Chiral phase-coexistence in compressed double-twist elastomers
CChiral phase-coexistence in compressed double-twist elastomers
Matthew P. Leighton a , b , Laurent Kreplak a , and Andrew D. Rutenberg a , ∗ February 5, 2021We adapt the theory of anisotropic rubber elasticity to model cross-linked double-twist liquid crystal cylinders such asexhibited in biological systems. In mechanical extension we recover strain-straightening, but with an exact expressionin the small twist-angle limit. In compression, we observe coexistence between high and low twist phases. Coexistencebegins at small compressive strains and is robustly observed for any anisotropic cross-links and for general double-twistfunctions – but disappears at large twist angles. Within the coexistence region, significant compression of double-twistcylinders is allowed at constant stress. Our results are qualitatively consistent with previous observations of swollen orcompressed collagen fibrils, indicating that this phenomenon may be readily accessible experimentally.
Chiral nematic (cholesteric) liquid crystals can exhibit a double-twist structure within a cylindrical geometry, in which a moleculardirector field ˆ n = − sin ψ ( r ) ˆ φ + cos ψ ( r ) ˆ z has a radius-dependenttwist angle ψ ( r ) with respect to the cylindrical axis ˆ z . Double-twist structures are observed in biological systems such as the ker-atin macrofibrils in hair or wool , or the collagen fibrils foundin skin, bone, tendon, and the cornea of the eye . They are alsofound within the “blue phases” of chiral liquid crystal systems .Biological tissues often have substantial amounts of intermolec-ular cross-links. Enzymatic cross-linking can mechanically andthermodynamically stabilize double-twist collagen fibrils – andis crucial for healthy tissue formation. Substantial disulfide cross-linking is seen in hair . Non-enzymatic cross-linking, due to ad-vanced glycation endproducts (AGE) , can also accumulate invarious tissues.The elastomeric theory developed by Warner et al enablesthe calculation of mechanical properties of anisotropically cross-linked nematic liquid crystals. Previous work has concentratedon bulk cholesteric liquid crystals, modelling longitudinal strainsapplied perpendicular to the initial molecular director field – par-allel to the cholesteric twist axis. These systems exhibit a discon-tinuous director-field reorientation under extension , which canindicate a phase transition. Subsequent treatments of cholestericsystems have considered mechanical response to axial strains andelectromagnetic fields within the limit of linear elasticity the-ory , as well as phase transition behaviour under extension andcompression with variable chiral solvents . Equilibrium phasetransitions of cylindrical double-twist elastomers have been con-sidered, but without consideration of mechanical strain effects.Other applications have included tunable optical or acoustic properties of these systems.Recent experimental work has demonstrated plastic torsionalbuckling in compressed cross-linked (ex-vivo) collagen fibrils. Since compression in elastomeric systems and the mechanicalproperties of elastomeric double-twist cylinders are relatively un-explored, we have explored whether elastomeric theory could a Department of Physics and Atmospheric Science, Dalhousie University, Halifax,Nova Scotia, B3H 4R2, Canada. b Department of Physics, Simon Fraser University, Burnaby, British Columbia, V5A1S6, Canada. ∗ Corresponding author, email: [email protected] help us understand the coupling between chiral structure and me-chanical strain in these systems. We find that it does, and thataxially compressed double-twist elastomers exhibit a novel chiralphase coexistence.Our approach is general. We first compute an expression forthe free energy density for a double-twist elastomer, which con-stitutes a full thermodynamic fundamental relation for the sys-tem. Using this fundamental relation as a model we explore themechanics of double-twist cylinders under both extension andcompression. We focus in particular on the changes in molecularorientation as well as the internal stresses within the elastomer.Under sufficient compression we find that large twist angles arealways observed – so we develop a general approach valid for alltwist-angle functions ψ ( r ) . We also develop a small-angle approx-imation that allows us to derive analytic expressions for manyproperties of collagen fibrils under both extension and compres-sion. We limit our numerical studies to two model twist functions– a rope-like constant twist ψ = ψ , or a linear twist ψ = ar thatcorresponds to a pure torsion of a cylinder. A constant twist anglehas been proposed for corneal collagen fibrils . A linear twiststructure has been proposed for the cores of blue phases , and hasbeen directly observed in hair or wool macrofibrils . Double-twist behavior that is close to either linear or constant twist isalso seen in both equilibrium and non-equilibrium models ofdouble-twist collagen fibrils. When the cross-link configurational entropy dominates, the free-energy density within a nematic liquid crystal elastomer understrain is : f = µ Tr ( (cid:96) λ (cid:62) (cid:96) − λ ) . (1)The energy scale µ = k B T ρ is proportional to both temperature T and the volumetric cross-link density ρ . The applied deformationgradient tensor is λ . The tensors (cid:96) and (cid:96) describe the initial andpost-strain structure of the elastomer in terms of the initial andpost-strain molecular director fields ˆ n and ˆ n : (cid:96) = δ + ( ζ − ) ˆ n ⊗ ˆ n , (2a) (cid:96) = δ + ( ζ − ) ˆ n ⊗ ˆ n , (2b)where ⊗ indicates a tensor product. The anisotropy parameter ζ is the ratio between the cross-link orientation in the directions a r X i v : . [ phy s i c s . b i o - ph ] F e b arallel to and perpendicular to the molecular director field ˆ n . ζ istypically taken to be greater than 1 in modelling approaches ,which is consistent with experimentally observed values of thecross-link anisotropy in nematic liquid crystals .We assume that both the post-strain and zero-strain directorfields retain a double-twist structure, so that ˆ n = − sin ψ ( r ) ˆ φ + cos ψ ( r ) ˆ z with strain and ψ ( r ) with zero-strain. We consider anextension or compression by a factor of λ along the cylinder axis.We assume that both ends of an incompressible cylinder are fullyclamped – with no shear or rotation. Our deformation is thendescribed by the coordinate transform z → λ z , r → λ − / r , and φ → φ . The deformation gradient tensor for this deformation is λ = √ λ √ λ
00 0 λ . (3)For small strains, the stress field σ , strain field ε , and Helmholtzfree energy density f are related by the thermodynamic rela-tion σ i j = (cid:0) ∂ f / ∂ ε i j (cid:1) T , N , where the infinitesimal strain tensoris ε = (cid:16) λ (cid:62) + λ (cid:17) − δ . We are specifically interested in axial (i.e.longitudinal) deformations. The axial stress within the cylinder(or, e.g., fibril) is then given by σ = ∂ f ∂ λ , (4)where we use a scalar σ for simplicity. Similarly we use the scalar ε ≡ ε zz , so that the axial strain is ε = λ − . Using the deformation gradient tensor eqn (3), for general doubletwist director fields ˆ n and ˆ n we evaluate the free energy densitydefined in eqn (1): f ( r ) = µ (cid:26) λ + λ [ + ( ζ − ) sin ψ ][ + ( ζ − − ) sin ψ ]+ λ [ + ( ζ − ) cos ψ ][ + ( ζ − − ) cos ψ ]+ λ / ( − ζ − ζ − ) sin (cid:0) ψ (cid:1) sin (cid:0) ψ (cid:1)(cid:27) . (5)The free energy density, f , constitutes a thermodynamic funda-mental relation for a strained double-twisted liquid-crystal elas-tomeric cylinder (fibril). The free energy is minimized in equilib-rium; this lets us determine the post-strain twist ψ ( r ) along withthe stress within the cylinder. Our expression for f applies forgeneral double-twist ψ ( r ) , extension λ , and anisotropy ζ .The post-strain twist angle function ψ ( r ) minimizes f , so that ∂ f / ∂ ψ = for all r . This gives ψ ( r ) =
12 cot − (cid:18) ( ζ + )( λ − ) + ( ζ − )( λ + ) cos ( ψ ) λ / ( ζ − ) sin ( ψ ) (cid:19) , (6)where we take ψ ∈ [ , π / ] . This applies for general ψ ( r ) , λ , and ζ . We can solve equations 4, 5, and 6 numerically. Common tan-gent constructions for phase diagrams are performed using cus- − − − −
5% 0% 5% 10%0 . . . . ψ A) ψ = 0 ψ = 0 . ψ = 0 . ψ = 0 . − − − −
5% 0% 5% 10%
Fibril Strain . . . . fµ B) Fig. 1
A) The twist angle ψ as a function of longitudinal (axial) strain ε = λ − , for various values of ψ . Positive and negative strains correspondto extension and compression of a cylindrical fibril, respectively. B) Thefree energy landscape f / µ as a function of strain for several values of ψ . Horizontal brackets indicate the coexistence region for correspondingvalues of ψ ; for ψ = . there is no coexistence region. In both A) andB) we use a default value of ζ = . . tom code. All of our numerical and plotting code is available onGitHub . While we only show numerical results below for ζ > ,we note that our analytical results above also apply for ζ < . Using eqn. (6), ψ is plotted as a function of strain ε = λ − forvarious values of constant twist ψ in Fig. 1A, with ζ = . . Wesee that compression monotonically increases ψ .Fig. 1B shows the corresponding value of the free energy f ,where we have used the post strain twist angle ψ that minimizes f . For sufficiently small ψ we find a substantial coexistenceregion between two compressional strains ( ε H and ε L ) that canbe identified by a standard common-tangent construction – with f (cid:48) H = f (cid:48) L and f L = f H + f (cid:48) H ( ε L − ε H ) . This coexistence allows thesystem to further reduce the free-energy of the system, and de-termines thermodynamic equilibrium wherever the free energy isnot a convex function of λ .We show the coexistence region in Fig. 2A for various values of ε , ψ , and ζ . We observe coexistence at sufficiently small ζ and ε < for all ψ (cid:46) . , while with ψ = we observe coexistenceat all ζ (cid:54) = and compressive strains ε < . One consequence ofthe coexistence across different strains is that the coexistence isalso across different twist-angles ψ – as determined by eqn (6). InFig. 2B we show coexistence curves on the ζ − σ plane for variousvalues of ψ . As ψ increases the extent of the coexistence regiondecreases – until it disappears at ψ (cid:39) . . − − −
5% 0%
Fibril Strain . . . . . ζ coexistenceregion (cid:15) L (cid:15) H A) − . − . − . − . − . − .
05 0 . Stress σ/µ . . . . . ζ B) ψ = 0 ψ = 0 . ψ = 0 . ψ = 0 . ψ = 0 . Fig. 2
A) Coexistence regions for various values of ψ , plotted versuscross-linking anisotropy ζ and strain ε . The smaller and larger strainboundaries of coexistence, ε L and ε H respectively, are indicated. Coex-istence is not observed for ψ (cid:38) . . B) Within the coexistence regionstress is constant. The coexistence regions are therefore lines when plot-ted versus ζ and stress σ / µ . When increasing compressive strains enter the coexistence re-gion, at e.g. ε L in Fig. 2A, a slowly increasing fraction φ of thesystem will have local strains ε H – while the remainder fraction − φ will have unchanging strains at ε L . This is the “lever rule” ofphase coexistence, and we have that φ = ( ε − ε L ) / ( ε H − ε L ) where ε is the average fibril strain. In other words, the free energy f is a linear function of ε within coexistence. This implies that thestress σ = ∂ f / ∂ λ is constant within the coexistence region. Thisconstant value was shown in Fig. 2B vs ζ . In Fig. 3 we show theconstant coexistence region in a plot of stress σ vs strain ε . To demonstrate that this phase transition behaviour extends tomore general double-twist director fields, we also consider a lin-ear twist field ψ ( r ) = ψ surf r / R – where ψ surf is the surface twistand R is the fibril radius. This is a model for, e.g., corneal collagenfibrils or keratin macrofibrils .For this inhomogeneous director field, minimizing the volumeaveraged free energy density (cid:104) f (cid:105) = (cid:82) R f ( r ) rdr / R still yieldseqn (6), since f has no dependence on the derivatives of ψ ( r ) .Fig. 4A shows (cid:104) f (cid:105) using eqn (6). We see that, like in the constant-twist case, the free energy is a non-convex function of the strainfor sufficiently small surface twist angles. Thus we still observephase coexistence in the case of a linear twist function.Phase-coexistence for linear twist may be unsurprising, given − − − −
5% 0% 5% 10%
Fibril Strain − . − . − . . . σµ (cid:15) L (cid:15) H ψ = 0 ψ = 0 . ψ = 0 . ψ = 0 . Fig. 3
Stress-strain curves of σ / µ vs ε for constant-twist elastomerswith various initial twist angles. For ψ (cid:46) . coexistence is observed,and a constant stress coexistence is observed between a high and lowcompressive strain, ε H and ε L respectively as indicated. Negative strainindicates compression. We use ζ = . . − − − −
5% 0% 5% 10%
Fibril Strain . . . h f i µ A) ψ surf = 0 . ψ surf = 0 . ψ surf = 0 . . . . . . . r/R . . . . ψ B) (cid:15) H = − . (cid:15) L = − . ψ ( r ) (cid:15) = +5% Fig. 4
For cylinders with an initially linear double-twist, ψ ( r ) = ψ sur f r / R .A) The volume-averaged free energy (cid:104) f (cid:105) / µ vs longitudinal strain ε forseveral values of ψ surf as indicated. B) The post-strain twist angle func-tion ψ ( r ) vs r / R for various values of applied strain ε , given an initiallinear twist function with ψ surf = . (corresponding to the solid blackcurves in A). Note that at ε H we observe a qualitative inversion of thetwist-function. We use ζ = . in both A) and B). coexistence is observed for similar constant twists. However, nearthe coexistence region the double-twist function ψ ( r ) that mini-mizes (cid:104) f (cid:105) is no longer linear in r – as shown by the ε = ε H curvein Fig. 4B. Furthermore, the twist angle function exhibits strikingchanges of both monotonicity and convexity under larger com-pressive strains. This can be clearly understood in the small ψ limit. .3 Small ψ limit The small ψ limit is useful for building intuition for the system,and is also observed in e.g. collagen fibrils . From eqn (6), wecan extract the leading behavior for small ψ : ψ ( r ) = (cid:40) ψ ( r )( ζ − ) λ / / (cid:0) ζ λ − (cid:1) , if ζ λ > , π / − ψ ( r )( ζ − ) λ / / (cid:0) ζ λ − (cid:1) , if ζ λ < ,(7)where the corrections are O ( ψ ) . This applies to any twist fieldas long as ψ ( r ) (cid:28) for all r , which explains the observed twistinversion seen in Fig. 4B.When ψ = , we can illustrate the phase-coexistence calcula-tion and explicitly show that coexistence is observed for all valuesof ζ (cid:54) = . From eqns (5) and (7), we have f / µ = (cid:40) / λ L + λ L / , if ζ λ > , ( + ζ − ) / ( λ H ) + λ H ζ / , if ζ λ < , (8)where we have used λ L for the low-twist small-strain branch and λ H for the high-twist large-strain branch. The common tangentconstruction for coexistence is given by f (cid:48) ( λ L ) = f (cid:48) ( λ H ) (whichalso determines the constant σ = f (cid:48) during coexistence), togetherwith f ( λ L ) − σλ L = f ( λ H ) − σλ H : − / λ L + λ L = − ( + ζ − ) / λ H + λ H ζ , / λ L − λ L = ( + ζ − ) / λ H − λ H ζ . (9)These equations are easily solved numerically, and corresponds tothe thin black curves in Fig. 2.We can take the ζ → ∞ limit in eqns (9) and obtain λ H = ( − + √ ) / ζ − / (cid:39) . ζ − / , λ L = λ H / ( −√ ) (cid:39) . ζ − / , and σ = − / λ L (cid:39) − . ζ / . This illustrates that coexistence can al-ways be observed under compression, even for ζ → ∞ , for smallenough ψ .Under extension (and with ζ > ), initial twist-angles alwaysdecrease as λ increases – they exhibit strain-straightening. Thiscan be seen generally in eqn (6), and also in the small-angle limitin eqn (7). We can therefore self-consistently take the small an-gle limit of f in eqn (5) and use eqn (4) to obtain the stress vsextensive strain: σ = µ (cid:16) λ − λ − (cid:17) − µ ( ζ − )( λ − )( λ + λ + ) (cid:0) λ [ ζ ( λ + ) − ] − (cid:1) λ ( ζ λ − ) ψ + O ( ψ ) . (10)We see that for small initial twist angles under extension, the lead-ing behavior agrees with standard isotropic rubber elasticity. We have considered the effects of axial strain on double-twistelastomeric cylinders. We have focused on the relatively simplecase of a constant twist angle, like a twisted rope, though ourresults also apply to twist angles that depend on radial distance from the cylinder center. Minimizing the standard entropic free-energy arising from anisotropic cross-linking, we obtain standardstrain-straightening under axial extension – with a simple analyticform at small twist angles.Under axial compression, we have identified a novel phase-coexistence between high and low twist double-twist phases thatbegins at small compressive strains. This phase-coexistence is ob-served for initial twist angles up to ψ (cid:39) . and for all non-zerovalues of the cross-link anisotropy parameter ζ . Notably, even aninitially achiral cylinder with ψ = will spontaneously exhibit astrong chiral rotation at moderate compressive strains. We wouldexpect physical achiral systems to exhibit spontaneous chiral sym-metry breaking as a result.The mechanical Euler buckling of elastic rods on compressionis well understood, and can be manipulated by micropatternedmaterials. Phase-coexistence under compression has also beenreported for nanostructured materials and for macrostructuredKirigami materials . Chiral-shape instabilities of axially com-pressed elastic rods can also be observed and are well under-stood. However, the chiral double-twist instabilities we describehere appear to be novel. In particular, they are evident even in aninitially achiral cylinder with ψ = .Mechanical transitions in double-twist elastomers have notbeen previously studied, to our knowledge. Xing and Baskaran showed that a double-twist configuration was a good candidateground state for an unstressed initially isotropic elastomeric cylin-der that was moved into a cholesteric phase – i.e. due to chang-ing Frank free-energy contributions. We have not included Frankfree-energy terms in our treatment.Mechanical transitions of strained elastomers have long beenstudied, but typically under extensional strain. Both axialextension and compression of bulk cholesteric elastomershave been studied. Singular transitions were predicted, thoughcoexistence was not explored. Since these transitions relied onrelatively weak Frank free-energy contributions, they may be lessrobust experimentally than the elastomeric transitions and coex-istence we report.Ex-vivo collagen fibrils appear to be well described by a double-twist configuration, and to be in the strongly cross-linked regimethat should be dominated by our elastomeric model . Recent ex-perimental studies of collagen fibrils under both extension andcompression appear to qualitatively support our results. Underaxial extension, Bell et al. observed strain-straightening of theaverage twist of corneal fibrils. This is consistent with Fig. 1 andeqn (7). With the assumption that the reported D-band strainequals the fibril strain, quantitative agreement with the Bell re-sults with anisotropy parameter ζ = . are obtained (not shown)– this motivated our choice of ζ in some figures. A more detailedanalysis of the differences between fibril strain and D-band strainin collagen fibrils under extension is in progress .Under compression, ex-vivo (cross-linked) collagen fibrils at-tached to elastic substrates exhibit buckled regions that coexistwith unbuckled regions along the fibril length . Qualitativelyconsistent with our results is that this coexistence starts at smallcompressional strains (as little as ). Also consistent with coex-istence, the amplitude of the buckled regions does not appear to ncrease with compressional strain but their frequency of appear-ance does. Furthermore, the buckled regions exhibit an increasedfibril diameter consistent with the smaller λ and increased trans-verse / √ λ predicted from our results in Fig. 2 and eqn (3).These AFM studies of compressed collagen fibrils were notable to resolve the fibril twist. However, much earlier EM studiesof swollen fibrils due to urea treatment did exhibit the coex-istence of strongly twisted swollen regions of the fibril with lesstwisted narrower regions. While these were not studies of com-pressed fibrils, the qualitative similarity to our proposed coexis-tence indicates that a similar phenomenon may be observed asosmotic pressure is varied. We do note that both of these exper-imental studies reported irreversible plastic damage of thefibrils. We may therefore expect quantitative differences with ourequilibrium results.Polarization-resolved second harmonic generation (P-SHG) mi-croscopy, a technique which takes advantage of non-linear opti-cal phenomena to measure volume-averaged anisotropy withina sample, could potentially be used to measure the change inmolecular twist within different types of collagen fibrils underextension or compression. P-SHG anisotropy measurements ofcollagen-rich tissues such as full tendons have been made in re-cent years , however no experimental measurements have yetbeen made at the single-fibril level. Measurements of twist withinstrained fibrils would allow our model to be tested further.Mechanical effects of coexistence may be easier to detect fromdetailed stress-strain curves. At coexistence, substantial compres-sive strains can be realized at constant stress – as detailed inFig. 2 and 3. Short biaxially strained cylinders embedded withinan elastic substrate provide an accessible geometry for explor-ing the compression effects we describe here. Long and unsup-ported double-twist cylinders under compression would exhibitEuler buckling, which might complicate axial strain applicationand estimation. However, lateral reinforcement can significantlyincrease the load that can be applied to long cylinders with-out buckling –– as could larger radius cylinders. As discussedabove, compressive studies have been done by attaching singlefibrils to pre-stretched elastic substrates though stress is noteasily measured in that configuration.While our focus is on thermodynamic equilibrium elastomericconfigurations, an interesting dynamical consequence would beobserved if the system was rapidly mechanically “quenched” to acompressive strain within the spinodal region, with ∂ f / ∂ λ < ,where the system will spontaneously nucleate both phases. Wewould expect interfaces between the coexisting phases to haveexcess free-energy, as described by a more detailed treatment thatincludes gradient terms in the director field and characterized byFrank elastic coefficients . To reduce such interfacial costs, thesystem would then slowly “coarsen” towards bulk coexistence.Such coarsening could be exceptionally slow since it would notbe driven by interfacial curvature but instead by exponentially-small interactions between distant interfaces . As such it may beexperimentally accessible, and could provide details of the non-elastomeric contributions to the free-energy.Two open questions remain in terms of our calculation. Thefirst is the effect of the Frank free-energy terms for a weakly cross-linked elastomer. We anticipate a rich phase-diagram un-der compression. The second is the role of shear deformationswhen strained fibrils are allowed to freely rotate – as might be ex-pected in some experimental setups. We will address this secondquestion in future work . Conflicts of interest
There are no conflicts to declare.
Acknowledgements
We thank the Natural Sciences and Engineering Research Coun-cil of Canada (NSERC) for operating Grants RGPIN-2018-03781(LK) and RGPIN-2019-05888 (ADR). MPL thanks NSERC for sum-mer fellowship support (USRA-552365-2020), and a CGS Mastersfellowship.
Notes and references
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