In this paper, we give an overview of our studies by static and time-resolved X-ray diffraction of inverse cubic phases and phase transitions in lipids. In §1, we briefly discuss the lyotropic phase behaviour of lipids, focusing attention on non-lamellar structures, and their geometric/topological relationship to fusion processes in lipid membranes. Possible pathways for transitions between different cubic phases are also outlined. In §2, we discuss the effects of hydrostatic pressure on lipid membranes and lipid phase transitions, and describe how the parameters required to predict the pressure dependence of lipid phase transition temperatures can be conveniently measured. We review some earlier results of inverse bicontinuous cubic phases from our laboratory, showing effects such as pressure-induced formation and swelling. In §3, we describe the technique of pressure-jump synchrotron X-ray diffraction. We present results that have been obtained from the lipid system 1 : 2 dilauroylphosphatidylcholine/lauric acid for cubic–inverse hexagonal, cubic–cubic and lamellar–cubic transitions. The rate of transition was found to increase with the amplitude of the pressure-jump and with increasing temperature. Evidence for intermediate structures occurring transiently during the transitions was also obtained. In §4, we describe an IDL-based ‘AXcess’ software package being developed in our laboratory to permit batch processing and analysis of the large X-ray datasets produced by pressure-jump synchrotron experiments. In §5, we present some recent results on the fluid lamellar–Pn3m cubic phase transition of the single-chain lipid 1-monoelaidin, which we have studied both by pressure-jump and temperature-jump X-ray diffraction. Finally, in §6, we give a few indicators of future directions of this research. We anticipate that the most useful technical advance will be the development of pressure-jump apparatus on the microsecond time-scale, which will involve the use of a stack of piezoelectric pressure actuators. The pressure-jump technique is not restricted to lipid phase transitions, but can be used to study a wide range of soft matter transitions, ranging from protein unfolding and DNA unwinding and transitions, to phase transitions in thermotropic liquid crystals, surfactants and block copolymers.
1. Lyotropic phase behaviour
Lyotropic liquid crystals of one-, two- or three-dimensional periodicity spontaneously assemble when biological amphiphiles are mixed with a solvent under various conditions of temperature, pressure and hydration (Seddon & Templer, 1993, 1995). The mesophases formed by double-chain lipids include the fluid lamellar (Lα), inverse hexagonal (HII) and inverse bicontinuous cubic phases (QII). Biologically, the fluid lamellar phase (figure 1a) is ubiquitous, being the structure upon which cell membranes are based. However, the inverse bicontinuous cubic phases have become increasingly accepted as not only being present in certain cell membranes, but as relevant to cell processes involving membrane fusion (Hyde et al. 1997). To understand how bicontinuous cubic phases may arise, it is helpful to consider the curvature frustration that may exist in a fluid lamellar Lα phase (Charvolin 1990). This arises because a build-up of conformational disorder in the chain region, for example upon heating, tries to bend each lipid monolayer away from the bilayer midplane, avoiding additional water–chain contact (figure 1b). However, this would create voids and is resisted by the hydrophobic effect.
One of the ways by which the system can relax is to undergo a transition to an inverse hexagonal HII phase, where the monolayers wrap around into inverse cylinders, which then pack onto a two-dimensional hexagonal lattice. This transition from a lamellar to an HII phase is very commonly observed in fully hydrated lipids upon heating (figure 2a). However, there is still a packing frustration in this phase (figure 2b), because circular cylinders fill space only to a packing fraction of 0.907, and this would leave voids comprising over 9% of the volume of the phase in the centre of the hydrocarbon region, which cannot be tolerated.
Thus, either the interface must develop a non-uniform mean curvature (by becoming faceted) or some of the chains must deviate away from their preferred conformational state, in order to fill the hydrophobic volume. For a given lipid, the effect of reducing the chain length is to make this packing frustration relatively more severe, and below a critical chain length, the system finds other ways of developing inverse interfacial mean curvature, without creating costly voids. This can be achieved by the bilayer deforming to a saddle surface, with negative Gaussian curvature everywhere (apart from special flat points, where K=0). For a symmetrical bilayer, the midplane should correspond to a minimal surface, having zero mean curvature at all points. Such surfaces can be extended to form three-dimensionally ordered arrays, known as infinite periodic minimal surfaces (IPMS). Three of the simplest IPMS are the cubic P, D and G surfaces. These are related to each other by a Bonnet transformation, which is one-to-one isometric mapping between identical surface patches. Draping a lipid bilayer onto these three IPMS leads to the three most common lipid inverse cubic phases (figure 3), of crystallographic space groups Im3m, Pn3m and Ia3d, respectively (Luzzati et al. 1993). The lattice parameters of these cubic phases are typically in the range of 100–300 Å.
When the underlying minimal surfaces are Bonnet-related, the lattice parameters of these three cubic phases should be in the ratios: aP : aD : aG=1.279 : 1.000 : 1.576 (Hyde et al. 1984). Experimental observation of these ratios implies that the three cubic phases have the same free energy (to second order in the principal curvatures), when the free energy is dominated by the curvature elastic energy and the two halves of the bilayer form parallel interfaces to the underlying minimal surface (Templer et al. 1998a). This degeneracy is broken if the two halves of the bilayer tend to form constant curvature interfaces instead. Since the three underlying minimal surfaces pack space differently (degree of compactness increasing from P–D–G), the degeneracy may also be broken by transverse interactions between the lipid layers across the water regions. The phase sequence with decreasing water content is thus expected to follow the order Im3m–Pn3m–Ia3d (Templer et al. 1998a; Schwarz & Gompper 2000), and this is observed experimentally (Templer et al. 1998b). The Ia3d (G) cubic phase is very common both in type I and type II versions in surfactants, lipids and block copolymers (Bates & Fredrickson 1999), whereas Pn3m (D) and Im3m (P) seem only to occur as type II (inverse) versions in lipid systems so far. We (and others) have found that for various lipid systems which exhibit Lα–HII transitions upon heating, reducing the chain length, typically to approximately C12, usually induces the formation of inverse bicontinuous cubic phases between the Lα and the HII phases, or may suppress the HII phase completely.
Many cellular processes including endocytosis, exocytosis and membrane budding involve changes in membrane topology. The inverse bicontinuous cubic phases may be relevant to, or may even act as intermediates in some of these processes, since it is likely that the mechanism of formation of the cubic phase from a corresponding fluid lamellar phase has much in common with the mechanism of cell membrane fusion and fission (Siegel & Banschbach 1990). Much is now known about amphiphile–water mixtures in equilibrium but, as yet, rather little is known about their non-equilibrium behaviour. This is surprising given that in all of the examples we have mentioned, the structural dynamics of phase transitions in the lyotropic mesophases play a profound role. Previous studies of lyotropic phase transitions have concentrated on transformations between different lamellar structures and from lamellar to inverse hexagonal phases, with remarkably little work being done on transitions involving cubic phases (Erbes et al. 1994; Qiu & Caffrey 2000; Zana 2005). However, a complete understanding of the physical processes governing such transitions, including the nature of any intermediates formed and the mechanistic routes taken, is essential if we are to further our knowledge of their possible roles in cellular processes involving membranes. Rather little is known yet about the mechanisms and pathways of transitions involving cubic phases. It was first thought that they might follow the same pathway as the Bonnet transformation, a view which was quickly realized to be unphysical, since it would necessarily involve tearing and self-intersection of the bilayer (Hyde et al. 1984). Later, it was suggested that a continuous P–D–G transition could occur by a stretching mechanism, without any tearing (Sadoc & Charvolin 1989; Benedicto & O'Brien 1997). Thus, a sixfold junction of water channels in the P (Im3m) phase can be converted into two fourfold junctions of the D (Pn3m) phase; each of these can then, in turn, be converted into two threefold junctions of the G (Ia3d) phase (figure 4).
Recently, it has been suggested that such continuous cubic transitions could involve non-cubic (tetragonal, rhombohedral) distortions of the underlying minimal surfaces, yet with the surfaces remaining minimal during the processes (Fogden & Hyde 1999; Schroeder et al. 2004). We can try to visualize this continuous process for the G–D phase transition that involves (figure 5), first, a tetragonal stretch, changing c/a=1 to c/a=2, followed by an ‘untwisting’ of the water channels with a merging of each pair of three-way junctions to form single four-way junctions.
2. Effects of hydrostatic pressure
In order to predict the pressure dependence of lipid phase transition temperatures Tt from the Clapeyron equation(2.1)it is necessary to determine the molar entropy ΔSm and molar volume ΔVm changes at the transitions. These can be conveniently measured using a combination of differential scanning calorimetry (DSC) for the enthalpy ΔHm (figure 6a) and the rather new technique of pressure perturbation calorimetry (PPC) for the volume change ΔVm (figure 6b). DSC measures heat flow to/from the sample upon a change of temperature, whereas PPC measures the heat flow induced by a small change in pressure, which is directly proportional to the coefficient of thermal expansion α=(1/V)(∂V/∂T)p (Lin et al. 2002). The DSC values from the scan shown in figure 6a for the transition temperature Tt and enthalpy change ΔHm at the gel–fluid (chain-melting) transition of the lipid dipalmitoylphosphatidylcholine (DPPC) in excess water are 41.5°C and 33.8 kJ mol−1, respectively, yielding a transition entropy ΔSm of 108 J K−1 mol−1. The PPC value from the scan shown in figure 6b for the volume change ΔVm is 23.5 cm3 mol−1. In combination, our data predict a pressure dependence of the gel–fluid transition of DPPC/water of 22°C kbar−1, in perfect agreement with the measured pressure dependence of the transition temperature for this lipid (Kusube et al. 2005). The Clapeyron equation predicts a linear relationship between pressure and transition temperature when ΔSm and ΔVm are independent of pressure, or have the same pressure dependence; in practice, the linearity is often maintained until at least 2 kbar.
There is an interesting consequence of the fact that the Clapeyron equation contains the ratio of the volume change and the entropy change; a phase transition with a smaller enthalpy (entropy) change, such as a fluid–fluid transition, also tends to have a correspondingly smaller volume change, and thus the pressure dependence of the transition temperature is little changed. In practice, for fluid lamellar–bicontinuous cubic, lamellar–hexagonal and hexagonal–inverse micellar cubic phases, the value of dTt/dp lies within the narrow range of 20–30°C kbar−1 (Duesing et al. 1997).
The possibility that pressure might induce topological transitions in membranes was suggested by Gruner and co-workers, although for the lipid system studied, they concluded that pressure increased the rate of formation of the cubic phase, rather than inducing its formation (So et al. 1993). For the phospholipid ditetradecylphosphatidylethanolamine/H2O, we found that and inverse bicontinuous cubic phases (space groups Pn3m and Im3m) were induced to form between the fluid lamellar Lα and HII phases when the pressure was increased above 700 bar (Duesing et al. 1997). On the other hand, for 1 : 2 phosphatidylcholine/fatty acid mixtures in water, having a range of symmetric chain lengths, pressure-induced formation of cubic phases was not observed (Winter et al. 1999).
The application of hydrostatic pressure to lyotropic phases in excess water tends to increase the lattice parameters of the phases. This effect is caused by the increase in conformational order of the hydrocarbon chains by pressure, and is the opposite effect to that of increasing temperature. The effect is generally small for lamellar phases (less than 2 Å kbar−1), slightly larger for the HII phase, but can be as much as 80 Å kbar−1 for inverse bicontinuous cubic phases, where the reduction in chain splay tends to reduce the magnitude of the (negative) interfacial curvature, thereby swelling the phase if it is in contact with an excess water phase (Duesing et al. 1997; Winter et al. 1999). This can provide a method for isothermally tuning the diameter of the water channels of lipid bicontinuous cubic phases, which could be useful for certain biotechnological applications.
To second order in the principal curvatures, the curvature elastic energy per unit area of a thin amphiphile film is given (Helfrich 1973) by(2.2)where H and K are the surface-averaged mean and Gaussian curvatures, H0 is the spontaneous mean curvature, and κ and κG are the mean curvature (bending) and Gaussian curvature (saddle-splay) moduli, respectively. κ is a measure of the energy required to bend the film where κG is a measure of the energy cost of changing its Gaussian curvature; from the Gauss–Bonnet theorem, this second term integrates to a constant value unless a change occurs in topology. A positive value of κG favours saddle deformations of the film, since these have negative Gaussian curvature K. For systems which tend to form inverse phases, for the lipid monolayer, the spontaneous curvature H0 is negative and κ is positive; typical values for monoolein are −0.025 Å−1 and 1.2×10−20 J (approx. 3 kT), respectively (Vacklin et al. 2000). On the other hand, κG for a lipid monolayer in an inverse cubic phase is negative, with typical values of −0.75κ (Templer et al. 1998c) or −0.83κ (Siegel & Kozlov 2004). For a symmetric lipid bilayer, H0 is zero (by symmetry), κb is twice the value for a single monolayer, but a more complex form is found for the Gaussian modulus,(2.3)where all the terms on the right-hand side, including the thickness , refer to the lipid monolayer. Since the second term on the right-hand side is overall positive for a lipid that tends to form inverse phases (negative monolayer H0), can be positive even when κG for a monolayer is negative. This is one of the main driving forces behind the formation of inverse bicontinuous cubic phases in lipid systems (Seddon & Templer 1993; Templer et al. 1998a).
Our results (Duesing et al. 1997; Winter et al. 1999) indicate that increasing pressure tends to reduce the magnitude of the spontaneous curvature H0, increase the bending rigidity κ and increase the rigidity of chain extension. We surmise from the pressure-induced formation of inverse bicontinuous cubic phases (Duesing et al. 1997) that pressure also tends to increase , and thereby stabilizing inverse bicontinuous cubic phases; although this is difficult to predict from equation (2.3), as the effect of pressure on κG is not known, and the effect of pressure on κ and H0 will tend to cancel out. In contrast, Mariani and co-workers (Mariani et al. 1996; Pisani et al. 2001, 2003) report similar effects on H0, but deduce that κ decreases with pressure, and that the ratio κG/κ is positive and increases strongly with pressure (from 0.14 to 1 by 1.8 kbar).
3. Pressure-jump X-ray diffraction
We have used the X-ray pressure-jump technique (Mencke et al. 1993; Österberg et al. 1994; Erbes et al. 1996; Winter et al. 1999; Winter 2002; Winter & Koehling 2004) to investigate the rate and mechanism of lyotropic phase transitions in monoglyceride/water and fatty acid/phospholipid/water systems, by monitoring the time evolution of these structural conversions using time-resolved X-ray diffraction. The use of pressure as a trigger mechanism has several advantages: (i) the solvent properties are not significantly altered, (ii) pressure propagates rapidly, indicating that equilibrium is achieved rapidly, and (iii) pressure-jumps can be both in the pressurization and depressurization directions. The Dortmund millisecond pressure-jump X-ray cell was used for our experiments (Woenckhaus et al. 2000). This cell has flat diamond windows that can be used at pressures up to 8 kbar, and the pressure-jumps occur in 5–7 ms. In order to reduce the attenuation of the X-ray beam by the diamond windows, it is best to carry out the experiments at short wavelengths, such as 0.75 Å (17 keV), which gives typically a transmission of 65%. Beamline ID02 at the ESRF is ideally suited to these experiments, as will be beamline I22 at the Diamond synchrotron, once this is operational.
We have used time-resolved X-ray diffraction to monitor transitions, induced by pressure-jumps, between various inverse lyotropic phases of a 1 : 2 dilauroylphosphatidylcholine/lauric acid (DLPC/LA) system in 50 wt% water (Squires et al. 2000, 2002, 2005; Seddon et al. 2003). The pressure–temperature (p–T) phase diagram (figure 7) of this sample was first assessed by static X-ray diffraction measurements.
We initially studied transitions between the G and D inverse bicontinuous cubic phases, and between D and the inverse hexagonal HII phase. Typical small-angle X-ray diffraction patterns from the G and D cubic phases are shown in figure 8. The effect of a pressure-jump from 600 to 240 bar at T=59.5°C is shown in figure 9.
The transition from the G to the D cubic phase is clearly seen from the changes in the diffraction patterns. The lattice parameters of the coexisting G and D phases when the D phase initially forms are in the ratio 1.58–1.59±0.015, very close to the Bonnet ratio of aG/aD=1.576. Plots of the peak intensities with time (not shown) are fitted quite well by single exponentials, indicating that the transition appears to follow first-order kinetics. For experiments with an end pressure of 290 bar, increasing the temperature from 57.5 to 62.5°C decreased the transition half-time from 4 to 0.3 s. Apparent activation energy can be calculated from the temperature dependence, but the interpretation of the resulting value is open to some questions. The transition kinetics are also sensitive to the magnitude of the pressure-jump. The half-time at T=59.5°C with a starting pressure of 600 bar increases from approximately 3 to 180 s on changing the final pressure from 240 to 360 bar. Thus, the rate becomes slower when we jump less deeply into the D phase. This is consistent with the thermodynamic driving force becoming larger; further, the final pressure takes the sample away from the phase boundary (Erbes et al. 2000; Squires et al. 2005).
Another interesting feature of the diffraction patterns is that they show evidence for transient, weak additional peaks, labelled (a) and (b) in figure 9, which are not associated with either the G or the D cubic phases. The first such peak (a), close to a spacing of 54 Å, appears 1–2 s after the pressure-jump and only survives for approximately 0.5 s. A long-lived intermediate peak (b) is observed at a spacing of 51 Å. We have now established that the latter is the first-order peak of a transient HII phase, but we have not yet been able to identify the initial, short-lived peak. There are two possible roles of these intermediate structures: (i) they could be true intermediates in some well-defined pathway between the initial and final cubic phases, or (ii) they could transiently form as coexisting phases, possibly in order to act as sources/sinks of water. This could facilitate the cubic–cubic transition by allowing it to occur initially at constant curvature (as previously mentioned, the G, D and P cubic phases fill space with differing degrees of compactness, and thus have different water contents at equivalent curvatures).
On carrying out pressure jumps from D to a region of coexisting (D+HII+excess water; see figure 7), we found evidence for the transient formation (surviving for some tens of seconds) of a P (Im3m) cubic phase, with a lattice parameter of 138.5 Å (Squires et al. 2000). The ratio of the P and D cubic phase lattice parameters was 1.276±0.015, which is very close to the Bonnet ratio aP/aD=1.279.
We have also studied the kinetics of the lamellar Lα–G (Ia3d) cubic phase transition of the same 1 : 2 DLPC/LA/50 wt% water system, induced by pressure-jumps (Squires et al. 2002). The time-scale of the transition was found to vary from 4 to 193 s for the small range of temperatures and pressure-jumps studied so far. During this transition, transient intermediate X-ray peaks were also observed, which we tentatively ascribed to an intermediate D cubic phase, but this interpretation is still somewhat speculative at this point in time.
4. Batch processing/analysis of time-resolved X-ray data
The IDL-based AXcess software package, developed in our laboratory, allows for batch processing and analysis of the large X-ray datasets produced by pressure-jump synchrotron experiments. It has a widget-based front end for user accessibility, and can process thousands of images sequentially, allowing for a large reduction in analysis time.
Figure 10a shows a small-angle diffraction pattern from a silver behenate test sample in a highly ordered lamellar mesophase. The program automatically finds the centre of the image and integrates over a butterfly-shaped area to produce a one-dimensional plot of intensity versus pixel number, as shown in figure 10b. Each peak in this one-dimensional plot may be fitted individually within pre-specified constraints to a specified functional form, for example a modified Gaussian. Indexing of these peaks allows us to identify the symmetry of the lipid mesophase, provided the system has been calibrated using a known standard, such as silver behenate (d=58.38 Å), to assign a d-spacing to the phase.
Following a pressure-jump, a sequence of diffraction images are generated. Such a sequence may be integrated sequentially by the program producing a ‘stacked’ plot of one-dimensional images with time. Figure 10c shows such a plot for a typical cubic–cubic transition, the transition in 1-monoolein (1-MO), following a pressure-jump from 470 to 1500 bar at 60.5°C. Such stacked plots can be viewed from a variety of orientations and are a useful visual aid when analysing changes in intensity and spacing occurring during the transition. Figure 10d shows the fitting of the √2, √3, √4 and √6 peaks characteristic of the cubic phase. Initial peak constraints must be manually set; however, if the lattice parameter changes with time, then the system may be set up so that the constraints follow the moving peaks. The d-spacing of each image is then automatically generated and an output file is created, showing the lattice parameter as a function of image number, along with the indexing of the phase and the fit of each peak analysed. This offers a very considerable reduction in the analysis time (figure 10e).
5. Lamellar Lα to Pn3m cubic phase transition
We have studied the fluid lamellar to Pn3m () inverse bicontinuous cubic phase transition, effected by means of jumps in temperature and pressure, for the single chain amphiphile monoelaidin in excess water (Conn et al. 2006). On increasing temperature or decreasing pressure, the system displays the phase sequence Lβ–Lα–––HII (Czeslik et al. 1995). In particular, the fluid lamellar to Im3m cubic () transition is interesting, as it is not well defined, proceeding via a kinetically hindered intermediate, termed X, of unknown structure and characterized by a broad featureless ring of scatter at low angle in the diffraction pattern (Caffrey 1987).
Figure 11 shows the structural changes following pressure-jumps from (a) 1110 to 240 bar and (b) 1110 to 260 bar, at T=46.7°C. The two jumps, carried out on the same sample, are remarkably similar and are observed to follow a common mechanistic route. The disappearance of the fluid lamellar phase is accompanied by the appearance of a broad, featureless ring of scatter (marked X in figure 11).
After a period of a few seconds, the diffuse scatter resolves itself into a set of peaks from intermediate inverse bicontinuous cubic phases, and , which replace it. These intermediate cubic phases are initially significantly more hydrated than expected from equilibrium results and dehydrate over time. In both jumps, a final phase forms at fairly constant lattice parameter.
A major and long-standing problem in the field of lipid phase transition kinetics is the lack of reproducibility in results. The requirement for intense synchrotron X-ray sources for adequate time resolution indicates that a transition may be observed only a few times, yet a number of studies have noted an increase in the speed of transition with successive jumps. We have overcome the variability in such dynamical measurements and, for the first time, this has allowed us to record the dynamical process reproducibly in temperature- and pressure-jump experiments. Figure 12 shows the lattice parameters of all phases present during the transformation, superimposed for both jumps (a) (filled symbols) and (b) (hollow symbols). With the exception of the (2) phase, which forms during jump (a) but not jump (b), the two transformations are virtually identical. We observe the transition to be highly reproducible not only in terms of the sequence of phases and their lattice parameters, but also in the kinetics of the process, i.e. the time at which each successive phase appears. We believe that the effect is owing to the homogenization of domain size on repeated thermal cycling and strongly linked to the transfer of water throughout the sample. The relative amounts of each phase present throughout the transformation may be approximated by the area under one of the characteristic diffraction peaks.
Immediately following the pressure-jump, the intensity of the lamellar phase begins to drop sharply (figure 13), the complete disappearance of this phase occurring within 2 s, by which point its layer spacing has dropped by nearly 4 Å. The (1) phase appears 1.4 s later and grows steadily with time until the appearance of the (1) phase at the point in which its intensity begins to drop sharply. This is strongly indicative of a direct transformation between these two cubic phases. However, the structural aspects of the transformation, specifically the fact that the (1) phase is initially swollen and decreases towards an equilibrium value, are more usually associated with the formation of a cubic phase directly from the fluid lamellar phase. In contrast, during a cubic–cubic transition, the growing cubic phase maintains a fairly constant lattice parameter. This is displayed here during the growth of the final two cubic phases, (2) and (2), which form directly from the (1) phase, both of which maintain a constant lattice parameter throughout the transformation. It is interesting to note that, even in such non-equilibrium conditions, both sets of cubic phases maintain a ratio of lattice parameters close to that predicted by the Bonnet ratio (1.279). The (1) : (1) ratio (estimated error ±0.015) is initially 1.317, but drops sharply to 1.287 as the transition proceeds, while the (2) : (2) ratio is maintained between 1.295 and 1.305.
We have achieved the same level of reproducibility with our temperature-jump experiments as that seen in the pressure-jump results previously. Figure 14 shows the changes following a temperature jump from 30 to 60°C, at a nominal jump rate of 50°C min−1.
In this case, the transition to the first () cubic phase, via the X phase, occurs much more slowly, and the appearance of the cubic phase does not occur until more than 1000 s have passed (the time for the temperature-jump to equilibrate at the new temperature is estimated to be 40 s). Thus, the temperature-jump seems to follow the equilibrium phase diagram but very slowly, whereas the pressure-jump induces a more complex pathway between the Lα phase and the final phase, but the various steps occur much more quickly.
The lattice parameters of all phases present during the transformations are shown in figure 15. Again, the disappearance of the lamellar phase is accompanied by the appearance of the intermediate phase X. However, some coexistence is observed between the fluid lamellar and phase X during a pressure-jump, whereas for temperature-jumps, the process is entirely discontinuous. Peaks characterizing a swollen phase grow directly from the diffuse scatter, bypassing the intermediate phase observed during pressure-jumps. We are actively investigating the dependence of the transition pathway on the thermodynamic parameters.
6. Future prospects
Time-resolved studies of lyotropic liquid crystal transitions are still at an early stage, but clearly will be invaluable in helping to clarify complex transition pathways and mechanisms. It would be particularly useful—although experimentally challenging—to carry them out on aligned samples, which would allow any epitaxial relationships between the phases to be determined at the same time. The pressure-jump technique is not restricted to lipid phase transitions, but can be used to study a wide range of soft matter transitions, ranging from protein unfolding and DNA unwinding, to phase transitions in liquid crystals, surfactants and block copolymers.
We anticipate that the most useful technical advance will be the development of X-ray pressure-jump apparatus on the microsecond time-scale, involving the use of piezoelectric stacks of pressure actuators. Such a device, for pressure-jumps of up to ±200 bar in 80 μs, has recently been developed for optical studies of protein solutions (Pearson et al. 2002). Beamline I22 at the new Diamond synchrotron, with its planned fast detectors with enhanced count-rate and energy-range capabilities, will be an ideal beamline at which to establish a UK-based microsecond X-ray pressure-jump facility.
H. F. Gleeson (School of Physics and Astronomy, University of Manchester, UK). Do you see flow as part of the p-jump or T-jump transition process? Work of Julia Yeoman (Oxford) simulating flow in cubic (blue phase) structures show structural changes in the unit cell—could this kind of process be contributing to the intermediate signals in the transitions from cubic to cubic?
J. M. Seddon. In our experiments, we have been careful to create sample environments, in which no shearing of the hydrated lipid samples can occur, nor is extraneous water able to flow in or out of the samples. However, in transforming from one phase to another there may indeed be local flows of water through the sample, related to the mechanism of the transition. As an example of this, we typically see swollen cubic phases that appear as intermediates as we make jumps from a lamellar to an inverse bicontinuous cubic phase. This evidence is consistent with the hypothesis that these out-of-equilibrium bicontinuous structures appear as a low-energy means of transporting water from one part of the sample to another, while the additional bicontinuous channels are being created in order to form the equilibrium structure.
V. Percec (Department of Chemistry, University of Pennsylvania, USA). Your pressure giving X-ray experiments were carried out on lyotropic liquid crystalline phases. What is the lowest lipid concentration on which these experiments can be carried out? Can single molecules in solution be analysed by p-jump X-ray experiments?
J. M. Seddon. The amphiphiles we study have very low critical micelle concentrations, and hence aggregate even at very low concentrations in aqueous solution. It is becoming feasible to carry out single molecule X-ray experiments, but, of course, the kind of information obtained will bear little or no relation to the structural data we are obtaining from collective, cooperative transitions between different self-assembled interfacial phase structures.
S. T. Lagerwall (The Royal Swedish Academy of Sciences, Sweden). When you heat the lamellar phase the layers curve in a certain fashion, giving the flexible carbon chains more space. Is this sign of bending a general feature for all molecules with a polar head?
J. M. Seddon. Yes, so long as they have a single polar headgroup, attached to one or more flexible chains.
P. Palffy-Muhoray (Liquid Crystal Institute, Kent State University, USA). What is the status of theory? Are there estimates for the time-scales on which pressure-jump-induced phase transitions occur?
J. M. Seddon. Theories are being developed to describe certain steps which may be involved in the transition process, for example the dynamics of hemifusion in lipid bilayers (Hed & Safran 2003). However, we are not aware of any theory that can successfully predict the time-scales of pressure-jump-induced lyotropic cubic phase transitions. We hope that our experimental results may stimulate such development.
C. R. Safinya (Department of Materials, University of California at Santa Barbara, USA). The Gaussian curvature modulus κG should be positive for the minimal surfaces to be preferred over spherical phases. You appeared to say that you had measured a negative κG. How is that possible?
J. M. Seddon. Although for a monolayer κG may be negative, however, one can show that for a bilayer κG can be positive, as is required for the thermodynamic stability of the inverse bicontinuous cubic phases. For the bending modulus, the bilayer value κb should be simply twice that of the monolayer bending modulus κ, and both should have positive values. However, as discussed in detail in a forthcoming review article (Shearman et al. 2006), a more complex situation exists when considering the link between the monolayer and bilayer Gaussian moduli, κG and . The monolayer Gaussian modulus κG has been found to be negative for systems forming inverse curved phases, and with a magnitude less than that of the bending modulus κ. The bilayer Gaussian curvature modulus is directly related to the monolayer Gaussian modulus, but also includes a term that contains the bending modulus κ. The explicit expression (Helfrich & Rennschuh 1990) is
where the bilayer is taken to be symmetric, H0 is the spontaneous mean curvature of the monolayer, and , the monolayer thickness. Since for inverse-phase forming amphiphiles, H0 is negative, the second term tends to make positive.
We have, however, found it more instructive to consider the energetics of inverse bicontinuous cubic phases from the point of view of the monolayer curvature elasticity. Here we have found that although saddle-shaped interfacial curvature is energetically more costly than cylindrical or spherical curvature, the cost of packing hydrocarbon chains into the saddle-shaped geometry of the inverse bicontinuous cubics is significantly less costly than in packings of cylindrical or spherical monolayers. When chain lengths are short this difference becomes great enough that the inverse bicontinuous cubic phases are the energetically preferred structure.
This work has been supported by Platform grant (GR/S77721) and Ph.D. studentships from the EPSRC. We thank Aurelien Huisman (Socrates student, Imperial College London) for measuring the DSC and PPC scans shown in figure 6, and Nick Brooks, Chandrashekhar Kulkarni, Sarra Sebai and Christina Turner for their help with carrying out the synchrotron experiments. We also thank Prof. R. Winter and Ms J. Kraineva (Dortmund University) and Drs S. Finet and T. Narayanan (ESRF) for their invaluable contributions.
↵† Present address: Cavendish Laboratory, University of Cambridge, Cambridge CB2 1TN, UK
One contribution of 18 to a Discussion Meeting Issue ‘New directions in liquid crystals’.
- © 2006 The Royal Society