Phases behaviour

The fact that microemulsions have gained increasing importance both in basic research and in industry is reflected in the large number of publications on microemulsions. A survey of paper titles reveals that the number of papers on the subject of microemulsions increased within the last 30 years from 474 in 1976-1985 to over 2508 in 1986-1995 and to 6691 in 1996-2005.1 The fact that micro emulsions also provide the potential for numerous practical applications is mirrored in the number of patents filed on this topic. A survey of patents on microemulsions2 shows an increase from 159 in 1976-1985 to over 805 in 1986-1995 and to 2107 in 1996-2005. In the following the basic properties of microemulsions will be presented concentrating on the close connection between the phase behaviour and the interfacial tensions as well as on the fascinating microstructure. 

The primary aim of microemulsion research is to find the conditions under which the surfactant solubilises the maximum amounts of water and oil, i.e. the phase behaviour has to be studied. As the effect of pressure on the phase behaviour is (in general) rather weak [30 ], it is sufficient to consider the effect of the temperature. Furthermore, it hasbeen shown that simple ternary systems consisting of water, oil and non-ionic n-alkyl polyglycol ethers (QEj) exhibit all properties of complex and technically relevant systems [6]. Therefore, we will first describe the phase behaviour of ternary non-ionic microemulsions. 

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The physical properties of hydrocarbon fluids General hydrocarbon phase behaviour... 

Phase behaviour describes the phase or phases in which a mass of fluid exists at given conditions of pressure, volume (the inverse of the density) and temperature (PVT). The simplest way to start to understand this relationship is by considering a single component, say water, and looking at just two of the variables, say pressure and temperature. 

So far we have considered only a single component. However, reservoir fluids contain a mixture of hundreds of components, which adds to the complexity of the phase behaviour. Now consider the impact of adding one component to the ethane, say n-heptane (C7H.,g). We are now discussing a binary (two component) mixture, and will concentrate on the pressure-temperature phase diagram. 

The example of a binary mixture is used to demonstrate the increased complexity of the phase diagram through the introduction of a second component in the system. Typical reservoir fluids contain hundreds of components, which makes the laboratory measurement or mathematical prediction of the phase behaviour more complex still. However, the principles established above will be useful in understanding the differences in phase behaviour for the main types of hydrocarbon identified. 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

The boundaries separating these principal types of phase behaviour are shown on X,C, diagram (for equalsized molecules) in figure A2.5.13. For molecules of different size, but with the approximation of equation (A2.5.10). more global phase diagrams were calculated using a third parameter,... 

Figure B3.3.1. Simulations as a bridge between the microscopic and the macroscopic. We mput details of molecular structure and interactions we obtain predictions of phase behaviour, structural and time-dependent properties.

Orkoulas G and Panagiotopoulos A Z 1999 Phase behaviour of the restrioted primitive model and square-well fluids from Monte Carlo simulations in the grand oanonioal ensemble J. Chem. Phys. 110 1581-90... 

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

A multitude of different variants of this model has been investigated using Monte Carlo simulations (see, for example [M])- The studies aim at correlating the phase behaviour with the molecular architecture and revealing the local structure of the aggregates. This type of model has also proven useful for studying rather complex structures (e.g., vesicles or pores in bilayers). 

For structures with a high curvature (e.g., small micelles) or situations where orientational interactions become important (e.g., the gel phase of a membrane) lattice-based models might be inappropriate. Off-lattice models for amphiphiles, which are quite similar to their counterparts in polymeric systems, have been used to study the self-assembly into micelles [ ], or to explore the phase behaviour of Langmuir monolayers [ ] and bilayers. In those systems, various phases with a nematic ordering of the hydrophobic tails occur. 

Since the amphiphilic nature is essential for the phase behaviour, systems of small molecules (e.g., lipid water mixtures) and polymeric systems (e.g., homopolymer copolymer blends) share many connnon features. 

These chain models are well suited to investigate the dependence of tire phase behaviour on the molecular architecture and to explore the local properties (e.g., enriclnnent of amphiphiles at interfaces, molecular confonnations at interfaces). In order to investigate the effect of fluctuations on large length scales or the shapes of vesicles, more coarse-grained descriptions have to be explored. 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

It has not proved possible to develop general analytical hard-core models for liquid crystals, just as for nonnal liquids. Instead, computer simulations have played an important role in extending our understanding of the phase behaviour of hard particles. Frenkel and Mulder found that a system of hard ellipsoids can fonn a nematic phase for ratios L/D >2.5 (rods) or L/D <0.4 (discs) [73] however, such a system cannot fonn a smectic phase, as can be shown by a scaling... 

Nuzzo R G, Korenic E M and Dubois L H 1990 Studies of the temperature-dependent phase behaviour of iong chain / -aikyi thioi monoiayers on goid J. Chem. Phys. 93 767-73... 

The remainder of this contribution is organized as follows. In section C2.6.2, some well studied colloidal model systems are introduced. Methods for characterizing colloidal suspensions are presented in section C2.6.3. An essential starting point for understanding the behaviour of colloids is a description of the interactions between particles. Various factors contributing to these are discussed in section C2.6.4. Following on from this, theories of colloid stability and of the kinetics of aggregation are presented in section C2.6.5. Finally, section C2.6.6 is devoted to the phase behaviour of concentrated suspensions. 

Altliough tire behaviour of colloidal suspensions does in general depend on temperature, a more important control parameter in practice tends to be tire particle concentration, often expressed as tire volume fraction ((). In fact, for hard- sphere suspensions tire phase behaviour is detennined by ( ) only. For spherical particles... 

Many properties of colloidal suspensions, such as their stability, rheology, and phase behaviour, are closely related to the interactions between the suspended particles. The background of the most important contributing factors to these interactions is discussed in this section. 

In the previous section, non-equilibrium behaviour was discussed, which is observed for particles with a deep minimum in the particle interactions at contact. In this final section, some examples of equilibrium phase behaviour in concentrated colloidal suspensions will be presented. Here we are concerned with purely repulsive particles (hard or soft spheres), or with particles with attractions of moderate strength and range (colloid-polymer and colloid-colloid mixtures). Although we shall focus mainly on equilibrium aspects, a few comments will be made about the associated kinetics as well [69, 70]. 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

As shown in section C2.6.6.2, hard-sphere suspensions already show a rich phase behaviour. This is even more the case when binary mixtures of hard spheres are considered. First, we will mention tire case of moderate size ratios, around 0.6. At low concentrations tliese fonn a mixed fluid phase. On increasing tire overall concentration of mixtures, however, binary crystals of type AB2 and AB were observed (where A represents tire larger spheres), in addition to pure A or B crystals [105, 106]. An example of an AB2 stmcture is shown in figure C2.6.11. Computer simulations confinned tire tliennodynamic stability of tire stmctures tliat were observed [107, 1081. 

Pusey P N and van Megen W 1986 Phase behaviour of concentrated suspensions of nearly hard colloidal spheres Nature 320 340-2... 

Lekkerkerker FI N W, Peon W C K, Pusey P N, Stroobants A and Warren P B 1992 Phase behaviour of colloid + polymer mixtures Europhys. Lett. 20 559-64... 

Rosenbaum D, Zamora P C and Zukoski C F 1996 Phase behaviour of small attractive colloidal particles Phys. Rev. Lett. 76 150-3... 

Oykstra M, van Roi] R and Evans R 1999 Direot simulation of the phase behaviour of binary hard-sphere mixtures test of the depletion potential desoription Phys. Rev. Lett. 82 117-20... 

This section will describe the current status of research in two different aspects of nanocrystal phase behaviour melting and solid-solid phase transitions. In the case of melting, thennodynamic considerations of surface energies can explain the reduced melting point observed in many nanocrystals. Strictly thennodynamic models, however, are not adequate to describe solid-solid phase transitions in these materials. 

The radial distribution function can also be used to monitor the progress of the equilibration. This function is particularly useful for detecting the presence of two phases. Such a situation is characterised by a larger than expected first peak and by the fact that g r) does not decay towards a value of 1 at long distances. If two-phase behaviour is inappropriate then the simulation should probably be terminated and examined. If, however, a two-phase system is desired, then a long equilibration phase is usually required. 

Amphiphiles often have a complex phase behaviour with several liquid crystalline phases These liquid crystalline phases are often characterised by long-range order in one directior together with the formation of a layer structure. The molecules may nevertheless be able tc move laterally within the layer and perpendicular to the surface of the layer. Structura information can be obtained using spectroscopic techniques including X-ray and neutror diffraction and NMR. The quadrupolar splitting in the deuterium NMR spectrum can be... 

R. G. Laughlin, TheMqueous Phase Behaviour of Sufactants Academic Press, Ltd., London, 1994, pp. 448—451. 

R. J. Sadus, High Pressure Phase Behaviour of Multicomponent Fluid Mixtures, Elsevier, Amsterdam, the Netherlands, 1992. 

The combination of non-ideal phase behaviour of solutions, the non-linearity of particle formation kinetics, the multi-dimensionality of crystals, their interactions and difficulties of modelling, instrumentation and measurement have conspired to make crystallizer control a formidable engineering challenge. Various aspects of achieving control of crystallizers have been reviewed by Rawlings etal. (1993) and Rohani (2001), respectively. 

Computer simulations therefore have several inter-related objectives. In the long term one would hope that molecular level simulations of structure and bonding in liquid crystal systems would become sufficiently predictive so as to remove the need for costly and time-consuming synthesis of many compounds in order to optimise certain properties. In this way, predictive simulations would become a routine tool in the design of new materials. Predictive, in this sense, refers to calculations without reference to experimental results. Such calculations are said to be from first principles or ab initio. As a step toward this goal, simulations of properties at the molecular level can be used to parametrise interaction potentials for use in the study of phase behaviour and condensed phase properties such as elastic constants, viscosities, molecular diffusion and reorientational motion with maximum specificity to real systems. Another role of ab initio computer simulation lies in its interaction... 

Small chemical changes to the tolane family of mesogenic molecules are also known to bring about major changes in phase behaviour [22]. Two examples are shown in Fig. 5 where subtle changes in the tail can eliminate the nematic phase. 

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