Table Of ContentPolydomain structure and its origins in isotropic-genesis nematic elastomers
Bing-Sui Lu1, Fangfu Ye1, Xiangjun Xing2, and Paul M. Goldbart1
1Department of Physics and Institute for Condensed Matter Theory,
University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801
2Institute of Natural Sciences and Department of Physics,
Shanghai Jiao Tong University, Shanghai 200240 China
(Dated: January 10, 2011)
We address the physics of nematic liquid crystalline elastomers randomly crosslinked in the
isotropic state. To do this, we construct a phenomenological effective replica Hamiltonian in terms
of two order-parameter fields: one for the vulcanization, the other for nematic alignment. Using
a Gaussian variational approach, we analyze both thermal and quenched fluctuations of the local
nematic order, and find that, even for low temperatures, the macroscopically isotropic polydomain
state is stabilized by the network heterogeneity. For sufficiently strong disorder and low enough
1
1 temperature, our theory predicts unusual, short-range oscillatory structure in (i.e., anti-alignment
0 of) the local nematic order. The present approach, which naturally takes into account the compli-
2 ant, thermally fluctuating and heterogeneous features of elastomeric networks, can also be applied
to other types of randomly crosslinked solids.
n
a
PACSnumbers: 64.70.pp,61.41.+e
J
6
I. INTRODUCTION A heuristic argument, of the type first developed by
]
Imry and Ma [19] for systems at low temperature hav-
t
f ingquenchedrandomfieldswithshort-rangecorrelations,
o Liquid crystal elastomers [1] are fascinating materials,
s predictsthatlong-rangeorderinvolvingcontinuoussym-
in which the liquid crystalline order is strongly coupled
. metry breaking is precluded in systems of spatial dimen-
at to the elasticity of the underlying elastomeric network. sions fewer than four. This argument surely applies to
m This gives rise to novel properties, such as soft [2, 3]
nematogens in the presence of an immobile random sub-
and semi-soft elasticity [4, 5], which is of interest in fun-
- strate, such as aerogel [20, 21]. However, for nematic
d damental research and, furthermore, has significant po-
elastomers, the network elasticity induces a long-range
n tential for technological applications. From the perspec-
interaction between nematic directors. Furthermore, it
o
tive of statistical physics, these materials, like any other
c is not a priori clear whether the effects of network het-
randomly crosslinked systems, pose important concep-
[ erogeneities can be properly modeled as quenched ran-
tual challenges, as their physical properties do not de-
dom fields. Hence, the original Imry-Ma type of argu-
1 pend just on the conditions under which various physi-
v ment does not necessarily apply. Nevertheless, absent
cal quantities are measured. They also depend on both
3 any convincing alternative, this traditional random-field
the conditions under which the system is prepared and
2 approach has been adopted by various authors [9–14].
thenetworkheterogeneitiesnecessarilygeneratedviathe
3
1 random crosslinking preparation process. A coherent Simulation work has been done by Uchida [10, 11]
. theory that takes all these factors into account requires on a simple, two-dimensional, random-field model of ne-
1
two statistical ensembles, i. e. the preparation ensemble matic elastomers. He found that IGNEs always exhibit
0
1 and the measurement ensemble. This essential point was the PD state at a sufficiently low temperature. On the
1 qualitatively understood by Deam and Edwards and by other hand, Terentjev et al. [9, 13, 14] have used Gaus-
: de Gennes as early as the 1970’s (see, e.g., Refs. [6, 7]). sian variational methods to study a similar model, and
v
Its full significance however has not been explored previ- they concluded that replica symmetry is broken, indica-
i
X ously. tive of glassy low-temperature properties in PD IGNEs.
r Xing and Radzihovsky [22, 23] start from a disorder-
Onesuchchallengeconcernstheorigins, structureand
a free model of a MD nematic elastomer, turn on an in-
elasticity of the polydomain state in isotropic-genesis
finitesimal quenched random field and study the con-
nematic elastomers (IGNEs; i.e., elastomers that were
sequent stability of the long-range nematic order, via
crosslinked in the isotropic state) [8–18]. It has always
renormalization-group techniques. They found that ne-
been observed experimentally that IGNEs at low tem-
maticorderpersistsevenbelowthreespatialdimensions.
perature are characterized by a polydomain (PD) state,
The random-field idea is also implicit in the more re-
i.e., a macroscopically isotropic state consisting of well-
cent works by Biggins et al. [17, 18], which employs a
defined domains of randomly oriented nematic order.
quasi-convexification method to construct textured, soft
The typical domain-size is of the order of microns. Fur-
deformationsofIGNEs, andthusallowstheseauthorsto
thermore, PD IGNEs exhibit much softer stress-strain
obtain sharp bounds on the stress-strain curve.
curves than PD nematic elastomers crosslinked in ne-
matic state, as demonstrated by recent experiments of The aim of the present paper is to explore the nature
Urayama’s group [16]. of quenched disorder in IGNEs, and its impact on local
2
a) b) is forbidden. In the framework of vulcanization theory,
this causality is naturally enforced via the replica trick,
inwhichthemeasurement-stateorderparameterisrepli-
cated n times, {Qα;α = 1,...,n}, and the number of
copies n is later taken to zero.
As IGNEs are quench-disordered systems, we need
to distinguish between thermal averaging (i.e., averag-
ing over thermal fluctuations), denoted (cid:104)···(cid:105), and disor-
der averaging (i.e., averaging over the realizations of the
crosslinking),denoted[···]. Assumingthatthesystemis
self-averaging, it follows that every experimentally mea-
sured quantity has already been disorder-averaged. Var-
ious disorder- and thermal-averaged quantities involving
the local nematic order in the measurement state Q can
FIG. 1. (Schematic) snapshots of nematogen orientations at
a particular instant (blue unshaded), and at a much earlier be constructed. To lowest order in Q, one can construct
instant (shaded). (a) A conventional liquid crystal in the the quantity [(cid:104)Q(r)(cid:105)]; however, this quantity trivially
isotropic state. Such systems do not memorize the local pat- vanishes, owing to the macroscopic isotropy of the PD
ternofnematogenalignmentindefinitely. Thereisnocorrela- state. At second order in Q, one can form the following
tionbetweentheorientationsofblueandshadednematogens two, distinct, averaged quantities: (i) the thermal corre-
that are depicted near one another. (b) A liquid cystalline lator
ealstomer in the macroscopically isotropic state. Such sys-
tems do memorize the local pattern of nematogen alignment CT(r−r(cid:48)):=Tr(cid:2)(cid:10)(cid:0)Q(r)−(cid:104)Q(r)(cid:105)(cid:1)(cid:0)Q(r(cid:48))−(cid:104)Q(r(cid:48))(cid:105)(cid:1)(cid:11)(cid:3),
indefinitely. Theorientationsofblueandshadednematogens (1a)
depicted near one another are correlated. and (ii) the glassy correlator
CG(r−r(cid:48)):=Tr[(cid:104)Q(r)(cid:105)(cid:104)Q(r(cid:48))(cid:105)]. (1b)
nematicorder,withintheframeworkaffordedbyvulcan- AlthoughwearefocusingontheCartesianscalaraspects
ization theory [24–27]. We find that the most relevant ofthecorrelators,itisstraightforwardtoreconstructthe
type of disorder does indeed resemble a quenched ran- fullstructureofthecorrespondingfourth-rankCartesian
dom field, but with an important difference which we tensors, by appending suitable isotropic tensor factors
shall discuss in detail below. By using the replica Gaus- constructed from Kronecker deltas.
sian variational approach, we calculate the thermal as The glassy correlator automatically vanishes for
wellassample-to-samplefluctuationsofthelocalnematic isotropic liquids. For IGNEs, however, it is nonzero and
order. Our principal findings are that the PD state is al- encodes the correlation in the randomly frozen nematic
ways stable, even at sufficiently low temperature and, pattern. Thedifference between thesetwostates isqual-
furthermore, that for sufficiently strong disorder, the lo- itatively indicated in Fig. 1. The value if the glassy
calfrozennematicorderhasanoscillatorycharacter(i.e., correlator at the origin, CG(r)|r=0, is the nematic ana-
anti-alignment) at short distances. logue of the Edwards-Anderson order parameter for spin
glasses [28], and measures the magnitude of the local
frozen nematic order. We shall call this the glassy or-
II. ORDER PARAMETERS, FREE ENERGIES der parameter. On the other hand, the thermal correla-
AND CORRELATORS tor can be used to detect the existence of a continuous
phase transition. If such a transition exists, the thermal
The physics of randomly crosslinked systems depends correlator would diverge at the critical temperature. If a
on both the state under which the system is crosslinked, discontinuous transition occurs, however, the correlator
and the state under which it is measured. It is therefore would diverge at the spinodal point.
of crucial importance to distinguish between these two Toaddresstheinterplaybetweennematicorderingand
states. We call the former the preparation state and the network heterogeneity in randomly crosslinked systems,
latter the measurement state. The local nematic order we shall also need the vulcanization order parameter
Q0 in the preparation state and its counterpart Q in the Ω(r0,r1,...,rn), which is a scalar function of 1 + n
measurement state are, in principle, distinct, and this replicated three-dimensional position vectors {rα;α =
distinctionisnaturallycapturedbytheformalismofvul- 0,1,...,n} (see, e.g., Refs. [24–27]). Up to an additive
canization theory. For IGNEs crosslinked at high tem- constant, it gives the joint probability density function
perature, Q0 is negligibly small. In contrast, although (pdf) for the results of the following 1 + n indepen-
the macroscopic average of Q vanishes in IGNEs, its lo- dent measurements: At a time just prior to crosslink-
cal value can be substantial; this would reflect random ing, a monomer is found at r0, whilst in n independent
local frozen nematic alignment. It is also important to measurements made long after crosslinking, the same
note the fundamental asymmetry in the causal ordering monomer is found at r1,...,rn. This order parame-
of Q0 and Q: whilst Q0 can influence Q, the converse ter detects and characterizes the positional localization
3
of monomers that emerges at the gelation/vulcanization free energy for local nematic order in the measurement
transition. Intheliquidphase,allmonomersaredelocal- state and the coupling term, Eqs. (4):
ized, so the joint pdf is a trivial constant, and hence the
orderparameterΩvanishesidentically. Inthegel/rubber H[Q]=(cid:90) d3r (cid:88)n (cid:26)TrTrQαQα + KTr∇ Qα∇ Qα
phase, a fraction G of monomers are localized. Their 2 2 i i
positions in the 1 + n distinct measurements are then α=1
(cid:27)
v w
correlated with one another, and consequently the order − Tr (Qα)3+ (TrQαQα)2 (5)
parameter has a nontrivial, mean-field profile, given by 3 4
Ω¯(rˆ)=G(cid:90)d3z(cid:90)dτp(τ)(cid:16) τ (cid:17)32(n+1e)−τ2(cid:80)nα=0|rα−z|2− G , −h2 (cid:88)n (cid:90)d3rd3r(cid:48)ω(r−r(cid:48))Tr Qα(r)Qβ(r(cid:48)).
2π Vn α,β=1
(2) (α(cid:54)=β)
where rˆ:= (r0,...,rn) and V is the volume of the sys- Here, T is the reduced temperature in the measurement
tem. Note that the saddle point order parameter Ω¯ is state. Irn addition, K is proportional to the Frank con-
invariantundersimultaneoustranslationofallreplicated stantintheone-constantapproximation. Itisimportant
vectorsrˆ:=(r0,...,rn). Theparameterτ hasthephysi- to note that any standard replica treatment of a phe-
calmeaningofthelocalinversesquarelocalizationlength nomenological theory in which the nematic order is cou-
for the monomers, and it fluctuates from monomer to pled linearly to a quenched Gaussian random field with
monomer with pdf p(τ). In this work, we shall make correlatorω(r)wouldnecessarilylead,inadditiontothe
the simplifying assumption that the localization length replica off-diagonal term in Eq. (5), to a nonvanishing
has a sharp value ξL rather than being distributed, i.e., replica-diagonaltermhavingpreciselythesamekernelas
p(τ) = δ(τ −ξ−2). This does not change our essential its off-diagonal counterpart. This absence of the replica-
L
results. diagonaltermisasalientfeatureofvulcanizationtheory.
It is rather significant that the heterogeneous nature Itdemonstratesthatthequencheddisordersinelastomers
of elastomers is captured already at the mean field level can not be simply modeled as random field. As we shall
of vulcanization theory. At long lengthscales, the most show below, it is also responsible for the oscillatory spa-
important coupling between the vulcanization order pa- tial structure for IGNEs in the strong-disorder regime.
rameter Ω¯ at the saddle point and the nematic order Standard random-field models would not yield such os-
parameters Qα reads cillatory structure.
(cid:90) n FocusingonthequadraticpartsoftheeffectiveHamil-
−h drˆ (cid:88)∇α∇αΩ¯(rˆ)∇β∇βΩ¯(rˆ)Qα(rα)Qβ(rβ),(3) tonian (5), we can obtain the bare thermal (CT) and
i j k l ij kl 0
α,β=0 glassy(CG)correlators[seeEqs.(1)]ofthenematicorder
0
wheredrˆindicatesmultipleintegrationoverallreplicated parameter in wave-vector space:
positionsrα,andasummationisimpliedovertheCarte-
R
sian coordinates i,j,.... In the above summation, the CT(p)= l , (6a)
0 t+κ2+∆e−κ2/2
replica-diagonaltermsandthereplica-off-diagonalterms
contribute qualitatively distinctly from one another. In CG(p)= Rl∆e−κ2/2 . (6b)
particular, the replica-diagonal terms can be omitted 0 (cid:0)t+κ2+∆e−κ2/2(cid:1)2
as they simply renormalize the quadratic term in the
Landau-de Gennes free energy [29]. We proceed to in- Here, R := ξ2/K, whilst t := T ξ2/K, ∆ :=
tegrate out the coordinates rγ (with γ (cid:54)= α,β). Hence, 32π3hG2/lKξ2 >L0 and κ := pξ are thre Ldimensionless
L L
Eq.(3)istransformedintothefollowing,essentiallynon- rescaled measurement-temperature, disorder strength,
local, quadratic term: and wavevector. The appearance of ∆ in the denomi-
h(cid:90) (cid:88)n nators, in combination with the scale-dependent factor
− d3rd3r(cid:48)ω(r−r(cid:48)) TrQα(r)Qβ(r(cid:48)), (4a) exp(−κ2/2), is a significant result of the present ap-
2
α,β=0 proach. Itfollowsdirectlyfromtheabsenceofthereplica-
(α(cid:54)=β)
√ diagonal contributions to the term (4), and therefore
2ω(r):=(4π/ξL2)7/2G2e−|r|2/2ξL2, (4b) cannot result from a standard random-field approach.
where G is the gel fraction, and ω is a kernel that rep- Originating in the random network, and proportional
resents the disordering effects of the random network on to the shear modulus of the network [30], it leads to
thenematicorder. Thecharacteristiclengthscalebeyond a scale-dependent, downward renormalization of the re-
which ω(r) is suppressed is ξ . This suppression reflects duced temperature.
L
theliquidnatureofthenetworkatshortlengthscalesbut The bare correlators exhibit instability at some neg-
not at lengthscales longer than ξ . ative reduced temperature t < 0. For ∆ < 2, i.e., at
L
ForIGNEscrosslinkedataveryhightemperature,the weakerdisorder,thedenominatorfactort+κ2+∆e−κ2/2
localnematicorderQ0 inthepreparationstateisnegligi- has a minimum at κ=0. Consequently, both of the cor-
bly small and can therefore be ignored. The full free en- relators given in Eqs. (6) diverge, at κ = 0, at a criti-
ergyofIGNEsisacombinationoftheLandau-deGennes cal(reduced, rescaled)temperature−∆. Thissuggestsa
4
continuoustransitiontoauniformlyordered(i.e.,macro- ByminimizingF withrespecttoCT andCG foragiven
var
scopically anisotropic) nematic state. By contrast, for value of Q¯, anddefining a renormalizedreduced temper-
∆ > 2, i.e., at stronger disorder, the denominator fac- ature t via
R
tors have a minimum at κ2 = 2ln(∆/2); therefore, both
bare correlators exhibit nonzero-wavelength divergences t =t+7wR (cid:90) (cid:18)CT +CG+ 1 TrQ¯2(cid:19), (11)
at a (reduced, rescaled) temperature −2ln(e∆/2). This R l 5V
p
suggests a continuous phase transition to a state having
periodic modulations. As we shall see shortly, however, weobtainfortherenormalized thermalandglassycorre-
neitheroftheseputativetransitionsactuallyoccurs,both lators the results
of them being precluded by the quenched randomness.
R
CT(p)= l , (12a)
t +κ2+∆e−κ2/2
R
III. GAUSSIAN VARIATIONAL APPROACH R ∆e−κ2/2
CG(p)= l . (12b)
TO THERMAL AND GLASSY CORRELATORS (cid:0)t +κ2+∆e−κ2/2(cid:1)2
R
To investigate the fate of the system at temperatures NotethatEqs.(12)arestructurallyidenticaltotheirbare
below these putative instabilities, we employ the Gaus- counterparts, Eqs. (6), the only difference being the re-
sian variational method (see, e.g., Ref. [31]) for systems placementofthebarereducedtemperaturetbyitsrenor-
havingquenchedrandomness[32–35]. Wechoosethecor- malized counterpart t .
R
responding quadratic trial Hamiltonian H0 to be The relation (11) between tR and t determines the ex-
istence, or lack thereof, of a transition out of the macro-
1 (cid:88)n (cid:90)
H = Γαβ(p ,p )× scopically isotropic state. Let us first discuss the high
0 2α,β=1 p1,p2 1 2 (measurement)temperatureregime, inwhichQ¯ =0and
Tr(cid:0)Qα(p )−Q¯(p )(cid:1)(cid:0)Qβ(p )−Q¯(p )(cid:1), (7) thecorrelatorsaresmall. Inthiscase,thesecondtermon
1 1 2 2 theRHSofEq.(11),proportionaltow,canbeneglected.
which generically features a nonvanishing mean nematic Thus, tR is essentially the bare (reduced, rescaled) tem-
orderparameterQ¯ togetherwithakernelΓαβ [36]. Here, perature t, and the renormalized correlators (12) reduce
the notation (cid:82) denotes (cid:82) d3p/(2π)3. Assuming that to the bare correlators (6), as expected.
p
But what happens at lower temperatures? As t is re-
replica permutation symmetry remains intact [37], we
may parametrize the replica-space inverse of Γαβ as duced,thecorrelatorsbecomelarger,andthecorrections
due to fluctuations, present in Eq. (11), increase. The
(cid:0)Γ−1(cid:1)αβ(p ,p ):=¯δ(12)(cid:0)CT(p )δαβ +CG(p )(cid:1), (8) essential question is then the following: Do the denomi-
1 2 1 1
nators in Eqs. (12) vanish at some wave-vector p for low
whereδ¯(12···m)denotes(2π)3δ(p +p +···+p ),and enough temperature? If this were indeed to happen, it
1 2 m
CT(p)andCG(p)aretherenormalized thermalandglassy wouldsignifyacontinuoustransitiontosomemacroscop-
correlators, in which fluctuations are approximately ac- icallysymmetry-brokenstate. Now,thelow-temperature
counted for. Both of them, together with Q¯, are to be physics of our model depends sensitively on the value of
determined self-consistently, by minimizing the resulting ∆. For ∆<2, the denominator factor in Eqs. (12) [viz.,
variational free energy t +κ2 +∆e−κ2/2] has a minimum at κ = 0. Assum-
R
ing that a continuous transition occurs at some finite
(cid:90) n
F :=(cid:10)H −H (cid:11) −ln (cid:89)DQαe−H0, (9) temperature tc, this denominator would have to vanish
var 0 H0 α=1 ftoermκs,=how0eavnedr,ttRhe+in∆teg=ral0.ofFtohretghlarsesey-dciomrerenlsaitoonra(cid:82)l sCysG-
p
where (cid:104)···(cid:105)H0 denotes averaging with respect to the in Eq. (11) has infrared divergence, and thus tR(tc) di-
Boltzmann weight e−H0. The explicit form of Fvar is verges, implying that the denominator would diverge,
5n2V Fvar = R1l(cid:90)p(cid:0)t+κ2+∆e−κ2/2(cid:1)(cid:0)CG+TrQ¯2/5V(cid:1) annadtorthviasncisohnetsr.adWicetsththereefaosrseumcopntciloundethtahtatthinistdherneeomdii--
mensions and for ∆ < 2 there is no continuous transi-
+ 1(cid:90) (cid:0)t+κ2(cid:1)CT + 7w(cid:90) TrQ¯2(cid:90) (CT +CG) tion to any state having long-range nematic order. For
Rl p 5V p p ∆ > 2, the factor t +κ2+∆e−κ2/2 has a minimum at
1 2 R
7w (cid:32)(cid:90) (cid:33)2 (cid:90) (cid:18) CG(cid:19) κ2 =2ln(∆/2). Inthiscase,tohaveacontinuoustransi-
+ (CT +CG) − lnCT + tion (towards a macroscopically periodically modulated
2 p p CT state) would require the vanishing of the denominator,
2v (cid:90) nowatκ2 =2ln(∆/2). However,onceagain,theintegral
+ ¯δ(123)TrQ¯3 inEq.(11)isdivergent,andthereforenocontinuoustran-
15V p1,p2,p3 sition occurs. Hence, owing to network heterogeneities,
+ w (cid:90) ¯δ(1234)(cid:0)TrQ¯2(cid:1)2. (10) the macroscopically isotropic phase is always locally sta-
10V p ,p ,p ,p ble, for any positive ∆. This result agrees with experi-
1 2 3 4
5
a) b)
FIG. 2. (a) Glassy order parameter CG(r)|r=0 (measured in
unitR)and(b)glassycorrelationlengthξ (inunitξ )as
l G,d L
a function of t for three cases of weak disorder: ∆=0.2 (red
solid),∆=0.7(greendashed)and∆=1.5(bluedot-dashed), FIG. 3. (a) Real-space optical microscopic image of polydo-
where we have set wRl =0.4π. main structure in IGNEs, obtained by the Urayama group.
Note the occurrence of a short-range ‘checkerboard’ pattern
indicative of oscillatory structure in the local nematic align-
mentalobservations,e.g.bytheUrayamagroup[16],and ment, most evident in the upper-right region of the panel.
the results of simulations performed by Uchida [10, 11]. (b) Noise-filtered (Log amplitude-squared) Fourier transfor-
mation of the optical image in panel (a). Note the occur-
rence of a central high-intensity (bright) peak accompanied
IV. CORRELATOR STRUCTURE AND by a high-intensity ring, which correponds to the checker-
LOW-TEMPERATURE PROPERTIES board pattern in real space. The shaded rectangular grid is
due to the denoising filter.
We now unfold the physics encoded in the correlators
Eqs. (12). First, we look at the glassy order parameter
In the strong-disorder regime (i.e., ∆ > 2), the ther-
CG(r)|r=0, which is given by (cid:82)pCG(p). From Eq. (12b), mal correlator CT and the glassy correlator CG are each
weseethatbelowt=0itincreasesrapidlywithdecreas- peaked on a spherical shell of wave-vectors: CT(p) peaks
ingtemperature,becoming,atlowtemperatures,asymp- for |p|2 = 2ln(∆/2)/ξ2, and CG peaks for |p|2 = ξ−2,
L G,o
totically linear in t, behaving as K|t|/7wξ2. We have with ξ obeying
L G,o
computed this order parameter: the results are shown
for three cases of weak disorder in Fig. 2a and one case t +4+(ξ /ξ )2−∆e−(ξL/ξG,o)2/2 =0. (14)
R L G,o
of strong disorder in Fig. 4b.
As the fluctuation-response theorem indicates, the We term ξG,o the glassy oscillation length. The peak-
staticsusceptibilityχ(p)forlocalnematicorderisrelated ing of CT(p) and CG(p) is associated with oscillatory
to the thermal correlator via χ(p) = T−1CT(p). The behaviour (i.e., anti-alignment) of the local nematic or-
peak value of CT, which for weak disorder (i.e., ∆ < 2) der, which is superposed on their overall decay in real
occurs at κ2 = 0 and for strong disorder (i.e., ∆ > 2)
occurs at κ2 = 2ln(∆/2), increases rapidly as the tem-
a) t t b)
perature is decreased, as we see from Eq. (12a). Due to
thecouplingbetweenlocalnematicalignmentandelastic
strain, we thus expect there to be a corresponding soft
elasticity. We speculate that this mechanism lies behind CG(κ) CG(r)
the supersoft elasticity of PD IGNEs (cf. Refs. [17, 18]).
A detailed analysis of elasticity of PD IGNEs is however
not possible using the current simple model which does
not involve the elastic degrees of freedom. We shall pur-
sue it in a separate publication.
κ r
In the weak-disorder regime (i.e. ∆ < 2), both corre-
lators have a finite peak at zero wave-vector, indicating
spatial decay (but without oscillation). The characteris-
tic lengthscales ξ and ξ over which the correlators FIG. 4. Glassy correlator CG (measured in unit Rl) at ∆ =
T,d G,d
10 as a function of reduced temperature t and (a) reduced
decay can be obtained by examining their small wave-
wave-vector κ; (b) separation distance r (in unit ξ ), with
vector behaviours: L
wR =0.4π. Note the progression in (a) of the peak location
l
(2−∆) fromzerowave-vectortononzerowave-vectorwithdecreasing
ξT2,d ≈ 2(t +∆)ξL2, ξG2,d ≈2ξT2,d+ 12ξL2. (13) t. The peak at nonzero wave-vector is associated with the
R spatialoscillationsobservedin(b)fortheglassycorrelatorin
Since t +∆ approaches zero towards low temperature, real-space. Note also in (b) that the glassy order parameter
both leRngthscales ξ grow rapidly (see Fig. 2b). CG(r)|r=0 rapidly increases as the temperature is reduced.
G/T,d
6
of nematic elastomers, in which nematic ordering—both
inthepreparationandthemeasurementstates—andnet-
workheterogeneityarenaturallyincorporated. Itisnote-
worthy that the quenched disorder naturally emerges
from this approach differs in essential ways from the tra-
ditional quenched random field approach. This differ-
ence evinces a strength of vulcanization theory. Using
a variational method, we show that the macroscopically
isotropic state of isotropic-genesis nematic elastomers is
FIG. 5. (a) Crossover diagram for the glassy and thermal
always thermodynamically stable and, furthermore, that
correlators,indicatingthethreequalitativelydistinctregimes
for sufficiently high disorder strengths, short-range oscil-
of behavior. Above the blue solid line, both correlators oscil-
latory structure of the local nematic alignment arises.
late and decay as a function of separation. Between the blue
solidandreddashedlines,bothcorrelatorsdecaybutonlythe
Apartfromitsrelevancetothespecificsubjectofliquid
thermal one also oscillates. Below the red dashed line, both
crystalline elastomers, the present work brings to light a
correlatorsdecaybutneitheroscillates. (b)Oscillationwave-
length ξ (measured in unit ξ ) as a function of reduced more general issue, viz., that the concept of a quenched
G,o L
temperature t for ∆ = 10. As t decreases, ξ eventually randomfieldshouldbebroadenedtoincorporatenotonly
G,o
saturatesatanonzerovalueoforderξ . Forbothpanels,we the traditional, ‘frozen’ type, which does not fluctuate
L
have set wR =0.4π. thermally, but also the type necessary for understand-
l
ing media such as liquid crystalline elastomers, in which
the frozen nature of the random field is present only at
space. Thisechosthepolydomainstructurewithwellde- longer lengthscales, fading out as the lengthscale pro-
fined length scale observed by the Urayama group [16]. gressesthroughacharacteristiclocalizationlength,owing
Indeed, as shown in Fig. 3, Fourier transform of their tothethermalpositionfluctuationsofthenetwork’scon-
real space image of the PD structure, after denoising stituents. The framework elucidated in the present work
and deblurring, shows clearly a ring of local maximum can be extended, with suitable modifications, to explore
at the scale of about half micron. In Fig. 4, we show the statistical physics of other randomly crosslinked sys-
examples of how the shell radius for the glassy correla- tems, such as smectic elastomers and various biological
tor progresses from the origin to a nonzero wave-vector materials.
magnitude and, furthermore, how the oscillatory decay
develops in real space, as the measurement temperature
is decreased. The crossover, from non-oscillatory to os-
cillatory decay as the disorder strength is increased, is
shown in Fig. 5a. It can be deduced from Eq. (14) that,
as the temperature is decreased, ξ decreases and ul-
G,o
timately saturates at the temperature-independent ther- ACKNOWLEDGMENTS
(cid:112)
maloscillationlength ξ (=ξ / 2ln(∆/2)). Provided
T,o L
thatξ exceedsξ ,thelengthξ willreflectthesize
G,d G,o G,o
ofthelocallyalignednematicdomains;therefore,thede- We thank Kenji Urayama for informative discus-
creaseofξG,o (showninFig.5b)willreflecttheshrinking sions on experimental results and Xiaoqun Zhang for
of the domain size with decreasing temperature. valuable assistance with image processing. This work
was supported by the National Science Foundation via
grants DMR06-05860 and DMR09-06780, the Institute
V. CONCLUSION AND OUTLOOK for Condensed Matter Theory and the Frederick Seitz
Materials Research Laboratory at the University of Illi-
Via a strategy rooted in vulcanization theory, we have nois at Urbana-Champaign, the Institute for Complex
developed a unified approach to the statistical physics Adaptive Matter, and Shanghai Jiao Tong University.
[1] Warner, M. & Terentjev, E. M. Liquid Crystal Elas- mol. 28, 4303–4306 (1995).
tomers. Oxford University Press (2003). [5] Ye,F.,Mukhopadhyay,R.,Stenull,O.&Lubensky,T.C.
[2] Olmsted, P. D. Rotational invariance and Goldstone Semisoft nematic elastomers and nematics in crossed
modes in nematic elastomers and gels. J. Phys. II electric and magnetic fields Phys. Rev. Lett. 98, 147801
(France) 4, 2215–2230 (1994). (2007).
[3] Golubovic,L.&Lubensky,T.C., Nonlinearelasticityof [6] Deam, R. T. & Edwards, S. F. The theory of rubber
amorphoussolids Phys. Rev. Lett. 63,1082-1085(1989). elasticity. Phil. Trans. R. Soc. A 280, 317–353 (1976).
[4] Verwey, G. & Warner, M. Soft rubber elasticity. Macro- [7] De Gennes, P. G. Scaling Concepts in Polymer Physics.
7
Cornell University Press, Ithaca (1979). [24] Goldbart, P. M., Castillo, H. E. & Zippelius, A. Ran-
[8] Clarke, S. M., Terentjev, E. M., Kundler, I. & Finkel- domly crosslinked macromolecular systems: vulcaniza-
mann, H. Texture evolution during the polydomain- tion transition to and properties of the amorphous solid
monodomain transition in nematic elastomers. Macro- state. Adv. Phys. 45, 393–468 (1996).
mol. 31, 4862–4872 (1998). [25] Peng, W., Castillo, H. E., Goldbart, P. M. & Zippelius,
[9] Fridrikh, S. V. & Terentjev, E. M. Polydomain- A. Universality and its origins at the amorphous solidi-
monodomain transition in nematic elastomers. Phys. fication transition. Phys. Rev. B 57, 839–847 (1998).
Rev. E 60, 1847–1857 (1999). [26] Broderix, K., Weigt, M. & Zippelius, A. Towards finite-
[10] Uchida, N. Elastic Effects in Disordered Nematic Net- dimensional gelation. Eur. Phys. J. B 29, 441–455
works Phys. Rev. E 60, R13-R16 (1999). (2002). DOI: 10.1140/epjb/e2002-00325-4
[11] Uchida, N. Soft and nonsoft structural transitions in [27] Xing, X., Pfahl, S., Mukhopadhyay, S., Goldbart, P. M.
disordered nematic networks. Phys. Rev. E 62, 5119– &Zippelius,A. Nematicelastomers: fromamicroscopic
5136 (2000). modeltomacroscopicelasticitytheory. Phys.Rev.E77,
[12] Selinger, J. V. & Ratna, B. R. Isotropic-nematic transi- 051802 (2008).
tion in liquid-crystalline elastomers: lattice model with [28] Edwards,S.F.&Anderson,P.W.Theoryofspinglasses.
quenched disorder. Phys. Rev. E 70, 041707 (2004). J. Phys. F: Metal Phys. 5, 965 (1975).
[13] Petridis,L.&Terentjev,E. Nematic-isotropictransition [29] For α = β, we can integrate out all xγ for γ (cid:54)= α.
withquencheddisorder. Phys.Rev.E74,051707(2006). The ∇α∇αΩ¯∇α∇αΩ¯ part in Eq. (3) then reduces to a
[14] Petridis,L.&Terentjev,E. Quencheddisorderandspin- function of xα only. Due to the invariance of the saddle
glass correlations in XY nematics. J. Phys. A: Math. point order parameter Ω¯ under simultaneous translation
Gen. 39, 9693–9708 (2006). of rˆ := (r0,...,rn), however, this function must be a
[15] Feio, G., Figueirinhas, J. L., Tajbakhsh, A. R. & Teren- trivial constant.
tjev, E. M. Critical fluctuations and random-anisotropy [30] At the mean field level, the shear modulus of a network
glass transition in nematic elastomers. Phys. Rev. B 78, isproportionaltoG2ξ−2,andsoisthedisorderstrength
L
020201 (2008). ∆.
[16] Urayama, K., Kohmon, E., Kojima, M. & Takigawa, T. [31] Feynman,R.P. StatisticalMechanics: ASetofLectures.
Polydomain-monodomain transition of randomly disor- Westview Press, 2nd Edn. (1998).
derednematicelastomerswithdifferentcross-linkinghis- [32] Shakhnovich, E. I. & Gutin, A. M. Formation of mi-
tories. Macromol. 42, 4084–4089 (2009). crodomains in a quenched disordered heteropolymer. J.
[17] Biggins,J.S.,Warner,M.&Bhattacharya,K. Supersoft Phys. (France) 50, 1843–1850 (1989).
elasticity in polydomain nematic elastomers. Phys. Rev. [33] M´ezard,M.&Parisi,G. Interfacesinarandommedium
Lett. 103, 037802 (2009). and replica symmetrybreaking. J. Phys. A: Math. Gen.
[18] Bhattacharya, K., Biggins, J. S. & Warner, M. 23, L1229–L1234 (1990).
Elasticity of polydomain liquid crystal elastomers. [34] M´ezard,M.&Parisi,G. Replicafieldtheoryforrandom
arXiv:0911.3513v1, 2009. manifolds. J. Phys. I (France) 1, 809–836 (1991).
[19] Imry, Y. & Ma, S.-K. Random-field instability of the [35] Stepanow, S., Dobrynin, A. V., Vilgis, T. A., & Binder,
ordered state of continuous symmetry. Phys. Rev. Lett. K. Copolymer melts in disordered media. J. Phys. I
35, 1399–1401 (1975). (France) 6, 837–857 (1996).
[20] Radzihovsky, L. & Toner, J. Smectic liquid crystals in [36] Strictly speaking, the structure of the kernel Γαβ should
randomenvironments. Phys. Rev. B60,206-257(1999). allow for anisotropy for the case of Q¯ (cid:54)= 0. Our main
[21] Feldman, D. E. Quasi-long-range order in nematics con- analysis, however, only concerns the isotropic state, for
finedinrandomporousmedia.Phys.Rev.Lett.84,4886– which Q¯ =0.
4889 (2000). [37] Accordingtothestandardunderstanding,areplicasym-
[22] Xing,X.&Radzihovsky,L. Universalelasticityandfluc- metry breaking atlow temperature implies that the sys-
tuations of nematic gels. Phys. Rev. Lett. 90, 168301 tem freezes into one of many nearly degenerated states.
(2003). Such a system should exhibit irreversible glassy proper-
[23] Xing, X. & Radzihovsky, L. Nonlinear elasticity, fluc- tiessuchashysteresisinstress-straincurve.Theseprop-
tuations and heterogeneity of nematic elastomers. Ann. erties have not been observed in polydomain nematic
Phys. 323, 105–203 (2008). elastomers.
