Unbinding transition

Fig. 6.22 Phase diagram for blends of PE and PEP homopolymers (A/j, - 392 and 409 respectively) with a PE-PEP diblock (iVc = 1925) (Bates et al. 1995). Open and filled circles denote experimental phase transitions between ordered and disordered states measured by SANS and rheology respectively. Phase boundaries obtained from self-consistent field calculations are shown as solid lines. The diamond indicates the Lifshitz point (LP), below which an unbinding transition (UT) separates lamellar and two-phase regions in mean field theory.

Fig. 6.33 Similar to Fig. 6.31, but for ft = (Matsen 19956). In this case the Helmholtz free energy curve indicates that macrophase separation does not occur, and so an unbinding transition occurs at the composition indicated by the dot. In the phase diagram, the diamond shows where the stability line for microphase separation meets the unbinding transition (Lifshitz point). 

Self-consistent field theory has recently been employed by Janert and Schick (1996,1997a) to analyse the swelling of diblock lamellar phases with homopolymer. It was shown that a complete unbinding transition, where added homopolymer swells the lamellae, finally leading to a transition to a disordered phase, is predicted by mean field theory. The swelling does not continue without limit. 

Fig. 6.36 Phase diagram calculated using SCFT for a blend of a symmetric diblock with a homopolymer with fl = 1 (see Fig. 6.32 for a blend with a diblock with / = 0.45) as a function of the copolymer volume fraction

Fig. 6.40 A phase diagram calculated using SCFT for a mixture containing equal amounts of two homopolymers and a symmetric diblock, all with equal chain length (Janert and Schick 1997a). A-rich and B-rich swollen lamellar bilayer phases are denoted LA and LH respectively whilst the corresponding disordered phases are denoted A and B. The con-solute line of asymmetric bilayer phases LA and Lu, shown dotted, is schematic.The dashed line is the unbinding line. The arrows indicate the locations of the unbinding transition X jN and multicritical Lifshitz point, cMiV " 6.0.

In a lipid/water system, the thickness of the bilayers is constant, and the unbinding transition can occur, in principle, by varying the Hamaker constant or temperature.22 For lyotropic lamellar liquid crystals, hyperswelling in a liquid of one kind might also occur if the lamellae of the other kind are thin enough and hence unbound. However, thin lamellae might lead to a positive contribution to the free energy (since the repulsive forces overcome at short distances the van der Waals attraction), and hence the lamellar phase can become unstable. 

The role of undulation on the equilibrium of lipid bilayers was also examined by Lipowsky and Leibler,18 who used a nonlinear functional renormalization group approach, and by Sornette,14 who employed a linear functional renormalization approach. It was theoretically predicted that a critical unbinding transition (corresponding to a transition from a finite to an infinite swelling) can occur by varying either the temperature or the Hamaker constant. However, the renormalization group procedures do not offer quantitative information about the systems, when they are not in the close vicinity of this critical point. 

In Figure 7 we present the free energy for an asymmetric Gaussian distribution (a = 1.4) as a function of distance for various values of the Hamaker constant (with all the other parameters unchanged). For H > 3.825 x 10-21 J, a stable minimum is obtained at a finite distance. For H < 3.825 x 10 21 J, the stable minimum is at infinite distance however, for 3.825 x 10-21c7 > H > 3.45 x 10-21 J, a local (unstable) minimum is still obtained at finite distance. For H = 3.825 x 10-21 J, a critical unbinding transition occurs, since the minima at finite and infinite distances become equal. However, these two minima are separated by a potential barrier, with a maximum height of 1.68 x 10 7 J/m2, located at a separation distance of 90 A. The results remained qualitatively the same for any combination of the interaction parameters. 

The role of thermal undulations of membranes in their stability [4] and particularly in the unbinding transition [5] has been early recognized. Helfrich derived the first analytical expression for the repulsion due to the entropic confinement of the membrane, by assuming hard-wall repulsions between membranes. When the membranes are far apart, they can undulate freely ... 

Halperin and Nelson [53,54] and Young [55] recognized that the vector character of dislocations must be taken into account in calculating the melting temperature, and also recognized that the dislocation-unbinding transition results in a sixfold bond orientationally ordered fluid phase, the hexatic phase, and that a second, discliriatiori- mbm mg transition is required to obtain an isotropic fluid. 

The quasi-long-range bond orientational order present in the hexatic phase is destroyed at a second, disclination-unbinding, transition, which occurs at a temperature T >T. In this transition the tightly bound... 

Deme, B., Dubois, M., Gulik-Krzywicki, T. and Zemb, Th. (2002) Giant collective fluctuations of charged membranes at the lamellar-to-vesicle unbinding transition. 1. Characterisation of a new lipid morphology by SANS, SAXS, and electron microscopy. Fangmuir, 18, 997-1004. 

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