1、Thermodynamics and Phase Diagrams from Cluster Expansions,Dane Morgan University of Wisconsin, ddmorganwisc.eduSUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE Hands-on introduction to Electronic Structure and Thermodynamics Calculations of Real Materials University of Illinois at Urbana-Champaign,
2、June 13-23, 2005,The Cluster Expansion and Phase Diagrams,a = cluster functions s = atomic configuration on a lattice,H. Okamoto, J. Phase Equilibria, 93,How do we get the phase diagram from the cluster expansion Hamiltonian?,Cluster Expansion,Outline,Phase Diagram Basics Stable Phases from Cluster
3、Expansion - the Ground State Problem Analytical methods Optimization (Monte Carlo, genetic algorithm) Exhaustive search Phase Diagrams from Cluster Expansion: Semi-Analytical Approximations Low-T expansion High-T expansion Cluster variation method Phase Diagrams from Cluster Expansion: Simulation wi
4、th Monte Carlo Monte Carlo method basics Covergence issues Determining phase diagrams without free energies. Determining phase diagrams with free energies.,Phase Diagram Basics,What is A Phase Diagram?,Phase: A chemically and structurally homogeneous portion of material, generally described by a dis
5、tinct value of some parameters (order parameters). E.g., ordered L10 phase and disordered solid solution of Cu-Au Gibbs phase rule for fixed pressure: F(degrees of freedom) = C(# components) - P(# phases) + 1 Can have 1 or more phases stable at different compositions for different temperatures For a
6、 binary alloy (C=2) can have 3 phases with no degrees of freedom (fixed composition and temperature), and 2 phases with 1 degree of freedom (range of temperatures). The stable phases at each temperature and composition are summarized in a phase diagram made up of boundaries between single and multip
7、le phase regions. Multi-phase regions imply separation to the boundaries in proportions consistent with conserving overall composition.,H. Okamoto, J. Phase Equilibria, 93,The stable phases can be derived from optimization of an appropriate thermodynamic potential.,2 phase,3 phase,1 phase,The stable
8、 phases minimize the total thermodynamic potential of the system The thermodynamic potential for a phase a of an alloy under atmospheric pressure: The total thermodynamic potential isThe challenges: What phases d might be present? How do we get the Fd from the cluster expansion? How use Fd to get th
9、e phase diagram? Note: Focus on binary systems (can be generalized but details get complex), focus on single parent lattice (multiple lattices can be treated each separately),Thermodynamics of Phase Stability,Stable Phases from Cluster Expansion the Ground State Problem,Determining Possible Phases,A
10、ssume that the phases that might appear in phase diagram are ground states (stable phases at T=0). This could miss some phases that are stabilized by entropy at T0. T=0 simplifies the problem since T=0 F is given by the cluster expansion directly. Phases d are now simply distinguished by different f
11、ixed orderings sd.So we need only find the s that give the T=0 stable states. These are the states on the convex hull.,H. Okamoto, J. Phase Equilibria, 93,The Convex Hull,a,b,d,Energy,CB,A,B,Convex Hull in blue,2-phase region,1-phase point,None of the red points give the lowest F=SFd. Blue points/li
12、nes give the lowest energy phases/phase mixtures. Constructing the convex hull given a moderate set of points is straightforward (Skiena 97) But the number of points (structures) is infinite! So how do we get the convex hull?,Getting the Convex Hull of a Cluster Expansion Hamiltonian,Linear programm
13、ing methods Elegantly reduce infinite discrete problem to finite linear continuous problem. Give sets of Lattice Averaged (LA) cluster functions LA(f) of all possible ground states through robust numerical methods. But can also generate many “inconstructable” sets of LA(f) and avoiding those grows e
14、xponentially difficult. Optimized searching Search configuration space in a biased manner to minimize the energy (Monte Carlo, genetic algorithms). Can find larger unit cell structures that brute force searching Not exhaustive can be difficult to find optimum and can miss hard to find structures, ev
15、en with small unit cells. Brute force searching Enumerate all structures with unit cells Nmax atoms and build convex hull from that list. Likely to capture most reasonably small unit cells (and these account for most of what are seen in nature). Not exhaustive can miss larger unit cell structures.,(
16、Zunger, et al., http:/www.sst.nrel.gov/topics/new_mat.html),(Blum and Zunger, Phys. Rev. B, 04),Phase Diagrams from Cluster Expansion: Semi-Analytical Approximations,Semi-Analytic Expressions for F (F),High-temperature expansion Low-temperature expansion Mean-field theory,From basic thermodynamics w
17、e can write F in terms of the cluster expansion Hamiltonian,But this is an infinite summation how can we evaluate F?,Cluster expansion,For a binary alloy on a fixed lattice the number of particles is conserved since NA+NB=N=# sites, thus we can write the semi-grand canonical potential F in terms of
18、one chemical potential and NB (Grand canonical = particle numbers can change, Semi-Grand canonical = particle types can change but overall number is fixed),High-Temperature Expansion,Assume x=b(E-mn) is a small number (high temperature) and expand the ln(exp(-x),Could go out to many higher orders ,H
19、igh-Temperature Expansion Example (NN Cluster Expansion),z = # NN per atom,So first correction is second order in bVNN and reduces the free energy,Low-Temperature Expansion,Start in a known ground state a, with chemical potentials that stabilize a, and assume only lowest excitations contribute to F,
20、This term assumed small,Expand ln in small term,Keep contribution from single spin flip at a site s,Low-Temperature Expansion Example (NN Cluster Expansion),Assume an unfrustrated ordered phase at c=1/2,So first correction goes as exp(-2zbVNN) and reduces the free energy,kBTc,Transition Temperature
21、from LT and HT Expansion,NN cluster expansion on a simple cubic lattice (z=6) VNN0 antiferromagnetic ordering,kBTc/|zV|=0.721 (0th), 0.688 (1st), 0.7522 (best known),Mean-Field Theory The Idea,The general idea: Break up the system into small clusters in an average “bath” that is not treated explicit
22、ly,For a small finite lattice with N-sites finding f is not hard just sum 2N terms,For an infinite lattice just treat subclusters explicitly with mean field as boundary condition,Mean field,Treated fully,Implementing Mean-Field Theory The Cluster Variation Method,Write thermodynamic potential F in t
23、erms of probabilities of each configuration r(s), Fr(s). The true probabilities and equilibrium F are given by minimizing Fr(s) with respect to r(s), ie, dFr(s)/dr(s)=0. Simplify r(s) using mean-field ideas to depend on only a few variables to make solving dFr(s)/dr(s)=0 tractable.,(Kikuchi, Phys. R
24、ev. 51),Writing fr(s).,Where,Factoring the Probability to Simplify r(s),Irreducible probabilities. Depend on only spin values in cluster of points h. Have value 1 if the sites in h are uncorrelated (even if subclusters are correlated),Cluster of lattice points.,Probability of finding spins sh on clu
25、ster of sites h.,Kikuchi-Barker coefficients,Has 2N values,Has 2NhM values much smaller,Maximal size cluster of lattice points to treat explicitly.,Truncating the Probability Factorization = Mean Field,aM,Setting,treats each cluster aM explicitly and embeds it in the appropriate average environment,
26、The Mean-Field Potential,F now depends on only,and can be minimized to get approximate probabilities and potential,The Modern Formalism,Using probabilities as variables is hard because you must Maintain normalization (sums = 1) Maintain positive values Include symmetry A useful change of variables i
27、s to write probabilities in terms of correlation functions this is just a cluster expansion of the probabilities,The CVM Potential,For a multicomponent alloy,Simplest CVM Approximation The Point (Bragg-Williams, Weiss Molecular Field),aM=Single point on lattice,For a disorderd phase on a lattice wit
28、h one type of site,CVM Point Approximation - Bragg-Williams (NN Cluster Expansion),Disordered phase more complex for ordered phases,Bragg-Williams Approximation (NN Cluster Expansion),Predicts 2-phase behavior,Criticaltemperature,Single phase behavior,Comparison of Bragg-Williams and High-Temperatur
29、e Expansion,Assume,High-temperature,Bragg-Williams,Optimize F over cB to get lowest value cB=1/2 ,Bragg-Williams has first term of the high-temperature expansion, but not second. Second term is due to correlations between sites, which is excluded in BW (point CVM),Critical Temperatures,HT/LT approx:
30、 kBTc/|zV|=0.721 (0th), 0.688 (1st),de Fontaine, Solid State Physics 79,Limitations of the CVM (Mean-Field), High- and Low-Temperature Expansions,CVM Inexact at critical temperature, but can be quite accurate. Number of variable to optimize over (independent probabilities within the maximal cluster)
31、 grows exponentially with maximal cluster size. Errors occur when Hamiltonian is longer range than CVM approximation want large interactions within the maximal cluster. Modern cluster expansions use many neighbors and multisite clusters that can be quite long range. CVM not applicable for most moder
32、n long-range cluster expansions. Must use more flexible approach MonteCarlo! High- and Low-Temperature Expansions Computationally quite complex with many terms Many term expansions exist but only for simple Hamiltonians Again, complex long-range Hamiltonians and computational complexity requires oth
33、er methods Monte Carlo!,Phase Diagrams from Cluster Expansion: Simulation with Monte Carlo,What Is MC and What is it for?,MC explores the states of a system stochastically with probabilities that match those expected physically Stochastic means involving or containing a random variable or variables,
34、 which is practice means that the method does things based on values of random numbers MC is used to get thermodynamic averages, thermodynamic potentials (from the averages), and study phase transitions MC has many other applications outside materials science, where is covers a large range of method
35、s using random numbers Invented to study the neutron diffusion in bomb research at end of WWII Called Monte Carlo since that is where gambling happens lots of chance!,http:/www.monte-carlo.mc/principalitymonaco/index.html,http:/www.monte-carlo.mc/principalitymonaco/entertainment/casino.html,MC Sampl
36、ing,Can we perform this summation numerically? Simple Monte Carlo Sampling: Choose states s at random and perform the above summation. Need to get Z, but can also do this by sampling at random This is impractically slow because you sample too many terms that are near zero,r(s) is the probability of
37、a having configuration s,Problem with Simple MC Sampling r(s) is Very Sharply Peaked,States s,r(s),Sampling states here contributes 0 to integral,Almost all the contribution to an integral over r(s) comes from here,E.g., Consider a non-interacting cluster expansion spin model with H=-mNB. For b=m=1
38、cB=1/(1+e)=0.27. For N=1000 sites the probability of a configuration with cB=0.5 compared to cB=0.27 is,P(cB=0.5)/P(CB=0.27)=exp(-NDcB)=10-100,Better MC Sampling,We need an algorithm that naturally samples states for which r(s) is large. Ideally, we will choose states with exactly probability r(s) b
39、ecause When r(s) is small (large), those s will not (will) be sampled In fact, if we choose states with probability r(s), then we can write the thermodynamic average as r(s) is the true equilibrium thermodynamic distribution, so our sampling will generate states that match those seen in an equilibri
40、um system, which make them easy to interpret The way to sample with the correct r(s) is called the Metropolis algorithm,Detailed Balance and The Metropolis Algorithm,We wants states to occur with probability r(s) in the equilibrated simulation and we want to enforce that by how we choose new states
41、at each step (how we transition). Impose detailed balance condition (at equilibrium the flux between two states is equal) so that equilibrium probabilities will be stable,r(o)p(on)=r(n)p(no),Transition matrix p(on) = a(on) x acc(on), where a is the attempt matrix and acc is the acceptance matrix. Ch
42、oose a(on) symmetric (just pick states uniformly): a(on)=a(no) Then,r(o)p(on)=r(n)p(no) r(o)xacc(on)=r(n)xacc(no) acc(on)/acc(no) = r(n)/r(o) = exp(-bF(n)/exp(-bF(o),So choose,acc(on) = r(n)/r(o) if r(n)=r(o),This keeps detailed balance (stabilizes the probabilities r(s) and equilibrates the system
43、if it is out of equilibrium this is the Metropolis Algorithm There are other solutions but this is the most commonly used,The Metropolis Algorithm (General),An algorithm to pick a series of configurations so that they asymptotically appear with probability r(s)=exp(-bE(s) Assume we are in state si C
44、hoose a new state s*, and define DE=E(s*)-E(s) If DE0 then accept s* with probability exp(-bDE) If we accept s* then increment i, set si=s* and return to 1. in a new state If we reject s* then return to 1. in the same state state si This is a Markov process, which means that the next state depends o
45、nly on the previous one and none before.,Metropolis Algorithm for Cluster Expansion Model (Real Space),We only need to consider spin states Assume the spins have value (s1 , sj , sN) Choose a new set of spins by flipping, sj* = -sj, where i is chosen at random Find DF=E(s1 , -sj , sN)-E(s1 , sj, sN)
46、-msj (note that this can be done quickly be only recalculating the energy contribution of spin i and its neighbors) If DF0 then accept spin flip with probability exp(-bDF) If we reject spin flip then change nothing and return to 1 The probability of seeing any set of spins s will tend asymptotically
47、 to,Obtaining Thermal Averages From MC,The MC algorithm will converge to sample states with probability r(s). So a thermal average is given by Note that Nmcs should be taken after the system equilibrates Fluctuations are also just thermal averages and calculated the same way,Energy Vs. MC Step,MC St
48、ep,Energy,Equilibration period: Not equilibrated, thermal averages will be wrong,Equilibrated, thermal averages will be right,Correlation length?,Measuring Accuracy of Averages,What is the statistical error in ?,Vl is the covariance and gives the autocorrelation of A with itself l steps later Longer
49、 correlation length less independent data Less accurate ,For uncorrelated data,But MC steps are correlated, so we need more thorough statistics,Example of Autocorrelation Function,D MC Step,Normalized autocorrelation,Correlation length 500 steps,Semiquantitative Understanding of Role of Correlation in Averaging Errors,This makes sense! Error decreases with sqrt of the number of uncorrelated samples, which only occur every L/Lc steps. As Lc1 this becomes result for uncorrelated data.,
