Bicontinuous cubic phases, composed of bilayers arranged in the geometries of periodic minimal surfaces, are found in a variety of different lipid/water systems.It has been suggested recently that these cubic structures arrive as the result of competition between two free-energy terms: the curvature energy of each monolayer and the stretching energy of the lipid chains. This scenario, closely analogous to the one that explains the origin of the hexagonal phases, is investigated here by means of simple geometrical calculations. It is first assumed that the lipid bilayer is of constant thickness and the distribution of the (local) mean curvature of the phospholipid-water interfaces is calculated. Then, assuming the mean curvature of these interfaces is constant, the distribution of the bilayer's thickness is calculated. Both calculations quantify the fact that the two energy terms are frustrated and cannot be satisfied simultaneously. However, the amount of the frustration can be smaller for the cubic phase than for the lamellar and hexagonal structures. Therefore, this phase can appear in the phase diagram between the other two, as observed in many recent experiments.
Surface micro and nanostructural modifications of dental and orthopaedic implants have shown promising in vitro, in vivo, and clinical results. Surface wettability has also been suggested to play an important role in osteoblast differentiation and osseointegration. However, the available techniques to measure surface wettability are not reliable on clinically-relevant, rough surfaces. Furthermore, how the differentiation state of osteoblast lineage cells impacts their response to micro/nanostructured surfaces, and the role of wettability on this response, remains unclear. In the current study, surface wettability analyses (optical sessile drop analysis, ESEM analysis, and the Wilhelmy technique) indicated hydrophobic static responses for deposited water droplets on microrough and micro/nanostructured specimens, while hydrophilic responses were observed with dynamic analyses of micro/nanostructured specimens. The maturation and local factor production of human immature osteoblast-like MG63 cells was synergistically influenced by nanostructures superimposed onto microrough titanium (Ti) surfaces. In contrast, human mesenchymal stem cells (MSCs) cultured on micro/nanostructured surfaces in the absence of exogenous soluble factors, exhibited less robust osteoblastic differentiation and local factor production compared to cultures on unmodified microroughened Ti. Our results support previous observations using Ti6Al4V surfaces showing that recognition of surface nanostructures and subsequent cell response is dependent on the differentiation state of osteoblast lineage cells. The results also indicate that this effect may be partly modulated by surface wettability. These findings support the conclusion that the successful osseointegration of an implant depends on contributions from osteoblast lineage cells at different stages of osteoblast commitment.
The structure of the isotropic L3 phase observed in many surfactant-water or surfactant-water-oil systems is analyzed. It is pointed out that the L3 phase generally appears in equilibrium with a dilute solvent phase on one hand and a lamellar liquid crystalline phase on the other. Irrespective of the detailed chemical nature of the system, the one-phase region is remarkably narrow in one direction, indicating that the thermodynamic degrees of freedom are effectively reduced by one due to an internal constraint in the phase. In accordance with previous work it is argued that the basic structural unit in the L3 phase is a surfactant bilayer. Furthermore, we conclude that the L3 phase appears when there is a spontaneous mean curvature toward the solvent at the polar/apolar interface. It is shown that, for a system which has such a curvature toward the solvent, the surface formed by the bilayer midplane has a negative average Gaussian curvature {K). By virtue of the Gauss-Bonnet theorem the bilayer under such circumstances has a multiply connected structure. The conclusion is then that, under conditions when there is a spontaneous mean curvature toward the solvent, it is possible to reach a low free energy state by forming multiply connected bilayer structures, as in many cubic phases, rather than planar bilayers. When interbilayer forces are weak, the structure tends to be disordered, leading to an isotropic solution (L3) rather than an ordered cubic structure. To minimize local variations in curvature at the polar/apolar interface, we demonstrate that the midplane surface should be close to a minimal surface. We then show that a certain dimensionless group associated with a given periodic minimal surface has approximately the same value for all of the well-known isotropic minimal surfaces. Assuming a minimal midplane surface, we can then show that, for a given thickness, a bilayer structure with a prescribed area-averaged mean curvature can only exist at a single volume fraction. This explains the internal constraint in the L3 phase, which is manifested in the narrow character of the L3 phase. Applying the equations that express this constraint, and using results from a theory due to Cantor to account for the effect of water/head-group interactions on water penetration, we present fits of these narrow phase-existence regions to the theory and rationalize the temperature dependence of the L3 phases in a variety of nonionic surfactant systems. For a microemulsion system the analysis shows that the spontaneous monolayer curvature'increases strongly on the addition of hydrocarbon. The emerging picture of the L3 phase is that the solution structure is characterized by a highly connected bilayer, extending in three dimensions, thus appearing bicontinuous in, e.g., NMR self-diffusion experiments, and having an average mean curvature at the polar/apolar interface toward the solvent. The basic driving force forming an L3 rather than a lamellar phase is thus not an entropy increase associated with disorder, as previously sug...
We derive analytical and numerical results for the geometric obstruction factor in the case of self-diffusion within model cubic-phase microstructures and apply the results to the analysis of literature self-diffusion data in cubic phases, bicontinuous microcmulsions, and L3 (or "L*") phases. Each model microstructure is defined by a dividing surface, which divides the polar rcgions-surfactant head groups. and usually water-from the nonpolar regions-surfactant tails and possibly oil. The polar-apolar dividing surfaces treated are ( I ) interconnected cylinders, and (2) smooth surfaces of constant mean curvature, recently computed by one of the authors, which are generalizations of periodic minimal surfaces of identically zero mean curvature. The surfactant self-diffusion can often be modeled as diffusion of a particle confined to the polar-apolar dividing surface, with a constant diffusion coefficient Do. Self-diffusion within the labyrinthine subvolumes created by each dividing aurfacc is solved by a three-dimensional finite element calculation, yielding curves of 0 = Dew/& (the "obstruction factor") vcrbus volume fraction. Then for the case of surface diffusion, i.c., surfactant self-diffusion, the surface diffusion equation is solved over the surfucc by a two-dimensional finite element method. It is also proven analytically that the effective diffusion cocflicicnt D,rr for a particle diffusing over any minimal surface of cubic symmetry is exactly ( 2 / 3 ) D 0 . Assuming an interconnected-cylinder microstructure leads to an apparently universal relation between volume fraction and obstruction factor. For the constant mean curvature models we find an approximate universal relation between the Obstruction factor and mean curvature in the regime of low (dimensionless) mean curvature, which quantifies the observation that higher coordination number leads to a steeper dccrease of the obstruction factor with volume fraction, for dividing surfaces of low iiican curvature. Application of thc results for cubic phascs in the DDAB/water/styrene system yields information on the dcgrcc of head-group hydration over a range of water contents from 17% to 58%. For L3 phases in the CI2E,-water system, results for thc "D" family of constant-mean-curvature surfaces provide a n excellent fit of the water self-diffusion data. However, for iiiicrw~iiulsion systcms. including systems based on C,E,, surfactants and on Aerosol OT, the slope of the measured obstruction factor as a function of volume fraction is in many cases significantly higher than the theoretical values for families of low or moderate coordination number, and even higher than thc valuc for the randomly decorated Voronoi model of fairly high coordination. This appears to indicate that as the oil:watcr ratio moves away from unity, obstructing "neck" regions, and pcrhapa cvcn topological disconncctions. develop due to a combination of nonzero spontaneous mean curvature and thermal fluctuation.
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