Khuếch tán hay khuếch tán phân tử là sự dao động nhiệt của tất cả các phân tử ở nhiệt độ lớn hơn độ không tuyệt đối. Tốc độ của chuyển động nhiệt là hàm số của nhiệt độ, độ nhớt của dòng chảy và kích thước của các phần tử nhưng không phải là hàm số của nồng độ.
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/340438781 lecture-diffusion in solids Chapter · April 2020 CITATIONS READS 51 authors, including: Adrees Edaan University of Mosul 13 PUBLICATIONS 0 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: material science View project SSNTDS View project All content following this page was uploaded by Adrees Edaan on 02 May 2020 The user has requested enhancement of the downloaded file Diffusion in Solids Diffusion refers to the transport of atoms in a medium along a given direction, It takes place in solid liquid or gas It occurs even in the absence of any external force Diffusion processes play a crucial role in many solid-state phenomena and in the kinetics of microstructural changes during metallurgical processing and applications typical examples include phase transformations : Nucleation Recrystallization Oxidation Creep Sintering ionic conductivity intermixing in thin film devices Direct technological uses of diffusion include solid electrolytes for advanced battery and fuel cell applications Semiconductor chip and microcircuit fabrication 10 Surface hardening of steels through carburization Types of Diffusion : (i) Self Diffusion : It is the transition of a thermally excited atom from a site of crystal lattice to an adjacent site or interstice (ii) Inter Diffusion : This is observed in binary metal alloys such as the (Cu-Ni) system (iii) Volume Diffusion : This type of diffusion is caused due to atomic movement in bulk in materials (iv) Grain Boundary Diffusion : This type of diffusion is caused due to atomic movement along the grain boundaries alone (v) Surface Diffusion : This type of diffusion is caused due to atomic movement along the surface of a phase Dr Edrees Edaan Al Obeidi Important definition Self-Diffusion : It is the transition of a thermally excited atom from a site of the crystal lattice to an adjacent site or interstice If the solid is composed of a single element (say pure copper), the movement of the atom is called self-diffusion because in this case the moving atom and the solid are the same chemical element The process of self-diffusion is very important for annealing and creep Use of radioactive tracers have found to be quite useful in determining self-diffusion coefficient Inter-Diffusion : It is contrary to self-diffusion and takes place in binary metallic alloys, e.g the Cu-Ni system If nickel had been plated onto the surface of copper, then atomic diffusion would bring about nickel homogenization within the copper, after a sufficient time, at elevated temperatures Diffusion mechanisms : i) Vacancy Mechanism : This mechanism is a very dominant process for diffusion in FCC, BCC and HCP metals and solid solution alloy The activation energy for this process comprises the energy required to create a vacancy and that required to move it In a pure solid, the diffusion by this mechanism is shown in below Diffusion by the vacancy mechanism can occur by atoms moving into adjacent sites that are vacant Concentration changes takes place due to diffusion over a period of time We must note that vacancies are continually being created and destroyed at the surface, grain boundaries and suitable interior positions, e.g dislocations Obviously, the rate of diffusion increases rapidly with increasing temperature Dr Edrees Edaan Al Obeidi If a solid is composed of a single element (pure metal) the movement of thermally excited atom from a site of the crystal lattice to an adjacent site or interstice is called self diffusion because the moving atom and the solid are the same chemical-element There are two type of vacancy mechanism for atomic diffusion : Pure solid solution Substitutional solid solutions (ii) The Interstitial Mechanism : The interstitial mechanism where an atom changes positions using an interstitial site does not usually occur in metals for self-diffusion but is favoured when interstitial impurities are present because of the low activation energy When a solid is composed of two or more elements whose atomic radii differ significantly, interstitial solutions may occur The large size atoms occupy lattice sites where as the smaller size atoms fit into the voids (called as interstices) created by the large atoms, figure below show this mechanism We must note that an activation energy is associated with interstitial diffusion because, to arrive at the vacant site, it must squeeze past neighbouring atoms with energy supplied by the vibrational energy of the moving atoms Dr Edrees Edaan Al Obeidi The interstitial mechanism process is simpler since the presence of vacancies is not required for the solute atom to move This mechanism is vital for the following cases: (a) The presence of very small atoms in the interstices of the lattice affect to a great extent the mechanical properties of metals (b) At low temperatures, oxygen, hydrogen and nitrogen can be diffused in metals easily (iii) Interchange Mechanism : In this type of mechanism, the atoms exchange places through rotation about a mid point The activation energy for the process is very high and hence this mechanism is highly unlikely in most systems Two or more adjacent atoms jump past each other and exchange positions, but the number of sites remains constant, as show in figures below This interchange may be two-atom or four-atom (Zenner ring) for BCC Due to the displacement of atoms surrounding the jumping pairs,interchange mechanism results in severe local distortion For jumping of atoms in this case, much more energy is required In this mechanism, a number of diffusion couples of different compositions are produced, which are objectionable This is also termed as Kirkendall’s effect Laws of Diffusion: FICK’S FIRST LAW (STEADY-STATE DIFFUSION): Diffusion can be treated as the mass flow process by which atoms (or molecules) change their positions relative to their neighbours in a given phase under the influence of thermal energy and a gradient It is necessary to know how fast diffusion occurs, or the rate of mass transfer This rate is frequently expressed as a diffusion flux (J), defined as the mass (number of atoms) (M) diffusing through and perpendicular to a unit crosssectional area of solid per unit of time Dr Edrees Edaan Al Obeidi J= 𝑁 ………… (1) 𝐴𝑡 Where (A) denotes the area across which diffusion is occurring and (t) is the elapsed diffusion time In differential form, this expression becomes J= 𝑑𝑁 ………… (2) 𝐴 𝑑𝑡 The units for (J) are kilograms or atoms per meter squared per second (kg/m2.sec or atoms/m2.sec) If the diffusion flux does not change with time, a steady-state condition exists The mathematics of steady-state diffusion in a single (x) direction is relatively simple, in that the flux is proportional to the concentration gradient, (dC/dx) through the expression: J = −𝐷 𝑑𝐶 ………… (3) 𝑑𝑥 Or J=− ∆𝑥 𝑑𝐶 2∆𝑡 𝑑𝑥 ………… (4) Substituting Eq.(2) to (3), we get : 𝑑𝑁 𝑑𝑡 = −𝐷𝐴 𝑑𝐶 𝑑𝑥 ………… (5) Or This equation is sometimes called Fick’s first law The constant of proportionality (D) is called the diffusion coefficient, which is expressed in (m2/sec) Dr Edrees Edaan Al Obeidi The negative sign in this expression indicates that the direction of diffusion is down the concentration gradient, from a high to a low concentration Fick’s first law may be applied to the diffusion of atoms of a gas through a thin metal plate for which the concentrations (or pressures) of the diffusing species on both surfaces of the plate are held constant This diffusion process eventually reaches a state where in the diffusion flux does not change with time—that is, the mass of diffusing species entering the plate on the high-pressure side is equal to the mass exiting from the low-pressure surface—such that there is no net accumulation of diffusing species in the plate This is an example of what is termed steady-state diffusion When concentration (C) is plotted versus position (or distance) within the solid (x), the resulting curve is termed the concentration profile; furthermore, concentration gradient is the slope at a particular point on this curve 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝑑𝐶 𝑑𝑥 = Δ𝐶 Δ𝑥 = 𝐶𝐴 − 𝐶𝐵 𝑥𝐴 − 𝑥𝐵 …… … (6) For diffusion problems, it is sometimes convenient to express concentration in terms of mass of diffusing species per unit volume of solid (kg/m3 or g/cm3) The driving force for diffusion reactions, according to equation (3) is the concentration gradient Dr Edrees Edaan Al Obeidi Q : What factors affect the diffusion rate in solid metal crystals? The diffusion rate in solid metal crystals is affected by five factors: Diffusion mechanism; Temperature of diffusion; Concentration of the diffusion species (concentration gradient); Type of crystal structure; (bcc > fcc) Type of crystal imperfections FICK’S SECOND LAW (NONSTEADY-STATE DIFFUSION): Most practical diffusion situations are nonsteady-state that is, the diffusion flux and the concentration gradient at some particular point in a solid vary with time This is illustrated in Figure below, which shows concentration profiles at three different diffusion times 𝑑𝐶 𝑑𝑡 = − 𝜕𝐽 𝜕𝑥 =− 𝜕 𝜕𝑥 (−𝐷 𝜕𝐶 𝜕𝑋 …… … (7) ) known as Fick’s second law, is used If the diffusion coefficient is independent of composition (which should be verified for each particular diffusion situation), Equation (7) simplifies to : 𝑑𝐶 𝑑𝑡 = 𝐷 𝜕2 𝐶 …… … (8) 𝜕𝑥 One practically important solution is for a semi-infinite solid in which the surface concentration is held constant The partial pressure is maintained at a constant value Solutions to above expression (concentration in terms of both position and time) are possible when physically meaningful boundary conditions are specified as follows : Initial condition Dr Edrees Edaan Al Obeidi For t = 0, C = Co at ≤ x ≤ ∞ Boundary conditions For t > 0, C = Cs (the constant surface concentration) at x = For t > 0, C = Co at x = ∞ Application of these conditions to Equation (8) yields the solution 𝐶𝑥 −𝐶𝑜 𝐶𝑠 −𝐶𝑜 = − 𝑒𝑟𝑓 ( 𝑒𝑟𝑓(𝑧) = Where; ( √ 𝑧 𝑥 2√𝐷𝑡 …… … (9) ) ∫ 𝑒 −𝑦 𝑑𝑦 𝜋 𝑥 2√𝐷𝑡 …… … (10) ) has been replaced by the variable (z) (Cx) : represents the concentration at depth (x) after time (t) (Cs) : represents the concentration element at the surface (Co) : represents the initial uniform concentration of element in Solid (x) : distance from surface (D) : diffusivity of diffusing solute element (t) : time (erf) : mathematical Gaussian error function.function ca Note: A bar of solid is considered to be semi-infinite if none of the diffusing atoms reaches the bar end during the time over which diffusion takes place A bar of length ( l ) is considered to be semi-infinite when ( l > 10 √𝐷𝑡 ) Dr Edrees Edaan Al Obeidi Suppose that it is desired to achieve some specific concentration of solute, (C1), in an alloy; the left-hand side of Equation (9) now becomes: 𝐶1 −𝐶𝑜 𝐶𝑠 −𝐶𝑜 …… … (11) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 the right-hand side of Equation (9) is also a constant, and subsequently: ( 𝑥 2√𝐷𝑡 …… … (12) ) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Or 𝑥2 …… … (13) ( ) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐷𝑡 For some diffusion situations where in time and temperature are variables and in which composition remains constant at some value of (x), Eq.(13) takes the form: …… … (14) (𝐷𝑡) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 FACTORS THAT INFLUENCE DIFFUSION: Diffusing Species : If the diffusing species is able to occupy interstitial sites, then it can easily diffuse through the parent matrix On the other hand if the size of substitutional species is almost equal to that of parent atomic size, substitutional diffusion would be easier Thus size of diffusing species will have great influence on diffusivity of the system Temperature : The temperature dependence of the diffusion coefficients is D = 𝐷𝑜 𝑒𝑥𝑝 (− 𝑄𝑑 𝑅𝑇 …… … (15) ) Where: Do = a temperature-independent preexponential (m2/s) Qd = the activation energy for diffusion (J/mol or eV/atom) R = the gas constant, 8.31 J/mol.K or 8.62 * 10-5 eV/atom.K T = absolute temperature (K) Dr Edrees Edaan Al Obeidi The activation energy may be thought of as that energy required to produce the diffusive motion of one mole of atoms Taking natural logarithms of Eq.(15) yields: 𝑄 𝑅 𝑇 𝑙𝑛D = 𝑙𝑛𝐷𝑜 − ( 𝑑 ) ( ) …… … (16) or, in terms of logarithms to the base 10: 𝑙𝑜𝑔D = 𝑙𝑜𝑔𝐷𝑜 − ( 𝑄𝑑 2.3𝑅 …… … (17) )( ) 𝑇 Because (Do), (Qd), and (R) are all constants, Equation (17) takes on the form of an equation of a straight line (Y = b + mx) typical Arrhenius plot of log10 of the reaction rate versus reciprocal absolute temperature 10 Dr Edrees Edaan Al Obeidi Lattice structure : Diffusion is faster in open lattices or in open directions than in closed directions Presence of defects : defects like dislocations, grain boundaries act as short-circuit paths for diffusing species, where the activation energy is diffusion is less Thus the presence of defects enhances the diffusivity of diffusing species Diffusion paths in solids: Diffusion of species in solids also depends on the path that it follows In a solid we can think of three distinct paths Metals are made of several crystals that meet along grain boundaries If a species has to move through this it could either move through the grain, the grain boundary or the top surface This is illustrated in Fig below The diffusivity through the grain is denoted by (Dg) This is often known as bulk diffusion coefficient The diffusivity through the grain boundary is denoted by (Dgb) The space between atoms because of irregular arrangement is more at the grain boundary than that within the grains This is why the mobility of atoms through the grain boundary is expected to be higher than that through the grain The same logic can be extended to the exposed top surface There is enough space to accommodate extra atoms at the free surface if required Therefore the mobility of the atoms along the free surface (Ds) is much higher The relation between the three could be described as (Ds > Dgb > Dg) 11 Dr Edrees Edaan Al Obeidi The Figure also illustrates that the temperature dependence of the diffusivity through three different paths The activation energy of surface diffusion is likely to be the lowest and that for the grain is the highest Mathematically this is denoted as (Qg > Qgb > Qs) Diffusion as a random walk process: The process of diffusion is governed by the movement of atoms In solids where atoms are closely packed such movements will be difficult in the absence of vacant sites At a given temperature several lattice sites are vacant If there are several sites around an atom how would an atom decide where to move ? An obvious option could be a random selection Fig below illustrates the difference between a normal and a random walk If a species moves through the lattice at a velocity (v), the distance between its initial and final position after time t is equal to (vt), provided with an arrow head denote the distance covered in this case However if the direction keeps changing randomly the distance between the initial and final position is much less This is shown with the help of a shorter red line with an arrow head The distance between the initial and the final position is best represented as a vector sum of each step (𝑟̅𝑖 ) The final location (𝑅̅𝑖 ) after (n) steps of movement is given by: …… … (18) The dot product of a vector with itself gives the square of its magnitude This is given by: …… … (19) 12 Dr Edrees Edaan Al Obeidi Each term of the second series in equation (19) denotes dot product of two vectors If the angle between the two is (Ɵij), and each step size is equal to (λ) the equation (19) on simplification becomes: …… … (20) Since the direction of motion is random the magnitude of (cosƟ) could be both and negative The sum total value is therefore likely to be zero Thus the average distance between the initial and final location in this case is (𝜆√𝑛) as against (λn) in the case of normal walk (𝑅̅𝑛 ) represents average root mean square distance In equation (4), (∆x/∆t) denotes the average velocity (v) of the diffusing species and (∆x) is the average step size which equivalent to (λ) The total distance covered by the species is (nλ) Therefore the time it takes to cover this distance is (nλ/v) Thus by a little algebraic simplification you get the following relations: …… … (20) This shows that average random walk distance is equal to (√2𝐷𝑡) Kirkendall Effect : When we consider in a binary solution of A and B, the rates at which A and B diffuse are not necessarily equal It is observed that, usually, the lower melting component diffuses much faster than the other This lead to certain effects which are interesting and first observed by Kirkendall 13 Dr Edrees Edaan Al Obeidi Inert markers, i.e thin rods of a high melting point substance which is insoluble in the diffusion matrix, are placed at the weld joint of the couple, prior to the diffusion of anneal It is found that these markers shift during the anneal in the same direction as the slower moving species The extent of this shift is reported to be proportional to the square root of the diffusion time This type of movement reveals that the net mass flow due to the difference in diffusivities is being compensated by a bulk flow of matter in the opposite direction within the diffusion zone Obviously, lattice planes are created on one side of the diffusion zone, while they are destroyed on the other side of the diffusion zone, and the resulting bulk flow carries the markers along We must note that the bulk flow occurs relative to the ends of the diffusion couple It is interesting to note that it is quite a different phenomenon from the diffusion process itself In several cases, one observes porosity on the lower-melting component side, indicating that the bulk flow does not fully compensate for the difference in diffusivities of the two species To understand the Kirkendall effect, one may consider the analogy of gaseous interdiffusion Let us consider that hydrogen and argon at the same pressure be kept in two chambers interconnected through a tube and a frictionless piston in the tube separates the gases When an orifice in the piston is opened, the gases interdiffuse Obviously, the lighter hydrogen will diffuse faster, resulting in a pressure difference that will tend to shift the piston in the same direction as the slower diffusion argon is moving Applications of Diffusion : Diffusion processes are the basis of crystallization, recrystallization, phase transformation and saturation of the surface of alloys by other elements Few important applications of diffusion are : (i) Oxidation of metals (ii) Doping of semiconductors (iii) Joining of materials by diffusion bonding, e.g welding, soldering, galvanizing, brazing, and metal cladding (iv) Production of strong bodies by sintering, i.e powder metallurgy (v) Surface treatment of steels, e.g case hardening (vi) Important in heat treatment, e.g homogenising treatment of castings, recovery, recrystallization and precipitation of phases (vii) Diffusion is fundamental to phase changes, e.g γ to α-iron 14 Dr Edrees Edaan Al Obeidi Summary Diffusion FASTER for open crystal structures lower melting (T) materials materials /secondary bonding cations smaller diffusing atoms lower density materials خالصة Diffusion SLOWER for close-packed structures higher melting (T) materials materials /covalent bonding anions larger diffusing atoms higher density materials References : Materials Science and Engineering; an introduction, WILLIAM D CALLISTER, JR and DAVID G RETHWISCH, John Wiley & Sons, 2014 Diffusion in solids I, Lecture 15, Kharagpur : Prof R N Ghosh, Dept of Metallurgical and Materials Engineering Materials Science, G K Narula، K S Narula، V K Gupta, Tata McGraw-Hill Education, 2007 Mass Transport–Induced Failure, Milton Ohring, Copyright 2020 Elsevier Journals & Books, book in Reliability and Failure of Electronic Materials and Devices, printed in 1998 15 Dr Edrees Edaan Al Obeidi View publication stats ... introduction, WILLIAM D CALLISTER, JR and DAVID G RETHWISCH, John Wiley & Sons, 2014 Diffusion in solids I, Lecture 15, Kharagpur : Prof R N Ghosh, Dept of Metallurgical and Materials Engineering Materials