Lecture 23: Spinodal Decomposition: Part 2: regarding free energy. change and interdiffusion coefficient inside the spinodal

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1 Leture 3: Spinodal eoposition: Part : regarding free energy hange and interdiffusion oeffiient inside the spinodal Today s topis ontinue to understand the basi kinetis of spinodal deoposition. Within the spinodal, d G/d < and interdiffusion oeffiient ( )<; how to interpret this regarding the free energy hange. adapted fro the ook "Phase Transforations in etals and lloys", by.. Porter, K. E. Easterling and. Y. Sherif, R Press, third edition. d G/d < or X here refers to the onentration of oponents. The starting point for the analysis is the ahn-hilliard equation for free energy of a solid solution. G = N [ g( ) + K( )] dv V or, in one diensional oposition variation (oposition gradient in one diension): d G = NV [ g() + K( )] d d where is the ross-setional area perpendiular to the -ais, N V is the nuber of atos/unit volue. 1

2 lso, g() and K are quantities defined on a per ato basis. If we define g() and K on a per unit volue basis, then we have d G = [ g() + K( )] d (per unit volue) (1) d In the following, onentration will be used to represent ato or olar fration. Note: The above equation was derived by loal density approiation --- that is, by epanding g as a funtion of, d d, d d, about a loally hoogeneous (unifor) solution, for whih = onstant, d d =, =,. d d Now onsider a hoogeneous solid solution of oposition, the free energy of a volue L (L = length along ) is d G ( ) = L g ( ) ( ) d =. Now, let us assue that soehow a oposition perturbation (or flutuation) is reated suh that the oposition is a funtion of position, although the average oposition is still. Then, we an epand g() about the average oposition. That is g 1 g ( ) = g ( ) + ( ) + ( ) +... Thus, Gibbs free energy for an alloy of average oposition, but with a spatially varying oposition is (Eq. (1) ) 1 G = [ g( ) + ( ) + ( ) K( ) ] d g () Note that any oposition variation an be given in ters of a Fourier series. We will siply assue that the entire oposition variation an be desribed by a single sinusoidal wave of wavelength. Speifially, let us assue that the oposition flutuation is given by π ( ) = os( ) = os β (3) where = aplitude, and β π = = wave nuber

3 Initial => g( ) Thus, when suh a oposition flutuation eists, the free energy per unit volue an be dedued as follows: First fro Eq. (), we have g ( ) [ g ( ) ( ) ( ) K( ) ] d G 1 g 1 g d = = d Seond, substitute for with Eq. (3), we get 1 g π 1 π π π g ( ) = [ g ( ) + os( ) + os ( ) + K( ) sin ( )] d g π π π π g( ) = g( ) + os( ) d + os ( ) d + K( ) sin ( ) d now, write π θ =, then, π dθ = d, d = d θ, π = θ = = θ = π Then, g 1 π g ( ) = g ( ) + osθdθ + os θdθ + K( ) sin θdθ π π π π π π Now π osθdθ =, π π os θdθ = sin θdθ = π K π g () = g ( ) + + ( ) Or 4 K β g g g K ( ) = ( ) + + = ( ) + { + β } 4 4 The above gives the free energy per unit volue of a solid solution having a oposition flutuation desribed by π ( ) = os( ) = os β (3) (4) 3

4 Eq.(3) tells us: For a hoogeneous solid solution of oposition, if a oposition perturbation (or flutuation) is reated suh that the oposition is a funtion of position, although the average oposition is still. Now, lets ontinue to analyze the above equation to interpret the spinodal deoposition, wherein g and interdiffusion oeffiient ( )<. With the oposition flutuation given by ( ) = + os β developed within the initially hoogeneous solid solution of oposition, the hange in free energy an be written (fro Eq. 4 above) as g= g g = g+ g = + K ( ) ( ) ( os β ) ( ) { β } 4 For the flutuation to be stable in relation to the original, hoogeneous solid solution, we ust have g = + { Kβ } 4 Now, K is always >, and β >, thus, Kβ >. lso, always >. Therefore, for g, it is neessary that < We have seen that inside the isibility gap (the diagra on page ), the free energy of the original solid solution is possible to have <.. The two points (S 1 and S ) at whih = are defined as spinodes (i.e., infletion points of the g urve), and the boundary of defined as the spinodal, as shown in the diagra above. = in the phase diagra is Thus, if the teperature is suh that <, i.e., inside the spinodal, and for the value of β to be suffiiently sall ( suffiiently large), it is possible to have g to be negative g = + < { Kβ } 4 4

5 Then, the flutuation is ore stable oposition opared to the hoogeneous solid solution, i.e., g <. Initial => g( ) The above equation shows that the sign of g does not depend on the agnitude of. The largest possible β (lowest ) suh that the flutuation is stable with respet to the hoogeneous solid solution (i.e., as required by g < ), is given by β, g + Kβ = 1 β = [ ( )] K 1/ or π 1 g = [ ( )] β = 8π K 1/ Positive, real values of β and are possible only inside the spinodal so that < The lower liitation (i.e., ) is due to the gradient energy ter (K), whih iplies that the flutuation (and eventual deoposition) annot our on too fine a sale. s we see later, there are ertain siilarities between the phenoenon of spinodal deoposition and ellular preipitation, ainly fro the standpoint of the role of bulk free energy and the interfaial energy (gradient energy). The Kinetis of Spinodal eoposition: Uphill iffusion (against onentration gradient), and understanding based on heial diffusion oeffiient ( ), and Inter-diffusion oeffiient ( ) s we learned before in Leture 5, the heial diffusion oeffiient (e.g., ) depends on, as epressed by the following equations: 5

6 ln γ G G = {1 + } = = ln RT RT ln γ G G = {1 + } = = ln RT RT (per ole) (per ole) Let = onentration of ato or ole fration, then the above equations ay also be written as (1 ) = = NkT NkT (1 ) = = NkT NkT g g (per unit volue) (per unit volue) Where N is the vogadro onstant, k is the oltzann onstant. Fro Nernst-Einstein Equation: kt = and kt =, where and are obilities of and, respetively. So, (1 ) N =, (1 ) N = Let us assue that the interdiffusion oeffiient ( ) obeys arken s equation (Leture 6), then, = + (1 ) (1 ) (1 ) = + N N (1 ) { (1 g ) } N = + write, = [ + (1 ) ] (1 ) so, = N 6

7 Inside the spinodal, < so, <. Now, as we learned in Leture 6, the Fik s first Law gives: J = -, J = - s the diffusion flu (J, J ) ust be positive, this eans (d /d, or d /d) ust be positive, i.e., the diffusion is up against onentration gradient, though still along heial potential or free energy gradient, G<. 7

Lecture 24: Spinodal Decomposition: Part 3: kinetics of the

Lecture 24: Spinodal Decomposition: Part 3: kinetics of the 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: