Impurity particles and the lateral growth of cadmium iodide

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a tentative explanation, assuming two additional conditions. Fisher et UP clamped the tin-plated specimen to apply the pressure on it. Considerable tin was extruded at the free side surface. They noticed, however, that whiskers seem to grow not from this extruded tin, but from the vicinity thereof. At the region where the tin extrusion is hindered, the applied pressure, P, may be maintained without relaxation up to the very vicinity of the free surface, as if it were a hydrostatic pressure in a closed vessel. The reason why this kind of hindrance of extrusion occurs is not known, but one of the necessary conditions for it may be the thinness of plated tin between the clamps. If a Frank’s spiral of whisker-producing dislocation2 is situated in such a region where the tin extrusion is hindered, a whisker will grow there at an accelerated rate, which depends on the applied pressure. This whisker growth will relieve the applied stress. The basic concept of the present writer is that the stress is relieved by means of whisker growth at the region where the stress is not relieved by means of tin extrusion. The concentration of atomic vacancies in a region of tin crystal, where the applied pressure, P, is maintained, is decreased from its normal value by a factor of exp(- Pu3/kT), where a is the atomic spacing, K is Boltzmann’s constant, and T is absolute temperature.3 The vacancy concentration at the Frank’s dislocation spiral is somewhat higher than the normal value. Then the concentration gradient of vacancies is (Z/Ra3)( l-exp(-Pu3/kT))

=ZP/RkT,

(1)

where Z is normal concentration of vacancies, and R is the distance between the dislocation spiral and the region where pressure, P, is maintained. Here the vacancy concentration at the dislocation spiral was assumed to have normal value. Now the distance, R, is assumed to be very small, say from 10 to lOO& because the pressure, P, is considered to be maintained up to the very vicinity of free surface. This is the first assumption. The concentration gradient as expressed in (1) must be kept constant to secure the constant diffusion current which enables whiskers to grow at a constant rate. The concentration gradient is assumed to be kept constant by means of absorption of vacancies by edge dislocations which are situated near the end of concentration gradient. This is the second assumption. Now the flux of vacancies is DP/RkT, if the approximate value of (1) is used. Here D is the self-diffusion coefficient of tin at the temperature T. Thus the rate of whisker growth G is G= DPu3/RkT

cm/set

(2)

Equation (2) gives a linear relation between G and P, which agrees with experimental results. Taking D= lV* cmz/sec, T=300”K, u3=30A3, and P=S,OOO psi, we

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obtain G- 26O&sec

(3)

G= 26&‘sec

(4)

for R= lOA, and

for R= lOOA. The value (3) is ten times smaller, and the value (4) is a hundred times smaller than the experimental value, which is about 26OO&sec. The calculated values of G may be increased by a factor of ten or more, if we consider two more conditions. (1) The concentration of vacancies is higher than the normal value near the Frank’s dislocation spiral. This makes the concentration gradient increase. (2) Although the tin extrusion is hindered around the whisker root, it may be possible that a slow creep occurs at the vicinity of the root. If this is the case, vacancy concentration will be increased there.4 This also makes the diffusion current increase. Fisher et al have shown that the whiskers may exhibit three stages of growth : (1) an induction period, (2) a period of constant growth rate, and (3) an abrupt transition to a much slower growth rate. The present writer’s explanation is consistent with these three stages. Induction period is explained in the same way as in Frank’s theory.2 The second stage is explained as mentioned above. After some period of secondstage growth, the stress will be relieved. This will result in a deceleration of growth rate. Thus the third stage is accounted for. R. R. HASIGUTI Department of Metallurgy Faculty of Engineering University of Tokyo Tokyo, Japan

R. M. Fisher, L. S. Darken, and K. G. Carroll, Acta Met. 2, 368 (1954). F. C. Frank, Phil. Mag. 44, 854 (1953). F. R. N. Nabarro, Rep. Conf. Strength of Solids, Phys. Sot. 1948, p. 75. F. S. Buffington and M. Cohen, J. Metals, 4, 8.59 (1952). * eRceived August 19, 1954.

Impurity Particles and the Lateral Growth of Cadmium Iodide* The engulfment of impurity particles during the lateral growth of cadmium iodide plates has recently been postulated to account for the development of screw dislocations in crystals, leading to growth in thickness.’ More detailed study of the growth of cadmium iodide from aqueous solution seems to indicate that the presence of impurities must also be taken into account in considering the process of lateral growth itself, through the advance of the (1010) faces. We have observed the growth of cadmium iodide plates of uniform thickness (evidenced by the uniformity of the interference colors) formed on depositing

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dynamic functions in the bulk, the increase in the chemical potential, p, over that for a face without contamination will be7 & &=+y~

--i

FIG. 1. Border formation in growth of cadmium iodide from aqueous soiution. Magnification 340X _

a solution saturated with reagent grade material at 35.O”C in a cell2 thermostated at this temperature, On lowering the temperature S-10 degrees (giving a supersaturation ratio ff = 1.02-1.04), many of the plates, originally in the form of irregular hexagons, were found to grow through the addition of a layer around the prism faces of the original crystal, with an interference color different from that of the ariginal formation (Figs. la, b). The step thus formed in the (0001) plane is filled in only after the border has grown to some extent, sometimes through the action of a spiral growth step.” A sequence leading to the formation of such a border is illustrated in Figs. 2a-f. Upon lowering the temperature of th.e cell 9 degrees, a blue crystal plate emanates from a corner of the original red crystal. All but the (0001) face of this addition advance immediately; the equivalent faces of the original platelet remain stationary, however, and are finally engulfed (Fig. 2f). A similar sequence is apparent in Figs, Ja-d ; here more than one source appears to be operating, however, sending out separate strips which join, as shown in Fig. 3b. Separate outgrowths thus formed have been found to show the same interference color, without any apparent material connection. The formation of these borders can be rationalized by postulating the existence of a fault plane in the original crystal plate, possibly separating a region in which the layers are randomly stacked from one in which adjoining layers of cadmium iodide are so arranged as to form a C27 structure, with two Cd12 sheets per unit cell, which has been found to predominate for slow crystallization4 and presumably minimizes the free energy. To account for development of crystal spokesPL we suggest that in the interval between crystal formation and the lowering of the temperature, small impurity particles present in solution are adsorbed on the platelets, primarily on the loosely packed prism faces, and that these can delay growth, as already shown by Zener for grain that boundary migration. 6 Assuming, for simplicity, such particles subdivide the prism faces into regions replicating the habit of the original crystal, and neglectbg the contribution of impurities to the thermo-

S

-

SIsin7.Vi(,-Coswi)t

where y is the specific volume of the solid, u is the surface tension of the uncontaminated face and wd the exterior angle between it and an adjoining face i, of surface tension Be. S and S’ denote, respectively, the area of the uncontaminated face and the average extent of the region delimited by impurities, kd and li’ the length of the edges in the two cases. For ca~ium iodide the individual terms in the summation are, under our assumptions, positive quantities, so that Ap>O. Even without taking account of the detailed mechanism by which prism faces advance, it follows that the supersaturation necessary for growth must be the greater the greater the number of impurities adsorbed on a surface. The formation of isolated outgrowths of the prism faces thus appears as a result

‘,e. ,... ,.

.;.

f

FIG. 2. Border formation in growth of cadmium iodide from aqueous solution. Magnification 340X,

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strength is not proportional to the quantity of dissolved nitrogen (see their Fig. 1). In contrast with this conclusion the damping theory of Snoek2 and Polde? anticipates proportionality between relaxation strength and N (or C) content for contents so small that interaction between interstitially dissolved atoms may be neglected. Dijkstra4 substantiated this anticipation experimentally for dilute solutions of carbon in a-iron. According to Astrom and Borelius’ Fig. 2, the maximum solubility of nitrogen in a-iron at the eutectoid temperature of 585°C is about 0, 1% by weight. According to their Fig. 1 this would mean that an iron wire containing nitrogen can never show an internal friction peak higher than QmaXW1=e”/2~‘=0.055 at -18°C after quenching from 580°C. Our experiments show that values of QmaX-1 higher than 0.070 can easily be obtained even at room temperature by quenching iron wires containing 0.1% nitrogen from 580°C. Still higher values can be obtained by measuring at lower temperatures. The greatest peak height obtainable in principle is difficult to measure because precipitation is so rapid in iron containing O.l7,N, that part of the nitrogen has already precipitated before the first value FIG. 3. Border formation in growth of cadmium iodide of internal friction can be measured. The foregoing from aqueous solution. Magnification 340X. means that at least one of Astrom and Borelius’ curves, the solubility or the relaxation strength curve, is of an unequal distribution of adsorbed impurities, incorrect. which leaves only those sections of the prism faces free Based on their experimental relation between to grow at low supersaturation which have the more relaxation strength and nitrogen content (their Fig. l), stable C27 structure and which are only sparsely Astrom and Borelius “correct” the solubility curve populated with impurities. The details of the mechanism given by Dijkstra.6 Theo “corrected” values agree well of lateral growth are not revealed by these experiments, with those obtained by Astrom and Borelius from their and do not affect these conclusions since the adsorption measurements on elastic after-effect and from their of impurities serves only to obstruct growth. calorimetric measurements. The latter deviate strongly G. EHRLICH from the older calorimetric measurements of Borelius, General Electric Research Laboratory Berglund and Avsan.6 It is to be remarked that in Schenectady, New York their “correction,” mentioned above, Astrom and Borelius do not start from the curve as given by References Dijkstra, but from a strongly deviating curve obtained 1. J. C. Fisher, R. L. Fullman, and G. W. Sears, Acta Met. In by giving special weight to some of Dijkstra’s values. In our laboratory both the relation between quantity 2. !?& Hartshorne and A. Stuart “Crystals and the Polarizing Microscope (L. Arnold, London,‘l950), p. 393. of dissolved nitrogen and internal friction, and the 3. Such bordered plates have also been reported by J. B. Newkirk, solubilities of nitrogen in a-iron in equilibrium with Acta Met. In press. Nz of 1 atm, Fe4N and “FesN” were determined with 4. Structure Reports 11,491 (1951). 5. This interpretation of lateral growth arose in discussion with great care, making use of a torsional pendulum system R. L. Fullman, for whose help I am greatly indebted. with iron wires of 0.7 mm diameter and 240 mm length 6. C. Zener, Trans. A.I.M.E. 175, 15 (1948). 7. The Collected Works of J. W. Gibbs, Yale University Press, as the suspension element. New Haven, Conn. Reprinted 1948, Vol. 1, p. 321. In order to find the solubility of nitrogen in a-iron * Received May 21, 1954. in equilibrium with Nz of 1 atm, fine-grained, textureless wires were heated at several temperatures in the range 700 to 900°C and at 1450°C in a gas stream of the composition 99 vol %Nz+l vol YcHz, until, after Solubility of Nitrogen in a-Iron* quenching in water, a constant height of the internal friction peak was obtained. In a recent “Letter to the Editor” Astrom and The solubility in equilibrium with Fe4N was deterBoreliusi report on new measurements of the solubility mined on wires that were previously loaded with about of nitrogen in a-iron in equilibrium with Fe4N and 0.1 wt 7c of nitrogen at 570°C in a mixture of NHs+Hz. “FegN.” From measurements of the elastic after-effect These wires were heated in a pure nitrogen atmosphere caused by nitrogen they deduce that the relaxation