The Young-Laplace, Kelvin, and Gibbs-Thomson equations form a cornerstone of colloidal and surface sciences and have found successful applications in many subfields of physics, chemistry, and ...biology. The Gibbs-Thomson effect, for example, predicts that small crystals are in equilibrium with their liquid melt at a lower temperature than large crystals and the positive interfacial energy increases the energy required to form small particles with a high curvature interface. In cases of liquids contained within porous media (confined geometry), the effect indicates decreasing the freezing/melting temperatures and the increment of the temperature is inversely proportional to the pore size. These phenomena can be reformulated for Gaussian maps of macromolecules and can be asked the following question: can one use the equations for predicting the melting temperature and shape of polymer chains in confined geometries? The answer is no, mainly because macromolecules form highly curved surfaces (Gaussian maps), and the equations hold only for simple geometries (sphere, plane, or cylinder). Here, we present general Young-Laplace, Kelvin, and Gibbs-Thomson equations for arbitrarily curved surfaces and apply them to predict temperature distribution on a few protein surfaces. Also, after increased interest toward liquid/liquid phase separation in biology, we derive generic Ostwald ripening and show that for shape-changing condensates, instead of a monotonic growing mechanism, a variety of processes are possible. Due to the generality of equations, we clarify that at appropriate internal/external pressure conditions systems, bounded by surfaces, may adopt any shape and thermal stability is strongly influenced by the geometries of confined spaces.
•We derive generalization of the Kelvin equation for arbitrarily curved surfaces.•The generalization problem is resolved without any prepositions and it holds for every generic surface tensions.•The ...equation holds for any surfaces from equilibrium to non-equilibrium at any scale: from nano to macro.•We find that increase of directional frictional forces decreases the curvature tensor pushing the surface toward the plane.•We explain why clouds look always curvy, frictional forces in the air are minimal.
Capillary condensation, which takes place in confined geometries, is the first-order vapor-to-liquid phase transition and is explained by the Kelvin equation, but the equation's applicability for arbitrarily curved surface has been long debated and is severe problem. Recently, we have proposed generic dynamic equations for moving surfaces. Application of the equations to the vapor/fluid interfaces in chemical equilibrium conditions nearly trivially solves the generalization problem for the Kelvin equation. The equations are universally true for any surfaces: atomic, molecular, micro or macro scale, real or virtual, Riemannian or pseudo-Riemannian, active or passive.
Oscillating shape motion of a freely falling and bouncing water droplet has long fascinated and inspired scientists. We propose dynamic non-linear equations for closed, two-dimensional surfaces in ...gravity and apply it to analyze shape dynamics of freely falling and bouncing drops. The analytic and numerical solutions qualitatively well explain why drops oscillate among prolate/oblate morphology and display a number of features consistent with experiments. In addition, numerical solutions for simplified equations indicate nonlinear effects of nonperiodic/asymmetric motion and the growing amplitude in the surface density oscillations and well agree to previous experimental data.
We present dynamic equations for two dimensional closed surfaces and analytically solve it for some simplified cases. We derive final equations for surface normal motions by two different ways. The ...solution of the equations of motions in normal direction indicates that any closed, two dimensional, homogeneous surface with time invariable surface energy density adopts constant mean curvature shape when it comes in equilibrium with environment. In addition, we show that the shape equation is an approximate solution to our equation of motion in the normal direction and is valid for stationary or near to stationary shapes. As an example, we apply the formalism to analyze equilibrium shapes of micelles and explain why they adopt spherical, lamellar, and cylindrical shapes. Theoretical calculation for micellar optimal radius is in good agreement with all atom simulations and experiments.
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