We calculate the stress tensor for a quasi-spherical vesicle and we thermally average it in order to obtain the actual, mechanical, surface tension $τ$ of the vesicle. Both closed and poked vesicles are considered. We recover our results for $τ$ by differentiating the free-energy with respect to the proper projected area. We show that $τ$ may become negative well before the transition to oblate shapes and that it may reach quite large negative values in the case of small vesicles. This implies that spherical vesicles may have an inner pressure lower than the outer one.
The dynamics of a single fluid bilayer membrane in an external hydrodynamic flow field is considered. The deterministic equation of motion for the configuration is derived taking into account both viscous dissipation in the surrounding liquid and local incompressibility of the membrane. For quasi-spherical vesicles in shear flow, thermal fluctuations can be incorporated in a Langevin-type equation of motion for the deformation amplitudes. The solution to this equation shows an overdamped oscillatory approach to a stationary tanktreading shape. Inclination angle and ellipiticity of the contour …
The stochastic motion of a two-dimensional vesicle in linear shear flow is studied at finite temperature. In the limit of small deformations from a circle, Langevin-type equations of motion are derived, which are highly nonlinear due to the constraint of constant perimeter length. These equations are solved in the low temperature limit and using a mean field approach, in which the length constraint is satisfied only on average. The constraint imposes non-trivial correlations between the lowest deformation modes at low temperature. We also simulate a vesicle in a hydrodynamic solvent by using t…
We consider a ubiquitous scenario where a fluctuating, semipermeable vesicle is embedded in solution while enclosing a fixed number of solute particles. The swelling with increasing number of particles or decreasing concentration of the outer solution exhibits a continuous phase transition from a fluctuating state to the maximum-volume configuration, whereupon appreciable pressure difference and surface tension build up. This criticality is unique to particle-encapsulating vesicles, whose volume and inner pressure both fluctuate. It implies a universal swelling behavior of such vesicles as the…
It has been known for long that the fluctuation surface tension of membranes $r$, computed from the height fluctuation spectrum, is not equal to the bare surface tension $σ$ introduced in the Helfrich theory. In this work we relate these two surface tensions both analytically and numerically and compare them to the Laplace tension $γ$, and the mechanical frame tension $τ$. Using one-loop renormalisation calculations, we obtain, in addition to the effective bending modulus $κ_{\rm eff}$, a new expression for the effective surface tension $σ_{\rm eff}=σ- εk_{\rm B}T/(2a_p)$ where $a_p$ the proje…
We show that a metastable multilamellar vesicle (`onion'), in contact with excess solvent, can spontaneously deplete solute molecules from its interior through an unusual, entropy-driven mechanism. Fluctuation entropy is gained as the uneven partition of solute molecules helps the onion relieve tension in its lamellae. This mechanism accounts for recent experiments on the interaction between uncharged phospholipid onions and dissolved sugars.
A continuous transformation from a closed vesicle to an open vesicle needs that the area of open hole enlarge from zero. Since the shape equation and boundary conditions of lipid open vesicles with free edges have been obtained, we want know whether this process can be archived with valid parameters. By study the boundary conditions in the axisymmetric case, the analytic expression of the boundary edges is obtained generally. It reveals that the radius and line tension of boundary edges are confined strongly by bending moduli. In some cases, there is the minimal nonzero boundary radius and the…
Onsager's irreversible thermodynamics is used to perform a systematic deduction of the kinetic equations governing the opening and collapse of transient pores in spherical vesicles. We show that the edge tension has to be determined from the initial stage of the pore relaxation and that in the final state the vesicle membrane is not completely relaxed, since the surface tension and the pressure difference are about $25\%$ of its initial value. We also show that the pore life-time is controlled by the solution viscosity and its opening is driven by the solution leak-out and the surface tension …
Non-uniform fields are commonly used to study vesicle dielectrophoresis and can be used to hitherto relatively unexplored areas of vesicle deformation and electroporation. A common but perplexing problem in vesicle dynamics is the cross over from the entropic to enthalpic (stretching) tension during vesicle deformation. A lucid demonstration of this concept is provided by the study of vesicle deformation and dielectrophoresis under axisymmetric quadrupole electric field. Small deformation theory incorporating the Maxwell stress approach is used (employing area and volume conservation constrain…
Motivated by a long-standing debate concerning the nature and interrelations of surface-tension variables in fluid membranes, we reformulate the thermodynamics of a membrane vesicle as a generic two-dimensional finite system enclosing a three-dimensional volume. The formulation is shown to require two tension variables, conjugate to the intensive constraints of area per molecule and volume-to-area ratio. We obtain the relation between these two variables in various scenarios, as well as their correspondence to other definitions of tension variables for membranes. Several controversies related …
Lipid vesicles are known to undergo complex conformational transitions, but it remains challenging to systematically characterize non-equilibrium membrane shape dynamics. Here, we report the direct observation of lipid vesicle relaxation from highly deformed shapes using a Stokes trap. Vesicle shape relaxation is described by two distinct characteristic time scales governed by the bending modulus and membrane tension. Interestingly, experimental results are consistent with a viscoelastic model of a deformed membrane. Overall, these results show that vesicle relaxation is governed by an interpl…
Recent experiments by Kantsler et. al. (2007) have shown that the relaxational dynamics of a vesicle in external elongation flow is accompanied by the formation of wrinkles on a membrane. Motivated by these experiments we present a theory describing the dynamics of a wrinkled membrane. Formation of wrinkles is related to the dynamical instability induced by negative surface tension of the membrane. For quasi-spherical vesicles we perform analytical study of the wrinkle structure dynamics. We derive the expression for the instability threshold and identify three stages of the dynamics. The scal…
Osmotic deflation of vesicles enclosing two liquid phases can lead to bulging of one of the phases from the vesicle body. This budding process is preceded by a complete to partial wetting transition of one of the liquid phases on the membrane and depends on the membrane tensions and the tension of the interface between the enclosed liquid phases. These tensions dominate in different morphology regimes, the crossover of which initiates the budding process. In addition, the degree of budding can be controlled by aspiration via micropipettes. We also demonstrate that the budding direction can be …
We study the nucleation of a single pore in a fluctuating lipid membrane, specifically taking into account the membrane fluctuations, as well as the shape fluctuations of the pore. For large enough pores, the nucleation free energy is well-described by shifts in the effective membrane surface tension and the pore line tension. Using our framework, we derive the stability criteria for the various pore formation regimes. In addition to the well-known large-tension regime from the classical nucleation theory of pores, we also find a low-tension regime in which the effective line and surface tensi…
We use computer simulations and a simple free energy model to study the response of a bilayer membrane to the application of a negative (compressive) mechanical tension. Such a tension destabilizes the long wavelength undulation modes of giant vesicles, but it can be sustained when small membranes and vesicles are considered. Our negative tension simulation results reveal two regimes - (i) a weak negative tension regime characterized by stretching-dominated elasticity, and (ii) a strong negative tension regime featuring bending-dominated elastic behavior. This resembles the findings of the cla…
The free energy of a crystalline domain coexisting with a liquid phase on a spherical vesicle may be approximated by an elastic or stretching energy and a line tension term. The stretching energy generally grows as the area of the domain, while the line tension term grows with its perimeter. We show that if the crystalline domain contains defect arrays consisting of finite length grain boundaries of dislocations (scars) the stretching energy grows linearly with a characteristic length of the crystalline domain. We show that this result is critical to understand the existence of solid domains i…
Recent studies of elasto-capillary phenomena have triggered interest in a basic variant of the classical Young-Laplace-Dupré (YLD) problem: The capillary interaction between a liquid drop and a thin solid sheet of low bending stiffness. Here, we consider a two-dimensional model where the sheet is subjected to an external tensile load and the drop is characterized by a well-defined Young's contact angle $θ_Y$. Using a combination of numerical, variational, and asymptotic techniques, we discuss wetting as a function of the applied tension. We find that, for wettable surfaces with $0<θ_Y<π/2$, co…
We develop a novel biased Monte-Carlo simulation technique to measure the force-extension curves and the distribution function of the extension of fluctuating filaments stretched by external force. The method is applicable for arbitrary ratio of the persistence length to the contour length and for arbitrary forces. The simulation results agree with analytic expressions for the force-extension curves and for the renormalized length-scale-dependent elastic moduli, derived in the rigid rod and in the strong force limits. We find that orientational fluctuations and wall effects produce non-Gaussia…
In many interfacial flow systems, variations of surface properties lead to novel and interesting behaviors. In this work a three-dimensional model of flow dynamics for multicomponent vesicles is presented. The surface composition is modeled using a two-phase surface Cahn-Hilliard system, while the interface is captured using a level set jet scheme. The interface is coupled to the surrounding fluid via a variation of energy approach. Sample energies considered include the total bending, variable surface tension energy, and phase segregation energy. The fully coupled system for surface inhomogen…
Biopolymer looping is a dynamic process that occurs ubiquitously in cells for gene regulation, protein folding, etc. In cellular environments, biopolymers are often subject to tensions which are either static, or temporally fluctuating far away from equilibrium. We study the dynamics of semiflexible polymer looping in the presence of such tensions by using Brownian dynamics simulation combined with an analytical theory. We show a minute tension dramatically changes the looping time, especially for long chains. Considering a dichotomically flipping noise as a simple example of the nonequilibriu…
In development and homeostasis, multi-cellular systems exhibit spatial and temporal heterogeneity in their biochemical and mechanical properties. Nevertheless, it remains unclear how spatiotemporally heterogeneous forces affect the dynamical and mechanical properties of confluent tissue. To address this question, we study the dynamical behavior of the two-dimensional cellular vertex model for epithelial monolayers in the presence of fluctuating cell-cell interfacial tensions, which is a biologically relevant source of mechanical spatiotemporal heterogeneity. In particular, we investigate the e…
Atomically thin sheets, such as graphene, are widely used in nanotechnology. Recently they have also been used in applications including kirigami and self-folding origami, where it becomes important to understand how they respond to external loads. Motivated by this, we investigate how isotropic sheets respond to uniaxial tension by employing the self-consistent screening analysis method and molecular dynamics simulations. Previously, it was shown that for freely suspended sheets thermal fluctuations effectively renormalize elastic constants, which become scale-dependent beyond a characteristi…
How long a fluid membrane vesicle stressed with a steady ramp of micropipette last before rupture? Or conversely, how high the surface tension should be to rupture a membrane? To answer these challenging questions we have developed a theoretical framework that allows description and reproduction of Dynamic Tension Spectroscopy (DTS) observations. The kinetics of the membrane rupture under ramps of surface tension is described as a combination of initial pore formation followed by Brownian process of the pore radius crossing the time-dependent energy barrier. We present the formalism and derive…
Three types of surface tensions can be defined for lipid membranes: the internal tension, $σ$, conjugated to the real membrane area in the Hamiltonian, the mechanical frame tension, $τ$, conjugated to the projected area, and the "fluctuation tension", $r$, obtained from the fluctuation spectrum of the membrane height. We investigate these surface tensions by means of a Monge gauge lattice Monte Carlo simulation involving the exact, nonlinear, Helfrich Hamiltonian and a measure correction for excess entropy of the Monge gauge. Our results for the relation between $σ$ and $τ$ agrees well with th…
Recently, Rozovsky et al. reported on the morphology and dynamics of superstructures in three-component lipid bilayers containing saturated and unsaturated lipids as well as cholesterol [J. Am. Chem. Soc. 127, 36 (2005)]. We comment that the observed sequence of the stripe to hexagonal morphological transition in mixed bilayers can be attributed to an enhanced membrane surface tension that is induced by the vesicle adhesion on the solid surface.
Nematic liquid crystals at rough and fluctuating interfaces are analyzed within the Frank elastic theory and the Landau-de Gennes theory. We study specifically interfaces that locally favor planar anchoring. In the first part we reconsider the phenomenon of Berreman anchoring on fixed rough surfaces, and derive new simple expressions for the corresponding azimuthal anchoring energy. Surprisingly, we find that for strongly aligning surfaces, it depends only on the geometrical surface anisotropy and the bulk elastic constants, and not on the precise values of the chemical surface parameters. In …
We present measurements of the fluctuation spectrum of giant vesicles containing bacteriorhodopsin (BR) pumps using video-microscopy. When the pumps are activated, we observe a significant increase of the fluctuations in the low wavevector region, which we interpret as due to a lowering of the effective tension of the membrane.
To describe the full spectrum of surface fluctuations of the interface between phase-separated colloid-polymer mixtures from low scattering vector q (classical capillary wave theory) to high q (bulk-like fluctuations), one must take account of the interface's bending rigidity. We find that the bending rigidity is negative and that on approach to the critical point it vanishes proportionally to the interfacial tension. Both features are in agreement with Monte Carlo simulations.
An exact description is provided of an almost spherical fluid vesicle with a fixed area and a fixed enclosed volume locally deformed by external normal forces bringing two nearby points on the surface together symmetrically. The conformal invariance of the two-dimensional bending energy is used to identify the distribution of energy as well as the stress established in the vesicle. While these states are local minima of the energy, this energy is degenerate; there is a zero mode in the energy fluctuation spectrum, associated with area and volume preserving conformal transformations, which brea…
Many vesicles have a spherical resting shape and exposure to fluid flows induces an exchange between sub-optical area and visible (systematic) deformation, while the total area is conserved. The dynamics which controls the exchange between sub-optical and visible area depends on membrane properties such as bending rigidity and initial tension. Conversely, observation of these dynamics can be used to determine the membrane properties. The goal of this work is to create a general numerical model which accounts for the exchange between sub-optical and visible area. Unlike prior modeling efforts, …
The height-height correlation function for a fluctuating interface between two coexisting bulk phases is derived by means of general equilibrium properties of the corresponding density-density correlation function. A wavelength-dependent surface tension $γ(\mathbf{q})$ can be defined and expressed in terms of the direct correlation function $c(\mathbf{r},\mathbf{r}')$, the equilibrium density profile $ρ_{\circ}(\mathbf{r})$ and an operator which relates density to surface configurations. Neither the concept of an effective interface Hamiltonian nor the difference in pressure is needed to deter…
Active solids consume energy to allow for actuation, shape change, and wave propagation not possible in equilibrium. Whereas active interfaces have been realized across many experimental systems, control of three-dimensional (3D) bulk materials remains a challenge. Here, we develop continuum theory and microscopic simulations that describe a 3D soft solid whose boundary experiences active surface stresses. The competition between active boundary and elastic bulk yields a broad range of previously unexplored phenomena, which are demonstrations of so-called active elastocapillarity. In contrast …
Theoretical studies of nearly spherical vesicles and microemulsion droplets, that present typical examples for thermally-excited systems that are subject to constraints, are reviewed. We consider the shape fluctuations of such systems constrained by fixed area $A$ and fixed volume $V$, whose geometry is presented in terms of scalar spherical harmonics. These constraints can be incorporated in the theory in different ways. After an introductory review of the two approaches: with an exactly fixed by delta-function membrane area $A$ [Seifert, Z. Phys. B, 97, 299, (1995)] or approximatively by mea…
The stochastic dynamics of inclusions in a randomly fluctuating biomembrane is simulated. These inclusions can represent the embedded proteins and the external particles arriving at a cell membrane. The energetics of the biomembrane is modelled via the Canham-Helfrich Hamiltonian. The contributions of both the bending elastic-curvature energy and the surface tension of the biomembrane are taken into account. The biomembrane is treated as a two-dimensional sheet whose height variations from a reference frame is treated as a stochastic Wiener process. The lateral diffusion parameter associated w…
The astronomically observed value of the cosmological constant is small but non-zero. This raises two questions together known as the cosmological constant problem a) why is lambda so nearly zero? b) why is lambda not EXACTLY zero? Sorkin has proposed that b) can be naturally explained as a one by square root N fluctuation by invoking discreteness of spacetime at the Planck scale due to quantum gravity. In this paper we shed light on these questions by developing an analogy between the cosmological constant and the surface tension of membranes. The ``cosmological constant problem'' has a natur…
We study elasticity-driven morphological transitions of soft spherical core shell structures in which the core can be treated as an isotropic elastic continuum and the surface or shell as a tensionless liquid layer, whose elastic response is dominated by bending. To generate the transitions, we consider the case where the surface area of the liquid layer is increased for a fixed amount of interior elastic material. We find that generically there is a critical excess surface area at which the isotropic sphere becomes unstable to buckling. At this point it adopts a lower symmetry wrinkled struct…
