Lifshitz-point

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

To find the actual transition we compare the temperatures determined for a given ps by (60) and (61) and choose the higher one. It turns out that for concentrations lower than at the Lifshitz point [13], in our case given by... 

G. Gompper, R. Holyst, M. Schick. Interfacial properties of amphiphilic systems the approach to Lifshitz points. Phys Rev A 45 3157-3160, 1991. 

The spinodal represents a hypersurface within the space of external parameters where the homogeneous state of an equilibrium system becomes thermodynamically absolutely unstable. The loss of this stability can occur with respect to the density fluctuations with wave vector either equal to zero or distinct from it. These two possibilities correspond, respectively, to trivial and nontrivial branches of a spinodal. The Lifshitz points are located on the hyperline common for both branches. 

A particular complex problem has been the modelling of Si/W(l 10) Amar et have included pairwise interactions up to the sixth nearest neighbor shell, as estimated experimentally from field-ion microscopic studies The predicted phase diagram (Fig. 30) exhibits (5 x 1), (6 x 1) and p(2 x 1) commensurate phases, as well as a broad regime of an incommensurate phase. In contrast to the ANNNI model the present model does seem to have a finite-temperature Lifshitz point, where the incommensurate, commensurate... 

Fig. 30. Phase diagram of a model for Si/W(110) in the temperature versus 9 plane. Experimentally determined interactions J Jj,are used. Full dots are from Monte Carlo calculations, while triangles are based on transfer matrix finite size scaling using strip widths of 8 and 12. The point labelled L indicates approximate location of Lifshitz point. The dotted line indicates the transition region between the (5 x l)and(6 x 1) phases. (From...

In a blend of immiscible homopolymers, macrophase separation is favoured on decreasing the temperature in a blend with an upper critical solution temperature (UCST) or on increasing the temperature in a blend with a lower critical solution temperature (LCST). Addition of a block copolymer leads to competition between this macrophase separation and microphase separation of the copolymer. From a practical viewpoint, addition of a block copolymer can be used to suppress phase separation or to compatibilize the homopolymers. Indeed, this is one of the main applications of block copolymers. The compatibilization results from the reduction of interfacial tension that accompanies the segregation of block copolymers to the interface. From a more fundamental viewpoint, the competing effects of macrophase and microphase separation lead to a rich critical phenomenology. In addition to the ordinary critical points of macrophase separation, tricritical points exist where critical lines for the ternary system meet. A Lifshitz point is defined along the line of critical transitions, at the crossover between regimes of macrophase separation and microphase separation. This critical behaviour is discussed in more depth in Chapter 6. 

Fig. 2.42 Spinodal lines for a random multiblock copolymer melt of variable X (Fredrickson el al. 1992). On cooling a melt with X > AL —0.268, the first instability is predicted to be phase separation into two homogeneous liquid phases (x = %m)- On further cooling to % = the two liquid phases become unstable with respect to formation of a microphase. In contrast, a melt with X < XL first becomes absolutely unstable to the formation of microphases (x = fom)- At the critical composition of /= j, the point (AL, Xi) is an isotropic Lifshitz point.

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. 6.23 Logarithmic plots of the correlation length ( ) and zero-angle scattering intensity (/(0)) as a function of temperature reduced with respect to the Lilshitz temperature Tlp) for a blend of PE and PEP homopolymers with a PE-PEP diblock (details as Fig. 6.22) at a copolymer volume fraction

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). 

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.

The phase behaviour of blends of homopolymers containing block copolymers is governed by a competition between macrophase separation of the homopolymer and microphase separation of the block copolymers. The former occurs at a wavenumber q = 0, whereas the latter is characterized by q + 0. The locus of critical transitions at q, the so-called X line, is divided into q = 0 and q + 0 branches by the (isotropic) Lifshitz point. The Lifshitz point can be described using a simple Landau-Ginzburg free-energy functional for a scalar order parameter rp(r), which for ternary blends containing block copolymers is the total volume fraction of, say, A monomers. The free energy density can be written (Selke 1992)... 

Now another multicritical point arises for the special case where K = 0 (cf. fig. 23), and then eq. (117) yields a Lifshitz point (Hornreich et al., 1975)... 

Fig. 23. Schematic phase diagram of a system where by a variation of a parameter p the coefficient K (p) of the gradient energy (1/2)Jf (p)(v0)2 vanishes at a Lifshitz point Kl = 0, r, (pL) = Tl. For p < pl one has a ferromagnetic structure, while for p > pi where fC (p) < 0 one has a modulated structure, with a characteristic wavenumber q describing the modulation. For p -> p from above one has — 0 along the critical line...

Again one concludes that the scaling relations eqs. (80), (90) and (91) are satisfied, while the hyperscaling relation [eq. (93)] would only be satisfied at d — 8. Indeed, using the Lifshitz exponents in the Ginzburg criterion [eqs. (52)—(55)] one does find that the Landau description of Lifshitz points becomes self-consistent only for d > 8. Thus it is no surprise that the behavior at physical dimensionalities (d = 2, 3) is very different from the above predictions. In fact, in d = 2 one does not have Lifshitz points at non-zero temperature (Selke, 1992). 

For ordinary critical phenomena, such a spatial anisotropy is not very important — it gives rise to an anisotropy of the critical amplitude, of the correlation length in different lattice directions ( i = r[-1, fx = xffl 1 ), while the critical exponent dearly is the same for all spatial directions. Of course, this is no longer necessarily true at Lifshitz points There is no reason to assume that both functions / (p), K (p) in eq. (122) vanish for p = pt Let us rather assume that only ih(pl) — 0 while K (pO > 0 this yields the uniaxial Lifshitz point (Homreich et al., 1975). We then have to add a term f Ki (p)[32 (x)/9x ]2 to eq. (122) to find... 

Earlier, Homreich, Luban and Shtrikman had proposed a new type of higher order critical point, which they called the Lifshitz point, by considering a Landau-Ginsburg free energy expansion in terms of a scalar order parameter M for a magnetic system... 

The phase diagram involves the ferromagnetic, paramagnetic and heli-coidal phases, the last being the modulated form of the ordered phase described by the gradient terms. The Lifshitz point occurs when the coefficients a and q vanish, and is characterized by the fact that the... 

Hence the = 0 point is a fluctuation cross-over point. The locus of such points (a > 0, Cj = 0) in a phase diagram, say the T-X diagram, gives the Lifshitz line, and the point where both a and are zero is the Lifshitz point. 

The transverse mass density fluctuations then become purely quartic. Thus the model makes the important prediction that there will be a change from Lorentzian to quartic in the scattering profile near the Lifshitz point. 

At the Lifshitz point itself, there is a drastic change and Sk oc while dk ... 

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