Smectic-C tilt under shear in Smectic-A elastomers
aa r X i v : . [ c ond - m a t . s o f t ] A ug Smectic- C tilt under shear in smectic- A elastomers Olaf Stenull and T. C. Lubensky
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
J. M. Adams and Mark Warner
Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom (Dated: October 30, 2018)Stenull and Lubensky [Phys. Rev. E , 011706 (2007)] have argued that shear strain and tilt ofthe director relative to the layer normal are coupled in smectic elastomers and that the impositionof one necessarily leads to the development of the other. This means, in particular, that a Smectic- A elastomer subjected to a simple shear will develop Smectic- C -like tilt of the director. Recently,Kramer and Finkelmann [arXiv:0708.2024, Phys. Rev. E , 021704 (2008)] performed shear experi-ments on Smectic- A elastomers using two different shear geometries. One of the experiments, whichimplements simple shear, produces clear evidence for the development of Smectic- C -like tilt. Here,we generalize a model for smectic elastomers introduced by Adams and Warner [Phys. Rev. E ,021708 (2005)] and use it to study the magnitude of Sm C -like tilt under shear for the two geometriesinvestigated by Kramer and Finkelmann. Using reasonable estimates of model parameters, we esti-mate the tilt angle for both geometries, and we compare our estimates to the experimental results.The other shear geometry is problematic since it introduces additional in-plane compressions in asheet-like sample, thus inducing instabilities that we discuss. PACS numbers: 83.80.Va, 61.30.-v, 42.70.Df
I. INTRODUCTION
Smectic elastomers [1] are rubbery materials with theorientational properties of smectic liquid crystals [2].They possess a plane-like, lamellar modulation of den-sity in one direction. In the smectic- A (Sm A ) phase, theFrank director n describing the average orientation ofconstituent mesogens is parallel to the normal k of thesmectic layers whereas in the smectic- C (Sm C ) phase,there is a non-zero tilt-angle Θ between n and k .Recently, there has been some controversy aboutwhether shear strain and tilt of the director relativeto the layer normal are coupled in smectic elastomersand whether the imposition of one necessarily leads tothe development of the other. Beautiful experiments byNishikawa and Finkelmann [3], where a Sm A elastomerwas subjected to extensional strain along the layer nor-mal, found a drastic decrease in Young’s modulus at athreshold strain of about 3% accompanied by a rota-tion of k through an angle ϕ that sets in at the samethreshold. Interpreting their x-ray data, the authors con-cluded that there was no Sm C -like order over the rangeof strains they probed, and that the reduction in Young’smodulus stems from a partial breakdown of smectic lay-ering. Recently, Adams and Warner (AW) [4] developeda model for Sm A elastomers which assumes that n and k are rigidly locked such that Θ = 0. This model producesa stress-strain curve and a curve for the rotation angle ϕ of k in full agreement with the experimental curvesbut without needing to invoke a breakdown of smecticlayering. Evidently, because of the assumption Θ = 0,the AW model predicts no Sm C -like order. More re-cently, Stenull and Lubensky [5] argued that shear strainand Sm C -like order are coupled and that the imposition of one inevitably leads to the development of the other.They developed a model based on Lagrangian elastic-ity that predicts, as does the AW model, a stress-straincurve and a curve for ϕ in full agreement with the experi-ment. In contrast to the interpretation of Nishikawa andFinkelmann of their data and to the central assumptionof the AW model, Ref. [5] found that the tilt angle ϕ n of n is not identical to that of the layer normal, ϕ , imply-ing that there is Sm C -like order with non-zero Θ abovethe threshold strain. However, estimates of ϕ and ϕ n based on reasonable assumptions guided by the availableexperimental data of the layer normal-director coupling,turn out the same order of magnitude. The upshot isthat Ref. [5] predicts Sm C -like tilt above the thresholdstrain but that the angle Θ is small. It is entirely possiblethat Θ is smaller than the resolution of the experimentsby Nishikawa and Finkelmann. In this case, there is nodisagreement between the predictions of Ref. [5] and theexperimental data by Nishikawa and Finkelmann.The arguments of Ref. [5] imply in particular thata Sm A elastomer subjected to a shear in the planecontaining k will develop Sm C -like tilt of the director.Very recently, Kramer and Finkelmann (KF) [6, 7] per-formed corresponding shear experiments using two differ-ent shear geometries. One geometry, which we refer to astilt geometry, imposes a shear that is accompanied by aneffective compression of the sample along the layer nor-mal. In this geometry, the elastomer ruptures at an im-posed mechanical shear angle φ of 13 deg [7] or 14 deg [6],and up to these values of φ no Sm C -like tilt is detectedwithin the accuracy of the experiments of about 1 deg,as was the case in the earlier stretching experiments ofNishikawa and Finkelmann. The other shear geometry,which we refer to as slider geometry, imposes simple shearand can be used to probe values of φ exceeding 20 deg.For this geometry, the KF x-ray data provides clear evi-dence for the emergence of Sm C -like tilt.In this communication, we generalize the model intro-duced by AW [4] and use it to study the magnitude ofSm C -like tilt under shear for the two geometries inves-tigated by KF. Using reasonable estimates of model pa-rameters, we estimate the tilt angle Θ, and we compareour estimates to the experimental results.The outline of the remainder of this communication isas follows. In Sec. II we develop our model, which gener-alizes the original AW model. In Sec. III we describe thetwo experimental setups that we consider. We discussthe respective deformation tensors for these setups andcalculate for both setups the Sm C -tilt angle Θ as a func-tion of the imposed mechanical tilt angle φ . We addresswhy the tilter apparatus is problematic for experimentswhere shear induces director tilt. In Sec. IV we discussour findings and we make some concluding comments.There are two appendices. In App. A, we comment onanalytical calculations of Θ for small φ . In App. B, webriefly discuss the Euler instability in the context of theexperiments by KF. II. GENERALIZING THE ADAMS-WARNERMODEL
In this section we devise a theory for smectic elas-tomers based on the neo-classical approach, developedoriginally by Warner and Terentjev and coworkers [1, 8]for nematic elastomers, and the subsequent extension ofthe neo-classical approach by AW to smectics. The neo-classical approach generalizes the classical theory of rub-ber elasticity [9] to include the effects of orientationalanisotropy on the random walks of constituent polymerlinks. It treats large strains with the same ease as theyare treated in rubbers in the absence of orientational or-der, it provides direct estimates of the magnitudes ofelastic energies, it is characterized by a small number ofparameters, and it accounts easily for incompressibility.In the neo-classical approach, one formulates elasticenergy densities in terms of the Cauchy deformation ten-sor Λ, defined by Λ ij = ∂R i /∂x j , where R ( x ) is thetarget space vector that measures the position in the de-formed medium of a mass point that was at position x in the undeformed reference medium. In this approach,a generic model elastic energy density for smectic elas-tomers that allows for a relative tilt between the directorand the layer normal can be written in the form f = f trace + f layer + f tilt + f semi . (2.1)Here and in the following, incompressibility of the mate-rial is assumed, i.e., the deformation tensor is subject tothe constraint det Λ = 1. f trace is the usual trace formulaof the neo-classical model with µ the shear modulus, f trace = µ Tr h Λ ℓ Λ T ℓ − i , (2.2) where ℓ = δ + ( r − n n , with n describing theuniaxial direction before deformation, is the so-calledshape tensor describing the distribution of conforma-tions of polymeric chains before deformation and ℓ − = δ − (1 − r − ) nn is the inverse shape tensor after defor-mation. δ denotes the unit matrix, and r denotes theanisotropy ratio of the uniaxial Sm A state. The contri-bution f layer = B "(cid:18) dd cos Θ (cid:19) − (2.3)describes changes in the spacing of smectic layers withlayer normal k = Λ − T k | Λ − T k | , (2.4)where k = n is the layer normal before deformationand B the layer compression modulus [10]. d and d are,respectively, the layer spacing before and after deforma-tion which are related via dd = 1 (cid:12)(cid:12) Λ − T k (cid:12)(cid:12) . (2.5)The tilt energy density, f tilt = a t sin Θ , (2.6)incorporates into the model the preference for the direc-tor to be parallel to the layer normal in the Sm A phase.To study the Sm C phase, we would have to include a termproportional to sin Θ, which, however, is inconsequen-tial for our current purposes. Without the contribution f semi , the model elastic energy density (2.1) is invariantwith respect to simultaneous rotations of the smectic lay-ers, the nematic director, and Λ in the target space. Tobreak this unphysical invariance, we include the semisoftterm [1] f semi = µα Tr[( δ − n n )Λ T nn Λ] , (2.7)where α is a dimensionless parameter.The relation of the model presented here to the AWmodel is the following: AW assume that the layer normaland the director are rigidly locked such that the angle Θ isconstrained to zero. Moreover, the semi-soft term f semi isabsent in the AW model. Essentially, we retrieve the AWmodel from Eq. (2.1) by setting Θ = 0 (or equivalently a t → ∞ ) and α = 0.As mentioned above, one of the virtues of the neo-classical approach is that it involves only a few param-eters, and for most of these there exist experimental es-timates, which we will now review briefly. The shearmodulus µ for rubbery materials is typically of the orderof 10 − Pa. A typical value for the smectic layercompression modulus in smectic elastomers, as observedin experiments with small strains along the layer normal, x φ z SampleFixed ClampSliding Clamp y FIG. 1: (Color online) Sketch of a slider apparatus where theupper clamp slides on a horizontal bar such that the extensionof the sample in the z -direction remains fixed. is B ∼ Pa, which is greater than the values in liquidsmectics. In previous experiments by the Freiburg group,the anisotropy ratio was approximately r ≈ . a t can be estimated from experiments by Brehmer, Zen-tel, Gieselmann, Germer and Zungenmaier [11] on smec-tic elastomers and by Archer and Dierking [12] on liquidsmectics. The former experiment indicates that a t is ofthe order of 10 Pa at room-temperature, and the latterexperiment produces a room-temperature value of the or-der of 10 Pa. We are not aware of any experimental datathat allows us to estimate α for smectic elastomers reli-ably. For nematic elastomers, it has been estimated fromthe Fredericks effect [13] and from the magnitude of thethreshold to director rotation in response to stretchesapplied perpendicular to the original director [14] that α ≈ .
06 or α ≈ .
1, respectively. For our argumentshere, we adopt the latter value acknowledging that α may be considerably larger in smectics than in nemat-ics. As we will see in the following, our findings do notdepend sensitively on this assumption. For the deforma-tions that we consider, α appears only in the combina-tion ζ = a t + αµ , that is, semi-softness simply adds to thedirector-layer normal coupling. Uncertainty in estimatesfor ζ is expected stem mainly from the spread in esti-mates for a t . We account for this spread by discussingseveral values of ζ or, more precisely, for several valuesof ζ/µ . III. EXPERIMENTAL SHEAR GEOMETRIESAND TILT ANGLES
In this section we apply the model defined in Sec. IIto study the behavior of Sm A elastomers in shear exper-iments. We consider two experimental setups. In thefirst setup, which is perhaps the first that comes to mindfrom a physicist’s viewpoint, a simple shear is applied,i.e., a shear strain in which the externally imposed dis-placements all lie in a single direction. Figure 1 showsa sketch of such an experiment, and we call the appa-ratus sketched in it the slider apparatus. The secondsetup, which is depicted in Fig. 2, is one in which op-posing surfaces of the sample remain essentially parallel φ z xy FIG. 2: (Color online) One of the two tilt apparatuses usedby KF. The photo has been taken from Ref. [6]. but in which the height of the sample (its extension inthe z -direction) decreases upon shearing. Hence, the ap-plied shear is, strictly speaking, not just simple shear. Inthe following we will refer to the this apparatus as a tiltapparatus. A. Slider apparatus
To make our arguments to follow as concrete and sim-ple as possible, we now choose a specific coordinate sys-tem. That is, we choose our z direction along the uniaxialdirection n of the unsheared samples, n = (0 , , x direction as the direction along whichthe sample is clamped, which is perpendicular to n , cf.Fig. 1. With these coordinates, the deformation tensorfor both the slider apparatus and the tilt apparatus is ofthe form Λ = Λ xx xz yy
00 0 Λ zz . (3.1)As can be easily checked, for this type of deformation,the layer normal lies along the z -direction for the un-sheared and for the sheared samples. Thus, the tilt angleΘ between n and k is identical to the tilt angle between n and the z -axis, and we can parametrize the director as n = (sin Θ , , cos Θ) . (3.2)It is worth noting that a deformation tensor of the formshown in Eq. (3.1) leads to a particularly simple ex-pression for the semi-soft contribution to f , f semi =(1 / αµ Λ xx sin Θ, which combines with the tilt energydensity to f tilt + f semi = (1 / ζ (Λ xx ) sin Θ. Thus, asindicated above, our model depends for the experimentalgeometries under consideration on α and a t through asingle effective parameter, viz. ζ (Λ xx ) = a t + Λ xx αµ .From the photos of the experimental samples pro-vided in Ref. [6], see Figs. 1 and 2, it appears as ifΛ xx does not deviate significantly from 1. In the fol-lowing, we set Λ xx = 1 for simplicity [15]. In this event (iv)Sliding (iii)(ii)(i) φ (deg) Θ ( d e g ) FIG. 3: The tilt angle Θ between the layer normal and thedirector as a function of the mechanical tilt angle φ in theslider apparatus for (i) ζ/µ = 0 .
1, (ii) ζ/µ = 1, (iii) ζ/µ =5, and (iv) ζ/µ = 20. The dashed line corresponds to theresolution for Θ in the experiments by KF. ζ (Λ xx ) = ζ ≡ a t + αµ . In the slider apparatus, the ex-tent of the sample in the z -direction is fixed, and henceΛ zz = 1. The incompressibility constraint det Λ = 1 thusmandates that Λ yy = 1. The remaining nonzero compo-nent of the deformation tensor is entirely determined bythe externally imposed shear, Λ xz = tan φ , where φ is themechanical tilt angle of the sample, see Fig. 1. The onlyremaining degree of freedom in the problem is, therefore,the angle Θ.To calculate Θ as a function of φ , we insert the justdiscussed deformation tensor into f and then minimize f over Θ holding φ fixed. For φ small, this can be doneanalytically by expanding f to harmonic order in Θ andby then solving the resulting linear equation of state forequilibrium value of Θ. This type of analysis is presentedin the appendix. To provide reliable predictions for largershears, one has to refrain from expanding in Θ and usenumerical methods instead. To this end, we minimize f numerically assuming, based on what we discussed at theend of Sec. II, that B/µ = 10 and ζ/µ ∈ { . , , , } .For this minimization, we use Mathematica’s FindMini-mum routine. Figure 3 shows the resulting curves for Θas a function of φ . The dashed horizontal line in Fig. 3 isa guide to the eye; it corresponds to the angle resolutionin the KF experiments which was about 1 degree [16].For the slider geometry, the KF data produces clear ev-idence for the development of Sm C -like tilt under simpleshear, and our theoretical estimates agree well with theexperimental data. However, given that thus far only twodata points are available for φ >
0, it cannot be judgedreliably whether our theoretical curve could be fitted toan experimental curve with more data points. It is en-couraging, though, that our curve for ζ/µ = 0 . B. Tilt apparatus
Now we turn to the tilt geometry. The essential differ-ence between the slider and the tilt geometry is that inthe former the height of the sample L z ( φ ) remains con-stant, L z ( φ ) = L z ( d times the number of smectic layers)whereas in an ideal tilter, L z ( φ ) is not constant but ratherdecreases as φ increases, L z ( φ ) = L z cos φ . This differ-ence has far reaching consequences. In the slider, thesample-height remains larger than the modified naturalheight of the sample created by the director tilt, L z cos Θ.In other words, the sample is under effective tension. Inthe tilter, on the other hand, the sample-height L z cos φ can be smaller than L z cos Θ, i.e., the sample can be un-der effective compression. For a sample under effective zz -compression, one has to worry, experimentally andtheoretically, about all sorts of complications. Most no-table are perhaps buckling and wrinkling.The theory of buckling of elastic sheets is well estab-lished [17], and we now briefly comment on buckling inthe context of the tilt geometry. As mentioned above,a sample clamped into the tilt apparatus can develop abuckling instability, if the height of the sample imposedby tilt, L z cos φ , is smaller than the natural height ofthe sample created by the director tilt, L z cos Θ. As wewill see below, our results for Θ as a function of φ implythat cos φ < cos Θ, i.e., buckling is possible. Alterna-tively, this can be seen by calculating the engineeringstress σ eng zz = ∂f /∂ Λ zz , which can be done using the nu-merical approach outline above. σ eng zz is positive for φ > φ > y -direction inthe latter. The angle φ c at which buckling sets in is ex-pected to be comparable to that for the well known EulerStrut instability [17], φ c ≈ φ Euler = arccos " − (cid:18) π L y L z (cid:19) , (3.3)where L y is the thickness of the sample in the y -direction,and where clamped (rather than hinged) boundary condi-tions are assumed. A brief derivation of Eq. (3.3) is givenin App. B. In the experiments of KF, L z = 5 . L y = 0 .
45 mm [16], which leads to φ Euler ≈
15 deg. Theexperimental samples buckle immediately before theyrupture [16] at φ = 13 deg or φ = 14 deg, which is veryclose to our estimate for φ c . The observation that buck-ling occurs immediately before rupturing might suggestthat the former actually triggers the latter.A detailed analysis of sample-wrinkling in the tilt ge-ometry is beyond the scope of the present paper. How-ever, a few comments about wrinkling are in order. Thewavelength of wrinkling is expected to be much shorterthan that of buckling. Thus, wrinkling can be harderto detect by visual inspection of an experimental sample L Lsin χ Lcos χ Lcos -2d χ d ddd χ φ FIG. 4: Schematic of shear and compression arising in a non-ideal tilter. than buckling, and there is a risk that it remains un-noticed. The main problem is, however, that wrinklingcan act as to effectively reduce the mechanical shear inthe sample. When designing a tilt experiment, one thushas be very careful to avoid wrinkling. If not, the effectsof wrinkling can bias the data, and there is the risk tounderestimate the Sm C -tilt significantly.Our calculation of the the Sm C -tilt will be for an idealtilter. Figure 4 shows the tilter of Fig. 2 schematicallyin a non-tilted and a tilted configuration to make clearthat shear and compression are complex in the non-idealcase because the frame-axles allowing angle change areoff-set from the corners of the sample. Using elementarytrigonometry, one can deduce for the shears and the angle φ λ xz = LL − d sin χ , λ zz = L cos χ − dL − d , (3.4)tan φ = tan χ − (2 d/L ) sec χ , (3.5)In particular, the relation between the shear angle φ andthe deformation components λ zz and λ xz is not simple.Returning to an ideal tilter, d = 0, χ ≡ φ , we calcu-late the Sm C -tilt as a function of the applied mechanicalshear, suppressing buckling and wrinkling. Then, thedeformation tensor is of the same form (3.1) as for theslider apparatus, but with the essential difference thathere Λ xz = sin φ and Λ zz = cos φ . As we did for the sliderapparatus, we assume Λ xx = 1. The incompressibilityconstraint then implies that Λ yy = 1 / cos φ , leaving theangle Θ as the only degree of freedom. We calculate Θ asa function of φ in exactly the same way as above, i.e., weminimize f numerically assuming the values of the modelparameters discussed in Sec. II. The resulting curves areshown in Fig. 5. The dip in Fig. 5 occurs because of acompetition between ζ which would prefer Θ = 0 and B which would prefer Θ = φ . If ζ > B then it is possible toget curves where Θ stays close to 0. The dashed horizon-tal line in Fig. 5 indicates the angle resolution of about1 degree of the KF experiments [16].As mentioned in Sec. III A, the data points for theslider are compatible with ζ/µ ≈ .
1. The elastomersused by KF in the slider and the tilter are identical, and,therefore, curve (i) of Fig. 5 should describe the Sm C -tilt if the tilter used by KF were ideal or nearly so. Notefrom Fig. 5, however, that this implies that the Sm C -tiltfor an ideal tilter at φ = 13 or 14 deg should be signifi- Tilting (iii)(ii)(i) (iv) φ (deg) Θ ( d e g ) FIG. 5: The tilt angle Θ between the layer normal and thedirector as a function of the mechanical tilt angle φ in the tiltapparatus for (i) ζ/µ = 0 .
1, (ii) ζ/µ = 1, (iii) ζ/µ = 5, and(iv) ζ/µ = 20. The dashed line corresponds to the resolutionfor Θ in the experiments by KF. cantly larger than the experimental resolution, which isincompatible with the findings of Refs. [6, 7].Since any effective compression of the sample along thelayer normal, as in a tilter, promotes rather than hampersdirector rotation, we obtain a Sm C -tilt at a given φ ina tilter if it occurs at the same angle in the slider. It islikely that the failure of KF to observe director rotation intheir tilter is due to mechanical instability that effectivelyreduces the mechanical shear of the sample and thus leadsto a systematic suppression of the Sm C -tilt in the tiltgeometry. IV. DISCUSSION AND CONCLUDINGREMARKS
In summary, we have developed a neo-classical modelfor smectic elastomers, and we used this model to studySm A elastomers under shear in the plane containing thedirector and the layer normal. In particular, we investi-gated the tilt angle Θ between the layer normal and thedirector for two different experimental setups as a func-tion of the mechanical tilt angle φ measuring the imposedshear.Our model builds upon the neo-classical model forsmectic elastomers by AW. In their original work, AWchose not to consider the possibility of Sm C order intheir theory: they forced the layer normal and the ne-matic director to be parallel. In our present theory, thefundamental tenet is different in that the nematic di-rector n and the deformation tensor Λ are independentquantities that must be allowed to seek their equilibriumin the presence of imposed strains or stresses. The layernormal k is determined entirely by Λ. Thus, n and k areindependent variables that rotate to minimize the freeenergy - they are not locked together.In the present paper, we have for simplicity focusedon idealized monodomain samples. Boundary conditionsin real experiments prevent this simple scenario and pro-duce a microstructure structure of the type that AW dis-cuss in Ref. [4] based on their original model that pro-hibits Sm C ordering. Our current theory, which admitsSm C ordering, predicts the same type of microstructure;stretching will produce a polydomain layer structure justas the AW theory does. In particular, the existence ofSm C ordering does not contradict the appearance of op-tical cloudiness. To avoid undue repetition, we refraindiscussing this here in more detail and refer the readerto Ref. [4].The main finding of our present work is that the tilt an-gle Θ between n and k is non-zero if a shear in the planecontaining k is imposed. This angle depends on materialparameters, the experimental setup, and the magnitudeof the imposed shear. Figures 3 and 5 depict our resultsfor the slider apparatus and the tilt apparatus used byKF. As expected, in both setups the angle Θ decreasesas the value of the parameter ζ/µ increases, at fixed φ .This is because the director is becoming more stronglyanchored to the layer normal. If one keeps ζ/µ fixed andvaries B/µ instead, then for
B/µ → ∞
Θ approaches φ because in this limit f layer locks Θ to Θ = φ . As B/µ isdecreased then Θ decreases at fixed φ , because the f tilt term starts to compete with the f layer term. In the op-posite limit of very small B/µ , Θ is locked to Θ = 0 byvirtue of the tilt and semi-soft contributions to f . Werefrain to show the corresponding curves, which are sim-ilar to those depicted in Figs. 3 and 5, to save space. Themain difference between our results for the two setups isthat for B > ζ and large φ , say 40 degrees or so, Θ isroughly one order of magnitude smaller than φ in theslider apparatus, whereas it is of the order of φ in the tiltapparatus.In closing, we would like to stress once more that theexperiment using the slider geometry produces clear ev-idence of the development of Sm C -like tilt under shearas predicted originally in Ref. [5]. Our theory developedhere allowed us to understand in some detail how themagnitude of this tilt depends on the shear geometryand on model/material parameters, such as ζ , which canbe reasonably estimated from nematic analogues. Giventhese estimates, the resulting estimates for the tilt anglevary over a range from being small enough to be essen-tially unobservable with the X-ray equipment used by theFreiburg group to exceeding the experimental resolution.While the latter is consistent with the shear experimentsin the slider geometry, the former is consistent with thestretching experiments by Nishikawa and Finkelmann inwhich no Sm C -like tilt was detected. We believe thatthe reason for the discrepancy between the rotations inthe slider and tilter apparatus lies in a possible wrin-kling of the samples in the tilt apparatus used by KF.Mechanical instability leads to an effective reduction ofthe mechanical shear of the sample causing a systematicunderestimation of the Sm C -like tilt in the tilt geometry.Another experimental approach that could be used, at least in principle, to investigate Θ is to measurechanges in the layer spacing, as was done by Nishikawaand Finkelmann [3]. It should be noted, however, thatin this approach one measures cos Θ rather than Θ di-rectly. For small values of Sm C tilt, say Θ = 1 de-gree, cos Θ ≈ . A order.For a refined comparison between experiment and the-ory it would be useful to critically evaluate and poten-tially improve the design of the tilt apparatuses regard-ing sample-wrinkling. Also, it would be interesting toperform shear and stretching experiments with an an-gle resolution better than 1 deg. We hope that our workstimulates the interest for such experiments. Acknowledgments
This work was supported in part by the National Sci-ence Foundation under grants No. DMR 0404670 and No.DMR 0520020 (NSF MRSEC) (T.C.L.). We are gratefulto D. Kramer and H. Finkelmann for discussing with ussome of the details of their experiments.
APPENDIX A: ANALYTICAL CONSIDERATIONFOR SMALL ANGLES
For arbitrary Sm C -tilt angle Θ, the total elastic energydensity f is, for the deformations discussed in Sec. III,a fairly complicated conglomerate of trigonometric func-tions (and powers thereof) of Θ and φ . Thus, we resortedin Sec. III to a numerical approach to determine the equi-librium value of Θ over a wide range of the imposed me-chanical tilt angle φ . Here, we focus on the regime of φ where Θ is small, such that it is justified to expand f to harmonic order in Θ. In this case, the equations ofstate are linear in Θ, of course, and are therefore readilysolved. For the slider apparatus we obtainΘ = ( r − rµ tan φrζ + µ (1 − r )[1 − r + r tan φ ] , (A1)and for the tilt apparatus we getΘ = 2 − ( r − rµ sin 2 φrζ − rB sin φ + µ (1 − r )[1 − r cos 2 φ ] . (A2)Equations (A1) and (A2) imply, that the Sm C -tilt inboth setups is identical for small φ ,Θ = ( r − rµ φrζ + µ (1 − r ) + O ( φ ) . (A3)This is consistent with Figs. 3 and 5, where the initialslopes are identical for identical values of ζ (though thisis perhaps somewhat hard to see because the ordinate-scales are different in the two figures).An interesting related question, which we have not ad-dressed in the main text because there is currently noexperimental data available, is that of the stress that iscaused by the imposed shear strains. From the above,it is straightforward to calculate this stress for small φ .Inserting Eq. (A3) into the aforementioned harmonic (inΘ) total elastic energy density provides us with an effec-tive f in terms of φ . From this effective f , we readilyextract the engineering or first Piola-Kirchhoff stress as σ eng xz = ∂f∂ Λ xz = r µζ φrζ + µ (1 − r ) + O ( φ ) . (A4)Equation (A4) highlights a problem that arises when thetilt and the semi-soft contributions f tilt and f semi to thetotal elastic energy density are missing, and one allows n and Λ to be independent quantities. Setting ζ = 0in Eq. (A4) leads to zero stress for non-zero φ , i.e., thistruncated model predicts soft elasticity. This soft elas-ticity, however, is not compatible with Sm A elastomerscrosslinked in the Sm A phase, such as the experimentalsamples of Refs. [3, 6], where the anisotropy direction ispermanently frozen into the system. APPENDIX B: EULER INSTABILITY IN THETILT APPARATUS
In this appendix, we give a brief derivation of Eq. (3.3).We are interested primarily in a rough estimate for φ c ,and therefore, for simplicity, we assume that we can ig-nore the effects of smectic layering. In the following, weemploy, for convenience, the Lagrangian formulation ofelasticity theory.In the Lagrangian formulation, the elastic energy den-sity of a thin elastomeric film with thickness L y andheight L z that is compressed along the z -direction canbe written as f film = κ (cid:0) ∂ z u y (cid:1) + Y d u zz , (B1)where the choice of coordinates is the same as depictedin Fig. 2. u y is the y -component of the elastic dis- placement u = R − x , and u zz is the zz -component ofthe Cauchy-Saint-Venant strain tensor, u ij = ( ∂ i u j + ∂ j u i + ∂ i u k ∂ j u k ). κ and Y d are, respectively, the bend-ing modulus and Young’s modulus of the film, which aregiven in the incompressible limit by κ = ( µ/ L y and Y d = 3 µL y [17]. At leading order, ∂ z u z = −| δL z /L z | ,where we have used that the height change δL z is nega-tive when the sample is effectively compressed as in thetilt apparatus. This leads to u zz = −| δL z /L z | + ( ∂ z u y ) , (B2)when we concentrate on the parts of u zz that are mostimportant with respect to buckling in the y -direction.Next, we substitute the strain (B2) into Eq. (B1) andswitch to Fourier space. To leading order in the elasticdisplacement, this produces˜ f film = 12 (cid:26) κ q z − Y d (cid:12)(cid:12)(cid:12)(cid:12) δL z L z (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) q z ˜ u y ( q )˜ u y ( − q ) , (B3)where q is the wavevector conjugate to x , ˜ u y is theFourier transform of u y , and so on. Equation (B3) makesit transparent that buckling occurs for (cid:12)(cid:12)(cid:12)(cid:12) δL z L z (cid:12)(cid:12)(cid:12)(cid:12) = κY d q z = L y q z . (B4)The smallest value of q z is determined by the specificsof the boundary conditions. From Fig. 2 it appears asif the sample of KF is clamped such that it prefers tostay parallel to the clamps in their immediate vicinity.In this case, the smallest value of q z is q z = 2 π/L z . Forhinged boundary conditions, in comparison, its smallestvalue would be q z = π/L z . Using the former value andexploiting that, approximately, | δL z /L z | = 1 − cos φ , weobtain the estimate (3.3) for the onset of the Euler insta-bility in the tilt apparatus. [1] For a review on liquid crystal elastomers see M. Warnerand E.M. Terentjev, Liquid Crystal Elastomers (Claren-don Press, Oxford, 2003).[2] For a review see P. G. de Gennes and J. Prost,
ThePhysics of Liquid Crystals (Clarendon Press, Oxford,1993); S. Chandrasekhar,