Equilibrium and flow properties of surfactants in solution
2 Phase diagrams and membrane elasticity
A classical binary phase diagram (surfactant
+
water) exhibits a succession of isotropic, liquid crystalline, and crystalline phases as a function of temperature and composition [l].Figure 1 shows such a phase diagram for the binary mixture of sodium laurate and wa- ter. Besides complicated phases at very low water concentration (which correspond to hydrated solids), the phase diagram contains three main phases: the lamellar phase which is a periodic stack of fluid membranes made of the surfactant molecules (lyotropic liquid crystal), the hexagonal phase made of infinite tubes placed on a two-dimensional tri- angular lattice (lyotropic liquid crystal) and the isotropic liquid phase made of spherical micelles. (Note that liquid crystal phases are called ‘lyotropic’ when controlled by con- centration as well as temperature.) Although these are the most common phases found in surfactant solutions, many other structures have also been described, certain of them very recently [2-51. most interesting ones, we should note the cubic phases [2] correspond- ing either to a crystal of spherical micelles or to more complex structures (e.g. bilayers wrapped on a triply periodic ‘minimal surface’). The most intriguing structures, and the
+Co-authors: Annie Colin, Jacques Leng and Anne-Sophie Wunenburger.
186 Didier Roux
I I I I I I I I I
0 100 %
@ Sodium Laurate
Figure 1. A typical phase diagram where, as a function of the surfactant concentration, one has successively the isotropic micellar phase (I), the hexagonal phase (H) and the lamellar phase (L). Other regions are biphasic domains.
ones which were the most difficult to characterise, are complex isotropic liquid phases such as the microemulsion phases [3), the sponge phase [4] and the phase of giant micelles [5].
The complexity of the structure and phase behaviour of these systems has been an experimental and a theoretical challenge. In the last 30 years, a tremendous amount of work has been done leading to a rather unified picture of the way these systems behave.
From a theoretical point of view, the direct relation between the microscopic properties of the surfactant and the phase diagram is not accessible. However, it has been very useful to introduce, as proposed by Canham [6] and Helfrich [7], an intermediate step in the statistical physics description of the properties of surfactants in solution. This intermediate description corresponds to the idea that the physics is dominated by the interfacial properties of the hydrophobic/hydrophilic domains. The microscopic structure of the various phases was then attributed to a competition between the curvature energy of the microscopic interface and the entropy (thermal fluctuations). The importance of the curvature energy arises because, for self-assembled structures which are at thermal equilibrium, there is no surface tension at the hydrophobic/hydrophilic interface. (Surface tension is instead the signature of bulk phase separation.) Despite this, the interfacial area is almost fixed: the forces opposing stretching and compression would rapidly become large if the area was changed. Consequently, the first term that matters in the small deformation of the interface is the curvature energy, whose energy scale is most often close to kBT. Therefore this term in practice controls most of the deformations of the interface.
The concept of curvature energy applied to surfactants in solution turns out to be remarkably efficient. While better adapted to describe dilute phases (mainly because, in that case, the interface becomes very thin compared to the characteristic length of the
Equilibrium and flow properties of surfactants in solution 187
structure), this concept remains qualitatively valid for more concentrated phases. In all cases, it is a good guideline in trying to understand the phase behaviour and general properties of surfactants in solution.
The main problem with this model is that it relates the stability and the structure of the different phases to three phenomenological parameters which do not have an obvious relationship with the surfactant molecules. These parameters are the spontaneous radius of curvature Q , the mean curvature rigidity IE and the Gaussian curvature rigidity &. The energy cost of deforming a surface S, describing the surfactant film, therefore reads:
where c1 and c2 are the two principal radii of curvature of the deformation. Some effort has been devoted to working out microscopic models that link the phenomenological elastic constants to the geometry of the surfactant molecules and these attempts are qualitatively successful. Mechanical models [8], models based on the microscopic description of the molecules (91, and simulations [lo] have all been quite useful in understanding the role of the surfactant geometry. Since the most common interfacial shapes are spheres (micelles), cylinders (hexagonal phase) and planes (lamellar phase), it is useful to calculate the elastic energy for these simple cases. One can also quite easily calculate the curvature energy for the unit cell of a cubic minimal surface, which we call a ‘cubic element’ below.
General properties emerge from such calculations. In particular, we can notice that if the spontaneous radius of curvature is zero (for surfactant bilayers, this is the case by symmetry), the elastic energy of a sphere is not a function of its size: Fsph. = 4n(2r;
+
E ) .This property can easily be generalised (for example to the cubic element) and in general, the bending energy of a finite object is invariant under a change of scale, whenever CO = 0.
Moreover, it has been known since the XIX century that the Gaussian curvature term, Js clczdS, is a function of the global topology of the surface and not directly dependent on the local curvatures (this is the Gauss-Bonnet theorem). Since for = 0 the curvature energy, Equation 1, is based on a quadratic expansion around a flat surface, one expects the energy of a sphere and of a cubic element both to be positive. This fixes a range of stability for the values of K and R , namely 2~ > -R > 0. If R becomes less than - 2 ~ , an instability towards very small spheres will develop. Otherwise, if il becomes larger than zero, there is instead an instability towards small cubic elements: a periodic surface of very small lattice constant will arise.
The effects of thermal fluctuations on the surfactant aggregates are different depending upon the shape of those aggregates. For spherical objects (micelles), the fluctuations will mainly stabilise the isotropic liquid phase of micelles against the ‘crystal’ of micelles or other more organised phases (liquid crystals). For cylindrical objects, depending upon the value of the elastic constant of these ‘rod-like’ micelles, one can have either flexible or rigid systems. If the persistence length (Khokhlov, this volume) of the rods is very large, one will only find cylindrical aggregates in a liquid crystalline arrangement such as the hexagonal phase. However, if the persistence length is small enough (typically smaller than 1000
A),
one can find a phase where the cylindrical micelles are disordered and polymer-like. Just as for regular polymers, this isotropic liquid phase can be found either in dilute or semi-dilute regimes. In such cases, the flexible cylindrical micelles can make a random walk in space and most of their static and dynamic properties can be understood within a model of so-called ‘living’ (i.e. self-assembled) polymers [ 5 ] .188 Didier Row
Similar ideas apply when surfactants instead form membranes (either monolayers or bilayers). Because K and il have the units of energy, it is possible to compare directly their value to kBT. Indeed, as shown first by de Gennes [ll], one can thereby define a persistence length for fluctuating membranes. This persistence length
E
varies expo- nentially with the ratio n/kBT and consequently, only values of K not much larger than k B T lead to a microscopic persistence length (lo25 5 5
104A). Relatively small changes in n differentiate between what we term ‘rigid’ systems, where the persistence length is quite large (> lOpm), and ‘flexible’ systems, where the persistence length is typically less than lpm.The easiest way to understand qualitatively the phase diagram of a rigid system is to realise that many of the properties come from a competition between the spontaneous radius of curvature and the geometrical length resulting from the choice of the concentra- tion of the species. Indeed, for a general system made of water, oil and surfactant, and making the reasonable assumption that all‘the surfactant lies at the water/oil interface, it is quite easy to show that in each case (spheres, cylinders and planes) the characteristic length of the structure (radius of the sphere, radius of the cylinder, or thickness of the oil and water layers) is completely determined by the respective concentrations.
The simplest model is then to take into consideration just the bending energy. It is possible to calculate the most stable structure depending upon the concentration (for a given spontaneous radius of curvature Q). The result is, naturally, found to be the struc- ture whose curvature best matches the spontaneous radius of curvature. Consequently, for this very simple model of ternary mixtures with no other term apart from the bending energy, one can already finds phase transitions from spherical micelles, to a hexagonal phase, and from there to a lamellar phase by changing the respective surfactant/oil con- centrations. Taking into account entropy of mixing and interactions can change the phase diagrams, but this simplest behaviour already gives some reasonable results [12].
Figure 2. Schematic drawing of a lyotropic lamellar phase. It consists of a periodic stacking of membranes (repeat distance d ) , each of a thickness 6 , separated by a solvent.
One of the key questions concerning the lamellar phase is to determine the capacity of this phase to be swollen, i.e. the capacity to change the repeat distance d of the lamellar phase (Figure 2) by adding more solvent. Experimentally, depending on the system, the maximum repeat distance can vary from 50 Angstroms to several thousand Angstroms.
In order to understand what happens to a lamellar phase when it is swollen with a solvent (either water or an organic solvent), we need a reasonable description of the interactions between the membranes. There are several reviews on membrane-membrane interactions which list all the attractive and repulsive interactions that have been calculated. Most of
Equilibrium and flow properties of surfactants in solution 189 them have also been measured (electrostatic, Van der Waals, steric.,.) [13].
One of the most interesting and quite recently discovered long range repulsive in- teractions comes from the thermal undulations of the membranes. Indeed, Helfrich in 1984 predicted that two membranes subjected to thermal fluctuations should develop a repulsive interaction coming from the multiple collisions they will develop one against the other [16]. This interaction, entropic in origin, has been quantitatively measured by sev- eral techniques in sufficiently flexible lamellar phases [13]. Competition between attractive and repulsive interactions can lead to phase transitions, explaining why a lamellar phase cannot be swollen indefinitely. Whereas the competition between electrostatic and Van der Waals interactions can be calculated following the DLVO theory (Frenkel, this volume), the competition between attractive Van der Waals interactions and repulsive undulation forces is much more complex to model, but leads to very interesting behaviours [14, 151.
The phase separation with excess solvent is not the only kind of phase transition that a lamellar phase can experience upon dilution. When a lamellar phase, made of flexible membranes, is swollen with a single solvent, the characteristic repeat, distance increases because of the undulation forces acting as a repulsive interaction on the membranes.
However, when the d-spacing of the lamellar ordering reaches a length which is comparable to the membrane persistence length [, the lamellar phase melts into a sponge phase. (The sponge phase contains a web of bilayer which divides space into two solvent domains. For swelling with equal amounts of oil and water, the an'alogous phase is the microemulsion.) This phase transition, which corresponds to a change in topology, is also influenced by the Gaussian curvature rigidity
( e )
which controls the energy cost of handle formation [4, 171.Upon adding more solvent the sponge phase itself swells and it eventually undergoes another phase transition to a vesicle phase [4].
The effect of fluctuations on flexible membranes, once analysed in detail, leads to a universal phase diagram where the lamellar, sponge, vesicle phases and their phase transitions can be understood in terms of a competition between curvature energy and entropy [18]. While a lot has been done and understood, some open questions remain and new systems have been studied showing interesting behaviours. We will just cite, as an example, the fact that an extremely dilute microemulsion phase, with a characteristic size of several thousands of Angstroms, has recently been found. This phase cystallises upon further swelling into a three dimensional ordered phase [19], unlike most ordered phases which melt upon swelling.