Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 2015
A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling in... more A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling instability, forming an inward hump. Our study shows that the strip morphology depends on the delicate balance between the compression energy and the bending energy. We find that this instability is a first-order phase transition between the adhered solution and the buckled solution whose main control parameter is related to the sheet stretchability. In the nearly unstretchable regime, we provide an analytic expression for the critical threshold. Compressibility is the key assumption which allows us to resolve the apparent paradox of an unbounded pressure exerted on the external wall by a confined flexible loop.
This paper derives governing equations of the interaction of finite samples of smectic-A liquid c... more This paper derives governing equations of the interaction of finite samples of smectic-A liquid crystals with an electrostatic field. Our model takes into account effects of the electric field on the layers distortion and vice-versa. The governing equations are obtained by the principle of virtual work. They are adapted for a particular problem in order to analyze the phase transition of Helfrich-Hurault type induced by an electrostatic field on a finite sample of smectic.
The symmetry breaking of the actin network from radial to longitudinal symmetry has been identifi... more The symmetry breaking of the actin network from radial to longitudinal symmetry has been identified as the major mechanism for keratocytes (fish cells) motility on solid substrate. For strong friction coefficient, the two-dimensional actin flow which includes the polymerisation at the edge and depolymerisation in the bulk can be modelled as a Darcy flow, the cell shape and dynamics being then modelled by standard complex analysis methods. We use the theory of active gels to describe the orientational order of the filaments which varies from the border to the bulk. We show analytically that the reorganisation of the cortex is enough to explain the motility of the cell and find the velocity as a function of the orientation order parameter in the bulk.
ABSTRACT a b s t r a c t In this paper we study theoretically the equilibrium configurations of t... more ABSTRACT a b s t r a c t In this paper we study theoretically the equilibrium configurations of two-dimensional nematic liquid crystals on a cylindrical surface. Configurations are described through a tensor-order parameter that provides information about the tangential optical axis and the dispersion of the molecules around it. Equilibrium equations are obtained by the surface Landau–de Gennes energy recently proposed in Napoli and Vergori [Physical Review E 061701 (2012)]. We show that, whenever free boundary conditions are applied, the extrinsic curvature induces the alignment of the optical axis along the cylinder axis. On the contrary, when strong planar anchoring is imposed on the boundaries, the curvature triggers a sort of Freé dricksz-like transition in the alignment. We provide the asymptotic solution of the non-linear problem with strong planar anchoring boundary conditions for elongated cylindrical shells.
We analyse the effects that a rigid inclusion induces on the stationary shapes of an impermeable ... more We analyse the effects that a rigid inclusion induces on the stationary shapes of an impermeable three-dimensional vesicle. Our study, performed via a numerical calculation, takes into account shapes which are not close to any reference configuration (neither spherical nor planar). The shape perturbations induced by the embedded inclusions are restricted within distances of the order of the inclusion size. Thus, inclusions do not interfere with global vesicle properties, such as budding transitions. The local character of the inclusion perturbation announces a fast distance decay of the membrane mediated elastic force between different proteins.
The growth of an elastic film adhered to a confining substrate might lead to the formation of del... more The growth of an elastic film adhered to a confining substrate might lead to the formation of delimitation blisters. Many results have been derived when the substrate is flat. The equilibrium shapes, beyond small deformations, are determined by the interplay between the sheet elastic energy and the adhesive potential due to capillarity. Here, we study a non-trivial generalization to this problem and consider the adhesion of a growing elastic loop to a confining circular substrate. The fundamental equations, i.e., the Euler Elastica equation, the boundary conditions and the transversality condition, are derived from a variational procedure. In contrast to the planar case, the curvature of the delimiting wall appears in the transversality condition, thus acting as a further source of adhesion. We provide the analytic solution to the problem under study in terms of elliptic integrals and perform the numerical and the asymptotic analysis of the characteristic lengths of the blister. Finally, and in contrast to previous studies, we also discuss the mechanics and the internal stresses in the case of vanishing adhesion. Specifically, we give a theoretical explanation to the observed divergence of the mean pressure exerted by the strip on the container in the limit of small excess-length.
The Helfrich-Hurault effect is a phase transition that occurs in samples of cholesteric or smecti... more The Helfrich-Hurault effect is a phase transition that occurs in samples of cholesteric or smectic liquid crystals subject to external electric or magnetic fields. In this paper we analyze the Helfrich-Hurault effect of smectic-A liquid crystals in an electrostatic field taking into account the complete electromechanical coupling. A comparison is made with the results already obtained for the partially coupled case where one takes into account only the effect of the field on the crystal configuration and considering that field unaffected.
Journal of Physics A: Mathematical and Theoretical, 2010
We study the equilibrium configurations of a nematic liquid crystal confined between two parallel... more We study the equilibrium configurations of a nematic liquid crystal confined between two parallel plates, when an electric field is applied. We take into account the mutual interaction of the field and the material. We also analyse the effects of two possibly different weak anchoring potentials at the plates. We use asymptotic methods to study in detail two different regimes of the applied voltage. The former concerns applied voltages close to the Freedericksz and the saturation critical thresholds; the latter is the case of high applied potentials. We discuss the new effects that arise with respect to the partial electric coupling and the strong anchoring cases.
When subjected to magnetic or electric fields, nematic liquid crystals confined between two paral... more When subjected to magnetic or electric fields, nematic liquid crystals confined between two parallel glass plates and initially uniformly oriented may undergo homogeneous one-dimensional spatial distortions (Fréedericksz and Zolina, Trans.
Motivated by the recent experimental findings by Kumar et al. (Phys. Rev. E., 82, 011701 (2010)) ... more Motivated by the recent experimental findings by Kumar et al. (Phys. Rev. E., 82, 011701 (2010)) in which the inverse Fréedericksz transition is observed, we have theoretically investigated the parity and the stability of the equilibrium configurations of a Fréedericksz cell with weak planar boundary conditions. Within the one-constant approximation of the Frank theory, the bulk equilibrium equation reduces to the nonlinear pendulum equation. Its solutions, when combined with boundary conditions deriving by the energy anchoring, lose uniqueness, exhibiting various symmetries. Thus, at a given anchoring strength and applied field, the cell becomes a system with metastable discrete energy levels. Our analysis proposes an explanation of the experimental results.
The equilibrium shapes of lipid vesicles are perturbed by rigid inclusions. In a two-dimensional ... more The equilibrium shapes of lipid vesicles are perturbed by rigid inclusions. In a two-dimensional vesicle, that may also model a cylindrically elongated tubule, the shape modifications can be determined analytically, and turn out to be significant even far from the inclusion. On the contrary, previous numerical work has given evidence that in the three-dimensional case the shape perturbations decay quite rapidly and are negligible a few inclusion radii away. In this paper, we use the tools of asymptotic analysis to derive analytically the shape of the boundary layer induced by the inclusion. As a result, we are able to determine the dominant part of the free-energy perturbation that, in turn, allows to identify the vesicle points where the inclusion prefers to sit.
We propose a continuum model to describe the molecular alignment in thin nematic shells. By contr... more We propose a continuum model to describe the molecular alignment in thin nematic shells. By contrast with previous accounts, the two-dimensional free energy, aimed at describing the physics of thin films of nematics deposited on curved substrates, is not postulated, but it is deduced from the conventional three-dimensional theories of nematic liquid crystals. Both the director and the order-tensor theories are taken into account. The so-obtained surface energies exhibit extra terms compared to earlier models. These terms reflect the coupling of the shell extrinsic curvature with the nematic order parameters. As expected, the shape of the shell plays a key role in the equilibrium configurations of nematics coating it.
Journal of Physics A: Mathematical and General, 2004
We analyse the effects of the impermeability constraint on the equilibrium shapes of a three-dime... more We analyse the effects of the impermeability constraint on the equilibrium shapes of a three-dimensional vesicle hosting a rigid inclusion.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009
We study the equilibrium shapes of a lipid membrane, attached to a fixed circular substrate. We s... more We study the equilibrium shapes of a lipid membrane, attached to a fixed circular substrate. We show how the weakening of the boundary conditions is able to break the axial symmetry of the optimal equilibrium configuration. We derive the critical threshold of the symmetry-breaking transition, and obtain the analytical expression of the free-energy minimizers in the quasi-planar approximation. Metastable states turn out to contain contributions only from the axisymmetric mode, and at most one single non-trivial Fourier mode.
Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The gro... more Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The ground state of most nematics is either isotropic or uniaxial, depending on the external temperature. Nevertheless, biaxial domains have been frequently identified, especially close to defects or external surfaces. In this paper we show that any spatially varying director pattern may be a source of biaxiality. We prove that biaxiality arises naturally whenever the symmetric tensor S = ͑١n͒͑١n͒ T possesses two distinct nonzero eigenvalues. The eigenvalue difference may be used as a measure of the expected biaxiality. Furthermore, the corresponding eigenvectors indicate the directions in which the order tensor Q is induced to break the uniaxial symmetry about the director n. We apply our general considerations to some examples. In particular we show that, when we enforce homeotropic anchoring on a curved surface, the order tensor becomes biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 2015
A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling in... more A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling instability, forming an inward hump. Our study shows that the strip morphology depends on the delicate balance between the compression energy and the bending energy. We find that this instability is a first-order phase transition between the adhered solution and the buckled solution whose main control parameter is related to the sheet stretchability. In the nearly unstretchable regime, we provide an analytic expression for the critical threshold. Compressibility is the key assumption which allows us to resolve the apparent paradox of an unbounded pressure exerted on the external wall by a confined flexible loop.
This paper derives governing equations of the interaction of finite samples of smectic-A liquid c... more This paper derives governing equations of the interaction of finite samples of smectic-A liquid crystals with an electrostatic field. Our model takes into account effects of the electric field on the layers distortion and vice-versa. The governing equations are obtained by the principle of virtual work. They are adapted for a particular problem in order to analyze the phase transition of Helfrich-Hurault type induced by an electrostatic field on a finite sample of smectic.
The symmetry breaking of the actin network from radial to longitudinal symmetry has been identifi... more The symmetry breaking of the actin network from radial to longitudinal symmetry has been identified as the major mechanism for keratocytes (fish cells) motility on solid substrate. For strong friction coefficient, the two-dimensional actin flow which includes the polymerisation at the edge and depolymerisation in the bulk can be modelled as a Darcy flow, the cell shape and dynamics being then modelled by standard complex analysis methods. We use the theory of active gels to describe the orientational order of the filaments which varies from the border to the bulk. We show analytically that the reorganisation of the cortex is enough to explain the motility of the cell and find the velocity as a function of the orientation order parameter in the bulk.
ABSTRACT a b s t r a c t In this paper we study theoretically the equilibrium configurations of t... more ABSTRACT a b s t r a c t In this paper we study theoretically the equilibrium configurations of two-dimensional nematic liquid crystals on a cylindrical surface. Configurations are described through a tensor-order parameter that provides information about the tangential optical axis and the dispersion of the molecules around it. Equilibrium equations are obtained by the surface Landau–de Gennes energy recently proposed in Napoli and Vergori [Physical Review E 061701 (2012)]. We show that, whenever free boundary conditions are applied, the extrinsic curvature induces the alignment of the optical axis along the cylinder axis. On the contrary, when strong planar anchoring is imposed on the boundaries, the curvature triggers a sort of Freé dricksz-like transition in the alignment. We provide the asymptotic solution of the non-linear problem with strong planar anchoring boundary conditions for elongated cylindrical shells.
We analyse the effects that a rigid inclusion induces on the stationary shapes of an impermeable ... more We analyse the effects that a rigid inclusion induces on the stationary shapes of an impermeable three-dimensional vesicle. Our study, performed via a numerical calculation, takes into account shapes which are not close to any reference configuration (neither spherical nor planar). The shape perturbations induced by the embedded inclusions are restricted within distances of the order of the inclusion size. Thus, inclusions do not interfere with global vesicle properties, such as budding transitions. The local character of the inclusion perturbation announces a fast distance decay of the membrane mediated elastic force between different proteins.
The growth of an elastic film adhered to a confining substrate might lead to the formation of del... more The growth of an elastic film adhered to a confining substrate might lead to the formation of delimitation blisters. Many results have been derived when the substrate is flat. The equilibrium shapes, beyond small deformations, are determined by the interplay between the sheet elastic energy and the adhesive potential due to capillarity. Here, we study a non-trivial generalization to this problem and consider the adhesion of a growing elastic loop to a confining circular substrate. The fundamental equations, i.e., the Euler Elastica equation, the boundary conditions and the transversality condition, are derived from a variational procedure. In contrast to the planar case, the curvature of the delimiting wall appears in the transversality condition, thus acting as a further source of adhesion. We provide the analytic solution to the problem under study in terms of elliptic integrals and perform the numerical and the asymptotic analysis of the characteristic lengths of the blister. Finally, and in contrast to previous studies, we also discuss the mechanics and the internal stresses in the case of vanishing adhesion. Specifically, we give a theoretical explanation to the observed divergence of the mean pressure exerted by the strip on the container in the limit of small excess-length.
The Helfrich-Hurault effect is a phase transition that occurs in samples of cholesteric or smecti... more The Helfrich-Hurault effect is a phase transition that occurs in samples of cholesteric or smectic liquid crystals subject to external electric or magnetic fields. In this paper we analyze the Helfrich-Hurault effect of smectic-A liquid crystals in an electrostatic field taking into account the complete electromechanical coupling. A comparison is made with the results already obtained for the partially coupled case where one takes into account only the effect of the field on the crystal configuration and considering that field unaffected.
Journal of Physics A: Mathematical and Theoretical, 2010
We study the equilibrium configurations of a nematic liquid crystal confined between two parallel... more We study the equilibrium configurations of a nematic liquid crystal confined between two parallel plates, when an electric field is applied. We take into account the mutual interaction of the field and the material. We also analyse the effects of two possibly different weak anchoring potentials at the plates. We use asymptotic methods to study in detail two different regimes of the applied voltage. The former concerns applied voltages close to the Freedericksz and the saturation critical thresholds; the latter is the case of high applied potentials. We discuss the new effects that arise with respect to the partial electric coupling and the strong anchoring cases.
When subjected to magnetic or electric fields, nematic liquid crystals confined between two paral... more When subjected to magnetic or electric fields, nematic liquid crystals confined between two parallel glass plates and initially uniformly oriented may undergo homogeneous one-dimensional spatial distortions (Fréedericksz and Zolina, Trans.
Motivated by the recent experimental findings by Kumar et al. (Phys. Rev. E., 82, 011701 (2010)) ... more Motivated by the recent experimental findings by Kumar et al. (Phys. Rev. E., 82, 011701 (2010)) in which the inverse Fréedericksz transition is observed, we have theoretically investigated the parity and the stability of the equilibrium configurations of a Fréedericksz cell with weak planar boundary conditions. Within the one-constant approximation of the Frank theory, the bulk equilibrium equation reduces to the nonlinear pendulum equation. Its solutions, when combined with boundary conditions deriving by the energy anchoring, lose uniqueness, exhibiting various symmetries. Thus, at a given anchoring strength and applied field, the cell becomes a system with metastable discrete energy levels. Our analysis proposes an explanation of the experimental results.
The equilibrium shapes of lipid vesicles are perturbed by rigid inclusions. In a two-dimensional ... more The equilibrium shapes of lipid vesicles are perturbed by rigid inclusions. In a two-dimensional vesicle, that may also model a cylindrically elongated tubule, the shape modifications can be determined analytically, and turn out to be significant even far from the inclusion. On the contrary, previous numerical work has given evidence that in the three-dimensional case the shape perturbations decay quite rapidly and are negligible a few inclusion radii away. In this paper, we use the tools of asymptotic analysis to derive analytically the shape of the boundary layer induced by the inclusion. As a result, we are able to determine the dominant part of the free-energy perturbation that, in turn, allows to identify the vesicle points where the inclusion prefers to sit.
We propose a continuum model to describe the molecular alignment in thin nematic shells. By contr... more We propose a continuum model to describe the molecular alignment in thin nematic shells. By contrast with previous accounts, the two-dimensional free energy, aimed at describing the physics of thin films of nematics deposited on curved substrates, is not postulated, but it is deduced from the conventional three-dimensional theories of nematic liquid crystals. Both the director and the order-tensor theories are taken into account. The so-obtained surface energies exhibit extra terms compared to earlier models. These terms reflect the coupling of the shell extrinsic curvature with the nematic order parameters. As expected, the shape of the shell plays a key role in the equilibrium configurations of nematics coating it.
Journal of Physics A: Mathematical and General, 2004
We analyse the effects of the impermeability constraint on the equilibrium shapes of a three-dime... more We analyse the effects of the impermeability constraint on the equilibrium shapes of a three-dimensional vesicle hosting a rigid inclusion.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009
We study the equilibrium shapes of a lipid membrane, attached to a fixed circular substrate. We s... more We study the equilibrium shapes of a lipid membrane, attached to a fixed circular substrate. We show how the weakening of the boundary conditions is able to break the axial symmetry of the optimal equilibrium configuration. We derive the critical threshold of the symmetry-breaking transition, and obtain the analytical expression of the free-energy minimizers in the quasi-planar approximation. Metastable states turn out to contain contributions only from the axisymmetric mode, and at most one single non-trivial Fourier mode.
Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The gro... more Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The ground state of most nematics is either isotropic or uniaxial, depending on the external temperature. Nevertheless, biaxial domains have been frequently identified, especially close to defects or external surfaces. In this paper we show that any spatially varying director pattern may be a source of biaxiality. We prove that biaxiality arises naturally whenever the symmetric tensor S = ͑١n͒͑١n͒ T possesses two distinct nonzero eigenvalues. The eigenvalue difference may be used as a measure of the expected biaxiality. Furthermore, the corresponding eigenvectors indicate the directions in which the order tensor Q is induced to break the uniaxial symmetry about the director n. We apply our general considerations to some examples. In particular we show that, when we enforce homeotropic anchoring on a curved surface, the order tensor becomes biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.
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Papers by G. Napoli