J. A. CAMPBELL Oberlin College, Oberlin, Ohio
teacher and student must continually bear in mind the inherent imperfection of all models lest these "crutches" for our thinking become rigid molds for our thoughts. Our ideas of the state of affairs in any system are abstractions, and the models second order abstractions of the ideas. Yet it is often convenient to construct models to serve as mnemonic devices on which to hang our interpretations. Models, even with their inevitable imperfections, are useful and even necessary to the majority of students. We shall make use in this article of a common atomic model-that of the hard sphere-fully realizing that atoms are not as sharply hard as the spheres used, that they may not be spheres at all, and that they are certainly not colored-as the spheres may be for differentiating. Yet the use of hard spheres for the constituent atoms allows the model to approximate many of the gross properties of a solid atomic aggregate. These hard sphere models for solids, more exactly for crystals, have been in use for a t least forty years with considerable success. Present crystalline models are often of great complexity; hormones and even more complicated proteins being approximated. Simple models, on the other hand, are of considerable use and have even found their way into elementary laboratory manuals. This article will restrict itself to very simple models indeed; models readily available to anyone with as little as ten dollars to i n v e ~ tmo?els ,~ which may be used readily in a classroom to interpret gross behavior of crystalline systems in terms of atomic arrangements and rearrangements. EVERY
CLOSEST PACKING OF SPHERES
The maximum coordination number of identically sized spheres may be easily demonstrated to be 12 using the frame pictured in the figures to contain a suitable number of spheres. Twenty-five spheres work very well as shown in Figure la. (The frame is made of four pieces of plywood, slotted as shown to allow close adjustment of frame dimensions to accommodate the spheres used, and bolted together a t the corners. The bolts serve as hinge fulcrums so that the frame can be forced into any desired parallelogram.) The frame dimensions and angles are adjusted so that the 25
'Presented as part of the Symposium on Use and Abuse of Models in Teaching before the Division of Chemical Education a t the 130th Meetinp of the American Chemical Society. Atlantic Citv, September, 1956. All the models discussed in this article may be constructed from 55 spheres 3 or 4 in. in diameter plus a. selection of smaller spheres which will fit into the octahedral, tetrahedral, and triangular holes formed between the large spheres. The cheapest available spheres seem to be styrofoam ones such as discussed in THE JOURNAL, 30, 503 (1953).
spheres are firmly in contact but still all in one plane. The coordination number in this single layer is readily seen to be six. We usually demonstrate this by presetting the frame and then just dropping spheres into i t in a random manner until i t is full. 4 second layer (of 16 sphere) is now rolled onto the first with minimum guidance by the demonstrator. Each ball will automatically fall into a "hole" made by three in the first layer. Enforcing only the single criterion that each sphere must be as close as possible to others in its own layer, one can now readily show that each sphere in the original layer is in contact with three in this second layer and vice versa. (See Figure lb.) Since there could, of course, be an identical layer in the plane below the first layer the maximum coordination of identically sized spheres must he twelve. THE "THIRD" LAYER-FACE CENTERED CUBIC OR HEXAGONAL CLOSE PACKING
A third layer may now be rolled on just as the second was. But a difficultyarises, or a t least an ambiguity. Two non-identical arrangements are formed depending on whether the third layer spheres are directly over the spheres in the first layer, or whether they are directly over the triangular holes formed by the first layer spheres. Each of these arrangements maintains the coordination number of twelve and an identical arrangement of spheres in any given layer, but the "long range order" is different. Figures id and l j demonstrate the difference when a total of five layers has been positioned. Most observers will readily agree that the external appearance of the model illustrated in Figure id is more symmetrical than that shown in Figure If. All three visible faces of the former are more regular than those of the latter. It is possible to describe this symmetry in many ways but the most common method is to refer to the structure shown in Figure Id as face centered cubic. Black spots have been placed on a set of spheres which may be considered to occupy the corners of one of these face centered cubes. The atoms in the centers of several of the faces may also be identified. Examination will show that this model was built up by having the third layer of spheres over the holes in the first layer. See also Figure lc where a third face of the cube is exposed. Seven of its eight corners are visible here. The model in Figure l j is usually called hexagonal close packed. I n this model the spheres in the third layer are immediately over those in the first layer as the model was constructed. The same spheres are spotted here as in Figure Id. Examination of the JOURNAL OF CHEMICAL EDUCATION
Filjiiic I " . r a v e C e n t r i d C u L l c Model ~ 8 t h C o r n e i Spheres of S i n g l e C u h r S ~ o t t e dlor Three faces a r e V i s i b l e
Clarity.
Fiqrirc.l h .
T W OS u p r r > m p o s c d Cloaa Packed Lilyrra o f Spheres
figures, or preferahly of an actual model, will show t,hat t,he face centered cubic model is readily converted into the hexagonal close packed one and vice versa, by sliding or rolliug every third layer of the former approximately one spherical radius over the layer below it. Compare Figures l c aud l e , and Figures id and 1.f. Planes of spheres within the model which can readily slide in this fashion, with little effort and no distortion of the rest of the model, are known as slip planes. They are often iuvoked in interpreting gross properties as we shall soon find. ACTUAL PACKING OF ATOMS
It is interesting t o note that most of the chemical elements crystallize in one or the other of the two closest packed structures outlined above, excluding from consideration those elements which form polyatomic molecules as do hydrogen, phosphorus, selenium, etc. Apparently one'of the main driving forces in the formatiori of these crystalline elements is toward the maximum coordination number of twelve. A theoretical interpretation of this is not, of course, far behind the experimental observations. High coordination numbers mean more bonds to neighboring atoms and more bonds mean a more stable crystal. This simple criterion will be insufficient if an atom has a marked tendency t o form bonds strongly oriented in space, as does carbon, or if an atom has a bonding strength which varies rapidly wit,h distance. I n the VOLUME 34. NO. S. MAY, 1957
Herasonal Clere P a c k e d Scquoncc
F L ~ Y I ICI S a m e N , > r n b c i of Spheres as I :, but H c r c Thry aro in H r ~ a q o n a l Close Packed Sequence. Note Shift of Every Third Layer from Bottom Compared to Figvro I l l Figvrc
latter case the atom may achieve a more stable arrangpment mith fewer than twelve bonds since the smaller number of bonded neighbors may shorten the internuclear distanre and give, for example, eight shorter bonds each sufficiently strouger than the twelve longer ones so that the over-all bonding is stronger mith eight, neighbors under certain conditions. But me are staying with very simple models so shall discuss only those systems in which closest packing. i.e., coordination numbers of 12, are found. Now face ceutered cuhic aud hexagoual close packing differ only in the arrangement of each third layer wit,h respect t o the first one. I n hexagonal close packing each atom in the third layer is directly over a corresponding atom in the first layer, while in face centered cubic the third layer atoms are over the triangular holes in the first layer. This raises the iuteresting question as to how the atoms "know" what is going on in a layer mith which they are not in contact. No real answer has heeu given t o this problem, but the situation becomes even more fascinating when real crystals are studied. As far as the hard sphere model goes, the layers could repeat indiscriminately so that some layers might have a face centered sequence to earh other while different layers in the same model have an hexagonal close packed sequence. I n nature this seldom happens. The great majority of real crystals in which a coordination number of 12 is achieved are 211
e ~ t h e rfare centered rubic or hexagonal close packed throughout. It is, perhaps, not too difficult to rationalize a crystal force effective over two or three atomic layers as in these two simple stmctures. Unfortunately for one who has succeeded in this rationalization, however, crystals are known in which the repeat distance is much greater. These crystals have only close packed layers hut the repeat distance may be as high as 12 to 15 layers. The nature of the crystal forces which can propagate though such long distances is not interpreted by any current theory. But again our model is becoming too complicated and we must return to simple experiments and their interpretation in terms of our simple model. However, the existence of these long range forces should always he kept in mind for they may be of great importance in crystals even when they do not evidence themselves by long range repeat distances. SOME SIMPLE EXPERIMENTS
The inert gases and most of the common metallir elements crystallize in the face centered cubic system. Any model, such as the ones we have constructed, ought to allow an interpretation of some of the gross properties of these substances in terms of the behavior of the model under analogous situations. Consider for example some experiments with an ordinary copper rod some sixteenth inch in diameter,
Fiourr 2,. Fa'" C e n t e r e d Cvbls Clon. P a c k r d M a d e l with Spheres at Cornors of a Sinrile C u b e Snotted for C l a r i t y
The Model in Figure Ze After ~ trrame ~ has i been ~ rorsed i ~into ~ a Square from its Original 60" Paralldopiped Shape Figure 2b.
the ~
212
though any convenient rlementary metal rod will do. The rod may be deformed a considerable distanre and yet spring back to its original shape. Application of more than a certain force in a given period of time will, however, lead to permanent deformation. h kink or bend will remain when the force is relieved. The same kinking or bending may be attained if less force is applied over a longer period of time. Repeated bending will crack the rod. N o ~ vtry to straighten the rod by again holding the ends and bending it in the reverse direction. Rathrr than straightening, two additional hends mill form, one on each side of the original one. If a well directed attempt is made to straighten t,he original bend it is found that considerably more force must he exerted t,o straighten the bent portion than to make the original bend. Our model must then account for both the rigidity and elasticity of an elementsly metal under limited stress, its deformation under greater stress, the increasing yield strength with increased bending (or vorking) and breakage under repeated bending. (If i t will also interpret other propert,ies, so much the better.) BENDING AND THE MODEL
The sequence of pictures in Figure 2 illustrates the effect of "bending" on the face centered cubic model
F,qure Zc. Model i n rioure211 w i t h Eight spheres~~~~~~d to s h o w the racecentered Cubic Close Sphe..s
rig"re2d.
Packed
Arrengamont
of
the
~ h M = O ~ i n~ figure I 2b ~ f tF~~~~ ~ hai been ~ ~~~~~d i ~~ ~ a into the Original 60" Parallelopiged Shape. Note High Dfsordo? Near Middle of Model
the ~
~
Figure 2e T h e M o d e l in F i q u r e 2 : A f t e r t h e Disordered Spheres have been Removed. the Rest being U n c h a n q r d . Note ths Roguler Face Centered Arrengement of the Two Ordared Ragionn
same
m rigure 2f. as rig"re 2 , ~ . nerefor k C o m p n ~ i s o nwith Figureh 211 and 2r
EW
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JOURNAL OF CHEMICAL EDUCATION
shown in Figure 2a. The corner spheres in a typical face centered cube in Figure 2a are shown with spots. The position of these and the other spheres may be followed in each of the other pictures. When a metal is subjected to a stress the stress will be relieved by a minimum movement of the constituent atoms and breaking of as few bonds as possible. I t is common to interpret such movement in terms of the models we have constructed by invoking movement along the slip planes, i.e., slippage of a plane of atoms with respect to an adjacent plane. Such motion is certainly possible, particularly when the stress is applied parallel to the slip planes. The ease of this motion is easily shown by moving the upper layer in a system such as shown in Figure l b with respect to the lower layer. Models with more layers tend to be unstable with respect to this type of motion. Motion along slip planes typically converts a face centered structure into one having many hexagonal close packed sequences (or vice versa) and eventually gives a highly disordered sequence of layers though each layer may, in principal, remain close packed (like the layer in Figure la) with respect to other atoms in its own layer. There is, furthermore, a high potential energy barrier to motion along a slip plane since a t least one bond per atom must be pretty well broken before its replacement can begin to form. A second type of motion is even more easily demonstrated with the model, one which approximates closely to a typical bending stress in an actual crystal. This motion is demonstrable by starting with the model in the position shown in Figure 2a and pushing gently on the long diagonal of the frame. Relatively slight effort will convert the model smoothly into the structnre shown in Figure 2b. Motion along slip planes, as we have pointed out, leads t o a high degree of disorder more or less automatically, but the motion described here has obviously produced a system still having a high degree of order. The original face centered cubes have lost their cubic symmetry as shown by examination of the new positions of the spotted spheres. Removal of five of the upper spheres, as shown in Figure 2c, however, reveals the interesting fact that the system is again face centered cubic. Note that the upper layer in Figure 2c is typically face centered and is, indeed, one face of a completely face centered cube whose face diagonals are perpendicular to the edges of the frame. (Actually the structure achieved is not quite face centered cubic since the frame dimensions are now slightly too long as shown by the fact that the spheres in the lowest layer are not quite in contact.) Let us now investigate what happens if we try to return the frame to its original position. Figure 2d indicates a typical result when a model of the form of Figure 26 is deformed by pushing slowly along one of the frame diagonals while the model is on a smooth surface, like a wooden table. (Excessive friction in the bottom layer causes trouble as anyone using the model will find out.) Parts of Figure 2d are identical to the original arrangement in Figure 2a (or Z f ) , but it is ciear that this earlier figure has not been reformed. There has developed, rather, a volume of considerable disorder near the middle of the model. VOLUME 34, NO. 5, MAY, 1957
(Sometimes this disorder appears only after several deformations.) If the spheres which are clearly not part of any ordered array are removed, without disturbing any of the ordered ones, the arrangement shown in Figure 2e is found. I n this particular case two different face centered fragments have been formed. The left-hand portion is identical with the original arrangement shown in Figure 2a or 2f while the smaller righthand fragment is a mirror image of the left-hand portion. The two portions share a common ordered row of atoms in the bottom layer. If the model of Figure 2d is now reconstructed and deformed once more by again pressing along the long diagonal of the frame, it will be almost impossible to accomplish any motion without rather complete disruption of the model. The disordered region does not distort easily. (A further disclaimer is needed a t this point to admit that some of the behavior of this model is due to the fortuitous way in which gravity cooperates with the wooden frame and the particular motion chosen to produce the smooth motion and order when moved in one direction and the disorder when a reciprocal motion is attempted. This does not seriously affect the argument from the model. The smooth transition from one face centered to another face centered system with differently oriented axes is quite possible and does not require passage over any high potential energy barriers. New bonds form as old ones are stretched.) We can now proceed to interpret the actual behavior of the metal under stress in terms of that of the model under similar stress. A rod of elementary metal is composed of single crystalline regions (usually small) in which the atomic order is well defined as in Figures 2a or 2f. Application of a slight stress gives a bend from which t,he metal will recover (spring back) just as application of a small stress to the model will let the spheres return automatically to their original positions when the stress is relieved. Try this with the top spheres in any model for instance. If sufficient stress is applied, however, the spheres move over a potential barrier toward another position and do not return when the stress is relieved. This is again readily demonstrable with the top spheres in any model. When a crystal, or the model, is bent in a very selected way as in going from Figure 2a to Figure 26 there may be little deformation so that little effort is needed. This type of bending will always occur first, when possible, since it involves overcoming the smallest available potential barriers. Any given sample soon runs out of these easy methods of deformation, however, and regions of gross disorder form. Both slip planes and easy deformations now are absent so that a marked increase in force is necessary to cause deformation. However, these disordered regions contain fewer bonds per unit of volume than ordered regions and are, indeed, incipient cracks in the substance. Repeated stress, enlarges these cracks by breaking more and more of the bonds in them so that eventually the sample separates into two distinct portions. It breaks. EFFECT OF TEMPERATURE
If our original, typical elementary metal rod (assumed to have been formed by cold working) is heated
before an attempt is made to bend it, one observes the bent form with much less stress than in the unheated rod. Furthermore, if a bent section is heated it becomes relatively easy to straighten that bend. Again the model gives a simple interpretation. If one slightly deforms the upper layer in a model such as shown in Figure l e and then shakes the frame gently, the spheres will return to the positions of maximum stability shown in the figure. I n the same way an increase in temperature raises the average kinetic energy of each atom and allows a greater freedom of motion toward the positions of minimum potential energy. This minimum is the close packed structure of the single crystal. As the disordered regions disappear with a corresponding growth of ordered regions the sample becomes easier and easier to deform by bending as already explained. EITECT OF ALLOYING
At least three types of alloys are known which show properties similar to the ordered single crystals we have been discussing: (1) interstitial alloys, (2) substitutional alloys, and (3) compounds. We shall not discuss class 3 though some of the properties of alloys of this class may be inferred from our models for the other two types. Hard steels are often cited as examples of interstitial alloys. They may be formed by dissolving elements such 'as carbon, nitrogen, or boron in iron. Such steels typically have a very high elastic limit, are hard, and brittle. They are difficult to bend, will spring back to the original &ape after a much greater deformation than pure iron could survive, and will crack rather than flow if subjected to an excessive stress. If we consider the processes of deformation found so far in the model we can, perhaps, see how this deformation can be effectively forbidden. None of the models shown in the figures are "solid." There are always interstices between spheres of like size no matter how closely they may be packed. As a matter of fact, easy deformation is only possible because of these interstices. Atoms do not readily interpenetrate and can move readily from one crystal site to another only if empty space is available into which they can move, creating more empty space behind them and making the deformation a cooperative phenomenon through large volumes of the crystal. If these interstitial holes were not present, motion would be most difficult. Simple trial will show that close packed models contain a t least three types of "holes." The holes on the closest face of the model shown in Figure 2a are known as tetrahedral. An atom fitting into one of these holes would have a possible coordination of sixthe four atoms in the face, the one visible behind the face, and the one that would be present in front of the face if the crystal were continued. These are the largest l'holes" in a close packed strncture and equal in number the number of close packed atoms in the structure. The tetrahedral holes are also readily visible in Figure 2c where the second layer of spheres is exposed. A second type of hole is the tetrahedral one. These
constitute the space between any three contiguous spheres arranged such as in Figure l a and a fourth sphere resting in the well formed by the first three. These holes are considerably smaller than the tetrahedral holes in the same model. The third type of hole is formed between any three contiguous spheres arranged as in Figure l a and in the same plane as the three spheres. These are known as triangular holes. It is quite possible to fill any of these types of holes without interfering with the holes of either of the other types. Which hole a particular size of second sphere will enter will depend on its relative size in comparison with the larger spheres. The radius ratio which is exactly right for the smaller sphere to fit snugly into the interstitial holes is readily calculable. If, now, smaller spheres do enter the octahedral or tetrahedral holes in a close packed strncture, some of the holes disappear. It becomes correspondingly more di5cult to deform the crystal. The new set of spheres blocks the motion necessary for a smooth deformation. I t also blocks the attainment of new stable positions after deformation so that gross deformation is apt to lead to cracking and breaking of the substance. It will be noted that the atoms which form interstitial alloys are of relatively small effective atomic radius and can thus occupy these interstitial holes with little distortion of the original structure. Substitutional alloys are those in which each second type of atom occupies a crystal position formerly occupied by one of the original atoms. The crystal structure is not appreciably changed, but two different kinds of atoms are distributed throughout the close packed strncture in terms of the model we are using. Substitutional alloys are typically somewhat harder to deform than the pure elements but are not as hard as the interstitial type. Some gold alloys are of this type, for instance. This lack of a great change in hardness fits well with the ideas developed so far since the crystal strncture is relatively unchanged. The increase which is noted is interpretable in terms of a slight decrease in the uniformity of the crystalline layers, since even incipient disorder mill lead to some hardness. Substitutional alloys Are usually found when the effective radii of the alloying elements are close to the same. CLOSE PACKING IN IONIC COMPOUNDS
The model thus far has been used to interpret the behavior of crystals made up of electrically neutral atoms whose packing is determined in a large measure by a tendency to conserve the available space through close packing. I n a very large number of ionic substances similar criteria hold with the additional complication that the atoms are now electrically charged. The resulting coulombic forces now have an effect on the crystalline properties. In a very large number of instances, however, the negative ions (which are usually larger than the positive ones) are essentially close packed with the positive ions occupying the octahedral and/or tetrahedral holes. Sodium chloride is a typical example. The sodium ions are in octahedral holes in cubic face centered (close packed) lattice of chloride ions. JOURNAL OF CHEMICAL EDUCATION
There are, of course, not only attractive forces between ions of opposite charges, but repulsive forces between ions of similar charge. Thus, while the crystals tend to be hard and brittle because of the interstitial nature of the packing, the hardness is decreased and the brittleness increased by the strong repulsions which set in if the atoms are moved with respect to one another. Under these conditions there will be a state where the like atoms come closer together than in the equilibrium crystal, and the solid will crack as a result.
are, as with all models, artifacts of its construction rather than reflections of the situation in crystals. We have not discussed the important role of faults and dislocations in determining deformation properties of crystals. But a sound foundation has been laid, little mill need to be unlearned, and the comprehension of more contemporary ideas vill be easier if this simple model and its behavior have been first understood by t,he student. Typical of the elementary discussions of dose packing and radius ratio effects are those available in the texts by Pauling3 and Steiner and CampbelL4
CONCLUSION
PAULIXG, L., "College Chemistry," 2nd ed., W. H. Freeman and Co., Snn Francisco, 1955. 4 S ~ L., ~A N D ~J. A.~ CAMPBELL, ~ ~ "General , Chemistry," Thc Macmillan Co., New York, 1955.
This model, then, is based on ideas long used to interpret crystalline behavior. Some of its successes
VOLUME 34, NO. 5, MAY, 1957
