Kinetics of Materials - R. Balluff S. Allen W. Carter (Wiley 2005) WW Part 7 pot

9.2: 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 tjt 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]. 222 CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS plot in Fig. 9.3 must be regarded as an approximation that yields an effective activation energy, EB, 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 9.4 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 pred~minant.~ 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 60ur 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. 12.1 224 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, 1957. 6. L.G. Harrison. Influence of dislocations on diffusion kinetics in solids with particular reference to the alkali halides. Trans. Faraday Soc., 57(7):1191-1199, 1961. 7. D. Turnbull. Grain boundary and surface diffusion. In J.H. Holloman, editor, Atom Movements, pages 129-151, Cleveland, OH, 1951. American Society for Metals. Spe- cial Volume of ASM. 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 AS. Nowick, editors, Diffusion in Crystalline Solids, pages 319-377, Orlando, FL, 1984. Academic Press. 3. A. Atkinson. Phys., 50(3):1339-1348, 1979. EXERCISES 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. 15. R.W. Balluffi. Eff., 75~183-192, 1983. EXERCISES 9.1 In a Type-A regime, short-circuit grain-boundary self-diffusion can enhance the effective bulk self-diffusivity according to Eq. 9.4. A density of lattice dislocations distributed throughout a bulk single crystal can have a similar effect if the crystal diffusion distance for the diffusing atoms is large compared with the dislocation spacing. 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 to the fraction of all available sites that are located in the dislocation cores. This fraction will be 7 = p7d2/4. The mean-square displace- ment due to self-diffusion along the dislocations is then *DDqt, while the corresponding displacement in the crystal is *DxL(l - 7)t. Therefore, (*D)t = *DXL(l - 7)t + *DD7t (9.17) and because 7 << 1, (9.18) p7rP (*D) = *DxL + - *DD 4 9.2 Exercise 9.1 yielded an expression, Eq. 9.18, for the enhancement of the ef- fective bulk self-diffusivity due to fast self-diffusion along dislocations present in the material at the density, p. Find a corresponding expression for the enhancement of the effective bulk self-diffusivity of solute atoms due to fast solute self-diffusion along dislocations. Assume that the solute atoms segre- gate to the dislocations according to simple McLean-type segregation where 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 cgL, and the number of 226 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, (*Dz)t = *D,””(l - v)t + *Dpqt (9.19) and thus (*D2) = *DfL + @ *Df (9.20) 4 9.3 For Type-B diffusion along a grain boundary, Eq. 9.9, which holds for self- diffusion, takes the form of Eq. 9.15 for solute diffusion when simple McLean- type segregation occurs with cf/cgL = k. Show that this causes Eq. 9.13, which holds for self-diffusion, to take the form (9.21) for solute diffusion. Solution. As indicated in the text, Eq. 9.9 must have the form of Eq. 9.15 in order to satisfy the segregation condition k = cf/c?” at the boundary slab. Equation 9.10 then becomes Equation 9.11 becomes [ - (A) Yl] B c2 (yi,ti) = exp (9.23) Equation 9.12 becomes cz XL (zl,yl,tl) = -exp 1 [- (A)”* YI] [1 -erf -$)I (9.24) k and, finally, Eq. 9.13 becomes (9.25) 9.4 As described in Section 9.2.2, grain-boundary diffusion rates in the Type-C diffusion regime can be measured by the surface-accumulation method illus- trated in Fig. 9.12. Assume that the surface diffusion is much faster than the grain-boundary diffusion and that the rate at which atoms diffuse from the %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 EXERCISES 227 and the parameter SDF that can be used to determine SDf experimentally. Assume that each grain is a square of side d in the plane of the surface. c Source surface Fil thi Accumulation surface Figure 9.12: diffusion. Transport 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) at 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, cB(I) = 0 and, therefore, to a good approximation, B Id dN 6D2 = - - 2cB(0) dt (9.27) All quantities on the right-hand side of Eq. 9.27 are measurable, which allows the determination of bDf [12]. 9.5 Using the result of Exercise 9.1 and data in Fig. 9.1, estimate the density of dissociated dislocations necessary to enhance the average bulk self-diffusivity by a factor of 2 at Tm/2, where T, is the absolute melting temperature of the material. Note: typical dislocation densities in annealed f.c.c. metal crystals are in the range 106-108 cm-2. 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 to diffusion under many common conditions. 228 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, e (9.29) lr 4 where b is the magnitude of the Burgers vector and 6’ is the tilt angle. sin 4 + cos 4 b *DB(para) = - *DD6 Use *DD >> *DL (9.30) Solution. As usual, take the boundary as a slab that is 6 thick. In considering diffusion along the tilt axis, any contribution of the crystal regions in the slab can be neglected and only the contributions of the dislocation pipes are included because *DD >> *DxL. The flux through a unit cross-sectional area of the boundary slab is then (9.31) where the first bracketed term is the flux along a single pipe and the second is the number of 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) *D (transv) (9.32) 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) - *DB(transv) *DxL (9.33) CHAPTER 10 DIFFUSION IN NONCRYSTALLINE M AT E R I A LS Noncrystalline materials exist in many different forms. A huge variety of atomic and molecular structures, ranging from liquids to simple monatomic amorphous structures to network glasses to dense long-chain polymers, are often complex and difficult to describe. Diffusion in such materials occurs by a correspondingly wide variety of mechanisms, and is, in general, considerably more difficult to analyze quantitatively than is diffusion in crystals. The 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. 230 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 ’ *D(V) = Cgeom a(V) (u) (10.1) 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 Vfree = NiV, (10.3) 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, N. is (10.4) i The entropy associated with the number of ways that the free volume can be distributed at constant energy is (10.5) [...]... Reptation of a polymer chain in the presence of fixed obstacles J Chem Phys., 55(2): 57 2-5 79 , 1 971 40 M Doi and S.F Edwards Press, Oxford, 1986 The Theory of Polymer Dynamics Oxford University 41 S.M Allen and E.L Thomas The Structure of Materials John Wiley & Sons, New York, 1999 PART II MOTION OF DISLOCATIONS AND INTERFACES A host of important kinetic processes in materials depends upon the motion of dislocations... Free-volume model of the amorphous phase glass transition J Chem Phys., 34(1):12 0-1 25, 1961 4 E.H Kennard Kinetic Theory of Gases McGraw-Hill, New York, 1938 5 G Frohberg Diffusion in liquid metals between the glass transition and the evaporation temperature Defect and Diffusion Forum, 14 3-1 47: 86 9-8 74 , 19 97 6 M.H Cohen and G.S Grest The nature of the glass transition J Non-Cryst Solids, 6 1-6 2: 74 9 -7 59,... DIFFUSION IN NONCRYSTALLINE MATERIALS 16 K Ratzke, P.W Huppe, and F Faupel Transition from single-jump type to highly cooperative diffusion during structural relaxation of a metallic-glass Phys Rev Lett., 68(15):234 7- 2 349, 1992 17 K Ratzke, A Heesemann, and F Faupel The vanishing isotope effect of cobalt diffusion in Fe39Ni40B21 glass J Phys Condens Matter, 7( 39) :76 6 3 -7 668, 1995 18 R.S Averback Defects and... 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 -1 1 -1 3 -5 -4 -3 -2 -1 log P Figure 10.5: Logarithm of the diffusivity of H in amorphous PdsoSizo as a function of the H concentration probability at different temperatures Points are experimental data The curves are the predictions of the... Minerals, Metals and Materials Society 28 W.D Kingery, H.K Bowen, and D.R Uhlmann Introduction to Ceramics John Wiley & Sons, New York, 1 976 29 G.N Greaves EXAFS and the structure of glass J Non-Cryst Solids, 71 ( 1-3 ):2032 17, 1985 30 C Huang and A.N Cormack The structure of sodium silicate glass J Chem Phys., 93(11):818 0-8 186, 1990 31 G.N Greaves, S.J Gurman, C.R.A Catlow, A.V Chadwick, S Houdewalter,... 50 Conformation of polyethylene (-CHz-CHz -) N Degree of polymerization 10.5:DIFFUSION OF POLYMER CHAINS 243 type of chain of fixed N in different theta solvents and in its melt, the magnitudes of for the chain will differ 10.5.2 Diffusion of Isolated Polymer Chains in Dilute Solutions From a hydrodynamical standpoint, a single isolated chain immersed in a liquid solvent consisting of relatively small... 16(11):4 7- 5 2, 1991 19 F Faupel Diffusion in noncrystalline metallic and organic media Phys Status Solidi A , 134(1): 9-5 9, 1992 20 U Stolz, R Kirchheim, J.E Sadoc, and M Laridjani Hydrogen in liquid-quenched and vapor-quenched amorphous PdgoSizo J Less-Common Muter., 103(1):8 1-9 0, 1984 21 R Kirchheim and U Stolz Modeling tracer diffusion and mobility of interstitials in disordered materials J Non-Cryst... molecules Multiplying Eq 10. 37 by X yields + d2 X dX r n x - dt2 = - 3 X - d t But d2X x-= 1 d dt2 and 2 dt [ +XF, ] (x) d(X2) dt - _ d x- X dX 1d ( X 2 ) = -dt 2 dt Putting these expressions into Eq 10.38 then yields [ md _ _ d (X2) 2 dt ~ dt dX Fd(X2) + =-2 dt XF, (10.38) (10.39) (10.40) (10.41) 244 CHAPTER 10: DIFFUSION IN NONCRYSTALLINEMATERIALS Next, the mean values of these terms over a long period... rootmean-square value of its end-to-end length, = f i b , will be small compared with its length if it were stretched out (i.e., N b , when N is large) Figure 10.9 shows a simulated molecule of polyethylene, (-CH2-CH2 -) N,which approximates a freely jointed configuration The relationship 0 N1/’ derived on the basis of the freely jointed model is : often quite satisfactory despite the approximate nature of. .. case of the glide and climb of an edge dislocation in Fig 11.1, The glide along IC in Fig 11.1 a and b is accomplished by the local conservative shuffling of atoms at the dislocaKinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter Copyright @ 2005 John Wiley & Sons, Inc 253 254 CHAPTER 11: MOTION OF DISLOCATIONS & 0 0 @ (b) Y t Figure 11.1: Glide and climb of . 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. somewhat larger inter-row distances, diffusion is usually easier parallel to the dense rows than across them. In some cases, it appears that the 60ur discussion follows reviews by of Shewmon [18]. 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