Smectic-C tilt under shear in smectic-a elastomers

Transcription

1 Smectic-C tilt uner shear in smectic-a elastomers Olaf Stenull an T. C. Lubensky Department of Physics an Astronomy, University of Pennsylvania, Philaelphia, Pennsylvania 19104, USA J. M. Aams an Mark Warner Cavenish Laboratory, JJ Thomson Avenue, Cambrige CB3 0HE, Unite Kingom Receive 27 February 2008; publishe 8 August 2008 Stenull an Lubensky Phys. Rev. E 76, have argue that shear strain an tilt of the irector relative to the layer normal are couple in smectic elastomers an that the imposition of one necessarily leas to the evelopment of the other. This means, in particular, that a smectic-a elastomer subjecte to a simple shear will evelop smectic-c-like tilt of the irector. Recently, Kramer an Finkelmann e-print arxiv: ; Phys. Rev. E 78, , performe shear experiments on smectic-a elastomers using two ifferent shear geometries. One of the experiments, which implements simple shear, prouces clear evience for the evelopment of smectic-c-like tilt. Here, we generalize a moel for smectic elastomers introuce by Aams an Warner Phys. Rev. E 71, an use it to stuy the magnitue of SmC-like tilt uner shear for the two geometries investigate by Kramer an Finkelmann. Using reasonable estimates of moel parameters, we estimate the tilt angle for both geometries, an we compare our estimates to the experimental results. The other shear geometry is problematic since it introuces aitional in-plane compressions in a sheetlike sample, thus inucing instabilities that we iscuss. DOI: /PhysRevE PACS number s : Va, v, Df I. INTRODUCTION Smectic elastomers 1 are rubbery materials with the orientational properties of smectic liqui crystals 2. They possess a planelike, lamellar moulation of ensity in one irection. In the smectic-a SmA phase, the Frank irector n escribing the average orientation of constituent mesogens is parallel to the normal k of the smectic layers, whereas in the smectic-c SmC phase, there is a nonzero tilt angle between n an k. Recently, there has been some controversy about whether shear strain an tilt of the irector relative to the layer normal are couple in smectic elastomers an whether the imposition of one necessarily leas to the evelopment of the other. Beautiful experiments by Nishikawa an Finkelmann 3, where a SmA elastomer was subjecte to extensional strain along the layer normal, foun a rastic ecrease in Young s moulus at a threshol strain of about 3% accompanie by a rotation of k through an angle that sets in at the same threshol. Interpreting their x-ray ata, the authors conclue that there was no SmC-like orer over the range of strains they probe, an that the reuction in Young s moulus stems from a partial breakown of smectic layering. Recently, Aams an Warner AW 4 evelope a moel for SmA elastomers that assumes that n an k are rigily locke such that =0. This moel prouces a stress-strain curve an a curve for the rotation angle of k in full agreement with the experimental curves but without neeing to invoke a breakown of smectic layering. Eviently, because of the assumption =0, the AW moel preicts no SmC-like orer. More recently, Stenull an Lubensky 5 argue that shear strain an SmC-like orer are couple an that the imposition of one inevitably leas to the evelopment of the other. They evelope a moel base on Lagrangian elasticity that preicts, as oes the AW moel, a stress-strain curve an a curve for in full agreement with the experiment. In contrast to the interpretation of Nishikawa an Finkelmann of their ata an to the central assumption of the AW moel, Ref. 5 foun that the tilt angle n of n is not ientical to that of the layer normal,, implying that there is SmC-like orer with nonzero above the threshol strain. However, estimates of an n base on reasonable assumptions guie by the available experimental ata of the layer normal-irector coupling turn out to have the same orer of magnitue. The upshot is that Ref. 5 preicts SmC-like tilt above the threshol strain but that the angle is small. It is entirely possible that is smaller than the resolution of the experiments by Nishikawa an Finkelmann. In this case, there is no isagreement between the preictions of Ref. 5 an the experimental ata by Nishikawa an Finkelmann. The arguments of Ref. 5 imply in particular that a SmA elastomer subjecte to a shear in the plane containing k will evelop SmC-like tilt of the irector. Very recently, Kramer an Finkelmann KF 6,7 performe corresponing shear experiments using two ifferent shear geometries. One geometry, which we refer to as tilt geometry, imposes a shear that is accompanie by an effective compression of the sample along the layer normal. In this geometry, the elastomer ruptures at an impose mechanical shear angle of 13 eg 7 or 14 eg 6, an up to these values of no SmC-like tilt is etecte within the accuracy of the experiments of about 1 eg, as was the case in the earlier stretching experiments of Nishikawa an Finkelmann. The other shear geometry, which we refer to as slier geometry, imposes simple shear an can be use to probe values of exceeing 20 eg. For this geometry, the KF x-ray ata provie clear evience for the emergence of SmC-like tilt. In this paper, we generalize the moel introuce by AW 4 an use it to stuy the magnitue of SmC-like tilt uner shear for the two geometries investigate by KF. Using reasonable estimates of moel parameters, we estimate the tilt /2008/78 2 / The American Physical Society

2 STENULL et al. angle, an we compare our estimates to the experimental results. The outline of the remainer of this paper is as follows. In Sec. II, we evelop our moel, which generalizes the original AW moel. In Sec. III, we escribe the two experimental setups that we consier. We iscuss the respective eformation tensors for these setups an calculate for both setups the SmC-tilt angle as a function of the impose mechanical tilt angle. We aress why the tilter apparatus is problematic for experiments where shear inuces irector tilt. In Sec. IV, we iscuss our finings an we make some concluing comments. There are two Appenixes. In Appenix A, we comment on analytical calculations of for small. In Appenix B, we briefly iscuss the Euler instability in the context of the experiments by KF. II. GENERALIZING THE ADAMS-WARNER MODEL In this section, we evise a theory for smectic elastomers base on the neoclassical approach, evelope originally by Warner an Terentjev an co-workers 1,8 for nematic elastomers, an the subsequent extension of the neoclassical approach by AW to smectics. The neoclassical approach generalizes the classical theory of rubber elasticity 9 to inclue the effects of orientational anisotropy on the ranom walks of constituent polymer links. It treats large strains with the same ease as they are treate in rubbers in the absence of orientational orer, it provies irect estimates of the magnitues of elastic energies, it is characterize by a small number of parameters, an it accounts easily for incompressibility. In the neoclassical approach, one formulates elastic energy ensities in terms of the Cauchy eformation tensor =, efine by ij = R i / x j, where R x is the target space vector that measures the position in the eforme meium of a mass point that was at position x in the uneforme reference meium. In this approach, a generic moel elastic energy ensity for smectic elastomers that allows for a relative tilt between the irector an the layer normal can be written in the form f = f trace + f layer + f tilt + f semi. 2.1 Here an in the following, incompressibility of the material is assume, i.e., the eformation tensor is subject to the constraint et = =1. f trace is the usual trace formula of the neoclassical moel with the shear moulus, f trace = 1 2 Tr = = 0 = T = 1, 2.2 where = 0 = = + r 1 n 0 n 0, with n 0 escribing the uniaxial irection before eformation, is the so-calle shape tensor escribing the istribution of conformations of polymeric chains before eformation, an = 1 = = 1 r 1 nn is the inverse shape tensor after eformation. = enotes the unit matrix, an r enotes the anisotropy ratio of the uniaxial SmA state. The contribution f layer = 2 B cos 2.3 escribes changes in the spacing of smectic layers with layer normal k = = T k 0 = T k 0, 2.4 where k 0 =n 0 is the layer normal before eformation an B is the layer compression moulus an are, respectively, the layer spacing before an after eformation, which are relate via 1 = 0 = T k The tilt energy ensity, f tilt = 1 2 a t sin 2, 2.6 incorporates into the moel the preference for the irector to be parallel to the layer normal in the SmA phase. To stuy the SmC phase, we woul have to inclue a term proportional to sin 4, which, however, is inconsequential for our current purposes. Without the contribution f semi, the moel elastic energy ensity 2.1 is invariant with respect to simultaneous rotations of the smectic layers, the nematic irector, an = in the target space. To break this unphysical invariance, we inclue the semisoft term 1 f semi = 1 2 Tr = n 0n 0 = T nn =, 2.7 where is a imensionless parameter. The relation of the moel presente here to the AW moel is the following: AW assume that the layer normal an the irector are rigily locke such that the angle is constraine to zero. Moreover, the semisoft term f semi is absent in the AW moel. Essentially, we retrieve the AW moel from Eq. 2.1 by setting =0 or equivalently a t an =0. As mentione above, one of the virtues of the neoclassical approach is that it involves only a few parameters, an for most of these there exist experimental estimates, which we will now review briefly. The shear moulus for rubbery materials is typically of the orer of Pa. A typical value for the smectic layer compression moulus in smectic elastomers, as observe in experiments with small strains along the layer normal, is B 10 7 Pa, which is greater than the values in liqui smectics. In previous experiments by the Freiburg group, the anisotropy ratio was approximately r 1.1 3, an we aopt this value for our arguments here. The value of a t can be estimate from experiments by Brehmer, Zentel, Gieselmann, Germer, an Zungenmaier 11 on smectic elastomers an by Archer an Dierking 12 on liqui smectics. The former experiment inicates that a t is of the orer of 10 5 Pa at room temperature, an the latter experiment prouces a room-temperature value of the orer of 10 6 Pa. We are not aware of any experimental ata that allow us to estimate for smectic elastomers reliably. For nematic

3 SMECTIC-C TILT UNDER SHEAR IN SMECTIC-A φ Sliing Clamp Sample Fixe Clamp FIG. 1. Color online Sketch of a slier apparatus where the upper clamp slies on a horizontal bar such that the extension of the sample in the z irection remains fixe. elastomers, it has been estimate from the Freericks effect 13 an from the magnitue of the threshol to irector rotation in response to stretches applie perpenicular to the original irector 14 that 0.06 or 0.1, respectively. For our arguments here, we aopt the latter value acknowleging that may be consierably larger in smectics than in nematics. As we will see in the following, our finings o not epen sensitively on this assumption. For the eformations that we consier, appears only in the combination =a t +, that is, semisoftness simply as to the irector-layer normal coupling. Uncertainty in estimates for is expecte to stem mainly from the sprea in estimates for a t. We account for this sprea by iscussing several values of or, more precisely, for several values of /. III. EXPERIMENTAL SHEAR GEOMETRIES AND TILT ANGLES In this section, we apply the moel efine in Sec. II to stuy the behavior of SmA elastomers in shear experiments. We consier two experimental setups. In the first setup, which is perhaps the first that comes to min from a physicist s viewpoint, a simple shear is applie, i.e., a shear strain in which the externally impose isplacements all lie in a single irection. Figure 1 shows a sketch of such an experiment, an we call the apparatus sketche in it the slier apparatus. The secon setup, which is epicte in Fig. 2, is one in which opposing surfaces of the sample remain essentially parallel but in which the height of the sample its extension in the z irection ecreases upon shearing. Hence, the applie shear is, strictly speaking, not just simple shear. In the following, we will refer to this apparatus as a tilt apparatus. φ FIG. 2. Color online One of the two tilt apparatuses use by KF. The photo has been taken from Ref. 6. z z y y x x A. Slier apparatus To make our arguments as concrete an simple as possible, we now choose a specific coorinate system. That is, we choose our z irection along the uniaxial irection n 0 of the unsheare samples, n 0 = 0,0,1, an we choose our x irection as the irection along which the sample is clampe, which is perpenicular to n 0 ; cf. Fig. 1. With these coorinates, the eformation tensor for both the slier apparatus an the tilt apparatus is of the form = xx 0 xz = 0 yy zz. 3.1 As can be easily checke, for this type of eformation, the layer normal lies along the z irection for the unsheare an for the sheare samples. Thus, the tilt angle between n an k is ientical to the tilt angle between n an the z axis, an we can parametrize the irector as n = sin,0,cos. 3.2 It is worth noting that a eformation tensor of the form shown in Eq. 3.1 leas to a particularly simple expression for the semisoft contribution to f, f semi = 1/2 2 xx sin 2, which combines with the tilt energy ensity to f tilt + f semi = 1/2 xx sin 2. Thus, as inicate above, our moel epens for the experimental geometries uner consieration on an a t through a single effective parameter, viz. xx =a t + 2 xx. From the photos of the experimental samples provie in Ref. 6, see Figs. 1 an 2, it appears as if xx oes not eviate significantly from 1. In the following, we set xx =1 for simplicity 15. In this event, xx = a t +. In the slier apparatus, the extent of the sample in the z irection is fixe, an hence zz =1. The incompressibility constraint et = =1 thus manates that yy =1. The remaining nonzero component of the eformation tensor is entirely etermine by the externally impose shear, xz =tan, where is the mechanical tilt angle of the sample; see Fig. 1. The only remaining egree of freeom in the problem is, therefore, the angle. To calculate as a function of, we insert the just iscusse eformation tensor into f an then minimize f over holing fixe. For small, this can be one analytically by expaning f to harmonic orer in an by then solving the resulting linear equation of state for an equilibrium value of. This type of analysis is presente in the Appenix. To provie reliable preictions for larger shears, one has to refrain from expaning in an use numerical methos instea. To this en, we minimize f numerically assuming, base on what we iscusse at the en of Sec. II, that B/ =10 an / 0.1,1,5,20. For this minimization, we use MATHEMATICA s FinMinimum routine. Figure 3 shows the resulting curves for as a function of. The ashe horizontal line in Fig. 3 is a guie to the eye; it correspons to the angle resolution in the KF experiments, which was about 1 egree 16. For the slier geometry, the KF ata prouce clear evience for the evelopment of SmC-like tilt uner simple

4 STENULL et al. Θ (eg) (i) Sliing (ii) (iii) (iv) L Lsinχ Lcosχ φ χ Lcosχ-2 FIG. 4. Schematic of shear an compression arising in a nonieal tilter φ (eg) FIG. 3. Color online The tilt angle between the layer normal an the irector as a function of the mechanical tilt angle in the slier apparatus for i / =0.1, ii / =1, iii / =5, an iv / =20. The ashe line correspons to the resolution for in the experiments by KF. shear, an our theoretical estimates agree well with the experimental ata. However, given that thus far only two ata points are available for 0, it cannot be juge reliably whether our theoretical curve coul be fitte to an experimental curve with more ata points. It is encouraging, though, that our curve for / =0.1 agrees with the available experimental points within their errors. B. Tilt apparatus Now we turn to the tilt geometry. The essential ifference between the slier an the tilt geometry is that in the former, the height of the sample L z remains constant, L z =L z 0 times the number of smectic layers, whereas in an ieal tilter, L z is not constant but rather ecreases as increases, L z =L z cos. This ifference has far-reaching consequences. In the slier, the sample height remains larger than the moifie natural height of the sample create by the irector tilt, L z cos. In other wors, the sample is uner effective tension. In the tilter, on the other han, the sampleheight L z cos can be smaller than L z cos, i.e., the sample can be uner effective compression. For a sample uner effective zz compression, one has to worry, experimentally an theoretically, about all sorts of complications. Most notable are perhaps buckling an wrinkling. The theory of buckling of elastic sheets is well establishe 17, an we now briefly comment on buckling in the context of the tilt geometry. As mentione above, a sample clampe into the tilt apparatus can evelop a buckling instability if the height of the sample impose by tilt, L z cos, is smaller than the natural height of the sample create by the irector tilt, L z cos. As we will see below, our results for as a function of imply that cos cos, i.e., buckling is possible. Alternatively, this can be seen by calculating the engineering stress eng zz = f / zz, which can be one using the numerical approach outline above. eng zz is positive for 0 in the slier apparatus, whereas it turns out to be negative for 0 in the tilt apparatus. Thus, the sample is effectively uner tension in the slier apparatus, whereas it is effectively uner compression in the tilt apparatus, opening the possibility for buckling in the y irection in the latter. The angle c at which buckling sets in is expecte to be comparable to that for the well-known Euler-Strut instability 17, c Euler = arccos 1 2 L y 3L z 2, 3.3 where L y is the thickness of the sample in the y irection, an where clampe rather than hinge bounary conitions are assume. A brief erivation of Eq. 3.3 is given in Appenix B. In the experiments of KF, L z =5.0 mm an L y =0.45 mm 16, which leas to Euler 15 eg. The experimental samples buckle immeiately before they rupture 16 at =13 or 14 eg, which is very close to our estimate for c. The observation that buckling occurs immeiately before rupturing might suggest that the former actually triggers the latter. A etaile analysis of sample wrinkling in the tilt geometry is beyon the scope of the present paper. However, a few comments about wrinkling are in orer. The wavelength of wrinkling is expecte to be much shorter than that of buckling. Thus, wrinkling can be harer to etect by visual inspection of an experimental sample than buckling, an there is a risk that it remains unnotice. The main problem is, however, that wrinkling can act to effectively reuce the mechanical shear in the sample. When esigning a tilt experiment, one thus has to be very careful to avoi wrinkling. If not, the effects of wrinkling can bias the ata, an there is the risk of unerestimating the SmC-tilt significantly. Our calculation of the SmC-tilt will be for an ieal tilter. Figure 4 shows the tilter of Fig. 2 schematically in a nontilte an a tilte configuration to make clear that shear an compression are complex in the nonieal case because the frame axles allowing angle change are offset from the corners of the sample. Using elementary trigonometry, one can euce for the shears an the angle L xz = sin, L 2 zz = L cos 2, 3.4 L 2 tan tan = 1 2/L sec. 3.5 In particular, the relation between the shear angle an the eformation components zz an xz is not simple. Returning to an ieal tilter, =0,, we calculate the SmC-tilt as a function of the applie mechanical shear, suppressing buckling an wrinkling. Then, the eformation tensor is of the same form 3.1 as for the slier apparatus, but with the essential ifference that here xz =sin an zz =cos. As we i for the slier apparatus, we assume xx =1. The incompressibility constraint then implies that yy

5 SMECTIC-C TILT UNDER SHEAR IN SMECTIC-A Θ (eg) (i) Tilting (ii) φ (eg) FIG. 5. Color online The tilt angle between the layer normal an the irector as a function of the mechanical tilt angle in the tilt apparatus for i / =0.1, ii / =1, iii / =5, an iv / =20. The ashe line correspons to the resolution for in the experiments by KF. =1/cos, leaving the angle as the only egree of freeom. We calculate as a function of in exactly the same way as above, i.e., we minimize f numerically assuming the values of the moel parameters iscusse in Sec. II. The resulting curves are shown in Fig. 5. The ip in Fig. 5 occurs because of a competition between, which woul prefer =0, an B, which woul prefer =. If B, then it is possible to get curves where stays close to 0. The ashe horizontal line in Fig. 5 inicates the angle resolution of about 1 egree of the KF experiments 16. As mentione in Sec. III A, the ata points for the slier are compatible with / 0.1. The elastomers use by KF in the slier an the tilter are ientical, an, therefore, curve i of Fig. 5 shoul escribe the SmC-tilt if the tilter use by KF were ieal or nearly so. Note from Fig. 5, however, that this implies that the SmC-tilt for an ieal tilter at =13or14eg shoul be significantly larger than the experimental resolution, which is incompatible with the finings of Refs. 6,7. Since any effective compression of the sample along the layer normal, as in a tilter, promotes rather than hampers irector rotation, we obtain a SmC-tilt at a given in a tilter if it occurs at the same angle in the slier. It is likely that the failure of KF to observe irector rotation in their tilter is ue to mechanical instability that effectively reuces the mechanical shear of the sample an thus leas to a systematic suppression of the SmC-tilt in the tilt geometry. IV. DISCUSSION AND CONCLUDING REMARKS In summary, we have evelope a neoclassical moel for smectic elastomers, an we use this moel to stuy SmA elastomers uner shear in the plane containing the irector an the layer normal. In particular, we investigate the tilt angle between the layer normal an the irector for two ifferent experimental setups as a function of the mechanical tilt angle measuring the impose shear. Our moel buils upon the neoclassical moel for smectic elastomers by AW. In their original work, AW chose not to consier the possibility of SmC orer in their theory: they force the layer normal an the nematic irector to be parallel. In our present theory, the funamental tenet is ifferent in (iii) (iv) that the nematic irector n an the eformation tensor = are inepenent quantities that must be allowe to seek their equilibrium in the presence of impose strains or stresses. The layer normal k is etermine entirely by =. Thus, n an k are inepenent variables that rotate to minimize the free energy they are not locke together. In the present paper, we have focuse for simplicity on iealize monoomain samples. Bounary conitions in real experiments prevent this simple scenario an prouce a microstructure structure of the type that AW iscuss in Ref. 4 base on their original moel that prohibits SmC orering. Our current theory, which amits SmC orering, preicts the same type of microstructure; stretching will prouce a polyomain layer structure just as the AW theory oes. In particular, the existence of SmC orering oes not contraict the appearance of optical clouiness. To avoi unue repetition, we refrain from iscussing this here in more etail an refer the reaer to Ref. 4. The main fining of our present work is that the tilt angle between n an k is nonzero if a shear in the plane containing k is impose. This angle epens on material parameters, the experimental setup, an the magnitue of the impose shear. Figures 3 an 5 epict our results for the slier apparatus an the tilt apparatus use by KF. As expecte, in both setups the angle ecreases as the value of the parameter / increases, at fixe. This is because the irector is becoming more strongly anchore to the layer normal. If one keeps / fixe an varies B/ instea, then for B/, approaches because in this limit f layer locks to =. AsB/ is ecrease, then ecreases at fixe, because the f tilt term starts to compete with the f layer term. In the opposite limit of very small B/, is locke to =0 by virtue of the tilt an semisoft contributions to f. We refrain to show the corresponing curves, which are similar to those epicte in Figs. 3 an 5, to save space. The main ifference between our results for the two setups is that for B an large, say 40 egrees or so, is roughly one orer of magnitue smaller than in the slier apparatus, whereas it is of the orer of in the tilt apparatus. In closing, we woul like to stress once more that the experiment using the slier geometry prouces clear evience of the evelopment of SmC-like tilt uner shear as preicte originally in Ref. 5. Our theory evelope here allowe us to unerstan in some etail how the magnitue of this tilt epens on the shear geometry an on moel an material parameters, such as, which can be reasonably estimate from nematic analogues. Given these estimates, the resulting estimates for the tilt angle vary over a range from being small enough to being essentially unobservable with the x-ray equipment use by the Freiburg group to exceeing the experimental resolution. While the latter is consistent with the shear experiments in the slier geometry, the former is consistent with the stretching experiments by Nishikawa an Finkelmann in which no SmC-like tilt was etecte. We believe that the reason for the iscrepancy between the rotations in the slier an tilter apparatus lies in a possible wrinkling of the samples in the tilt apparatus use by KF. Mechanical instability leas to an effective reuction of the mechanical shear of the sample causing a systematic unerestimation of the SmC-like tilt in the tilt geometry

6 STENULL et al. Another experimental approach that coul be use, at least in principle, to investigate is to measure changes in the layer spacing, as was one by Nishikawa an Finkelmann 3. It shoul be note, however, that in this approach one measures cos rather than irectly. For small values of SmC tilt, say =1 egree, cos , which is practically inistinguishable from the cos =1 pertaining to SmA orer. For a refine comparison between experiment an theory, it woul be useful to critically evaluate an potentially improve the esign of the tilt apparatuses regaring samplewrinkling. Also, it woul be interesting to perform shear an stretching experiments with an angle resolution better than 1 eg. We hope that our work stimulates interest in such experiments. ACKNOWLEDGMENTS This work was supporte in part by the National Science Founation uner Grants No. DMR an No. DMR NSF MRSEC T.C.L.. We are grateful to D. Kramer an H. Finkelmann for iscussing with us some of the etails of their experiments. APPENDIX A: ANALYTICAL CONSIDERATION FOR SMALL ANGLES For arbitrary SmC-tilt angle, the total elastic energy ensity f is, for the eformations iscusse in Sec. III, a fairly complicate conglomerate of trigonometric functions an powers thereof of an. Thus, we resorte in Sec. III to a numerical approach to etermine the equilibrium value of over a wie range of the impose mechanical tilt angle. Here, we focus on the regime of where is small, such that it is justifie to expan f to harmonic orer in. In this case, the equations of state are linear in, of course, an are therefore reaily solve. For the slier apparatus, we obtain r 1 r tan = r + 1 r 1 r + r tan 2, an for the tilt apparatus we get A1 2 1 r 1 r sin 2 = r rb sin r 1 r cos 2. A2 Equations A1 an A2 imply that the SmC-tilt in both setups is ientical for small, r 1 r = r + 1 r 2 + O 3. A3 This is consistent with Figs. 3 an 5, where the initial slopes are ientical for ientical values of though this is perhaps somewhat har to see because the orinate scales are ifferent in the two figures. An interesting relate question, which we have not aresse in the main text because there are currently no experimental ata available, is that of the stress that is cause by the impose shear strains. From the above, it is straightforwar to calculate this stress for small. Inserting Eq. A3 into the aforementione harmonic in total elastic energy ensity provies us with an effective f in terms of. From this effective f, we reaily extract the engineering or first Piola-Kirchhoff stress as eng xz = f xz = r 2 r + 1 r 2 + O 3. A4 Equation A4 highlights a problem that arises when the tilt an the semisoft contributions f tilt an f semi to the total elastic energy ensity are missing, an one allows n an = to be inepenent quantities. Setting =0 in Eq. A4 leas to zero stress for nonzero, i.e., this truncate moel preicts soft elasticity. This soft elasticity, however, is not compatible with SmA elastomers crosslinke in the SmA phase, such as the experimental samples of Refs. 3,6, where the anisotropy irection is permanently frozen into the system. APPENDIX B: EULER INSTABILITY IN THE TILT APPARATUS In this appenix, we give a brief erivation of Eq We are intereste primarily in a rough estimate for c, an therefore, for simplicity, we assume that we can ignore the effects of smectic layering. In the following, we employ, for convenience, the Lagrangian formulation of elasticity theory. In the Lagrangian formulation, the elastic energy ensity of a thin elastomeric film with thickness L y an height L z that is compresse along the z irection can be written as f film = zu y Y 2u 2 zz, B1 where the choice of coorinates is the same as epicte in Fig. 2. u y is the y component of the elastic isplacement u =R x, an u 2 zz is the zz component of the Cauchy-Saint- Venant strain tensor, u ij = 1 2 iu j + j u i + i u k j u k. an Y 2 are, respectively, the bening moulus an Young s moulus of the film, which are given in the incompressible limit by = /3 L 3 y an Y 2 =3 L y 17. At leaing orer, z u z = L z /L z, where we have use that the height change L z is negative when the sample is effectively compresse as in the tilt apparatus. This leas to u zz = L z /L z zu y 2, B2 when we concentrate on the parts of u zz that are most important with respect to buckling in the y irection. Next, we substitute the strain B2 into Eq. B1 an switch to Fourier space. To leaing orer in the elastic isplacement, this prouces f film = 2 q 1 2 z Y 2 L z L z z q 2 ũ y q ũ y q, B3 where q is the wave vector conjugate to x, ũ y is the Fourier transform of u y, an so on. Equation B3 makes it transparent that buckling occurs for

7 SMECTIC-C TILT UNDER SHEAR IN SMECTIC-A L z L z = q 2 Y z = L 2 y 2 9 q 2 z. B4 The smallest value of q z is etermine by the specifics of the bounary conitions. From Fig. 2 it appears as if the sample of KF is clampe such that it prefers to stay parallel to the clamps in their immeiate vicinity. In this case, the smallest value of q z is q z =2 /L z. For hinge bounary conitions, in comparison, its smallest value woul be q z = /L z. Using the former value an exploiting that, approximately, L z /L z =1 cos, we obtain the estimate Eq. 3.3 for the onset of the Euler instability in the tilt apparatus. 1 For a review on liqui crystal elastomers, see M. Warner an E. M. Terentjev, Liqui Crystal Elastomers Clarenon, Oxfor, For a review, see, P. G. e Gennes an J. Prost, The Physics of Liqui Crystals Clarenon, Oxfor, 1993 ; S. Chanrasekhar, Liqui Crystals Cambrige University Press, Cambrige, E. Nishikawa an H. Finkelmann, Macromol. Chem. Phys. 200, J. M. Aams an M. Warner, Phys. Rev. E 71, O. Stenull an T. C. Lubensky, Phys. Rev. E 76, D. Kramer an H. Finkelmann, e-print arxiv: D. Kramer an H. Finkelmann, preceing paper, Phys. Rev. E 78, P. Blaon, E. Terentjev, an M. Warner, J. Phys. II 4, L. R. G. Treloar, The Physics of Rubber Elasticity Clarenon, Oxfor, Note that the layer compression term formulate by AW is of the form / 0 1 2, whereas setting =0 in Eq. 2.3 prouces / This form is chosen for consistency with the Chen-Lubensky moel of smectics. This term can be factorize into / / / for 0. Thus the ifference is merely a factor of 4 in the efinition of the smectic moulus B in the two representations. 11 M. Brehmer, R. Zentel, F. Gieselmann, R. Germer, an P. Zungenmaier, Liq. Cryst. 21, P. Archer an I. Dierking, Phys. Rev. E 72, E. M. Terentjev, M. Warner, R. B. Meyer, an J. Yamamoto, Phys. Rev. E 60, H. Finkelmann, I. Kunler, E. M. Terentjev, an M. Warner, J. Phys. II 7, We checke the valiity of this assumption by allowing xx to relax to its equilibrium value, i.e., we repeate our analysis using the xx that minimizes f rather than xx =1. The results obtaine by using this more involve approach o not iffer significantly from those obtaine using our simplifie approach with xx =1. 16 D. Kramer an H. Finkelmann private communication. 17 L. D. Lanau an E. M. Lifshitz, Theory of Elasticity, 3r e. Pergamon, New York,