Curvature by design and on demand in liquid crystal elastomers

Active Textile Fabrics from Weaving Liquid Crystalline Elastomer Filaments

Active fabrics, responding autonomously to environmental changes, are the "Holy Grail" of the development of smart textiles. Liquid crystal elastomers (LCEs) promise to be the base materials for large-stroke reversible actuation. The mechanical behavior of LCEs matches almost exactly the human muscle. Yet, it has not been possible to produce filaments from LCEs that will be suitable for standard textile production methods, such as weaving. Based on the recent development of LCE fibers, here, the crafting of active fabrics incorporating LCE yarn, woven on a standard loom, giving control over the weave density and structure, is presented. Two types of LCE yarns (soft and stiff) and their incorporation into several weaving patterns are tested, and the "champions" identified: the twill pattern with stiffer LCE yarn that shows the greatest blocking force of 1-2 N cm^-1 , and the weft rib pattern with over 10% reversible actuation strain on repeated heating cycles. Reversible 3D shape changes of active fabric utilize the circular weaving patterns that lead to cone shapes upon heating. The seamless combination of active LCE yarns into the rich portfolio of existing passive yarns can be transformative in creating new stimuli-responsive actuating textiles.

Discontinuous Metric Programming in Liquid Crystalline Elastomers

Liquid crystalline elastomers (LCEs) are shape-changing materials that exhibit large deformations in response to applied stimuli. Local control of the orientation of LCEs spatially directs the deformation of these materials to realize a spontaneous shape change in response to stimuli. Prior approaches to shape programming in LCEs utilize patterning techniques that involve the detailed inscription of spatially varying nematic fields to produce sheets. These patterned sheets deform into elaborate geometries with complex Gaussian curvatures. Here, we present an alternative approach to realize shape-morphing in LCEs where spatial patterning of the crosslink density locally regulates the material deformation magnitude on either side of a prescribed interface curve. We also present a simple mathematical model describing the behavior of these materials. Further experiments coupled with the mathematical model demonstrate the control of the sign of Gaussian curvature, which is used in combination with heat transfer effects to design LCEs that self-clean as a result of temperature-dependent actuation properties.

Leaping liquid crystal elastomers

Snap-through mechanisms are pervasive in everyday life in biological systems, engineered devices, and consumer products. Snap-through transitions can be realized in responsive materials via stimuli-induced mechanical instability. Here, we demonstrate a rapid and powerful snap-through response in liquid crystalline elastomers (LCEs). While LCEs have been extensively examined as material actuators, their deformation rate is limited by the second-order character of their phase transition. In this work, we locally pattern the director orientation of LCEs and fabricate mechanical elements with through-thickness (functionally graded) modulus gradients to realize stimuli-induced responses as fast as 6 ms. The rapid acceleration and associated force output of the LCE elements cause the elements to leap to heights over 200 times the material thickness. The experimental examination in functionally graded LCE elements is complemented with computational evaluation of the underlying mechanics. The experimentally validated model is then exercised as a design tool to guide functional implementation, visualized as directional leaping.

Multiphysics modelling of light-actuated liquid crystal elastomers

Liquid crystalline elastomers (LCEs) represent a promising class of responsive polymers whose physical properties are peculiar to both fluids and solids. Thanks to their microscale structure made of elongated rigid molecules (mesogens)—characterized by their capability to reversibly switch from an isotropic to an ordered state—LCEs exhibit a number of remarkable physical effects, such as self-deformation and mechanical actuation triggered by external stimuli. Efficient and physics-based modelling, aimed at designing and optimizing LCE-based devices (such as artificial muscles, deployable structures, soft actuators, etc.), is a fundamental tool to quantitatively describe their mechanical behaviour in real applications. In the present study, we illustrate the multi-physics modelling of light-driven deformation of LCEs, based on the photo-thermal energy conversion. The role played by the light diffusion and heat transfer within the medium is considered and their effect on the obtainable actuation is studied through numerical simulations based on the multi-physics theory developed.

Formation of rolls from liquid crystal elastomer bistrips

Formation of desired three-dimensional (3D) shapes from flat thin sheets with programmed non-uniform deformation profiles is an effective strategy to create functional 3D structures. Liquid crystal elastomers (LCEs) are of particular use in programmable shape morphing due to their ability to undergo large, reversible, and anisotropic deformation in response to a stimulus. Here we consider a rectangular monodomain LCE thin sheet divided into one high- and one low-temperature strip, which we dub a 'bistrip'. Upon activation, a discontinuously patterned, anisotropic in-plane stretch profile is generated, and induces buckling of the bistrip into a rolled shape with a transitional bottle neck. Based on the non-Euclidean plate theory, we derive an analytical model to quantitatively capture the formation of the rolled shapes from a flat bistrip with finite thickness by minimizing the total elastic energy involving both stretching and bending energies. Using this analytical model, we identify the critical thickness at which the transition from the unbuckled to buckled configuration occurs. We further study the influence of the anisotropy of the stretch profile on the rolled shapes by first converting prescribed metric tensors with different anisotropy to a unified metric tensor embedded in a bistrip of modified geometry, and then investigating the effect of each parameter in this unified metric tensor on the rolled shapes. Our analysis sheds light on designing shape morphing of LCE thin sheets, and provides quantitative predictions on the 3D shapes that programmed LCE sheets can form upon activation for various applications.

Interfacial metric mechanics: stitching patterns of shape change in active sheets

A flat sheet programmed with a planar pattern of spontaneous shape change will morph into a curved surface. Such metric mechanics is seen in growing biological sheets, and may be engineered in actuating soft matter sheets such as phase-changing liquid crystal elastomers (LCEs), swelling gels and inflating baromorphs. Here, we show how to combine multiple patterns in a sheet by stitching regions of different shape changes together piecewise along interfaces. This approach allows simple patterns to be used as building blocks, and enables the design of multi-material or active/passive sheets. We give a general condition for an interface to be geometrically compatible, and explore its consequences for LCE/LCE, gel/gel and active/passive interfaces. In contraction/elongation systems such as LCEs, we find an infinite set of compatible interfaces between any pair of patterns along which the metric is discontinuous, and a finite number across which the metric is continuous. As an example, we find all possible interfaces between pairs of LCE logarithmic spiral patterns. By contrast, in isotropic systems such as swelling gels, only a finite number of continuous interfaces are available, greatly limiting the potential of stitching. In both continuous and discontinuous cases, we find the stitched interfaces generically carry singular Gaussian curvature, leading to intrinsically curved folds in the actuated surface. We give a general expression for the distribution of this curvature, and a more specialized form for interfaces in LCE patterns. The interfaces thus also have rich geometric and mechanical properties in their own right.

Metric mechanics with nontrivial topology: Actuating irises, cylinders, and evertors

Liquid crystal elastomers contract along their director on heating and recover on cooling, offering great potential as actuators and artificial muscles. If a flat sheet is programed with a spatially varying director pattern, then it will actuate into a curved surface, allowing the material to act as a strong machine such as a grabber or lifter. Here we study the actuation of programed annular sheets which, owing to their central hole, can sidestep constraints on area and orientation. We systematically catalog the set of developable surfaces encodable via axisymmetric director patterns and uncover several qualitatively new modes of actuation, including cylinders, irises, and everted surfaces in which the inner boundary becomes the outer boundary after actuation. We confirm our designs with a combination of experiments and numerics. Many of our actuators can reattain their initial inner or outer radius upon completing actuation, making them particularly promising, as they can avoid potentially problematic stresses in their activated state even when fixed onto a frame or pipe.

Light-Actuated Liquid Crystal Elastomer Prepared by Projection Display

Soft materials with programmability have been widely used in drug delivery, tissue engineering, artificial muscles, biosensors, and related biomedical engineering applications. Liquid crystal elastomers (LCEs) can easily morph into three-dimensional (3D) shapes by external stimuli such as light, heat, and humidity. In order to program two-dimensional (2D) LCE sheets into desired 3D morphologies, it is critical to precisely control the molecular orientations in LCE. In this work, we propose a simple photopatterning method based on a maskless projection display system to create spatially varying molecular orientations in LCE films. By designing different synchronized rotations of the polarizer and projected images, diverse configurations ranging from individual to 2D lattice of topological defects are fabricated. The proposed technique significantly simplified the photopatterning procedure without using fabricated masks or waveplates. Shape transformations such as a cone and a truncated square pyramid, and functionality mimicking the responsive Mimosa Pudica are demonstrated in the fabricated LCE films. The programmable LCE morphing behaviors demonstrated in this work will open opportunities in soft robotics and smart functional devices.

Multivalued Inverse Design: Multiple Surface Geometries from One Flat Sheet

Designing flat sheets that can be made to deform into three-dimensional shapes is an area of intense research with applications in micromachines, soft robotics, and medical implants. Thus far, such sheets were designed to adopt a single target shape. Here, we show that through anisotropic deformation applied inhomogeneously throughout a sheet, it is possible to design a single sheet that can deform into multiple surface geometries upon different actuations. The key to our approach is development of an analytical method for solving this multivalued inverse problem. Such sheets open the door to fabricating machines that can perform complex tasks through cyclic transitions between multiple shapes. As a proof of concept, we design a simple swimmer capable of moving through a fluid at low Reynolds numbers.

Reversible Curvature Reversal of Monolithic Liquid Crystal Elastomer Film and Its Smart Valve Application

Beyond a traditional stimuli-responsive soft actuator that shows a single motion by a stimulus, multidirectional actuation reversal with a single stimulus is highly required in applications such as shape morphing sensors and soft robotics. Liquid crystal elastomers (LCEs) are one of the most attractive candidates for the soft actuator due to their capability of stimuli-responsive shape changing in 3D, which is programmable with local orientation of LC mesogens. Here, a simple but effective method to fabricate monolithic LCE actuators that are capable of reversible curvature reversal in bending and twisting deformation by a single stimulus-heat-is reported. The curvature reversal of the LCE film can be programmed by means of asymmetric crosslinking density along the thickness and the orientation of the LC mesogens. The curvature reversal of the monolithic LCE film exhibits highly reversible (more than 100 times) and fast actuation (≈3-5 s) by heating and cooling, allowing new concept of a practical application using LCE material: a self-regulated smart valve that is capable of qualitatively sorting liquids by temperature. It is believed that this system is potentially applied to a self-regulated sorting platform for various endothermic and exothermic chemical or biological reactions.

Liquid Crystal Elastomers for Biological Applications

The term liquid crystal elastomer (LCE) describes a class of materials that combine the elastic entropy behaviour associated with conventional elastomers with the stimuli responsive properties of anisotropic liquid crystals. LCEs consequently exhibit attributes of both elastomers and liquid crystals, but additionally have unique properties not found in either. Recent developments in LCE synthesis, as well as the understanding of the behaviour of liquid crystal elastomers—namely their mechanical, optical and responsive properties—is of significant relevance to biology and biomedicine. LCEs are abundant in nature, highlighting the potential use of LCEs in biomimetics. Their exceptional tensile properties and biocompatibility have led to research exploring their applications in artificial tissue, biological sensors and cell scaffolds by exploiting their actuation and shock absorption properties. There has also been significant recent interest in using LCEs as a model for morphogenesis. This review provides an overview of some aspects of LCEs which are of relevance in different branches of biology and biomedicine, as well as discussing how recent LCE advances could impact future applications.

Ridge energy for thin nematic polymer networks

Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations: we consider ridged isometric immersions, which can cause a sharp ridge in the immersed surfaces. We propose a model to compute the extra energy distributed along such ridges. This energy comes from bending; it is shown under what circumstances it scales quadratically with the sheet's thickness, falling just in between stretching and bending energies. We put our theory to the test by studying the spontaneous deformation of a disk on which a radial hedgehog was imprinted at the time of crosslinking. We predict the number of folds that develop in terms of the degree of order induced in the material by external agents (such as heat and illumination).

Boundary effect on the spontaneous deformation of a liquid crystal elastomer plate with arbitrary director orientation

Except for director orientation, the deformation modes of constrained liquid crystal elastomer thin plate display specimen geometry size dependence due to the boundary effect. In this paper, the effect of plate geometry size on the spontaneous deformation of a simply supported liquid crystal elastomer plate is studied. The relation between the deformation modes with director orientation and plate geometry size are investigated. Results show that the deformation modes are decided by the director orientation for a certain liquid crystal elastomer, but the geometry size affects the mode transformation with respect to the director. These results are supposed to be used in the design and application of liquid crystal elastomer-based smart actuators or sensors.

Torque-dense photomechanical actuation

Contactless actuation powered using light is shown to generate torque densities approaching 10 N m kg-1 at angular velocities ∼102 rad s-1: metrics that compare favorably against tethered electromechanical systems. This is possible even though the extinction of actinic light limits the characteristic thickness of photoresponse in polymers to tens of μm. Confinement of molecularly patterned developable shells fabricated from azobenzene-functionalized liquid crystalline polymers encodes torque-dense photoactuation. Photostrain gradients from unstructured irradiation segment this geometry into two oppositely curved regions connected by a curved crease. A monolithic curved shell spontaneously bifurcates into a jointed, arm-like mechanism that generates flexure over sweep angles exceeding a radian. Strain focusing at the crease is hierarchical: an integral crease nucleates at smaller magnitudes of the prebiased curvature, while a crease decorated with point-like defects emerges at larger curvatures. The phase-space of morphogenesis is traceable to the competition between stretch and bending energies and is parameterizable as a function of the geometry. The framework for generating repetitive torque-dense actuation from slender light-powered actuators holds broader implications for the design of soft, remotely operated machines. Here, it is harnessed in illustrative mechanisms including levers, lifters and grabbers that are powered and regulated exclusively using light.

Defective nematogenesis: Gauss curvature in programmable shape-responsive sheets with topological defects

Flat sheets encoded with patterns of contraction/elongation morph into curved surfaces. If the surfaces bear Gauss curvature, the resulting actuation can be strong and powerful. We deploy the Gauss-Bonnet theorem to deduce the Gauss curvature encoded in a pattern of uniform-magnitude contraction/elongation with spatially varying direction, as is commonly implemented in patterned liquid crystal elastomers. This approach reveals two fundamentally distinct contributions: a structural curvature which depends on the precise form of the pattern, and a topological curvature generated by defects in the contractile direction. These curvatures grow as different functions of the contraction/elongation magnitude, explaining the apparent contradiction between previous calculations for simple +1 defects, and smooth defect-free patterns. We verify these structural and topological contributions by conducting numerical shell calculations on sheets encoded with simple higher-order contractile defects to reveal their activated morphology. Finally we calculate the Gauss curvature generated by patterns with spatially varying magnitude and direction, which leads to additional magnitude gradient contributions to the structural term. We anticipate this form will be useful whenever magnitude and direction are natural variables, including in describing the contraction of a muscle along its patterned fiber direction, or a tissue growing by elongating its cells.

A blend of stretching and bending in nematic polymer networks

Nematic polymer networks are (heat and light) activable materials, which combine the features of rubber and nematic liquid crystals. When only the stretching energy of a thin sheet of nematic polymer network is minimized, the intrinsic (Gaussian) curvature of the shape it takes upon (thermal or optical) actuation is determined. This, unfortunately, produces a multitude of possible shapes, for which we need a selection criterion, which may only be provided by a correcting bending energy depending on the extrinsic curvatures of the deformed shape. The literature has so far offered approximate corrections depending on the mean curvature. In this paper, we derive the appropriate bending energy for a sheet of nematic polymer network from the celebrated neo-classical energy of nematic elastomers in three space dimensions. This task is performed via a dimension reduction based on a modified Kirchhoff-Love hypothesis, which withstands the criticism of more sophisticated analytical tools. The result is a surface elastic free-energy density where stretching and bending are blended together; they may or may not be length-separated, and should be minimized together. The extrinsic curvatures of the deformed shape not only feature in the bending energy through the mean curvature, but also through the relative orientation of the nematic director in the frame of the directions of principal curvatures.

Tapered elasticæ as a route for axisymmetric morphing structures

Transforming flat two-dimensional (2D) sheets into three-dimensional (3D) structures by combining carefully made cuts with applied edge-loads has emerged as an exciting manufacturing paradigm in a range of applications from mechanical metamaterials to flexible electronics. In Kirigami, patterns of cuts are introduced that allow solid faces to rotate about each other, deforming in three dimensions whilst remaining planar. In other scenarios, however, the solid elements bend in one direction. In this paper, we model such bending deformations using the formulation of an elastic strip whose thickness and width are tapered (the 'tapered elastica'). We show how this framework can be exploited to design the tapering patterns required to create planar sheets that morph into desired axisymmetric 3D shapes under a combination of horizontal and vertical edge-loads. We exhibit this technique by recreating miniature structures with positive, negative, and variable apparent Gaussian curvatures. With sheets of constant thickness, the resulting morphed shapes may leave gaps between the deformed elements. However, by tapering the thickness of the sheet too, these gaps can be closed, creating tessellated three-dimensional structures. Our theoretical approaches are verified by both numerical simulations and physical experiments.

Light-induced spontaneous bending of a simply supported liquid crystal elastomer rectangular plate

Designing the director alignment of a liquid crystal elastomer (LCE) is a key tuning approach for LCE based smart devices. In this paper, the spontaneous strain of the LCE with arbitrary three-dimensional director orientation is derived, and the governing equation for a simply supported LCE rectangular plate is established. By using the finite difference method, the bending configurations are obtained. Different from the freestanding case, three bending modes, that is, unimodal, bimodal, and trimodal modes, are observed for the simply supported rectangular LCE plate. The relation between the bending modes and the director orientation is established. This paper enhances the understanding and facilitates the design of LCE based intelligent light driven devices.

Blueprinting Photothermal Shape-Morphing of Liquid Crystal Elastomers

Liquid crystal elastomers (LCEs) are an attractive platform for dynamic shape-morphing due to their ability to rapidly undergo large deformations. While recent work has focused on patterning the director orientation field to achieve desired target shapes, this strategy cannot be generalized to material systems where high-resolution surface alignment is impractical. Instead of programming the local orientation of anisotropic deformation, an alternative strategy for prescribed shape-morphing by programming the magnitude of stretch ratio in a thin LCE sheet with constant director orientation is developed here. By spatially patterning the concentration of gold nanoparticles, uniform illumination leads to gradients in photothermal heat generation and therefore spatially nonuniform deformation profiles that drive out-of-plane buckling of planar films into predictable 3D shapes. Experimentally realized shapes are shown to agree closely with both finite element simulations and geometric predictions for systems with unidirectional variation in deformation magnitude. Finally, the possibility to achieve complex oscillatory motion driven by uniform illumination of a free-standing patterned sheet is demonstrated.

Dynamically Crystalizing Liquid-Crystal Elastomers for an Expandable Endplate-Conforming Interbody Fusion Cage

Degenerative disc disease (DDD) is the leading cause of low back pain and radiating leg pain. DDD is commonly treated surgically using spinal fusion techniques, but in many cases failure occurs due to insufficient immobilization of the vertebrae during fusion. The fabrication and demonstration of a 3D-printed semi-crystalline liquid crystal elastomer (LCE) spinal fusion cage that addresses these challenges in particular subsidence are described. During implantation of the fusion cage, the LCE is rubbery and capable of deforming around and conforming to delicate anatomy. In the hours following implantation, the device crystallizes into a rigid, structural material with the modulus increasing tenfold from 8 to 80 MPa. In the crystalline regime, a 3D-printed prototype device is capable of enduring 1 million cycles of physiologic compressive loading with minimal creep-induced ratcheting. Effects of LCE molecular architecture on the rate and magnitude of modulus increase, material processability, and mechanical properties are explored. This fundamental characterization informs a proof-of-concept device-the first bulk 3D printed LCE demonstrated to date. Moreover, the novel deployment strategy represents an exciting new paradigm of spinal fusion cages, which addresses real clinical challenges in expandable interbody fusion cages.

Tunable wrinkling of thin nematic liquid crystal elastomer sheets

Instabilities in thin elastic sheets, such as wrinkles, are of broad interest both from a fundamental viewpoint and also because of their potential for engineering applications. Nematic liquid crystal elastomers offer a new form of control of these instabilities through direct coupling between microscopic degrees of freedom, resulting from orientational ordering of rodlike molecules, and macroscopic strain. By a standard method of dimensional reduction, we construct a plate theory for thin sheets of nematic elastomer. We then apply this theory to the study of the formation of wrinkles due to compression of a thin sheet of nematic liquid crystal elastomer atop an elastic or fluid substrate. We find the scaling of the wrinkle wavelength in terms of material parameters and the applied compression. The wavelength of the wrinkles is found to be nonmonotonic in the compressive strain due to the presence of the nematic. Finally, due to soft modes, the critical stress for the appearance of wrinkles can be much higher than in an isotropic elastomer and depends nontrivially on the manner in which the elastomer was prepared.

Curved Geometries from Planar Director Fields: Solving the Two-Dimensional Inverse Problem

Thin nematic elastomers, composite hydrogels, and plant tissues are among many systems that display uniform anisotropic deformation upon external actuation. In these materials, the spatial orientation variation of a local director field induces intricate global shape changes. Despite extensive recent efforts, to date there is no general solution to the inverse design problem: How to design a director field that deforms exactly into a desired surface geometry upon actuation, or whether such a field exists. In this work, we phrase this inverse problem as a hyperbolic system of differential equations. We prove that the inverse problem is locally integrable, provide an algorithm for its integration, and derive bounds on global solutions. We classify the set of director fields that deform into a given surface, thus paving the way to finding optimized fields.

Direct shape programming of liquid crystal elastomers

Liquid crystal elastomers (LCEs) are shape morphing materials promising for many applications including soft robotics, actuators, and biomedical devices, but current LCE synthesis techniques lack a simple method to program new and arbitrary shape changes. Here, we demonstrate a straightforward method to directly program complex, reversible, non-planar shape changes in nematic LCEs. We utilize a double network synthesis process that results in a competitive double network LCE. By optimizing the crosslink densities of the first and second network we can mechanically program non-planar shapes with strains between 4-100%. This enables us to directly program LCEs using mechanical deformations that impart low or high strains in the LCE including stamping, curling, stretching and embossing methods. The resulting LCEs reversibly shape-shift between the initial and programmed shape. This work widens the potential application of LCEs in biomedical devices, soft-robotics and micro-fluidics where arbitrary and easily programmed shapes are needed.

Universal inverse design of surfaces with thin nematic elastomer sheets

Programmable shape-shifting materials can take different physical forms to achieve multifunctionality in a dynamic and controllable manner. Although morphing a shape from 2D to 3D via programmed inhomogeneous local deformations has been demonstrated in various ways, the inverse problem-finding how to program a sheet in order for it to take an arbitrary desired 3D shape-is much harder yet critical to realize specific functions. Here, we address this inverse problem in thin liquid crystal elastomer (LCE) sheets, where the shape is preprogrammed by precise and local control of the molecular orientation of the liquid crystal monomers. We show how blueprints for arbitrary surface geometries can be generated using approximate numerical methods and how local extrinsic curvatures can be generated to assist in properly converting these geometries into shapes. Backed by faithfully alignable and rapidly lockable LCE chemistry, we precisely embed our designs in LCE sheets using advanced top-down microfabrication techniques. We thus successfully produce flat sheets that, upon thermal activation, take an arbitrary desired shape, such as a face. The general design principles presented here for creating an arbitrary 3D shape will allow for exploration of unmet needs in flexible electronics, metamaterials, aerospace and medical devices, and more.

Layered liquid crystal elastomer actuators

Liquid crystalline elastomers (LCEs) are soft, anisotropic materials that exhibit large shape transformations when subjected to various stimuli. Here we demonstrate a facile approach to enhance the out-of-plane work capacity of these materials by an order of magnitude, to nearly 20 J/kg. The enhancement in force output is enabled by the development of a room temperature polymerizable composition used both to prepare individual films, organized via directed self-assembly to retain arrays of topological defect profiles, as well as act as an adhesive to combine the LCE layers. The material actuator is shown to displace a load >2500× heavier than its own weight nearly 0.5 mm.

Nematic director fields and topographies of solid shells of revolution

We solve the forward and inverse problems associated with the transformation of flat sheets with circularly symmetric director fields to surfaces of revolution with non-trivial topography, including Gaussian curvature, without a stretch elastic cost. We deal with systems slender enough to have a small bend energy cost. Shape change is induced by light or heat causing contraction along a non-uniform director field in the plane of an initially flat nematic sheet. The forward problem is, given a director distribution, what shape is induced? Along the way, we determine the Gaussian curvature and the evolution with induced mechanical deformation of the director field and of material curves in the surface (proto-radii) that will become radii in the final surface. The inverse problem is, given a target shape, what director field does one need to specify? Analytic examples of director fields are fully calculated that will, for specific deformations, yield catenoids and paraboloids of revolution. The general prescription is given in terms of an integral equation and yields a method that is generally applicable to surfaces of revolution.