Lifshitz Line Research Articles

Dynamic light scattering from ternary polymer blends: critical behavior and bicontinuous microemulsions

We have used dynamic light scattering to study the dynamics of ternary polymer blends consisting of poly(dimethylsiloxane) (PDMS) and poly(ethylethylene) (PEE) homopolymers and a PDMS-PEE diblock copolymer nearly symmetric in composition. The intensity autocorrelation functions for the binary blend are single-exponential, and the associated correlation length ξ scales with reduced temperature ϵ in accordance with the Ising universality class (i.e., ξ ∼ ϵ−ν, with ν = 0.63). An addition of copolymer depresses the critical temperature, but also increases the magnitude of ν. For compositions within the microemulsion channel, ξ exhibits a distinct maximum with decreasing temperature, near the Lifshitz line obtained from the static structure factor. For a particular composition, there is a “re-entrant” microemulsion, as the system passes into and then out of the phase-separated region upon cooling.

Partial integration and local mean-field approach for a vector lattice model of microemulsions

A vector model on the simple cubic lattice, describing a mixture of water, oil, and amphiphile, is considered. An integration over the amphiphile orientational degrees of freedom is performed exactly in order to obtain an effective Hamiltonian for the system. The resulting model is a three-state (spin-1) system and contains many-site interaction terms. The analysis of the ground state reveals the presence of the water--oil-rich phase as well as the amphiphile-rich and the cubic phases. The temperature phase diagram of the system is analyzed in a local mean-field approach, and a triple line of water-rich, oil-rich, and microemulsion coexistence is obtained. For some values of the model parameters, lamellar phases also appear in the system, but only at finite temperature. The Lifshitz line is determined in a semianalytical way in order to locate the microemulsion region of the disordered phase.

The structure and phase transitions in polymer blends, diblock copolymers and liquid crystalline polymers: The Landau‐Ginzburg approach

AbstractThe polymer systems are discussed in the framework of the Landau‐Ginzburg model. The model is derived from the mesoscopic Edwards Hamiltonian via the conditional partition function. We discuss flexible, semiflexible and rigid polymers. The following systems are studied: polymer blends, flexible diblock and multi‐block copolymer melts, random copolymer melts, ring polymers, rigid‐flexible diblock copolymer melts, mixtures of copolymers and homopolymers and mixtures of liquid crystalline polymers. Three methods are used to study the systems: mean‐field model, self consistent one‐loop approximation and self consistent field theory. The following problems are studied and discussed: the phase diagrams, scattering intensities and correlation functions, single chain statistics and behavior of single chains close to critical points, fluctuations induced shift of phase boundaries. In particular we shall discuss shrinking of the polymer chains close to the critical point in polymer blends, size of the Ginzburg region in polymer blends and shift of the critical temperature. In the rigid‐flexible diblock copolymers we shall discuss the density nematic order parameter correlation function. The correlation functions in this system are found to oscillate with the characteristic period equal to the length of the rigid part of the diblock copolymer. The density and nematic order parameter measured along the given direction are anticorrelated. In the flexible diblock copolymer system we shall discuss various phases including the double diamond and gyroid structures. The single chain statistics in the disordered phase of a flexible diblock copolymer system is shown to deviate from the Gaussian statistics due to fluctuations. In the one loop approximation one shows that the diblock copolymer chain is stretched in the point where two incompatible blocks meet but also that each block shrinks close to the microphase separation transition. The stretching outweights shrinking and the net result is the increase of the radius of gyration above the Gaussian value. Certain properties of homopolymer/copolymer systems are discussed. Diblock copolymers solubilize two incompatible homopolymers by forming a monolayer interface between them. The interface has a positive saddle splay modulus which means that the interfaces in the disordered phase should be characterized by a negative Gaussian curvature. We also show that in such a mixture the Lifshitz tricritical point is encountered. The properties of this unusual point are presented. The Lifshitz, equimaxima and disorder lines are shown to provide a useful tool for studying local ordering in polymer mixtures. In the liquid crystalline mixtures the isotropic nematic phase transition is discussed. We concentrate on static, equilibrium properties of the polymer systems.

Effect of capillary wave fluctuations on wetting transitions in balanced amphiphilic systems

Wetting transitions in balanced amphiphilic systems containing only short-range forces are studied. In the absence of fluctuations, a weak continuous wetting transition would take place at three-phase coexistence at the disorder line. We argue that capillary wave fluctuations cause the transition to become first order and to occur on the structured side of the disorder line. With reasonable model parameters, the transition occurs in the vicinity of the Lifshitz line where it is observed experimentally.

Effects of fluctuations on the wetting transition in amphiphilic systems

The wetting of the oil–water interface by the middle phase in an amphiphilic system is considered for both balanced and unbalanced systems within the context of a one-component Ginzburg–Landau theory. Whereas mean field theory predicts a continuous transition in the balanced system at the disorder line, the effect of capillary fluctuations is to cause the transition to be first order, and to occur on the microemulsion side of the disorder line. With reasonable parameters, the location is shifted beyond the Lifshitz line, as is found in experiment. When the wetting transition is brought about in the unbalanced system, we find that the middle phase is more structured at the transition than in the balanced system. This occurs because the fluctuations become greater as the critical end point is approached, and are able to bring about the unbinding of interfaces in the presence of even stronger binding potentials. Experiments are consistent with this prediction.

The Lifshitz line in binary systems: Structures in water/C4E1 mixtures

In a recent publication we compared phase behavior and scattering data obtained from SANS in water, n-alkane, n-alkyl polyglycol ether (CiEj) mixtures. By analyzing the scattering spectra using the Teubner–Strey formula, it is possible to determine a measure for the amphiphilic strength of each system called the amphiphilicity factor, fa. It was demonstrated that the amphiphilicity factor is constant on a variety of surfaces within the three-dimensional space of composition and temperature. For instance, the Lifshitz surface, where fa=0, depends on composition and temperature and may be a precursor for the Lα phase. Here we show that the same sequence exists in the binary system water/C4E1, where the surfaces become lines in the two-dimensional composition-temperature planes. Variations in amphiphile concentration and solution temperature allow one to reach almost the entire accessible amphiphilicity scale (∞≳fa≳−1) within this simple binary mixture. The occurrence of the Lifshitz surface at high C4E1 concentrations and its disappearance as a function of temperature give more evidence that the Lifshitz region can be treated as a precursor of the lamellar liquid crystalline phase. While earlier investigations of water/C4E1 solutions in the dilute regime support the existence of micellar aggregates, the current work provides the first definitive evidence of the presence of supramolecular aggregates and short range order in the water/C4E1 system in the amphiphile-rich region.

Small angle neutron scattering near Lifshitz lines: Transition from weakly structured mixtures to microemulsions

We have studied the phase behavior, wetting transitions, and small angle neutron scattering (SANS) of water, n-alkane, and n-alkyl polyglycol ether (CiEj) systems in order to locate the transition between weakly structured mixtures and microemulsions, and to provide a measure for the transition. We first determined the wetting transition by macroscopic measurements and then measured the location of the Lifshitz lines by SANS. Starting with well-structured mixtures (exhibiting nonwetting middle phases and well-expressed scattering peaks, features that qualify them as microemulsions) the wetting transition was induced by increasing the chain length of the alkane or by changing the oil/water volume ratio, and then the Lifshitz line was crossed. Further, starting with systems past the disorder line (weakly structured mixtures that display wetting middle phases and no scattering peaks), local structure was induced by either increasing the surfactant concentration or decreasing the oil/water volume ratio or the temperature. In each case a Lifshitz line was crossed. Analyzing the scattering experiments quantitatively, allows determination of the amphiphilicity factor, which is a measure of the strength of the surfactant. The results suggest there is a sequence of roughly parallel surfaces within the three-dimensional composition-temperature space. As the amphiphilicity factor increases, first a disorder surface is encountered, then a Lifshitz surface, and finally a wetting transition surface. How and to what extent these surfaces move in the one-phase region toward smaller surfactant concentrations, and intersect there with the body of heterogeneous phases, depends on a number of factors that are discussed in some detail.

Scattering properties of the triangular Ising antiferromagnet: Disorder and Lifshitz lines.

We study an anisotropic nearest-neighbor Ising antiferromagnet on the triangular lattice, using Monte-Carlo simulations and random-phase (or related) approximations. Our main interest concerns the high-temperature magnetic scattering. Exact results are available at the disorder line (Stephenson; Welberry and Galbraith). Of greater experimental significance is the Lifshitz line where (i) splitting of the scattering peaks occurs and (ii) dislocationlike defects appear in the (instantaneous) spin configurations. A good analytical fit of the Lifshitz line is given by the Bethe-Peierls approximation. These results could be relevant to other systems with disorder lines, e.g., the ANNNI model and microemulsions.

A triangular lattice model for binary and ternary mixtures containing surfactants

The phase equilibria of ternary mixtures of water, oil and surfactants and binary mixtures of water and surfactants are studied using a triangular lattice-gas model that is an extension of the spin-1 Blume-Capel model with the addition of the orientational degrees of freedom for surfactants. The phase diagram of the model is investigated via Monte Carlo simulations in conjunction with the extrapolation technique of Ferrenberg and Swendsen combined with the finite-size scaling method of Lee and Kosterlitz. Five phases are found for ternary mixtures in the case of equal water and oil concentrations: a water-oil coexistence region, a lamellar phase, a liquid crystalline phase, which has the symmetry of a rhombic lattice, and an extended disordered phase, which is divided by a Lifshitz line into disordered fluid and microemulsion regions. In the limiting case of a binary mixture of water and surfactants, we found a rich phase diagram corresponding to a lamellar phase, a hexagonal phase and a disordered phase. which is again divided by a Lifshitz line into disordered fluid and structured fluid regions. The latter can be considered as the analogue of the microemulsion found for the ternary systems.

Monte Carlo study of interfacial properties in an amphiphilic system.

A three-component model representing a balanced, amphiphilic system at a three-phase coexistence of oil-, water-rich, and disordered phases is simulated, and interfacial tensions between all phases are obtained from histogram methods. We find that the disordered phase wets the oil-water interface, and that its tension is reduced due to the presence of the amphiphile by a factor of 93. The determination of the structure function shows that the system is between disorder and Lifshitz lines. These three factors are all consistent with the simulated amphiphile being weak. Fluctuations do not reduce the tension significantly from mean-field estimates, but do alter the location of the wetting transition. The latter is in accord with the expected behavior of a system in which the interfacial tension is small.

On the Stability Range of Microemulsions: From the Tricritical Point to the Lamellar Phase in Water/Formamide‐Octane‐CiEj Systems

AbstractSymmetric microemulsions containing equal volume fractions of water/formamide, n‐octane and a sufficient amount of n‐alkyl polyglycol ether (CiEj) were studied for various surfactant chain lengths ranging from C4E1 to C14E5 in pure water, as well as for given C4E1 with increasing relative amounts of formamide in the water/formamide mixture (ψ in wt%). In both cases the amphiphilic strength (the amphiphilicity) is changed. Increasing the amphiphilicity one observes, as is well known, the occurrence of the lamellar phase. For the balanced state the lamellar phase occupies a large portion of the isothermal Gibbs triangle as is demonstrated with C10E5. We argue that with further increasing amphiphilicity the three phase body disappears at a four phase point. Decreasing the amphiphilicity the three phase body disappears passing a sequence of lines. First a nonwetting‐wetting transition can macroscopically be observed. Then, a disorder line is passed at which the behavior of the real space correlation function changes from a damped oscillatory behavior to a monotonically decreasing one. We determined previously that this line has been passed by small‐angle neutron scattering. Further, a Lifshitz line as predicted by Landau‐Ginzburg theories occurs until, as the amphiphilicity is even further reduced, a tricritical point is reached at which the three phase body disappears altogether. The results presented may serve as a basis for further experimental research and also stimulate the theoretical description of such systems.

Copolymers as amphiphiles in ternary mixtures: An analysis employing disorder, equimaxima, and Lifshitz lines

We study the phase diagram and scattering functions of a ternary mixture of A and B homopolymers, in equal concentrations, and AB diblock copolymer. We locate the disorder line, which marks the appearance of damped oscillatory decaying correlation functions in the system, and also the Lifshitz and equimaxima lines of the structure functions. At these lines, the oscillatory component becomes dominant in a given correlation so that the peak in scattering intensity is no longer at the zero wave vector. We find that while the Lifshitz line of the structure function of all A monomers is quite close to the disorder line, that of A monomers in the homopolymer only is far from it. This shows that the copolymer orders the homopolymers inefficiently, a behavior which is also reflected in the copolymer’s weak ability to solubilize them, and which contrasts with the amphiphilic solubilizers of oil and water. The disorder line and all Lifshitz lines meet at a Lifshitz tricritical point. As these ternary mixtures provide a rare opportunity to study this unusual point, we discuss its effects on the structure functions and surface tensions.

One-dimensional model for microemulsions.

In this paper, a model for water-oil-surfactant mixtures, which we have previously studied on two- and three-dimensional lattices, is now studied on a one-dimensional lattice. In this case we are able to obtain exact results, whereas on the higher-dimensional lattices it was necessary to use approximations. This one-dimensional model produces correlation and structure functions that are similar to those obtained for the disordered phase on the two- and three-dimensional lattices. The disorder line is obtained from the water-water correlation function and the Lifshitz line is derived from the water-water structure function. One or the other of these lines is typically used to divide the disordered phase into a region of ordinary disordered fluid and a microemulsion region. Both these lines calculated exactly for the one-dimensional lattice behave similarly to their counterparts on the two- and three-dimensional lattices calculated by various approximations.

Phase diagram, fluctuations, and phase transitions near the liquid-crystal nematic-smectic-A-smectic-C multicritical point.

We have performed high-resolution x-ray scattering studies of the binary liquid-crystal system of octyloxycyanobiphenyl and heptyloxypentylphenylthiolbenzoate (80CB+7\ifmmode\bar\else\textasciimacron\fi{}S5) near the nematic--smectic-A--smectic-C (NAC) multicritical point. We find that the phase diagram and smectic-order fluctuations are well described by a Lifshitz-point model. Above the nematic--smectic-C(N-SmC) phase boundary, we find a line along which the fluctuations change from being SmA-like to SmC-like; this is signaled by the vanishing of the transverse quadratic term in the expression for the x-ray scattering intensity; that is, the transverse fluctuations scale like ${q}_{\ensuremath{\perp}}^{\mathrm{\ensuremath{-}}4}$. The geometry of this Lifshitz line appears to mirror that of the SmA-SmC boundary. A much more extended SmC fluctuation region is observed for lower concentrations of 80CB. Thus the N-SmC pretransitional area is divided into two regions: a pure N-SmC region and a multicritical region. The tilt angle exhibits a first-order transition at the N-SmC boundary. Along the SmA-SmC phase boundary, the tilt is well described either by a power law with an average exponent \ensuremath{\beta}=0.43\ifmmode\pm\else\textpm\fi{}0.03 or by mean-field theory including a sixth-order correction. No crossover to critical or tricritical behavior is observed as the NAC point is approached. Along the N-SmA phase boundary, we observe second-order N-SmA transitions with average values of the exponents in this region of ${\ensuremath{\nu}}_{?}$=0.90, ${\ensuremath{\nu}}_{\ensuremath{\perp}}$=0.75, and \ensuremath{\gamma}=1.55 for the parallel and perpendicular correlation lengths and susceptibility, respectively. Very near the NAC concentration, we observe multicritical behavior, with the above exponents renormalized by the crossover exponent of the N-SmA phase boundary.