## Abstract

Nematic elastic bodies can develop a gradient of response to heat, light and other stimuli. They then bend and develop curvature in a complex manner depending on director field distributions, on whether they are monodomain or polydomain structures and on linear or nonlinear light absorptive processes. In each case, we derive the general weak response where bend in each direction is treated independently of that in others. In a subsequent paper, we address the reverse phenomenon, that is of strong spontaneous distortion leading to curvature suppression.

## 1. Introduction

Nematic solids, both glassy and elastomeric, offer large, anisotropic, spontaneous mechanical deformation in response to temperature change, illumination or solvent. This makes them potential actuators relying upon stimuli different from conventional actuation. Typically, one achieves contraction along and elongation perpendicular to the director on heating or illumination and vice versa on cooling or on recovery in the dark. With uniform contraction and elongation, one achieves a lineal actuation. However, bend is sometimes of greater use, for instance in micro-fluidic mixers and pumps where artificial cilia are of interest (van Oosten *et al.* 2009). Bend is achievable with a spatial variation of mechanical response through the thickness of a sheet or cantilever of the active material: (i) If light is the driving field, then it is absorbed by the material and the reduction in intensity from the entry face to deeper into the cantilever gives a gradient of response and hence also bend (Warner & Mahadevan 2004; Jin *et al.* 2006, 2007). Likewise, solvent concentration in the material resulting from exposure to vapour at one face, or a gradient in temperature through the cantilever, will cause bend. All these gradient-of-stimulus methods will work for uniform director conformations. Considerable additional mechanical control can be achieved by using a polydomain rather than a uniform director nematic solid and exploiting the specificity of response to the polarization of light (Yu *et al.* 2003; Corbett & Warner 2006). Strains can be large, up to hundreds of per cent, but even when strains are only a few per cent, deformations can still be enormous (figure 1). (ii) Non-uniform liquid crystal director conformation in nematic solids allows a spatially varying mechanical response (since response depends on the director orientation). Such response results even in the presence of uniform stimulus fields, for instance heating, or with light that is not appreciably decaying with penetration. Enormous control of director fields is possible, through surface anchoring, electric and magnetic fields and holography. Thus by setting up the desired director configurations, then cross-linking the progenitor liquid crystal to form a solid, one can obtain more subtle actuation effects than by simply relying on gradients of the stimulation fields.

There are additional ways one can achieve out-of-plane displacement, namely by having strain spatially varying in-plane. A highly interesting case is that of strain non-uniform in plane (but uniform through the thickness), for instance in plastically stretched solids or in botanical systems (Dervaux & Amar 2008; Müller *et al.* 2008; Liang & Mahadevan 2009), or for nematic solids responsive to heat or light as in this paper, but where there is in-plane director variation (Modes *et al.* 2010), for instance that of a topological defect. In these cases, curvature is achieved (flat surfaces even converted to ones with a Gaussian curvature), but it is not suppressed and we do not treat it here. Another possibility is where strain is localized, for instance the photo-response to a localized light beam falling on a solid sheet, which leads to surface topography change (Warner & Mahadevan 2004; Wei & He 2006; Chen & He 2008). Here, we are concerned rather with bend and curvature of the elastic body as a whole.

To make a robust solid from a liquid crystal, one cross-links the polymer liquid crystal melt or solution to form a stable network, which is still liquid crystalline in its ordering. At low cross-link density, one achieves a highly extensible elastomeric network where the chains are still very mobile and the director is still able to move relatively freely with respect to the solid matrix of the rubber. At low temperatures, these elastomeric solids become glassy by virtue of the usual mechanism of freezing in chain motions, rather than by any localizing effect of interchain cross-linkage. At high cross-link density, one achieves a high-modulus solid (with modulus at least 1 GPa, rather than the MPa and lower moduli typical of rubber). Chain motion is highly limited by the cross-links so that one can think of these materials as in effect nematic glasses throughout their range, even though they will also have a lower temperature transition to a conventional glass phase associated with an additional classical, temperature-inspired freezing-in of chain motions. Such nematic glasses are characterized by a Poisson ratio less than 1/2 (since the shear modulus is high—comparable to the bulk modulus), by the director not being independently mobile from the elastic matrix as it is in elastomers and by their (anisotropic) spontaneous elongations and contractions with temperature change and illumination being of a few per cent rather than the tens of per cent to hundreds of per cent of nematic elastomers. We are primarily concerned with nematic glasses in this paper and thus the director distribution changes only through convection by mechanical deformations. Strains are small, but deformations still large.

In a related paper (Warner *et al.* 2010), hereafter denoted by WMC-II, we address the converse problem that also arises in classical elasticity for imposed deformations: with large spontaneous deformations, because of Poisson effects or from spontaneous deformations existing in more than one dimension, there is the Gaussian curvature. As a result, stretches arise that can be large if the body is large—curvature in one direction means that parts of the body can be far removed from the neutral plane associated with curvature in the other direction. Bodies then react by flattening in one direction. Such suppression is determined by the underlying curvatures calculated in this paper and will develop differently in the varied cases we analyse. Here, we set up the general form for weak bending (§2) and then examine (§3) how bend develops in cases of experimental interest and new possibilities we speculate about.

## 2. Weak spontaneous bending

Having achieved a gradient in natural length by either a gradient of stimulus or a gradient of the director field, a sheet of material will bend, perhaps in more than one dimension, in order to minimize the elastic energy penalties associated with deviations from these induced changes in natural length. We analyse here the changes when spontaneous distortion is small. The spontaneous or natural strain (denoted by ^{s}), , is that which would be attendant on a body deforming as if uniformly exposed to the conditions currently pertaining at depth *z*, see figure 2*a*,*b*. It is locally the change in the natural shape away from which there is then an elastic energy cost if there is further distortion. The effective strain is the geometric strain minus the spontaneous strain:
2.1
a simple subtraction since strains are small. We shall need the geometric strain in the material mid-plane (initially at *z*=0) and denote it by . The geometric strain has a curvature-induced part, *z*/*R*, where *z* is the distance from the mid-plane and 1/*R* is a curvature. The spontaneous (thermal, optical, solvent-induced) strain is diagonal in a principal frame determined by the director , for instance with elements *ϵ*_{⊥} in the *zz*- and *xx*-directions and *ϵ*_{∥} in the *yy*-direction for a specific case, where is aligned along the *y*-axis. In a frame-independent form, this part of the strain is
2.2
Values of order −1 per cent for *ϵ*_{∥} and +0.3 to 2 per cent for *ϵ*_{⊥} are typical (Mol *et al.* 2005) for heating or illumination. We introduce a thermal (or optical) Poisson ratio, *ν*_{th}. Analogous to the elastic Poisson ratio, in our locally uniaxial solids, it relates *ϵ*_{⊥} in the perpendicular directions to *ϵ*_{∥} along the director: *ϵ*_{⊥}=−*ν*_{th}*ϵ*_{∥}, where *ν*_{th}∼+0.3 to 2 would be typical for nematic glasses (Mol *et al.* 2005).

For bending, the relevant in-plane elements of are as follows:
2.3
*R*_{xx} and *R*_{yy} are the radii of curvature in the *xx*- and *yy*-directions and *e*_{xx} and *e*_{yy} are the mid-plane *xx*- and *yy*-strains in the appropriate sections of the cantilever (see equation (2.1) and figure 2).

For not along a principal axis of cantilever bend, then in equation (2.2) has off-diagonal response, for instance in the case of a cantilever with a twist director field treated specifically below, and which has similarities to the spontaneous distortions of a cholesteric elastomer (Warner 2003). We show in twist cantilevers that it is the response along the principal curvature directions that is important.

We analyse bend in the approximation of isotropic elasticity, rather than that of a uniaxial solid defined by the local nematic director field. Elsewhere, we show how a full analysis proceeds with only non-essential complications that arise from the fourth-rank nature of the elasticity tensor. The linear stress–strain relations give
2.4
2.5
where *E* is Young’s modulus and *ν* is the Poisson ratio. We make the usual assumption that the cantilever is sufficiently thin that *σ*_{zz} is constant through the thickness. Furthermore, if the top and bottom of the cantilever are free surfaces, then *σ*_{zz} vanishes on the boundaries in *z* and thus *σ*_{zz}(*z*) is identically zero (as are also *σ*_{xz} and *σ*_{yz} by the same argument).

For spontaneous rather than the usual imposed deformations, we have bend with no net force or torque acting through any section of the cantilever, in particular through the *xx*- and *yy*-sections (Warner & Mahadevan 2004; Corbett & Warner 2007; van Oosten *et al.* 2007; Jin *et al.* 2010):
2.6
where *σ* is *σ*_{xx} or *σ*_{yy} and the thickness is 2*h*. The four conditions arising from equation (2.6) are as follows:
2.7
2.8
2.9and
2.10
where a bar denotes an average . The former two equations depend on first moments of , the latter two on average spontaneous strains. In pairs, these equations yield the following:
2.11
and thus the curvatures and mid-plane distortions are functionals of the choice of . The superscript ^{o} on the radii of curvature denotes these small strain forms generated by . The relative signs and magnitudes of and will depend on the underlying response and also the orientation of , the latter giving rich possibilities, see examples below. The response differs markedly from the response of cantilevers to imposed torques, say imposed *yy*-bending, where 1/*R*_{xx}=−*ν*/*R*_{yy} is required by Poisson and the two directions are completely coupled. The apparent independence of the two directions in this case is unfamiliar because there are no external torques. When curvature gives rise to stretch, the small strain results will be radically altered (WMC-II), but the results are still only functions of this underlying response (2.11). Classical analysis (Timoshenko 1925) of curvature of bimetallic strips estimates this weak bend analysis to be valid for deflections less than about half the cantilever thickness, an estimate we return to (WMC-II).

## 3. Possibilities of geometry and response mechanisms for cantilever bend

We now examine geometry and response mechanisms for the induction of bend by external agents.

The strategy of director variation to obtain bend was introduced by Broer *et al.* in thermal (Mol *et al.* 2005) and in optical (van Oosten *et al.* 2007) cases. We show that no anti-clastic response arises for splay–bend, and maximal anti-clasticity arises for twist variation. Curling also results (Harris *et al.* 2005) when the phase of the twist is adjusted. A bi-glass, a nematic cantilever bonded to a non-nematic one, see figure 1*b*, is analogous to a bi-metallic strip and is easily analysed.

Bend from the spatial variation of stimulus is most directly accessed with light. Photo-elasticity and its application to the systems of this paper has been recently reviewed (Corbett & Warner 2009). There are two cases of absorption profiles—Beer (weak absorption; exponential profiles) and non-Beer (photo-bleaching leading to initially linear and then finally exponential variation of intensity with penetration (Corbett & Warner 2007)). Each case is explicitly shown to have quite different weak curvatures. In either case, the curvature is non-monotonic in the cantilever thickness in relation to the underlying (Beer) absorption length (Warner & Mahadevan 2004; Jin *et al.* 2006); very deep and very shallow penetration give little bend and for some intermediate penetration the curvature is maximal. It is in fact possible to observe saddles in elastomer analogues of the nematic glasses we have considered, for instance in light-driven ‘swimmers’ Camacho-Lopez *et al.* (2004)—flexing in and out of a saddle shape drives the swimming action.

Our examples are mostly of anti-clastic response, but we show some cases where one would expect a syn-clastic response where the curvatures are of the same sign and hence spherical caps are formed in the weak case. We suggest (Corbett & Warner 2008) a syn-clastic response when the director is normal to the cantilever or when the cantilever is made of a polydomain nematic and is illuminated by unpolarized light. In general, the mechanics of polydomain nematics is determined by the polarization of light (Yu *et al.* 2003; Corbett & Warner 2006; Harvey & Terentjev 2007; White *et al.* 2009) and subtle control is thereby afforded.

Other stimuli can be employed as well. For instance, gradients of solvent vapour cause bend, as do those of temperature (Hon *et al.* 2008), which is discussed below. Heat release is also important in photo-effects (Hogan *et al.* 2002; Jiang *et al.* 2009), but the dependence of polydomain mechanics on light polarization proves that effects are dominated by optical response, and we limit ourselves to either light, or purely thermal effects.

### (a) Director gradient cantilevers

The simplest way to get an asymmetric response is to rotate the director by 90^{°} on going from the bottom to the top surface of the cantilever, see figure 3, either by a splay–bend or by a twist conformation.

#### (i) Splay–bend

The splay–bend configuration (figure 3*a*) gives a greater response than that of twist. One does not need a gradient of stimulation to get response—heating or cooling give bend (of opposite signs) and illumination will give bend in the same sense irrespective of which side it is incident from.

For splay–bend, the director rotates in the *yz*-plane from parallel to the bottom boundary (*z*=−*h*) through to normal on the top (*z*=*h*). As an example, take the angle *θ*(*z*) the director makes with the *y*-axis to be
3.1
The zeroth and first moments of are easily evaluated and give
3.2
For this geometry, there is no asymmetry of the *xx*-thermal/optical strains ( for all *z*) and hence no drive to bend in the *xz*-plane, that is . No saddle is created and no suppression needs to take place in the strong limit. Accordingly, this must be the most effective geometry to choose since suppression costs, via elastic Poisson effects, cause a reduction in the principal, *yy*, curvature. Figure 4 shows both clamped and unclamped splay–bend cantilevers bending in response to temperature changes (Mol *et al.* 2005). Bend can be seen to be eventually strong and saddle response is not prominent in the photographs.

Variants of this choice of director field can be taken. For instance, the director could execute splay–bend in the *zx*-plane (we take the *x*-direction to be shorter generically than the *y*). Curvature would then be purely *xx* instead of *yy*. Since there is no saddle development, this interchange can be made without complication even though *L*>*b*, which (WMC-II) can preferentially select out *y* for bending and *x* for suppression in the strong case with saddles and their suppression. Equally, one could take a director variation over a range either less than or greater than *π*/2. One can see that the asymmetry about *z*=0 is only thereby reduced and so too is the bend response. Also one could change the phase of the variation while keeping the range at *π*/2. This too reduces the response. A critical example would be adopting the variation
where the director splay-bends from −*π*/4 to *π*/4. Asymmetry about *z*=0 is lost altogether and no curvature is induced.

#### (ii) Twist

The second possibility to rotate the director through the cantilever is the twist configuration (figure 3*b*), explored thermally (Mol *et al.* 2005) and optically (van Oosten *et al.* 2007) by the Broer group. As before, one does not need a gradient of stimulation to get response. The director twists uniformly in the *xy*-plane from being parallel to *y* at the bottom boundary to parallel to *x* on the top. The angle *ϕ*(*z*) the director makes with the *y*-axis is given as for *θ* in equation (3.1). The spontaneous strain, equation (2.2), now has off-diagonal components:
3.3
The and components contribute as usual as in equation (2.11) for the weak curvatures and mid-plane strains.

The zeroth and first moments of are, on evaluating etc.,
3.4
For this geometry, there is maximal asymmetry also of the *xx*-thermal/optical strains and the equal and opposite drive to bend in the *xz*-plane as in the *yz*-plane, that is . Indeed the *x*- and *y*-directions are equivalent, and maximal saddle (anti-clastic) response is created. Figure 1 shows a twisted nematic cantilever bending in response to UV illumination (van Oosten *et al.* 2007).

Note that we do not allow extensions in directions that rotate on traversing the thickness of the sample since the shears resulting from compatibility would give energy densities diverging as (*ϵ*_{∥}−*ϵ*_{⊥})^{2}(*b*/*h*)^{2}≫1. To see this, consider two material points at (*x*=*l*,*y*=0,*z*=0) and (*x*=*l*,*y*=0,*z*=*δz*), where without loss of generality let be along , and where *l* is a characteristic extent of the body. Applying of equation (2.2) to each of these points with rotated by *δϕ* for *z*=*δz* would convert them to (*l*(1+*ϵ*_{∥}),0,0) and (*l*(1+*ϵ*_{∥})+*O*(*δϕ*)^{2},*l*(*ϵ*_{∥}−*ϵ*_{⊥})*δϕ*+*O*(*δϕ*)^{2},*δz*(1+*ϵ*_{⊥})). The shear gives the energy density just cited. Hence, such *ϵ*_{iz} shears are suppressed. The suppression of spontaneous extensions along the director (the local principle material frame), that is the suppression of and in the ribbon frame, must generate shear stresses *σ*_{xy} and *σ*_{yx} that are, as usual, equal. They accordingly generate no net *z*-torque, as in the spontaneous distortions of a cholesteric slab (Warner 2003).

Appending an extra, additive constant angle, *ϕ*_{o}, to *ϕ* allows the orientation of the twisted-nematic texture relative to the cantilever to change since now the director at the lower surface is offset from *y*. The result is principal curvatures now also offset in direction from *y* and *x*. A cantilever will now appear as a twisted ribbon.

#### (iii) Bi-glass cantilevers

The simplest nematic cantilevers are perhaps the nematic glass analogues of bimetallic strips (Timoshenko 1925). A nematic glass with the director along *y* in the thickness interval (*h*_{1},*h*) is bonded to a conventional glass, where we ignore the thermal or optical response, occupying the remainder (−*h*,*h*_{1}) of the cantilever thickness. Evaluating the appropriate moments of , one obtains
3.5
Curvature can be extreme and suppression is certainly very important; see figure 1*b* which is in fact a bi-rubber (from Prof. E. M. Terentjev) but where the principles are similar to a putative bi-glass.

### (b) Stimulus gradient inducing bend in cantilevers

Bending behaviour may also be induced in both simple monodomains and in polydomains if there is spatial variation of the stimulating field through the material. In optical stimulation, the gradient comes about via absorption by the very dye molecules one is exciting for the mechanical response. Absorption is either linear (Beer–Lambert) or, more likely, nonlinear.

#### (i) Weak optical absorption

One convenient way to achieve a variation of stimulus is by normal illumination of one surface (at *z*=*h*) with light, so that the intensity, *I*, decays inside the sample as the material absorbs the light weakly according to Beer’s law: *I*=*I*_{0}*e*^{(z−h)/d}, where *d* is the Beer absorption length scale inherent to the material. Strain arises from illumination if suitable chromophores (dye molecules) are present in the solid. Some dyes bend at the molecular scale (photoisomerize) on absorbing a photon. This transition from a straight ground state (*trans*) to a bent excited state (*cis*) is reversible (thermally or by optically stimulated decay) and disrupts molecular packing—both increases of volume and reductions of the nematic order are possible; see a recent review (Corbett & Warner 2009) of photo-elasticity. The dye molecules can be either guests in the network, or chemically bonded to it. We make the simplest assumption, namely that the spontaneous strains follow the *cis* fraction of the chromophores present which, in the limit of weak absorption, is proportional to the intensity. Thus, contraction along the director is proportional to the optical intensity in this limit. When intensity varies through the thickness of the solid, so will the spontaneous strain and thus bend is induced. Consider, for example, a cantilever illuminated in the Beer regime in which the director is everywhere along the long direction, *y*, of the material. The resultant spontaneous strains are and , where *ϵ*_{∥} and *ϵ*_{⊥} here relate the reduced optical intensity *I*_{0}/*I*_{c} to parallel and perpendicular optical strain, respectively, and *I*_{c} is a material constant discussed below. Evaluating the moments as before gives the associated curvatures and mean strains. We now have another length, *d*, in addition to the thickness *h*. Thus, the radius of curvature reduced by *h* is now a function of both *ϵ*_{∥} and *ϵ*_{⊥}, and of the reduced thickness :
3.6
where *ν*_{op} is the optical equivalent of the thermal Poisson ratio. The curvature is non-monotonic in (Warner & Mahadevan 2004; Jin *et al.* 2006). Penetration *d* much less than the thickness *h* means that insufficient volume of the solid has a natural contraction compared with the volume that is essentially un-irradiated and does not want to change length; thus bend vanishes as . Conversely, when penetration is very deep, *d*/*h*≫1, then there are weak gradients in response and thus little bend, only some uniform contraction. Optimal bend in this model arises when *d*/*h*∼1/3. There are two neutral planes in this limit of weak spontaneous bend. Note that rotating the director field and the polarization of the light in the plane of the cantilever simply rotates the directions of the principal curvatures relative to the material. A vivid example of curvature and saddle formation can be found in elastomers (Camacho-Lopez *et al.* 2004), see figure 2*c*, systems that bend in tens of milliseconds to light and which swim by virtue of their flexing into saddles while resting on the surface of water.

#### (ii) Strong absorption

If, on the other hand, the incident light intensity *I*_{0} is high, then the *cis* (excited) population becomes high and the material becomes bleached, that is the absorbing *trans* population is depleted. The material’s ability to absorb light is reduced and the intensity profile through the thickness changes from exponential to linear, at least until at depths where the beam is sufficiently attenuated that Beer’s law decay is restored. For nematic photo-glasses with high dye loadings (usually the case), the Beer length *d* used above is as little as 60 nm. For nematic solids that may be hundreds of micrometres thick, the observed bending means that a layer of material much thicker than a Beer length is responding and, therefore, the intensity profile must extend much more deeply than given by Beer’s law. This was the initial motivation for considering strong absorption (Corbett & Warner 2007).

For a given material, the combination of the forward cross-section for absorption of light to give *trans* *cis* with the natural back-reaction rate (with which it is in competition) gives the characteristic intensity, *I*_{c}, used above. One can compare the incident intensity with this intensity, *α*=*I*_{0}/*I*_{c}, to quantify the notion of strong (*α*≫1) and weak (*α*≪1) beams. For beams incident with general reduced incident intensity *α*, the profile ℐ(*z*)=*I*(*z*)/*I*_{c} takes the form of the Lambert-*W* function *W*_{α}(*h*−*z*), which is linear until *h*−*z*∼*αd*, that is ℐ(*z*)≃1−(*h*−*z*)/(*αd*), and thereafter is Beer-like. The *cis* fraction of molecules, *n*_{c}, is now *n*_{c}=*α*ℐ/(1+*α*ℐ) and in general requires the *W* function. The spontaneous strains generated in a cantilever with a uniform director along the *y*-direction will be of the form and . In the extreme non-Beer regime, we take the intensity through the thickness to vary linearly with depth, that is we require 2*h*<*αd*. The mean strain and curvatures can easily be calculated and are as follows:
3.7
The limit of very large *α*≫1 gives a fraction *n*_{c}∼1 of the *cis* form of the dyes and hence a saturated response at the values *ϵ*_{∥} and *ϵ*_{⊥} essentially uniformly through the thickness. One expects these to be the values of the mid-plane strains, which is indeed the case: *e*_{yy}∼*ϵ*_{∥}(1−1/(1+*α*)). With nearly uniform optical strains, one expects the curvature to vanish and indeed , or equivalently *d*/*R*^{o}_{∥}∼*ϵ*_{∥}/(1+*α*)^{2}, that is independent of the thickness since we have a linear variation with a characteristic length *d* through the thickness. Now, there can be either two or three neutral surfaces.

In both light absorption cases, we have ignored a small source of nonlinearity. As there is an in-plane contraction, so the sample becomes thicker. The decay owing to propagation in the *z*-direction of an absorbing medium becomes greater as a result. We have neglected this effect as being a small modification of the exponential variation or a renormalization of the linear variation in the strong case.

#### (iii) Syn-clastic response

If is aligned along the film normal, , one can envisage normal, unpolarized (or circularly polarized) illumination so that , say. Circular symmetry is preserved. If there is a gradient of optical intensity through the thickness, and hence also of *ϵ*^{s}*n*_{c}, then curvature will result. However, it will be the same in each direction, that is , say (and *e*_{yy}=*e*_{xx}). Thus, spherical (syn-clastic) curvature, that is positive Gaussian curvature, develops. The curvature results of equations (3.6) and (3.7) for weak and strong absorption can be taken over with *ϵ*_{∥} replaced by *ϵ*^{s}. The sign of 1/*R*^{o} is not obvious. Normally, incident light has its electric vector in the plane and hence rods that are outliers in the nematic distribution of molecules about are preferentially bent. The bending of this part of the rod population may well cause in-plane contractions, more pronounced where the light is more intense, and hence curvature upwards towards the direction of beam incidence. On the other hand, illumination might have the more usual, opposite effect of inducing lower order parameters and thus in-plane elongations leading to downward curvature. We believe this latter response to be more likely.

#### (iv) Polydomain photo-solid response

A polydomain elastomer does not respond mechanically to heat since each domain would prefer to contract in a random direction and overall response is reduced. By contrast, light on polydomains offers great subtlety in response (Yu *et al.* 2003; Harvey & Terentjev 2007; White *et al.* 2009). Domains are selected for response according to their orientation with respect to the optical electric vector , while others misaligned with are initially inert and deform sympathetically to allow at least a partial response of those domains that are effectively irradiated (Corbett & Warner 2007). Overall, the photo-induced strain is contractile along and of uniaxial symmetry—this polydomain mechanics is dictated by the optical polarization (but see below) and changes as is changed. The response should be non-monotonic with increasing intensity since eventually at high enough intensity even domains significantly misaligned with also suffer molecular photoisomerization (as in §3*a*(iii) where and are actually perpendicular). With all domains irradiated, one returns to null response as in the case of heating. The curvature and mid-plane strains are as in the weak and strong cases §3*a*(i) and (ii) above where the *y*-direction of principal response is now established by rather than .

Another possibility has been theoretically envisaged (Corbett & Warner 2008) where circularly or unpolarized light is normally incident on a polydomain sample. Now all domains in the sample plane are most directly irradiated to give in-plane contraction and overall there is likely to be elongation along , the propagation direction. There is again uniaxial response, but along rather than along . The in-plane contractions are weaker at greater depth, thus there should be upward syn-clastic curvature of magnitude given by equations (3.6) and (3.7) for weak and strong absorption.

#### (v) Curvature from thermal gradients

A uniform director sample (, say) will contract along *y* and not bend on heating. However, if heating produces a temperature gradient along *z*, then the contraction is greater nearer the hotter surface and there is bend towards that surface. The possibility has been examined theoretically and experimentally in nematic elastomers (Hon *et al.* 2008), which have certain similarities with nematic glasses in this regard. For a linear temperature gradient, and in a temperature regime where the variation of length along with temperature is also linear, then anti-clastic bends will be induced as in the extreme strong absorption case of §3*a*(ii) (*α*≫1). When elongation is not linear with temperature, then the bend response is more complicated, much as in the weak and strong absorption cases for optical bend. An additional complication indirectly associated with thermal effects is the release of heat during the recovery photo-excited states. This has been recognized experimentally (Hogan *et al.* 2002) and theoretically (Jiang *et al.* 2009). Syn-clastic response would be expected with thermal gradients if were normal to the cantilever, that is along *z*.

## 4. Summary and conclusions

We have shown that asymmetry in spontaneous strains about the mid-plane of a cantilever causes bend, equation (2.11), which depends on the first moment of the spontaneous strains, and . Inhomogeneous responses arise either from director distributions or from gradients in stimulation (temperature, light, solvent, etc.). We evaluated the first moments to give the weak curvatures in each of the principal cases, including some that are suggestions for future experimental investigation, for instance syn-clastic response.

Bend is important in applications of light and heat-driven cantilevers, particularly in micromechanics. For example, such cantilevers are used in microfluidic mixers. In an associated paper, we investigate the converse phenomenon, namely the suppression of such bend owing to nonlinear coupling between perpendicular bending modes arising from geometrically inspired stretch (WMC-II). In practice, such suppression is important in applications of these complex cantilevers.

## Acknowledgements

We are grateful for guidance from Professors Broer and Bastiaansen and Dr van Oosten, and for permission to use their illustrations in this paper, and also to Professors P. Palffy-Muhoray and E.M. Terentjev for the use of figures. D.C. thanks the EPSRC for funding under grant EP/F013787/1 and M.W. and C.D.M. under EP/E051251/1.

- Received March 5, 2010.
- Accepted March 29, 2010.

- © 2010 The Royal Society