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Phase Diagrams and Phase Separation Phase Diagrams and Phase Separation Books MF Ashby and DA Jones, Engineering Materials Vol 2, Pergamon P Haasen, Physical Metallurgy, G Strobl, The Physics of Polymers, Springer Introduction Mixing two (or more) components together can lead to new properties: Metal alloys e.g. steel, bronze, brass…. Polymers e.g. rubber toughened systems. Can either get complete mixing on the atomic/molecular level, or phase separation. Phase Diagrams allow us to map out what happens under different conditions (specifically of concentration and temperature). Free Energy of Mixing Entropy of Mixing nA atoms of A nB atoms of B AM Donald 1 Phase Diagrams Total atoms N = nA + nB Then Smix = k ln W N! = k ln nA!nb! This can be rewritten in terms of concentrations of the two types of atoms: nA/N = cA nB/N = cB and using Stirling's approximation Smix = -Nk (cAln cA + cBln cB) / kN mix S AB0.5 This is a parabolic curve. There is always a positive entropy gain on mixing (note the logarithms are negative) – so that entropic considerations alone will lead to a homogeneous mixture. The infinite slope at cA=0 and 1 means that it is very hard to remove final few impurities from a mixture. AM Donald 2 Phase Diagrams This is the situation if no molecular interactions to lead to enthalpic contribution to the free energy (this corresponds to the athermal or ideal mixing case). Enthalpic Contribution Assume a coordination number Z. Within a mean field approximation there are 2 nAA bonds of A-A type = 1/2 NcAZcA = 1/2 NZcA nBB bonds of B-B type = 1/2 NcBZcB = 1/2 NZ(1- 2 cA) and nAB bonds of A-B type = NZcA(1-cA) where the factor 1/2 comes in to avoid double counting and cB = (1-cA). If the bond energies are EAA, EBB and EAB respectively, then the energy of interaction is (writing cA as simply c) 1/2 NZ [cEAA + (1-c)EBB + c(1-c) (2EAB - EAA – EBB)] energy of 2 separate starting materials ∴ Umix = 1/2 NZ c(1-c) (2EAB – EAA – EBB) AM Donald 3 Phase Diagrams The term (2EAB – EAA – EBB) determines whether mixing or demixing occurs. Define χkT = Z/2 (2EAB – EAA – EBB) where χ is known as the interaction parameter, and is dimensionless. (The definition of χ in this way, including the kT term, is for historical reasons). Fmix/ kT = c ln c + (1-c) ln (1-c) + χ kT c(1-c) Three cases to consider Þ χ =0 athermal or ideal solution case. / kT mix F Fmix = -TSmix Fmix is always negative. AM Donald 4 Phase Diagrams Þ χ>0 i.e. 2EAB> EAA + EBB χc(1-c) / kT mix F -TSmix Resultant Fmix 2 local minima result Þ χ< 0 χc(1-c) -TSmix / kT mix F Fmix One deep minimum AM Donald 5 Phase Diagrams These curves correspond to the case for one particular T. In order to plot out a phase diagram, we need to be able to see how temperature affects these curves. And we need to know whether or not these curves imply phase separation. Finally we need to know, if phase separation occurs, how much of each phase is present. Lever Rule for Determining Proportions of Phases Imagine we have a system with 2 phases present (labelled 1 and 2), and two types of atoms A and B. In phase 1 concentration of A = c1 In phase 2 concentration of A = c2 and that there is x of phase 1 present and (1-x) of phase 2 present. Total of N atoms, overall concentration of A = c ∴ no of A atoms is Nc = Nxc1 + N(1-x) c2 n c − c n m Þ x = 2 = − c1 c2 l c c c 1 2 Þ Lever Rule AM Donald l 6 Phase Diagrams c − c m x c − c n Similarly 1-x= 1 = and = 2 = − − − c1 c2 l 1 x c c1 m Thus, when one wants to find the average of some quantity such as F, it is usually sufficient (neglecting surface effects) to take a weighted average. Consider the free energy F as F(c) F S F1 F2 PQR c1 c c2 c F(c) = F2 + (F1 – F2) QR/PR By similar triangles, Þ F(c ) = SQ in the diagram. Therefore one can find free energy of any intermediate composition by drawing straight line between the free energies of the two constituent phases. Can now use this to interpret free energy curves. Consider cases of χ=0 or negative; free energy curve had a single minimum. AM Donald 7 Phase Diagrams As pure components (A and B) F(c ) = F F FB F F 1 A B1 A1 Fn A C B This can be lowered by going to compositions A1,B1 to give free energy F1 etc And as A and B continue to dissolve more and more of each other, free energy continues to drop. Minimum energy for homogeneous single phase, energy Fn i.e no phase separation occurs in this case. Composition overall determines the state of the mixture. AM Donald 8 Phase Diagrams Minimum free energy will not be given (in general) by the minimum in the free energy curve. e.g. starting with composition c, if this were to result in composition corresponding to the minimum on the curve cmin, would necessarily also have phase with composition c' present. c' c cmin Necessary condition for homogeneous mixing to occur is for d 2 f ≥ 0 everywhere dc2 However when χ is positive, this inequality does not hold, and the behaviour is very different. Start with homogeneous free energy F, for concentration c. AM Donald 9 Phase Diagrams FB F B1 A1 F1 FA A 3 B3 F3 A cA c cB B This can split into A1 and B1; energy drops to F1. Minimum in energy occurs at F3 when the representative points on the free energy curve are joined by the lowest straight line: common tangent construction. In this case, have phase separation into compositions cA and cB, with phases α and β. For compositions c < cA have α phase cA < c < cB have α + β cB<c have just β. Proportions of α and β given by Lever rule. For c<cA A dissolves B For cB<c B dissolves A cA and cB define solubility limits. AM Donald 10 Phase Diagrams This common tangent construction can be extended to quite complicated situations, with several minima, which then give rise to quite complicated free energy curves and hence phase diagrams. For plotting a phase diagram we need to know how solubility limits (as determined by the common tangent construction) vary with temperature. Have seen that if d2F/dc2 everywhere ≥0 have a homogeneous solution. Phase separation occurs when free energy curve has regions of negative curvature. This permits us to evaluate the limits of solubility in terms of χ. For the symmetric case (i.e FA = FB i.e terms involving EAA and EBB are assumed equal) Fmix/ kT = c ln c + (1-c) ln (1-c) + χc (1-c) per site d2 F 1 1 ∴ = kT( + − 2χ) dc2 c 1− c Critical value when d2F/dc2 = 0 1 1 2 Þ c = ± 1− 2 2 χ For a regular solution χ ∝ 1/T = A/T AM Donald 11 Phase Diagrams Then limits of solubility vary with temperature according to 1 1 2T c = ± 1− 2 2 A i.e. as T raised the two compositions of phases α and β converge, until at Tcrit = A/2 there is no further separation and a homogeneous mixture results. In general, critical value of χ = 2. There is no solution for c for χ < 2 so there is no phase separation. Also, for the symmetrical case, the common tangent construction reduces to the condition df/dc = 0 (i.e. horizontal tangent) This equation defines the binodal or coexistence curve. 1 c χ = ln( ) binodal 2c −1 1− c For the regular solution case, with χ∝ 1/Τ this allows us to plot out how the binodal behaves. Some comments on this Approach AM Donald 12 Phase Diagrams This approach of a 'symmetrical' AB mixture is very similar to the Ising model, where the 2 states correspond to opposite spins (although of course A cannot transform to B as spins can). The approach used has been a mean field theory. This will fail: Þ near a critical point: will give the wrong critical point and the wrong critical exponents. In practice most of the phase diagram will be well away from this point, so this is not too severe. Þ for strongly negative χ. In this case there are strong attractions between unlike atoms and the idea of random distributions breaks down. This can lead to order-disorder transitions. In general it is difficult to calculate χ from 1st principles. Picture described here works best for liquid-liquid phase separation. Melting may complicate matters. For solid-solid transitions have additional problems, leading to very complicated phase diagrams, due to Strain energy Other intermediate compounds eg AB2. However for the regular solution model we now can construct a phase diagram – which contains all the essential physics. AM Donald 13 Phase Diagrams χ<2 , only a single minimum Þ homogeneous mixture 0 -0.2 -0.4 chi= 2.5 -0.6 chi= 2.0 per site /kT chi= 0 -0.8 chi=-1.0 T Free energy of mixing energy of mixing Free -1 decreasing 0 0.2 0.4 0.6 0.8 1 concentration χ>2 , two minima develop Þ phase separation Free energy of mixing per site /kT T decreasing concentration AM Donald 14 Phase Diagrams Knowing the form of the free energy curves as a function of T means that we can map out the limits of solubility etc.
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