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A relaxed vacancy model for diffusion in crystalline metals
A RELAXED VACANCY MODEL FOR DIFFUSION IN CRYSTALLINE METALS* N. H. NACHTRIEB
and G. S. HANDLERi
A mechanism for diffusion in solid metals is proposed in which the rate-limiting atom movements occur within small regions of disorder in the crystal. The disordered regions consist, on the average, of 12 to 14 atoms which have relaxed inward around a lattice vacancy, and have an energy content about the same as the equivalent number of atoms in the liquid state. UN MODELE
DE LA DIFFUSION DANS DES METAUX CRISTALLINS, LACUNES RETICULAIRES, RELACHEES
BASE
SUR
DES
On propose un mCcanisme de diffusion dans des metaux solides, dans lequel les mouvements atomiques qui limitent la vitesse ont lieu au sein de petites regions desordonnees du cristal. Ces regions desordonnees consistent, en moyenne, en 12 a 14 atomes qui ont et6 rellches vers l’interieur autour d’une lacune reticulaire, et dont l’energie est approximativement la m&me que celle d’un nombre equivalent d’atomes B l’etat liquide. EIN
SPANNUNGSFREIES
LEERSTELLENMODELL KRISTALLINEN METALLEN
DER
DIFFUSION
IN
Es wird ein Diffusionsmechanismus in festen Metallen vorgeschlagen, bei dem die geschwindigkeitsbegrenzenden Atombewegungen innerhalb kleiner fehlgeordneter Bereiche des Kristalls erfolgen. Diese fehlgeordneten Bereiche bestehen im Durchschnitt aus 12 bis 14 Atomen, die urn eine Gitterleerstelle gruppiert sind und sich in diesem Komplex spannungsfrei anordnen. Der Energieinhalt dieser Anordnung ist etwa gleich dem der gleichen Anzahl von Atomen im fliissigen Zustand.
Introduction Atom movements in crystalline solids have been best explained in terms of lattice imperfections, notably vacant lattice sites and interstitial atoms. The latter are believed to be responsible for the diffusion of small atoms in a host lattice of atoms of large radius, while the former provide the widely accepted mechanism for self-diffusion in pure metals and in alloys in which the atom radii are not widely different (i.e., substitutional solid solutions). The vacancy model is apparently fundamentally correct for such cases, and its essential fe$ures will doubtless be retained in more elaborate theories of solid state self-diffusion. It shares with other mechanisms a rather serious defect, however; it offers no ready means of predicting the rates and activation energies for diffusion or of relating these kinetic quantities to the bulk physical properties of metals. Its real value has been qualitative . . . conceptual, rather than quantitative. Even the most thorough-going discussion of the energy of formation and movement of vacancies in copper by Huntington and Seitz [l] disagrees with experiment by about 35 per cent. The point of view advanced in this paper is that it may be more consistent with experimental evidence to consider a model for diffusion in which the elementary acts are not the creation and move*Received tInstitute ACTA
February 4, 1954. for the Study of Metals,
METALLURGICA,
VOL.
University
2, NOV.
1954
of Chicago.
ment of a simple vacancy, but rather the creation of a small region of disorder whose movement through the crystal is the origin of exchange. Two essential differences are implied in this notion: (1) that the energy of formation of the imperfection must take into account a larger region of the crystal, and (2) that the elementary atom movements are not isolated &ents, but cooperative motions involving a number of atoms. The basis for regarding diffusion as a kind of small-scale cooperative phenomenon is a relation which has been found to exist between the activation energies for diffusion in cubic metals and their latent heats of fusion [2]. The simple relation (1)
AH = 16.5
L,
obtained from a study of the effect of hydrostatic pressure on the rate of self-diffusion in sodium, applies to six f.c.c. and b.c.c. metals and to a-white phosphorous within experimental error, as Table I shows. The relation does not apply to lead with the same precision, although our recent measurements on this metal give an activation energy more nearly in agreement with equation (1) than does the less precise determination of Hevesy, Seith, and Keil [3]. It is interesting to note that no similar relation exists between the activation energy for diffusion and the latent heat of vaporization. Taken at face value, equation (1) suggests that diffusion in cubic metals is in some way related to fusion. The simple vacancy model, on the other hand, would lead one to expect a connection with sublimation.
ACTA
798 TABLE OBSERVED
AND
I
CALCULATED ENTHALPIES OF ACTIVATION FOR DIFFUSION Structure
Substance
METALLURGICA,
L,
AH
(obs)
AH (talc)
Cal/g atom
Ag cu
f.c.c. f.c.c. f.c.c.
3060 2730 3110
co Fe Pb
f.c.c. f.c.c. f.c.c.
3700 3630 1190
P4 (white) Na
b.c.c. b.c.c.
Au
*Data of the authors,
601 636.2
51,000 45,950 48,000 c 54,000 1 61,909 59,700 ( 27,900 \ 24,500* > 9,360* 10,450
50,280 44,850 50,920 60,790 59,640 19,650 9,920 10,450
not yet published.
The Enthalpy
of Activation
On any lattice imperfection model of diffusion we may regard the “experimental” activation energy as consisting of two terms: (1) an enthalpy of formation, AHr, of the particular lattice defect which is antecedent to diffusion, and (2) an enthalpy of activation, AH*, for the movement of atoms involved in measurable diffusion over macroscopic distances. AHI is the difference in heat content of an infinitely large crystal containing N lattice defects of the appropriate kind and the same crystal devoid of such defects. We assume that the equilibrium density of such imperfections in a pure substance is determined solely by the temperature and pressure. In the simple vacancy theory the lattice imperfections are considered to be discrete voids whose spherically inscribed volume is about equal to the atom volume. The energy of formation of such a void would be twice the latent heat of vaporization per atom if the atom were taken to the vapor. It is here that the present model departs from the discrete vacancy theory, by the proposal that the vacancy volume is locally distributed. In effect, the vacancy dissolves in the lattice as a result of the inward relaxation of the neighboring atoms. We shall refer to such dispersed vacancies and to the complex of atoms most closely associated with the excess volume as “relaxions,” and retain the term “vacancy” in its familiar sense to denote a discrete atom
void.
suggested.
What is than an vacancy? relaxion?
Several
questions
are
immediately
How many atoms comprise a relaxion? the basis for expecting an inward, rather outward, relaxation of atoms around a What is the energy of formation of a What is the configuration of atoms within
VOL.
2, 1954
a relaxion? Finally, what is the equilibrium concentration of relaxions? In an attempt to answer the first question, let us note two inferences which may be drawn from equation (1) : 1. The invariance of AH over a wide range of temperatures implies that the average number of atoms associated with a relaxion is constant. 2. The coefficient 16.5 is somewhat larger than the number of nearest neighbors in a f.c.c. lattice (12) and of the sum of the nearest and next nearest neighbors in a b.c.c. lattice (14). Let us draw the naive conclusion that the number of atoms most closely associated in a relaxion is of the order of 12 to 14, realizing that it is somewhat arbitrary to set any definite boundary on the disordered region,* We may expect the energy changes associated with these displacements to be related to the latent heat of fusion and the volume change to be related to the volume change on melting (see later). The independence of relaxion size with temperature implied by (1) is at first surprising. One might have expected the size distribution as well as the number distribution to depend exponentially upon temperature, according to such heterophase fluctuation theories as Frenkel [4] has discussed. The reason for the invariance of relaxion size with temperature appears to be the short-range character of the interatomic forces. Relaxation of atoms about a vacancy is accompanied by a decrease in free energy, and most of the energy change arises from the movement of the nearest (and in the case of b.c.c. lattices, next nearest) neighbors. That an inward, rather than an outward, displacement of the atoms surrounding a vacancy is to be expected may be argued on logical grounds if a suitable potential function can be found. No rigorous method is known, however, for computing the potential energy of an atom in a metallic crystal. Pair potential functions are incorrect in principle, although useful for rough estimates if the forces are sufficiently short range. Regardless of the exact form of the potential function, we recognize that it must have the property of repulsion for sufficiently close interatomic distances and attraction for great separations. For any crystal of minimum energy there is a balance *We neglect in this approximation contributions to the relaxion energy which might be described as interfacial energy or strain energy. Their effect would be to decrease the number of atoms which are regarded to comprise a relaxion. Properly, of course, there is no interface and the use of an interfacial energy term is somewhat artificial.
NACHTRIEB
AND
HANDLER:
between the forces of attraction and repulsion appropriately summed over all atoms in the crystal. The last two statements can be true only if the net forces between nearest neighboring atoms are repulsive. It then follows that removal of an atom from the interior of a metallic crystal must result in an inward movement of its nearest former neighbors.* Removal of an ion from the interior of an ionic crystal, on the other .hand, would result in outward relaxation of the neighboring ions because of the repulsion of their Coulomb fields.
Energy of Formation
of Relaxions
An exact calculation of the energy of a relaxion relative to the perfect crystal would require a detailed balance of all interactions and is not possible with approximate potential functions. A rough calculation with an admittedly imperfect function is nevertheless worth while, merely to indicate that large relaxation energies are involved. Slater [5] has represented the energies of the alkali metals by means of a Morse function: (2)
u0 = L(e2a(r-r0) _ 2e-““-‘“‘)
where UO is the energy of the crystal at O’K, L is the latent heat of vaporization (26.2 kcal/g atom for sodium), ro is the nearest interatomic distance (3.72A), and a is a parameter derived from compressibility measurements (0.67 for sodium). Such a function disregards all interactions between atoms farther apart than nearest neighbors, and in so doing ignores the fact that the net force between nearest neighbors is repulsive. The energy per atom pair for a b.c.c. structure will then be given by (3)
E. = + (e-za+-r.)
-
2e++)
).
799
DIFFUSION
is as good as can be expected in view of the crude assumptions which underlie the calculation. Equation (l), first obtained for b.c.c. sodium, applies equally well to f.c.c. metals, as Table I indicates. The f.c.c. lattice is close-packed, however, and no relaxation about a vacancy is possible if the atoms are regarded as hard spheres. Presumably, considerable polarization must take place if inward relaxation is to occur in such structures.
Structure
of Relaxations
In the foregoing we have discussed in rough terms the energy, AHi, required to form a lattice imperfection involving 12 to 14 atoms by a symmetrical relaxation about a vacancy and have proposed that it amounts to about 12 to 14 times the latent heat of fusion per gram atom. Such a region would have a density about 7 to 8 per cent lower than that of the normal crystal if the vacancy volume were entirely localized within it. We believe that the density is only 3 to 4 per cent lower, for reasons to be discussed in the last section. If this is so, such a region would have about the same heat content as the same number of atoms in the liquid state. A symmetrical relaxion would then be one of the configurations of the liquid state. Atom movements within a relaxion resulting from the absorption of phonons would cause the configuration to progress through less symmetrical atom arrangements more typical of the liquid state. Within a relaxion atoms are considered to squeeze or roll past one another, but the movement of any one atom would entail the correlated movement of the remaining atoms.? In effect, atom movements within a relaxion are considered to take place with a rate and activation energy comparable to that observed for diffusion in liquids.
Relaxation of the eight nearest atoms surrounding Enthalpy for Atom Movement the vacancy and of the six next nearest atoms to a Activation If the enthalpy of relaxion formation is taken closest-packed configuration gives - 13.0 kcal. Combined with the 26.2 kcal required to produce a to be AH1 = 14 L, for sodium, there remains for the enthalpy of activation for atom movements, vacancy and a surface atom gives 13.2 kcal as the AHz, about 2.5 L,. This amounts to about 1600 energy of formation of a relaxion. The agreement with the observed activation energy (10.45 kcal) ,cal in the case of sodium, in reasonably good agreement with the experimental value (2580 cal) for the activation energy for diffusion in liquid *It is worth calling attention to the very instructive magnet model of a two-dimensional crystal which Hilsch sodium. With such comparative freedom for atoms has described. Rod-form magnets axially suspended by to pass from one configuration in a relaxion to threads from a central point take up a close-packed hexagonal array in a plane. The repulsive forces are due to the magnetic dipole interaction, and vary as l/r’; the simulated attractive forces are the restoring gravitational component, and vary as I for small disolacements. Such a model shows comnlete inward relaxationbf the surrounding magnets when a central one is released from the array to simulate a vacancy.
tin this connection it is interesting to call attention to Zener’s ring diffusion model (Acta Cryst. 3, 1950, 346), wherein a ring of 4 atoms cooperatively rotates through an angle of 90 degrees in the elementary diffusion act.
ACTA
800
METALLURGICA,
another, there would still be no macroscopic diffusion if the relaxion were anchored to the particular lattice site where it was formed. Propagation of a symmetrical relaxion would constitute a vacancy wave which would produce macroscopic diffusion in the same manner as the movement of a discrete vacancy, however. We assume that a relaxion moves through the crystal with a Brownian-like course because of scattering by phonons. The process may be likened to “melting” and “freezing” one or two atoms at a time with an activation energy of the order of Lm. Since the activation energy for diffusion in monatomic liquids is of comparable magnitude, we think it likely that the cooperative movement* of atoms within a relaxion is a process of similar probability. In effect, this is to propose that the various diffusion mechanisms . . . vacancy, interstitial, direct exchange, and ring rotation . . . are partial truths and that an eclectic approach may be nearer to the true situation.
Concentration
VOL.
2,
extrapolation of the data to the undercooled range. Figure 1 shows a plot of log D versus l/T for liquid and solid sodium. Table II lists the relevant TABLE
CONCENTRATION OF RELAXIONS IN SODIUM
D, (ohs)
V.7
0.3 39.4 69.3 94.2 97.5
9.22x10-‘0 1.31X10-s 5.26X10-s 1.43X10-’ 1.65X10-’
DI (talc)* 1.15x10-s 2.08X10-’ 3.00X10-’ 3.88X 1O-6 4.OOX1O-6
X 8.04x10-s 6.29X10-” 1.76X10-” 3.68X10-* 4.13X10-*
?I 5.74x10-s 4.49X10-s 1.26X10-’ 2.63X lo-’ 2.95X10+
*Calculated from D1 = 1.33 X 10-s exp(-2530/2PT) (unpublished data of Mr. R. E. Meyer for self-diffusion in liquid sodium). The results are not materially altered if DI is calculated from viscosity data with the Stokes-Einstein equation.
1
1.6
from
the Stokes-Einstein
D=$
For sodium, however, unpublished experimental diffusion coefficients in the liquid state are available,? together with the activation energy for *Observation of the magnet model is instructive in this. connection. Readjustments involving many magnets occur when a vacancy is created. The motions are so rapid and complex that it has not been possible to observe a relaxion wave. tMr. Robert E. Meyer has measured the rate of selfdiffusion in sodium between 101.7”C and 176.5”C in this laboratory. The measured Dr is 4.15 X 10-s cm’ se@ at 101.7” and the activation energy is 2530 Cal/g atom. These values may be compared with 4.17 X lo+ cm* seer and 2470 Cal/g atom, obtained by use of Andrade’s data (Proc. Roy. Sot. Lond. A157, 1936, 264) for the viscosity of liquid sodium at the same temperature and with the ionic radius (0.95A) for I in equation 5.
i?
3 f
FIGURE 1.
(5)
II
of Relaxions
If we assume that relaxions move through a metal lattice by a process which is comparable to small-scale melting, it is possible to make a simple estimate of their average concentration. Let X represent the fraction of all atoms which are present in relaxions at a given temperature and pressure, and let the average number of atoms per relaxion be denoted by g. Thep if D, and D1 are the self-diffusion coefficients for the solid and liquid at a given temperature, the fractional number of relaxions is simply
D1 can be estimated equation
1954
Diffusion
x io3
4
A61W
in solid and liquid sodium.
data, together with the fractional number of relaxions calculated for five temperatures. The values range from 5.74 X 10M6 at 0.3”C to 2.95 X 10m4 (extrapolated to the melting point), and are consistent with general estimates of the fractional numbers of vacancies present in metals correspondingly near to their melting points.
Activation
Volume for Diffusion
Further evidence that diffusion in solid metals may be regarded as a small scale cooperative phenomenon having some of the characteristics of melting may be deduced from the effect of hydrostatic pressure on the rate of self-diffusion. The activation volume for diffusion is defined in a completely general way by the relation =
AJ’wt
NACHTRIEB
AND
HANDLER:
where AF is the activation free energy. AVaot, like the activation free energy, enthalpy, and entropy, is a composite quantity and can be regarded as consisting of a volume change for the formation of N relaxions, AVI, and a volume change for their movement and the movement of atoms within them, AVZ. Following Zener [6] we may define AF by means of the expression (7)
D = y a2 y e-AFIRT
where v is the average lattice vibrational frequency, a is the lattice parameter, and y is a constant determined by the lattice geometry and the assumed jump mechanism. Its value is unity for b.c.c. lattices for a vacancy mechanism. The limitations of equation (7) must now be noted. It was derived for mechanisms in which the elementary jump distance is well defined and geometrically related to the lattice parameter. Moreover, it is implicit that a single characteristic lattice frequency may be used with sufficient approximation to the truth. In the relaxion model, however, many atom movements take place on a scale small compared to the lattice parameter and unrelated to it in any simple manner. Further, the dominant vibration frequency in a relaxion must be somewhat lower than in a flawless portion of a crystal. In the face of these apparent objections, we nevertheless use equation (7) for the following reasons. Measureable diffusion occurs only when interchanged atoms are left in the wake of a moving relaxion. Only those movements of atoms within a relaxion contribute to diffusion which involve displacements amounting to mass rotation by one atom distance. Lesser movements are merely anharmonic vibrations which restore the atoms to their original sites on the perfect lattice. Such relaxions with sub-marginal heat content do not contribute to diffusion until an energy fluctuation (absorbed phonon) occurs. y is a constant, and its particular value will not affect the value of the derivative
(8)
T
Finally, it is the variation of v with P which concerns us, not v, itself. Grtineisen’s relation shows that for sodium v increases only 16 per cent for a 20 per cent increase in crystal density, and since a2 and v vary in opposite ways with pressure, their product is constant to within 4 per cent for pressures up to 12,000 atmospheres. Without serious error we could have taken A V,,, to be - RT
DIFFUSION
801
[d log D/dP]T. I n other words, this amounts to ignoring the change in the entropy of activation with pressure. Figure 2 shows the fractional variation ot AV.ot and AVtisloo as a function of pressure up to 12,000 atmospheres at 90°C for sodium. We note that equation (8) gives the activation volume referred to one mole of relaxions, and that for comparison with the volume change on fusion per
030
DIFFUSION
DO01 0
4
PRESSURE
8 (ATM.
90’ G
12
x 10-31
1613-1
FIGURE 2. Volume changes (fusion and diffusion) sodium as a function of pressure.
for
gram atom it must be divided by the number of atoms in a relaxion. The interesting point to note is that the two quantities are of similar magnitude, and their variation with pressure is also similar. We should expect AVaE, to be larger than AVtilto,,, since the former includes AV2, the activation volume for the movement of atoms and relaxions. An experimental determination of the activation volume for liquid diffusion would now be interesting.
Acknowledgements It is a pleasure to acknowledge the many helpful suggestions and criticisms offered by members of the Institute for the Study of Metals. The authors are particularly indebted to Professor Cyril Stanley Smith and to Professor Charles S. Barrett for their encouragement and advice. This research was supported by the United States Air Force under Contract No. AF33(616)2090.
ACTA
METALLURGICA,
References 1. HUNTINGTON,H. B. and SEITZ, F.
VOL.
2, 1954
3. HEVESY, G. V., SEITH, W., and KEIL, A. Phys. Rev. 61 (1942)
315. 2. NACHTRIEB, N. H., WEIL, J. A., CATALANO, E., and LAWSON, A. W. J. Chem. Phys. 20 (1952) 1189.
(1932) 197. 4. FRENKEL, J.
2. Physik 79
Kinetic Theory of Liquids (Oxford Univer-
sity Press, 1946), p. 383. 5. SLATER, J. C. Introduction to Chemical Physics (New York, McGraw-Hill, 1939), p. 452. 6. ZENER, C. J. Appl. Phys. 22 (1959) 373.
and G. S. HANDLERi
A mechanism for diffusion in solid metals is proposed in which the rate-limiting atom movements occur within small regions of disorder in the crystal. The disordered regions consist, on the average, of 12 to 14 atoms which have relaxed inward around a lattice vacancy, and have an energy content about the same as the equivalent number of atoms in the liquid state. UN MODELE
DE LA DIFFUSION DANS DES METAUX CRISTALLINS, LACUNES RETICULAIRES, RELACHEES
BASE
SUR
DES
On propose un mCcanisme de diffusion dans des metaux solides, dans lequel les mouvements atomiques qui limitent la vitesse ont lieu au sein de petites regions desordonnees du cristal. Ces regions desordonnees consistent, en moyenne, en 12 a 14 atomes qui ont et6 rellches vers l’interieur autour d’une lacune reticulaire, et dont l’energie est approximativement la m&me que celle d’un nombre equivalent d’atomes B l’etat liquide. EIN
SPANNUNGSFREIES
LEERSTELLENMODELL KRISTALLINEN METALLEN
DER
DIFFUSION
IN
Es wird ein Diffusionsmechanismus in festen Metallen vorgeschlagen, bei dem die geschwindigkeitsbegrenzenden Atombewegungen innerhalb kleiner fehlgeordneter Bereiche des Kristalls erfolgen. Diese fehlgeordneten Bereiche bestehen im Durchschnitt aus 12 bis 14 Atomen, die urn eine Gitterleerstelle gruppiert sind und sich in diesem Komplex spannungsfrei anordnen. Der Energieinhalt dieser Anordnung ist etwa gleich dem der gleichen Anzahl von Atomen im fliissigen Zustand.
Introduction Atom movements in crystalline solids have been best explained in terms of lattice imperfections, notably vacant lattice sites and interstitial atoms. The latter are believed to be responsible for the diffusion of small atoms in a host lattice of atoms of large radius, while the former provide the widely accepted mechanism for self-diffusion in pure metals and in alloys in which the atom radii are not widely different (i.e., substitutional solid solutions). The vacancy model is apparently fundamentally correct for such cases, and its essential fe$ures will doubtless be retained in more elaborate theories of solid state self-diffusion. It shares with other mechanisms a rather serious defect, however; it offers no ready means of predicting the rates and activation energies for diffusion or of relating these kinetic quantities to the bulk physical properties of metals. Its real value has been qualitative . . . conceptual, rather than quantitative. Even the most thorough-going discussion of the energy of formation and movement of vacancies in copper by Huntington and Seitz [l] disagrees with experiment by about 35 per cent. The point of view advanced in this paper is that it may be more consistent with experimental evidence to consider a model for diffusion in which the elementary acts are not the creation and move*Received tInstitute ACTA
February 4, 1954. for the Study of Metals,
METALLURGICA,
VOL.
University
2, NOV.
1954
of Chicago.
ment of a simple vacancy, but rather the creation of a small region of disorder whose movement through the crystal is the origin of exchange. Two essential differences are implied in this notion: (1) that the energy of formation of the imperfection must take into account a larger region of the crystal, and (2) that the elementary atom movements are not isolated &ents, but cooperative motions involving a number of atoms. The basis for regarding diffusion as a kind of small-scale cooperative phenomenon is a relation which has been found to exist between the activation energies for diffusion in cubic metals and their latent heats of fusion [2]. The simple relation (1)
AH = 16.5
L,
obtained from a study of the effect of hydrostatic pressure on the rate of self-diffusion in sodium, applies to six f.c.c. and b.c.c. metals and to a-white phosphorous within experimental error, as Table I shows. The relation does not apply to lead with the same precision, although our recent measurements on this metal give an activation energy more nearly in agreement with equation (1) than does the less precise determination of Hevesy, Seith, and Keil [3]. It is interesting to note that no similar relation exists between the activation energy for diffusion and the latent heat of vaporization. Taken at face value, equation (1) suggests that diffusion in cubic metals is in some way related to fusion. The simple vacancy model, on the other hand, would lead one to expect a connection with sublimation.
ACTA
798 TABLE OBSERVED
AND
I
CALCULATED ENTHALPIES OF ACTIVATION FOR DIFFUSION Structure
Substance
METALLURGICA,
L,
AH
(obs)
AH (talc)
Cal/g atom
Ag cu
f.c.c. f.c.c. f.c.c.
3060 2730 3110
co Fe Pb
f.c.c. f.c.c. f.c.c.
3700 3630 1190
P4 (white) Na
b.c.c. b.c.c.
Au
*Data of the authors,
601 636.2
51,000 45,950 48,000 c 54,000 1 61,909 59,700 ( 27,900 \ 24,500* > 9,360* 10,450
50,280 44,850 50,920 60,790 59,640 19,650 9,920 10,450
not yet published.
The Enthalpy
of Activation
On any lattice imperfection model of diffusion we may regard the “experimental” activation energy as consisting of two terms: (1) an enthalpy of formation, AHr, of the particular lattice defect which is antecedent to diffusion, and (2) an enthalpy of activation, AH*, for the movement of atoms involved in measurable diffusion over macroscopic distances. AHI is the difference in heat content of an infinitely large crystal containing N lattice defects of the appropriate kind and the same crystal devoid of such defects. We assume that the equilibrium density of such imperfections in a pure substance is determined solely by the temperature and pressure. In the simple vacancy theory the lattice imperfections are considered to be discrete voids whose spherically inscribed volume is about equal to the atom volume. The energy of formation of such a void would be twice the latent heat of vaporization per atom if the atom were taken to the vapor. It is here that the present model departs from the discrete vacancy theory, by the proposal that the vacancy volume is locally distributed. In effect, the vacancy dissolves in the lattice as a result of the inward relaxation of the neighboring atoms. We shall refer to such dispersed vacancies and to the complex of atoms most closely associated with the excess volume as “relaxions,” and retain the term “vacancy” in its familiar sense to denote a discrete atom
void.
suggested.
What is than an vacancy? relaxion?
Several
questions
are
immediately
How many atoms comprise a relaxion? the basis for expecting an inward, rather outward, relaxation of atoms around a What is the energy of formation of a What is the configuration of atoms within
VOL.
2, 1954
a relaxion? Finally, what is the equilibrium concentration of relaxions? In an attempt to answer the first question, let us note two inferences which may be drawn from equation (1) : 1. The invariance of AH over a wide range of temperatures implies that the average number of atoms associated with a relaxion is constant. 2. The coefficient 16.5 is somewhat larger than the number of nearest neighbors in a f.c.c. lattice (12) and of the sum of the nearest and next nearest neighbors in a b.c.c. lattice (14). Let us draw the naive conclusion that the number of atoms most closely associated in a relaxion is of the order of 12 to 14, realizing that it is somewhat arbitrary to set any definite boundary on the disordered region,* We may expect the energy changes associated with these displacements to be related to the latent heat of fusion and the volume change to be related to the volume change on melting (see later). The independence of relaxion size with temperature implied by (1) is at first surprising. One might have expected the size distribution as well as the number distribution to depend exponentially upon temperature, according to such heterophase fluctuation theories as Frenkel [4] has discussed. The reason for the invariance of relaxion size with temperature appears to be the short-range character of the interatomic forces. Relaxation of atoms about a vacancy is accompanied by a decrease in free energy, and most of the energy change arises from the movement of the nearest (and in the case of b.c.c. lattices, next nearest) neighbors. That an inward, rather than an outward, displacement of the atoms surrounding a vacancy is to be expected may be argued on logical grounds if a suitable potential function can be found. No rigorous method is known, however, for computing the potential energy of an atom in a metallic crystal. Pair potential functions are incorrect in principle, although useful for rough estimates if the forces are sufficiently short range. Regardless of the exact form of the potential function, we recognize that it must have the property of repulsion for sufficiently close interatomic distances and attraction for great separations. For any crystal of minimum energy there is a balance *We neglect in this approximation contributions to the relaxion energy which might be described as interfacial energy or strain energy. Their effect would be to decrease the number of atoms which are regarded to comprise a relaxion. Properly, of course, there is no interface and the use of an interfacial energy term is somewhat artificial.
NACHTRIEB
AND
HANDLER:
between the forces of attraction and repulsion appropriately summed over all atoms in the crystal. The last two statements can be true only if the net forces between nearest neighboring atoms are repulsive. It then follows that removal of an atom from the interior of a metallic crystal must result in an inward movement of its nearest former neighbors.* Removal of an ion from the interior of an ionic crystal, on the other .hand, would result in outward relaxation of the neighboring ions because of the repulsion of their Coulomb fields.
Energy of Formation
of Relaxions
An exact calculation of the energy of a relaxion relative to the perfect crystal would require a detailed balance of all interactions and is not possible with approximate potential functions. A rough calculation with an admittedly imperfect function is nevertheless worth while, merely to indicate that large relaxation energies are involved. Slater [5] has represented the energies of the alkali metals by means of a Morse function: (2)
u0 = L(e2a(r-r0) _ 2e-““-‘“‘)
where UO is the energy of the crystal at O’K, L is the latent heat of vaporization (26.2 kcal/g atom for sodium), ro is the nearest interatomic distance (3.72A), and a is a parameter derived from compressibility measurements (0.67 for sodium). Such a function disregards all interactions between atoms farther apart than nearest neighbors, and in so doing ignores the fact that the net force between nearest neighbors is repulsive. The energy per atom pair for a b.c.c. structure will then be given by (3)
E. = + (e-za+-r.)
-
2e++)
).
799
DIFFUSION
is as good as can be expected in view of the crude assumptions which underlie the calculation. Equation (l), first obtained for b.c.c. sodium, applies equally well to f.c.c. metals, as Table I indicates. The f.c.c. lattice is close-packed, however, and no relaxation about a vacancy is possible if the atoms are regarded as hard spheres. Presumably, considerable polarization must take place if inward relaxation is to occur in such structures.
Structure
of Relaxations
In the foregoing we have discussed in rough terms the energy, AHi, required to form a lattice imperfection involving 12 to 14 atoms by a symmetrical relaxation about a vacancy and have proposed that it amounts to about 12 to 14 times the latent heat of fusion per gram atom. Such a region would have a density about 7 to 8 per cent lower than that of the normal crystal if the vacancy volume were entirely localized within it. We believe that the density is only 3 to 4 per cent lower, for reasons to be discussed in the last section. If this is so, such a region would have about the same heat content as the same number of atoms in the liquid state. A symmetrical relaxion would then be one of the configurations of the liquid state. Atom movements within a relaxion resulting from the absorption of phonons would cause the configuration to progress through less symmetrical atom arrangements more typical of the liquid state. Within a relaxion atoms are considered to squeeze or roll past one another, but the movement of any one atom would entail the correlated movement of the remaining atoms.? In effect, atom movements within a relaxion are considered to take place with a rate and activation energy comparable to that observed for diffusion in liquids.
Relaxation of the eight nearest atoms surrounding Enthalpy for Atom Movement the vacancy and of the six next nearest atoms to a Activation If the enthalpy of relaxion formation is taken closest-packed configuration gives - 13.0 kcal. Combined with the 26.2 kcal required to produce a to be AH1 = 14 L, for sodium, there remains for the enthalpy of activation for atom movements, vacancy and a surface atom gives 13.2 kcal as the AHz, about 2.5 L,. This amounts to about 1600 energy of formation of a relaxion. The agreement with the observed activation energy (10.45 kcal) ,cal in the case of sodium, in reasonably good agreement with the experimental value (2580 cal) for the activation energy for diffusion in liquid *It is worth calling attention to the very instructive magnet model of a two-dimensional crystal which Hilsch sodium. With such comparative freedom for atoms has described. Rod-form magnets axially suspended by to pass from one configuration in a relaxion to threads from a central point take up a close-packed hexagonal array in a plane. The repulsive forces are due to the magnetic dipole interaction, and vary as l/r’; the simulated attractive forces are the restoring gravitational component, and vary as I for small disolacements. Such a model shows comnlete inward relaxationbf the surrounding magnets when a central one is released from the array to simulate a vacancy.
tin this connection it is interesting to call attention to Zener’s ring diffusion model (Acta Cryst. 3, 1950, 346), wherein a ring of 4 atoms cooperatively rotates through an angle of 90 degrees in the elementary diffusion act.
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800
METALLURGICA,
another, there would still be no macroscopic diffusion if the relaxion were anchored to the particular lattice site where it was formed. Propagation of a symmetrical relaxion would constitute a vacancy wave which would produce macroscopic diffusion in the same manner as the movement of a discrete vacancy, however. We assume that a relaxion moves through the crystal with a Brownian-like course because of scattering by phonons. The process may be likened to “melting” and “freezing” one or two atoms at a time with an activation energy of the order of Lm. Since the activation energy for diffusion in monatomic liquids is of comparable magnitude, we think it likely that the cooperative movement* of atoms within a relaxion is a process of similar probability. In effect, this is to propose that the various diffusion mechanisms . . . vacancy, interstitial, direct exchange, and ring rotation . . . are partial truths and that an eclectic approach may be nearer to the true situation.
Concentration
VOL.
2,
extrapolation of the data to the undercooled range. Figure 1 shows a plot of log D versus l/T for liquid and solid sodium. Table II lists the relevant TABLE
CONCENTRATION OF RELAXIONS IN SODIUM
D, (ohs)
V.7
0.3 39.4 69.3 94.2 97.5
9.22x10-‘0 1.31X10-s 5.26X10-s 1.43X10-’ 1.65X10-’
DI (talc)* 1.15x10-s 2.08X10-’ 3.00X10-’ 3.88X 1O-6 4.OOX1O-6
X 8.04x10-s 6.29X10-” 1.76X10-” 3.68X10-* 4.13X10-*
?I 5.74x10-s 4.49X10-s 1.26X10-’ 2.63X lo-’ 2.95X10+
*Calculated from D1 = 1.33 X 10-s exp(-2530/2PT) (unpublished data of Mr. R. E. Meyer for self-diffusion in liquid sodium). The results are not materially altered if DI is calculated from viscosity data with the Stokes-Einstein equation.
1
1.6
from
the Stokes-Einstein
D=$
For sodium, however, unpublished experimental diffusion coefficients in the liquid state are available,? together with the activation energy for *Observation of the magnet model is instructive in this. connection. Readjustments involving many magnets occur when a vacancy is created. The motions are so rapid and complex that it has not been possible to observe a relaxion wave. tMr. Robert E. Meyer has measured the rate of selfdiffusion in sodium between 101.7”C and 176.5”C in this laboratory. The measured Dr is 4.15 X 10-s cm’ se@ at 101.7” and the activation energy is 2530 Cal/g atom. These values may be compared with 4.17 X lo+ cm* seer and 2470 Cal/g atom, obtained by use of Andrade’s data (Proc. Roy. Sot. Lond. A157, 1936, 264) for the viscosity of liquid sodium at the same temperature and with the ionic radius (0.95A) for I in equation 5.
i?
3 f
FIGURE 1.
(5)
II
of Relaxions
If we assume that relaxions move through a metal lattice by a process which is comparable to small-scale melting, it is possible to make a simple estimate of their average concentration. Let X represent the fraction of all atoms which are present in relaxions at a given temperature and pressure, and let the average number of atoms per relaxion be denoted by g. Thep if D, and D1 are the self-diffusion coefficients for the solid and liquid at a given temperature, the fractional number of relaxions is simply
D1 can be estimated equation
1954
Diffusion
x io3
4
A61W
in solid and liquid sodium.
data, together with the fractional number of relaxions calculated for five temperatures. The values range from 5.74 X 10M6 at 0.3”C to 2.95 X 10m4 (extrapolated to the melting point), and are consistent with general estimates of the fractional numbers of vacancies present in metals correspondingly near to their melting points.
Activation
Volume for Diffusion
Further evidence that diffusion in solid metals may be regarded as a small scale cooperative phenomenon having some of the characteristics of melting may be deduced from the effect of hydrostatic pressure on the rate of self-diffusion. The activation volume for diffusion is defined in a completely general way by the relation =
AJ’wt
NACHTRIEB
AND
HANDLER:
where AF is the activation free energy. AVaot, like the activation free energy, enthalpy, and entropy, is a composite quantity and can be regarded as consisting of a volume change for the formation of N relaxions, AVI, and a volume change for their movement and the movement of atoms within them, AVZ. Following Zener [6] we may define AF by means of the expression (7)
D = y a2 y e-AFIRT
where v is the average lattice vibrational frequency, a is the lattice parameter, and y is a constant determined by the lattice geometry and the assumed jump mechanism. Its value is unity for b.c.c. lattices for a vacancy mechanism. The limitations of equation (7) must now be noted. It was derived for mechanisms in which the elementary jump distance is well defined and geometrically related to the lattice parameter. Moreover, it is implicit that a single characteristic lattice frequency may be used with sufficient approximation to the truth. In the relaxion model, however, many atom movements take place on a scale small compared to the lattice parameter and unrelated to it in any simple manner. Further, the dominant vibration frequency in a relaxion must be somewhat lower than in a flawless portion of a crystal. In the face of these apparent objections, we nevertheless use equation (7) for the following reasons. Measureable diffusion occurs only when interchanged atoms are left in the wake of a moving relaxion. Only those movements of atoms within a relaxion contribute to diffusion which involve displacements amounting to mass rotation by one atom distance. Lesser movements are merely anharmonic vibrations which restore the atoms to their original sites on the perfect lattice. Such relaxions with sub-marginal heat content do not contribute to diffusion until an energy fluctuation (absorbed phonon) occurs. y is a constant, and its particular value will not affect the value of the derivative
(8)
T
Finally, it is the variation of v with P which concerns us, not v, itself. Grtineisen’s relation shows that for sodium v increases only 16 per cent for a 20 per cent increase in crystal density, and since a2 and v vary in opposite ways with pressure, their product is constant to within 4 per cent for pressures up to 12,000 atmospheres. Without serious error we could have taken A V,,, to be - RT
DIFFUSION
801
[d log D/dP]T. I n other words, this amounts to ignoring the change in the entropy of activation with pressure. Figure 2 shows the fractional variation ot AV.ot and AVtisloo as a function of pressure up to 12,000 atmospheres at 90°C for sodium. We note that equation (8) gives the activation volume referred to one mole of relaxions, and that for comparison with the volume change on fusion per
030
DIFFUSION
DO01 0
4
PRESSURE
8 (ATM.
90’ G
12
x 10-31
1613-1
FIGURE 2. Volume changes (fusion and diffusion) sodium as a function of pressure.
for
gram atom it must be divided by the number of atoms in a relaxion. The interesting point to note is that the two quantities are of similar magnitude, and their variation with pressure is also similar. We should expect AVaE, to be larger than AVtilto,,, since the former includes AV2, the activation volume for the movement of atoms and relaxions. An experimental determination of the activation volume for liquid diffusion would now be interesting.
Acknowledgements It is a pleasure to acknowledge the many helpful suggestions and criticisms offered by members of the Institute for the Study of Metals. The authors are particularly indebted to Professor Cyril Stanley Smith and to Professor Charles S. Barrett for their encouragement and advice. This research was supported by the United States Air Force under Contract No. AF33(616)2090.
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References 1. HUNTINGTON,H. B. and SEITZ, F.
VOL.
2, 1954
3. HEVESY, G. V., SEITH, W., and KEIL, A. Phys. Rev. 61 (1942)
315. 2. NACHTRIEB, N. H., WEIL, J. A., CATALANO, E., and LAWSON, A. W. J. Chem. Phys. 20 (1952) 1189.
(1932) 197. 4. FRENKEL, J.
2. Physik 79
Kinetic Theory of Liquids (Oxford Univer-
sity Press, 1946), p. 383. 5. SLATER, J. C. Introduction to Chemical Physics (New York, McGraw-Hill, 1939), p. 452. 6. ZENER, C. J. Appl. Phys. 22 (1959) 373.