Mathematical models of polyelectrolyte gels are often simplified by assuming the gel is electrically neutral. The rationale behind this assumption is that the thickness of the electric double layer (EDL) at the free surface of the gel is small compared to the size of the gel. Hence, the thin-EDL limit is taken, in which the thickness of the EDL is set to zero. Despite the widespread use of the thin-EDL limit, the solutions in the EDL are rarely computed and shown to match to the solutions for the electrically neutral bulk. The aims of this paper are to study the structure of the EDL and establish the validity of the thin-EDL limit. The model for the gel accounts for phase separation, which gives rise to diffuse interfaces with a thickness described by the Kuhn length. We show that the solutions in the EDL can only be asymptotically matched to the solutions for an electrically neutral bulk, in general, when the Debye length is much smaller than the Kuhn length. If the Debye length is similar to or larger than the Kuhn length, then phase separation can be initiated in the EDL. This phase separation spreads into the bulk of the gel and gives rise to electrically charged layers with different degrees of swelling. Thus, the thin-EDL limit and the assumption of electroneutrality only generally apply when the Debye length is much smaller than the Kuhn length.

We derive and analyse an energy to model lipid raft formation on biological membranes involving a coupling between the local mean curvature and the local composition. We apply a perturbation method recently introduced by Fritz, Hobbs and the first author to describe the geometry of the surface as a graph over an undeformed Helfrich energy minimising surface. The result is a surface Cahn–Hilliard functional coupled with a small deformation energy. We show that suitable minimisers of this energy exist and consider a gradient flow with conserved Allen–Cahn dynamics, for which existence and uniqueness results are proven. Finally, numerical simulations show that for the long-time behaviour raft-like structures can emerge and stabilise, and their parameter dependence is further explored.

Under conventional solidification conditions, immiscible alloy melt would undergo large-scale composition segregation after liquid–liquid phase separation, resulting in the loss of properties and application value. In the present study, the ternary immiscible Al70Bi10Sn20 alloy was chosen to study the effect of cooling rate on its resultant microstructure by casting the melt under different cooling conditions. The results indicated that the Al–Bi–Sn alloy with a slow cooling rate exhibits a strong spatial phase separation trend during solidification. However, as the cooling rate increases, the decreasing volume fraction of the segregated Bi–Sn-rich regions indicates the efficient suppression of spatial phase separation. The relatively dispersed distribution of Bi–Sn phase in the Al-rich matrix can be obtained by quenching the melt into water. The influence mechanism of cooling rate on the microstructure of the alloy is also discussed. The present study is beneficial to further tailoring the microstructure of immiscible alloys.

We provide a detailed mathematical analysis of a model for phase separation on biological membranes which was recently proposed by Garcke, Rätz, Röger and the second author. The model is an extended Cahn–Hilliard equation which contains additional terms to account for the active transport processes. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate two different asymptotic regimes. We study the case of large cytosolic diffusion and investigate the effect of an infinitely large affinity between membrane components. The first case leads to the reduction of coupled bulk-surface equations in the model to a system of surface equations with non-local contributions. Subsequently, we recover a variant of the well-known Ohta–Kawasaki equation as the limit for infinitely large affinity between membrane components.

In the design of high-entropy alloys (HEAs) with desired properties, identifying the effects of elements plays an important role. HEAs with eutectic microstructure can be obtained by judiciously modifying the alloy compositions. In this study, the effect of Nb addition to FeCoNiCuNbx (x = 0.5, 5, 7.5, 11.6, 15) alloys was studied by varying the Nb concentration (at.%). FeCoNiCuNb0.5 HEA shows liquid phase separation to form Cu-rich and FeCoNiCu-rich phases. Detailed solidification paths are proposed for these alloys, which show eutectic, peritectic, and pseudo quasi-peritectic reactions. Increasing Nb content promotes the liquid phase separation tendency and causes the formation of Cu-rich spheres. The effect of Nb on the FeCoNiCu-rich phase was studied based on the nanoindentation and correlated with nanohardness. The compressive deformation properties of these alloys are studied at room temperature and high temperature and correlated with microstructure. Fractography results show the mode of fracture and are correlated with the microstructure obtained.

Cell membrane fatty acids influence fundamental properties of the plasma membrane, including membrane fluidity, protein functionality, and lipid raft signalling. Evidence suggests that dietary n-3 PUFA may target the plasma membrane of immune cells by altering plasma membrane lipid dynamics, thereby regulating the attenuation of immune cell activation and suppression of inflammation. As lipid-based immunotherapy might be a promising new clinical strategy for the treatment of inflammatory disorders, we conducted in vitro and in vivo experiments to examine the effects of n-3 PUFA on CD4+ T cell membrane order, mitochondrial bioenergetics and lymphoproliferation. n-3 PUFA were incorporated into human primary CD4+ T cells phospholipids in vitro in a dose-dependent manner, resulting in a reduction in whole cell membrane order, oxidative phosphorylation and proliferation. At higher doses, n-3 PUFA induced unique phase separation in T cell-derived giant plasma membrane vesicles. Similarly, in a short-term human pilot study, supplementation of fish oil (4 g n-3 PUFA/d) for 6 weeks in healthy subjects significantly elevated EPA (20 : 5n-3) levels in CD4+ T cell membrane phospholipids, and reduced membrane lipid order. These results demonstrate that the dynamic reshaping of human CD4+ T cell plasma membrane organisation by n-3 PUFA may modulate down-stream clonal expansion.

A lattice Boltzmann method is utilized for governing equations which control phase separation of binary fluids with reversible chemical reaction in presence of a shear flow in this paper. We first present the morphology modeling of sheared binary fluids with reversible chemical reaction. We then validate the model by taking the unsheared binary fluids as an example. It is found that the results fit well with the references. The paper shows structures of the sheared system and gives the detailed analysis for the morphology of sheared binary fluids with reversible chemical reaction. The phase separation of the domain structures with different chemical reaction rates is discussed. Through simulations of the sheared binary fluids, two interesting phenomena are observed, which do not exist in a binary mixture without reversible chemical reaction. One is that the same results appear in both low and high viscosity, and the other is that the domain growth exponent with both low and high viscosities presents wave due to the competition of the viscosity and phase separation. In addition, we find that the finite size effects resulting in the growth exponent decreasing appear faster than that of the unsheared blend at a large time when the size of domains is comparable with the lattice size.

Diblock copolymers (BCPs) show phase separation on mesoscopic length scales and form ordered morphologies in both bulk and thin films, the latter resulting in nanostructured surfaces. Morphologies in thin films are strongly influenced by film parameters, the ratio of film thickness and bulk domain spacing. Laterally structured polymer surfaces may serve as templates for controlled assembly of nanoparticles (NPs). We investigated the BCP of poly(n-pentyl methacrylate) and poly(methyl methacrylate) which show bulk morphologies of stacked lamellae or hexagonally packed cylinders. Thin films were investigated by atomic force microscopy and grazing-incidence small-angle X-ray scattering. For film thicknesses f well below d bulk, standing cylinder morphologies were observed in appropriate molar ratios, while film thicknesses around and larger than d bulk resulted in cylinders arranged parallel to surface. To alter and/or improve the morphology also in presence of different NPs (e.g., silica, gold), solvent vapour annealing (SVA) was applied. The BCP morphology usually remains unchanged but periodicities change depending on type and amount of incorporated NPs. It was found that silica clusters enlarge lateral distances of cylinders, whereas Au NPs reduce it. The effect of SVA is weak. The quality of morphology is slightly improved by SVA and lateral distances remain constant or are slightly reduced.

In this study, a novel hybrid block copolymer containing POSS (BCP), poly(methacrylisobutyl-POSS)-b-poly(methylmethacrylate) (PMAiBuPOSS-b-PMMA) was synthesized via reversible addition-fragmentation chain transfer (RAFT) polymerization. The structure and molecular weight were characterized via 1H NMR and GPC. BCP was creatively used as the compatibilizer to overcome the bad compatibility of epoxy and POSS in their blend system. SEM and dynamic mechanical thermal analyses (DMTA) were used to observe the surface morphology and thermal–mechanical behaviors of the resultant products. We found that the amount of microaggregation domains of POSS decreased, while the nano ones increased, when BCP content increased. All the aggregation domains were distributed in epoxy matrix uniformly at nanoscale with the addition of 10 phr BCP and 5 phr POSS monomers. The results indicated that BCP could effectively improve the compatibility between epoxy resin and POSS owing to its amphiphilicity in DGEBA. The fracture behavior of products transformed from brittle fracture to ductile fracture gradually with the increase of BCP, whereas the Tg and E′ decreased.

Semifluorinated (SF) side chain polymers show phase separation between polymer backbone and SF side chains. Due to strong interaction between SF segments the side chains determine the structure behaviour strongly, often resulting in layered structures in which backbones and layers of SF side chains alternate. The interest in this work was directed to find out the dependence of these structures on concentration of SF side chains. Thin films of random copolymers consisting of methylmethacrylate (MMA) and semifluorinated side chain methacrylate (SFMA) segments and with different fluorine content in the perfluoroalkyl side chains (abbreviated as H10F10 and H2F8) were prepared by spin-coating. Phase separation and structure changes were initiated by external subsequent annealing. Corresponding bulk material served as basic information. Generation of ordered structures and variation of film parameters were observed using different X-ray scattering methods (XRR, GIWAXS, and GISAXS). The phase behaviour in bulk is governed by the SF side chain amount and their specific fluorine content which control the self-organization tendency of SF side chains. Additionally, the confinement in thin films generates an orientation of side chains normally to film surface.

In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫Ω½Γ∇c : ∇c + ½|∇z|2+Wch(c)+Wel(e,c,z)dx with a polynomial or logarithmic chemical energy density Wch, an inhomogeneous elastic energy density Wel and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.

This paper discusses the development of an invariant finite difference scheme to simulate the microphase separation of copolymers in one-dimensional thin liquid films. The film phenomena are modelled using two-phase shallow water equations and the Ohta-Kawasaki potential, which governs the phase separation of the copolymer. Non-positive volume fractions and spurious oscillations are eventually eliminated, in simulating the one-dimensional phase separation lamellar pattern.

We show that an enslaved phase-separation front moving with diffusive speeds can leave alternating domains of increasing size in their wake. We find the size and spacing of these domains is identical to Liesegang patterns. For equal composition of the components we are able to predict the exact form of the pattern analytically. To our knowledge this is the first fully analytical derivation of the Liesegang laws. We also show that there is a critical value for C below which only two domains are formed. Our analytical predictions are verified by numerical simulations using a lattice Boltzmann method.

A modified Verlet method which involves a kind of mid-point rule is constructed and applied to the one-dimensional motion of elastic balls of finite size, falling under constant gravity in space and then under the chemical potential in the interface region of phase separation within a two-liquid film. When applied to the simulation of two balls falling under constant gravity in space, the new method is found to be computationally superior to the usual Verlet method and to Runge–Kutta methods, as it allows a larger time step for comparable accuracy. The main purpose of this paper is to develop an efficient numerical method to simulate balls in the interface region of phase separation within the two-liquid film, where the ball motion is coupled with two-phase flow. The two-phase flow in the film is described via shallow water equations, using an invariant finite difference scheme that accurately resolves the interface region. A larger time step in computing the ball motion, more comparable with the time step in computing the two-phase flow, is a significant advantage. The computational efficiency of the new method in the coupled problem is demonstrated for the case of four elastic balls in the two-liquid film.

We determine the steady-state structures that result from liquid-liquid demixing in afree surface film of binary liquid on a solid substrate. The considered model correspondsto the static limit of the diffuse interface theory describing the phase separationprocess for a binary liquid (model-H), when supplemented by boundary conditions at thefree surface and taking the influence of the solid substrate into account. The resultingvariational problem is numerically solved employing a Finite Element Method on an adaptivegrid. The developed numerical scheme allows us to obtain the coupled steady-state filmthickness profile and the concentration profile inside the film. As an example wedetermine steady state profiles for a reflection-symmetric two-dimensional droplet forvarious surface tensions of the film and various preferential attraction strength of onecomponent to the substrate. We discuss the relation of the results of the present diffuseinterface theory to the sharp interface limit and determine the effective interfacetension of the diffuse interface by several means.

La rhéologie sanguine est complexe et non-linéaire. Elle présente, dans les plus petits vaisseaux,des variations de viscosité effective importantes liées au confinement des globules rouges dans les capillaires et des dissymétries de répartition des globules rouges aux bifurcations. Ces deux effets, appelés effet Fåhræus-Lindquist et ségrégation de phase, conduisent à une modélisation non-linéaire du flux sanguin en réponse à une différence de pression imposée.Le calcul numérique complet de différents modèles non-linéaires de rhéologie sanguine, dans des géométries réelles tridimensionnelles de micro-réseaux vasculaires, permet d'exhiber certaines propriétés remarquables de ces écoulements et de ces modèles.D'une part, nous constatons que différents modèles conduisent à des répartitions de la pressionsanguine très proches, pour une gamme d'hématocrite systémique imposée physiologique. D'autre part, différents modèles de ségrégation de phase conduisent à des répartitionsd'hématocrite très différentes. Toutefois, la répartition de l'hématocrite n'affecte que très faiblement la répartition de la pression. Ce constat nous montre que c'est la micro-structure vasculaire qui impose la distribution de pression, alors que les variations de viscosité sanguine avec l'hématocrite impactent finalement faiblement sur cette distribution.

Differences in structure between egg constituent proteins influence their ability to interact, upon heating, in gel systems or to compete for space at the interface of oil droplets in oil/water (o/w) emulsions. Phase separation phenomena could be the rule in protein network systems where mixtures of egg yolk with egg white or other protein containing materials are employed. In emulsions, apolipoproteins of LDL micelles appear to play a dominant role in adsorption and interfacial membrane formation around the oil droplets. Yolk granular proteins also have the ability to contribute in the building of the adsorbed layer although their adsorption and stabilizing potential seem to depend on factors as such as NaCl content and pH and also on the extent of granular structure disorganization. Egg white and milk whey proteins are competitively displaced from emulsion oil interfaces by yolk lipoproteins due to the globular and rigid structure of the former. Casein proteins also appear to adsorb less effectively in the presence of yolk apolipoproteins in spite of their rather disorganized molecular structure.