Paolo Biscari | Politecnico di Milano (original) (raw)

Papers by Paolo Biscari

International Journal of Non-Linear Mechanics, 1997

ABSTRACT We consider a nematic liquid crystal confined between two cylinders of radii r1 >... more ABSTRACT We consider a nematic liquid crystal confined between two cylinders of radii r1 > 0 and r2 > r1. We suppose that the lateral surfaces induce a uniaxial anchoring with the optic axis aligned along the radial direction, and we study the equilibrium configurations which minimize the free energy functional.This problem has been already studied under the assumption that the nematic remains uniaxial in the whole tube, with fixed degree of orientation s. We allow the nematic to become biaxial between the surfaces, and include in the free-energy functional an internal potential which favours uniaxiality.We prove that, if the internal potential is neglected (i.e. if biaxiality can arise at no cost), the free energy minimizer is biaxial in the whole volume (except, of course, on the lateral surfaces, where it must be uniaxial). The minimizer is unique, and no bifurcation arises for any value of ϱ: = r1r2.We arrive at new results also when the internal potential is at work: an exact solution, obtained in a special case, proves the existence of a bifurcation at a critical value of ϱ; approximate minimizers show how biaxiality fades away in the bulk as the potential is magnified, and numerical studies illustrate the features of the most general minimizers.

Mathematical Models and Methods for Smart Materials, 2002

ABSTRACT Perfect nematic order may be incompatible with strong anchoring boundary conditions in c... more ABSTRACT Perfect nematic order may be incompatible with strong anchoring boundary conditions in certain geometries. Partial liquid crystalline order can be described in terms of a second-order tensor, denoted here as nematic temperance tensor. Two quantities, prolation and biaxiality, are introduced in terms of the eigenvalues of temperance tensor, and the situations of an isotropic phase, a uniaxial phase and a biaxial phase are characterized. The temperance tensor is compared to inverse absolute temperature. Negative values of temperance may occur under simple conditions; such an example is considered in the paper.

Using a model of a nematic liquid crystal which extends Ericksen’s model [J. L. Ericksen, Arch. R... more Using a model of a nematic liquid crystal which extends Ericksen’s model [J. L. Ericksen, Arch. Ration. Mech. Anal. 113, No. 2, 97-120 (1991; Zbl 0729.76008)] and allows for biaxiality, we solve two simple problems for a slab of a nematic with strong anchoring conditions on the boundary planes. We show that, as the anchoring angle changes, a first- order transition between two solution types would be predicted on the basis of the Frank’s and Ericksen’s models, whereas, when biaxiality is allowed, the transition predicted is second-order, but with a non-smooth transition mode of the chevron type.

We evaluate the 1/N contributions to the mass gap and to the magnetic susceptibility for the two-... more We evaluate the 1/N contributions to the mass gap and to the magnetic susceptibility for the two-dimensional O (N) nonlinear a-model with standard lattice action. By a new asymptotic expansion of the effective propagator we obtain explicit representations of the scaling (field-theoretical) parts of these quantities and analytic expressions for their ratios to the A parameter. Renormalization group functions can be reconstructed and shown to be consistent with universality. Numerical evaluation of the effective Feynman diagrams is performed in a range including the onset of scaling. By comparing the two approaches we recognize that the breakdown of scaling is due to irrelevant dimension four operators contained in the lattice action. The onset of scaling occurs at fl=-1/NT=0.8.

Soft Matter, 2014

Despite the fact that quantitative experimental data have been available for more than forty year... more Despite the fact that quantitative experimental data have been available for more than forty years now, nematoacoustics still poses intriguing theoretical and experimental problems. In this paper, we prove that the main observed features of acoustic wave propagation through a nematic liquid crystal cell -namely, the frequency-dependent anisotropy of sound velocity and acoustic attenuation -may be plausibly explained by a first-gradient continuum theory characterized by a hyperelastic anisotropic response from an evolving relaxed configuration. The latter concept -new in liquid crystal modeling -provides the first theoretical explanation of the structural relaxation process hypothesized long ago by Mullen et al. . We compare and contrast our proposal with a competing theory where the liquid crystal is modeled as an isotropically compressible, anisotropic second-gradient fluid. * paolo.biscari@polimi.it arXiv:1311.1802v2 [cond-mat.soft]

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.

We analyze the interaction between a nematic liquid crystal and an electric field, in a cell in w... more We analyze the interaction between a nematic liquid crystal and an electric field, in a cell in which splay Freedericksz geometry is enforced. Equilibrium configurations are explored both close to the Freedericksz threshold and in the limit of strong applied voltages. We frame within de Gennes' order-tensor theory, which allows us to detect the effects of a variable degree of orientation on critical fields and bifurcation shapes. The applied voltage induces nontrivial effects on the degree of orientation as well. Up to the Freedericksz transition, the degree of orientation decreases, whereas ordering is recovered when the applied voltage drop increases. We also stress the role played by the dielectric anisotropy. In particular, the limit in which the dielectric anisotropy approaches the dielectric permittivities deserves attention, since the order-tensor theory regularizes some of the critical phenomena exhibited by classical Frank solutions.

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.

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 = (∇n)(∇n) 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 become biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.

The European Physical Journal Plus, 2011

UNITEXT, 2013

ABSTRACT Preliminary review / Publisher’s description: Il presente testo di Meccanica Razionale è... more ABSTRACT Preliminary review / Publisher’s description: Il presente testo di Meccanica Razionale è concepito in vista del rinnovamento dell’organizzazione e dei contenuti dei corsi offerti dalle Facoltà di Ingegneria, dove il ruolo della Meccanica Razionale non è solo quello di introdurre alla modellizzazione fisico-matematica rigorosa, ma anche di propedeuticità all’insegnamento di specifiche applicazioni ingegneristiche.In particolare, il testo introduce i concetti fondamentali a partire da esempi e problemi concreti, anche comuni ad altre discipline, in vista di sinergie didattiche a volte favorite dalla presenza di corsi integrati. Il libro possiede una impostazione il più possibile coerente con questa finalità, soprattutto in alcune sezionitradizionalmente caratterizzate da una trattazione forse più astratta: dai vincoli al Principio dei lavori virtuali, dal Principio di d’Alembert alla Meccanica Analitica.

SeMA Journal, 2011

... Leonardo da Vinci 32, 20133 Milano, Italy 2 MOX-Laboratorio di Modellistica e Calcolo Scienti... more ... Leonardo da Vinci 32, 20133 Milano, Italy 2 MOX-Laboratorio di Modellistica e Calcolo Scientifico, Dipartimento di Matematica Francesco Brioschi, Politecnico di Milano. Piazza Leonardo da Vinci 32, 20133 Milano, Italy email: paolo. biscari@ polimi. it, sara. minisini@ polimi. ...

SIAM Journal on Applied Mathematics, 2005

An individual defect in a nematic liquid crystal moves not only in response to its interaction wi... more An individual defect in a nematic liquid crystal moves not only in response to its interaction with other defects but also in response to an external field. We analyze the motion of a wedge disclination in the presence of an applied field of strength H. We neglect backflow and seek steadily travelling patterns. The stationary picture yields a semi-infinite wall of strength π, bounded by the defect line. We find that the disclination advances into the region containing the wall at velocity v(H), where v scales as H/| log H| as long as the magnetic coherence length is greater than the core radius. When the external field is applied in the presence of a pair of disclinations, their dynamics is strongly influenced. We compute the expected relative velocity of the disclinations as a function of distance and field. The natural tendency for the disclinations to annihilate each other can be overcome by a sufficiently strong field suitably directed.

International Journal of Non-Linear Mechanics, 1997

ABSTRACT We consider a nematic liquid crystal confined between two cylinders of radii r1 >... more ABSTRACT We consider a nematic liquid crystal confined between two cylinders of radii r1 > 0 and r2 > r1. We suppose that the lateral surfaces induce a uniaxial anchoring with the optic axis aligned along the radial direction, and we study the equilibrium configurations which minimize the free energy functional.This problem has been already studied under the assumption that the nematic remains uniaxial in the whole tube, with fixed degree of orientation s. We allow the nematic to become biaxial between the surfaces, and include in the free-energy functional an internal potential which favours uniaxiality.We prove that, if the internal potential is neglected (i.e. if biaxiality can arise at no cost), the free energy minimizer is biaxial in the whole volume (except, of course, on the lateral surfaces, where it must be uniaxial). The minimizer is unique, and no bifurcation arises for any value of ϱ: = r1r2.We arrive at new results also when the internal potential is at work: an exact solution, obtained in a special case, proves the existence of a bifurcation at a critical value of ϱ; approximate minimizers show how biaxiality fades away in the bulk as the potential is magnified, and numerical studies illustrate the features of the most general minimizers.

Mathematical Models and Methods for Smart Materials, 2002

ABSTRACT Perfect nematic order may be incompatible with strong anchoring boundary conditions in c... more ABSTRACT Perfect nematic order may be incompatible with strong anchoring boundary conditions in certain geometries. Partial liquid crystalline order can be described in terms of a second-order tensor, denoted here as nematic temperance tensor. Two quantities, prolation and biaxiality, are introduced in terms of the eigenvalues of temperance tensor, and the situations of an isotropic phase, a uniaxial phase and a biaxial phase are characterized. The temperance tensor is compared to inverse absolute temperature. Negative values of temperance may occur under simple conditions; such an example is considered in the paper.

Using a model of a nematic liquid crystal which extends Ericksen’s model [J. L. Ericksen, Arch. R... more Using a model of a nematic liquid crystal which extends Ericksen’s model [J. L. Ericksen, Arch. Ration. Mech. Anal. 113, No. 2, 97-120 (1991; Zbl 0729.76008)] and allows for biaxiality, we solve two simple problems for a slab of a nematic with strong anchoring conditions on the boundary planes. We show that, as the anchoring angle changes, a first- order transition between two solution types would be predicted on the basis of the Frank’s and Ericksen’s models, whereas, when biaxiality is allowed, the transition predicted is second-order, but with a non-smooth transition mode of the chevron type.

We evaluate the 1/N contributions to the mass gap and to the magnetic susceptibility for the two-... more We evaluate the 1/N contributions to the mass gap and to the magnetic susceptibility for the two-dimensional O (N) nonlinear a-model with standard lattice action. By a new asymptotic expansion of the effective propagator we obtain explicit representations of the scaling (field-theoretical) parts of these quantities and analytic expressions for their ratios to the A parameter. Renormalization group functions can be reconstructed and shown to be consistent with universality. Numerical evaluation of the effective Feynman diagrams is performed in a range including the onset of scaling. By comparing the two approaches we recognize that the breakdown of scaling is due to irrelevant dimension four operators contained in the lattice action. The onset of scaling occurs at fl=-1/NT=0.8.

Soft Matter, 2014

Despite the fact that quantitative experimental data have been available for more than forty year... more Despite the fact that quantitative experimental data have been available for more than forty years now, nematoacoustics still poses intriguing theoretical and experimental problems. In this paper, we prove that the main observed features of acoustic wave propagation through a nematic liquid crystal cell -namely, the frequency-dependent anisotropy of sound velocity and acoustic attenuation -may be plausibly explained by a first-gradient continuum theory characterized by a hyperelastic anisotropic response from an evolving relaxed configuration. The latter concept -new in liquid crystal modeling -provides the first theoretical explanation of the structural relaxation process hypothesized long ago by Mullen et al. . We compare and contrast our proposal with a competing theory where the liquid crystal is modeled as an isotropically compressible, anisotropic second-gradient fluid. * paolo.biscari@polimi.it arXiv:1311.1802v2 [cond-mat.soft]

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.

We analyze the interaction between a nematic liquid crystal and an electric field, in a cell in w... more We analyze the interaction between a nematic liquid crystal and an electric field, in a cell in which splay Freedericksz geometry is enforced. Equilibrium configurations are explored both close to the Freedericksz threshold and in the limit of strong applied voltages. We frame within de Gennes' order-tensor theory, which allows us to detect the effects of a variable degree of orientation on critical fields and bifurcation shapes. The applied voltage induces nontrivial effects on the degree of orientation as well. Up to the Freedericksz transition, the degree of orientation decreases, whereas ordering is recovered when the applied voltage drop increases. We also stress the role played by the dielectric anisotropy. In particular, the limit in which the dielectric anisotropy approaches the dielectric permittivities deserves attention, since the order-tensor theory regularizes some of the critical phenomena exhibited by classical Frank solutions.

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.

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 = (∇n)(∇n) 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 become biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.

The European Physical Journal Plus, 2011

UNITEXT, 2013

ABSTRACT Preliminary review / Publisher’s description: Il presente testo di Meccanica Razionale è... more ABSTRACT Preliminary review / Publisher’s description: Il presente testo di Meccanica Razionale è concepito in vista del rinnovamento dell’organizzazione e dei contenuti dei corsi offerti dalle Facoltà di Ingegneria, dove il ruolo della Meccanica Razionale non è solo quello di introdurre alla modellizzazione fisico-matematica rigorosa, ma anche di propedeuticità all’insegnamento di specifiche applicazioni ingegneristiche.In particolare, il testo introduce i concetti fondamentali a partire da esempi e problemi concreti, anche comuni ad altre discipline, in vista di sinergie didattiche a volte favorite dalla presenza di corsi integrati. Il libro possiede una impostazione il più possibile coerente con questa finalità, soprattutto in alcune sezionitradizionalmente caratterizzate da una trattazione forse più astratta: dai vincoli al Principio dei lavori virtuali, dal Principio di d’Alembert alla Meccanica Analitica.

SeMA Journal, 2011

... Leonardo da Vinci 32, 20133 Milano, Italy 2 MOX-Laboratorio di Modellistica e Calcolo Scienti... more ... Leonardo da Vinci 32, 20133 Milano, Italy 2 MOX-Laboratorio di Modellistica e Calcolo Scientifico, Dipartimento di Matematica Francesco Brioschi, Politecnico di Milano. Piazza Leonardo da Vinci 32, 20133 Milano, Italy email: paolo. biscari@ polimi. it, sara. minisini@ polimi. ...

SIAM Journal on Applied Mathematics, 2005

An individual defect in a nematic liquid crystal moves not only in response to its interaction wi... more An individual defect in a nematic liquid crystal moves not only in response to its interaction with other defects but also in response to an external field. We analyze the motion of a wedge disclination in the presence of an applied field of strength H. We neglect backflow and seek steadily travelling patterns. The stationary picture yields a semi-infinite wall of strength π, bounded by the defect line. We find that the disclination advances into the region containing the wall at velocity v(H), where v scales as H/| log H| as long as the magnetic coherence length is greater than the core radius. When the external field is applied in the presence of a pair of disclinations, their dynamics is strongly influenced. We compute the expected relative velocity of the disclinations as a function of distance and field. The natural tendency for the disclinations to annihilate each other can be overcome by a sufficiently strong field suitably directed.