacancy exchange actually occurs, the rigid-lattice approximation was also discarded, and neighbouringD. H . Smith 1387 molecules in the lattice were allowed to move as part of the diffusion mechanism. The results for potentials calculated in this way are summarized in tables 3 and 4: Energy of Vacancy Formation, AUf The calculated energy required to form vacancies in naphthalene is shown in table 4, along with the experimental enthalpy of sublimation. The average of five experimental determinations of AH,,,,, which range from 65.6 to 82.0 kJ mol-l, is 72.8 kJ mol-1.33-37 The computed value of AU, is 7 1 .O kJ mol-l.The quantities AU, and AH,,,, are not strictly comparable, since the enthalpy includes the additional term PAV,.23 In the present model, however, the change of volume of the crystal due to vacancy formation is identically zero. Temperature correction terms also occur in comparisons of calculated and measured values in table 4,23 since the heats of ~ u b l i m a t i o n , ~ ~ - ~ ~ crystal s t r u ~ t u r e s ~ ~ ~ l6 and rates of self-diff~sion~~-~~ were measured at different temperatures. However, the corrections are ca. 5% (or approximately equal to the uncertainties in the measured and computed values), so that in numerical comparisons they can be neglected. The agreement between the calculated AU, and the measured heat of s ~ b l i m a t i o n ~ ~ - ~ ~ is essentially an indication of the validity of the assumptions made in treating the lattice energy as the sum of atomic pair potentials, since the heat of sublimation of naphthalene was one of the 81 fitting equations used by Williamsloa to obtain the nine parameters listed in table 1.Heights of Potential Barriers, AU, in the Rigid Lattice Model Nearest Neigh bows It is generally assumed that self-diffusion in atomic solids takes place between nearest- neighbour sites. In naphthalene this diffusion mechanism is complicated by the fact that it requires molecular rotation. The four nearest-neighbour sites to a molecule at the origin lie 0.51 nm distant in the ab basal plane at (k a/2, b/2,0). Hence this mechanism would produce virtually identical rates of diffusion parallel to the a and b axes, but would not contribute to diffusion normal to the ab plane.[In terms of eqn (3), AH,,, would be identical for the two directions, and the ratio D t / D i would be a2/b2 = 1.9; given the accuracy with which Do can be measured, these values are essentially equal.] It appears from the calculations that there must be no significant nearest-neighbour diffusion in which the molecular trajectory crosses a saddle point in the potential surface: achieving the saddle-point orientation requires a rather complicated set of rotations by the diffusing molecule, and the saddle-point potential yields a barrier height (labeled AU* in table 3) which is ca. 60 kJ mol-l smaller than the experimental value (AH1 z 106 kJ mol-’, table 4).Because of this latter fact we are forced to conclude that the diffusing molecule must choose, instead, a dynamically simple trajectory which does not cross the saddle point. Since nearest-neighbour sites belong to different sublattices, a molecule diffusing between such sites must rotate while translating if it is to arrive at its new site with the ‘correct’ orientation.? Geometrically, there are two ways in which a molecule can 1 It is well known from theory and n.m.r. experiments that naphthalene molecule in an equilibrium orientation at a lattice site can undergo an in-plane rotation with an activation energy of CQ. 100 kJ mol-l.ll This motion occurs in the absence of any nearby vacancy, and a vacancy slightly reduces the activation energy.However, the barriers to out-of-plane rotations are extremely large, and the molecular planes of s- and t- molecules are not parallel. Thus the requirement of rotation-with-translation as stated above is not affected, except that it might not be necessary for the diffusing molecule to arrive with exactly the ‘correct’ orientation within the required plane. When the statistical aspects of the jumps are omitted, as here, the effect of the latter possibility is to increase slightly the uncertainty in the calculated value of AUS. If the diffusing molecule exchanged sublattices with no rotation whatsoever, its potential at the new site would be 550 kJ mol-l larger than the equilibrium potential - a very unlikely event. Hence, rotation-with-translation is a quite rigid requirement for any mechanism in which the molecu!e changes sublattices.1388 Self-dflusion in Naphthalene exchange sublattice orientations with a simple uniaxial rotation: it can either rotate 53" in one direction or 127" in the opposite sense, about a 'diagonal' molecular axis passing approximately through the numbers 2 and 6 carbons (IUPAC numbering).(The near coincidence of the rotational axis and the line through the two carbons is 'accidental' and not required by symmetry.) When the molecular potentials for nearest-neighbour diffusion incorporating these rotations are calculated, it is found that the potential barrier for the smaller-angle rotation is prohibitively large. AUI for the 127" rotation, however, is 230 kJ mol-l. (It is assumed that the midpoints of the translation and rotation are reached simultaneously.) Hence, from eqn (9, AU,,, for this motion is 301 kJ mol-l.This value is larger than the experimental one, but it neglects possible motions of other nearby molecules which would decrease AUI. Next-nearest Neighbours Translation without Rotation. The next -neares t -neigh bo ur distance is one unit -cell dimension (0.60 nm) along the b axis; intuitively it might be expected that vacancy interchange between these sites would constitute an important self-diffusion mechanism, since molecular rotation is not required. The potential barrier to this pathway is surprising, however: the value of AUI for a simple translational jump along the b axis is 470 kJ mol-l. From comparison with experiment (see below), the rate of self-diffusion by this mechanism must be completely negligible.Translation with Rotation. Curiously, although molecular rotation is not required since the initial and final sites belong to the same sublattice, the molecule can, in principle, greatly reduce the value of AUt by combining rotation with its translational jump. This is discussed as another example of the many molecular pathways having surprisingly low activation energies which exist in the crystal, but which make no appreciable contribution to the self-diffusion. The postulated mechanism involves a 'sideways slipping' motion in which the molecule rotates about its lengthwise axis (which lies approximately in the bc plane) while translating along the b axis. (see fig.1). The molecule must rotate ca. 40" by the time it reaches the jump mid-point at (0, b / 2 , 0 ) to minimize the repulsive potential from the two nearest molecules at (k a / 2 , b/2, 0). On the second half of the jump the molecule must rotate in the reverse direction, making a -40" rotation to its original orientation. [This is true whether the molecule jumps forward into the original vacancy at (0, b/2, 0) or backwards into the vacancy created at the lattice site it left.] This particular rotation produces a value of AU* of only 30 kJ mol-l, which is ca. 80 kJ mol-1 smaller than the potential barriers of the mechanisms which are actually operative in the crystal (AH1 x 106 kJ mol-l). Self-diflusion to Third-, Fourth- and Fifth-nearest-neighbour Sites From the experimental results it might be expected that the sole diffusion mechanism in naphthalene would be vacancy exchange between third-nearest-neighbour sites.This would account for the apparently isotropic diffusion, since the translational vector has approximately equal components in the three different lattice directions. However, the sites belong to separate sublattices, which makes the mechanisms intuitively less plausible. The calculations are in agreement with this latter expectation : the lowest barrier possible for third-nearest-neighbour diffusion is over 370 kJ mol-1 larger than the experimental value. No other reasonably close pair of sites would produce molecular displacement along all three crystal axes. Hence, there must be at least two separate mechanisms involvingD.H. Smith 1389 different pairs of sites, and the apparent isotropy of the diffusion coefficient must be accidental. No rotation is required by the site symmetries for a molecular jump into a fourth- nearest-neighbour vacancy, which is 0.81 nm distant on the short diagonal of the ac plane ; however, the potential barrier is prohibitively large for any molecular orientation. The situation is similar for diffusion into a fifth-nearest-neighbour vacancy : the site symmetries require only a simple translational motion 0.81 nm along the a axis, but AUI is extremely large regardless of the molecule’s orientation. Hence, significant rates of self-diffusion between third-, fourth- and fifth-nearest- neighbour sites are all ruled out by the intermolecular potentials.Self-diflusion Along the c Axis Sixth-nearest neighbours occur 0.867 nm apart (the unit-cell length) along the c axis. The potential with a molecule at (0, 0, 4) when (0, 0, 0) and (0, 0, 1) are unoccupied can be as little as - 100 kJ mol-l, which corresponds to AU* = 40 kJ mol-1 (table 3). This is a third case in which the height of the potential barrier for the saddle-point trajectory is ca. 70_f 10 kJ mol-1 smaller than the experimental value, and the trajectory must therefore be considered inoperable. This U* is produced by a rotation of M. Since the initial and final sites belong to the same sublattice, it is plausible to think that diffusion between them might occur without rotation of the diffusing molecule. For this particular molecular orientation, the potential passes through its maximum midway between the lattice sites, and the value of AUI is 210 kJ mol-l, which is about twice the experimental value.This point is neither the absolute nor a local sub-saddle point in full six-dimensional space. Summary of Results for the Rigid-lattice Model The experimental enthalpy of activation for self-diffusion is the sum of the heat of vacancy formation, AH,, and the enthalpy barrier, AH$. In the present model, however, both A& and AVJ are identically zero. Accordingly, the equations AUf = AH,, AUI = AH$, and thus AU,,, = AH,,,, should obtain. Early attempts to measure intrinsic self-diffusion in molecular crystals were often thwarted by extrinsic diffusion due to sub-grain boundaries, impurity-induced defects or damage produced in the crystal during preparation of the ample.^^-^^? 35-40 Sherwood, however, has obtained excellent data for self-diffusion normal to the ab cleavage plane of naphthalene.l9 Although the theoretical value of the energy of vacancy formation (AUf = 71 kJ mol-l) agrees well with the experimental heat of sublimation (AHsubl = 73 kJ mol-l, none of the heights of the various potential barriers calculated with the rigid lattice model agrees really well with the experimental value of AH$. While one would expect that the theoretical values should be too large owing to neglect of lattice relaxation, surprisingly all of the trajectories which cross saddle points give theoretical values which are either much smaller than the measured barrier height or else very much larger.In principle, the smaller values could simply indicate that the mechanisms are operative but the atomic pair potentials are too ‘ soft ’. However, the same atomic pair potentials have been used to calculate activation energies in the rigid-lattice approximation for rotational diffusion of a large number of compounds, and it has invariably been found that the theoretical activation energies are larger than the experimental values. This contradicts the suggestion that the potential functions are too soft. Furthermore, in all three saddle-point mechanisms for which AU < AH$ the diffusing molecule must make complicated rotatians about its molecular axes in order to attain the saddle-point orientation. For example, in the second-nearest-neighbour mechanism, the simplest to1390 Self-diflusion in Naphthalene describe, the diffusing molecule must reverse the sign of its angular momentum at the jump mid-point. In a dynamic model it would be implausible that the molecule could do so without being reflected back to its initial lattice site.Together, the evidence indicates that the molecular trajectories of the operative mechanisms do not cross saddle-points in the potential surfaces, and that the transition- state entropy cannot be neglected a priori. Additional evidence has been generated by investigation of two postulated non-saddle-point mechanisms with a relaxed lattice model. Potential Barrier Heights in the Relaxed-lattice Model Whenever it is concluded that the saddle point is not crossed in an operative mechanism, a large number of trajectories can be described which yield the experimental value of the barrier height; the only requirement imposed by the condition that the theoretical and experimental barrier heights agree is that the maximum potential encountered along the trajectory fall somewhere on the contour line in the potential surface defined by AUt = AH$.Despite the apparent limitations imposed by the conclusion that AUt = AH1 is not a sufficient condition to define a unique trajectory, it is still possible to test certain diffusion trajectories which for intuitive or other reasons somehow seem particularly likely. If applied to a variety of plausible trajectories for several different compounds, this procedure may make it possible to build up a detailed and reliable picture of the diffusion mechanisms.From table 3 it may be noted that the smallest value of AUt listed which meet the condition AUt 2 AH1 are for first and sixth-nearest-neighbour mechanism. For both of these trajectories the molecule's angular and spatial coordinates pass simultaneously through the midpoints between their initial and final values, and the orientation angles change either monotonically (nearest neighbour) or not at all (sixth-nearest-neighbour). In other words, changes in direction of the naphthalene molecules' angular momenta during these translational jumps do not occur even though such changes would allow the molecules to cross saddle points. In this section it is hypothesized that these apparent requirements are, in fact, necessary conditions; and the values of AUI are recalculated in a 'relaxed lattice' model which allows other nearby molecules to move.Sixth-nearest-neighbour Mechanism For sixth-nearest-neighbour diffusion the initial and final sites belong to the same sub-lattices, and this hypothesis requires that the diffusing molecule not rotate. Examin- ation of the individual molecular-pair interactions between M and the rest of the molecules in the crystal reveals that even at the barrier maximum in the rigid lattice model only four of these potentials are positive, i.e. repulsive. These four pair potentials are with the neighbours at (-4, -4, 0), (4, #, l), (-4, $, 0) and (+, -4, l), and are all equal to 12.6 kJ mol-1 when M is at (0, 0, +). Furthermore, most of the repulsion comes from just one pair of hydrogens per molecular pair.This suggests that rotation of these four molecules will play a major role in the mechanism and that relaxation of other molecules in the lattice may be minimal. When these four neighbouring molecules are allowed to relax it is found that rotations of ca. 12" (10" for one pair, 15" for the other; the four neighbours are not all equivalent) decrease their molecular pair potential with M at the barrier maximum by 27 and 31 kJ mol-l, respectively. The theoretical barrier for the mechanism in which all four neighbours relax in this manner is AUt = 95 kJ mol-l, in excellent agreement with the experimental value of ca. 106 kJ mol-l. (The agreement would be even better if weD.H . Smith 1391 -801 I I I I 1 I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Clnm Fig. 2. Potential for self-diffusion of a naphthalene molecule along the c direction to a sixth-nearest-neighbour vacancy (relaxed lattice model). assumed that for some reason only three of the neighbours rotated.) Rotations of this magnitude are physically quite reasonable; P a ~ l e y , ~ ~ for example, found that the root-mean-square amplitudes of rotational vibration of a molecule in a perfect region of the crystal were ca. 5". Fig. 2 shows the potential barrier for the relaxed lattice model. Nearest-neighbour Mechanism For the nearest-neighbour mechanism, in which M attempts to jump from (0, 0, 0) to (i, f, 0), only two of the neighbours repel the jump in the rigid-lattice model.One of these is at (0, 0, - 1) and has the orientation with which M ends its diffusional jump; the other is at (4, i, 1) and has the orientation with which M starts. Their molecular pair potentials with M at the jump mid-point, (f, f, 0), are 61 and 37 kJ mol-l, respectively. Rotations by ca. 15" of these two molecules (for one the axis is nearly orthogonal to the molecular plane; for the other it is colinear with the central carbon<arbon bond) decrease the net barrier height by 102 and 44 kJ mol-l, respectively, however. Thus, relaxation of these neighbours reduces the calculated barrier height to 85 kJ mol-1 and the value of AU,,, to 155 kJ mol-l. Nearest-neighbour vacancy exchange produces diffusion in both the a and b directions. The experimental enthalpies of activation for these directions are approximately equal, i.e.for both AHact = 180 kJ mol-l; however, the uncertainties in these values are significantly greater than they are for the perpendicular to the ab plane.22 Furthermore, diffusion along the a direction also occurs by a sixth-nearest-neighbour mechanism, so that the experimental value for this direction represents an unresolved average from two different mechanisms. Because of these uncertainties plus those in the calculations, the agreement between the relaxed lattice model and experiment must be considered reasonable for the nearest-neighbour mechanism. Activation-volume Eflects In discussions of diffusion mechanisms it is often suggested that there must be some 'collapse' of the surrounding molecules into a vacancy within a crystal.From a molecular potential standpoint, this effect is defined by the shifts in position of the minima of the potential wells in which the nearby molecules vibrate when a vacancy is introduced into the lattice.1392 -50 - -55- - 8 -60 -65 -70 Self-difusion in Naphthalene - - - - 1 1 I I I I I I I I -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 C/nm Fig. 3. Comparison of the potential wells for vibration of a molecule along the c axis when a sixth-nearest-neighbour site is (a) occupied (filled circles) and (b) vacant (open circles). If the degree of relaxation is small, the changes of position of all other molecules can be treated as small perturbations affecting the change in equilibrium position of any given molecules, and cooperative relaxation effects can be neglected.Fig. 3 compares the potentials for a naphthalene molecule vibrating towards a sixth-neighbour site when the site is (a) occupied and (b) vacant. The coordinates of all other molecules were held constant. Clearly the vacancy produces changes in the vibrational levels due to the increased width and shallower depth of the potential well, but the shift in the position of the potential minimum is negligible. The amplitude of the vibration also increases in the direction away from the vacancy. This happens, of course, because the attractive as well as the repulsive forces of a molecule at the vacancy site are missing. The molecular pair potential with a sixth-nearest neighbour is just 3 % of a molecule’s total lattice potential.The largest molecular pair potential, between nearest neighbours, is four times as large. Even for a nearest-neighbour to a vacancy, however, there is no indication of any significant ‘collapse’ of the molecule into the vacant site. In large part Ayf and AVI can be neglected simply because their contribution to the enthalpy must be very small compared to the potential-energy terms. The calculations indicate that the volume of vacancy formation is a real, but small, effect. On the other hand, it is implausible that the whole lattice can ‘breathe’ during a diffusional jump; at the position of the diffusing molecule, where its potential reaches its maximum, intermolecular distances are short and interatomic repulsions are sensitive functions of any changes of interatomic distances associated with AVt.Hence any expansion of the lattice would lead to substantially lower values of A V t than those calculated. However, excellent agreement with experiment is obtained for the theory in which AVI = 0. Summary of Results for the Relaxed-lattice Model The emergent picture of the mechanisms in the relaxed-lattice model is reminiscent of that envisioned by Rice,26 in which out-of-phase translational lattice vibrations of a pair of neighbours located between the vacancy and the diffusing molecule allow the diffusing molecule to jump between the neighbours and into the vacancy. An important motivation for Rice’s treatment of diffusion26 was the lack of any evidence that at its jump mid-point the diffusing molecule can attain thermodynamic equilibrium with the lattice as assumedD.H. Smith 1393 by the transition-state theory; it was hoped that the pseudo-thermodynamic basis of transition-state theory could be replaced by a completely dynamic model. In the mechanism described here the orientations to .which the neighbouring molecules rotate as M approaches the jump mid-point are, in fact, minima in the potential wells in which the neighbours would vibrate indefinitely if M were somehow fixed at the jump mid-point. Thus in a ‘real’ example similar to a dynamic model previously described in general terms for all crystals, we still find that a quasi-thermodynamic description of the mechanism is possible. Conclusion It has been concluded that diffusion normal to the ab plane occurs only by sixth- nearest-neighbour vacancy exchange, that diffusion along the a-direction occurs by this mechanism and also by nearest-neighbour vacancy exchange, and that diffusion along the b direction also occurs by the latter mechanism.Previous use of the rigid lattice model for rotational diffusion has yielded theoretical activation energies which were too large, but for translational diffusion in naphthalene the model gives saddle-point activation energies which are appreciably smaller than the experimental values. Furthermore, the saddle-point trajectories require complicated rotations of the diffusing molecule; and it appears qualitatively that the transition state entropies of the saddle-point trajectories may be unfavourable. For these reasons, it is tentatively concluded that the molecular trajectories do not cross saddle-points.A relaxed-lattice model, in which the diffusing model either undergoes a direct flip between equilibrium orientations or no rotation at all and in which only two to four lattice molecules undergo substantial rotational relaxation (ca. 10-1 5 O ) , gives activation energies which agree well with experiment. This is further evidence for rejecting the saddle-point trajectories The latter model is reminiscent of Rice’s dynamic theory of atomic diffusion, in which an out-of-phase lattice vibration of a pair of intervening atoms allows the diffusing atom to jump between them and into the vacancy. In the present case, however, the relaxed orientations of the intervening molecules would be their equilibrium orientations if the diffusing molecule were fixed at the potential maximum; hence the language of transition state theory may still be useful.The finding that the saddle-point trajectories are not operative is of considerable theoretical interest and should be further examined to determine if it can be substantiated. I thank Dr W. R. Busing for invaluable discussions and Prof. J. N. Sherwood for providing unpublished data which made possible a more detailed investigation of the diffusion in the ab plane. References 1 2 3 G. Hevesy and F. Paneth, Anorg. Chem., 1913,82, 323. P. G. Shewmon, Diffusion in Solids (McGraw-Hill, New York, 1963). Interatomic Potentials and Simulation of Lattice Defects, ed. P. Gehlen, J. R. Beeler Jr and R. I. Jaffee (Plenum Press, New York, 1972).4 R. Fieschi, G. F. Nardelli and A. Repenci-Chiarotti, Phys. Reo., 1961, 123, 141. 5 J. J. Burton and G. 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