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Unformatted text preview: g(c) miscibility gap spinodal stable nonlinearly unstable linearly unstable 0 c- c s- c s+ c + 1 VIII. Phase Transformations Lecture 38: Nucleation and Spinodal Decomposition MIT Student In this lecture we will study the onset of phase transformation for phases that differ only in their equilibrium composition, keeping in mind that it is possible for phases to differ in other physical quantities such as density, crystal structure, magnetization, etc. Consider a regular solution model for free energy, illustrated in figure 1. Suppose a homogeneous system is in region of its phase diagram (P,T,c) Regular solution model Figure 1 where the homogeneous phase is unstable thermodynamically. The system can lower its free energy by separating into two phases with an interface in between. In figure 1, c and c + are common tangent points that mark the- boundaries of the miscibility gap. For any composition inside the miscibility gap (between c and c + ), a phase separated system is energetically favorable.- 1 Lecture 38: Nucleation and spinodal decomposition 10.626 (2011) Bazant Outside the miscibility gap, the system remains homogeneous at equilibrium no matter what the composition. In his classical treatment of phase transformations, Gibbs distinguished between two types of transformations: those that are small in degree and large in extent (spinodal decomposition, linear instability), and those that are large in degree and small in extent (nucleation, nonlinear instability). Lets examine both in a bit more detail. 1 Spinodal decomposition Compositions between c s and c s + lie within the chemical spinodal, a re- gion of linear instability where g ( c ) < and small uctuations will grow spontaneously through a process called spinodal decomposition. Previously we showed that g ( c ) < is the condition for linear instability by graphical construction (ignoring variations of with c ). Spontaneous phase sepa ration is favored when the chord to a free energy curve is lower everywhere than the curve itself: (a) g ( c ) < promotes spinodal decom position. Spontaneous separation into c + c and c- c lowers the total free energy ( c << c ). (b) g ( c ) > does not promote spon taneous phase separation. Small com position uctuations raise the total free energy. 2 Classical nucleation Nucleation is a nonlinear instability that requires the formation of a large enough nucleus of the nucleating phase. The creation of a nucleus of the low-energy phase with concentration c + form a matrix of concentration c * (where c- < c * < c s ), is illustrated in figure 2a. There is a decrease in free- energy g associated with the conversion of c to c + ,...
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