Equilibrium morphology

In all considered above models, the equilibrium morphology is chosen from the set of possible candidates, which makes these approaches unsuitable for discovery of new unknown structures. However, the SCFT equation can be solved in the real space without any assumptions about the phase symmetry [130], The box under the periodic boundary conditions in considered. The initial quest for uy(r) is produced by a random number generator. Equations (42)-(44) are used to produce density distributions T(r) and pressure field ,(r). The diffusion equations are numerically integrated to obtain q and for 0 < s < 1. The right-hand size of Eq. (47) is evaluated to obtain new density profiles. The volume fractions at the next iterations are obtained by a linear mixing of new and old solutions. The iterations are performed repeatedly until the free-energy change... 

The equilibrium shape of a crystal is, as described above, a polyhedron where the size of the crystal facets is inversely proportional to their surface energy, ysg. In the present section we will consider other types of interfaces as well and we will show that the interface energies determine the equilibrium morphology of interfaces in general. 

Crystallization from the melt often leads to a distinct (usually lamellar) structure, with a different periodicity from the melt. Crystallization from solution can lead to non-lamellar crystalline structures, although these may often be trapped non-equilibrium morphologies. In addition to the formation of extended or folded chains, crystallization may also lead to gross orientational changes of chains. For example, chain folding with stems parallel to the lamellar interface has been observed for block copolymers containing poly(ethylene), whilst tilted structures may be formed by other crystalline block copolymers. The kinetics of crystallization have been studied in some detail, and appear to be largely similar to the crystallization dynamics of homopolymers. 

We will briefly discuss the molecular dynamics results obtained for two systems—protein-like and random-block copolymer melts— described by a Yukawa-type potential with (i) attractive A-A interactions (saa < 0, bb = sab = 0) and with (ii) short-range repulsive interactions between unlike units (sab > 0, aa = bb = 0). The mixtures contain a large number of different components, i.e., different chemical sequences. Each system is in a randomly mixing state at the athermal condition (eap = 0). As the attractive (repulsive) interactions increase, i.e., the temperature decreases, the systems relax to new equilibrium morphologies. 

The aim of fusion and controlled solidification of a catalytic material is the generation of a metastable catalytic material. The thermodynamic instability can be caused by a nonequilibrium composition, by a non-equilibrium morphology, or by a combination of both. In the case of the SLP catalysts the desired effect is to avoid the formation of solidification in order to maintain a structureless state of the active material. 

It is an important aspect that block-copolymer micelles are characterized by much longer relaxation times than compared to low molecular surfactants. Non-equilibrium morphologies can easily be obtained in a vitrified state due to the efficient suppression of structural reorganization, because of the corresponding very slow response of the micelles to changes of temperature, solvent and concentration. In the case of a block-ionomer, i.e. a diblock copolymer where one block consists of ionic units, it was observed that micelles which formed in non-polar solution needed weeks to re-equilibrate after dilution of the solvent [226-228]. 

However, in the case of thermal aging where annealing effects are relatively fast, one can expect that all the samples would rapidly tend towards a pseudo equilibrium morphological state. A point trajectory representative of thermal aging would have the shape shown in Fig. 8. 

Barnard, A. S., Lin, X. M. and Curtiss, L. A. (2005). Equilibrium morphology of face-centered cubic gold nanoparticles >3 nm and the shape changes induced by temperature. J. Phys. Chem. B 109 24465-24472. 

Fig. 8. Equilibrium morphologies attainable in di-block copolymers as a function of weight fraction and molecular weight of species...

Figure 1 shows the now familiar sequence of equilibrium morphologies (spheres, rods, lamellae) as a function of copolymer composition also included is the newly proposed bicontinuous double-diamond morphology which exists in certain cases in a narrow composition range between the rod-like and lamellar morphologies. 

In Section 3.2 both external interfaces confining binary polymer mixture in a thin film geometry are explicitly considered. These interfaces specify the equilibrium morphologies of the coexisting phases. Finite size effects relevant for thin films with reduced thickness are also described. 

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