= −J ij Si Sj − h i=1 Si (1) - The spins Si can take values ±1, - ij implies nearest-neighbor interaction only, - J > 0 is the strength of exchange interaction, - h is the magnetic field. In equilibrium at T < Tc the system magnetises. The system undergoes a 2nd order phase transition at Tc. 3
growth while second-order phase transitions proceed smoothly. • Nucleation is the process whereby new phases appear at certain sites within a metastable phase • Homogeneous nucleation - occurs spontaneously and there is no preferred nucleation site but it requires superheating or supercooling of the medium • Heterogeneous nucleation - occurs at preferential sites such as container surfaces, impurities, grain boundaries, dislocations. The effective surface area is lower here, diminishing the free energy barrier and hence facilitating nucleation. • Spinodal decomposition is more subtle than nucleation and occurs uniformly throughout. 4
a solution of two or more components can separate into distinct phases • Mechanism of phase separation in SD differs from nucleation as it happens uniformly and throughout the system and not just at the nucleation sites. • In spinodal region ∂2F ∂c2 < 0, and hence there is no thermodynamic barrier to the growth of a new phase, i.e., the phase transformation is solely diffusion controlled. • Phase separation usually occurs by nucleation and spinodal decomposition will not be observed. To observe SD, a very fast transition, a quench, is required to move from the stable to the spinodally unstable region. 5
spin in the Hamiltonian by a spatially uniform magnetization, S = m. The energy can thus be written as E(m) −J ij Si Sj − h i Si = − NqJ 2 m2 − Nhm (2) The entropy, S, can be calculated exactly S(m) = k ln N N↑ = k ln N N(1 + m)/2 (3) (4) = −Nk 1 + m 2 ln 1 + m 2 + 1 − m 2 ln 1 − m 2 − ln2 (5) where N↑ is number of up spins and N = N↑ + N↓ is total number of sites in the lattice. 6
is f (T, m) = (E − TS)/N = − NqJ 2 m2 − NkT 1 + m 2 ln 1 + m 2 + 1 − m 2 ln 1 − m 2 − ln2 The expression can be expanded in the powers of m to obtain a simplified expression of free energy, f. f = k(T − Tc) 2 m2 + kT 12 m4 − kTln2 + O(m6) (6) where Tc = qJ k for T > Tc, f has a positive curvature at origin and negative curvature for T < Tc. 7
written in the form f (m) = F(m) N = 1 2 (kT−qJ)m2−hm+ kT 12 m4−kTln2+O(m6) (9) This form of free energy makes contact with the Landau functional L = a 2 m2 + u 4 m4 (10) Ginzburg-Landau functional considers spatial variation of order parameter as well, G = a 2 m2 + u 4 m4 + K 2 (∇m)2 (11) 9
dynamics. For kinetics we assume that an associated heat bath generates spin flip (Si → −Si ). • Purely dissipative and stochastic models are ofter referred to as Kinetic Ising models. • Conserved and non-conserved cases can be describe as below: • The spin system. At the microscopic level, spin-flip Glauber model is used to describe the non-conserved kinetics of the paramagnetic to ferromagnetic transition. • The binary (AB) mixture or Lattice Gas. The spin-exchange Kawasaki model is used to describe the conserved kinetics of binary mixtures at the microscopic level. • At the coarse-grained level the respective order parameters, φ(r, t) are used to describe the dynamics. 10
a paramagnetic phase is quenched below the critical temperature Tc. • The paramagnetic state is no longer the preferred equilibrium state. • The far-from-equilibrium, homogenous, state evolves towards its new equilibrium state by separating in domains. • These domains coarsen with time and are characterized by length scale L(t). • A finite system becomes ordered in either of two equivalent states as t → ∞. • The simplest kinetics Ising model for non-conserved scalar field φ(r) is the time dependent Ginzburg- Landau (TDGL) model. 11
as: ∂φ ∂t = −Γ δF δφ + θ(r, t) (12) where δF δφ denotes functional derivative of free-energy functional F(φ) = F(φ) + 1 2 K(∇φ)2 (13) Typical form of the free energy F(φ) is given in eqn 6. The noise term has zero mean and has a white noise spectrum θ(r, t)θ(r , t ) = 2TΓδ(r − r )δ(t − t ) (14) 12
6) we arrive at the TDGL equation ∂φ ∂t = Γ a(Tc − T)φ − bφ3 + k∇2φ + θ(r, t) (15) • It is evident that φ = 0 is unstable for T < Tc and stable for T > Tc. • For T < Tc we can write TDGL in terms of rescaled variables as: ∂φ ∂t = φ − φ3 + ∇2φ (16) 13
i.e, φ = φ∗ + δφ. Plugging it back in TDGL equation and retaining only linear terms in δφ, we get ∂δφ ∂t = φ∗ + δφ − φ∗3 − φ∗2δφ + ∇2δφ (17) = (1 − 3φ∗2)δφ + ∇2δφ (18) Doing a Fourier transform we get ∂δφ ∂t = (1 − 3φ∗2 − k2)δφ (19) So, for k=0, fluctuations along φ = 0 will keep growing unless higher order terms stabilizes them. 14
∂φ ∂t = φ − φ3 + ∇2φ (20) Interface or kink can be obtained by steady state d2φ dz2 = φ − φ3 (21) The kink solution is φs(z) = tanh ± (z − z0) √ 2 (22) where z0 is center of the kink. Thus φ = ±1 except in the inter-facial region. 15
in terms of inter-facial coordinates (n, a) ∇φ = ∂φ ∂n t ˆ n (23) ∇2φ = ∂2φ ∂n2 t ˆ n · ˆ n + ∂φ ∂n t ∇ · ˆ n (24) Finally, we use the identity ∂φ ∂t n ∂t ∂n φ ∂n ∂φ t = −1 (25) in the TDGL equation, − ∂n ∂t φ ∂φ ∂n t = φ − φ3 + ∂2φ ∂n2 t ˆ n · ˆ n + ∂φ ∂n t ∇ · ˆ n (26) ∂φ ∂n t ∇ · ˆ n (27) 16
identification that ∂n ∂t φ = v(a) is normal inter-facial velocity which yields the Allen-Cahn equation v(a) = −∇ · ˆ n = −K(a) (28) where the curvature goes as K ∼ 1/L and v ∼ dL/dt, which gives the diffusive growth law for non-conserved scalar fields L(t) ∼ t1/2 (29) Here, L(t) is the typical domain size. 17
be modeled using Ising model as follows • Here nα i = 1 or 0 is occupation number of species α. • nA i + nB i = 1 for all the sites. The dynamics is conserved as numbers of A and B species are constant. • So we can identify these numbers with Si in the Ising Hamiltonian, i.e., Si = 2nA i − 1 = 1 − 2nB i . • And hence all the analysis of critical temperature goes through. • Order parameter, φ = nA(r, t) − nB(r, t), is conserved as it satisfies the continuity equation. 18
∂t = −∇ · J(r, t) J is current (30) J = −D∇µ(r, t) µ is chemical potential (31) • The chemical potential is determined as µ(r, t) = δF δφ (32) • Plugging this back in continuity equation gives the Cahn-Hilliard (CH) equation for phase separation of binary mixture. ∂φ ∂t = D∇2 δF δφ (33) 20
6), CH equation is ∂φ ∂t = ∇ · D∇[−a(Tc − T)φ + bφ3 − k∇2φ] (34) • Typical chemical potential of a domain of size L is µ ∼ σ L . • The concentration current is D|∇µ| ∼ Dσ L2 , where D is the diffusion constant. • So domains grow as dL dt ∼ Dσ L2 L(t) ∼ (Dσt)1/3 (35) 21
state to its preferred equilibrium state as parameters like temperature, etc. are changed. • Initially homogenous phase separates in phases rich in one of the constituents after quenching below Tc which is marked by emergence and growth of domains. • The domain growth law depends critically on: • conservation law governing the coarsening. • nature of defects and dimensionality (d). • relevance of hydrodynamic flow fields • The domain growth law for diffusive regime scales as: L(t) ∼ tη (36) η = 1/2: for d ≥ 2 and non-conserved order parameters. η = 1/3: for d ≥ 2 and conserved order parameters. 22
total magnetisation is time dependent on account of single-spin-flip processes. The probablity of a state {Si } can be found using conditional probabilities of ith spin being in state {Si } at time t, given that it was in state {S0 i } at time t=0. Thus we can write the master equation: dP({Si }, t) dt = N j=1 W (...Sj , ...|... − Sj , ...)P({Si }, t) (37) − N j=1 W (... − Sj , ...|...Sj , ...)P({Si }, t) The above equation is of the form Gain-Loss. Moreover the underlying stochastic process is Markovian. 27
has to be modeled in a way such that ensemble approaches the equilibrium distribution, Peq({Si }) as t → ∞ Peq({Si }) = exp[−β(H)] Z (38) where, Z is the partition function defines as, Z = {Si } exp[−β(H)] (39) Also, detailed balance demands that W (Sj |Sj )P({Si }) = W (Si |Sj )P({Si }) (40) 28
matrix W (Sj |Sj ) = λ 2 {1 − tanh[ −β H 2 ]} (41) where λ−1 sets timescale of the non-equilibrium process. Using this we obtain d S dt = − S + tanh βJ Lk SLk + βh (42) The steady state solution will have, S eq = tanh βJ Lk SLk + βh (43) These equations are often referred to as mean-field dynamical models. 29
atoms of A or B-type at lattice site is modeled by Ising model. As order parameter is conserved we can only exchange the particles. Here Spin-Exchange Kawasaki Model is being considered to write the master equation dP({Si }, t) dt = N j=1 K Lj W (...Sj , Sk, ...|...Sk, Sj , ...)P({Si }, t) (44) − N j=1 K Lj W (...Sk, Sj , ..|...Sj , Sk, ...)P({Si }, t) where K Lj means nearest neighbors The above equation is of the form Gain-Loss. Moreover the underlying stochastic process is Markovian. 30
form for the transition probability. • Finally, we arrive at what is called the Cahn-Hilliard (CH) equation. 2λ−1 ∂φ ∂t = −a2∇2 Tc T − 1 φ− 1 3 Tc T 3 φ3+ Tc qT a2∇2φ+... (45) where a is the lattice spacing • Growth law in the diffusive regime turns out to be: L(t) ∼ (t)1/3 (46) 31
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