Table Of ContentDynamics, dynamic soft elasticity and rheology of smectic-C elastomers
Olaf Stenull
Fachbereich Physik, Universit¨at Duisburg-Essen, Campus Essen, 45117 Essen, Germany
T. C. Lubensky
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
(Dated: February 6, 2008)
6
0 We present a theory for the low-frequency, long-wavelength dynamics of soft smectic-C elastomers
0
with locked-in smectic layers. Our theory, which goes beyond pure hydrodynamics, predicts a
2
dynamic soft elasticity of these elastomers and allows us to calculate the storage and loss moduli
n relevant for rheology experimentsas well as themode structure.
a
J PACSnumbers: 83.80.Va,61.30.-v,83.10.-y
3
1
Smectic elastomers [1] are rubbery materials that have displacement variable, labels the position of the mass
] the macroscopic symmetry properties of smectic liquid point x in the deformed (target) material. Lagrangian
t
f crystals [2]. They are sure to have intriguing properties, elasticenergiesareformulatedintermsofthestrainten-
o
some of which have already been studied experimentally sor u which, in its linearized form, has the components
s
. and/or theoretically [1]. Very recently, seminal progress u = 1(η +η ), where η =∂ u are the components
t ij 2 ij ji ij j i
a has been made on smectic-C (SmC) elastomers forming of the displacement gradient tensor η.
m
spontaneouslyfromasmectic-A(SmA)phaseuponcool-
The elastic energy density f of the SmC elastomersof
- ing. Hiraoka et al. [3] for the first time produced a mon-
interest here can be divided into two parts [4]
d
odomain sample of such a material and carried out ex-
n
o periments demonstrating its spontaneous and reversible f =fu+fu,n, (1)
c deformation in a heating an cooling process. Also very
[ recently, it was discovered theoretically that such a ma- wherefudependsonlyonuandfu,ndescribesthedepen-
1 terialexhibitsthefascinatingphenomenonofsoftelastic- dence off onthe Frankdirectornincluding its coupling
v ity[4],i.e.,certainelasticmodulivanishasaconsequence to the displacement variable. In the following we choose
7 of the spontaneoussymmetry breakingwhereforestrains thecoordinatesystemsothatthez-axisisparalleltothe
8 along specific symmetry direction cost no elastic energy director of the initial SmA phase and the x-axis is par-
2
and thus cause no restoring forces. allel to the direction of tilt in the resulting SmC phase
1
0 On one hand, due to the aforementioned experimen- so that the equilibrium directorcharacterizingthe unde-
6 tal advances, dynamical experiments on soft SmC elas- formed SmC phase is of the form n0 = (c,0,√1 c2)
−
0 tomers,such as rheologyexperiments of storageandloss withcbeingthe orderparameterofthe transition. With
t/ moduliorBrillouinscatteringmeasurementsofsoundve- these conventions, fu can be written in the same form
a
locities, seem within reach. On the other hand, there as the elastic energy density of conventional monoclinic
m
exists, to our knowledge, no dynamical theory that solids [6],
-
nd cmoeunldts.beHehreelpwfuelpirnesienntteraprtehteionrgytfhoersethkeinlodws-forfeqeuxepnecryi-, fu = 21Cxyxyu2xy+Cxyzyuxyuzy+ 12Czyzyu2zy
o long-wavelength dynamics of soft SmC elastomers with + 1C u2 + 1C u2 + 1C u2 + 1C u2
c 2 zzzz zz 2 xxxx xx 2 yyyy yy 2 xzxz xz
locked-in layers that goes beyond pure hydrodynamics.
: +C u u +C u u +C u u
v As in standard elastic media and nematic elastomers [5] zzxx zz xx zzyy zz yy xxyy xx yy
Xi a purely hydrodynamical theory of SmC elastomers in- +Cxxxzuxxuxz+Cyyxzuyyuxz+Czzxzuzzuxz, (2)
volves only a displacement field u and not the Frank di-
r but with constraints relating the three elastic constants
a rectorn,whichrelaxestothe localstrainin anonhydro-
in the first row. These latter constants can be expressed
dynamic time τn. We go beyond hydrodynamics, by in- in terms of an overall elastic constant C¯ and an angle
cluding nin ourtheory,because dynamicalexperiments,
θ, which depends on the order parameter c, as C =
likerheologymeasurements,typicallyprobeawiderange C¯cos2θ,C =C¯cosθsinθ,andC =C¯sin2xθy.xyNe-
offrequenciesthatextendsfromhydrodynamicregimeto xyzy zyzy
frequencies well above it. glectingcontributionsfromtheFrankenergy,fu,n canbe
stated as
Smectic elastomers are, like any elastomers, perma-
nently crosslinkedamorphous solids whose static elastic- fu,n = 21∆[Qy+αuxy+βuyx]2 (3)
ity is most easily described in Lagrangiancoordinates in
which x labels a mass point in the undeformed (refer- where ∆ is a coupling constant and where α and β are
ence) material and R(x) = x+u(x), where u(x) is the dimensionless parameters. The variable Q stands for
y
2
Q = n cη √1 c2η , where η and η As mentioned above,Eq.(3) omits contributions from
y y Ayx Ayz Ayx Ayz
− − −
are components of the antisymmetric part of η. the Frank elastic energy for director distortions, which
Equations (1) to (3) imply that SmC elastomers are are higher order in derivatives than those arising from
soft under static conditions. Imagine that Q has re- network elasticity. Without the Frank energy, our dy-
y
laxed locally to Q = αu βu so that f is ef- namicaltheorymissesdiffusivemodesalongcertainsym-
y xy yx
fectively reduced to fu.−Then,−due to the above rela- metrydirectionswheresoundvelocitiesvanish. Including
tions among the elastic constants, deformations charac- theFrankenergy,ontheotherhand,makestheequations
terized by Fourier transformeddisplacements u eˆ and of motion considerably more complicated. To keep our
y
wavevectors q eˆ2 = ( sinθ,0,cosθ) or, alterknatively, presentation as simple as possible, we will therefore, for
u eˆ2 and q ekˆy cost n−o elastic energy and hence cause the mostpart, exclude the Frank energy. Whenit comes
nokrestoringfkorces. Theeffectsofsoftelasticityaremore tostatingresultsfortheaforementioneddiffusivemodes,
evident in a coordinate systems rotated though θ about however, we will include Frank contributions in order to
the y-axis in which eˆx′ = (cosθ,0,sinθ), eˆy′ = eˆy, and present complete results.
eˆz′ = eˆ2. In this system, Cx′y′z′y′ and Cz′y′z′y′ vanish FromEq.(4a)wecanderiveanequationofmotionfor
impyling there is no energy cost for shears uz′y′ in the Qy. In frequency space, this equation can be written as
′ ′
z y -plane. If the elastomer is crosslinked in the SmC
1+iωτ3 1+iωτ2
phase, these moduli become nonzero. Here, we will not Q = α u β u (6)
y yx yz
consider such semi-soft SmC elastomers. − 1−iωτ1 − 1−iωτ1
Now, let us formulate our dynamical theory. Dynami- where we have introduced the relaxation times τ1 =
cal equations for u and n can be derived using standard 1/(Γ∆), τ2 = λ√1 c2/(Γ∆β), τ3 = λc/(Γ∆α). As we
Poisson-bracketapproaches [7], with the result [5] will see further belo−w, our dynamical equations predict
δ nonhydrodynamic modes with a decay time (“mass”) τ1
n˙i =λijk∂ju˙k−ΓδnH , (4a) which implies τ1 =τn.
i WithhelpofEqs.(6)and(4b)wederiveeffectiveequa-
δ δ
ρu¨ =λ ∂ H H +ν ∂ ∂ u˙ , (4b) tionsofmotionsintermsofthedisplacementsonlywhich
i kji j ijkl j l k
δn − δu
k i can be cast as
where is the elastic energy of the system, ν is the
ijkl 2
H ρω u = ∂ σ (ω) (7)
viscositytensor,Γisadissipativecoefficientwithdimen- i − j ij
sions of an inverse viscosity, and
with a symmetric stress tensor σ given by
λ = 1λ δT n +δT n + 1 δT n δT n , (5)
ijk 2 (cid:0) ij k ik k(cid:1) 2 (cid:0) ij k− ik k(cid:1) σ (ω)=C (ω)u +C (ω)u , (8a)
µ µν ν µxz xz
where δiTj =δij−ninj. As they stand, Eqs. (4) are valid σxz(ω)= 12Cxzν(ω)uν + 12Cxzxz(ω)uxz, (8b)
for any liquid crystal elastomer with a defined director, σ (ω)= 1CR(ω)u , (8c)
likee.g.,nematic,SmAandSmC elastomers. Todescribe ξ 2 ξχ χ
SmC elastomers we have have to specify and νijkl ac- where we use a compact notation with indices µ, ν run-
cordingly. From the above it is clear thaHt = d3xf ning over xx, yy and zz and indices ξ, χ running over
H R
with f as given in Eq. (1). The viscosity tensor entering xy and zy. C (ω) with no superscript R stands for
ijkl
here is that of a monoclinic system. It has 13 indepen- C (ω) = C iων . The superscript R indicates
ijkl ijkl ijkl
dent components and it can be parameterized, as we do, that certain elast−ic moduli are renormalized by the re-
so that the entropy production density Ts˙ takes on the laxation of the director. These are
same form as Eq. (2) with the elastic constants C re-
ijkl
pTlhaeceνd bydveipsecnosditoienstνhiejkolrdanerdpwairtahmuetijerrecp.lνaced,bνy u˙ij., CξRχ(ω)=Cξχ−iωνξχ− 1 iωiτω1τ1 ∆Aξχ
ijkl xxxz yyxz −
and ν vanish at the SmC to SmA transition and are =C iωνR +O(ω2), (9)
zzxz ξχ− ξχ
therefore expected being smaller thanthe remaining vis-
cosities for c small [8]. with renormalized viscosities νξRχ = νξχ +Γ−1Aξχ, and
A smectic elastomer is characterized in general by re- whereAxyxy =α2(1+τ3/τ1)2,Axyzy =αβ(1+τ3/τ1)(1+
laxation times associated with director relaxation and τ2/τ1) and Azyzy =β2(1+τ2/τ1)2.
with other modes, which we will simply refer to as elas- The frequency dependence of the elastic moduli in
tomer modes. For frequencies ω such that ωτ 1, Eq.(8)canbedetermined,inprinciple,byrheologymea-
E
≪
whereτ isthelongestelastomertime,theviscositiesand surements of the corresponding storage and loss moduli.
E
Γarepracticallyfrequencyindependent. Whenωτ 1, TheunrenormalizedmoduliC (ω)leadtoconventional
E ijkl
≥ ′ ′′
however,the viscosities ν and Γ develop a non-trivial storage and loss moduli C =C and C =ων
ijkl ijkl ijkl ijkl ijkl
frequency dependence. In the following we will consider that are, as in conventional rubbers, respectively con-
in detail only the case τ τ and ωτ 1. stant and proportional to ω at low frequencies. The
n E E
≫ ≪
3
z′
uli x′
od 102
m
oss 101
l
d
n
a 1
e
g
a
Stor10-110-3 10-2 10-1 1 101 102 103 (a) y′ (i)
ωτ −
n
y′ x′
FIG. 1: Log-log plot of the reduced storage and loss moduli
C′ (ω)/C (solidline)andC′′ (ω)/CR (dashedline)versus
ξχ ξχ ξχ ξχ
the reduced frequency ωτn as given respectively by the real
andnegativeimaginary partsofEq.(9)for τn≫τE. Forthe
purpose of illustration we have set, by and large arbitrarily,
2
νξχ/(τnCξχ)=1 and ∆Aξχ/Cξχ =10 . z′ z′
(i) (ii)
renormalizedmoduli, onthe otherhand,havethepoten- (i)
(ii)
tial for much more interesting rheology behavior. One
consequence of Eq. (9) is that SmC elastomerscould ex- (b) (iii) (c) (iii)
hibit so-called dynamic soft elasticity [9]. To highlight
this phenomenon,let usswitchbriefly tothe rotatedref- FIG.2: Schematicplotsofsoundvelocities: (a)sphericalplot
erence space coordinates x′,y′,z′. Then, Eq. (9) implies ofmodepair (i)only; (b)and(c) polarplots inthey′z′-and
′ ′
that xz - planes, respectively,of all 3 sound-mode pairs.
CyR′z′y′z′(ω)=−iωνy′z′y′z′ − 1 iωiτω1τ1 ∆Ay′z′y′z′ (10) nates. Their soundvelocities C(ϑ,ϕ), as calculatedfrom
−
Eq.(7) with the viscosities ν setto zero,are depicted
with νy′z′y′z′ = sin2θνxyxy sin2θνxyyz + cos2θνyzyz in Fig. 2. There are 3 pairisjkolf sound modes. One of
−
and an analogous expression for Ay′z′y′z′. CxR′y′y′z′(ω) these pairs (i) is associated with the soft deformations
is of the same form as Eq. (10). Thus, CyR′z′y′z′(ω) and discussed above. Its velocity vanishes for q along eˆy and
CxR′y′y′z′(ω) vanish in the limit ω → 0 where we recover eˆ2 so that when viewed in the y′z′-plane it has a clover
true soft elasticity. At non-vanishing frequency the sys- leave like shape. The remaining 2 pairs are associated
tem cannot be ideally soft but it can be nearly so for ω with non-soft deformations. In the incompressible limit,
small. This type of behavior was first predicted for ne- thesepairsbecomepurelytransverse(ii)andlongitudinal
′ ′ ′ ′
maticelastomers,whereithasbeentermeddynamicsoft (iii),respectively. Inthey z andxz -planes,theirveloc-
elasticity[9]. Thestoragemoduliforu andu strains, ities are non-vanishing in all directions. Note that, since
xy yz
Eq. (9), in the originalcoordinate system are nonzero at thevelocityofpair(i)vanishesindirectionswheretheve-
zero frequency. Their behavior for ω > 0, like that of locitiesoftheothermodesremainfinite,SmC elastomers
semi-soft nematic elastomers [5, 9], depends on τ /τ . are, like nematic elasomers [9], potential candidates for
n E
If τ τ , the storage moduli exhibit a step and the applications in acoustic polarizers.
n E
≫
correspondinglossmodulianassociatedpeakatωτ 1 Havingfoundthe generalsound-modestructure inthe
n
∼
as shownin Fig.1; if τ τ ,or τ <τ , this is notthe nondissipativelimit, we now turnto the full mode struc-
n E n E
case. The storagemodul≈i C′ (ω)andC′ (ω)in ture in the incompressible limit. In the softness-related
y′z′y′z′ x′y′y′z′
the rotated frame are zero at zero frequency. In a semi- symmetrydirectionsthemodesareofthefollowingtypes:
soft SmC, however, they will exhibit behavior similar to (i) non-hydrodynamic modes with frequencies
that Fig. 1 for τ τ . In nematic elastomers, there is
n ≫ E ω = iτ−1+iD q2 (11)
still some controversy [10] about whether the τ τ m 1 m
n E −
≫
regime has actually been observed in experiments. It
with a zero-q decay time τ1 and diffusion constants Dm,
would be interesting to see if it might exist in SmC elas-
(ii) propagating modes with frequencies
tomers,where τ mightbe shorterthanit is in nematics
E
because of the smectic layers. ω = Cq iD q2 (12)
p p
± −
To assess the mode structure of SmC elastomers, we
startwithananalysisofpropagatingsoundmodesinthe with sound velocities C and diffusion constants Dp, and
dissipationless limit. The sound modes have frequencies (iii) diffusive modes with frequencies
ω(q) = C(ϑ,ϕ)q, where q = q and where ϑ and ϕ are
| | ω = iD q2 (D q2)2+Bq4 (13)
the azimuthal and polar angles of q in spherical coordi- d − d ±p− d
4
withdiffusionconstantsDdandbendingtermsBthatare whereνzR′y′z′y′ istherenormalizedversionofνz′y′z′y′ and
missed if the Frank energy is neglected. For B/D2 1 where the K¯’s are once more bending moduli depending
d ≪
the diffusive modes split up into slow and fast modes on the Frank constants, c, λ, α, and β. Finally, there is
ω = iB/(2D )q2, ω = i2D q2. (14) a pair of propagating modes polarized along eˆx′ with
s d f d
− −
Specifics of the sound velocities, the diffusion constants Cx′ =qC¯/(4ρ), (21a)
anFdirtshte,bleetnduisngcotnersmidserartehegicvaesneinthtahteqfollileoswiinng.the xz- Dp,x′ =(8ρ)−1νxR′y′x′y′, (21b)
plane. In this case the equation of motion for uy decou- with νxR′y′x′y′ =cos2θνxRyxy+sin2θνxRyzy+sin2θνzRyzy.
ples from the equations of motion for ux and uz. This In summary, we have presented a theory for the low-
equationproduces a setoftransversemodeswith u eˆy. frequency, long-wavelength dynamics of soft SmC elas-
k
There is a non-hydrodynamic mode with tomers. Thistheorypredictsthat,atleastinanidealized
2 limit,SmC elastomerspossessthefascinatingpropertyof
D =(4ρ)−1 νR ν qˆ + νR ν qˆ , dynamic soft elasticity. Though the equations of motion
m,y hq xyxy− xyxy x q zyzy− zyzy zi
arecomplicated, the resulting mode structure is with re-
(15)
spect to the softness related symmetry directions nicely
where qˆ =q /q, and there are propagating modes with symmetric and it has, when visualized, a certain beauty.
i i
Wehavecalculatedvariousdynamicalquantities,suchas
Cy =qC¯/(4ρ) cosθqˆx+sinθqˆz =qC¯/(4ρ)qˆx′ , storageand loss moduli and sound velocities that are, in
| | | |
(16a) principle, accessible by experiments and we hope, that
our theory encourages such experimental work.
D =(8ρ)−1 νR qˆ2+2νR qˆ qˆ +νR qˆ2 . (16b)
p,y (cid:2) xyxy x xyzy x z zyzy z(cid:3) Support by the National Science Foundation under
In the soft direction, i. e. for q eˆz′, these propagating grant DMR 0404670 (TCL) is gratefully acknowledged.
k
modes become diffusive with D =D and
d,y p,y
By =ρ−1(cid:2)K¯1qˆx4+K¯2qˆz4+K¯3qˆx2qˆz2+2K¯4qˆx3qˆz +2K¯5qˆxqˆz3(cid:3),
(17)
[1] For a review on liquid crystal elastomers see W. Warner
wheretheK¯’sarebendingmodulithatarecombinations andE.M. Terentjev,LiquidCrystal Elastomers(Claren-
ofthe usualFrankelasticconstants,the orderparameter don Press, Oxford,2003)
c as well as λ, α, and β. The equations of motion for [2] For a review on liquid crystals see P. G. de Gennes
and J.Prost,The Physics of Liquid Crystals(Clarendon
u and u can be solved by decomposing (u ,u ) into a
x z x z Press, Oxford,1993); S. Chandrasekhar, Liquid Crystals
longitudinal part u along q and a transversal part u .
l T (Cambridge University Press, Cambridge, 1992).
In the incompressible limit u vanishes. The equation of
l [3] K. Hiraoka, W. Sagano, T. Nose and H. Finkelmann,
motionforuT producesapairofpropagatingmodeswith Macromolocules 38 7352 (2005).
[4] O. Stenull and T. C. Lubensky, Phys. Rev. Lett. 94,
018304 (2005);forthcomingpaper.SeealsoJ.M.Adams
2 2
CT =p1/ρ(cid:8)[Cxxxx+Czzzz −2Cxxzz]qˆxqˆz (18a) and M. Warner, Phys.Rev. E 72, 011703 (2005).
+[C C ]qˆ qˆ (qˆ2 qˆ2)+ 1C (qˆ2 qˆ2) 1/2 [5] O.StenullandT.C.Lubensky,Phys.Rev.E69,051801
xxxz− zzxz x z z − x 4 xzxz z − x (cid:9) (2004).
D =(2ρ)−1 [ν +ν 2ν ]qˆ2qˆ2 (18b) [6] See, e.g., N.W. Ashcroft and N.D.Mermin, Solid State
p,T (cid:2) xxxx zzzz − xxzz x z Physics, (Saunders, Philadelphia, 1976).
+[ν ν ]qˆ qˆ (qˆ2 qˆ2)+ 1ν (qˆ2 qˆ2) ,
xxxz− zzxz x z z − x 4 xzxz z − x (cid:3) [7] D. Forster, T. C. Lubensky, P. C. Martin, J. Swift,
and P. S. Persham, Phys. Rev. Lett. 26, 1016 (1971);
and u eˆ where eˆ =(qˆ ,0, qˆ ).
k T T z − x D. Forster, Hydrodynamic Fluctuations, Broken Symme-
Finally, we turn to the case q eˆy. There is a pair try, and Correlation Functions (Addison Welsey, Read-
k
of longitudinal propagating modes with u eˆy that is ing, Mass., 1983); D. Forster, Phys. Rev. Lett. 32, 1161
k
suppressed in the incompressible limit. There is a non- (1974).
hydrodynamic mode with [8] For more detailed theoretical estimates of the νijkl one
hastoresorttomolecular-basedtheories,aswasdonefor
D =(4ρ)−1 νR ν +νR ν , (19) conventional SmC’s in M. A. Osipov, T. J. Sluckin, and
m,xz (cid:2) xyxy− xyxy zyzy− zyzy(cid:3) E. M. Terentjev, Liq. Cryst. 19, 197 (1995).
whereulaysinthexz-planewithux =α(τ1+τ3)/[β(τ1+ [9] E. M. Terentjev, I. V. Kamotski, D. D. Zakharov, and
τ2)]uz. There is a pair of elastically soft diffusive modes L. J. Fradkin, Phys. Rev. E 66, 052701(R) (2002). L.
with polarization ukeˆz′ with JZ.akFhraadrokvin,,PIr.ocV..RK.aSmoco.tsLkoi,ndE..AM4.5T9e,re2n6t2j7ev(2a0n0d3)D. . D.
Dd,z′ =(8ρ)−1νzR′y′z′y′, (20a) [10] P. Martinoty et al., Eur. Phys. J. E 14, 311 (2004); E.
Bz′ =ρ−1(cid:2)sin2θK¯6−sin2θK¯8+cos2θK¯7(cid:3), (20b) TanedrenTt.jeCv.aLnudbMen.sWkya,rinbeidr,.i1b4id,.31343,(322030(42)004);O.Stenull
