J. Phys. Chem. 1986, 90, 5329-5333
5329
A Simulation Study on the Topochemical and Statistical Distribution of the Activity in Particulate Solids Masataka Machida and Mamoru Senna* Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama 223, Japan (Received: January 22, 1986)
A method for evaluating the local irregularity of an atomic arrangement and excess energy distribution in particulate solids was developed by using the concept of the “local potential energy (LPE)+’ as well as the “local radial distribution function (LRDF)”. Various two-dimensional assemblages of model disks from a hexagonal closest packing to an amorphous state were examined. The value of the local potential energy, EL, obtained by summing the pair potentials was used to evaluate the energetical fluctuation. The average value, serves as a measure of overall activity, comparable with a measurable excess enthalpy. The mean standard deviation, $, of the LRDF from that for perfect orderliness was used as a measure of local irregularity. Similarities and differences in topochemical as well as statistical distribution of LPE and LRDF were discussed.
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1. Introduction Crystalline imperfections are known to play a key role in solid-state rate A number of rate equations have been proposed either on a theoretical or on an empirical basis in order to describe solid-state rate proce~ses.~JIt is nevertheless often the case that none of such rate equations are applicable to the actual rate process, particularly when the starting materials have been preliminary activated. One of the reasons for such inapplicability seems to be attributed to the difference in the local reactivity. The local reactivity, in turn, is dominated by the local concentration of the structural imperfections, Le., active or potential centers6 The authors introduced the concept of the distribution of reactivity in the Johnson-Mehl-Avrami rate equation, a typical nuclei-growth kinetic equation.’ It turned out that extraordinarily small values of the apparent exponent, it, observed in the thermal transformation of a mechanically activated material could be explained by taking such a distribution into account. A peculiar rate process of polymorphic transformation of mechanically activated materials could also be attributed to such a distribution of chemical potentials.* It is extremely difficult to verify the existence of such a distribution, as long as we adhere to fine powdery materials. One of the main reasons is the restriction of the resolution of local analysis. A similar study using a single crystal is now in progress, but a smooth and direct connection between the results of such a model experiment and those of mechanically activated fine powders seems to be accomplished only with enormous difficulty. The present problem would be circumvented, however, by a deductive study using a computer simulation. As a first step in the deduction, we concentrate in the present study on the description of the local disturbance of the atomic arrangement within a single model particle. The main purpose of the present study is thus to develop a method for evaluating the fluctuation of the potential energy as well as the local concentration of structural imperfection, in order to verify and quantify the local distribution of chemical properties in solid particulate materials. (1) H+vall, J. A. In R e a k f i o ~ f ~ h i g k e i t f e sSroffe; f e r Johann Ambrosius Barth: Leipzig, 1938. (2) Hauffe, K. In Reakrionen in und anfesten Stoffen; Springer: Berlin, 1966 ___“.
(3) Schmalzried, H. In Solid State Reactions, Alper, A. M., Margrave, J. L., Nowick, A. S., Eds.; Academic: New York, 1974. (4) Hulbert, S. F. J . Brit. Ceram. SOC.1969, 6, 11. (5) Hancock, J. D.; Sharp, J. H. J . Am. Ceram. SOC.1972, 55, 74. (6) Steinike, U.; Linke, E. Z . Chem. 1982, 22, 397. (7) Machida, M.; Senna, M. J. Phys. Chem. 1985,89, 3134. (8) Imamura, K.; Senna, M. J . Chem. SOC.,Faraday Trans. 1 1982, 78, 1131.
0022-3654/86/2090-5329$01.50/0
2. Calculation 2.1. Models. An equisized hard-disk model in a two-dimensional assemblage was used for the purpose of visualization and evaluation of the local potential energy and atomic arrangement. Starting with hexagonal closest packing, different disturbances were introduced into this “perfect” assemblage, as is shown in Figure 1. Finally, we used also a highly disordered system symbolizing an apparent amorphous state. Since the disks cannot penetrate each other, the present model realizes the repulsive force in a simple manner. The effect of the attractive force, on the other hand, was neglected in preparing the model arrangements. 2.2. Calculation of Local Potential Energy. The potential energy, E(r), between two atoms was assumed to be represented by an equation in the form of a Morse function
E ( r ) = De[l - exp(-2a(r - r J l z
(1)
where De is an equivalent of the dissociation energy, a a constant, r the interatomic distance, and re the equilibrium interatomic distance. The value of re is set equal to the diameter of the disk. If all atoms are infinitely far apart, corresponding to complete dissociation, they are at the state of highest potential energy, which is assumed to be zero in the present study. The local potential energy of atom A decreases by an amount De - E(r), when atom A has a neighboring atom B at a distance r. The value of the local potential energy, EL, of atom A is obtained by summing all the possible pair potentials under the assumption that the potential energy does not have any angular preference, i.e. N
EL = E [ E ( r i )- Del i= 1
(2)
where ri is the distance between atom A and i , and N the number of atoms counted. In calculating EL, the shielding effect of other atoms was neglected. The values for De, a, and re were taken as unity for simplicity. All these simplification can be accepted without losing any generality of the model calculation. 2.3. Concept and Calculation of Local RDF. For a quantitative description of the local atomic arrangement, we introduced the concept of a “local radial distribution function (LRDF)”, Le., a radial distribution function with respect to each atom. If we know the LRDF, then we can describe the local irregularity of the assemblage, which in turn would be combined with the local reactivity. For the purpose of calculating the LRDF, the center of the “scanning circle” was laid on an atom under observation. A scanning circle was divided into concentric annuli with an equal breadth, Ar. From the number density of the atoms in each annulus, the LRDF for one particular atom was obtained. A 0 1986 American Chemical Society
5330 The Journal of Physical Chemisfry. Vol. 90, No. 21. 1986
Machida and Senna
similar calculation was repeated for all atoms in the assemblage. 2.4. Evaluation of Randomness from LRDF. If atoms are ordered in the hexagonal closest configuration as in a perfect crystal, the resulting LRDF is tentatively called the perfect local radial distribution function, PRDF. The difference between LRDF and PRDF serves as a measure of the disorderliness around the particular atom under observation. For a quantitative expression, the variance or the mean square deviation, 2,of the LRDF from the PRDF was calculated. A larger value of means a more severe disturbance of the local atomic arrangement. 2.5. Topochemical Representation of the LF’E and LRDF. The topochemical distribution of the LPE or LRDF was graphically visualized by dividing the total range of EL or $ into five regions. An atom lying in a region of higher irregularity, i.e., possessing a higher E , or larger 2,is represented by a darker circle. The division and symbols are given in the legend for Figures I and
6. 3. Local Potential Energy 3.1. Size Effect an the LPE. In calculating the LPE, it is convenient to know whether and to what extent the number of neighboring atoms taken into account influences the value of E,. For this purpose, the number N in eq 2 was changed by varying the radius of a scanning circle whose center was laid on the particular atom A. E , decreased rapidly with increasing relative radius, R,, i.e., the radius of the scanning circle divided by atomic radius, until &approached IO, above which EL remained almost constant. The scale of the present model assemblage was much larger than that corresponding to & = IO,so that the present result could not be unrealistically restricted by the size of the model. Since there was no computational problem, we took all the atoms in a model into account in calculating E , with respect to each individual atom. 3.2. Topochemical Distribufion of fhe LPE. The topochemical distribution of the LPE is represented in Figure I . For model A corresponding to hexagonal closest packing (Figure IA), most of the atoms belong to the lowest level of EL, except for those located on the margin of the assemblage representing a higher potential at the %‘surface”. For model B (Figure IB), atoms located in the vicinity of the unidimensional imperfections (“dislocations”) have higher values of EL,although they are located in the “interior” of the solid. With a further increase in disorderliness from model C to E (Figure I, C-E), the number of darker atoms increase. The value of the LPE at the surface would seem to possess the same value of E, throughout models A to E. In actual fact, however, it is not the case. This apparent similarity of the surface atoms comes only from the manner of the simplified division of E , into only five ranges, whose highest rank is above -8. As will be shown in the next section, the energy state of the surface atom differs considerably depending on the model. 3.3. Statistical Distribution of E,. E , distribution curves for models A to E are shown in Figure 2, A-E. In calculating the distribution, the entire region of E , was divided into 160 ranges in contrast to 5 for a topochemical representation, in order to obtain a curve as smooth and continuous as possible. Nevertheless, model A shows a rather discrete distribution as shown in Figure 2A. This is attributed to the effect of the relatively small number ofsurface atoms on the otherwise identical energy state of perfect packing. Peaks at the highest energy region belong to the surface atoms themselves. With increasing extent of disorderliness from models B to E (Figure 2, B-E), the continuity and breadth of the distribution curve increase and the peaks shift to the higher energy side (smaller absolute value of E,). The foot of the higher energy side at the same time becomes longer with increasing disturbance in atomic arrangement. A bimodal curve in model D (Figure 2D) is obviously attributed to the existence of a relatively wide region of orderliness surrounded by a highly disordered ensemble. 3.4. h e r a l l Excess Energy. EL for all the atoms included in the model were summed and averaged to obtain is., a measure
E,
,..
Figure 1. (A-E. lop 10 bollom)
tential energy for models A 5EL
