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Kinetics and Diffusion Basic concepts in kinetics Kinetics of phase transformations Activation free energy barrier Arrhenius rate equation Diffusion in Solids - Phenomenological description Flux, steady-state diffusion, Fick’s first law Nonsteady-state diffusion, Fick’s second law Atomic mechanisms of diffusion How do atoms move through solids? ● Substitutional diffusion ● Interstitial diffusion ● High diffusivity paths, diffusion along grain boundaries, free surfaces, dislocations Factors that influence diffusion ● Diffusing species and host solid (size, bonding) ● Temperature ● Microstructure MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Kinetics: basic concepts Thermodynamics can be used to predict what is the equilibrium state for a system and to calculate the driving force (ΔG) for a transformation from a metastable state to a stable equilibrium state. How fast the transformation occurs is the question addressed by kinetics. Let’s consider transition from a metastable to the equilibrium state. The transformation between the initial and final states involves rearrangement of atoms – the system should go through a transformation (or reaction) path. Since the initial and final states are metastable or stable ones, the energy of the system increases along any transformation path between them MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Kinetics: basic concepts G Transition path ΔG a G1 ΔG G2 Initial state (metastable) Final state Activated state (equilibrium or another metastable) G1 and G2 are the Gibbs free energies of the initial and final states of the system ΔG = G2 - G1 is the driving force for the transformation. ΔGa is the activation free energy barrier for the transition - the maximum energy along the transformation path relative to the energy of the initial state. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei The concept of thermal activation In order for a system to proceed through the transition path to the equilibrium state, it has to obtain the energy that is sufficient to overcome the activation barrier. The energy can be obtained from thermal fluctuation (when the thermal energy is “pooled together” in a small volume). Statistical mechanics can be used to predict the probability that a system gets an energy that exceeds the activation energy. This process is called thermal activation. The probability of such thermal fluctuation or the rate at which a transformation occurs, depends exponentially on temperature and can be described by equation that is attributed to Swedish chemist Svante Arrhenius*: a ⎞ ⎞ ~ exp⎛ − ΔH a ⎞~ exp⎛ ΔS a ⎞ exp⎛ − ΔH a rate~ exp⎛⎜ − ΔG ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ k BT ⎠ kB ⎠ k BT ⎠ k BT ⎟⎠ ⎝ ⎝ ⎝ ⎝ ΔGa = ΔHa -TΔSa Arrhenius equation can be applied to a wide range of thermally activated processes, including diffusion that we consider next. * Arrhenius equation was first formulated by J. J. Hood on the basis of his studies of the variation of rate constants of some reactions with temperature. Arrhenius demonstrated that it can be applied to any thermally activated process. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei What is diffusion? “Diffusion” is transport through “random walk” - atoms, molecules, electrons, phonons, etc. are moving around randomly in a crystal. This random motion can lead to mass, heat, or charge transport. We will consider atomic diffusion that is involved in most phase transformations. Most kinetic processes in materials involve diffusion. Inhomogeneous materials can become homogeneous by diffusion, compositions of phases can change by diffusion, etc. For an active diffusion to occur, the temperature should be high enough to overcome energy barriers to atomic motion. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Phenomenological description of diffusion: diffusion flux The flux of diffusing atoms, J, is used to quantify how fast diffusion occurs. The flux is defined as either number of atoms diffusing through unit area and per unit time (e.g., atoms/m2second) or in terms of the mass flux - mass of atoms diffusing through unit area per unit time, (e.g., kg/m2-second). A J Let’s consider steady state diffusion - the diffusion flux does not change with time. Example: diffusion of gas molecules through a thin metal plate. gas at pressure P1 gas at pressure P1 < P2 C1 C2 concentration profile MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Steady-State Diffusion: Fick’s first law Fick’s first law: the diffusion flux along direction x is proportional to the concentration gradient dC J = −D dx where D is the diffusion coefficient C dC dx A J x Concentration gradient: dC/dx (Kg m-4) is the slope at a particular point on concentration profile. The minus sign in the equation means that diffusion is down the concentration gradient. Fick’s first law describes steady state flux in a uniform concentration gradient. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Nonsteady-State Diffusion: Fick’s second law (I) In most practical cases steady-state conditions are not established, i.e. concentration gradient is not uniform and varies with both distance and time. Let’s derive the equation that describes nonsteady-state diffusion along the direction x. Consider an element of material with dimensions dx, dy, and dz dV =dx dy dz dAx = dy dz Jx Jx+dx dV x J x = −D ∂ C (x, t ) ∂x x+dx J x + dx = J x + ∂J x dx ∂x The number of particles that diffuse into the volume dV during time dt is JxdAxdt from the left and -Jx+dxdAxdt from the right. From the balance of incoming and outgoing particles: (J x - J x + dx )dA x dt = dC(x, t )dV and using expressions for Jx and Jx+dx we have ∂C(x, t ) dx = J x − J x + dx ∂t ∂ C (x, t ) ∂ ⎛ ∂ C (x, t ) ⎞ = ⎟ ⎜D ∂t ∂x ⎝ ∂x ⎠ MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Nonsteady-State Diffusion: Fick’s second law (II) ∂ ⎛ ∂ C (x, t ) ⎞ ∂ C (x, t ) = ⎜D ⎟ ∂x ⎝ ∂x ∂t ⎠ If dependence of D on x (and C !) is ignored, ∂ 2 C (x, t ) ∂ C (x, t ) = D ∂x 2 ∂t [m-3t-1] = [m2t-1]×[m-3m-2] Fick’s second law relates the rate of change of composition with time to the curvature of the concentration profile: C C x C x x Concentration increases with time in those parts of the system where concentration profile has a positive curvature. And decreases where curvature is negative. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Nonsteady-State Diffusion: Fick’s second law (III) ∂ 2 C (x, t ) ∂ C (x, t ) = D ∂x 2 ∂t C Can we apply this equation to steady state diffusion? MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei x Microscopic picture of diffusion (I) At the microscopic (atomic) level, diffusion is defined by random movement (“random walk”) of the diffusing species (atoms, molecules, Brownian particles). The mobility of the diffusing species can be described by their mean square displacement: r MSD ≡ s ≡ Δr (t)2 2 s r ri (0) r ri (t) i 1 ≡ N N r r 2 ( r (t) − r ( 0 )) ∑ i i i =1 How to relate this microscopic characteristic (the mean square distance of atomic migration) to the macroscopic transport coefficient D used in the phenomenological Fick’s laws? Let’s first consider an intuitive (not rigorous) “derivation” of the relationship between s and D. Suppose that in time t, the average particle moved a distance sx along the direction in which diffusion is occurring. X0 JL CL JR sx CR sx Assuming that the movement of particles is random, half of the particles moved to the left, half to the right. The total flux of particles from left to right is JL×t. If CL is the average concentration of diffusing particles in the left zone, than the total flux per unit area is JL×t = (sxCL)/2 - half of the particles will cross the plane X0 from left to right. The 3050, samePhase for Diagrams the fluxand from rightLeonid to left, JR×t MSE Kinetics, Zhigilei = (sxCR)/2 Microscopic picture of diffusion (II) Since JL×t = (sxCL)/2 and JR×t = (sxCR)/2, the net flow across X0 is J = JL – JR = sx(CL – CR)/2t We can express (CL–CR) in terms of concentration gradient dc/dx: (CL – CR)/sx = -(CR – CL)/sx = - dc/dx Therefore, X0 J = sx(CL – CR)/2t = -sx2/2t dc/dx CL From the Fick’s law we also have J = -D dc/dx Thus, D = sx2/2t or JL JR sx sx2 = 2tD CR sx for 1D diffusion For 3D diffusion s2 = sx2 + sy2 +sz2 = 3sx2, and D = s2/6t In general, D = s2/2dt where d is the dimensionality of the system This expression is called Einstein relation since it was first derived by Albert Einstein in his Ph.D. thesis in 1905. It relates macroscopic transport coefficient D with microscopic information on the mean square distance of molecular migration. We will consider a more rigorous derivation of this relation after we talk about the analytical solutions of the diffusion equation MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Atomic mechanisms of diffusion Two main mechanisms of atomic diffusion in crystals: Atoms located at the crystal lattice sites, usually diffuse by a vacancy mechanism. ¾ Substitutional impurities ¾ Substitutional self-diffusion – can be studied by depositing of a small amount of radioactive isotope of the element (tracer diffusion) Interstitial atoms diffuse by jumping from one interstitial site to another interstitial site without permanently displacing any of the matrix/solvent atoms: interstitial mechanism In both cases the moving atom must pass through a state of high energy – this creates energy barrier for atomic motion. The phenomenological description in terms of 1st and 2nd Fick’s laws is valid for any atomic mechanism of diffusion. Understanding of the atomic mechanisms is important, however, for predicting the dependence of the atomic mobility (and, therefore, diffusion coefficient) on the type of interatomic MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei bonding, temperature, and microstructure. Diffusion Mechanisms: Vacancy diffusion (I) To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump. This energy necessary for the jump, ΔGmv, is called the activation free energy for vacancy motion. It comes from the thermal energy of atomic vibrations (thermal energy of an atom in a solid <Uatom> ≈ 3kT). G ΔGmv Atom Vacancy Distance The average thermal energy of an atom (3kBT ≈ 0.08 eV at room temperature) is usually much smaller that the activation free energy ΔGmv (~ 1 eV/vacancy) and a large thermal fluctuation is needed for a jump. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Diffusion Mechanisms: Vacancy diffusion (II) For a simple one-dimensional case, the probability of such fluctuation or frequency of jumps, Rj, can be described by the Arrhenius equation: v ⎛ Δ Gm ⎞ R j = υ0 exp⎜ − ⎟ k T B ⎠ ⎝ where ν0 is an attempt frequency related to the frequency of atomic vibrations. The value of ν0 is of the order of the mean vibrational frequency of an atom about its equilibrium site (usually taken to be equal to the Debye frequency). Rj = frequency of atom vibrations in the diffusion direction ν0 × probability that a given oscillation will move the atom to an adjacent site To relate this to the diffusion of atoms we have to consider the jump frequency of a given atom in a 3D crystal. Moreover, for an atom to jump, there must be a vacancy next to it MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Diffusion Mechanisms: Vacancy diffusion (III) The probability for any atom in a solid to move is the product of the probability of finding a vacancy in n ⎛ ΔG vf ⎞ eq ⎜ ⎟ exp z = z − an adjacent lattice site (fraction of atoms ⎜ k BT ⎟ N ⎝ ⎠ that have a vacancy as a neighbor): × the rate of jumps of a vacancy (defined ⎞ by a probability of a thermal fluctuation R = υ exp⎛ − ΔGmv ⎜ ⎟ 0 j k T B needed to overcome the energy barrier ⎝ ⎠ for vacancy motion): The rate at which atom jumps from place to place in the crystal is therefore ⎛ ΔG v ⎞ ⎛ ΔG v 1 ⎞ atom Rj = ≈ υ0 z exp⎜⎜ − τj ⎝ f ⎟exp − k BT ⎟ ⎜⎝ ⎠ m k BT ⎟⎠ where τj is the average time between jumps for atoms. If the distance atoms cover in each jump is a, the Einstein r relation Δ ri(t) 2 i = 6 Dt can be used to estimate the diffusion coefficient from the average time between jumps: ⎛ ΔG vf + Δ G mv ⎞ a2 a 2 υ0 z ⎟= D= = exp ⎜ − ⎜ ⎟ 6τ j 6 k BT ⎝ ⎠ ⎛ Δs vf + Δs mv ⎞ ⎛ Δ h vf + Δ hmv a 2 υ0 z ⎟ exp ⎜ − exp ⎜ ⎜ ⎟ ⎜ 6 kB k BT ⎝ ⎠ ⎝ ⎞ ⎛ ⎟ = D0 exp ⎜ − E d ⎜ k T ⎟ B ⎝ ⎠ ⎞ ⎟⎟ ⎠ where D0 is a parameter of material (both matrix and diffusing species) and is independent of temperature, Ed is activation energy for diffusion: E = Δh v + Δ h v m MSE 3050, Phase Diagrams anddKinetics,f Leonid Zhigilei Diffusion Mechanisms: Vacancy diffusion (IV) ⎛ Δ s vf + Δ s mv a 2 υ0 z a2 D= = exp ⎜ ⎜ kB 6τ j 6 ⎝ ⎞ ⎛ Δ h vf + Δ hmv ⎟ exp ⎜ − ⎟ ⎜ k BT ⎠ ⎝ ⎞ ⎛ ⎟ = D0 exp ⎜ − E d ⎜ k T ⎟ B ⎝ ⎠ ⎞ ⎟⎟ ⎠ Let’s perform an order of magnitude estimate of the average time between jumps and the diffusion coefficient for self-diffusion in aluminum ⎛ Δs vf + Δsmv ⎞ ⎛ Δhvf + Δhmv ⎞ 1 ⎟ exp⎜ ⎟ τj = exp⎜ − ⎜ ⎟ ⎜ ⎟ υ0 z kB ⎝ ⎠ ⎝ kBT ⎠ ⎛ Δs vf + Δs mv a 2 υ0 z D= exp ⎜ ⎜ kB 6 ⎝ Δhfv = 0.72 eV Al: z ≈ 12 ⎛ Δs vf + Δsmv ⎞ ⎟ ~1 exp⎜ ⎜ ⎟ kB ⎝ ⎠ Δhmv = 0.68 eV ⎞ ⎛ Δ h vf + Δhmv ⎟ exp ⎜ − ⎟ ⎜ k BT ⎠ ⎝ ⎞ ⎟ ⎟ ⎠ υ0 ≈ 1013 s -1 a ≈ 3 × 10 −10 m - e.g., Shewmon, Diffusion in solids, Ch. 2.4-2.7 T = 0ºC T = 650ºC τj ≈ 6 ×1011s (less than one jump in 20000 years) τj ≈ 4 × 10-7s (2.5 million jumps per second) D ≈ 3×10-32 m2/s D ≈ 4×10-14 m2/s 17 orders of magnitude difference! MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Substitutional self-diffusion For a given crystal structure and bonding type Ed/RTm is roughly constant → D(T/Tm) ≈ const Tm K D0 10-6 m2/s Ed kJ/mol Ed eV Ed/RTm Al 933 170 142 1.47 18.3 1.9 Cu 1356 31 200.3 2.08 17.8 0.59 Ni 1726 190 279.7 2.90 19.5 0.65 γ-Fe 1805 49 284.1 2.94 18.9 0.29 Cr 2130 20 308.6 3.20 17.4 0.54 bcc V 2163 28.8 309.2 3.20 17.2 0.97 Nb 2741 1240 439.6 4.56 19.3 5.2 transition metals K 337 31 40.8 0.42 14.6 15 Na 371 24.2 43.8 0.45 14.2 16 Li 454 23 55.3 0.57 14.7 9.9 Ge 1211 440 324.5 3.36 32.3 4.4×10-5 Si 1683 900000 496.0 5.14 35.5 3.6×10-4 D(Tm) 10-12 m2/s fcc metals bcc alkali metals diamond cubic semicond. from Porter and Easterling textbook Not surprisingly, we see a correlation between Tm and Ed → stronger interatomic bonds make it more difficult to melt material and increase both Δh vf and Δhmv MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Diffusion of interstitial atoms Interstitial diffusion also involve transition through the energy barrier and can be discussed in a manner similar to the vacancy diffusion mechanism. The difference is that there are always sites for an interstitial atom to jump to. ⎛ Δ G mi a2 a 2 υ0 p D= = exp ⎜⎜ − 6τ j 6 ⎝ k BT ⎞ ⎟⎟ = ⎠ ⎛ Δ s mi a 2 υ0 p exp ⎜⎜ 6 ⎝ kB ⎛ Δ hmi ⎞ ⎟⎟ exp ⎜⎜ − ⎝ k BT ⎠ ⎛ E ⎞ ⎟⎟ = D0 exp ⎜⎜ − d ⎝ k BT ⎠ ⎞ ⎟⎟ ⎠ i where p is number of neighbor interstitial sites and E d = Δhm Small interstitial atoms of a foreign (extrinsic) type, e.g., C in Fe or O in Si may diffuse directly through the lattice (i.e., without the help of vacancies) and play an important role in defining properties of materials. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Diffusion of interstitial atoms Impurity D0, mm2/s-1 Ed, kJ/mol C in FCC Fe 23.4 148 C in BCC Fe 2 84.1 N in FCC Fe 91 168.6 N in BCC Fe 0.3 76.1 H in FCC Fe 0.63 43 H in BCC Fe 0.1 13.4 vacancy mechanism → from Porter and Easterling textbook & Smithells Metals Reference Book ← interstitial impurities ⎛ E D = D0 exp ⎜⎜ − d ⎝ k BT ⎞ ⎟⎟ ⎠ D0, mm2/s-1 Ed, kJ/mol Fe in γ-Fe 49 284 Fe in α-Fe 276 250.6 Fe in δ-Fe 201 240.7 Fe in Cr 47 332 Au in Ag 85 202.1 Si in Si 146000 484.4 • Smaller atoms cause less distortion of the lattice during migration and diffuse more readily than big ones (the atomic diameters decrease from C to N to H). • Diffusion is faster in more open lattices Diffusion of interstitials is typically faster as compared to the vacancy diffusion mechanism (self-diffusion or diffusion of substitutional atoms). MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Diffusion of self-interstitials Intrinsic interstitials, also called “self-interstitials” are interstitials atoms of the same kind as the atoms of the crystal. Self-interstitials in most materials introduce strong deformations into the lattice and have very high formation energy, Δhfi ≈ 3Δhfv for metals. The number of equilibrium interstitials can be estimated by an equation similar to the one derived for vacancies: ⎛ Δhi f ⎞ We can estimate that at room i ⎟⎟ neq = N exp ⎜⎜ − temperature in copper there is less k BT ⎠ ⎝ than one interstitial per cm3, whereas just below the melting point there is one interstitial for every 1012 atoms – there are virtually no “equilibrium” interstitials in metals and most other elemental crystals. In Si, however, intrinsic interstitials play an important role in diffusion and formation of defect structures. Non-equilibrium self-interstitials in most materials are very mobile, e.g., Δhmi ≈ 0.5Δhmv for metals and they quickly diffuse out of the bulk of the crystal after being formed. Self-interstitials can move through formation of intermediate low-energy configurations, e.g., dumbbells (two atoms share the space of one), and crowdions. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Diffusion of clusters of interstitials One-dimensional motion of an almost isolated ½[111] loop at 575 K. A loop continuously moves in a direction parallel to its Burgers vector Arakawa et al., Science 318, 956, 2007 Schematic view of the observation of the 1D glide motion of a interstitial-type prismatic perfect dislocation loop by TEM. Point defects in atomistic simulations [Lin et al., Phys. Rev. B 77, 214108, 2008] 175 ps 200 ps 400 ps MSE 3050, Phase Diagrams<110>-dumbbell and Kinetics, Leonid Zhigilei vacancy interstitial 450 ps cluster of 4 interstitials <111>-crowdion Diffusion – Temperature Dependence ⎛ Ed ⎞ D = D0 exp⎜ − ⎟ ⎝ RT ⎠ D0 – T-independent pre-exponential (m2/s) Ed – activation energy for diffusion (J/mol) R – the gas constant (8.31 J/mol-K) T – absolute temperature (K) The above equation can be rewritten as ln D = ln D0 − Ed ⎛ 1 ⎞ ⎜ ⎟ R ⎝T ⎠ or Ed ⎛ 1 ⎞ log D = log D0 − ⎜ ⎟ 2.3R ⎝ T ⎠ The activation energy Ed and pre-exponential D0, therefore, can be estimated by plotting lnD vs. 1/T or logD vs. 1/T. Such plots are called Arrhenius plots. b = logD0 y = ax + b Graph of log D vs 1/T has slop of –Ed/2.3R, intercept of ln Do Ed a=− 2.3R x = 1/T [( ) (1 T1 − 1 T2 )] Ed = −2and .3RKinetics, × log D D2 MSE 3050, Phase Diagrams Leonid Zhigilei 1 − log Diffusion of interstitial and substitutional impurities log D = log D0 − Ed ⎛ 1 ⎞ ⎜ ⎟ 2.3R ⎝ T ⎠ Diffusion of interstitials is typically faster as compared to the vacancy diffusion mechanism (self-diffusion or diffusion of substitutional atoms). MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Fast diffusion paths (I) More open atomic structure at defects (grain boundaries, dislocations) can result in a significantly higher atomic mobility along the defects. r MSD = Δri(t)2 = 6 Dt i Mean-square displacement of all atoms in the system (B), atoms in the grain boundary region (C), and bulk region of the system (A). The plots are from the computer simulation by T. Kwok, P. S. Ho, and S. Yip. Initial atomic positions are shown by the circles, trajectories of atoms are shown by lines. We can see the difference between atomic mobility in the bulk crystal and in the MSE Phase Diagrams and Kinetics, Leonid Zhigilei grain3050, boundary region. Fast diffusion paths (II) Diffusion coefficient along a defect (e.g. grain boundary) can be also described by an Arrhenius equation, with the activation energy for grain boundary diffusion significantly lower than the one for the bulk. ⎛ E ddef ⎞ def def ⎟⎟ D = D0 exp ⎜⎜ − ⎝ k BT ⎠ However, the effective cross-sectional area of the defects is only a small fraction of the total area of the bulk (e.g., an effective thickness of a grain boundary is ~0.5 nm). The diffusion along defects is less sensitive to the temperature change → becomes important at low T. Self-diffusion coefficients for Ag. The diffusivity if greater through less restrictive structural regions – grain boundaries, dislocation cores, external surfaces. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Fast diffusion paths: Diffusion in nanocrystalline materials image by Zhibin Lin et al. J. Phys. Chem. C 114, 5686, 2010 Arrhenius plots for 59Fe diffusivities in nanocrystalline Fe and other alloys compared to the crystalline Fe (ferrite). [Wurschum et al. Adv. Eng. Mat. 5, 365, 2003] MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Fast diffusion paths: Diffusion in nanocrystalline materials High-resolution electron micrograph (left, [Acta Mater. 56, 5857, 2008]) and computed atomic structure (right, [Acta Mater. 45, 987, 1997]) of nanocrystalline Si. Wurschum et al., Defect Diffus. Forum 143-147, 1463, 1997] volume fraction of grain boundary regions: ~50%. diffusivity is enhanced by ~30 orders of magnitude. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Summary on the Diffusion Mechanisms Make sure you understand language and concepts: ¾ ¾ ¾ ¾ ¾ ¾ ¾ Mobility of atoms and diffusion Activation energy High diffusivity path Arrhenius equation Interstitial diffusion Self-diffusion Vacancy diffusion Connection between the microscopic picture of diffusion (mean square displacement of atoms) and the diffusion coefficient Factors that Influence Diffusion: ¾ Temperature - diffusion rate increases very rapidly with increasing temperature (Arrhenius dependence) ¾ Diffusion mechanism - interstitial is usually faster than vacancy ¾ Diffusing and host species - Do, Ed is different for every solute - solvent pair ¾ Microstructure – low-temperature diffusion is faster in polycrystalline vs. single crystal materials because of the accelerated diffusion along grain boundaries and dislocation cores. MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei