9.2: GRAIN BOUNDARIES 221 9.2.3 The mechanisms by which fast grain-boundary diffusion occurs are not well estab- lished at present.. Calculations showed that vacancies were more numerou
Trang 19.2: GRAIN BOUNDARIES 221 9.2.3
The mechanisms by which fast grain-boundary diffusion occurs are not well estab- lished at present There is extensive evidence that a net diffusional transport of atoms can be induced along grain boundaries, ruling out the ring mechanism and implicating defect-mediated mechanisms as responsible for grain-boundary diffu- sion [13] Due to the small amount of material present in the grain boundary, it has not been possible, so far, to gain critical information about defect-mediated processes using experimental techniques Recourse has been made to computer simulations which indicate that vacancy and interstitial point defects can exist in the boundary core as localized bona fide point defects (see the review by Sutton and Balluffi [4]) Calculations also show that their formation and migration energies are often lower than in the bulk crystal Figure 9.9 shows the calculated trajectory of
a vacancy in the core of a large-angle tilt grain boundary in b.c.c Fe Calculations showed that vacancies were more numerous and jump faster in the grain boundary than in the crystal, indicating a vacancy mechanism for diffusion in this particular boundary However, there is an infinite number of different types of boundaries, and computer simulations for other types of boundaries indicate that the dominant mechanism in some cases may involve interstitial defects [4, 121
During defect-mediated grain-boundary diffusion, an atom diffusing in the core will move between the various types of sites in the core Because various types of jumps have different activation energies, the overall diffusion rate is not controlled
by a single activation energy Arrhenius plots for grain-boundary diffusion therefore should exhibit at least some curvature However, when the available data are of only moderate accuracy and exist over only limited temperature ranges, such curvature may be difficult to detect This has been the case so far with grain-boundary diffusion data, and the straight-line representation of the data in the Arrhenius
Mechanism of Fast Grain-Boundary Diffusion
Boundary midplane
[ooi]
Figure 9.9: Calculated atom jumps in the core of a C5 symmetric (001) tilt boundary in b.c.c Fe A pair-potential-molecular-dynamics model was employed For purposes of clarity the scales used in the figure are [I301 : [310] : [OOT] = 1 : 1 : 5 All jumps occurred in the fast-diffusing core region Along the bottom, a vacancy was inserted at B and subse uently executed the series of jumps shown The tra'ectory was essentially parallel to the t j t axis Near the center of the figure, an atom in a b site jumped into an interstitial site at I At the top an atom jumped between B , I and B' sites From Balluffi et al [14]
Trang 2222 CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS
plot in Fig 9.3 must be regarded as an approximation that yields an effective activation energy, E B , for the temperature range of the data Some evidence for curvature of Arrhenius plots for grain-boundary diffusion has been reviewed [4]
9.3 DIFFUSION ALONG DISLOCATIONS
As with grain boundaries, dislocation-diffusion rates vary with dislocation struc- ture, and there is some evidence that the rate is larger along a dislocation in the edge orientation than in the screw orientation [15] In general, dislocations in close- packed metals relax by dissociating into partial dislocations connected by ribbons
of stacking fault as in Fig 9.10 [16] The degree of dissociation is controlled by the stacking fault energy Dislocations in A1 are essentially nondissociated because
of its high stacking fault energy, whereas dislocations in Ag are highly dissociated because of its low stacking fault energy The data in Fig 9.1 (averaged over the available dislocation orientations) indicate that the diffusion rate along dislocations
in f.c.c metals decreases as the degree of dislocation dissociation into partial dislo- cations increases This effect of dissociation on the diffusion rate may be expected because the core material in the more relaxed partial dislocations is not as strongly perturbed and “loosened up’’ for fast diffusion, as in perfect dislocations
In Fig 9.1, *DD for nondissociated dislocations is practically equal to *DB, which indicates that the diffusion processes in nondissociated dislocation cores and large- angle grain boundaries are probably quite similar Evidence for this conclusion also comes from the observation that dislocations can support a net diffusional transport
of atoms due to self-diffusion [15] As with grain boundaries, this supports a defect- mediated mechanism
The overall self-diffusion in a dislocated crystal containing dislocations through- out its volume can be classified into the same general types of regimes as for a polycrystal containing grain boundaries (see Section 9.2.1) Again, the diffusion may be multiple or isolated, with or without diffusion in the lattice, and the dis- locations may be stationary or moving However, the critical parameters include
*DD rather than *DB and the dislocation density rather than the grain size The multiple-diffusion regime for a dislocated crystal is analyzed in Exercise 9.1 Figure 9.11 shows a typical diffusion penetration curve for tracer self-diffusion into a dislocated single crystal from an instantaneous plane source at the sur- face [17] In the region near the surface, diffusion through the crystal directly from the surface source is dominant However, at depths beyond the range at
,Stacking fault
ribbon
Partial f 2 Partial dislocation 1 dislocation 2
Figure 9.10:
partial dislocations separated by a ribbon of stacking fault
Dissociated lattice dislocation in f.c.c metal The structure consists of two
Trang 39.4 FREE SURFACES 223
Dislocation pipe diffusion
C
e
Penetration depth -w
Figure 9.11: Typical penetration curve for tracer self-diffusion from a free surface at
tracer concentration csurf into a single crystal containing dislocations Transport near the
surface is dominated by diffusion in the bulk; at greater depths, dislocation pipe diffusion is the major transport path
which atoms can be delivered by crystal diffusion alone, long penetrating “tails” are present, due to fast diffusion down dislocations with some concurrent spreading into the adjacent lattice and no overlap of the diffusion fields of adjacent dislo- cations This behavior corresponds to the dislocation version of the B regime in Fig 9.4
9.4 DIFFUSION ALONG FREE SURFACES
The general macroscopic features of fast diffusion along free surfaces have many
of the same features as diffusion along grain boundaries because the fast-diffusion path is again a thin slab of high diffusivity, and a diffusing species can diffuse in both the surface slab and the crystal and enter or leave either region For example,
if a given species is diffusing rapidly along the surface, it may leak into the adjoining crystal just as during type-B kinetics for diffusion along grain boundaries In fact, the mathematical treatments of this phenomenon in the two cases are similar The structure of crystalline surfaces is described briefly in Sections 9.1 and 12.2.1 and in Appendix B All surfaces have a tendency to undergo a “roughening” tran- sition at elevated temperatures and so become general Even though a considerable effort has been made, many aspects of the atomistic details of surface diffusion are still unknowns6
For singular and vicinal surfaces at relatively low temperatures, surface-defect- mediated mechanisms involving single jumps of adatoms and surface vacancies are
p r e d ~ m i n a n t ~ Calculations indicate that the formation energies of these defects are of roughly comparable magnitude and depend upon the surface inclination [i.e.,
(hkl)] Energies of migration on the surface have also been calculated, and in most cases, the adatom moves with more difficulty Also, as might be expected, the diffusion on most surfaces is anisotropic because of their low two-dimensional symmetry When the surface structure consists of parallel rows of closely spaced atoms, separated by somewhat larger inter-row distances, diffusion is usually easier parallel to the dense rows than across them In some cases, it appears that the
6 0 u r discussion follows reviews by of Shewmon [18] and Bocquet et al [19]
7Adatoms, surface vacancies, and other features of surface structure are depicted in Fig 12.1
Trang 4224 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS
transverse diffusion occurs by a replacement mechanism in which an atom lying between dense rows diffuses across a row by replacing an atom in the row and pushing the displaced atom into the next valley between dense rows Repetition of this process results in a mechanism that resembles the bulk interstitialcy mechanism described in Section 8.1.3 In addition, for vicinal surfaces, diffusion rates along and over ledges differs from those in the nearby singular regions
At more elevated temperatures, the diffusion mechanisms become more complex and jumps to more distant sites occur, as do collective jumps via multiple defects
At still higher temperatures, adatoms apparently become delocalized and spend significant fractions of their time in “flight” rather than in normal localized states
In many cases, the Arrhenius plot becomes curved at these temperatures (as in Fig 9.1), due to the onset of these new mechanisms Also, the diffusion becomes more isotropic and less dependent on the surface orientation
The mechanisms above allow rapid diffusional transport of atoms along the sur- face We discuss the role of surface diffusion in the morphological evolution of
surfaces and pores during sintering in Chapters 14 and 16, respectively
Bibliography
1 N.A Gjostein Short circuit diffusion In Diffusion, pages 241-274 American Society for Metals, Metals Park, OH, 1973
2 I Herbeuval and M Biscondi Diffusion of zinc in grains of symmetric flexion of
aluminum Can Metall Quart., 13(1):171-175, 1974
Diffusion in ceramics In R.W Cahn, P Haasen, and E Kramer, editors, Materials Science and Technology-A Comprehensive Treatment, volume 11,
pages 295-337, Wienheim, Germany, 1994 VCH Publishers
4 A.P Sutton and R.W Balluffi Interfaces in Crystalline Materials Oxford University Press, Oxford, 1996
5 E.W Hart On the role of dislocations in bulk diffusion Acta Metall., 5(10):597,
8 J.W Cahn and R.W Balluffi Diffusional mass-transport in polycrystals containing stationary or migrating grain boundaries Scripta Metall Mater., 13(6):499-502, 1979
9 I Kaur and W Gust Fundamentals of Grain and Interphase Boundary Diffusion Ziegler Press, Stuttgart, 1989
10 J.C Fisher Calculation of diffusion penetration curves for surface and grain boundary diffusion J Appl Phys., 22(1):74-77, 1951
11 J.C.M Hwang and R.W Balluffi Measurement of grain-boundary diffusion at low- temperatures by the surface accumulation method 1 Method and analysis J Appl
12 Q Ma and R.W Balluffi Diffusion along [OOl] tilt boundaries in the Au/Ag system
1 Experimental results Acta Metall., 41(1):133-141, 1993
13 R.W Balluffi Grain boundary diffusion mechanisms in metals In G.E Murch and
A S Nowick, editors, Diffusion in Crystalline Solids, pages 319-377, Orlando, FL,
1984 Academic Press
3 A Atkinson
Phys., 50(3):1339-1348, 1979
Trang 5EXERCISES 225
14 R.W Balluffi, T Kwok, P.D Bristowe, A Brokman, P.S Ho, and S Yip Deter- mination of the vacancy mechanism for grain-boundary self-diffusion by computer simulation Scripta Metall Mater., 15(8):951-956, 1981
On measurements of self diffusion rates along dislocations in f.c.c metals Phys Status Solidi, 42(1):11-34, 1970
16 R.E Reed-Hill and R Abbaschian Physical Metallurgy Principles PWS-Kent, Boston, 1992
17 Y.K Ho and P.L Pratt Dislocation pipe diffusion in sodium chloride crystals Radiat
18 P Shewmon Diffusion in Solids The Minerals, Metals and Materials Society, War- rendale, PA, 1989
19 J.L Bocquet, G Brebec, and Y Limoge Diffusion in metals and alloys In R.W Cahn and P Haasen, editors, Physical Metallurgy, pages 535-668 North-Holland, Amsterdam, 2nd edition, 1996
Derive an equation similar to Eq 9.4 for the effective bulk self-diffusivity, (*D),
in the presence of fast dislocation diffusion Assume that the dislocations are present at a density, p, corresponding to the dislocation line length in a unit volume of material
Solution During self-diffusion, the fraction of the time that a diffusing atom spends
in dislocation cores is equal t o the fraction o f all available sites that are located in
the dislocation cores This fraction will be 7 = p7d2/4 The mean-square displace- ment due t o self-diffusion along the dislocations is then *DDqt, while the corresponding displacement in the crystal is *DxL(l - 7)t Therefore,
(*D)t = * D X L ( l - 7)t + *DD7t (9.17) and because 7 << 1,
cf/cf" = k = constant, where cf is the solute concentration in the disloca-
tion cores and cfL is the solute concentration in the crystal
Solution Because the fraction of solute sites in the dislocations is small, the number
of occupied solute-atom sites (per unit volume) in the crystal is c g L , and the number of
Trang 6226 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS
occupied sites in the dislocations is pd2kc?XL/4 The fraction of time that a diffusing solute atom spends in dislocation cores is then 17 = p7d2k/4 Therefore, following the same argument as in Exercise 9.1,
(9.21) for solute diffusion
Solution As indicated in the text, Eq 9.9 must have the form of Eq 9.15 in order
t o satisfy the segregation condition k = cf/c?” at the boundary slab Equation 9.10
%ource” surface to the “accumulation” surface is controlled by the diffusion rate along the transverse boundaries If the diffusant, designated component
2, is initially present on the source surface and absent on the accumulation surface and the specimen is isothermally diffused, a quasi-steady rate of ac- cumulation of the diffusant is observed on the accumulation surface after a short initial transient Derive a relationship between the rate of accumulation
Trang 7Transport of diffusant through a thin polycrystalline film by grain-boundary
Solution Because of the fast surface diffusion, the concentrations of the diffusant
on both surfaces are essentially uniform over their areas After the initial transient, the quasi-steady rate (per unit area of surface) a t which the diffusant diffuses along the transverse boundaries between the two surfaces is
Here, d is the average grain size of the columnar grains, JB is the diffusional flux along the grain boundaries, dcB/dx = [cB(0) - cB(I)] / I , where cB(0) and cB(I) are the diffusant concentrations in the boundaries at the source surface and accumulation surface, respectively, and I is the specimen thickness In the early stages, c B ( I ) = 0
and, therefore, t o a good approximation,
Solution Equation 9.18 may be solved for p in the form
(9.28)
It is estimated from Fig 9.1 that *DD(dissoc)/*DXL = 3 x lo6 at Tm/T = 2.0 Also,
6 % 6 x lo-* cm-* Using these values and (*D)/*DxL = 2 in Eq 9.28,
p E 10' cmP2 Therefore, it appears that the dislocations could make a significant contribution t o diffusion under many common conditions
Trang 8228 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS
9.6 The asymmetric small-angle tilt boundary in Fig B.5a consists of an array
of parallel edge dislocations running parallel to the tilt axis During diffusion they will act as fast diffusion “pipes.” Show that fast self-diffusion along this boundary parallel to the tilt axis can be described by an overall boundary diffusivity,
(9.31) where the first bracketed term is the flux along a single pipe and the second is the number o f pipes per unit area of the boundary slab The desired expression is obtained
by equating this result with J = - *DB(para) &/ax and solving for *DB
9.7 Self-diffusion along the boundary in Exercise 9.6 is highly anisotropic because
diffusion along the tilt axis (parallel to the dislocations) is much greater than diffusion transverse to it (i.e., perpendicular to the dislocations but still in the boundary plane) Find an expression for the anisotropy factor,
*D (para)
where *DB (transv) is the boundary diffusivity in the transverse direction Solution The transverse diffusion rate is controlled by the relatively slow crystal diffusion rate because the diffusing atoms must traverse the patches of perfect crystal between the dislocation pipes Therefore, when the dislocations are discretely spaced,
a good approximation is the simple result
*DB (para) - - *DB (para)
Trang 9The understanding of diffusion in many noncrystalline materials has lagged be- hind the understanding of diffusion in crystalline material, and a unified treatment
of diffusion in noncrystalline materials is impossible because of its wide range of mechanisms and phenomena In many cases: basic mechanisms are still controver- sial or even unknown We therefore focus on selected cases, although some of the models discussed are still under development and not yet firmly established
10.1 FREE-VOLUME MODEL FOR SELF-DIFFUSION IN LIQUIDS
Self-diffusion in simple monatomic liquids at temperatures well above their glass- transition temperatures may be interpreted in a simple manner.' Within such liquids, regions with free volume appear due to displacement fluctuations Occa- sionally, the fluctuations are large enough to permit diffusive displacements
'This section closely follows Cohen and Turnbull's original derivation [l] The original paper should be consulted for further details
Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter 229
Copyright @ 2005 John Wiley & Sons, Inc
Trang 10230 CHAPTER 10: DIFFUSION IN NONCRYSTALLINE MATERIALS
The hard-sphere model for the liquid serves as a reasonably good approximation for the atomic interactions [2] Here, the potential energy between any pair of approaching particles is assumed to be constant until they touch, at which point it becomes infinite On average, the particles in the liquid maintain a volume larger than that which they would have if they all touched; the resulting volume difference
is the free volume Each particle effectively traverses a small confined volume within which the interatomic potentials are essentially flat [3] The average velocity of a particle in the region of flat potential inside the confining volume is the same as the velocity of a gas particle Most of the time a particular particle is confined
to a particular region However, there will occasionally be a fluctuation in local density that opens a space large enough to permit a considerable displacement of the particle If another particle jumps into that space before the displaced first particle returns, a diffusive-type jump will have occurred Diffusion therefore occurs as a result of the redistribution of the free volume that occurs at essentially constant energy because of the flatness of the interatomic potentials
According to the kinetic theory of gases, the self-diffusivity of a hard-sphere gas is given by *DG = (2/5)(u)L, where (u) is the average velocity and L is the mean free path [4] Because the mean free path of a confined particle in the liquid is about equal to the diameter of its confining volume, the contribution of the confined particle to the self-diffusivity of the liquid may be written
’
where u ( V ) is the diameter of the confining volume, V is the free volume associ- ated with the particle, (u) is the average velocity of the particle, and C,,,, is a geometrical constant.It is reasonable to assume that the diffusivity is very small,
*D(V) = 0, unless the local free volume V exceeds a critical volume, Vcrit There- fore, the overall diffusivity may be expressed
(10.2)
where p(V) dV is the free volume’s probability that it lies between V and V + dV
To determine this probability distribution, consider a system containing n/ particles and divide the total range of possible free volumes for a particle into bins indexed
by i Let Ni(V,) be the number of particles with free volume V, If Vfree is the total free volume, the condition
i
must hold The factor y accounts for all free-volume overlap between adjacent particles y lies between zero and one because of the physical limits of complete and no overlap; its value is probably closer to one The total number of particles,
Trang 1110 1 FREE-VOLUME MODEL FOR LIQUIDS 231
for bin populations given by Ni The equilibrium probability distribution in Eq 10.2
is the continuum limit of the bin populations Ni that maximize Sconf subject to constraints, Eqs 10.3 and 10.4 Introducing Lagrange multipliers p and X for the total free volume and fixed-number constraints, the extremal conditions are
which, using Stirling's formula Ni! x Ni In Ni and the limit of Ni >> 1, reduces to
shows that the probability distribution p(V) must be proportional to y exp[-ypV/k]
= y exp[-yV/(Rfree)] The proportionality factor can be determined by setting the sum of probabilities equal to one, and
(10.11)
The probability distribution, Eq 10.11, can be used in Eq 10.2 as an estimate for evaluating *DL Above the critical free volume Vcrit, *D(V) is probably nearly constant; therefore,
(10.12)
*D(vcrit) e - y ~ C P ' t / ( ~ f r e e ) = c geom U(vcrit)(u) e- y ~ c r i t /(afree)
Equation 10.12 matches diffusivities measured in simple liquids if the character- istic "cage" diameter, u(Vcrit), is approximately the particle diameter and yVcrit is approximated by the particle volume [l] *DL is not thermally activated-it does not exhibit Arrhenius behavior as does, for example, the diffusivity in crystals, because (u) 0; T1/*and (afree) increases approximately linearly with T [4] Less
approximate models for diffusion in liquids have been reviewed by Frohberg [5]
Trang 12232 CHAPTER 10 DIFFUSION IN NONCRYSTALLINE MATERIALS
10.2 DIFFUSION IN AMORPHOUS METALS
Amorphous metallic alloys (metallic glasses) can be produced by rapid cooling (quenching) from the liquid phase If the initially stable liquid avoids solidification
by crystallization by being quenched rapidly below its ordinary melting tempera- ture, T,, it first becomes a supercooled liquid, and then, at a still lower tempera- ture, it undergoes a glass transition to an amorphous glassy state as in Fig 10.1 Occurring over a range of temperatures that is dependent upon the cooling rate, the glass transition is characterized by an abrupt change in the rate at which the volume and other physical properties change with decreasing temperature The glass transition temperature, T,, which occurs at a given cooling rate, is obtained from the intersection of the extrapolated cooling curves from well above and well below the transition Because the glass transition occurs at a higher temperature during rapid cooling than during slow cooling, less free volume remains in glasses formed at low temperatures Below the glass transition temperature, the combined effects of the low temperature and the loss of free volume cause the initially liquid material to lose its characteristic fluidity and become relatively rigid and unable to reorganize itself quickly as the temperature is decreased further (i.e., it becomes a frozen-in glass)
system attempts to relax and equilibrate without crystallizing The free volume
is mobile and is presumably annihilated when it encounters regions of higher than average atomic density [6, 71 The self-diffusivity that is measured during such annealing decreases initially However, it eventually reaches an asymptotic value and becomes time independent, as in Fig 10.2 The asymptotic value of the dif-
fusivity is then that of the relaxed glassy state in which the supersaturated excess volume has annealed out This dense structure is,randomly packed, and the atoms are arranged with the highest density compatible with their hard-sphere radii and
Trang 1310.2: DIFFUSION IN AMORPHOUS METALS 233
Figure 10.2: Self-diffusion coefficient of 5gFe in amorphous Fe40Ni40B20 during isothermal annealing below T, after rapid quenching from liquid state as in Fig 10.1 Arrows indicate different time scales used at each temperature Reprinted from "Tracer Diffusion of Fe-
59 in Amorphous F e m N i m B m , " J Horvath and H Mehrer, 1986, Crystal Lattice Defects and Amorphous Materials, Taylor and Francis, http://www.tandf.co.uk/journals/ [8]
lack of translational symmetry Locally, the atoms form various polyhedral units in definite ratios with neither microcrystallites nor large holes present Even though relaxed, this structure is still metastable with respect to the crystalline state Extensive measurements show that self-diffusivities in the relaxed glassy state are time independent and closely exhibit Arrhenius behavior (i.e., ln*D, vs 1/T plots appear as essentially straight lines) [8-111 The diffusion therefore is ther- mally activated (in contrast to self-diffusion in the liquid above Tg as described in Section 10.1)
The mechanism by which the self-diffusion in the relaxed state occurs is not firmly established at present However, there are reasons to believe that for certain atoms in glassy systems, self-diffusion occurs by a direct collective mechanism and
is not aided by point defects in thermal equilibrium as in the vacancy mechanism for self-diffusion in crystals (Section 8.2.1) .2 These reasons include:
0 Sudden changes in temperature during diffusion cause instantaneous changes
in the diffusivity [9, 121 This result is unexpected if diffusion occurs by a point-defect mechanism because significant time is required to obtain the new equilibrium defect concentrations corresponding to the temperature changes
0 The activation volume for diffusion, as measured by the pressure dependence
of the diffusivity, is zero to within experimental accuracy [13, 141 This is unexpected for defect-mediated diffusion, as in such cases, the activation vol- ume for diffusion should consist of the sum of the volume of formation of the defect and the activation volume for the defect migration, and this is usually measurable
0 Computer simulations of the diffusion process in relaxed F e Z r glasses re- veal diffusion which takes place directly via thermally activated displacement
2The ring mechanism in Section 8.1.1 is an example of a direct mechanism
Trang 14234 CHAPTER 10 DIFFUSION IN NONCRYSTALLINE MATERIALS
chains like that in Fig 10.3 [7, 9, 151 These chains do not start at localized
Figure 10.3:
displacement chains
hlechanism of diffusion in amorphous glasses by thermally activated
point defects but in regions where the initial density deviations are small Fur- thermore, when the displacement sequences are completed, any large-density deviations disperse gradually and do not leave behind localized point defects The entire displacement process, from beginning to end, involves a relatively large number of atoms and, therefore, is of a collective nature Such a direct collective diffusion process, which is spread over a considerable volume and involves relatively little ion-core overlap and repulsion, presumably occurs with relatively little volume change, in agreement with the small activation volume cited above
0 The observation that the self-diffusion exhibits Arrhenius behavior is consis- tent with a direct collective mechanism because the thermally activated dis- placement chains are spread over a considerable number of atomic distances Irregularities in the disordered glassy structure are therefore averaged in the activated state, and all activation energies for displacements are then closely the same
0 No isotope effect is observed (see Eq 8.31) during self-diffusion in relaxed glasses [16, 171 In tracer self-diffusion studies of crystalline materials, where the atomic displacements that lead to vacancy migration and diffusion are highly localized, the harmonic model for the isotope effect is justified How- ever, if the migration process involves a relatively large number of atoms and is highly collective, this estimate of the effective attempt frequency is
no longer valid Instead, it is expected that two isotopes diffuse at close to the same rates because the mass difference of the two isotopes hardly affects their jump frequencies when relatively large numbers of atoms are strongly involved in the activated state
Further discussion of self-diffusion in relaxed metallic glasses and other disor- dered systems may be found in key articles [7, 10, 14, 18, 191
10.2.2 Diffusion of Small Interstitial Solute Atoms
Small solute atoms in the interstices between the larger host atoms in a relaxed metallic glass diffuse by the direct interstitial mechanism (see Section 8.1.4) The host atoms can be regarded as immobile A classic example is the diffusion of H solute atoms in glassy Pd80Si20 For this system, a simplified model that retains the essential physics of a thermally activated diffusion process in disordered systems is used to interpret experimental measurements [20-221
Trang 1510.2: DIFFUSION IN AMORPHOUS METALS 235
W
Because many different types of interstitial sites exist in the disordered glassy structure, the energy of the system varies as an interstitial atom jumps between the sites The trace of the energy during successive jumps has the general form illustrated in Fig 10.4a, where, for simplicity, the energy at each saddle point is assumed to be the same [20, 22, 231 This approximation has the realistic feature that a diffusing interstitial encounters sites of varying energy and jump barriers of various heights
The following quantities will be of use in describing the interstitial self-diffusion and intrinsic chemical diffusion:
W
N = total number of interstitial sites
p = fraction of all interstitial sites that are occupied
*p = fraction of all sites that are tracer-interstitial occupied
p i = fraction of all sites that are type k sites (Fig 10.4a)
pk = fraction of all sites that are occupied type k sites
*pk = fraction of all sites that are tracer-occupied type k sites
p ( k ) = fraction of type k sites that are occupied
(10.14)
Figure 10.4: (a) The energy variation of an amorphous glass with the displacement
of a diffusing interstitial atom as it jumps between successive interstitial sites (b) A plot similar to (a) for interstitial jumping in a hypothetical material containing only sites of the reference state and having activation energies corresponding to E"
Trang 16236 CHAPTER 10: DIFFUSION IN NONCRYSTALLINE MATERIALS
and the partial concentration, p k , can be written
( 10.15) Also,
k
A model for the tracer self-diffusivity of the interstitials is now developed for a system in which the total concentration of inert interstitials and chemically similar radioactive-tracer interstitials is constant throughout the specimen but there is a gradient in both concentrations Since the inert and tracer interstitials are randomly intermixed in each local region,
In a typical tracer self-diffusion experiment, the tracer concentration probability,
*p, depends upon position, whereas the total interstitial concentration probability,
p , does not
An expression for the tracer self-diffusivity, of the interstitials, *D, can be de- rived by employing the same basic method applied to a crystalline material to obtain the self-diffusivity given by Eq 8.19 This involves finding the net flux of tracer interstitials jumping through a unit cross-sectional area in the diffusion zone perpendicular to the concentration gradient For a crystalline material, this flux is found by considering the jumping of atoms between well-defined adjacent atomic planes lying parallel to the unit cross section This approach, however, cannot be applied to a glassy material because of the disorder that is present, and therefore the flux must be determined by a slightly modified method Consider two thin slabs
in the material, each of thickness Ax and having unit area, lying perpendicular to the concentration gradient along x Slab 1 extends from zo - Ax to 20, and slab 2 extends from xo to zo + Ax Let I?;, be the jump rate of a tracer interstitial from
an i site to an adjacent empty i site According to Fig 1 0 4 ~ ~ the activation energy for such a jump will be Go + Eo - Gk, so
r/ kz - - ve-(Go+Eo-Gk)/(kT) (10.19) The rate of k-to-i site jumping originating in slab 1 is proportional to the quantity
(P/(Q))Ax *pk(pp - p z ) q Z In this expression, (a) is the average atomic volume
in the glass, p is the ratio of interstitial sites to atoms, and @/(a)) is the number
of interstitial sites per unit volume (p," - p z ) is the probability that a site is an empty i site Making the approximation that all jump distances in the disordered material are of the same magnitude and equal to Ax, the net number of jumps of all types crossing the x = zo plane per unit time in the x direction is
(10.20)
Trang 1710.2 DIFFUSION IN AMORPHOUS METALS 237
where g is a purely geometrical constant and the double summation ensures that all types of different jumps between the various sites are included The first term represents the jumps that originate in slab 1 and cross x = xo in the x direction, while the second term represents the jumps that originate in slab 2 and cross x = xo
in the -x direction During tracer self-diffusion, the total concentration of inert and tracer interstitial atoms is constant, so both p and (pg - pi) are independent
of 2 Making the usual Taylor expansion to evaluate the small difference between the terms and using Eqs 10.18 and 10.19,
Using Eq 10.16 and the fact that z i p ; = 1, c i ( p g -pi) = 1 - p Also, using this result and Eq 10.15,
Putting these results into Eq 10.21 then yields
Equation 10.23 can be put into the simpler form
(10.24)
where *D; = g (Az)' vexp[-E"/(kT)] is the self-diffusivity in a hypothetical ma-
terial that contains only sites of the reference state with the energy Go, and in
which jumps may occur between them with the activation energy E", as illustrated
in Fig 10.4b Equation 10.24 is a Fick's-law equation with a tracer interstitial self-diffusivity corresponding to
(10.25)
Having this result, an expression can be obtained for the "intrinsic" chemical diffusivity, D I , which describes the diffusion arising from an inert-interstitial con- centration gradient According to Eqs 3.35 and 3.42, the flux in such a system
of position Putting Eq 10.27 into Eq 10.26 leads to the Fick's-law-type expression
( 10.28)
Trang 18238 CHAPTER 10: DIFFUSION IN NONCRYSTALLINE MATERIALS
and, therefore,
For tracer self-diffusion, a similar initial equation for the flux is
(10.29)
(10.30) However, in this system, the ideal free energy of mixing of the inert and tracer interstitials is the only component that varies with x By taking the derivative of the free energy to obtain the chemical potential, the x-dependent component of the chemical potential of the tracer interstitials is simply kT ln(*p/p), and there- fore, because p is constant, (a*p/dx) = (kT/*p)(d*p/dx) Putting this result into
Eq 10.30,
(10.31) which is a Fick’s-law-type expression with an interstitial tracer self-diffusivity given
Neglecting any small isotope effect, MI =*MI, and comparing Eqs 10.29 and 10.32,
bY
(10.33)
which is of the same form as Eq 3.13
The model above has been compared to experimental results for the diffusion of
H in glassy PdsoSizo by Kirchheim and coworkers [21, 221 DI increases strongly with increasing H concentration as seen in Fig 10<5 By assuming that the energies
of the interstitial sites follow a Gaussian distribution around a mean value, good agreement was obtained between the model and experiment The increase of DI
Trang 1910 3 SMALL ATOMS (MOLECULES) IN GLASSY POLYMERS 239
with p arises from the successive saturation of the lower-energy sites as the concen-
tration is increased This causes a progressive decrease of the activation energy and
a corresponding increase in the diffusivity For example, at very low concentrations, essentially all of the interstitials become trapped at the lowest-energy sites and they engage in long-range diffusion only with difficulty Further aspects are discussed elsewhere [22]
Figure 10.6 plots the tracer diffusivity data for a number of solute species in glassy Ni80Zr50 as a function of their metallic radius The diffusivity increases rapidly as the metallic radius decreases The relatively rapid diffusion of the small atoms in this case may result from the fact that they diffuse by the interstitial mechanism [lo, 181
of their size (as measured by their metallic radii) [25] Reprinted, by permission, from H Hahn and R.S Averback "Dependence of tracer diffusion on atomic size in amorphous Ni-Zr," Phys Rev B, Vol 37,
p 6534 Copyright 01988 by the American Physical Society
Some small atoms and molecules, such as He, Ar, COz, and Nz, dissolve in glassy polymers from the gas phase These particles then diffuse in the bulk polymer presumably by occupying interstices in the glassy structure and jumping between them by the direct interstitial mechanism The solubilities increase with increasing partial pressure, and the behavior observed can be well explained on the basis of
a model in which the dissolved species occupy interstitial sites, the site occupancy obeys Fermi-Dirac statistics, and the site energies are distributed about a mean value in the form of a Gaussian distribution [26, 271 The corresponding diffusivities
of these species increase with increasing concentration, in a manner similar to the diffusion of small solute atoms in amorphous metals This behavior can be explained
by the same interstitial diffusion model Here, the diffusing particles must again occupy progressively higher-energy sites as their concentration increases, causing the average activation energy for diffusion to decrease and the diffusivity to increase The diffusion of small particles in glassy polymers therefore appears to be quite similar to that in glassy metals
Trang 20240 CHAPTER 10 DIFFUSION IN NONCRYSTALLINE MATERIALS
10.4 DIFFUSION OF ALKALI IONS I N NETWORK OXIDE GLASSES
The structure of a pure oxide network glass having stoichiometry G203, free of any alkali ions, is illustrated in Fig 10.7a [28] In this structure, cations are three- coordinated and the oxygen anions are two-coordinated In three-dimensional silica glass, each glass-forming Si4+ cation is enclosed in a polyhedron of oxygen anions, and these polyhedra are arranged in a network lacking special symmetry and peri- odicity The oxygen polyhedra share corners, not edges or faces, and each oxygen ion is covalently bonded to no more than two cations
The oxide glass structure changes significantly when modzfyzng alkali ions are added, as in Fig 10.7b where the G203 glass has been altered by adding a signifi- cant amount of the network modifier M 2 0 The structure accommodates the net- work modifier M+ ions by substitution of three one-coordinated modifier cations for one three-coordinated glass-forming ion In three-dimensional silica glass, the ad- dition of Na ions (e.g., via Na2O) causes oxygen ions previously covalently bonded
to two of the glass-forming Si4+ cations between which it formed a brzdge to reduce this bonding so that they become bonded to only one glass-forming cation These oxygen ions, called nonbrzdgzng oxygens, possess an effective negative charge The corresponding positively charged Na’ ions are then ionically bonded to the non- bridging oxygens, resulting in a partly covalent and partly ionic overall structure Studies show that in silica glasses with low concentrations of NazO, the ionically bound material exists in the form of small isolated patches or lakes As the concen- tration increases, these patches link and eventually form a network of continuous channels [29-321 Continuous percolation networks are present at and above a per- colation threshold of about 16 vol % of modifier
Na’ ions are highly mobile compared to the glass-forming components and pos- sess a diffusivity which follows Arrhenius behavior [21, 26, 29, 31, 331 Furthermore, the activation energy for diffusion decreases markedly (and the diffusivity increases correspondingly) as the modifier concentration is increased, as in Fig 10.8 The
Figure 10.7: (a) Two-dimensional schematic of pure, oxide network glass of composition
Gz03 Small open circles are glass-forming cations G 3 + Large open circles represent oxygen anions From Kingery et a1 [28] (b) Schematic of glass as modified by the addition of alkali hl+ cations (filled small circles) At high modifier-ion content, the modifier ions aggregate and form high-diffusivity “lakes” or channels in the glass Adapted from Greaves [29]