Phase ordered

Most theoretical models that have been proposed are effectively based on hard core interactions by implementing the incompressibility constraint where the 

(19%) Introduction to Polymer Physics, Clarendon Press, Oxford. 

2 Flory, P. (1953) Principles of Polymer Chemistry, Cornell University Press, 

3 Higgins, J. and Beno, H. (1994) Polymers and Neutron Scattering, Clarendon Press, Oxford. 

4 deCennes, P. (1970) Theory of x-ray scattering by liquid macromolecules with heavy atom label Le J. Phys-Paris, 

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Stigter and Dill [98] studied phospholipid monolayers at the n-heptane-water interface and were able to treat the second and third virial coefficients (see Eq. XV-1) in terms of electrostatic, including dipole, interactions. At higher film pressures, Pethica and co-workers [99] observed quasi-first-order phase transitions, that is, a much flatter plateau region than shown in Fig. XV-6. 

Wlien 2 g > (Eaa BB binary alloy corresponds to an Ismg ferromagnet (J> 0) and the system splits into two phases one rich in A and the other rich in component B below the critical temperature T. On the other hand, when 2s g < (Eaa+ bb > system corresponds to an antiferromagnet the ordered phase below the critical temperature has A and B atoms occupying alternate sites. 

It has long been known from statistical mechanical theory that a Bose-Einstein ideal gas, which at low temperatures would show condensation of molecules into die ground translational state (a condensation in momentum space rather than in position space), should show a third-order phase transition at the temperature at which this condensation starts. Nonnal helium ( He) is a Bose-Einstein substance, but is far from ideal at low temperatures, and the very real forces between molecules make the >L-transition to He II very different from that predicted for a Bose-Einstein gas. 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

Consider simulating a system m the canonical ensemble, close to a first-order phase transition. In one phase, is essentially a Gaussian centred around a value j, while in the other phase tlie peak is around Ejj. 

In the microcanonical ensemble, the signature of a first-order phase transition is the appearance of a van der Waals loop m the equation of state, now written as T(E) or P( ). The P( ) curve switches over from one... 

Berg B A and Neuhaus T 1992 Multicanonical ensemble—a new approach to simulate Ist-order phase transitions Phys. Rev.L 68 9-12... 

Lovett R 1995 Can a solid be turned into a gas without passing through a first order phase transition Observation, Prediction and Simuiation of Phase Transitions in Compiex Fiuids vol 460 NATO ASi Series O ed M Baus, L F Rull and J-P Ryckaert (Dordrecht Kluwer) pp 641-54... 

Frenkel D 1986 Free-energy computation and first-order phase transitions Moiecuiar Dynamics Simuiation of Statisticai Mechanicai Systems ed G Ciccotti and W G Hoover (Amsterdam North-Holland) pp 151-88... 

Brown F R and Yegulalp A 1991 Microcanonical simulation of Ist-order phase transitions in finite volumes Phys. Lett. A 155 252-6... 

Binder K and Landau D P 1984 Finite size scaling at Ist-order phase transitions Phys. Rev. B 30 1477-85... 

Most characteristics of amphiphilic systems are associated with the alteration of the interfacial stnicture by the amphiphile. Addition of amphiphiles might reduce the free-energy costs by a dramatic factor (up to 10 dyn cm in the oil/water/amphiphile mixture). Adding amphiphiles to a solution or a mixture often leads to the fomiation of a microenuilsion or spatially ordered phases. In many aspects these systems can be conceived as an assembly of internal interfaces. The interfaces might separate oil and water in a ternary mixture or they might be amphiphilic bilayers in... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

Odi]k T 1996 Ordered phases of elongated micelles Curr. Opin. Colloid Interface Sol. 1 337-40... 

Magnussen O M, Hageboeck J, Hotios J and Behm R J 1992 In s/fu scanning tunneiing microscopy observations of a disorder-order phase transition in hydrogensuiphate adiayers on Au(111) Faraday Discuss. 94 329-38... 

Huckaby D A and Blum L 1991 A model for sequential first-order phase transitions occuring in the underpotential deposition of metals J. Eiectroanai. Chem. 315 255-61... 

Several years ago Baer proposed the use of a mahix A, that hansforms the adiabatic electronic set to a diabatic one [72], (For a special twofold set this was discussed in [286,287].) Computations performed with the diabatic set are much simpler than those with the adiabatic set. Subject to certain conditions, A is the solution of a set of first order partial diffei ential equations. A is unitary and has the form of a path-ordered phase factor, in which the phase can be formally written as... 

Prenkel, D. Pree energy computation and first order phase transitions. In Molecular Dynamic Simulation of Statistical Mechanical Systems, Enrico Fermi Summer School, Varenna 1985, G. Ciccotti and W. Hoover, eds. North Holland, Amsterdam (1986) 43-65. 

A given molecular dynamics trajectory can be divided into three seqneiitlally-ordered phases ... 

Figure 4.14 Behavior of thermodynamic variables at Tg for a second-order phase transition (a) volume and fb) coefficient of thermal expansion a and isothermal compressibility p.

The separation of Hquid crystals as the concentration of ceUulose increases above a critical value (30%) is mosdy because of the higher combinatorial entropy of mixing of the conformationaHy extended ceUulosic chains in the ordered phase. The critical concentration depends on solvent and temperature, and has been estimated from the polymer chain conformation using lattice and virial theories of nematic ordering (102—107). The side-chain substituents govern solubiHty, and if sufficiently bulky and flexible can yield a thermotropic mesophase in an accessible temperature range. AcetoxypropylceUulose [96420-45-8], prepared by acetylating HPC, was the first reported thermotropic ceUulosic (108), and numerous other heavily substituted esters and ethers of hydroxyalkyl ceUuloses also form equUibrium chiral nematic phases, even at ambient temperatures. 

Construct, using asymptotes and standard second-order phase diagrams, the Bode diagrams for... 

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