Order parameter field

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Here the functions g(0) and /(0) are defined in a suitable way to produce the desired phase behavior (see Chapter 14). The amphiphile concentration does not appear expHcitly in this model, but it influences the form of g(0)— in particular, its sign. Other models work with two order parameters, one for the difference between oil and water density and one for the amphiphile density. In addition, a vector order-parameter field sometimes accounts for the orientional degrees of freedom of the amphiphiles [1]. 

The order parameter field (r) characterizes the local structure of the phases we have investigated. The most interesting is the topology of the phases,... 

The mean-field SCFT neglects the fluctuation effects [131], which are considerably strong in the block copolymer melt near the order-disorder transition [132] (ODT). The fluctuation of the order parameter field can be included in the phase-diagram calculation as the one-loop corrections to the free-energy [37,128,133], or studied within the SCFT by analyzing stability of the ordered phases to anisotropic fluctuations [129]. The real space SCFT can also applied for a confined geometry systems [134], their dynamic development allows to study the phase-ordering kinetics [135]. 

Let us consider a dynamically symmetric binary mixture described by the scalar order parameter field < )(r) that gives the local volume fraction of component A at point r. The order parameter < )(r) should satisfy the local conservation law, which can be written as a continuity equation [143] ... 

Let now consider a system where in addition to the diffusion flux due to the chemical potential differences, there is also a certain flow field v(r, t). The equation for the temporal change of the order parameter field in this case is [1,4,157]... 

Model energy functionals will be obtained through consideration of the energetic contribution of order parameter fields, and this is preceded by a survey of order parameters. 

The local diffusion potential for a transformation, 4>(r), at a time t = t0, can be determined from the rate of change of total free energy, f, with respect to its current order-parameter field, f (r,t0). At time t = ta, the total free energy is... 

Numerical models of conserved order-parameter evolution and of nonconserved order-parameter evolution produce simulations that capture many aspects of observed microstructural evolution. These equations, as derived from variational principles, constitute the phase-field method [9]. The phase-field method depends on models for the homogeneous free-energy density for one or more order parameters, kinetic assumptions for each order-parameter field (i.e., conserved order parameters leading to a Cahn-Hilliard kinetic equation), model parameters for the gradient-energy coefficients, subsidiary equations for any other fields such as heat flow, and trustworthy numerical implementation. 

In this section we derive the effective Hamiltonian which will be the starting point for our further treatment. The strategy of the calculation is therefore separated into two steps. In the first step the system is treated in a mean-field-(MF) type approximation applied to a microscopic Hamiltonian. This leaves us with a slowly varying complex order parameter field for which we derive an effective Hamiltonian. The second step involves the consideration of the fluctuations of this order parameter. 

Using the potential V(h)°ch2 in Eq. (110), one recognizes that Eq. (110) is formally identical to a Ginzburg-Landau theory of a second-order transition for T>TC(D), with h(x,y) the order parameter field [186,216]. Therefore, it is straightforward to read off the correlation length , associated with this transition at TC(D), namely... 

Fig. 10.47. Temporal evolution of mean grain size in phase field model of grain growth (adapted from Chen and Yang (1994)). Plots are of logarithm of average grain area as a function of time, with the two curves corresponding to four (crosses) and thirty-six (circles) different order parameter fields.

Free energy functionals like those described above have been associated with several important classes of evolution equation. In the context of an order parameter field r] that are conserved, one useful approach is to posit that the dynamics of the field r] satisfies an equation of the form... 

Fig. 12.5. Illustration of the spatio-temporal evolution of the order parameter field for a system described by the Allen-Cahn equation (courtesy of W. Craig Carter). The temporal sequence runs from left to right starting with the first row.

In this subsection we will consider two extremely different approaches to the same general class of problems, namely, the development of two-phase microstructures in three dimensions. We first consider a scheme which features a combination of first-principles analysis with Monte Carlo techniques. This is followed by a phase field analysis which includes the important coupling between the order parameter field and elastic deformations. 

Cahn-Hilliard Treatment of Phase Field Evolution The temporal evolution of the order parameter field for phase separation (i.e. the concentration) is given by the Cahn-Hilliard equation... 

The rotational symmetry breaking cannot be detected in such a way there are no orientational Goldstone modes. One could look at Raman spectra or neutron diffraction experiments that are sensitive to the molecular orientations. The order parameter field for an orientation order in molecular systems can be chosen to be a three-component field of the cosine distribution of the mutual orientations of molecular axes. This index reveals the continuous, low-temperature transition. 

In this section the continuum and order parameter fields of our simple mesoscopic approach is presented. The free energy of the mixtures and a review of the most important known facts relevant to our investigation are introduced. 

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