Elastic effects in a model of disordered nematic elastomers are numerically
investigated in two dimensions. Networks crosslinked in the isotropic phase
exhibit unusual soft mechanical response against stretching. It arises from
gradual alignment of orientationally correlated regions that are elongated
along the director. A sharp crossover to a macroscopically aligned state is
obtained on further stretching. The effect of random internal stress is also
discussed.Comment: 5 pages, 5 figure

Uchida, Nariya

English

arXiv

Institutional Repositories DataBase (IRDB)

Elastic effects in disordered nematic networks

application/pdf学術論文 (Article)174186 bytesjournal articl

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

Elastic effects in disordered nematic networks著者 内田  就也journal orpublication titlePhysical review. Evolume 60number 1page range R13-R16year 1999URL http://hdl.handle.net/10097/35295doi: 10.1103/PhysRevE.60.R13RAPID COMMUNICATIONSPHYSICAL REVIEW E JULY 1999VOLUME 60, NUMBER 1Elastic effects in disordered nematic networksNariya UchidaDepartment of Physics, Kyoto University, Kyoto 606, Japan~Received 23 February 1999!Elastic effects in a model of disordered nematic elastomers are numerically investigated in two dimensions.Networks crosslinked in the isotropic phase exhibit an unusual soft mechanical response against stretching. Itarises from a gradual alignment of orientationally correlated regions that are elongated along the director. Asharp crossover to a macroscopically aligned state is obtained on further stretching. The effect of randominternal stress is also discussed. @S1063-651X~99!50107-4#PACS number~s!: 61.30.Cz, 61.41.1e, 64.70.MdNematic elastomers and gels exhibit rich mechanical ef-fects due to elasticity-orientation coupling @1,2#. While aconsiderable number of theoretical studies has been directedto homogeneous systems, nematic elastomers are often in ahighly nonuniform polydomain state, in which the correla-tion length for the director orientation is typically of micronscales. Polydomain networks show unusual nonlinear elasticresponse against stretching @3–8#, often with an extremelylow stress over a sizable interval of strain. As the strain isincreased, the directors gradually rotate toward the directionof stretching until a macroscopically aligned state is attained.This structural change is called the polydomain-monodomain~P-M! transition. Attempting to describe the presumablyequilibrium polydomain textures, Terentjev and co-workers@9–11# proposed a random-field model analogous to thosefor random anisotropy magnets. They argued that crosslink-ers of anisotropic shapes act as sources of quenched disorder.On the other hand, the mechanical response is not yet wellunderstood. It is known that elasticity-mediated long-rangeinteractions among spatial inhomogeneities are crucial insystems such as metallic alloys @12,13# and gels @14#. Forpolydomain networks, the role of elastic interactions amongorientationally correlated regions ~‘‘domains’’! is yet to beclarified. In this Rapid Communication, we numerically in-vestigate the mechanical response and the domain structureof model nematic networks incorporating both rubber elas-ticity and quenched random anisotropy. Unusual soft re-sponse is obtained and is explained in terms of the elasticinteraction. We briefly discuss the effect of random internalstress as another kind of quenched disorder that can destroylong-range orientational order @15#.The total free energy of our model system is of the formF5Fel1FR1FF , where Fel , FR , and FF are, respectively,the rubber-elastic, random disorder, and Frank contributions.We assume networks brought deep into the nematic phaseafter crosslinking in the isotropic phase, and apply the affine-deformation theory of nematic rubber elasticity due toWarner et al. @16#. Then Fel is written in terms of the sym-metric deformation tensor Wi j5(]ri /]rk0)(]r j /]rk0), whereri0 and ri are the Cartesian coordinates of the material pointat the moment of crosslinking and after deformation, respec-tively. Summation over repeated indices is implied through-out unless otherwise stated. It is convenient to rewrite theoriginal form of Fel @16#, using the tensor Qi j5nin j2d i j /d , where n is the director, to obtain @17#PRE 601063-651X/99/60~1!/13~4!/$15.00Fel5m2 E dr~Wii2aQi jWi j!. ~1!The dimensionless coupling constant a(.0) is determinedby chain anisotropy and does not exceed d/(d21). Themodulus m is given by kBT multiplied by the crosslink num-ber density and a numerical prefactor (;1), which weaklydepends on the temperature. We consider the incompressiblelimit and impose the constraint detW51. The disorder freeenergy is assumed in the form given in @9–11#, and is rewrit-ten asFR5E drPi jQi j , ~2!where Pi j is a symmetric, traceless, Gaussian random tensorwith vanishing quenched average (^Pi j(r)&50) and withvariance^Pi j~q!Pi8 j8~2q!&5US d ii8d j j81d i j8d j i822d d i jd i8 j8D .~3!For the Frank free energy we assume the formFF5K2 E dr~n!2. ~4!Here we treat the two-dimensional case for numerical andanalytical advantages. Then, in the absence of elasticity, ourmodel reduces to the random-anisotropy XY model by re-garding the unit vector m5(2Qxx,2Qxy)5(cos 2u,sin2u) asthe spin variable, where u is the director orientation definedby n5(cos u,sin u). We consider deformations of the formri5l ir i01ui ~assuming no summation!, where lx5l andly51/l express the average deformation, and u5u(r) is theinternal displacement. Cooling into the nematic phase tendsto induce spontaneous elongation along the director @1,2#. Ifthe directors are uniformly aligned along the x axis, the elas-tic free energy ~1! is minimized at l5lm and u50 withlm5S 11a/212a/2D1/4. ~5!On the other hand, if there is no macroscopic deformation, orl51, the ground state is polydomain. Our questions concernR13 ©1999 The American Physical SocietyRAPID COMMUNICATIONSR14 PRE 60NARIYA UCHIDAhow domains spontaneously deform and to what degree theelastic free energy is reduced in such a highly nonuniformstate.The mechanical response was numerically simulated byvarying the macroscopic strain l and minimizing the freeenergy with respect to n and u. We solved the Langevinequation for the director,]n]t5~I2nn!S 2L dFdn 1hD , ~6!on a lattice. Here h is an uncorrelated Gaussian thermalnoise introduced to facilitate structural evolution. Withoutthe noise the minimization process would stop at one of thelocal minima close to the initial configuration. After ap-proaching the global minimum we turned off the noise asexplained below. The displacement u was determined bysolving the nonlinear equation d(Fel1Fv)/du50 with a re-laxation method, where Fv is an artificial free energy func-tional of u, which penalizes volume change. With thismethod the local volume was kept constant with errors below1% throughout the runs. Periodic boundary conditions wereimposed on n and u. The simulation was performed on a1282 square lattice with the grid size Dx51. We set K54and U51 for all of the runs, whereas m and a were variedfor different runs. In each run the external strain l wasslowly increased after an initial equilibration stage at l51.Occasionally, we stopped the increase of l and turned off thethermal noise for an interval of time. Thus a single run con-sisted of alternating periods of annealing ~with increasingstrain! and quenching. In each quench period we computedthe spatially averaged free energy density f 5 f el1 f R1 f Fand the orientation S5^2Qxx&5^cos 2u&. This procedure en-abled us to approximately minimize the free energy at nu-merous values of l in reasonable computational time. For afurther check, we then decreased l back from a large valuein a similar manner. A small hysteresis was obtained but itdoes not affect the description below.FIG. 1. Top, dimensionless total stress m21] f /]l; bottom, ori-entation S5^2Qxx&5^cos 2u&. Cases with different couplingstrengths a50.2,0.4,0.6,0.8 from left to right are compared withma254 fixed. The arrows indicate the corresponding values of lm .In Fig. 1 we show the strain-stress and strain-orientationrelations for several values of a with ma254 fixed. In bothcurves we can see a sharp crossover around l5lm(a). Be-low lm the total stress ] f /]l is vanishingly small andslightly positive. The average orientation increases almostlinearly in the same region. The free energy densities areplotted in Fig. 2. The elastic free energy has a slightly nega-tive slope in the region l,lm , while the disorder free en-ergy has a positive slope and makes the total stress slightlypositive. We chose the parameters so that the Frank contri-bution is much smaller than ma2, which is considered to bethe case in typical experiments. We also studied a few caseswith stronger or weaker elastic effects. For larger values ofma2 the shapes of the strain-elastic stress and strain-orientation curves were almost unchanged. For cases withma2&0.2, these two curves exhibited less sharp crossovers.In order to discuss the origin of the soft response it isuseful to examine the structure of the polydomain state atl51, for which an analytical treatment is possible in theweak coupling case a!1. We expand DFel5Fel@u#2Fel@0# with respect to u to obtainDFel5mE drF14 S ]ui]r j 1 ]u j]ri D22aQi j]ui]r jG . ~7!Eliminating the elastic field using the conditions of mechani-cal equilibrium dDFel /du50 and incompressibility u50, we have a nonlocal elastic interaction among orienta-tional inhomogeneities. We define new variables Q1(r) andQ2(r) through their Fourier transforms,Q1~q!5sin~2w!Qxx~q!2cos~2w!Qxy~q!, ~8!Q2~q!5cos~2w!Qxx~q!1sin~2w!Qxy~q!, ~9!where w is the azimuthal angle of the wave vector q5q(cos w,sin w). Then the average free energy density reads@17#f elul515mS 12 a22 ^Q12& D ~10!to order a2. Note that Q1 and Q2 satisfy ^Q121Q22&5^Qxx21Qxy2 &51/4. In the absence of the elastic coupling we haveFIG. 2. Free energy densities for ma254 and a50.4. The totalfree energy in the polydomain regime has a positive but small slopedue to the disorder contribution. The value m is subtracted from theelastic free energy density.RAPID COMMUNICATIONSPRE 60 R15ELASTIC EFFECTS IN DISORDERED NEMATIC NETWORKS^Q12&5^Q22&51/8. In its present asymmetry, ^Q12&.^Q22&arises to reduce the elastic free energy ~10!. In the elasticity-dominated limit where ma2 is much larger than the disorderand the Frank free energy densities, we expect ^Q12&!1/4,^Q22&!0, and f elul51!m(12a2/8). Indeed these are nu-merically confirmed as shown in the next paragraph. On theother hand, the elastic free energy density under the uniformdeformation with l5lm is also given by m(12a2/8) toorder a2. Thus, in the above limit, the P-M transition accom-panies only a small change of order a3 in the elastic freeenergy. To see how each domain is deformed at l51, weconsider the local elastic stress, which is given as s i j5m(] iu j1] jui2aQi j) from Eq. ~7!. After some calcula-tion, its variance in the mechanical equilibrium is obtained as^s i j2 &52m2a2^Q22&. ~11!In the elasticity-dominated limit, the variance of the quantitym21s i j5] iu j1] jui2aQi j vanishes due to the factor ^Q22&in Eq. ~11!, which means that each part of the system iselongated by 11a/4 (.lm) times along the local director.This, together with the numerical result on the mechanicalresponse, supports the following simple picture: In the poly-domain state each domain is uniaxially elongated by lmtimes along the local director, and thus the elastic free energyis equal to that for the monodomain state at l5lm ~Fig. 3!.The P-M transition in the region 1,l,lm proceeds viarotation of domains and does not change the elastic free en-ergy.Next we present numerical results on the polydomainstructure at l51, which was studied through the correlationfunction G(r)52^Qi j(r)Qi j(0)& and the degree of structuralasymmetry A5^Q12&2^Q22&. To accelerate the computationof the elastic field we assumed a weak coupling a50.1, andsolved dDFel /du50 under the constraint u50 using fastFourier transform instead of the relaxation method above.The amplitude of the thermal noise was set constant in aninitial stage and then gradually reduced to zero at a constantrate. The correlation function is computed for the final stateand averaged over 20 independent runs for each set of pa-rameters. Runs were sufficiently long to insure that the initialconfigurations with uniform and random orientations giveindistinguishable results for G(r). Shown in Figs. 4 and 5are the correlation function and the correlation length R de-fined by G(R)/G(0)51/2. The elastic coupling increases thecorrelation length without qualitatively affecting the form ofFIG. 3. Schematic illustration of the P-M transition. The ellipsesrepresent domains under spontaneous deformations ~from circles atthe moment of crosslinking!, and the arrows in them indicate thelocal director orientations. The transition from polydomain at l51 ~left! to monodomain at l5lm ~right! does not change theelastic free energy if every domain is elongated by lm times alongthe local director.the correlation function. We could not deduce a quantitativedecay law for G(r) from the relatively small number ofsamples, but the decay was slightly faster than exponentialnear the origin. For the nonelastic case the same feature wasobtained in the Monte Carlo simulation by Gingras and Huse@18# in the presence of thermal noise, while Yu et al. @11#obtained exponential decay using free boundary conditions.Another important factor affecting G(r) is the disorderstrength. More systematic study of the decay law is left tofuture work. In Fig. 5 the degree of asymmetry A is alsoshown. With increasing the magnitude of the elastic interac-tion it approaches to the upper limit 1/4 as expected.Finally we discuss the effect of random internal stressarising from microscopic heterogeneities in the networkstructure, which are intrinsic to gels @19#. We restrict ourdiscussion to the case l51 with small internal deformations.In the expansion of the elastic free energy with respect tou, there will arise an additional term,DFel ,R5E drRi j ]ui]r j , ~12!where Ri j is the Gaussian random stress with ^Ri j(r)&50and withFIG. 4. Correlation functions G(r) as a function of the scaleddistance r/R . It is insensitive to the elastic interaction. Inset, semi-logarithmic plot.FIG. 5. Top, correlation length R; bottom, structural asymmetryA5^Q12&2^Q22&.RAPID COMMUNICATIONSR16 PRE 60NARIYA UCHIDA^Ri j~q!Ri8 j8~2q!&5V1d i jd i8 j81V2~d ii8d j j81d i j8d j i8!.~13!Eliminating the elastic field from DFel1DFel ,R we have anew interaction term a*drR1SQ1, where R1S is defined usingthe shear component Ri jS 5Ri j2Rkkd i j /d by an equation par-allel to Eq. ~8! asR1S~q!5sin~2w!RxxS ~q!2cos~2w!RxyS ~q!. ~14!Treating this interaction as a weak perturbation as in @20#, wecan see that it renders the equilibrium correlation length fi-nite even in the absence of the disorder free energy ~3!. Wemention that Golubovic´ and Lubensky @15# discussed an-other mechanism of long-range–orientational-order breakingdue to random stress. Their argument is based on the obser-vation that the amplitude of thermal fluctuations around auniformly aligned state diverges. Its relevance to the presentcase of nematic networks is limited in that their free energydoes not explicitly include the orientational degree of free-dom.To summarize, we have numerically obtained a soft me-chanical reponse during the P-M transition. It originates fromstructural self-organization of domains due to the long-rangeelastic interaction, and should be distinguished from the softelasticity @2,21# of uniformly oriented networks. The elasticcontribution to the stress is slightly negative in the transitionregion. We have found a positive disorder contribution to thestress. The elastic interaction is found to increase the corre-lation length. We have demonstrated that random internalstress acts as a random field on the director. Further experi-mental and theoretical studies are necessary to examine itsrelevance to real polydomain textures.The author gratefully acknowledges Professor A. Onukifor helpful discussions and a critical reading of the manu-script, and Dr. E. M. Terentjev for valuable comments on ourrelated work.@1# P. G. de Gennes, C. R. Seances Acad. Sci., Ser. B 281, 101~1975!; in Liquid Crystals of One- and Two-Dimensional Or-der, edited by W. Helfrich and G. Heppke ~Springer, Berlin,1980!, p. 231.@2# M. Warner and E. M. Terentjev, Prog. Polym. Sci. 21, 853~1996!, and references cited therein.@3# J. Scha¨tzle, W. Kaufhold, and H. Finkelmann, Macromol.Chem. 190, 3269 ~1989!.@4# J. Ku¨pfer and H. Finkelmann, Macromol. Chem. Phys. 195,1353 ~1994!.@5# G. H. F. Bergmann, H. Finkelmann, V. Percec, and M. Zhao,Macromol. Rapid Commun. 18, 353 ~1997!.@6# E. R. Zubarev, R. V. Talroze, T. I. Yuranova, N. A. Plate, andH. Finkelmann, Macromolecules 31, 3566 ~1998!.@7# S. M. Clarke, E. M. Terentjev, I. Kundler, and H. Finkelmann,Macromolecules 31, 4862 ~1998!.@8# S. M. Clarke and E. M. Terentjev, Phys. Rev. Lett. 81, 4436~1998!.@9# S. M. Clarke, E. Nishikawa, H. Finkelmann, and E. M. Ter-entjev, Macromol. Chem. Phys. 198, 3485 ~1997!.@10# S. V. Fridrikh and E. M. Terentjev, Phys. Rev. Lett. 79, 4661~1997!.@11# Y. -K. Yu, P. L. Taylor, and E. M. Terentjev, Phys. Rev. Lett.81, 128 ~1998!.@12# J. W. Cahn, Acta Metall. 9, 795 ~1961!.@13# A. Onuki, J. Phys. Soc. Jpn. 58, 3065 ~1989!; 58, 3069 ~1989!.@14# A. Onuki, Adv. Polym. Sci. 109, 63 ~1993!, and referencescited therein.@15# L. Golubovic´ and T. C. Lubensky, Phys. Rev. Lett. 63, 1082~1989!.@16# M. Warner, K. P. Gelling, and T. A. Vilgis, J. Chem. Phys. 88,4008 ~1988!.@17# N. Uchida and A. Onuki, Europhys. Lett., 45, 341 ~1999!.@18# M. J. P. Gingras and D. A. Huse, Phys. Rev. B 53, 15 193~1996!.@19# E. Mendes, P. Lindner, M. Buzier, F. Boue, and J. Bastide,Phys. Rev. Lett. 66, 1595 ~1991!.@20# Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 ~1975!.@21# P. D. Olmsted, J. Phys. II 4, 2215 ~1994!.

Physical Review E

https://tohoku.repo.nii.ac.jp/?action=repository_action_common_download&item_id=4799&item_no=1&attribute_id=18&file_no=1

Elastic Effects in Disordered Nematic Networks

Abstract

Similar works

Full text

Available Versions

Institutional Repositories DataBase (IRDB)

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ