Decomposition spinodal

Phase diagram (left hand side) and spinodal (right hand side) for a completely immiscible system derived from free energy considerations. Mixtures cooled into the spinodal separate spontaneously whereas mixtures between the spinodal and the region of immiscibiUty are metastable and require a nucleation event to separate. 

From this discussion of the concept of spinodal decomposition it follows that, for a description of the kinetics of the spinodal decomposition process, it is necessary to formulate 

N% = Starting concentration of homogeneous solid solution in the metastable range. 

AN = Concentration fluctuation starting from iVj, not leading to nucleation. AN - Concentration fluctuation starting from TVb, leading to nucleation of precipitate. 

In order to allow a simple treatment, the decomposition problem will be discussed for a binary, one-dimensional system. A one-dimensional decomposition process in reality always occurs if the lattice parameter is strongly composition dependent and if the crystal exhibits elastical anisotropy. In this case, the strain energy during decomposition is strongly dependent upon the crystallographic direction in the crystal. The extension of the one-dimensional treatment to a three-dimensional case, however, offers no difficulties in principle. 

The formal treatment of spinodal decomposition starts with the flux equations (5-34) which are of the form  

The points of these segments represent the AmixG of two-phase alloys. In the composition range between the maximum and the spinodal (xs) a two-phase alloy, such as a mixture xul + xu2, has therefore an overall free energy lower than that of any single-phase alloy of an intermediate composition, which is therefore unstable. 

The energy increase related to a compositional fluctuation resulting in a two-phase splitting may be considered as an energy threshold of activation of the de-mixing process. The spinodal curve thus defines a kinetic limit, not a phase boundary line. 

From this descriptive introduction, it follows that the coherent spinodal decomposition is a continuous transport process occurring in a supersaturated matrix. It is driven by chemical potential gradients. Strain energy and concentration gradient energy have to be adequately included in the component chemical potentials. We expect that the initial stages of decomposition are easier to treat quantitatively than the later ones. The basic result will be the (directional) build-up of periodic variations in concentration [J.W. Cahn (1959), (1961), (1968)]. 

The third contribution to the chemical potential is due to strain. If A and B atoms (ions) have different size, clustering results in elastic lattice distortions. By making a Fourier transformation, one can decompose the concentration profile into harmonic plane waves [D. DeFontaine (1975)]. The elastic energy contributions of these concentration waves are additive in the Unear elastic regime and yield Ea. Therefore, we may write 

Equations of type (12.30) can be used to describe the kinetics of the spinodal decomposition process. If an arbitrary, spontaneous concentration fluctuation is Fourier analyzed, one finds that for 

Furthermore, for A = l/2-Amin, the term in brackets in Eqn. (12.30) has a maximum. Therefore, after a sufficient time of continuous spinodal decomposition, zones with the Amax periodicity will predominate. With other words, the decomposed solid solution exhibits periodicity in the direction, and the period length is Amax- min and max increase with strain, as can be seen from Eqns. (12.29) and (12.31). If the strain energy is high enough, Amax may become sufficiently large so that spinodal decomposition does not take place any more. 

According to Eqn. (12.30), r = [ ]mi n is the characteristic relaxation time of the decomposition process ([ ] represents the term in brackets). Let us assume D, = (ibj/RT) to be on the order of 10 10 cm2/s. r is then on the order of a second or less. This means that in-situ observations of the spinodal process are hardly possible. If the sample has been quenched to room temperature, the decomposition has often already reached its final stage. The continuous spinodal decomposition for which the early stages are the pertinent ones cannot be verified in this way. 

In 1965, Cahn [10] laid a basis for the mathematical treatment of spinodal decomposition, starting with an expression for the free energy of an inhomogeneous solution, the composition of which always differed only slightly from the average composition, and with small composition gradients. 

f c) is the free-energy density of homogeneous material of composition c, and KiycY is the additional free-energy density if the material is in a gradient in composition. In the development of the theory it became convenient to consider the Fourier components of the composition rather than the composition. Because of the orthogonality of the Fourier components, AF is the sum of contributions from each Fourier component separately. 

The kinetics of the initial stages of phase separation can be obtained by solving the diffusion equation. It is necessary to define a mobility M vhich is (minus) the ratio of diffusional fiux to the gradient in chemical potential 

Simple thermodynamic considerations show that M must be positive if diffusion which results spontaneously from the chemical potential gradient is to result in a decrease in free energy. One obtains — /Ug from the variational derivative of F in Eq. (5.8), 

Since the coefficient of V c may be identified with a diffusion coefficient, it can be seen that at the spinodal the diffusion coefficient changes sign. The solution to Eq. (5.11) is 

It was already mentioned that the thermodynamic stabiUty of the multi-component systems is determined by the concentration dependence of the Gibbs free energy. If 0 (as in the region between spinodal and bin- 

Cahn s theory was developed for the initial stages of phase separation (remember that the final result of the equilibrium phase separation does not depend on the mechanism and is determined only by the phase diagram of the system). 

We can apply this theory to our case provided phase separation takes no more than 200-300 s (the first linear branch). In our case, during the reaction time when phase separation was observed, the composition of the system changes only slightly and can be considered as constant. This fact gives us the right to apply Cahn s theory for describing the phase separation in our system. However, we believe that Cahn s theory can also be formally applied to the second linear branches in Fig. 9. 

the value of Am characterizes the size of the microheterogeneity regions. The value of Am is calculated from the dependence mentioned above. The interdiffusion coefficients and the sizes of the microheterogeneity regions for the systems with 10, 15, and 20 mass % of PBMA are given in Table 1 [88]. 

The data on the microphase structure of the semi-IPN show that the phase separation proceeds according to the spinodal mechanism in spite of the simultaneously proceeding chemical reaction. Thus, the process is a nonequilibrium one. The peculiarity of this process is in its two-stage nature. It 

Below the critical temperature, in the two-phase region, fluids may exhibit two distinctly diflerent types of dynamic behaviour. If a quench (rapid cooling) drives the system into a state between the coexistence curve and the spinodal (the absolute limit of thermodynamic stability), then the system will be in a metastable state and will phase separate through homogeneous nucleation. However, if the quench drives the system inside the spinodal, then the system will exhibit spinodal decomposition .  

When the system is quenched into the metastable region between the coexistence curve and the spinodal, the resulting metastable phase may not spontaneously decay into two-phase equilibrium. The transformation must be activated by some perturbation, such as thermal fluctuations. Consider a spherical liquid droplet, of radius R, immersed in the uniform metastable vapour. The diflerence of the chemical potential in the metastable state and the chemical potential in equilibrium is A/x = p-pcxc- The total Gibbs energy of the droplet is given by the sum of the bulk and surface energy contributions. 

The chemical potential difference in the inhomogeneous liquid mixture is the derivative of the Landau-Ginzburg functional, which in this case describes the local Gibbs energy. 

Substituting this expression into eq 7.98 we arrive at the Cahn equation for spinodal decomposition  

Taking a Fourier transform of the Cahn equation, one finds that the fluctuations grow exponentially as exp(2Tt) with a wave number-dependent rate given by 

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The unstable situation caused when a spread him begins to dewet the surface has been studied [32, 33]. IDewetting generally proceeds from hole formation or retraction of the him edge [32] and hole formation can be a nucleation process or spinodal decomposition [34]. Brochart-Wyart and de Gennes provide a nice... 

Gonnella G, Orlandini E and Yeomans J M 1997 Spinodal decomposition to a lamellar phase effect of hydrodynamic flow Phys. Rev. Lett. 78 1695... 

Binder K 1983 Collective diffusion, nucleation, and spinodal decomposition in polymer mixtures J. Chem. Phys. 79 6387... 

Therefore, the locus of the values ( ) with a vanishing second derivative of A delimits the region of the miscibility gap in which spinodal decomposition occurs. This locus is referred to as the spinodal (figure C2.1.10 (bl). The length scale of the concentration fluctuations at the beginning of the separation process is controlled by... 

Verhaegh NAM, Asnaghi D, Lekkerkerker FI N W, Giglio M and Cipelletti L 1997 Transient gelation by spinodal decomposition in colloid-polymer mixtures Phys/ca A 242 104-18... 

ALnico 5, with added ductihty. The very Low Co alloys, however, require extremely long he at-treatment times because of the decreased kinetics of the spinodal decomposition. Deformation aged 23%Cr—23%Co—2%Cu exhibits (BH) of 78 kJ/m (9.75MG - Oe) (85). 

In the examples given below, the physical effects are described of an order-disorder transformation which does not change the overall composition, the separation of an inter-metallic compound from a solid solution the range of which decreases as the temperature decreases, and die separation of an alloy into two phases by spinodal decomposition. 

The kinetics of spinodal decomposition is complicated by the fact that the new phases which are formed must have different molar volumes from one another, and so tire interfacial energy plays a role in the rate of decomposition. Anotlrer important consideration is that the transformation must involve the appearance of concenuation gradients in the alloy, and drerefore the analysis above is incorrect if it is assumed that phase separation occurs to yield equilibrium phases of constant composition. An example of a binary alloy which shows this feature is the gold-nickel system, which begins to decompose below 810°C. 

Figure 6.5 The appearence of spinodal decomposition as the temperature is lowered from a range of complete solubility, to the separation of two phases. In the range of composition between the inflection points, the equilibrium spinodal phases should begin to separate...

Lipson (1943, 1944), who had examined a copper-nickeMron ternary alloy. A few years ago, on an occasion in honour of Mats Hillert, Cahn (1991) mapped out in masterly fashion the history of the spinodal concept and its establishment as a widespread alternative mechanism to classical nucleation in phase transformations, specially of the solid-solid variety. An excellent, up-to-date account of the present status of the theory of spinodal decomposition and its relation to experiment and to other branches of physics is by Binder (1991). The Hillert/Cahn/Hilliard theory has also proved particularly useful to modern polymer physicists concerned with structure control in polymer blends, since that theory was first applied to these materials in 1979 (see outline by Kyu 1993). 

Thermodynamics and kinetics of phase separation of polymer mixtures have benefited greatly from theories of spinodal decomposition and of classical nucleation. In fact, the best documented tests of the theory of spinodal decomposition have been performed on polymer mixtures. 

Phase transitions in two-dimensional (adsorbed) layers have been reviewed. For the multicomponent Widom-Rowlinson model the minimum number of components was found that is necessary to stabilize the non-trivial crystal phase. The effect of elastic interaction on the structures of an alloy during the process of spinodal decomposition is analyzed and results in configurations similar to those found in experiments. Fluids and molecules adsorbed on substrate surfaces often have phase transitions at low temperatures where quantum effects have to be considered. Examples are layers of H2, D2, N2, and CO molecules on graphite substrates. We review the PIMC approach, to such phenomena, clarify certain experimentally observed anomahes in H2 and D2 layers and give predictions for the order of the N2 herringbone transition. Dynamical quantum phenomena in fluids are also analyzed via PIMC. Comparisons with the results of approximate analytical theories demonstrate the importance of the PIMC approach to phase transitions, where quantum effects play a role. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

K. Binder. Spinodal decomposition. In P. Haasen, ed.. Materials Science and Technology, Vol. 5. Weinheim VCH-Verlag, 1990. 

Fig. 2 illustrates the ordering process after a quench of a disordered alloy below the ordering spinodal. As it was mentioned by AC, the primary ordered domains are formed after few atomic exchanges A.t 1, while further evolution corresponds to the growth of these domains. Fig. 3 shows that in the absence of APBs the microstructure evolution under spinodal decomposition with ordering is similar to that for disordered... 

Fig. 3. Temporal evolution of Cj under spinodal decomposition of a single domain ordered state, at T = 0.42, c = 0.325, and following t (a) 500, (b) 2000, (c) 3000, and (d) 10000. The grey level in Figs. 3-5 linearly varies with Ci between Cj = 0 and c = 1.

Replication of interphase boundaries under spinodal decomposition... 

In Fig. 12 we present some results of MFKEbbased simulation of spinodal decomposition with the vacancy-mediated exchange mechanism. We use the same 2D model on a square lattice with the nearest-neighbor interaction and Fp = 0 as in Refs., ... 

Since the prefactor in Eq. (17) is a universal constant of order unity, the barrier AF / kaT is large only very close to the coexistence curve, i.e. for 5v / v /coex, while for larger 5v / the smallness of the barrier implies a very grad il transition from nucleation to spinodal decomposition.Conversely, for N x 1 where Eq. (16) holds the transition is very sharp since the barrier stays large right up to the spinodal for qo. 

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