Leture 4: Spinodal Deoposition: Part 3: kinetis of the oposition flutuation Today s topis Diffusion kinetis of spinodal deoposition in ters of the onentration (oposition) flutuation as a funtion of tie: t (, ) = A (,) ep[ R( ) t] os, where M g R( ) [ ] N = + Learn how to derive above equation fro the Fik s seond law, for whih how to dedue the ter g is ritial, g = g ( ) d. d Understand the ritial wavelength (or wave nuber ) for the oposition flutuation: for C < (or > C ), then ( ) R >, the oposition flutuations (aplitude) will grow --- therodynaially favorable. Understand the aial wavelength (or wave nuber ), at whih the oposition flutuations (aplitude) will grow the fastest: when << (or >> ), the growth beoes diffusion liited. Phase diagra and free energy plot of Spinodal Deoposition As a speial ase of phase transforation, spinodal deoposition an be illustrated on a phase diagra ehibiting a isibility gap (see the diagra below). Thus, phase separation ours whenever a aterial transitions into the unstable region of the phase diagra. The boundary of the unstable region, soeties referred to as the binodal or oeistene urve, is found by perforing a oon tangent onstrution of the free-energy diagra. Inside the binodal is a region alled the spinodal, whih is found by deterining where the urvature of the free-energy urve is negative. The binodal and spinodal eet at the ritial point. It is when a aterial is oved into the spinodal region of the phase diagra that spinodal deoposition an our. If an alloy with oposition of X is solution treated at a high teperature T 1, and then quenhed (rapidly ooled) to a lower teperature T, the oposition will initially be the sae everywhere and its free energy will be G on the G urve in the following diagra. However, the alloy will be iediately unstable, beause sall flutuation in oposition that produes A-rih and B-rih regions will ause the total free energy to derease. Therefore, up-hill diffusion (as shown) takes plae until the equilibriu 1
opositions X 1 and X are reahed. How suh sall oposition flutuation leads to the spinodal phase separation is today s and net two letures topis. The free energy urve is plotted as a funtion of oposition for the phase separation teperature T. Equilibriu phase opositions are those orresponding to the free energy inia. Regions of negative urvature (d G/d < ) lie within the infletion points of the urve (d G/d = ) whih are alled the spinodes (as arked as S 1 and S in the diagra above). For opositions within the spinodal, a hoogeneous solution is unstable against irosopi flutuations in density or oposition, and there is no therodynai barrier to the growth of a new phase, i.e., the phase transforation is solely diffusion ontrolled. adapted fro the Book "Phase Transforations in Metals and Alloys", by D. A. Porter,. E. Easterling and M. Y. Sherif, CRC Press, third edition. In last two letures (#, 3): For a hoogeneous solid solution of oposition C, when a oposition perturbation (or flutuation) is reated suh that the oposition is a funtion of position (although the average oposition is still ), the spinodal deoposition will initiated. The oposition flutuation an be desribed as ( ) = A os( π ) = A os
where A = aplitude, and π = = wave nuber Let onentration (olar fration) of B ato is, the onentration of A ato is (1-), Then the interdiffusion oeffiient an be epressed as: (1 ) g D = { M A + (1 ) M B } N (N is the Avogadro onstant, k B is the Boltzann onstant.) Where D M A A kt = and B B B D M kt = B, M A and M B are obilities of A and B, respetively. Defining M = [ M + (1 ) M ] (1 ) A Then, M g D = (Inside the spinodal, D <.) N B Today s Leture: Fik s first law gives. M g J = D = N Rewrite the above as M g J = D = ( ) N M g = ( ) N Then, Fik s seond law beoes J M g = = [ ( )] t N In general, M depends on the oposition, but here for sipliity, we assue suh oposition dependene is ignorable, so that M reains onstant. Thus. [ ( )] ( ) M g M g = = t N N 3
Fro Leture 3, we know G() per unit volue an be epressed as g 1 g d G() = g() d = [ g( ) + ( ) + ( ) + ( )] d d Taking a variational derivative g δg() = δ d g = + d d [ ( ) δ δ( )] d (1) d Now, let us evaluate the ter δ ( ) d d d d d d Note δ( ) = δ( ) d = ( ) δ( d) d d d d Thus, d d δ d d d d () ( ) = ( ) δ( ) Integration by parts gives d d d ( ) δ( d) = ( ) d ( ) dδ d d d Then, Eq. () an be re-written as: d d d d (3) δ( ) d = ( ) δ( d) = ( ) d ( ) dδ d d d d ********************************** What is integration by parts: In alulus, and ore generally in atheatial analysis, integration by parts is a rule that transfors the integral of produts of funtions into other (ideally sipler) integrals. The rule arises fro the produt rule of differentiation. If u = f(), v = g(), and the differentials du = f '() d and dv = g'() d, then the produt rule in its siplest for is: ********************************* 4
d At the very early stage of spinodal deoposition, oposition variation is very sall, d So, Eq. (3) beoes d d ( ) ( ) (4) δ d = δ d d d Thus, Eq. (1) beoes δ g g d G() d [ ] δ = = d δ ( ) d (5) That is d g = g ( ) d Now, Fik s first law above an be written as J M g M g 3 3 N N = ( ) = ( ) (6) And, Fik s seond law above an be written as { } 4 J M g = = 4 t N (7) The above is known as Cahn s diffusion equation, whih also aounts for the pseudo interfaial energy, the ter ontaining (see Leture 3). This ter (as disussed for the nuleation) opposes phase separation, but favors onentration aplitude growth in spinodal deoposition as disussed below. We have assued that the oposition flutuation is of the type ( ) = A os where A = aplitude, and π = = wave nuber, and is the wavelength. Assuing the wavelength to be independent of tie, then the tie dependene ust be in the aplitude, that is t (,) = A(,)os t By inspetion, it is seen that a solution to the diffusion equation (Eq. 7) has the following for: t (, ) = A(,) ep[ R( ) t] os (8) 5
where M g R( ) [ ] N = + (9) R( ) is tered aplifiation fator. As long as the ter inside the parentheses is negative (note: g < in the spinodal), R( ) >, and the aplitude will grow (see the diagra below) --- the ritial g is thus defined as: = ( / ) 1/, and =π/, or π 1 g = = [ ( )] 8π i.e., the largest (or sallest ) possible for the oposition ( ) at a teperature to vary. ---this is onsistent with what we learned in Leture 3, where To have A g g = + { } 4, i.e., to assure spontaneous proess, 1/ g We dedued the sae = ( / ) 1/, and π 1 g = [ ( )] = 8π 1/ Clearly, inside the spinodal, when < C (or > C ), then R( ) >, the onentration flutuations (aplitude) will grow; when > C (or < C ), then R( ) <, the onentration flutuations will deay away. grow deay < > As learly seen fro Eq. (8), the value of whih aiizes R() will grow the fastest. Fro Eq. (9), the aiu R() is given by dr 3 M = = [ g 8 ] + d N 6
so, g 1 = = 4 or, = --- R() is aiu at = / or at =. In other words, oposition flutuation of wavelength (or wave nuber ) very lose to ( ) grow uh ore rapidly than the rest (i.e., very uh kinetially favorable). As a result, the irostruture fored in spinodal deoposition is very unifor and fine. The typial values of are on the order of to 1Å in the early stage. Sall angle -ray sattering and eletron diffration are suitable tehniques to study the irostruture fored in spinodal deoposition. As shown in the plots of R() vs. or R() vs. below: the aial growth rate ours as a oproise between the therodynai fator and kineti fator. R() ineti (diffusion) Liitation Therodynai Liitation R() Therodynai Liitation ineti (diffusion) Liitation = 1 = Note the siilarity with ellular preipitation (see Leture ). In ellular preipitation, the laellar spaing orresponding to the aiu growth rate is twie that of the iniu possible. As learned fro Leture 3, for the flutuation to be stable (therodynaially favorable) in relation to the original, hoogeneous solid solution, we ust have A g g = + { } 4 This also gives the ritial and, 1 g 1/ = [ ( )], π 1 g = [ ( )] = 8π Meaning, when < C, or > C, g will beoe >, not therodynaially favorable for the flutuation to be stable, i.e., for the Spinodal deoposition to proeed. 1/ 7
Leture : Eutetoid phase transforation (a typial ellular preipitation): γ-iron α-iron + eentite y γ C 3 C 4 C 1 C α Fe 3 C Fe 3 C This gives * = in orresponding to the aiu growth rate as depited below. v Insuffiient driving fore: Therodynai liitation G= ineti Liitation Diffusion distane too long in *= in 8