Introduction to the Science and Engineering of Materials
Introduction to the Science and Engineering of Materials MSE 2090
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Introduction To Materials Science Chapter 5 Diffusion Chapter Outline Diffusion how do atoms move through solids I Diffusion mechanisms gt Vacancy diffusion gt Interstitial diffusion gt Impurities 39 The mathematics of diffusion gt Steadystate diffusion Fick s rst law gt NonsteadyState Diffusion F ick s second law I Factors that in uence diffusion gt Diffusing species gt Host solid gt Temperature gt Microstructure 54 NonsteadyState Di usz on 7 Not Covered Not Tested University of Virginia Dept of Materials Science and Engineering lnlrm luclinnTn Materials SciznmV Cluptars Di usinn What is diffusion Diffusion is material transport by atomic motion Atoms of type A Atoms of type B w 7 5 zworamare A Inhomogeneous materials can become homogeneous by diffusion For an active dif ision to occur the temperature should be high enough to overcome energy barriers to atomic motion Universitynrvngjniabgpt nszverials Sciencezn l Engineering 2 Introduction To Materials Science Chapter 5 Diffusion Interdiffusion and Selfdiffusion Interdiffusion or impurity diffusion occurs in response to a concentration gradient Generic 01 0 quot4 CJ I39d 990999990909 oneoaeoonooo 999909999999 999909999909 000099090909 W F39caitinn Before u h Fusc I cf CJ arc n Eu Culs allzn39 Ni Dilsticr 9f Ji Jms 139er OQDOUDUOUDDO DDOODDQGODDG onoanancunnc nanannnunnna unnauoncunna After Selfdiffusion is diffusion in onecomponent material when all atoms that exchange positions are of the same type University of Virginia Dept of Materials Science and Engineering 3 Introduction To Materials Science Chapter 5 Di 39usion Diffusion Mech anisms I Vacancy diffusion Atom migration Vacancy migration 0 Before After 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 comes from the thermal energy of atomic Vibrations Eav N kT Materials ow the atom is opposite the vacancy ow direction University at Virginia Dept 0 Materials Science and Engineering 4 lnunrlunjnn Tn Materials Science Chapter 5 Di usinn Diffusion Mechanisms II Interstitial diffusion Interstitial atom Interstitial atom before diffusion a er diffusion 0L9 a o 9 or 9 Q 0 Q 9 3 a 9 f3 0 9 GI Interstitial diffusion is generally faster than vacancy diffusion because bonding of interstitials to the surrounding atoms is normally weaker and there are many more interstitial sites than vacancy sites to jump to Requires small impurity atoms eg C H O to t into interstices in host University anirgjniaDqtnszt2rials Summeandtngimmng 5 Introduction To Materials Science Chapter 5 Diffusion Diffusion Flux The ux of diffusing atoms J is used to quantify how fast diffusion occurs The ux is de ned as either in number of atoms diffusing through unit area and per unit time eg atomsmZsecond or in terms of the mass ux mass of atoms diffusing through unit area per unit time eg kgmzsecond J M At E lA dMdt Kg m392 s1 Where M is the mass of atoms diffusing through the area A during time t University of Virginia Dept of Materials Science and Engineering 6 Intrndllctinn Tn Materials Scimce Chang 5 Muslim SteadyState Diffusion Steady state diffusion the diffusion flux does not change with time Concentration pro le concentration of atomsmolecules of interest as function of position in the sample Concentration gradient dCdx Kgm 3 the slope at a particular point on concentration profile 10 CB FMS WWW dx Ax xA XB and mustard Gas M mm PA m m Pusmnn x m University nl39Virg39niz Dan nflVLaterials Science and Enginwing 7 Introduction To Materials Science Chapter 5 Diffusion SteadyState Diffusion Fick s rst law Fick s rst law the diffusion ux along direction X is proportional to the concentration gradient dC J D d Where D is the diffusion coef cient X J ux of atom5 across plane with areaA The concentration gradient is often called the driving force in diffusion but it is not a force in the mechanistic sense The minus sign in the equation means that diffusion is down the concentration gradient University of Virginia Dept of Materials Science and Engineering 8 Intrndllctinn Tn lVIztErials Science Chapter 5 Di nsinn NonsteadyState Diffusion Fick s second law not tested In most real situations the concentration pro le and the concentration gradient are changing with time The changes of the concentration pro le is given in this case by a differential equation Fisk s second law 9Ci ac 92C BtBX 3x 3x2 Solution of this equation is concentration pro le as function of time CXt t3gtt2gtt1 Cancenlralmn a muesmg segues Dlstance University nf Virginia Dept nf lVLaterials Science and Engineering 9 Introduction To Materials Science Chapter 5 Diffusion Diffusion Thermally Activated Process 1 not tested In order for atom to jump into a vacancy site it needs to posses enough energy thermal energy to to break the bonds and squeeze through its neighbors The energy necessary for motion Em is called the activation energy for vacancy motion Energy Distance 0000 0000 0080 00 0 0 0 0 0 0 0000 0000 0000 Schematic representation of the diffusion of an atom from its original position into a vacant lattice site At activation energy EIn has to be supplied to the atom so that it could break interatomic bonds and to move into the new position University of Virginia Dept of Materials Science and Engineering 10 Introduction To Materials Science Chapter 5 Diffusion Diffusion Thermally Activated Process 11 not tested The average thermal energy of an atom kBT 0026 eV for room temperature is usually much smaller that the activation energy EIn 1 eVatom and a large uctuation in energy when the energy is pooled together in a small volume is needed for a jump The probability of such uctuation or frequency of jumps Rj depends exponentially from temperature and can be described by equation that is attributed to Swedish chemist Arrhenius R R m j Z 0 6Xp 413T Where R0 is an attempt frequency proportional to the frequency of atomic vibrations University of Virginia Dept of Materials Science and Engineering 11 Introduction To Materials Science Chapter 5 Diffusion Diffusion Thermally Activated Process 111 not tested For the vacancy diffusion mechanism the probability for any atom in a solid to move is the product of the probability of finding a vacancy in an adjacent lattice site see Chapter 4 Q P CNexp T B and the probability of thermal uctuation needed to overcome the energy barrier for vacancy motion E RJ 2 RO exp BT The diffusion coefficient therefore can be estimated as D x CNROa2 6Xp 13 19 Q BTj 2 DO exp Em QBT D0 6Xp QBTJ Temperature dependence of the diffusion coefficient follows the Arrhenius dependence University of Virginia Dept of Materials Science and Engineering 12 Introduction To Materials Science Chapter 5 Diffusion Diffusion Temperature Dependence 1 dc Diffusion coefficient is the measure of dX mobility of diffusing species D 2 D0 6Xp Qd RT D0 temperatureindependent preexponential mZs Q01 the activation energy for diffusion Jmol or eVatom R the gas constant 831 JmolK or 862gtlt10395 eVatomK T absolute temperature K The above equation can be rewritten as lnDlnD0 lj or logDlogD0 Qd R T 23R T The activation energy Q01 and preexponential D0 therefore can be estimated by plotting lnD versus lT or logD versus lT Such plots are Arrhenius plots University of Virginia Dept of Materials Science and Engineering 13 Intrndllctinn Tn NLaterials Science Chapter 5 Dimshin Diffusion Temperature Dependence II 1042 7 E w 5 5 Dimsmn coemmem NZ5 107 m w w w 07 08 09 10 11 12 Reclpvoca tempevatuve IOODK Graph oflog D vs 11quot has slop ofiQdZBR intercept of In DO Q 1 10 D10 D g g 23R T Qd 2I3R10gD1 10gDZ lTl 1T2 University nf Virginia Dept nflVLaterials Science and Enginwing 14 Intrm hlctinn Tn Niaterials Science Chapter 5 Di usinn Diffusion Temperature Dependence III Tern peyemre Dc 500 12001000 BOO 600 500 400 300 1039 mn e W mm 7 we anCu 1044 7 7 Fe n39yiFe 10457 Femer A m A Cu m Cu 107m 7 men x 0 5 10 Recwpmcm Lemperamre moom AIrhenius plot of dif lsivity data for some metallic systems University nf Virginia Dept nf NIzterials Science and Engineering 15 Table 52 Introduction To Materials Science Chapter 5 Diffusion 11 39l ulmllminn of Diffusion Dam Diffusion of different species 31mmquot rm Ac m an Enemy yd hlcufm39ed Veri es Sfl ll hquot Siftmi Dbufn fs U 139 ma v17qu quottquot I1rr 139 Fl uFLquot 23 36 11F 2395 1111 SD11 31 v 11139 EEC39C39 39J l 15 M 11 F1 TFI 51 gt IfFi 234 2134 lJIIIEI 11 N 111quot IIFfL39Equot mm 75 hf 11Fquot Ii uFu 153 Zr 11quot SI 1133 SIZIfJ 14 X 11139 lJIJEI 17 X 11 1quot I T rFr 23 r HI 118 153 39JIII 551 gt1 It 1101 5 x 111 In u 3911 2111 son 43 39x 11F 211 Cu 1811 1 1 SD13 41 91 It A A1 144 149 SUB 413 X It In Al 136 141 5131 41 3 11F Mg Al 13 135 SIZITJ 11 X 11 Cu Ni 3 HP 3925 2135 501 l 11139 Somme E A Branch and G H Bmuk Edi1or5 Sm dm39b Metals Rtf l t39l lt e 13mm T111 cdiliun 11 unexworlh Helium51111 Oxford 1997 Smaller atoms diffuse more readily than big ones and diffusion is faster in open lattices or in open directions University of Virginia Dept of Materials Science and Engineering 16 lntrnrlllctjnn Tn Materials SciEncE Chapter 5 Di 39usinn Diffusion Role of the microstructure I UK 2000 1500 1000 5 D l l M 9 Grain boundary quotE g 1042 E Volume u 1346 l l 05 10 15 20 1 fl 7400 K Selfdiffusion coef cients for Ag depend on the diffusion path e r the diffusivity if greater through less restrictive structural regions 7 grain boundaries dislocation cores external surfaces Univa39s39ty anirg39nia Dept nfMatErialsSEiEnceand Engneaing 17 Introduction To Materials Science Chapter 5 Diffusion Diffusion Role of the microstructure II The plots below are from the computer simulation by T Kwok P S H0 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 grain boundary region University of Virginia Dept of Materials Science and Engineering 18 Introduction To Materials Science Chapter 5 Diffusion Factors that In uence Diffusion Summary gt Temperature diffusion rate increases very rapidly with increasing temperature gt Diffusion mechanism interstitial is usually faster than vacancy gt Diffusing and host species D0 Qd is different for every solute solvent pair gt Microstructure diffusion faster in polycrystalline vs single crystal materials because of the accelerated diffusion along grain boundaries and dislocation cores University of Virginia Dept of Materials Science and Engineering 19 Introduction To Materials Science Chapter 5 Diffusion Summary Make sure you understand language and concepts VVVVVVVVVVVV University of Virginia Dept of Materials Science and Engineering Activation energy Concentration gradient Diffusion Diffusion coefficient Diffusion flux Driving force Fick s first and second laws Interdiffusion Interstitial diffusion Selfdiffusion Steadystate diffusion Vacancy diffusion 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