A MODEL ILLUSTRATING THE EFFECT OF THERMAL AGITATION' ROSCOE H. WOOLLEY and DAN McLACHLAN, IR. University of Utah, Salt Lake City, Utah
A
MOST instructive method of conveying to students the means by which metals form into crystals and the manner in which they undergo deformation by slip, along definite slip-planes, was exhibited by Sir Lawrence Bragg (I) in a motion picture of a tvo-dimensional model described in his publication, "A Dynamical Model of a Crystal Structure." The model used for his two-dimensional metal consisted of a single layer of soap bubbles floating on the surface of water. These bubbles, which represented the atoms, showed such a reluctance to burst that the surface of the water could be agitated to produce a state of chaos among them. But, upon standing undisturbed, a rearrangement gradually took place resulting in a perfectly regimented distribution of the bubbles analogous to a "crystalline arrangement." Special methods were developed for assuring that all the bubbles were of the same size, and an oversized bubble intentionally placed among the others served to illustrate how space lattice distortions can occur. With the work of Bragg as a stimulus, the present authors conceived the idea that still further information might be conveyed to those studying the structure of matter if the effects of thermal agitation could also be simulated in a two-dimensional model of atomic structure. It is the purpose of this paper to describe the construction and operation of such a model and also the methods of securing a motion picture record of the model in action. It is desired also, to describe some of the physical phenomena which it demonstrates.
l a shows the relative position of the parts and Figure l b is a cross-section view of the assembly. In both sketches, P and P' are 9- X 12-in. sheets of doublestrength window glass, G is a cardboard gasket of such thickness that it separat,esthe glass plates by a distance slightly greater than the diameter of the spheres R. Thus the spheres are able to roll about freely within the confines of the cardboard gasket, but are prevented from rolling over one another even when the glass plates are THE APPARATUS tilted out of a horizontal plane. The parts of the assemSome preliminary experiments were conducted in an bly are held together by a strip of adhesive tape T, bound effortto determine the best means of illustrating mhat is across the edges. I is a small section of the gasket cut believed to be the actions of atoms and molecules under out and left as a loose insert to provide an opening various physical conditions. Since the simulated molecu- through which spheres may be added or removed. The spheres used to represent the atoms and molelar actions were to take place in two dimensions only, it was found that the desired results could be obtained cules in this apparatus are of three kinds: (1) For by letting small metal spheres or shot represent the those atoms which are to exhibit no attraction for atoms. Also the effect of thermal agitation could be each other, number nine lead bird-shot, about 0.075 imitated by mechanical vibrations produced by a small inch in diameter, is used (2) In those cases where a electric motor. To confine the spheres to two dimen- mutual attraction between atoms of the same kind is to sions, a single layer of them mere placed between two be represented, magnetized steel shot is used. To make sheets of glass as shown in Figure l a and lb. Figure the steel shot easily distinguishable from the darkcolored lead shot, the steel shot is copper plated by immersing it in a 10 per cent copper sulfate solution 1 A motion picture of this ariiole may he obtained free upon request by writing ta Professor Roscoe H. Woolley, Engineering for about ten seconds. Also, in some cases the copperplated steel shot is used in the unmagnetized state when Experiment &ation, University of Utah, Salt Lake City, Utah.
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it is desirable to represent two kinds of ponattracting atoms intermixed on the glass plates. The steel shot is of the kind commonly used for cleaning castings by the shot-blasting method and is obtainable from any modern foundry. I t is necessary to screen this steel shot several times to obtain a quantity for use that is uniform in size and of about the same diameter as the lead shot. (3) For those instances where molecules are to be represented, particularly long-chain molecules, various length pieces of brass bead chain may he used. This chain may be of the kind used on pullswitches for electric light fixtures. The glass-plate assembly is spring-mounted in a wooden frame as shown in Figure 2. Here, F is the
wood frame, 11 X 14 X 3 in. inside dimensions, made of half-inch thick material. &Sf are the suspending springs which run from gussets fastened to the upper and lower corners of the frame to metal clips C slipped over each corner of the glass plate assembly P. These springs all incline slightly inward toward the middle of the glass plates to keep the corner clips from working off when the plates are vibrated. Several holes through the side members of the frame, as shown in the sketch, are to provide access to wires W, used to manipulate the groups of metal balls between the glass plates as will be described later. The large hole in the left end of the frame provides an opening through which projects the aluminum link by which the vibrations from the electric vibrator V are transmitted to the glass plates. The vibrator is a small electric massage vibrator made by the A. C. Gilbert Company of New Haven, Connecticut. The original rnbber massage cup is replaced by a short, pointed brass rod that fits into a cone-shaped hole in the aluminum driving link. A rheostat placed in the power line makes it possible to adjust the vibration rate to give the best effect. The action of this particular type of vibrator is such as to give a lateral swinging motion to the glass plates rather than a vertical bouncing motion. This lateral vibration causes the freely rolling shot to rebound violently from the cardboard gasket and proceed in nearly straight paths until they cross the glass plate or collide with each other. The camera suggested for the photographic work is a 16-mm. Eastman Cinekodak Special equipped with an f 1.9 lens and loaded with Eastman Super XX film. This camera should be mounted above the work table on a bracket supported by a vertical column so that the
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optical axis of the lens is perpendicular to the plane of the metal spheres being photographed. It will be found that the camera field of view just covers the clear area inside the binding tape around the glass plates when the distance from lens to glass plates is two and one-half feet. The camera should be operated at an exposure rate of thirty-two frames per second, to give a slow-motion effectwhen the finished film is projected a t the normal rate of sixteen frames per second. This smooths out the action and makes it easier to follow the motions of the spheres on the screen. Light may be provided by two General Electric 300-watt Reflector Photoflood lamps. Where only one kind of atom is to be represented it is found that the action can be most clearly portrayed by using bottom illumination to produce a silhouette effect in the picture. Figure 3a shows the relative positions of the camera C, the plate suspension frame A, the tracing paper lightdiffusing screen DS, and the lamps L. Even illumination is obtained with the diffusing screen about 1 foot below the plate being photographed, the lamps
Figure 3 la, above;
h below)
being about 18 inches below the screen and offset about a foot each side of the center of the screen. With such a setup the exposures may be made with a lens diaphragm setting of f/8. Illumination from above is recommended in those cases where shot of contrasting colors is being photo-
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graphed or where auxilliary apparatus is to be placed helow the glass plates. In such cases a piece of black paper is pasted on to the bott,om plate so that the shot appears against a dark background. Figure 3b shows the general arrangement of the parts. As before, the camera C is 21/2 feet above the plates A . The flood lights L are suspended about 18 inches above the plates and offset from the center just enough to prevent any direct reflections of the lightsfromshowing in the pictures. Exposures with top illumination may be made with the lens stopped down to f/l0. The use of this apparatus should be made clear by the following discussion of demonstrations which are made possible by its use and recorded on motion picture film. A R A m I E D GAS
Figure 4
(1e f t t o r i d ~ t : a and 6 , above;
When several dozen of the lead shot are placed between the horizontal glass plates and the assembly set in agitation, the small spheres are reflected from the inner surface of the gasket as though it were a hot vall and, in their recoil, proceed across the field of view. Frequent collisions with one another occur and their actions are such as one might imagine the atoms of a "perfect monatomic gas" to perform. Although the demonstration is relatively realistic, especially when viewed on the screen, it has two objectionable features: in their passage over the plate, the spheres are subject to some friction and therefore a definite decrease in their velocities is noticeable on long free paths. Also, those spheres which are by chance moving a t the lowest velocities exhibit a visible response to the vibration of the assembly. Therefore, those that are nearly a t rest appear to be "shimmering." However, the latter objection is not as serious as the former. Bottom lighting gives the best results for this demonstration. DIFNSION OF GASES
For this demonstration, as well as others to follow, it is necessaly to devise some means of putting partitions in the two-dimensional gas space. Figure 4a shows how this is accomplished by threading straight pieces of stiff wire or welding rod W through slots cut in the spacing gasket between the glass plates. The wire is of such a diameter that insufficient space is left between it and the glass to permit the shot to pass by. When the shot is placed all on one side of the wire par-
c and
d, b r l o ~ ~ l
tition, as shown in Figure 4a, agitation causes the spheres to bounce against the wire as well as the gasket and the appearance of a barrier is created. By withdrawing the wires a short distance to produce a gap, as shown in Figure 4b, further agitation forms the illusion of gas molecules diffusing through a small orifice or pinhole. To demonstrate the diffusion of two unlike gas molecules into each other, the glass plate with its &e partition is set up as shown in Figure 4c. Here lead shot is placed on one side of the closed partition and nnmagnetized copper-plated steel shot on the other side. After the vibrator has been started, the wires are partly withdrawn and the two kinds of shot allowed to intermingle as shown in Figure 4d. This demonstration is very effective in emphasizing to the student the role of mean free path in the interdiffusion process. When the experiment is repeated with increasing concentrations of the two kinds of shot on each side of the barrier, it becomes apparent that even though the "apparent" distance from one side of the barrier to the other is relatively short, the shot takes a very deviated route because the mean free paths become increasingly short and the direction of travel between collisions is more random as the concentration is increased. In fact, a t very high concentrations, one is impressed with the slowness of mixing even when the barriers are completely removed. Even more startling is the effect of a little stirring in bringing into intimacy the two kinds of shot. Stirring of a kind very commonly used in
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clusters do not retain their identity, but are in a constant process of dispersing, reforming, and exchanging members of the sets. A similar study of mixtures of two kinds of shot demonstrates the expected limitations of so-called "uniform" mixing and the errors in sampling. Here, as in the preceding demonstration, top lighting should he used for the photography. THE CRYSTALLIZATION OF A METAL
The contrast between the effects of quenching a molten metal and cooling it slowly may be demonstrated by suddenly shutting off the vibrator and tilting the glass plate assembly off the horizontal while it is about
Flgure
S
laboratory practice is closely imitated by giving the assembly a gently rolling motion off of the true horizontal. Since the effectiveness of this demonstration depends on the contrast between the two kinds of shot, top illumination and a black background are used when photographing the action.
. A further principle may be brought out by examining in more detail the effects exhibited by a high concentration of two kinds of shot in the assembly. It will be observed in Figure 5 that, although there is no attraction bet~veenthe shot and in spite of the constant agitation by the vibrator, there is still a tendency for clusters or aggregates of the spheres to be spontane&sly formed. This adds credence to the explanation that an unevenness of distribution is the mathematically most probable configuration and that thermal fluctuations (9) such as cause the bluenessof the sky (3) and turbidity in polymer solutions (4) are all natural effects. The motion picture reveals on the screen the fact that the COMPRESSED GASES AND FLUCTUATIONS
Figure 6 (o,
left; b, middle: c right)
Figure 1
a third full of shot. If the assembly is held undisturbed in the tilted position it may be seen, as in Figure 6a, that the shot is in a generally disordered array and only in very small domains are the spheres in any regular order. This corresponds to quenching and the small ordered domains are analogous to the crystallites produced on rapid cooling. If, however, the experiment is repeated and the agitation decreased gradually after tilting the assembly, much
stage shown in Figure 6a. In thisevent, thecrystallites already formed increase in
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size as shown in the two stagesillustrated by Figures 6b and 6c. This photographed for the motion picture record as a continuously progressive process and emphasizes the very slight movement of each individual sphere or "atom" necessary for an impressive rearrangement of the mass collectivity. The use of but one kind of shot for this demonstration permits the silhouette effect to be used and hence is photographed by bottom illumination. THE THREE PHASES-SOLID, LIQUID, AND GAS
In discussing the three phases of matter, the authors have found some convenience in opening the subject by stating that there are three kinds of space to be considered: ( a ) space filled by ordered material, (b) space filled with disordered material and, ( c ) empty space. Now, while these three kinds of space may be mixed in any proportions, the most important criterion for the study of the physical properties of such a mixture is the determination of which of these kinds of space in the mixture occupies the continuum. Figure 7 illustrates the meaning of a continuum as contrasted to the islands shown in black. With this thought in mind, one may sav that a solid is a mixture wherein the ordered mace is the continuum while a proportionately small fraction of the material occupies islands of disorder created by thermal fluctuations (5) and (6) and a very small part of the space is "occupied" by voids. The liquid state is characterized by the fact that the disordered state is the continuum and islands or nuclei of ordered space float about in it, but are always in the process of dispersing, reform in^, and exchanging atoms with the melt (continuum). The gaseous phase is Figure 9 characterized by the fact that empty space is the continuum. lo, above: b, below) These ideas may he demonstrated with the twodimensional apparatus by starting with the assembly in the stage shown in Figure 6c and gra21ually increasing the violence of the agitation (with the frame slightly tilted) until the aggregate of ordered spheres becomes chaotic and finally disperses against the force of gravity into a "gas." Figure 8 shows an intermediate stage in which the solid, liquid, and gaseous phases coexist. The above demonstration was recorded with bottom lighting. ~
~
.
THE CONTRACTION RESULTING FROM THE CLOSE PACKING OF ATOMS
The demonstration of this principle requires a glass plate assembly with mire partitions bent in such a shape as to form a sort of "thermometer bulb and stem." Figure 9a shows the disordered array of shot confined within the "bulb" and extending nearly to the top of the "stem." This condition is attained by holding the assembly level and applying a rather vigorous agitation. By tilting the assembly very slightly away from the "stem" and applying a very gentle agitation, the shot falls into regular close packed order over most of the area of the "bulb" as may he seen in Figure 96. The decrease in volume (area in this case since a twodimensional model is being used) is made evident by the
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"cold-shear" just previously illustrated. The fact is brought out that thermal agitation causes the slippage to occur more easily and the disorder and formation of small crystallites from the larger crystals is a more local process. (See Figure lob.) Also, the imperfections, when formed, are thermally mended rather quickly. It will also be observed by one actually operating the shear wires in the assembly that much less force is required to perform the shearing operation with the vibrator running than with it shut off as it is in the "cold shear" demonstration. Here, again, bottom illumination is used to obtain the photographic record. SUBSTANCES THAT CRYSTALLIZE WITH DIFFICULTY
The true metals crystallize rather easily because the mobile units which must undergo rearrangement in a transformation from the chaotic to the crystalline state are single atoms of a spherical nature. These atoms may migrate by rotation over one another and form attachments from any angle of approach. There are, however, a number of substances such as those that form glasses (10) (the metallic oxides for example) which crystallize very slowly. This is largely because there may be many choices of attachment between ueighboring units or molecules that are chemically satisfactory from a local point of view but which would form arrangements not compatible with their neighbors in the formation of long range order. To demonstrate the effect of nonspherical units on the ease of crystallization, enough shot is glued together in sets of three to partly fill the glass plate assembly. (See Figure 11.) When the assembly is tilted, either Figure 10
(a.above: b. below)
descent of the shot in the "stem." Bottom illumination for the photography is used in this case since the silhouette effect best brings out the contrast between the ordered and disordered states of the shot. HOT AND COLD SHEARING OF METALS
The deformation of metals, such as that accomplished by mechanical shears, is illustrated by preparing an assembly containing movable barriers formed by bent wires as shown in Fignre 10a. The wires are arranged so that they can be moved in opposite directions and thus the shot trapped between them are forccd to slide over one another until the point of complete separation is reached. With little or no agitation, the process corresponds to cold shear, and during the process the shot in the immediate vicinity of the barrier edges forces the neighboring ordered domains into disorder as shown in Figure 10a. When the continuous action is followed on the screen, it is seen that the changes in shape of the mass are accomplished by slip (7) along the slip planes, dislocations (S),and by the formation of holes (Q), some of which discontinuously migrate. When the demonstration is repeated with more vigorous agitation the effect is analogous to the hot-shearing of metals and, when viewed on the screen, is in sharp contrast with the
Figure 11
with or without agitation, the mass of shot "triplets" migrates to one side against the gasket. Figure 12 indicates the disordered arrangement into which the "triplets" invariably fell. And, even though the individual sets of three are each a small part of a close packed order, the remaining amount of rearrangement necessary to form continuous order is so difficult to attain that even prolonged and careful agitation fails to produce more than a few small scattered areas of
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Figure 12
regular or "crystalline" arrangement. Thus one is impressed with the details involved in the natural growth of crystals from complex units or molecules. The shot used for this demonstration shows up best when silhouetted against a light background; hence, bottom illumination is used for making the photographic record.
that crystals grow from solutions largely through the migration of single atoms or molecules to the main body of the growing crystal, rather than by the migration of pre-regimented clusters through the solution and a suhsequent welding together of the parts. By lowering the magnets from beneath the assembly and applying a vigorous agitation, the "crystals" may be dispersed or "dissolved." One watching the process gets the impression that in certain cases, where there are natural planes of weakness inherent in the crystal structure, it is possible that some crystals do not dissolve a t the surface, one atom a t a time, but that dispersion may occur by the growth of nuclei of chaos within the crystal and it comes apart by pieces. Also, it is interesting to observe that after a "crystal" has been "melted," by removing the magnets, a surprising amount of agitation is required before the "atoms" have disbursed from the sites of the crystals and diffused among the atoms of the solvent. In other words, points of high concentration of solute persist in the neighborhood of the old crystal sites and two possible results might he predicted: (a)
CRYSTALLIZATION FROM SOLUTION3
The demonstrations so far described have neglected any attraction between the atoms other than that simulated by the force of gravity during the tilting of the assembly. In order to demonstrate the crystallization of metals from solution, i t is necessary to have a t least two kinds of atoms, one of which will separate from the other and crystallize while thermal agitation keeps the second kind in a chaotic or liquid state. This is accomplished for these demonstrations by inducing a preferential attraction between the spheres of one kind, strong enough to withstand a certain amount of agitation. Letting lead shot represent atoms of one kind and copper-plated steel shot the atoms of the second kind, the desired effect is creat,edby placing several small "Alnico" magnets beneath the assembly, under the concealment of a sheet of black paper. As may he seen in Figure 13a, the steel shot collects over the magnets and gives the effect of a crystal growing from a nucleus or "seed." It is to be observed that the steel shot "atoms" are collected on the "crystals" a t the points of growth accessible by the shortest path from the "solution," and that branched or dendritic growth results. Upon further growth the "precipitated crystals" lose their dendritic character, and fill in to take on a massive form as shown in Figure 13b. It is interesting to observe on the screen the high degree of mobility of single atoms as compared to aggregates of two or more. Although this is to be expected, from the relationship between mass and velocity established by the kinetic theory, yet, the demonstration is useful in suggesting that it is likely
Figure 13
(a. above; b, below)
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'.
-9, rigurs14 (a, left;
b, middle; c, right)
in certain polymers as presented by Herrmann and Gerngross (I,?). By cementing side chains, of three or four links in length, at intervals of eight or ten diameters along the length of each chain, a means is had for demonstrating the hindrance to crystallization of high polymers caused by the presence of inert side chains. Figure 14c shows the relative scarcity of large mycelles with the side chains present.
ACKNOWLEDGMENTS
the local points of high concentration should have a The authors are indebted to the Utah Engineering higher crystallization temperature than a solution Experiment Station, under the direction of Dr. J. having a concentration of the average bulk material Hugh Hamilton, and to the Department of Metallurgy and, therefore, cyclic heating and cooling of a solution of the University of Utah, headed by Dr. John R. might give anticipated crystallization, i. e., abnormally Lewis, for support and materials for this investigation. high precipitation temperatures (11); (b) the crystals Also, we are indebted to the Office of Naval Research formed on each cooling cycle may show a memory for partial use of a grant made for "The Study of Creep effect by forming a t the sites of the crystals in the Flow and Failure of Metals," directed by Deans Henry previous cooling. Obviously, top lighting is used for Eyring and Carl J. Christensen. filming this demonstration, both to bring out the contrast between the two kinds of shot and because an LITERATURECITED opaque background is required to conceal the magnets. Bmcu, W. H., IWD J. F. xYE, PTOc.R ~sOC. ~ ( . L ~ ~A ~ ) , LONG-CHAIN MOLECULES IN NONMETALLIC SYSTEMS
With the object of showing that similar techniques may be used for systems that are not metallic, a description is given here of a representation of long-chain molecules by the use of brass beaded pull-switch chain. As mentioned previously, in the description of apparatus, this chain is cut into pieces of varying length and placed in a glass plate assembly with a spacing gasket of appropriate thickness. Figure 14a shows the random structure brought about by agitation. When the agitation is stopped and the frame tilted so that the chains slide to one side of the frame, crystalline domains are produced apparently by chance, as shown in Figure 14b. These have a resemblauce to figures applicable to the theories of crystalline domains or mycelle formations
190, 474-81 (1947). (2) EPSTEIN,PAULS., "Textbook of Thermodynamics," John Wiley and Sons, 1937, p. 389. (3) RAYLEIQH, LORD,Phil. Mag., 41, 107 (1871); 47, 375 (1899). (4) DEBYE,P., "Light scattering in solutions," J . Applied Phys., 15,338 (1944). (5) E ~ I N GH., , J. W. FREDR~CKSON, D. G. MCLACAUN, Proc. Nat. A d . Sci., 34,295 (1948). (6) FRENKEL, J., "Kinetic Theory of Liquids," Oxford Press, 1946, p. 112. C. S., "Structure of N~cleus," MeGrew-Hill, (7) BARRETT, 1942, p. 332. (8) TAYLOR, G. I., P~OC. ROY. Soe. (London), A 145,362 (1934). (9) GLASSMNE, S., K. J. LAIDLER, AND H. EYRINQ, "Theory of Rate Processes," McGrm-Hill, 1941, p. 477. (10) ZACEAEIA~EN, W. H., J . Am. them. SOC., 54, 3841 (1932). JOmR,, priv~tttecommunioation. (11) (12) HOUWINK, R., "El&sticity,Plasticity and the Structure of Matter," Cambridge Press, 1940, p. 25.
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