## Abstract

We discuss methods of reversibly inducing non-developable surfaces from flat sheets of material at the micro-scale all the way to macroscopic objects. We analyse the elastic ground states of a nematic glass in the membrane approximation as a function of temperature for disclination defects of topological charge +1. An aim is to show that by writing an appropriate director field into such a solid, one could create a surface with Gaussian curvature, dynamically switchable from flat sheets while avoiding stretch energy. In addition to the prospect of programmable structures, such surfaces offer actuation via stretch in thin systems since when illumination is subsequently removed, unavoidable stretches return.

## 1. Introduction

Geometry dictates that if a flat sheet is deformed so that it is curved in more than one direction at a point, stretches necessarily develop. For a membrane where bend is elastically cheap compared with stretch, deformation is isometric, that is the in-surface distance between adjacent points is preserved. Thus, at least one of the principal curvatures must vanish at each point. These geometric necessities and their elastic consequences are directly encountered in the crumpling of sheets into the third dimension and is reviewed and developed by Witten (2007). In particular, ‘d-cones’ (developable cones) (Ben Amar & Pomeau 1997; Cerda & Mahadevan 2005) emerge as structures with these singular points to accommodate a sheet being confined to a smaller volume. As the singularities emerge, so also must singular lines (folds) that connect them. Similarly, such response arises when attempting to wrap a sphere with a flat sheet, with the resultant folds and cusps accommodating the Gaussian curvature without large-scale in-surface stretches and compressions.

We are concerned here with materials that will spontaneously and reversibly deform under light or heat to surfaces that have localized or delocalized Gaussian curvature (that is, are non-developable surfaces), without the need to form d-cones and with no large-scale stretch from their new natural state. Our ultimate aim is to reversibly switch flat sheets into arbitrary desired surfaces and back without an elastic stretch cost by combinations of the effects we describe below.

In generating Gaussian curvature, we consider materials that can deform inhomogeneously. Of course, simple bend can occur if deformation is non-uniform through the sheet’s thickness. Typically, though the bend is in one direction only, with limited saddle or spherical response, a wide range of new possibilities are reviewed and modelled (Warner *et al.* 2010*a*). However, Gaussian curvature is soon suppressed (Warner *et al.* 2010*b*) by the stretches mentioned above. We overcome any suppressive stretches by having spontaneous deformations uniform through the thickness but inhomogeneous in the plane. In particular, we consider liquid crystalline glasses, which have a natural state of elongation along their director, and contraction perpendicular, that depends on their orientational order. Accordingly, they suffer large, reversible length change with either heating (Tajbakhsh & Terentjev 2001; Mol *et al.* 2005) or illumination (Finkelmann *et al.* 2001; Harris *et al.* 2005). For glasses, these strains can be 2–3% and of opposite sign (without conserving volume) along and perpendicular to the director (figure 1); in elastomers, the stretches are 100s per cent with perpendicular strains preserving volume. Bending actuation for these materials is reviewed in this context (Corbett & Warner 2009; Modes *et al.* 2010). In addition to the relative weakness of their strain response, glasses are distinguished from elastomers by being so heavily cross-linked that the director field only changes as a result of advection owing to material shape change from the elastic strain. Nematic glass directors do not rotate relative to the matrix as in nematic elastomers—they are (Harris *et al.* 2005) conventional, uniaxial elastic solids, with moduli about ×10^{4} higher than those of elastomers. We introduce the energy function for such glasses that will specify the spontaneous distortions and also the elastic energy cost associated with imposed changes away from the new natural shapes—we find global, three-dimensional deformation fields making these thermo-/optical strains compatible and thus of zero energy in the membrane limit we take. The energy function is thus only required when we go beyond the membrane approximation, for instance in calculating bend energies and the cap energies of any singular points.

Director fields can be established in the nematic liquid progenitor phase before cross-linking and are permanently recorded in the solid state achieved after linkage. Complex, three-dimensional director fields for subtle mechanical response can be achieved in nematic glasses (Serrano-Ramon *et al.* 2008) via holography and surface preparation. The ability to ‘write’ an initial director field into a solid so that it distorts in a predictable (and reversible) way into a new shape with applied temperature or light is thus conceivable. In particular, the ability to take flat, thin sheets of material and turn them into prescribed, non-developable shapes—on any length scale larger than a nematic defect core—is highly sought after. A key observation here is that thin sheets can accommodate many more potential inhomogeneous changes of stretch compared with bulk three-dimensional specimens by deforming out of plane. Further, the energy and forces associated with the membrane mode (in-plane stretch) scales as the thickness while those associated with the bending mode (differential in-plane stretch) scales as the third power of thickness. Hence, the bending energy is negligible compared with the in-plane stretches. All of this leads to a new paradigm for actuation (Bhattacharya & James 2005): if we are able to find a director arrangement that leads to stretches that are incompatible in bulk but compatible in sheets, then the change of order can be used to generate out-of-plane deformations with a very large blocking force (one that scales with thickness). This is in contrast with small blocking forces (third power of thickness) associated with bending cantilevers. We show here that +1 nematic disclinations are indeed such arrangements.

Disclination defect textures are not extensively studied, experimentally or theoretically, in nematic solids. However, it is known that the energy-minimizing deformation associated locally with changes in nematic order is not compatible in some three-dimensional disclinations (Fried & Sellers 2006). We analyse the mechanical response and elastic ground state of the most easily writable disclination defect in two dimensions, that with topological charge +1, and find that shrinking in-material circumferences and growing in-material radii (or vice versa) affect a relative ratio other than 2*π* and hence necessitate the development of Gaussian curvature. Figure 1 shows examples of such initially flat, defected sheets after heating or cooling. We also discuss how delocalized Gaussian curvature can be thermo-optically induced.

Response closely analogous to ours has been analysed by Ben Amar *et al.* in the elasticity of botanical systems and Sharon *et al.* in NIPA gel sheets, when anisotropic growth creates internal stresses and forces planar systems into the third dimension (Klein *et al.* 2007; Dervaux & Ben Amar 2008; Müller *et al.* 2008). They and we deal with localized Gaussian curvature and hence localized elastic stretch. But our cones and anti-cones are elementary, not ‘d-cones’ as discussed above. However, except for very weak spontaneous distortions, we also find our tip extent is of the order of the sheet thickness. By finding the core/far-field, bend/stretch energy balance, we also discuss the energies arising in liquid nematic membranes and show why we can ignore them.

## 2. Spontaneous distortions and the elastic energy function

Consider a nematic glass fabricated at temperature *T*_{0} with a director oriented along *n*_{0} (the latter may be heterogeneous). Suppose the temperature is changed to *T* (or there is an illumination change) and the glass experiences a deformation gradient . The free energy density of the glass may be written by adapting that for nematic elastomers (Bladon *et al.* 1994):
2.1
where and , and *α*_{0},*β*_{0},*α*,*β* depend on the order parameter and hence are temperature related. The modulus *E* reduces to three times the shear modulus *μ* for the case of elastomers (a special, volume-preserving example), and so for an exact correspondence for that case, one should really have *E*/6. Since in glasses the director does not rotate relative to the material, the deformation gradient advects to . To explore this, recall the polar decomposition theorem that decomposes the deformation into a stretch (symmetric, positive-definite) followed by a rotation . It follows that . Moreover, must be uniaxial in a frame based on since thermal or optical response is relative to this ordering direction. We may now rewrite the free energy density as
where . The ground state is obtained by noting that the argument of the trace has to be a constant (*γ*, which reflects the change in the ground-state volume) times identity. It follows that a ground-state stretch is given by
where and . In short, the ground states involve a stretch by *λ* along and stretch *λ*^{−ν} transverse to it. One can think of *ν* as a thermo-optical Poisson ratio to characterize volume change. It is known to take values in the range *ν* = 1/3−2 (Harris *et al.* 2005).

Any subsequent deformations away from the initial spontaneous distortions to the new natural shape cost an elastic energy of a form conventional for uniaxial solids. Indeed, it is known that without director rotation relative to the solid matrix, one can approximate such materials as isotropic. We characterize them, in those cases where we need to consider energies, in infinitesimal elasticity by a Young’s modulus, *E*, and an (elastic) Poisson ratio, *ν*_{el}. For glasses, *E*∼10^{9} J m^{−3} and *ν*_{el} < 1/2. For larger deformations, we take as an illustration the classical free energy density for elastomers:
2.2
where *Λ* is a stretch with respect to the new natural shape, and where this illustration has *ν*_{el} = 1/2.

The elastic deformations following spontaneous shape change may (cap formation) or may not (simple bends) involve further advection of the director from to an .

## 3. Cone and anti-cone formation from *m* = 1 defects

We consider +1 disclinations. In the membrane limit in which we choose to work, and with strong planar surface anchoring when in the liquid progenitor phase, escape into the third dimension (Meyer 1973) of the director field is not possible except perhaps near the core. These disclinations are then true topological defects. For +1 defects, many different textures, though topologically equivalent (figure 2), differ non-trivially in their mechanical response. We here analyse azimuthal and radial textures, and spirals in §5.

### (a) Cones

Consider a thin sheet of nematic glass whose director field is azimuthal around a +1 disclination defect, as in the left of figure 2, and which is flat at some reference temperature, *T*_{0}. As the sample is heated above *T*_{0}, the decline in nematic order will cause a contraction of the natural length along the nematic directors with a local elongation of the natural length owing to Poisson effects normal to them. In a free, uniform glass, these natural length changes would be manifested by actual mechanical strains so that the elastic ground state is achieved. Since the chosen director field is circularly symmetric, with integral curves simply concentric circles centred on the defect, clearly the sheet has a problem accommodating this change in the natural lengths as circumferences shrink while the corresponding natural radii grow.

In the membrane approximation, neglecting bending energies, there is an obvious geometric solution that allows the nematic glass to respond to the imposed thermal strain without the high energies of elastic compressions and expansions relative to this changed state—a cone. Consider a circle of material centred on the defect: at *T*_{0}, the sheet of nematic glass is flat, and the circle’s perimeter-to-radius ratio is 2*π*. However, as the temperature rises, the perimeter wants to change by a factor of the thermal deformation gradient along the director, , where here *λ* < 1. Meanwhile (figure 1), the material (in-plane) radius is changing as well owing to the thermal/optical Poisson effects associated with the perimeter’s change, . Together, these transformations imply that upon heating, a circle of material on the sheet remains circular, but adopts a new in-material perimeter-to-radius ratio, that of a circle enclosing a cone’s tip (figure 1, first panel):
3.1
with *ϕ* the cone opening angle. One can think of as the embedded radius. The localized Gaussian curvature associated with the cone tip is thus ; circles enclosing the tip of a non-developable cone no longer have the ratio 2*π* of the perimeter to the in-plane radius, unlike circles on the cone but not enclosing the tip. Away from the singularity, there is no Gaussian curvature and hence no shape-induced elastic compressions or extensions. Therefore, our sheet of nematic glass responds to temperatures above *T*_{0} by deforming out of plane, breaking up–down symmetry in the process. The opening angle varies sensitively with small strains; a 2–3% contraction, with *ν*∼2, achieved over 90^{°}C (Mol *et al.* 2005) gives *ϕ*∼70–66^{°}, such strains also being common for modest illuminations in photo-glasses (van Oosten *et al.* 2007).

### (b) Anti-cones

But what if we cool the azimuthal sample below *T*_{0}, increasing the nematic order relative to the reference state? The arguments relating the local changes in length along and normal to the nematic director go through unchanged; however, now *λ* > 1, invalidating our previous ansatz cone solution—now the perimeter is ‘too long’ for the in-plane radius and a more complex deformation must result.

Consider deformed surfaces where the height varies linearly with the distance from the centre or the defect. A circle of radius *r*_{0} centred at the defect in the flat undeformed reference plane becomes after deformation a curve described by {*r*(*ϕ*),*ϕ*,*h*(*ϕ*)} in cylindrical coordinates (*r*,*ϕ*,*h*) in the deformed configuration and parametrized by the azimuthal angle *ϕ*. If the material is in the minimal energy state unstretched from its new natural configuration, this curve has to satisfy two constraints. First, it has to have a constant distance *R* = *λ*^{−v}*r*_{0} from the origin (defect) and second, its length has to be equal to *P* = 2*πλr*_{0} = 2*πλ*^{1+ν}*R*. Thus, the curve has to lie on a sphere of radius *R*; see the trajectory in the last panel of figure 1. Further, since the length of the curve *P* is greater than the length 2*πR* of the great circle of the sphere, this curve has to oscillate. In other words, the deformed surfaces oscillate azimuthally and these oscillations grow linearly radially as shown in figure 1. We call them anti-cones. We also note,
3.2
and
3.3
The simplest possibilities for *h* are
3.4
for anti-nodal line angle to the initially flat plane of (the amplitude is *A* in effect; figure 1) and with integer *n* (so that the curve is closed). In equation (3.2), this ansatz gives the relationship between the *r* coordinate and *ϕ* at constant *R* needed for the perimeter:
3.5
Returning *r* and the form of *h* to equation (3.3) for the perimeter gives, grouping factors of *R* and simplifying,
3.6
where *I*(*n*,*A*) depends only on the scale *A* and the state *n*:
Connecting the new natural radius and perimeter as in equation (3.1), avoiding stretch requires *I*(*n*,*A*) = *λ*^{1+ν}(*T*−*T*_{0}); as temperature and hence spontaneous distortion changes, so does the character (that is, *A* and *n*) of the anti-cone. The negative Gaussian curvature localized at the apex of the anti-cone is −2*π*(*I*−1). As appropriate, *I* = 1 for *n* = 0—the surface is a flat plane. Otherwise, *I* ranges from 1 to |*n*| for (figure 3).

The analogous cosine solutions simply give rise to rotated versions of the same surfaces for all *n*≠0, recovering the conical solution discussed earlier for *n* = 0 and *λ* < 1. In this case, *I*(*A*;*n* = 0) ranges from 1 at *A* = 0 to 0 at , as required. The behaviour of *I* for *n*≠0 is encouraging—our trial solutions yield precisely the geometries that accommodate, at zero stretch energy, a cooling of our azimuthal +1 defect below *T*_{0}. However, since each individual surface is limited to a maximum perimeter-to-radius ratio of 2*π*|*n*|, we would expect interesting transition behaviour as cooling leads to strains requiring ever more crumpled geometries, with transition states characterized by simple Fourier combinations of the ‘pure’ surfaces or, in extreme cases, exotic surfaces that are multiply re-entrant in *ϕ*. Figure 1 shows an *n* = 3 anti-cone where the crumples take up more perimeter than the more slowly varying *n* = 2 anti-cone. See Müller *et al.* (2008) for an analysis of large-amplitude anti-cones where the functions *h*(*ϕ*),*r*(*ϕ*) may not be single valued. We conjecture that all minimal membrane energy solutions of these defects are anti-cones.

For radial textures (figure 2), the roles of the direct thermal strain and Poisson strain are swapped. Heating above *T*_{0} now requires a shortening of the radial length owing to decreased nematic order while the azimuthal direction expands from the corresponding Poisson effect—equivalent to *lowering* the temperature in the azimuthal texture. Hence, each of these textures behaves as the other under the mapping .

Although we are dealing with thin sheets in the membrane approximation where shape change has low bending energy, large forces can be exerted by switching on and off Gaussian curvature via stretch modes that arise if the natural and imposed geometries are in conflict.

### (c) Truncated cones

An annulus with inner and outer radii *R*_{a} and *R*_{1}, respectively, is cut out of a flat sheet co-centrically with an azimuthal +1 defect (figure 4*a*). The annulus encompasses the defect, and we show it suffers the conflict between radial and azimuthal distortions on heating (*λ* < 1) as before. Consider the material at the inner edge to distort to a distance *R*_{a}′ from the putative tip of a cone (figure 4*b*). Then, perimeters corresponding to the initial radii *R*_{a} < *r*_{0} < *R*_{1} distort as and radii as since we can only say with certainty that the material section (*R*_{a},*r*_{0}) distorts with *λ*. The perimeter/radius ratios in the heated body for two different initial radii, say *r*_{0} and *R*_{1}, should be the same, that is
a relation that can only be satisfied if *R*_{a}′ = *λ*^{−ν}*R*_{a}, that is the annulus distorts to a truncated cone, as one expects. However, this cone lacks a tip (of significance for the energy balance as we see in §6). It is also a developable surface, but it retains the memory of the centre of the topological defect that it encloses, and of its charge. One might use these structures in microfluidics as their truncated form would allow axial liquid flow on removal of illumination whereupon deviation from flatness carries a stretch penalty.

## 4. Creation of surfaces with delocalized Gaussian curvature

Surfaces of delocalized Gaussian curvature can be created from flat sheets if the, for instance azimuthal, director field has a radially varying associated order parameter or cross-link density, thus giving rise to *λ*(*r*_{0}) and *λ*^{−ν}(*r*_{0}) response tangentially and radially. Spatially varying order and linkage is exploited in nematic glass sheets used in reflectors for LC displays (Broer *et al.* 1999). Alternatively, one could have partial escape into the third dimension, the extent of which depends on *r*_{0} so that the effective azimuthal response would be also a function of *r*_{0}. As an example of this strategy, we find the *λ*(*r*_{0}) required to form a spherical cap, of radius of curvature *R*, at zero stretch cost. The in-material radial distance to the point that was initially at *r*_{0} in the flat disc determines (figure 4*c*) the angle *θ*(*r*_{0}) it sits at on the cap, given *R* its final curvature:
4.1
where *r*_{0}/*R* = *u*, that is distances *r*_{0} are reduced by *R* that is as yet undetermined. The corresponding in-material circumference arises from deforming that in the flat disc, 2*πr*_{0}, by *λ*.
4.2
Injecting this *λ* into the differential form of equation (4.1) gives
4.3
The solutions *θ*(*u*), with boundary condition *θ*(*u* = 0) = 0, give *λ*(*u*) when they are returned to equation (4.2). In general, the solutions are powers of and times the hypergeometric function_{2}*F*_{1}, but some powers *ν* give simple results. For instance, *ν* = 1 is a trivial, linear case yielding and thus . The spherical radius can then be determined: if a particular has a *λ**, then the *u** is and hence . For instance, if at the , then , and the original point deforms to being at angle *θ* = *π*/4 on the cap. Other values give solutions in closed form for the non-linear differential equation (4.3). For instance, *ν* = 2 gives *θ*(*u*) in the form , which is equivalent to solving the rubber case, *ν* = 1/2, for radial directors and cooling.

A naturally spherical cap, that is one without stretch, could terminate a cone if, for instance for *ν* = 1, the *λ*(*u*) varied initially from *λ* = 1, *u* = 0, as until it reached the value *λ*(*u*) = *λ* of the cone. Since at common tangency where the cap meets the cone *θ* = *π*/2−*ϕ*, the . This point in the initial disk, at *r*_{0} = *R*_{0} say, must have *λ*(*R*_{0}) = *λ*. From *u* = *r*_{0}/*R* we can obtain the radius of curvature of the cap as . See however further remarks in §6*a* about far-field bend energy having an effect on delineating the region 0 < *r*_{0} < *R*_{0} that becomes the cap. Figure 6*a* sketches this geometry.

The general problem of what shells of revolution arise from a given *λ*(*r*_{0}) can be straightforwardly reduced to a quadrature: with cylindrical coordinates (*z*,*ρ*) specifying the final, cylindrically symmetric state, the initial point (0,*r*_{0}) on the flat disc transforms to (*z*(*r*_{0}),*ρ*(*r*_{0})). To avoid stretch from the new natural state, the perimeter 2*πr*_{0} becomes 2*πρ* = 2*πr*_{0}*λ*, that is no azimuthal strain means *ρ* = *r*_{0}*λ*(*r*_{0}). Likewise, the in-material radial element d*r*_{0} must become d*r* = *λ*^{−ν} d*r*_{0}. Also (d*r*)^{2} = (d*ρ*)^{2}+(d*z*)^{2}, whence re-arranging for d*z*, using *ρ* from above, and integrating, one obtains
4.4
Clearly one needs gradients of *λ* such that *λ*^{−ν} > (*r*_{0}*ρ*)′ for the whole disc 0 < *r*_{0} < *R*_{1} in order to avoid the need for azimuthally varying shapes. Regions of negative Gaussian curvature can arise in the deforming shell for some forms of *λ*(*r*_{0}) by rendering it bell-shaped, which may be useful for welding without strain mismatch the responsive inner region onto an outer region (*r*_{0} > *R*_{1}) that is not optically responsive and hence remains flat.

An arbitrary (i.e. not spherical) cylindrically symmetric cap results when the spontaneous contraction *λ*(*r*_{0}) varies with *r*_{0} consistently with equation (4.4) but does not obey equation (4.3). We require the angle of the inner region to attain that of the cone to which it is joined, that is . From the azimuthal and radial changes just discussed, one has the derivatives and hence at the junction of the cap and cone, *r*_{0} = *R*_{1}. Using , with *λ* its (constant) value on the conical skirt side of the junction, this condition becomes *λ*^{1+ν} = *λ*^{ν}(*λ*+*r*_{0}*λ*′). It is clearly only satisfied if the derivative *λ*′(*r*_{0})|_{R1} = 0 also; the *λ*(*r*_{0}) variation must attain *λ* smoothly.

## 5. Distortions from spiral *m* = 1 defects

Generically a +1 disclination defect has an intermediate angle *δ* of the director with respect to the radial vector from the defect core. As a result, the effect of the direct and Poisson strains is now mixed along circles and radii centred on the defect. Curves along which the material feels the maximal effect of the direct strain and none of the Poisson strain, and vice versa, are the integral curves of the director field and its normal complement, respectively. For +1 textures with 0 < *δ* < *π*/2, the integral curves are logarithmic spirals instead of simple circles and radii (figure 5*a*). A logarithmic spiral is with *r* = *a*e^{bθ}, where *a* scales *r* and *θ* is the usual angular coordinate. We can write *a* = *e*^{−bφ} with and where . Figure 5*b* shows how coordinates based on spirals can be set up: the unit circle intercepts spirals with decreasing *φ* values marked. A curve of constant *r* is an arc of radius *r* with increasing *φ*, and arc length *r*Δ*φ*; see for instance proceeding from points *a* to *b* in the figure. To see this, note that , and at constant *r* we have Δ*θ* = Δ*φ*. The curve of constant *φ* is the logarithmic spiral itself. Spirals are advanced by increments of *φ*. This *θ*−*φ*−*r* connection allows us to parametrize the plane in terms of *r*,*φ*, so that a point is given by
5.1
This parametrization is one-to-one if we choose .

Now consider the deformation of the plane through the map
5.2
where satisfies
5.3
so that describes a curve on a unit sphere with arc-length parametrization. In addition to , the deformation is described by three parameters, *α*,*β*,*γ*. To understand these parameters, note that a curve of constant *r* is a curve on the sphere of radius *αr* and arc length *αγr*Δ*φ*. Thus, *α* is the effective radius stretch and *αγ* is the effective azimuthal stretch. Further, note that a curve of constant *φ* is a logarithmic spiral with modified angle on a generalized conical surface described by . Finally, the deformation is compatible (continuous) if is periodic with period 2*γπ*.

To see whether such a deformation corresponds to the ground state of a nematic glass membrane, we need to verify that the stretch along the director is *λ*, the stretch normal to the director is *λ*^{−ν} and that these are the principal stretches. The last condition is equivalent to requiring that the convected normal and convected director remain orthogonal (zero shear in the director–normal pair).

We begin with the first, the stretch induced by this deformation along the director. We note that the integral curve of the director is a logarithmic spiral with angle *δ* and thus the curve generated by *φ* = *φ*^{☆} (constant). Along this curve, the derivative (∂/∂*r*)_{φ☆} is the non-unit tangent:
5.4
and the corresponding derivative of equation (5.2) gives the new tangential element of the deformed space
5.5
Therefore, the tangential stretch (that is, along the director) is the ratio of their moduli:
5.6
In forming above, we have used the fact that .

Now consider the stretch normal to the director. The complementary integral curve to the director is again a logarithmic spiral, but with an angle , with −*b* since the rotation is in the opposite sense. Corresponding to equation (5.1), we have
5.7
where is the particular *φ*-parameter of the logarithmic spiral complementary to that above. Taking the variation with *r* of along the complementary spiral, equation (5.7) directly gives with , analogously to equation (5.4). To find it is more straightforward to use the original spirals as coordinates since we have already assumed how the primary spirals’ parameters scale on distortion. Requiring the complementary and primary spirals to pass through the same point means that *φ*^{☆} and are related by
5.8
Variation of with *r* now requires passing from one primary spiral with *φ* to another with a different *ϕ*. Scaling etc. in equation (5.2) and using equation (5.8) now for *φ* gives the distorted position and its tangent, , as
5.9
and
5.10
Taking the ratio of to , the stretch normal to the director is
5.11

Finally, the angle of shear between the director and normal is given by since and are orthogonal by construction.

Now, for this deformation to be a low-energy deformation of the nematic glass, we need to set the stretches equal to their spontaneous values *λ* and *λ*^{−ν}, that is
5.12
The final condition ensures that the director and its normal are the principal directions of stretch. This gives us three conditions to determine the three parameters *α*,*β*,*γ*. We find:
5.13
5.14
and
5.15

Note that in the limit (radial director), the radial scaling must be by *λ* and indeed with the azimuthal scaling as expected. Similarly, in the limit (azimuthal director), and as expected.

The spiral angle *δ* evolves with spontaneous deformation *λ* since . For instance, for heating where *λ* < 1, then the angle *δ* decreases—the spiral tends more to the radial direction. Also a mismatch between radius and perimeter as considered above arises: cones or anti-cones form depending on whether the radius or perimeter grows relative to the other. The perimeter/radius ratio is 2*πγ*, and hence the deformed shape is a ‘cone’ if *γ* < 1, and an ‘anti-cone’ or saddle if *γ* > 1. We obtain a flat surface when *γ* = 1. For instance, for an initial *δπ*/4 and hence *b* initially < 1, the spiral gives a conical response for *λ*∈(*b*^{2/(1+ν)},1) and anti-cones otherwise. The cone angle is .

Because of the spiral angle of the director relative to the radial direction, on thermo-optical change, material undergoes motion with an azimuthal component. Furthermore, if *δ* and *λ* are such that an out-of-plane deformation into a cone is required, then this rotation, combined with the material’s spontaneous choice to form an ‘upward’ or ‘downward’ pointing cone as the material moves from two to three dimensions, leads to a spontaneously broken chiral symmetry. This symmetry-breaking does not occur for the anti-cone solutions, as they do not break the up–down symmetry.

## 6. Tip structure of cones and energies of distortions

The cones considered thus far have Gaussian curvature localized at the tip. In the membrane approach, we have neglected the bend cost of forming the cone because the stretch/compression energy arising were the cone not to be formed would be expensive. However, the simple curvature of the cone diverges on approaching the tip and the bend (areal) energy density is greater than that arising from converting the cone’s tip to a cap when some stretch energy would be incurred. We now calculate the bend energy, put bounds on the cost of caps and estimate the resulting size of caps relative to both the thickness and the entire size of the solid sheet. For concreteness, we use the volume-preserving distortions of an elastomer. We then calculate conversely what energy is available in actuation, and finally compare our energies with those arising in other kinds of nematic sheets.

### (a) Cone bend and cap stretch energies

Consider a cone the tip of which we distort to a spherical cap (figure 6*a*) to delocalize its Gaussian curvature. We have seen in §4 how this can be done without energy cost if the spontaneous distortion varies radially. Here we find the energy cost if the cone-forming *λ* attains everywhere. Points in the initially flat disc at radii *R*_{1} > *r*_{0} > *R*_{0} distort to the cone, adopting an in-plane distance *λ*^{−ν}*r*_{0} from the putative tip. For points initially at *r*_{0} < *R*_{0}, we consider distorting first to the light region (figure 6*a*) of the cone, which will then be further distorted to the cap, the cost of which is estimated in the next subsection. The principal radius of curvature of the cone is (the other, principal direction, down the cone skirt, being of course flat) for points initially at *r*_{0}. One then has an energy per unit surface area of the case at the radius *r*_{0} of
6.1
where *h* = *λ*^{−ν}*h*_{0} is the current thickness with *h*_{0} the initial thickness before thermo-optical changes, and . We have taken *Λ* close to 1, that is, bend distortions are small, and we have ignored *Λ* terms in equation (6.1). Young’s modulus is *E*, which for instance in a rubbery material with *ν*_{el} = 1/2 is *E* = 3*μ*, with *μ* the shear modulus. The total bend energy associated with the initial disc’s region *r*_{0} > *R*_{0} is
6.2
with, as ever, the cone opening angle given by .

The bend energy density on the conical region diverges as on approaching the tip and eventually it is cheaper to instead invoke stretches and terminate the cone with a cap. For simplicity, take the cap to be spherical; the cone’s Gaussian curvature is then evenly delocalized. The cap’s radius of curvature is clearly (figure 6*a*). The bend energy of the cap is which depends only on the cone opening angle *ϕ*, that is, on what fraction of a hemisphere is required for the cap, but does not depend on the cap size, and we can neglect this energy in the following.

A simple estimate of the cap’s stretch energy takes the tip region arising from *r*_{0} ≤ *R*_{0}, that is, the part of the cone inside the in-material radius *R* = *λ*^{−ν}*R*_{0} in figure 6*a*, and compresses it uniformly in the radial sense to sit on the spherical cap below it, that is, the distortion is , the numerator being the in-material extent of the cap. For brevity, we denote *Λ*_{rr} simply by *Λ*, which is a function of *ϕ*. The concomitant *Λ*_{θθ} is determined geometrically. Note that a point initially at radius *r*_{0} in the flat disc is at a distance *r*_{0}*λ*^{−ν} from the cone’s tip and then, on cap formation, is *r*_{0}*λ*^{−ν}*Λ* from the sphere’s pole. It is accordingly at an angle . The perimeter on the sphere at angle *α* is and the perimeter on the cone was 2*πr*_{0}*λ*, whereupon taking the ratio of these two lengths:
6.3
The latter form clearly shows that *λ*_{θθ} ≥ 1 is an elongation.

As an illustration, we insert these deformations into the energy density (2.2), taking *ν*_{el} = 1/2 and noting that *Λ*_{θθ} is along and that, to preserve volume, the distortion in the thickness direction is *Λ*_{hh} = 1/(*ΛΛ*_{θθ}). We integrate over the cone tip region with d*r* = *λ*^{−ν} d*r*_{0}, *h* = *λ*^{−ν}*h*_{0} and perimeter (2*πr*_{0})*λ*:
6.4
where the −3 gives us *U*_{c} = 0 as the reference energy when no stretch is invoked, and where we take a general modulus *E* since we are mostly dealing with glasses rather than elastomers.

The total energy is
where
6.5
and
6.6
are identified from equations (6.2) and (6.4). The functions *f*_{1,2}(*ϕ*) are the integrals in equation (6.4) and can be written in closed forms, for instance , where *β* = *π*/2−*ϕ* and Ci is the cosine integral function. These forms are not very illuminating, and we illustrate them numerically. Minimizing over *R*_{0}, the energy becomes
6.7
The final capped cone energy, despite being the sum of bend and stretch contributions, scales purely with , the scale of the total bend energy. The region in the initially flat disc that becomes the cap is within the radius *R*_{cap} = *h*_{0}*g*(*ϕ*). Figure 6*b* shows *g*(*λ*) = *R*_{cap}/*h*_{0}, which derives straightforwardly from equation (6.7) since . The cap size would diverge as (no thermal strain)—there is no energy cost and — but of course *R*_{cap} is bounded by *R*_{1}. In these initial stages of distortion, expansion in *β* = *π*/2−*ϕ*∼0 gives *A*∝*β*^{2} and *B*∝(10/27)*β*^{4} at lowest orders. Recalling that , the region transformed to the cap is of radius
6.8
Here we define the small contractile strain along as *ϵ* = 1−*λ*. Since , then . The limit in equation (6.8) is for *ν* = *ν*_{el} = 1/2. Thus, the cap region should extend over an appreciable fraction of the disc, *R*_{cap}∼*R*_{1}, at small thermo-optical strains such that —the sample would be nearly flat with Gaussian curvature completely delocalized.

However, for small thermo-optical strains *ϵ* < *ϵ*_{c} there are alternatives. The disc can remain flat, paying the energetic cost of radial compressions and azimuthal elongations away from the new natural state, or can form a pure cap (not a cap with cone-line flanks) with both bend and stretch energy.

The energy of the disc remaining flat is , where *f*∼1, arising because elongating radii and contracting perimeters cannot remain in the correct ratio while staying in-plane. It can be relieved by forming a cap that we assume spherical with radius of curvature *R*_{c}. For instance, a perimeter at an in-material radius *r*_{0} can reduce from 2*πr*_{0} to on cap formation and become closer to its new natural length (1−*ϵ*)2*πr*_{0} via the cap azimuthal strain arising on curvature. For small curvatures, the flat strain is reduced to on expanding sine. Ignoring the concomitant radial change of strain gives a lower bound on the energy reduction from curving. Instead of terms like in the energy, we square the new, lower strain, the cross terms giving the reduction in the energy at the lowest order in the curvature:
6.9
with *g*∼1 again a harmless function of the Poisson ratios.

But the bend energy areal density (6.1), suppressing the *λ*∼1 factors and taking a factor of 2 since bend is in two dimensions, gives a bend energy
6.10

Both *U*_{b} and Δ*U* vary as and hence there is only an energy reduction on cap formation when (1/9)(*h*/*R*_{c})^{2}−(1−*λ*)(*R*_{1}/*R*_{c})^{2}*g* changes sign, taking here *ν* = *ν*_{e} = 1/2. Initially, flat discs remain flat until the strain is *ϵ*∼(*h*/*R*_{1})^{2}/(9*g*). This strain is comparable to *ϵ*_{c}. Thus, initially flat discs with *m* = +1 defects remain flat until they jump to cones dominated by caps, the caps thereafter becoming less significant.

On the other hand, for any appreciable deviation of *λ* away from 1, the cap is localized to an extent of the order of the thickness. The energy is vastly less than the stretch energy that would arise from the region (*R*_{cap},*R*_{1}) if the cone were not to form. The cap energy rises drastically as since the cone is then very sharp and our estimates of the energy via a very large uniform radial compression () deteriorate; we do not pursue this limit further.

### (b) Energy available in cone-based actuation

Consider a cone that has been formed from the natural shape changes induced by illumination. With removal of illumination, the material should return to being a flat disc. If this return is impeded, for instance by a load or a filling fluid, then the stretch energy density is
6.11
Again we adopt *ν*_{el} = 1/2 for simplicity but take a general modulus *E* which would be more characteristic of a glass. We have assumed that the stretch cost incurred in the cap region on not returning to the flat state is comparable to that if a cap had not been formed, a slight over-estimate since the cap region had already incurred some stretch energy in its formation. We can see that this energy is very large. If a similar piece of material were to be actuated in a purely bend mode with no Gaussian curvature unlike in the cone case, then the stored (bend) energy when a load impedes straightening on the removal of illumination would be for *R*_{c}∼*R*_{1}, in other words for displacements of the same order as the system size as we have in cones. The ratio of the actuation energies available for stretch and bend mechanisms is ∼(*R*_{1}/*h*_{0})^{2}, as one would expect.

The same analysis shows why the cost of the cap stretch plus cone bend in cone formation is not an impediment to this type of strong actuation. The energy (6.7) is what is incurred if there is change of natural dimensions but the cone shape is not accessible. Thus, the ratio of the stretch energy thereby incurred to the cap/bend energy that is avoided by staying flat is ∼(*R*_{1}/*h*_{0})^{2}, and cones are formed unless the thermal strain is very small (see above for this limit).

### (c) Energy in other types of distorting, nematic-based membranes

Frank & Kardar (2008) have investigated theoretically out-of-plane distortions about director defects in nematic *liquid* membranes. The competing influences are Frank director gradient energies competing with surface tension and with bend energy penalties eventually governing the development of very fine-scale bend. Likewise, Uchida (2002) has considered fluid nematic membranes where Frank and curvature elastic energies arise. Our solid sheets with their spontaneous stretches accommodated by shape changes are in the opposite limit. They are driven by higher energy processes—the avoidance of mechanical stretch with high associated elastic moduli in the range of 10^{6}–10^{9} N m^{−2}. The conditions for ignoring these other energy influences in our problem can be easily set: Frank and stretch energy densities are comparable when director variation is over a length scale *ξ* such that *K*/*ξ*^{2}∼*E*(*λ*−1)^{2}, that is where *K*∼10^{−11} N is a Frank constant. The length *ξ* arising from this competition has been termed the nematic penetration depth in studies of textures in nematic elastomers (Verwey *et al.* 1996). For elastomers with *E*∼10^{6} N m^{−2} and *λ*∼2 or for glasses with *E*∼10^{9} N m^{−2} and *λ*∼1.03, the cross-over length scale is *ξ*∼5×10^{−9} m. Length scales typically encountered in mechanical problems with these types of solids for actuation are and (Broer & Mol 1991; Harris *et al.* 2005; Mol *et al.* 2005; van Oosten *et al.* 2007; Serrano-Ramon *et al.* 2008), and Frank effects can be safely neglected. Surface energies for rubber and glassy networks are in the range of *σ*∼3×10^{−2} J m^{−2}, see for instance Johnson *et al.* (1971), which must be multiplied by the fractional area change (*λ*^{1−ν}−1) on a spontaneous shape change. The relevant length scale then arising from elastic and surface energy competition is *σ*(*λ*^{1−ν}−1)/[*E*(*λ*−1)^{2}]∼*σ*/[*E*(*λ*−1)], which is in the range 10^{−9}–10^{−8} m for glasses and rubbers, respectively. Again these effects play no role in our problem.

## 7. Summary and outlook

We have described the development, from flat sheets, of surfaces with Gaussian curvature but without stretch energy by heating or illumination. The mechanism is by the response of topological defects with charge +1. One goal is of stretch (rather than bend) actuation in thin sheets that would arise if the curved structure is then cooled or put in the dark while blocked. Elsewhere (Modes *et al.* 2010) we consider putting cone or anti-cone defects into cantilevers to stiffen and cusp them depending on the orientation of the cantilever with the defect. These strips could possibly be used as a light-activated stirrer, actuator, swimmer or perhaps as thermally sensitive, simple machines.

Even more ambitiously, one may create arbitrarily curved surfaces to achieve stretch-based actuation in thin systems normally otherwise switched into a simple bend (weak actuation). Rigid, disclinated director fields in responsive nematic glasses and elastomers show promise. To exploit them to their fullest, however, will rely on understanding the mechanical response of the *m*≠1 defect, along with how interacting multiple defects influence the resultant strain-mediated shape change. Such understanding would allow blueprinting an arbitrary three-dimensional shape in a flat sheet and switching it on (and off again) at will.

## Acknowledgements

We are grateful for guidance from Professors Broer, Bastiaansen and Sanchez. C.D.M. and M.W. thank the EPSRC for funding under grant EP/E051251/1 and K.B. under EP/F060033/1.

- Received July 6, 2010.
- Accepted September 16, 2010.

- This journal is © 2011 The Royal Society