Uniaxial and Biaxial Soft Deformations of Nematic Elastomers

arXiv:cond-mat/0111538v2 [cond-mat.soft] 18 Feb 2002 ieto[] h reeeg density, energy free The direction[4]. tr hne1 rb luiain2,cue ag ( large temper causes by illumination[2], either by anisotropy, or shape change[1] on molecular ature switching this The of elastomer. off the up and make that anisotr polymers the shape in T induces bodies. orientation elastic molecular ordinary of of anisotropy those to addition in freedom of tensors. nature of the roots and square tensors, of for rotatio Theorem and Decomposition shears Polar symmetric the into strains questio finite o related breaking the elasticity of discussing the by in conclude we role rubber, essential matic elas an conventional play biaxial to rotations of contrast modes solids, in soft Since the discuss elastomers. geomet- then nematic a fracti We give using no tensors. We described of or modes paper. powers soft little this the with of of interpretation change subject ric shape the is last, cost, The energy elasticity. respon sponta- soft optical-mechanical large and large - very elasticity deformations, conventional neous in found not nomena ehnclsaechanges. shape mechanical h rgnlyipsdsri srnee ozr hsi wh is This elasticity[3]. . soft zero call to we rendered is strain imposed cost originally the the distributions, molecular of ac changes which the contractions commodate Whe and shears observed. subsidiary is by that accompanied effect opto-mechanical extreme the is in,ta swti h iercniumter fauniaxi a of theory director ro mobile continuum a and linear with body the deformations within small is for that tions, initially elasticity soft and eai lsoeshv nitra,oinainldegree orientational internal, an have elastomers Nematic phe- related and unique three display elastomers Nematic oaino hsaiorp,b moe ehnclstrains mechanical imposed by anisotropy, this of Rotation n a xlr h eae da faiorp,rotation anisotropy, of ideas related the explore can One F = + + 1 2 I OTESI IERCONTINUA LINEAR IN SOFTNESS II. B ˜ 2 2 1 1 ( D µ ASnmes 14.ePlmr,eatmr,adplastics and elastomers, constants Polymers, 61.41.+e numbers: PACS exp finding of that and context this Olmsted). in discussed a is rotations rotations tensors body of of into role deformations The general decomposing theorem). for Lubensky rotation and under Golubovic sh invariance We (the the order. how nematic and biaxial elasticity classical and uniaxial both for examples, Tr 1 1 [ egv emti nepeaino h oteatcdefor elastic soft the of interpretation geometric a give We [ [( n ε ˜ Ω ]) × 2 .INTRODUCTION I. aeds aoaoy nvriyo abig,Madingley Cambridge, of University Laboratory, Cavendish − ε + × ω 1 2 n ) κ nailadbailsf eomtoso eai elastome nematic of deformations soft biaxial and Uniaxial ] × 2 Tr + n n [ ] htcaatrssteanisotropy the characterises that ε ˜ 1 2 2 ]( µ + n 2 ([ 1 2 · n ε D · × 2 n F n + ) ε is: , · ] ε · n · 1 2 [( ) 2 µ Ω 0 ( − n · ω .Wre n .Kutter S. and Warner M. ε Dtd coe 8 2018) 28, October (Dated: ) ≈ · × n ) 400%) n 2 ne- f ] onal , opy (1) ns, ta- tic he ns of se at al n - - , 12.xPlmrlqi rsas 22.cEatct,e Elasticity, 62.20.Dc crystals, liquid Polymer 61.20.Vx , fterfrneadtre ttsgvssf elasticity soft gives states target and reference the of s hti,i tl a ouecagsTr changes volume has still it is, that vectors of Rotation 1: FIG. where elc h rttotrsi qain()snete involve they since (1) equation in terms two and first volume the constant at neglect are deformations its material, soft a ymti at(h oyrotation), field. body (the part symmetric strain metric pnst rotation a to sponds n h oyt ufrtechange the suffer to body the aallto parallel Ω of iial,asalrotation small a Similarly, codnl ie eaiecag in change relative a gives accordingly ietrrtto,btceryms ei h eaierotat relative the in be in must terms coupling clearly sm for but (1) equation rotation, of terms director first the in distinguished not are directors The dsmercsris h oeo h qaeroots square the of role The strains. symmetric nd opig o ml oain,tedrco aito corr variation director the rotations, small for coupling: ainmdso eai lsoes ihexplicit with elastomers, nematic of modes mation − h atrtrsaetoeo h eaierotation relative the of those are terms latter The h tensor The n ii om o otdfrain teapoc of approach (the deformations soft for forms licit Æ ª ª Ò Ú Ú Æ wteiprac fbd oain nti non- this in rotations body of importance the ow ihrsett h arxiflce yterltv rotation relative the by inflicted matrix the to respect with ω Ò Ú ae h oa eopsto hoe vital Theorem Decomposition Polar the makes ¿ ½ ¼ ¿ ¾ = ε v ftemti ( matrix the of = = Only . = n od abig B H,U.K. 0HE, CB3 Cambridge Road, Ò . Æ ª ª Ò Ú Ú Æ ε ˜ ª ! − Ú Ò ¿ ½ ε ¼ ˜ ¢ ! Æ Æ ª ª Ò Ú Ú ε ¿ ¾ ˜ j i = n 3 1 Ω ¢ ¢ Ú Ò stesri eoei a enmd traceless, made been has it before strain the is = = ¿ ½ = Tr ¿ ¾ and 2 Æ = n Ò F and [ = Ò = Ò 2 1 Ú ε ˜ δ ª bu naxis an about ! . ( 1 ] ω n n ∂ ¢ Ò δ Ω 0 ª Ω ! j ( = = ¢ u ¢ stetaeespr ftelna sym- linear the of part traceless the is 3 eaiet h hnigdrco ( director changing the to relative ) eoeadatrapiaino strain of application after and before 1 ¢ i Æ n + ntson aeayefc nvectors on effect any have shown) (not ¢ Ω ¢ Ω Ò Ò Ú × ∂ Æ − 2 ftemti assavector a causes matrix the of i δ u Ò Ò Ú δ n ω j v ) ω ω Æ Æ ª ª Ò Ú Ú Ω ) and , n oyrotations body and , rs × Ò Ú or ¿ ½ ¼ = ª ª Ò Ú Ú Æ Æ ¿ ¾ = u n Ò Ú ω δ ¿ ½ = Ω = ¼ ε ¿ ¾ Ω ˜ n en h displacement the being . n = Ò ω × = = of: = ! Æ Æ ª ª Ò Ú Ú ni.Snerbe is rubber Since it. in ª ! Æ Æ ª ª Ò Ú Ú ! = Ò v ª ª Ò Ú Ú Æ Æ ! Ú Ò ¢ ¿ ½ ¼ Ò Ú 1 2 lastic h e rotation net The . ¿ ½ ª ! ¿ ¾ ¿ ¾ Ò Ú ω curl ¿ ½ = ¢ ¢ = ¿ ¾ ¢ = = = Æ = = × ¢ = = ¢ Ò Ò Ò Ú u n Æ Ò ª ! Ò ª ! e g 1. fig. see , Ò ( Ò Ú steanti- the is ª ! ¢ Ω Ω ¢ ¢ ¢ ¢ ¢ ¢ − rotating ¢ Æ ¢ Æ ω Ò Ò Ú Æ Ò Ò Ú v ) ion Ò Ò we Ú ω all in e- × ) 2 volume change. The geometry of the remaining strains is vital

and shown in ﬁg. 2. Ò n Æ εnk

εnm

¼ Ò

n z k y m x k

FIG. 3: Symmetric shear induces director rotation toward the elongation diagonal. The respective elongations and compressions along m ε the diagonals are shown. The chain shape distribution also rotates. mk When mechanical shape changes of a network accommodate the rotations of the chain distribution without distortions, such shape changes can take place at minimal energy cost. FIG. 2: The elements of strain in a uniaxially anisotropic medium. Dashed lines show the undistorted state, whereas the strained elements are shown by shaded areas. They divide into stretches ε nn later discuss. along n, stretches εmm and εkk and distortions εmk in the plane per- An effective shear modulus arises for an imposed elastic pendicular to n, and distortions εnm and εnk encompassing n and the perpendicular plane. strain, when the nematic director is free to evolve optimally. The elastic modulus µ2 is reduced; by the symmetry argument The de Gennes [4] relative rotation couplings are unique to it must vanish to give no overall energy cost. If the local ro- Ω nematic networks because they require an independent, rota- tational component of the deformation of the elastic ma- Ω tional degree of freedom, i.e., the rotations ω of the director trix and the rotation of the director are coaxial, i.e. and ω n. Its motion in the medium is coupled to the body rotation are parallel and uniform, the argument is simple. Min- Ω. Hence, the elastic energy unusually involves antisymmet- imising the relative-rotation part of the energy density (1), 1 Ω ω 2 1 ε Ω ω ric components of shear strain. In section VI, we accordingly 2 D1[( ) n] + 2 D2 n [( ) n], one obtains the − × · · − × ε show how to extract the rotational component from any fi- optimal relative rotation for a given shear strain : nite shear. The first coupling, D , purely resists the rotation 1 D D of the director relative to a rotating, undeformed matrix. The [(Ω ω) n]= 2 (n ε) or (Ω ω)= 2 [n ε n] . D term couples n to the symmetric part of shear in the plane − × −2D1 · − −2D1 · × 2 (2) that involves n (e.g. εzx if n0 = nz in equilibrium). These ε ε are the shears nm and nk in fig. 2. Since infinitesimal, in- We now substitute this back into the energy density and obtain compressible symmetric shear is equivalent to stretch along for the rotation-strain terms: one diagonal and compression along another, it is reasonable 1 2 1 that prolate molecules will reduce the cost of distortion by D1[(Ω ω) n] + D2 n ε [(Ω ω) n] (3) 2 − × 2 · · − × rotating their ordering direction to being as much as possi- 2 2 D2 2 D2 2 2 ble along the elongation diagonal, depicted in fig. 3. Oblate = [n ε n] εxz + εyz . elastomers rotate their director toward the compression diag- −8D1 · × ≡−8D1 onal to achieve the appropriate elastic accommodation. The The last expression is written in the specific coordinate frame spheroid represents the anisotropic shape distribution of the where the initial director n0 is parallel to z-axis. We can now crosslinked polymers from which the network is composed. unite this expression with the rest of the elastic energy density, A theorem of Golubovic and Lubensky (GL) [5] shows that equation (1) and thus obtain the effective rubber-elastic energy on symmetry grounds any solid with an internal degree of which depends only on strains; the director does not appear. freedom which is also capable of reaching an isotropic ref- For instance, in the specific coordinates of fig. 2: erence state must be invariant under the double set of rotations of both the reference and target states when considering 1 ε 2 1 ε 2 ε 2 ε 2 F = 2 µ0 zz + 2 µ1( xx + 2 xy + yy ) elastic deformations. This has the remarkable consequence 2 1 D2 2 2 that some elastic deformations must be soft. This was discov- + µ2 (εxz + εyz ) . (4) 2 − 4D1 ered independently and in a superficially different form when 2 studying at finite deformations the elastic response of nematic The modulus µ2 is renormalised to µ2 D2/(4D1) which elastomers [3]. We sketch how this can be understoodfor elas- by the GL theorem must be zero, thus establishing− a relation tomers in order to explain the complex soft modes we shall between the constants µ2, D1, and D2. The molecular model, 3 required below for finite deformations, produces linear con- measure of r. Indeed spontaneous elongations in the range of tinuum limiting values which also give: λm 1.03 4.00 are observed. Now∼ distortions− λ are no longer small, but must continue 2 R D2 to respect volume conservation, Det(λ)= 1 in the non-linear µ2 = µ2 0 . (5) − 4D1 → regime. The directors n0 and n, of the initial and current nematic states, may be greatly rotated from each other. Olmsted [6] first proposed this continuum mechanism be- The remainder of this paper is concerned with explaining hind the Golubovic–Lubenskytheorem: shape depends on the the character of the soft modes within finite elasticity theory orientation of an internal (nematic) degree of freedom, the ro- and extending this picture to that of soft modes in biaxial ne- tation of which causes a natural shape change at zero cost matic elastomers. We explore the role of rotations and the for suitable solids. A general discussion of the GL argument connections between the isolation of rotational components and its extension to semi-softness and thresholds to rotation is of strain at finite amplitude (the spherical decomposition the- given in [7]. The experimental evidence for mechanically soft orem) and the relation of the roots of tensors to this picture. distortions is also discussed there. One can picture the continuous rotation generating mechanical distortion at low energy cost, see Fig. 3. The natural long IV. UNIAXIAL SOFTNESS axis of the body, as defined by the principal axis of molecular shape, rotates in an attempt to follow the apparent extension Consider the strain [6]: axis. To the extent that macroscopic shape change can thus be imitated, there is no real accompanying distortion of poly- 1/2 1/2 λ = ℓ W ℓ− , (7) mer shape. We quantify this picture below when considering · α · 0 non-linear elasticity theory. α where W α is an arbitrary rotation by an angle . Therearetwo continuous degrees of freedom describing the rotation connecting n and n0 and three degrees of freedom for the rotation III. FINITE SOFT ELASTICITY λ W α. Hence, the strain is described by five continuous degrees of freedom and thus represents a large set of deforma- A simple extension of classical rubber elasticity theory to tions. nematic elastomers gives for the free energy density: If we insert such a strain into the Trace formula (6), as well T 1/2 T 1/2 as its transpose λ (equivalent to ℓ− W ℓ since the ℓ 1 T 1 0 α F = µTr ℓ λ ℓ− λ (6) are symmetric) we obtain: · · 2 0 · · · 1 1/2 T 1/2 1 1/2 1/2 where µ is the rubber elastic shear modulus in the isotropic F µTr − W W − el = 2 ℓ0 ℓ0 α ℓ ℓ− ℓ α ℓ0 phase and λi j = ∂Ri/∂R0 j is the homogeneous deformation · · · · · · · (gradient). If we take λ to be λ = δ + u, then the symmetric 1 δ 3 2 µTr = 2 µ . (8) part of u is identical to ε˜ of (1) in the infinitesimal limit. ≡ The chains are no longer characterised by spherical Gaus- Cancelling the middle section, ℓ1/2 ℓ 1 ℓ1/2 = δ, allows the sian distributions as in the classical case, but in general by − T · δ · anisotropic distributions. Thus the effective step length ten- W-terms to meet and gives W W = . Likewise disposing of · 3 sors initially and currently after a distortion λ are prolate the ℓ terms, one obtains the final expression Fel = µ. This ℓ0 ℓ 0 2 (or oblate) spheroids defining the second moments that char- is identical to the free energy of an undistorted network. The acterise the Gaussian distribution of chain spans R in the the non-trivial set of distortions λ of this particular form, equation 1 network, that is RiR j = ℓi jL where L is the arc length of (7), have not raised the energy of nematic elastomer. From the 3 1 1 a chain. Thus a measure of the mean size of a chain is ℓ1/2 results Det(A B)= Det(A)Det(B), Det(A− ) = (Det(A))− · λ where we shall soon define the roots of tensors more care- and Det(W)= 1 for all rotations W, onecan see that Det( )= 1/2 1/2 fully. The tensor ℓ has one principal value ℓ along n and ℓ Det(ℓ Wα ℓ0− )= 1, that is these soft modes are volume- perpendicular to n; thus k ⊥ preserving.· · We saw pictorially in fig. 3 that, on applying a stretch per- ℓ 0 0 ⊥ 100 pendicular to the initial director, rotation of the chain distribu- ℓ = 0 ℓ 0 = ℓ 010 , 0 0⊥ ℓ ⊥ 0 0 r tion is accommodated by the very elongation we have applied, k together with a shear. Two remarkable consequences immedi- where r . The extracted factor from cancels = ℓ /ℓ ℓ ℓ0 ately follow from nematic elastomer response via rotation: k ⊥ ⊥ 1 λ with the 1/ℓ factor extracted from ℓ− when both tensors ap- Impose an x-extension xx. All the accompanying distor- pear together⊥ in the Trace formula (6) and we can henceforth tions must be in the plane of rotation, that is a transverse con- just consider the ℓ tensors in their reduced form that simply traction λzz and shears λxz and λzx accommodate the rotation depends on the intrinsic anisotropy r. Anisotropy varies be- of the distribution. No shears perpendicular to this plane, that tween r = 1.1 60. In this model of rubber elasticity the is involving y-direction (λyy,λyx,... etc.) are needed. For a → spontaneous elongation, λm, on going from the isotropic to classical isotropic elastomer λyy = λzz = 1/√λxx is demanded 1/3 the nematic phase turns out to be λm = r and is thus a direct by incompressibility, whereas in soft elasticity there is no 4

½ of λ (using the notation s sinθ and c cosθ):

= ´` =` µ ?

ÜÜ soft k ≡ ≡ 1 λ zˆzˆT 1 1 s2 xˆxˆT 1 √r 1 s2

soft = ( ( ) )+ ( + ( ) ) =

ÞÞ − − √r −

½=¾ ½=¾

` `

´` =` µ ? k T T 1 T +yˆyˆ + xˆzˆ (1 )sc + zˆxˆ (√r 1)sc − √r −

=¾

½ 1 + (√r 1)s 0 (1 1/√r)sc `

k 0− 1− 0 . (10)

½=¾

` ≡  2 ? (√r 1)sc 0 1 (1 1/√r)s − − −   FIG. 4: Chain shape distribution before and after rotations. The ex- The soft modes are neither simple nor pure shear, but a mix- tent of soft extensions (λxx) perpendicular to the initial director is set ture of the two. Accordingly, the soft modes have an element by the anisotropy of the molecular shape distribution. The macro- of body rotation in addition to elongations, compressions and scopic dimensions are shown changing affinely with the distribution, pure shears. The importanceof body rotations is discussed be- for instance the x dimension changing from √ℓ to ℓ . fore and after (2). The degree of body rotation is given in sec- ⊥ k q tion VI. Note that the extensional and compressional strains 2 (λxx 1) and (1 λzz) are both proportional to sin θ. Thus − − the infinitesimal strains uzz and uxx, at small rotations θ, are 2 proportional to θ . By contrast λxz and λzx are proportional λ shrinkage in the y direction ( yy = 1) and the appropriate Pois- to sinθ cosθ and hence the infinitesimals uxz and uzx are pro- son ratio is zero. portional to θ – a lower order than uxx and uzz. There is no relaxation along y, perpendicular to the plane of rotation n, Fig. 4 shows the initial and final states of which the that is λyy = 1. Also note that in the isotropic limit (r = 1) this shear and partial rotation of fig. 3 is an intermediate step. particular strain λ = δ while the general soft deformation Softness must come to an end when the rotation is com- soft (7) reducesto the null strain, W α, a simple bodyrotation. Both plete and the z dimension has diminished in the proportion are the trivial cases evidently preserving the elastic energy at λzz = ℓ /ℓ and the x dimension extended in the proportion its minimum. The soft modes become non-trivial deforma- ⊥ k tions when the material becomes a nematic elastomer. λ q xx = ℓ /ℓ . The original sizes ℓ and √ℓ have trans- The soft modes start at no strain, λ δ, and as the director k ⊥ k ⊥ = formedq to √ℓ and ℓ respectively. Thus softness would θ rotates from 0 to π/2, they eventually end at ⊥ k p λ 1/2 λ3/2 cease and director rotation be complete at xx = r m . √r 0 0 p 1/3 ≡ The strain λm = (ℓ /ℓ ) is the spontaneous extension suf- λ = 01 0 , k ⊥ 0 01/√r fered on cooling to the nematic phase. Likewise one can imagine from an oblique form of figure 4 that if the initial o that is an extension λxx = √r and a transverse contraction director n0 (long axis of the shape ellipsoid) is not at 90 to λ the imposed strain, then rotation and softness is complete ata zz = 1/√r. The director rotation is taken up by a shape change so that smaller λ < λ . xx m there is no entropically expensive deformation of the chain We give as a concrete example the set of soft distortions distribution as when a conventional elastomer deforms. The 1/2 1/2 shape tensor’s anisotropy r = ℓ /ℓ characterises the ratio of λ = ℓ ℓ− , simplified by the absence of the arbitrary soft θ 0 the mean square size along thek director⊥ to that perpendicular rotation matrix· W . They are simply characterised (paramet- α to the director. The square root of this ratio, √r, gives the rically) by the angle θ by which ℓ is rotated to ℓ , that is by 0 θ characteristic ratio of average dimensions of chains in the net- 1/2 which n0 is rotated to n. Putting in the dyadic forms for ℓ0− work. During a soft deformation, the solid must change shape 1/2 such that the rotating ellipsoid ℓ1/2, characterising the physi- and ℓθ into λ gives: soft cal dimensions of the distribution of chains, is accommodated 1 without distortion. The ellipsoid is R ℓ− R = 1, or in a · · principal frame, x2/r +z2 = 1, that is in section the ellipse has T 1 T semi-major axes of √r and 1. Fig. 5 illustrates soft deforma- λ = (δ + (√r 1)nn ) (δ + ( 1)n0n ) soft − · √r − 0 tions of a nematic elastomer with chains of anisotropy r = 2. T T θ = δ + (1/√r 1)n n + (√r 1)nn The distortions are parameterised by the director rotation, , 0 0 π − −T which ranges between 0 and /2. The chain shape tensor, +(n.n0)(2 √r 1/√r)nn . (9) 1/2 − − 0 ℓ , rotates without distortion, just fitting into the solid into which it embedded. The soft deformations of fig. 5 and equations (9) and (10) of the body are in general non-symmetric. Below we decompose these into pure shears plus rotations. If n0 is along zˆ and is rotated by θ toward xˆ, it becomes n = 1/2 1/2 zˆ cosθ+xˆ sinθ. We can write down a particular representation One can think of λ = ℓθ ℓ− as converting the initial soft · 0 5

Ô Ô

½ · Ô=¾ ½ Ô=¾

k m

FIG. 5: Soft deformations of a nematic elastomer with anisotropy r = 2. The deformations correspond to director rotations of θ = 0, FIG. 6: The shape ellipsoid for a biaxially nematic polymer. The π/6, π/4, π/3, 5π/12 and π/2 which parametrically generate the section perpendicular to n is not circular, but has semi-axes of length distortions as discussed above. Note that the prolate spheroid char- 1 p 2. Rotations φ about n n generate distortions in the xy acterising the distribution of chains, embedded in the distorting solid / 0 plane± that softly accommodate the≡ non-circular shape as it rotates. (shaded and shown in section), when rotated by θ can be accommo- p dated without distortion. The reference, undeformed body is shown in outline; the original distribution shown dashed. With fixed n, in the biaxial case one can rotate the anisotropic transverse section mk of the square root of the 1/2 shape tensor, ℓ1/2, and accommodate it without distortion by ellipsoid of fig. 5 to a sphere by the action of the inverse ℓ0− and then recreating an ellipsoid at an angle θ with the action inducing shape changes in the mk plane. These distortions are soft for exactly the same reasons as in the uniaxial case when of ℓ1/2. This is precisely the scheme of [8, 9] who consider θ n suffered rotations in the nk or nm planes, see figs. 5. Given an isotropic reference state (the intermediate created after the λ 1/2 that there are (now in the Lab. frame) xy soft modes as well action of − ). We discuss this rotational invariance again ℓ0 as the λzx and λzy forms (which are now no longer equivalent below. modes), the order of softness has become very much greater.

V. BIAXIAL SOFTNESS A. Explicit examples of soft modes arising from biaxiality

Biaxial nematic phases are rare. They have been found The situation is exactly parallel to that of the soft shears (9) (Finkelmann et al [10, 11]) in nematic polymers since one 1/2 1/2 and (10). Now the soft modes are given by ℓ ℓ− where φ has more complex molecular structural possibilities. In prin- φ 0 is instead the angle of rotation about the principal director n. ciple such polymers could be used to make biaxially nematic 1/2 elastomers. They would have rich mechanical properties. The tensor ℓ0 is The shape tensor of a biaxial polymer is √1+p/2 0 0 1/2 ℓ1 0 0 ℓ = 0 √1 p/2 0 , (11) 0 − ℓ = 0 ℓ2 0 . 0 0 √r ! 0 0 ℓ k which will then be rotated by φ. By analogy with ﬁg. 5, the Now the step lengths in the two directions perpendicular to anisotropy in the xy part of the matrix which is to be rotated is n, ℓ1 and ℓ2, are distinguished. The tensor is reduced by 1+p/2 √r = ℓ1 . This measures the anisotropy of ac- taking out a factor of the mean perpendicular step length, ⊥ 1 p/2 ≡ ℓ2 1 tual (root mean− square) chain dimensions in the plane perpen- ℓ = 2 (ℓ1 + ℓ2), to give: q q ⊥ dicular to the director. The resulting soft shears in the xy-plane 1+p/2 0 0 arising from rotations φ about z = n0 are: ℓ = 0 1 p/2 0 0− 0 r 2 1 (1 1/√r )sφ (1 √r )sφcφ 0 − − ⊥ − ⊥ T p T p T λ 2 rnn + (1 + )mm + (1 )kk = (1/√r 1)sφcφ 1+(√r 1)sφ 0 . (12) biaxial ⊥− ⊥− ≡ 2 − 2 0 01 ! T p T p T δ + (r 1)nn + mm kk ≡ − 2 − 2 More complicated soft shears are possible if these shears are combined with those previously found in the uniaxial case. where the axes of the ellipsoid are n, m and k (with the latter For example, in the uniaxial case, section (IV) above, subse- a third perpendicular axis given by k = n m), see ﬁg. (6). quent rotation about n after the rotation of n itself had no ef- ℓ ℓ ℓ ℓ × The biaxiality is p = 2 1− 2 = 1− 2 . ℓ1+ℓ2 ℓ fect since there was symmetry about n. Now that this is lost, ⊥ 6 such rotations will reorient the perpendicular directors m and k and thereby induce additional soft shears.

z B. General biaxial softness x y If, however, the axis of rotation does not coincide with a principal axis as in the previous examples, we discover soft modes of lower symmetry, which only lead back to an undeformed body after the chain distribution has been rotated by 2π. Consider as an example of reduced symmetry the situation when the chain distribution is rotated around a general axis in the xy plane at an angle β to the x axis (fig. 7): a rotation of π (fig. 8) transfers the director n (along the z-axis) into n, but the unit vectors m and k along the other two principal −axes end up in a general position in the xy-plane. The same final position of the ellipsoid of chain distribution can be ob- tained by a rotation around the z-axis by an appropriate angle. The corresponding body deformation leaves all z-coordinates invariant, but transforms the the xy-coordinates as if they have suffered a body rotation of 2β about z. If we were to choose the axis of rotation in an arbitrary way, we would not observe this feature of invariant z-components after a rotation of π. Only a full rotation of 2π would lead back into a state of higher symmetry, in this case, of course, the identity. By rotating the ellipsoid of chain distribution, we create soft FIG. 7: Rotation of the chain distribution around an axis (arrow) in modes, which are closely related to the underlying symmetry, the xy-plane at an angle of π/3 with the x-axis. The figure shows which is the crystal class of the biaxial ellipsoid, the dipyra- π 1 π 1 π 1 π 2 π intermediate states of rotation at an angle of 0, , 6 , 3 , 2 , 3 and midal orthorhombic class. 5 π π 6 . Fig. (8) shows the final stage of a rotation by an angle of . The Comparing with equation (7), we see that so far we have fine outline shows the initial body shape, whereas the highlighted set the rotation W α to be the identity. If, however, we were to outline illustrates the body deformation as the the ellipsoid of the allow a general rotation, we would find an even larger set of internal chain distribution rotates. Here, we have chosen p = 6/5 deformations, in fact described by six degrees of freedom: the and r = 200/45, or, equivalently, the principal axes of the ellipsoid to be 6, 3 and 10 respectively. additional rotations W α and the rotations connecting the principal frames of the initial and final ellipsoids each introduce three degrees of freedom. Note that in the infinitesimal case, we have S = 1 + δS and T T O = 1 + δO, hence δS = δS and δO = δO . Analogous − VI. ROTATIONS, SYMMETRIC STRAINS AND THE statements hold for the matrices S and O . This shows that ROOTS OF TENSORS ′ ′ the rotation matrices O and O′ are related to the antisymmetric part of the deformation in the infinitesimal limit. We have seen in equations (1)–(4) yielding softness that ro- Applied to the deformation λ, we write: tations are vital to understanding soft elasticity. The cartoon of fig. 5 shows that soft shears are not symmetric and hence λ = λL V or U λR, (13) there must be a component of body rotation present. Here · · we explore the role of rotations in symmetry requirements. We then explicitly extract the rotational component from gen- where the rotations are denoted by V or U depending upon eral distortions and from soft deformations in particular. The whether they act on the reference or target spaces respectively requirements for extracting the rotational components of the of the deformation. The form of the accompanying symmetric λR strain tensor are intimately related to finding the square roots deformationswill depend on the order; they are denoted by of tensors which we discuss here since they are used in con- and λL respectively and yield the Cauchy-Green tensors C and T T structing soft deformations. B respectively: C = λ λ and B = λ λ . Any general matrix M can be broken down into products We summarise classical· elasticity· theory to highlight the S O or O S′ of symmetric matrices S and S′ and orthogonal differences with nematic rubber elasticity. The deformation · · matrices O and O′, which can be restricted to proper rotations. (gradient) is: In effect, one has a pure shear preceded or followed by a body rotation (the Polar Decomposition Theorem [12]). λi j = ∂Ri/∂R0 j . (14) 7

seen that rotations and symmetric shears enter, both separately (through the de Gennes D1 term) and coupled (through the D2 term). In ﬁnite elasticity, we see in the Trace formula that the initial and ﬁnal orientations of the solid and its directors en- 0 1 ters via the tensors ℓ and ℓ− and there are not combinations like B and C which eliminate rotations. For instance, inserting U λR into the trace result, one obtains: ·

z 1 R T T 1 R F = µTr ℓ λ [U ℓ− U] λ (19) x 2 0 · · · · · x y y where the [...] have been inserted to emphasise that a body rotation of ST effectively adds to the rotation of n. That is, 1 T 1 FIG. 8: Rotation of the chain distribution by π around an axis in a new ℓ′ evolves: ℓ′− = U ℓ− U. (The additional rotation π the xy-plane at an angle of /3 with the x-axis. The figure shows is not necessarily coaxial with that which took n0 to n.) Not two views of the same situation: the final ellipsoid together with the unexpectedly, we see the effect of U compounded with the ro- outlines of the bodies which indicate the corresponding soft mode 1 deformation. The outline of the deformed body is highlighted. The tations implicit in ℓ− since they both live in the target space. parameters are the same as in fig. (7). Other approaches have been taken [8, 9] to soft elasticity which apparently circumvent the necessity to follow orientations in both spaces, and restore objectivity. One can measure If the targetspace (ST ) deforms under rotations represented by all deformations λ from an isotropic reference state, that is the matrix U, as R = U R, and the reference space (SR) de- there is encoded into the λ first a spontaneous deformation ′ · forms under rotations V as R0′ = V R0, then the deformation to current conditions of temperature, then a deformation im- tensor deforms as · posed with respect to this intermediate state. Under these conditions the free energy must automatically be invariant under λ ∂ ∂ T i′ j = Uik Rk/ R0lVl j (15) operations of V, since the reference state is isotropic and with- T T out a director n to keep track of. Difficulties arise then when λ′ = U λ V or λ = U λ′ V . (16) 0 · · · · nematic elastomers are only semi-soft, that is, they do not de- Thus λ records the character of both the target and refer- form entirely at constant free energy, because they do not have ence states, a property that will be essential in non-ideal ne- a high temperature isotropic reference state. However they do matic elastomers where an isotropic reference state cannot be suffer director rotation, very low energy trajectories in λ space reached. The connection with both spaces is quite different in and a lack of λyy relaxation perpendicular to the plane of n’s character from that of the Cauchy tensors. Thus the combina- rotation [13]. In these cases, the complete cancellation render- 2 R D2 tion ing µ = µ2 0 fails, but nevertheless deformations are 2 − 4D1 → T T T T qualitatively soft and one has to keep track of both directions C = λ λ = V λ′ U U λ′ V · · · · · · n0 and n as strain evolves. T T T = V λ′ λ′ V = V C′ V (17) · · · · · is manifestly invariant under body rotations U of the final (target) space ST and transforms as a second rank tensor in A. Square roots of tensors and the Polar Decomposition SR. Since isotropic systems are invariant to rotations V of SR, Theorem the system’s final energy must be invariant to rotations of SR. Thus F is a function of the rotational (in SR) invariants of C We quote the classical conditions[12] for the square roots of and is assured by the above of being invariant under rotations tensors, that arise too in the all-important condition for polar of ST . As an example, the isotropic rubber elastic free energy, decomposition that is required for nematic elastomers. 0 1 δ setting ℓ = ℓ− = in equation (6) is If A is non-singular, with µ distinct eigenvalues and with ν Jordan blocks, then A has 2µ and 2ν non-singular square 1 1 T ≥ ≤ F = µTr C = µTr V C′ V roots. At least one of these roots is a polynomial in A. 2 2 · · 1 T 1 The proof of the Polar Decomposition Theorem (PDT) also = µTrC′ V V = µTr C′ (18) 2 · · 2 offers a practical algorithm for decomposition: take a non- T T singular λ and construct the manifestly symmetric λ λ. Take (by cyclical properties of the trace). Likewise, B = λ λ , is λTλ λTλ invariant to rotations of the reference state and transform· s like the square root, G, of that is a polynomial in . Since T T a second rank tensor in the target state. λ λ is symmetric, then so is G = λ λ. Then define Q = Thus we see that “objectivity”, frame indifference, is built 1λ T δ into classical elasticity theory from the outset. Nematic elas- G− . Clearly Q Q = , that is Q representsq rotations. From tomers are much more subtle. In continuum theory we have this one recovers λ = G Q. 8

B. Symmetric strain and body rotation of soft modes which, for small distortions is also proportional to θ, bearing in mind that ∆ is constant at first order in θ: We investigate the special case of uniaxial soft modes in the 1 θ(r 1) xz plane. They are achieved by rotating the chain distribution d = ∆ − . (24) around the the y axis. The resulting deformation keeps the y- √r components of the body constant. Hence these soft modes are As we have seen in fig. 5, it also vanishes at θ π 2, where effectively 2 2 and can be easily related to equation (10) and = / no further accommodation of the shape tensor by rotation is fig. 5. × possible. In fig. 9, we show the rotation and off-diagonal An explicit example of the PDT for decomposing a general element of the symmetric shear occurring during soft defor- xz-distortion λ into a combination of symmetric distortion, λR, mations between θ = 0 and π. Notice that both quantities are followed by a body rotation, U, about the y-axis by an angle α is: 0.6 λ δ λ xx c s a d λR = δ λ = U . 0.4 soft ′ zz s c d b ≡ · − (with s = sinα and c = cosα). One can confirm that the rota- 0.2 tion is: π/2 π

tanα = (δ δ′)/(λxx + λzz) (20) − -0.2 α and thus for instance for sin : -0.4

1 2 2 2 sinα = (δ δ′) with ∆ = (λxx + λzz) + (δ δ′) . -0.6 ∆ − −

Of course there is no body rotation for λ symmetric, that is FIG. 9: Off-diagonal element d of the symmetric shear λR, and an- α δ = δ′. One can also confirm that the symmetric shear tensor gle of the rotation U (dotted line) plotted against director rotation is : θ for the soft deformations of fig. 5 for a nematic elastomer with anisotropy r = 20. R 1 λxx(λzz+λxx) δ (δ δ ) λxxδ+δ λzz λ = − ′ − ′ ′ . (21) ∆ λxxδ+δ λzz λzz(λzz+λxx)+δ(δ δ ) ′ − ′ θ linear in for small distortions. As the director of the chain The results (20) and (21) break the soft modes down into a distributions is rotated beyond π/2, the solid body rotation symmetric shear λR followed by a body rotation U through an α and the off-diagonal element of the shear d become nega- π angle α about the y-axis. Thus the soft mode is λ U λR. tive. At a director rotation of , the original body shape is soft = We continue to parameterise them with the director rotation· θ. recovered, as it should be in an uniaxial nematic system under The body rotation is through an angle α given by reflection of the director n n. Similarly, the behaviour→− of the pure shear is instructive: for sinθ cosθ(√r 1)2 this purpose, we plot in fig. 10 the two eigenvalues of λR, tanα = − 2√r + sin2 θ(√r 1)2 which serve as a good measure of the net presence of shear. − Note that, due to incompressibility, the product of the two tanθ(√r 1)2 = − . (22) 2√r + tan2 θ(r + 1) 4 For small shear (small director rotation θ), the component of body rotation is small too and proportional to θ: 3 (√r 1)2 α θ − . (23) ≈ 2√r 2 For large rotations, θ π/2, the rotation α vanishes as we π/2 π have seen in fig. 5. Thus≈ it is only at first that body rotation plays a part in accommodating the rotating chains. As the rotation of chains approaches π/2, the body is simply extended 0 or compressed along the principal axes of the original chain distribution. FIG. 10: The two eigenvalues of the pure shear λR for a nematic The corresponding symmetric shear strain d, the off- elastomer with anisotropy r = 20. For a director rotation of π/2, diagonal component of λR, is the soft mode deformation becomes a pure shear with extension and compression ratio √r or 1/√r respectively. 1 sinθ cosθ(r 1) d = − ∆ √r eigenvalues is constant to 1. Both eigenvalues are linear for 9 small θ, saturate at values √r and 1/√r respectively for a di- both cases we show geometrically what these deformations rector rotation of π/2, where the soft mode becomes a simple look like. They correspond to rotations of prolate distribu- compression and extension along the principal axes. At a ro- tions being accommodated by elastic strains of the body they tation of π, we recover the eigenvalues of the identity matrix. are inscribed into. ¿From equation (10), we see that the compression and ex- The elasticity of nematic elastomers depends, unlike in tension in the x and y directions are quadratic in θ for small classical elastomers, on body rotations (since these can be director rotations. In other words, uzz and uxx too are propor- with respect to the underlying nematic director). At finite tional to θ2 ∝ α2 ∝ d2 (by equations (24) and (23)). strains it is important to isolate the rotational component of the generally non-symmetric deformations. We make contact with the Polar Decomposition Theorem in this context. In do- VII. CONCLUSIONS ing so we discuss the roots of tensors that are employed in finding the manifold of soft deformation tensors. Soft deformations of nematic elastomers result from the rotation of the anisotropic chain shape distribution without distortion and therefore without rubber elastic free energy cost. Acknowledgments This is by contrast to classical rubber, where distorsion of the distribution lowers entropy and raises the free energy. Ex- S. K. acknowledges the support of an Overseas Research plicit forms of these soft modes are derived in the case of Scholarship, of the Cavendish Laboratory and of Corpus biaxial nematic elastomers using the Olmsted method of the Christi College. M. W. thanks the A. v. Humboldt Founda- square roots of tensors developed for the uniaxial case. For tion for the award of a Humboldt Research Prize.

[1] J. K¨upfer and H. Finkelmann, Macromol. Chem. Phys. 195, [8] A. deSimone and G. Dorfmann, Physica D 136, 175 (2000). 1353 (1994). [9] T. C. Lubensky, R. Mukhopadhyay, L. Radzihovsky, and [2] H. Finkelmann, E. Nishikawa, G. Pereira, and M. Warner, Phys. X. Xing, Phys. Rev. E (submitted) ??, ??? (2002). Rev. Lett. 87, 015501 (2001). [10] F. Hessel and H. Finkelmann, Makromol. Chem. 188, 1597 [3] M. Warner, P. Bladon, and E. M. Terentjev, J. Phys. II France 4, (1987). 93 (1994). [11] H. Leube and H. Finkelmann, Makromol. Chem. 192, 1317 [4] P. de Gennes, in Liquid Crystals of One and Two-Dimensional (1991). Order, edited by W. Helfrich and G. Heppke (Springer, Berlin, [12] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cam- 1980). bridge University Press, Cambridge, 1991). [5] L. Golubovic and T. C. Lubensky, Physical Review Letters 63, [13] G. Verwey, M. Warner, and E. Terentjev, J. de Phys. II 6, 1273 1082 (1989). (1996). [6] P. D. Olmsted, J. Phys II France 4, 2215 (1994). [7] M. Warner, J. Mech. Phys. Solids 47, 1355 (1999).