Equilibrium Domain Size Research Articles

Shape morphology of dipolar domains in planar and spherical monolayers.

We present a continuum theory for predicting the equilibrium shape and size of dipolar domains formed during liquid-liquid phase coexistence in planar and spherical monolayers. Our main objective is to assess the impact of the monolayer surface curvature on domain morphology. Following previous investigators, we base our analysis around minimizing the free energy, with contributions from line tension and electrostatic dipolar repulsions. Assuming a monodisperse system of circularly symmetric domains, we calculate self-energies and interaction energies for planar and spherical monolayers and determine the equilibrium domain size from the energy minima. We subsequently evaluate the stability of the circularly symmetric domain shapes to an arbitrary, circumferential distortion of the perimeter via a linear stability analysis. We find that the surface curvature generally promotes the formation of smaller, circularly symmetric domains instead of larger, elongated domains. We rationalize these results by examining the effect of the curvature on the intra- and inter-domain dipolar repulsions. We then present a phase diagram of domain shape morphologies, parameterized in terms of the domain area fraction and the monolayer curvature. For typical domain dimensions of 1-30 µm, our theoretical results are relevant to monolayers (and possibly also bilayers) in liquid-liquid phase coexistence with radii of curvature of 1-100 µm.

Effects of Hybrid Lipid Concentration on Equilibrium Domain Size in a Lipid Bilayer Immersed in Water

The effects of introducing hybrid lipids to a lipid bilayer containing saturated and unsaturated lipids immersed in water were studied. The lipid and water molecules were modeled as coarse-grained particles. All particles were simulated by using the dissipative particle dynamics method. The results showed that the hybrid lipids accumulated at the interface between the saturated and the unsaturated lipid domains. The relation between the hybrid lipid concentration and the equilibrium domain size was obtained. Moreover, the sizes of the simulated lipid domains are consistent with that given by the lipid raft definition.

Twisted domain walls and skyrmions in perpendicularly magnetized multilayers

We present an analytical theory to describe three-dimensional magnetic textures in perpendicularly magnetized magnetic multilayers that arise in the presence of magnetostatic interactions and the Dzyaloshinskii-Moriya interaction (DMI). We demonstrate that domain walls in multilayers develop a complex twisted structure, which persists even for films with strong DMI. The origin of this twist is surface-volume stray field interactions that manifest as a depth-dependent effective field whose form mimics the DMI effective field. We find that the wall twist has a minor impact on the equilibrium skyrmion or domain size, but can significantly affect current-driven dynamics. Our conclusions are based on the derived analytical expressions for the magnetostatic energy and confirmed by micromagnetic simulations.

Insight into the Microscopic Structure of Module-Assembled Thermoresponsive Conetwork Hydrogels

The microscopic structure of module-assembled thermoresponsive conetworks was systematically investigated as a function of both temperature T and the mole fraction of the thermoresponsive modules r using small-angle neutron scattering (SANS). The conetworks were prepared by end-linking of hydrophilic modules and LCST-type thermoresponsive modules in water by the molar ratio of (1 – r):r. When the hydrogels with 0.02 ≤ r ≤ 0.10 were heated above certain T, nanometer-scale spherical domains were formed by aggregation of several prepolymer modules, whereas for the hydrogel with r = 0.01 such domain formation was not detected in the T range investigated. The size of spherical domains increased with increasing r and T. The observed r dependence of the domain size was theoretically explained by considering the free energy of domain formation, from which we concluded that the equilibrium domain size was determined mainly by the balance between two free energy contributions: the interfacial free energy of domain–...

Diffusion mediated coagulation and fragmentation based study of domain formation in lipid bilayer membrane

We estimate the equilibrium size distribution of cholesterol rich micro-domains on a lipid bilayer by solving Smoluchowski equation for coagulation and fragmentation. Towards this aim, we first derive the coagulation kernels based on the diffusion behaviour of domains moving in a two dimensional membrane sheet, as this represents the reality better. We incorporate three different diffusion scenarios of domain diffusion into our coagulation kernel. Subsequently, we investigate the influence of the parameters in our model on the coagulation and fragmentation behaviour. The observed behaviours of the coagulation and fragmentation kernels are also manifested in the equilibrium domain size distribution and its first moment. Finally, considering the liquid domains diffusing in a supported lipid bilayer, we fit the equilibrium domain size distribution to a benchmark solution.

Domain size in systems with canted magnetization

The equilibrium domain size in ferromagnetic single-layer films with canted magnetization, as observed in the spin-reorientation transition, is analytically described. It is demonstrated that for canted magnetization in the case of narrow domain walls the size of the magnetic domains depends solely on the second-order anisotropy constant and not on the first-order anisotropy. The single-layer model is extended to the application to multilayers. An approximation for the domain size of multilayers is developed and used for the analysis of data obtained via soft x-ray holography. Our analysis, including the canting angle, allows for the determination of the relative anisotropy constants to second order.

Phase-Field Crystal Modeling of Compositional Domain Formation in Ultrathin Films

Bulk-immiscible binary systems often form stress-induced miscible alloy phases when deposited on a substrate. Both alloying and surface dislocation formation lead to the decrease of the elastic strain energy, and the competition between these two strain-relaxation mechanisms gives rise to the emergence of pseudomorphic compositional nanoscale domains, often coexisting with a partially coherent single phase. In this work, we develop a phase-field crystal model for compositional patterning in monolayer aggregates of binary metallic systems. We first demonstrate that the model naturally incorporates the competition between alloying and misfit dislocations, and quantify the effects of misfit and line tension on equilibrium domain size. Then, we quantitatively relate the parameters of the phase-field crystal model to a specific system, CoAg/Ru(0001), and demonstrate that the simulations capture experimentally observed morphologies.

Magnetic domain replication in interacting bilayers with out-of-plane anisotropy: Application toCo∕Ptmultilayers

The effects of domain replication resulting from interlayer magnetostatic coupling have been investigated in trilayers which consist of two $\mathrm{Co}∕\mathrm{Pt}$ multilayers with out-of-plane anisotropy separated by a thick Pt layer. For demagnetized films, the strong correlation between the domain configurations of the two multilayers gives rise to an evolution of the equilibrium domain size with the spacer layer thickness. A good correspondence is found between our numerical calculations and the experimental measurements. These results are an extension of Kaplan's theory of domain size in single ferromagnetic films with out-of-plane magnetization, to the case of interacting ferromagnetic films.

Vortex chain states in a ferromagnet/superconductor bilayer

The discrete vortex lattices in a ferromagnet/superconductor bilayer are studied when the ferromagnet has periodic stripe domains with an out-of-plane magnetization. The vortices are assumed to be situated periodically on chains in the stripe domains. Only up to two vortex chains per domain configurations are considered. When the domain period is fixed, the threshold magnetization is calculated at which the transition from the Meissner state to the mixed state occurs. When the domain period is not fixed, the equilibrium domain size and vortex positions are calculated, depending on the domain's magnetization and the domain wall energy. In equilibrium, the vortices in the neighbor domains are half-way shifted, while they are next to each other in the same domain.

Nanoscale Domain Stability in Organic Monolayers on Metals

Certain organic molecules, such as alkanethiols, can adsorb on metals to form monolayers. Sometimes domains appear in the monolayers. For example, an incomplete monolayer may form islands, and a mixed-composition monolayer may separate into distinct phases. During annealing, the molecules diffuse on the metal surface. The domain boundary energy drives the domains to coarsen. The contact potential between the dissimilar domains drives the domains to refine. On the basis of existing experimental information, we suggest that the competition between coarsening and refining should stabilize certain domain patterns. We formulate a free energy functional to include the effects of mixed species, domain boundary, and contact potential. An approximate energy minimization estimates the equilibrium domain size. We derive a diffusion equation consistent with the free energy functional. The numerical solution of the diffusion equation follows the evolution of the monolayers from a random initial concentration field to patterns of dots and stripes. We also discuss the practical implications of the theory and, in particular, the possibility of guided self-assembly.

Domain Dynamics in a Ferroelastic Epilayer on a Paraelastic Substrate

This paper models the domain dynamics in a ferroelastic epilayer within the time-dependent Ginzburg-Landau (TDGL) framework. Constrained on a paraelastic substrate of square symmetry, the epilayer has rectangular symmetry, and forms domains of two variants. The domain wall energy drives the domains to coarsen. The spontaneous strains induce an elastic field, which drives the domains to refine. The competition between coarsening and refining selects an equilibrium domain size. We model the epilayer-substrate as a nonequilibrium thermodynamic system, evolving by the changes in the elastic displacements and the order parameters. The free energy consists of two parts: the bulk elastic energy, and the excess surface energy. The elastic energy density is taken to be quadratic in the strains. The surface energy density is expanded into a polynomial of the order parameters, the gradients of the order parameters, and the strains. In this expansion, the surface stress is taken to be quadratic in the order parameters. The evolution equations are derived from the free energy variation with respect to the order parameters. The elastic field is determined by superposing the Cerruti solution. Examples of computer simulation are presented.

Nonresonant dielectric hole-burning spectroscopy on a titanium-modified lead magnesium niobate ceramic

Nonresonant dielectric hole-burning experiments were performed on the titanium-modified relaxor ferroelectric lead magnesium niobate around the diffuse maximum in the dielectric permittivity. After applying large alternating electric pump fields we monitored the polarization response to small field steps for times between 0.3 ms and 100 s. Depending on the frequency of the pump oscillation a speedup of the polarization response was observed with a maximum located around times corresponding to the inverse pump frequency. The refilling of the dielectric holes was investigated for several temperatures, pump frequencies, and pump field amplitudes. It proceeded always slower than the time scale set by the pump frequencies. Additionally, we observe a significant increase of the refilling times for increasing pump field amplitudes. This finding can be interpreted to indicate that increasingly large pump fields enable the domain walls to cross larger and larger pinning barriers. The subsequent recovery process, which leads back to the equilibrium domain size distribution, proceeds in the absence of an external electrical field. This rationalizes that recovery is slowed down significantly by application of large pump field amplitudes since then the pinning barriers that have to be traversed back are larger.

Influence of domain boundary width on the statics of 90° domains in epitaxial ferroelectric thin films

The influence of the domain boundary width on the statics of single 90° elastic domains (twins) in epitaxial ferroelectric tetragonal films grown on a cubic substrate is theoretically investigated. The inhomogeneous internal stresses arising in polydomain epitaxial systems are calculated by the effective dislocation method. The elastic energy stored in the heterostructure is determined. The equilibrium domain size is found and the stability diagram for single domains at different wall widths is constructed by minimizing the total internal energy of the system. It is demonstrated that, as the domain boundary width 2w increases, the stability region of 90° domains increases and qualitatively changes for ultrathin films when the parameter 2w exceeds the specific critical value 2wcr. The equilibrium width 2w* of domain walls in thin films is predicted to be larger compared to the width 2w0 of domain boundaries in a macroscopic crystal.

Magnetic force microscopy of high-density perpendicular magnetic recording media

Multilayer Pt/Co and Pt/CoCrTa thin film structures have been fabricated that display different levels of intergranular exchange coupling. Perpendicular magnetic recordings of continuous wave patterns have been made using single pole contact heads and examined using a magnetic force microscope (MFM). Analysis of the MFM images show that the recording capabilities of the media are correlated inversely with both the degree of intergranular coupling and the equilibrium domain size. With exchange isolation of the grains, transitions may be written at linear densities > ̃ 300 kfci with good fidelity.

Entropy, Free Volume, and Cooperative Relaxation.

The high frequency end of the relaxation spectrum for polymer molecules involves the rotation of the segmental bonds. This fast relaxation process, however, cannot take place easily in the condensed state crowded by the densely packed conformers, necessitating the slower cooperatively synchronous relaxation. As the temperature is lowered, the domain of cooperativity grows towards the infinite size at the Kauzmann zero entropy temperature, though actually the system deviates from the equilibrium as the glass transition intervenes typically at 50 K above that temperature. The excess enthalpy and entropy drop faster than predicted by the rotational isomeric states which would reach zero only at 0 K. The real ΔCP is greater than that of the RIS value. The actual volume in excess of the crystalline lattice volume, however, points towards zero at 0 K. Thus, a polymer with higher Tg typically exhibits a lower density and modulus in the glassy state. Since the configurational entropy associated with the free volume is proportional to the logarithm of the latter, the Kauzmann temperature can be scaled by ln M, where M is the algebraic average of the conformer molecular weight. The temperature dependence of the most dominant, i.e., the largest equilibrium domain size will result in the Adam-Gibbs and Vogel equations for the characteristic relaxation time. The cooperative domain distribution leads to the relaxation spectrum that follows a power law. The relationship between the characteristic relaxation time and the rate of physical aging is derived.

Equilibrium Thermodynamics of Lipid Monolayer Domains

Coexisting thermodynamic phases of lipid molecules at the air−water interface exhibit domains with a variety of sizes and shapes. Both the sizes and shapes of these domains are affected by a competition between a line tension and long-range dipole−dipole repulsion. In the case of coexisting liquid phases, line tension favors large, circular domains, whereas the dipolar repulsions favor small and/or highly elongated domains. The present work discusses the thermodynamics of the equilibrium size and shape of these domains. It is shown that whereas there is only one stable equilibrium domain size, there is an infinite number of sizes that represent metastable equilibrium at different monolayer pressures. Circular domains that are far enough apart as to not interact with one another electrostatically have the same radii under this condition of metastable equilibrium. It is shown that, in principle, the metastable equilibrium radius reached in a monolayer can depend on the initial distribution of nonequilibrium...

The effects of diluent molecular weight on the structure of thermally-induced phase separation membrane

The effects of molecular weight of a homologous series of diluent on the droplet size of the thermally-induced phase separation membranes were investigated for the mixtures of poly(ethylene glycol) used as a diluent and nylon 12 as a matrix. All the phase diagrams obtained for different molecular weights of the diluent ranging from 200 to 600 showed the upper critical solution temperature type phase boundaries. When molecular weight of PEG was higher than 1000, nylon 12 did not form miscible blends with PEG below the thermal degradation temperature. Interaction energy denssty for each pair was evaluated from the phase separation temperature by the Flory-Huggins theory. The chain end group effects on the interaction energy were also considered using a binary interaction model and a Massa plot. The equilibrium domain size increased as the molecular weight of the diluent increased and then approached an asymptotic value. The increase of the average domain size with the increase of the diluent molecular weight was suppressed to a great extent for a higher content of nylon 12 in the mixture suggesting that there exist specific interactions between PEG and nylon 12.

Very low‐grade metamorphism of the Taveyanne greywacke, Glarus Alps, Switzerland

ABSTRACTDetailed textural and chemical data for mineral assemblages on a regional scale are presented for the metaandesitic Eocene‐Oligocene Taveyanne greywacke of the Glarus Alps, Eastern Switzerland. Presented data indicate an increase of metamorphic grade from zeolite facies to prehnite‐pumpellyite and pumpellyite‐actinolite facies. Low‐grade outcrops contain laumontite, minor corrensite and pumpellyite (assemblage type 1), whereas outcrops of higher metamorphic grade contain prehnite and two populations of pumpellyite (type 2), prehnite—pumpellyite‐(Al)—white mica (type 3), a single outcrop shows pumpellyite‐actinolite facies (type 4). From the zeolite to prehnite‐pumpellyite/pumpellyite‐actinolite facies there are indications for an increase of the chemical equilibrium domain size for the critical paragenesis from a single detrital grain ≤1 mm) in type 1, to a few millimetres in type 2, and to a whole thin section in type 3. MetamorphicP‐Tconditions were determined by a combination of chlorite thermometry, fluid inclusion and vitrinite reflectance data. Peak temperatures range from 170‐190d̀ C for zeolite facies to 270‐310d̀ C for prehnite‐pumpellyite and pumpellyite‐actinolite facies. For the higher temperature range, pressures of 2‐3 kbar are derived indicating a geothermal gradient of 24‐32d̀ C km‐1. The well‐constrained temperature estimations derived for the assemblages provide a useful test of the different empirical calibrations of chlorite thermometers recently proposed. The best correspondence to the temperatures determined here is for the Cathelineau calibration. In addition, in the lower grade samples differences in textures and calculated temperatures provide a mean to distinguish between detrital and newly formed chlorites.

Shock deformation of k-state in Ni-Cr alloys

Abstract The change of electrical resistivity was used to study the ordering kinetics and the anomalous resistivity behaviour in Ni-22 wt. % Cr and Ni-30 wt. % Cr (Ni-22 Cr and Ni-30 Cr) before and after shock deformation. Ordering was found to occur in two stages: the attainment of an equilibrium domain size, and the increase of long-range order (LRO) within the domains. Specimens of Ni-22 Cr and Ni-30 Cr in the ordered and disorderd states were subjected to explosive shock loading at shock pressures between 90 and 300 kbars. In the pressure range of 200 to 300 kbars, the resistivity of the ordered alloys decreased sharply. The resistivity of the shock-deformed initially quenched alloys decreased continuously between 90 and 300 kbars. The decrease in resistivity with shock deformation for the quenched alloys was related to the destruction of short-range order (SRO).

Magnetostatic effects in grain-oriented silicon iron

The function of the free-surface poles appearing on oriented silicon-iron strips in controlling the domain structure is discussed. The energy of a simple 180° domain structure is given as a function of the dimensions, angle of dip, and applied field; this model is shown to give high values of the magnetostatic energy for angles of misorientation greater than 2°. On general theoretical grounds, the proportion of 90° domains is calculated as a function of the misorientation and is shown to agree with experiment. A detailed model of the 90° domain structure is discussed; its magnetostatic energy is estimated and predictions of the equilibrium domain size are in reasonable agreement with observation.