Statistical physics of isotropic-genesis nematic elastomers: I. Structure and correlations at high temperatures
SStatistical physics of isotropic-genesis nematic elastomers:I. Structure and correlations at high temperatures
Bing-Sui Lu , Fangfu Ye , Xiangjun Xing , and Paul M. Goldbart Department of Physics and Institute of Natural Sciences, Shanghai Jiao Tong University,800 Dongchuan Road, Minhang District, Shanghai, China and School of Physics, Georgia Institute of Technology,837 State Street, Atlanta, GA 30332-0430, USA (Dated: September 18, 2018)Isotropic-genesis nematic elastomers (IGNEs) are liquid crystalline polymers (LCPs) that havebeen randomly, permanently cross-linked in the high–temperature state so as to form an equilib-rium random solid. Thus, instead of being free to diffuse throughout the entire volume, as theywould be in the liquid state, the constituent LCPs in an IGNE are mobile only over a finite, seg-ment specific, length–scale controlled by the density of cross-links. We address the effects that suchnetwork–induced localization have on the liquid–crystalline characteristics of an IGNE, as probedvia measurements made at high temperatures. In contrast with the case of uncross-linked LCPs,for IGNEs these characteristics are determined not only by thermal fluctuations but also by thequenched disorder associated with the cross-link constraints. To study IGNEs, we consider a micro-scopic model of dimer nematogens in which the dimers interact via orientation-dependent excludedvolume forces. The dimers are, furthermore, randomly, permanently cross-linked via short Hookeansprings, the statistics of which we model by means of a Deam-Edwards type of distribution. We showthat at length–scales larger than the size of the nematogens this approach leads to a recently pro-posed, phenomenological Landau theory of IGNEs [Lu et al.,
Phys. Rev. Lett. , 257803 (2012)],and hence predicts a regime of short–ranged oscillatory spatial correlations in the nematic align-ment, of both thermal and glassy types. In addition, we consider two alternative microscopic modelsof IGNEs: (i) a wormlike chain model of IGNEs that are formed via the cross-linking of side-chainLCPs; and (ii) a jointed chain model of IGNEs that are formed via the cross-linking of main-chainLCPs. At large length–scales, both of these models give rise to liquid–crystalline characteristics thatare qualitatively in line with those predicted on the basis of the dimer-and-springs model, reflectingthe fact that the three models inhabit a common universality class.
PACS numbers: 61.30.Vx,61.30.-v,61.43.-j
I. INTRODUCTIONA. Nematic elastomeric materials
Nematic elastomers are fascinating materials in whichthe liquid crystalline order is strongly coupled to the elas-ticity of the underlying elastomeric network (see, e.g.,Refs. [1–6]). This strong nemato-elastic coupling givesrise to novel, emergent properties in nematic elastomersthat are found neither in liquid crystal nematics nor or-dinary rubbery materials. One well-known example ofsuch properties is the soft elasticity characteristic of mon-odomain nematic elastomers (see, e.g., Ref. [7]) that areformed via the K¨upfer-Finkelmann procedure [8].Not only are nematic elastomers fascinating but alsothey have proven to be a challenging subject for the-oretical investigation. Part of the challenge originatesin the dependence of the physical characteristics of ne-matic elastomers on the conditions under which the elas-tomers are prepared. For example, isotropic-genesis ne-matic elastomers (or IGNEs)—nematic elastomers cross-linked in the isotropic state—exhibit the so-called super-soft version of elastic response at sufficiently low tem-peratures (see, e.g., Refs. [9–11]), unlike their nematic-genesis nematic elastomer (or NGNE) counterparts. Fur-thermore, in thermal equilibrium the nematic alignment in IGNEs exhibits a polydomain structure (see, e.g.,Refs. [3, 12–16, 18]) characterized by a length-scale deter-mined solely via thermodynamic quantities such as thedensity of cross-links.Another challenge to theorists comes from the factthat nematic elastomers possess a multi-level hierarchyof interdependent elements of randomness. First, thereis quenched disorder in its conventional form, associatedwith the permanent chemical structure that originatesin the cross-linking process. Second, as a result of suf-ficient cross-linking there arise the mean positions andr.m.s. displacements of the spatially localized polymersthat constitute the elastomeric network, both of theseelements being random. Third, there is the thermal dis-order associated with the Brownian motion of the posi-tional and orientational (i.e., nematogenic) freedoms inthe state of the system just prior to the instant of cross-linking ; in part, this thermal disorder is frozen in viathe process of cross-linking. Fourth, there is also thethermal disorder associated with the Brownian motionof the nematogens in the state of the system long afterthe instant of cross-linking. It is not a priori evidenthow the interplay between the various types of random-ness present in nematic elastomers resolve themselves,and thus determine the equilibrium structure and elasticresponse of such media. As we shall try to make clear a r X i v : . [ c ond - m a t . s o f t ] A p r in subsequent sections of the present Paper, in order tounderstand nematic elastomers it is valuable to go be-yond the conventional notion of quenched disorder and,instead, to consider an amalgam of the second, third andfourth types of randomness composed of a “frozen” part(due to the mean positions of the network constituents)and a “molten” part (due to the thermal fluctuations ofthe network constituents). B. Overview
In the present Paper we investigate the static structureof nematic alignment in IGNEs that can be probed inthe high-temperature regime. This investigation is partlymotivated by the experiments on polydomain structurereported in Ref. [9]. We focus our considerations here onsystems in the high-temperature regime and which arenot subject to externally applied deformations.Two types of nematic fluctuations are present inIGNEs: (i) those that are frozen in, however imperfectly,by the network during the process of cross-linking; and(ii) thermally driven departures away from the mean lo-cal alignment pattern that is frozen in. In order to char-acterize such fluctuations, we make use of (i) the correla-tor of the local nematic order that was frozen in duringthe cross-linking process, this correlator being appropri-ately averaged over realizations of the quenched disorder(and termed the glassy correlator); and (ii) the corre-lator of the thermally driven departures of the nematicorder from the mean frozen-in local alignment pattern,appropriately averaged over realizations of the quencheddisorder (and termed the thermal correlator).We have already mentioned that the physical proper-ties of IGNEs depend on the conditions under which theywere prepared. To reflect this fact, we make a careful dis-tinction between two thermodynamic ensembles in ourtheoretical approach: the first, in which the IGNE was prepared , which we term the preparation ensemble ; andthe second, in which the system is measured , which wecall the measurement ensemble . This structure enablesour theory to capture the ability of IGNEs to “remember,”at least to some degree, the local nematic alignment pat-tern at the moment of preparation, and in addition en-ables the determination of the dependence of the strengthof this memorization on (i) the temperature at which thesystem was cross-linked, and (ii) the average number ofcross-links per polymer.We note in passing that having both preparation andmeasurement ensembles places us in a family of dis-ordered systems for which spontaneous replica symme-try breaking is expected to be irrelevant, in contrastwith settings that do not feature a preparation ensem-ble; cf. Ref. [20]. (Technically, this expectation shows upin the need to investigate coupled replicas of the physicalsystem in the neighborhood of not zero but one replica.)Thus, we do not expect our approach to yield glassy phe-nomena such as hysteresis in the stress-strain behavior of IGNEs—which is not unreasonable, given the absence, todate, of experimental observations of such phenomena.The fact that the elastomer network is thermally fluc-tuating means that any prospective theory of IGNEsshould feature a typical localization length-scale, belowwhich the polymer constituents of the network are effec-tively delocalized. Our approach features such a length-scale, which leads to the possibility that nematic correla-tions undergo a novel, oscillatory form of decay with dis-tance in a certain regime. Prior theoretical approachesbuilt on conventional random-field models do not fea-ture the thermal fluctuations of the elastomer network(see, e.g., Refs. [12, 13, 19–21]), and thus do not cap-ture this intriguing phenomenon. A phenomenologicalrandom-field-type model that does take the thermal fluc-tuations of the elastomer network into account was pre-sented in Ref. [22]. In the present Paper we derive theLandau-type free energy for IGNEs that was presented inRef. [22], doing so via a microscopic model that involvesdimers that are randomly and permanently connected byHookean springs. The present Paper thus provides a mi-croscopic justification for the ideas and results presentedin Ref. [22].The outline of the Paper is as follows. In Sec. II wepresent a microscopic dimer-spring model of IGNEs. InSec. III we apply the replica technique and implementthe Hubbard-Stratonovich scheme to decouple the in-teracting microscopic degrees of freedom. In Sec. IVwe derive a Landau-Wilson type of free energy for theIGNE, which involves an order parameter field Q forthe isotropic-to-nematic phase transition as well as anorder parameter field Ω for the vulcanization/gelationtransition. In Sec. IV A we make an expansion of theLandau-Wilson free energy for small Q and Ω, whichis appropriate for exploring the physics in the vicinity ofthe transition to the random solid state. In Sec. V we de-termine the stationary states of the Landau-Wilson freeenergy. In Sec. VI we derive an effective replica Hamil-tonian describing local nematic order in IGNEs by set-ting Ω to its stationary value but retaining fluctuationsof Q to quadratic order. In Sec. VII we then comparethis effective Hamiltonian with that arising from the phe-nomenological Landau free energy considered in Ref. [22]and show that they are equivalent. In Sec. VIII we usethe effective Hamiltonian to derive the glassy and ther-mal correlators. In Sec. IX we describe two alternativemicroscopic models of the IGNE, viz. (i) a worm-likechain model of side-chain nematic polymer networks; and(ii) a jointed chain model of main-chain nematic polymernetworks. As we shall see, at length-scales larger thanthe size of a nematogen, both of these models give riseto liquid–crystalline characteristics similar to those re-sulting from the dimer-and-springs model, reflecting thefact that the three models inhabit a common universalityclass. In Sec. X we make some concluding remarks. II. INGREDIENTS OF THE MODEL
We model an IGNE microscopically as a system of N dimers in D spatial dimensions that are randomly, per-manently linked via springs (see Fig. 1 and Ref. [23]).We envision the springs as mimicking the flexible con-stituents of liquid crystalline polymers whilst also serv-ing as cross-links; the dimers mimic the stiff constituentsof liquid crystalline polymers. Each dimer (labeled by j ,where j = 1 , . . . , N ) consists of two particles at positionvectors c j, and c j, − separated by a fixed distance (cid:96) .The orientation of the j -th dimer is specified by the unitvector n j = c j, − c j, − | c j, − c j, − | . (2.1)The dimers interact via three types of forces. First, thereis an orientational interaction between dimers that pro-motes parallel or antiparallel alignment. We model thisinteraction via a potential of the Maier-Saupe type, viz., H nem = − V N N (cid:88) i,j =1 J ( c i − c j )( n i · n j ) , (2.2)where c j ≡ ( c j, + c j, − ) / j -thdimer’s center of mass. We assume that the aligning in-teraction is short-ranged, and model the interaction po-tential J ( c ) by the form (cid:0) J / (2 πa ) D/ (cid:1) exp( − c / a ).In Fourier space, the potential is given by J p = J exp( − p a / a specifies the range of the inter-action between dimers and J characterizes its strength.In addition to the orientational interaction, the dimersexperience a positional excluded-volume interaction be-tween particles belonging to any pair of dimers, which wemodel via an Edwards-type pseudo-potential [24, 25]: H ev = λ N (cid:88) i,j =1 (cid:88) s,t =1 , δ ( c i,s − c j,t ) , (2.3)where λ is the strength of the excluded-volume interac-tion [26]. The presence of sufficiently strong excluded-volume forces stabilizes the system against collapse to aglobule, even when well cross-linked. Third, any giventwo dimers that are connected by a spring are taken tointeract additionally via a harmonic potential associatedwith the spring, which we take to have zero rest-lengthand native mean-square end separation b , characteristicof Gaussian molecular chains. These springs and, specifi-cally, the architectural information indicating which pairsof rod ends are connected to one another by springs con-stitutes the quenched randomness χ of any given real-ization of the system. This information takes the form χ ≡ { ie, s e , j e , t e } Me =1 , where M is the total number ofsprings and spring e connects end s e of rod i e to end t e of rod j e . These springs result in the following Hookeanterm in the Hamiltonian: H xlink = T b M (cid:88) e =1 | c i e ,s e − c j e ,t e | , (2.4) FIG. 1: A model of dimers (indicated by dumbbells) cross-linked via harmonic springs. The dimers have length (cid:96) andthe springs have r.m.s. length b . where T denotes the temperature and we have adoptedunits in which Boltzmann’s constant is unity.The total Hamiltonian for the dimers-and-springsmodel is then given by H χ = H nem + H xlink + H ev . (2.5)For a given realization χ of the quenched disorder, thecorresponding partition function takes the form Z χ = (cid:90) (cid:32) N (cid:89) i =1 d c i, − d c i, δ ( | c j, − c j, − | − (cid:96) ) (cid:33) e − H χ /T , (2.6)and the free energy F χ is given by − T ln Z χ . As is wellknown [27], it is appropriate to average the free energyof the system over realizations of the quenched disorder.Denoting this average by the square brackets [ · · · ], onehas [ F χ ] = (cid:88) χ P χ F χ = − T (cid:88) χ P χ ln Z χ . (2.7) III. REPLICAS AND COLLECTIVE FIELDSA. Statistics of quenched disorder
What statistical distribution should one use to com-pute the average of the free energy over the quencheddisorder? In common with other elastomers such asisotropic rubbery systems, the quenched-disorder averagefor IGNEs can be performed via a variant of the Deam-Edwards distribution P ( χ ) [28]. Such distributions re-flect situations in which systems undergo instantaneous cross-linking: one begins with a melt or solution at equi-librium and—so rapidly that hardly any relaxation hastime to occur—one introduces permanent bonds betweensome random fraction of the pairs of dimers that hap-pen, at the instant of cross-linking, to be nearby oneanother. To construct the associated P ( χ ), one shouldrespect the causal order of the cross-linking process (see,e.g. Ref. [30]). Formally, this amounts to the specifica-tion P ( χ ) = (cid:90) D c P ( χ |{ c i,s } ) P ( { c i,s } ) , (3.1)where the multiple integral (cid:82) D c is defined via (cid:90) D c ≡ N (cid:89) i =1 (cid:90) d c i, − d c i, , (3.2) the distribution P ( { c i,s } ) describes the statistics of theliquid system at the instant of cross-linking, so that P ( { c i,s } ) ≡ e − ( H nem ( { n i } )+ H ev ( { c i,s } ) ) /T (cid:81) Ni =1 δ ( | c i, − c i, − | − (cid:96) ) (cid:82) D c e − ( H nem ( { n i } )+ H ev ( { c i,s } ) ) /T (cid:81) Ni =1 δ ( | c i, − c i, − | − (cid:96) ) , (3.3)and P ( χ |{ c i,s } ) is the conditional probability that cross-links are formed between pairs of dimer-ends at positions { ( c i e ,s e , c j e ,t e ) } Me =1 , given that the constituents of the liq-uid at the instant of cross-linking are at { c i,s } . It is givenby [30] P ( χ |{ c i,s } ) = 1 M ! (cid:32) ˜ V η N (cid:33) M e H norm ( { c i,s } ) /T × M (cid:89) e =1 e −| c ie,se − c je,te | / b . (3.4)Here, ˜ V is the dimensionless volume of the system, i.e., V / (2 πb ) D/ . The term H norm ( c ) arises from the re- quirement that P ( χ |{ c i,s } ) be properly normalized over χ , i.e., (cid:80) χ P ( χ |{ c i,s } ) = 1; it is given by H norm ( { c i,s } ) = − T ˜ V η N N (cid:88) i,j =1 1 (cid:88) s,t = − e −| c i,s − c j,t | / b . (3.5)The probability P ( χ |{ c i,s } ) features a dimensionless pa-rameter η that controls the likelihood that cross-links areactually formed. In App. A we show that it is related tothe average number of cross-linking springs per dimer viathe formula [ M ] /N ≈ η . B. Disorder averaging the free energy
Now that we have constructed a suitable disorder distribution P χ , we use it to perform the disorder average ofthe logarithm of the partition function [ln Z χ ], doing this indirectly, using the replica technique (see, e.g., Ref. [31]).Thus, we represent the logarithm in Eq. (2.7) as a limit to obtain[ F ] = − T lim n → [ Z nχ ] − n , (3.6)where we have interchanged the order of taking the replica limit and performing the disorder average on going fromEq. (2.7) to Eq. (3.6). We then insert the Deam-Edwards distribution (3.1) to obtain[ Z nχ ] = ∞ (cid:88) M =0 N (cid:88) i ,j =1 . . . N (cid:88) i M ,j M =1 1 (cid:88) s ,t = − . . . (cid:88) s M ,t M = − M ! (cid:32) ˜ V η N (cid:33) M Z e H norm ( { c i,s } ) /T × n (cid:89) α =0 (cid:90) D c α e − (cid:80) nα =0 ( H nem ( { n αi } )+ H ev ( { c αi,s } ) ) /T M (cid:89) e =1 e − (cid:80) nα =0 | c αie,se − c αje,te | / b × n (cid:89) α =0 N (cid:89) i =1 δ ( | c αi, − c αi, − | − (cid:96) ) . (3.7)Here, α = 0 , , . . . , n labels the replicas, and Z is de-fined via Z ≡ (cid:90) D c e − ( H nem ( { n i } ) − H ev ( { c i,s } ) ) /T × N (cid:89) i =1 δ ( | c i, − c i, − | − (cid:96) ) . (3.8) Note that in Eq. (3.7) the term H norm ( { c i,s } ) limits per-mutation invariance to replicas α = 1 , . . . , n , with replica α = 0 being excluded. This is consistent with our phys-ical expectation that the preparation and measurementensembles play distinct roles.Next, we complete the computation of the disorderaverage in Eq. (3.7), which yields [ Z nχ ] in terms of thereplica partition function Z n , i.e.,[ Z nχ ] = Z n Z , (3.9) Z n ≡ (cid:90) (cid:32) n (cid:89) α =0 N (cid:89) i =1 d c αi, d c αi, − δ ( | c αi, − c αi, − | − (cid:96) ) (cid:33) e − H rep , (3.10)in which (cid:82) d c ≡ (cid:81) Dd =1 (cid:82) dc d , where the effective replicaHamiltonian H rep is given by H rep = − n (cid:88) α =0 N (cid:88) i,j =1 J ij T α (cid:0) ( n αi · n αj ) − D (cid:1) + 12 n (cid:88) α =0 N (cid:88) i,j =1 1 (cid:88) s,t = − λT α δ ( c αi,s − c αj,t ) − V η N (2 πb ) D/ N (cid:88) i,j =1 1 (cid:88) s,t = − (cid:16) e − b (cid:80) nα =0 | c αi,s − c αj,t | − e − b | c i,s − c j,t | (cid:17) . (3.11)We note that in H rep the replicas are coupled as a result of the disorder averaging. See App. B for details of thederivation of Eqs. (3.10,3.11). C. Collective fields and their physical meaning
We note that in H rep , Eq. (3.11), the N interact-ing (replicated) dimers are coupled with one another,which means that the trace over microscopic variables c i,s in Z n [Eq. (3.10)] cannot be straightforwardly car-ried out. We decouple these dimers by defining micro-scopic collective fields ω ˆ k and q p , and making a Hubbard-Stratonovich transformation involving fluctuating auxil-iary fields conjugate to the collective fields. Thus, wearrive at a description in terms of N uncoupled copiesof a single replicated dimer, for which the trace over c i,s can be readily performed, order by order in an expansionin the auxiliary fields. The collective fields are given by ω ˆ k ≡ N N (cid:88) i =1 (cid:88) s = − , n (cid:89) α =0 e − i ˆ k · ˆ c i,s , (3.12a) q d d ( p ) ≡ N N (cid:88) i =1 e i p · c i ( n i d n i d − D − δ d d ) . (3.12b)Here, we have used the symbol ˆ k to denote the n + 1-fold replicated wave-vector ( k , k , . . . , k n ), and have re- stricted the value of ˆ k in ω ˆ k to the “higher-replica sector”(HRS), viz., the set of replicated vectors each having atleast two non-zero vector entries. Details of the Hubbard-Stratonovich transformation are given in App. B. The re-sult is that the replica partition function Z n becomesa functional integral over the auxiliary fields Ω ˆ k and Q p ,whose expectation values are related to the expectationvalues of ω ˆ k and q p via (cid:104) Ω ˆ k (cid:105) f = (cid:104)(cid:10) ω ˆ k (cid:11) χ (cid:105) ; (3.13a) (cid:104) Q α p (cid:105) f = (cid:104)(cid:10) q α p (cid:11) χ (cid:105) , (3.13b)as we demonstrate in App. C. Note that (cid:104)· · · (cid:105) f indicatean expectation value taken with respect to the Landau-Wilson free energy f n discussed in Sec. IV and definedin Eq. (4.2), and (cid:104)· · · (cid:105) χ denotes an expectation valuetaken with respect to the Hamiltonian H χ , Eq. (2.5).To gain some intuition for the physical significance of [ (cid:104) Ω (cid:105) f ] and [ (cid:104) Q α (cid:105) f ], we transform Eqs. (3.13) to replicatedreal space: [ (cid:104) ω (ˆ r ) (cid:105) χ ] = (cid:34)(cid:42) N N (cid:88) i =1 1 (cid:88) s = − n (cid:89) α =0 δ (cid:0) r α − c αi,s (cid:1)(cid:43)(cid:35) − V n = [ (cid:104) Ω(ˆ r ) (cid:105) f ] ; (3.14a)[ (cid:104) q d d ( r ) (cid:105) χ ] = (cid:34)(cid:42) N N (cid:88) i =1 ( n i d n i d − D − δ d d ) δ ( r − c i ) (cid:43)(cid:35) = [ (cid:104) Q d d ( r ) (cid:105) f ] . (3.14b)Thus, except for a trivial constant, we can interpret (cid:104) Ω(ˆ r ) (cid:105) f as the joint probability that a given dimer endis found at position r at the instant of cross-linking,and that the same dimer end would be found at n sub-sequent widely-separated time instants at the positions r , . . . , r n [23]. (cid:104) Ω(ˆ r ) (cid:105) f vanishes if all dimers are delo-calized, and has a nonzero value if a fraction of them arelocalized. Thus, (cid:104) Ω(ˆ r ) (cid:105) f serves as the order parameterthat detects the phase transition from the liquid stateto the random solid state. Similarly, (cid:104) Q ( r ) (cid:105) f is the ne-matic order parameter for the preparation state, whilst (cid:104) Q α ( r ) (cid:105) f (for α = 1 , . . . , N ) is the nematic order param-eter for the measurement state.We note that when defining Ω we have chosen to ex-clude the components associated with the one-replica sec-tor (denoted “1RS”), i.e., the set of replicated wave-vectors that each have only one non-zero vector entry[i.e., ˆ k = ( , . . . , k α , . . . , )]. This is because this sectorof the field corresponds to fluctuations in the macroscopicdensity of dimers, and is strongly stabilized as a result ofthe excluded volume interactions, regardless of the extentof the cross-linking. The physical content of the decom-position of fields into higher and lower replica- sectorsis that if condensation occurs in the higher sector only,this implies the random localization of particles. On theother hand, if condensation occurs in the 1RS as well,this indicates the formation of a state having some kindof spatially modulated density structure. We are focus-ing on highly incompressible systems, for which densityfluctuations are negligibly small. Incompressibility is en-forced by taking λ to have a large value, so that fluctu-ations of 1RS counterpart to Ω are strongly suppressed.We also define the zero-replica sector (denoted “0RS”)to be the set whose only member is the replicated wave-vector that has zero for every entry. The lower-replicasector (denoted “LRS”) would then refer to the union ofthe one-replica sector and the zero-replica sector. D. Intermezzo on replicas
Compared with the more familiar example of spinglasses (see, e.g., Ref. [31]), the effective replica Hamilto-nian (3.11) in our theory contains an extra replica. Whatmeaning can we ascribe to this “zeroth” replica as wellas to the other n replicas? Physically, the zeroth replica,which originates in the Deam-Edwards distribution, cor- responds to the state of the system at the instant ofpreparation, whilst the other n replicas correspond to thestate of the system when it is measured. The dependenceof the measured properties on the state of the system atpreparation corresponds, operationally, to the couplingin the effective replica Hamiltonian (3.11) between thefreedoms belonging to the zeroth replica and those be-longing to the other n replicas. On the other hand, sucha coupling does not imply that the measured propertiesinfluence the preparation state, as one can show via acareful consideration of the n → n replicas, it is well adapted to investi-gations of nematic elastomers, whose measured proper-ties are known to depend on their preparation histories;see, e.g., Ref. [9]. Such a dependence was already un-derstood by Deam and Edwards and by de Gennes morethan thirty years ago (see, e.g., Refs. [28, 33]). IV. LANDAU-WILSON FREE ENERGY
As we demonstrate in App. B, the local incompress-ibility of IGNEs allows us to express the effective replicatheory in terms of the auxiliary fields Ω and Q α as Z n ∝ (cid:90) D Ω ˆ k n (cid:89) α =0 D Q α p exp (cid:0) − N f n (Ω , Q α ) (cid:1) , (4.1)where the Landau-Wilson free energy per dimer f n isgiven by f n (Ω , Q ) = ˜ η V n (cid:88) ˆ k ∆ ˆ k | Ω ˆ k | + 12 n (cid:88) α =0 (cid:88) p J p T α { Q α p Q α − p }− ln (cid:104) exp (cid:0) G (Ω) + G ( Q ) (cid:1) (cid:105) , n . (4.2)We have introduced the notation G (Ω) ≡ ˜ η V n (cid:88) ˆ k ∆ ˆ k Ω ˆ k (cid:88) s = − e − i ˆ k · ˆ c s , (4.3) G ( Q ) ≡ n (cid:88) α =0 (cid:88) p J p T α Q αd d ( p ) e − i p · c α × (cid:18) n αd n αd − δ d d (cid:19) , (4.4)and have specialized to three spatial dimensions (i.e., D = 3). The curly braces { S S (cid:48) } in Eq. (4.2) denotethe trace of the product of the tensors S and S (cid:48) , i.e., (cid:80) Dd ,d =1 S d d S (cid:48) d d . The symbols ˜ η , ∆ k and (cid:104)· · · (cid:105) , n are defined via˜ η ≡ η , (4.5a)∆ k ≡ exp( − b k / , (4.5b) (cid:104)· · · (cid:105) , n ≡ n (cid:89) α =0 (cid:90) d c α d c α − πV (cid:96) δ ( | c α − c α − | − (cid:96) ) . (4.5c) Moreover, (cid:80) ˆ k denotes the sum over replicated wave-vectors that are restricted to the higher-replica sector. A. Expanding the Landau-Wilson free energy
To develop the expansion of the Landau-Wilson free energy we expand the log-trace term in Eq. (4.2) in powers of Q and Ω to obtain (see App. D for details): f n (Ω , Q ) ≈ f Ω (Ω) + f Q ( Q ) + f C (Ω , Q ) . (4.6)The first term on the right hand side, f Ω (Ω), describes the vulcanization/random solidification transition for anisotropic elastomer [29, 34, 37] and is given by f Ω (Ω) = ˜ η (cid:88) ˆ k (cid:16)(cid:0) − ˜ η (cid:1) + 12 (cid:0) b + (cid:96) (cid:1) | ˆ k | (cid:17) | Ω ˆ k | − ˜ η (cid:88) ˆ k , ˆ k , ˆ k δ ˆ k +ˆ k +ˆ k , ˆ0 Ω ˆ k Ω ˆ k Ω ˆ k . (4.7)It exhibits a linear instability at the critical value ˜ η c = 1, [36] which reflects the destabilization of the liquid state withrespect to a gel/random solid state when the average number of cross-links per dimer is increased beyond a certaincritical value. As IGNEs are random solids, they must have undergone a vulcanization transition (controlled by thedensity of cross-links) from the liquid state to the random solid state. Thus, we keep the Ω-only terms to cubic order;this is in line with the approach adopted in Ref. [34]. The second term, f Q , is given by f Q ( Q ) = n (cid:88) α =0 (cid:88) p T α M ( T α , p ) { Q α p Q α − p } , (4.8)These terms describe the free-energy cost of inducing nematic alignment from the unaligned state. As the focus of thepresent Paper is on the liquid-crystalline properties of IGNEs at high preparation and measurement temperatures,we retain the Q -only terms to quadratic order. This collection of terms is an (1 + n )-fold replicated version ofthe (quadratic part of the) Landau-de Gennes free energy for a liquid of nematogens at high temperatures [35]. InEq. (4.8), T is the preparation temperature (i.e., the temperature at which the system was cross-linked), and T α (for α = 1 , . . . , n ) is the measurement temperature (i.e., the temperature at which the system is measured, long aftercross-linking). As replicas α = 1 , . . . , n are actually copies of a single system measured at one temperature, we have T α = T for α = 1 , . . . , n . The kernel M ( T α , p ) is given by M ( T α , p ) ≡ J e − p a / (cid:18) − J T α e − p a / (cid:19) , (4.9)which, expanded in powers of wave-vectors, yields [32] M ( T α , p ) ≈ A α t α + L α p , (4.10)where t is the reduced preparation temperature ( T − T ∗ ) /T ∗ (where T ∗ has the value 4 J / t α (for α =1 , . . . , n ) is reduced measurement temperature, ( T − T ∗ ) /T ∗ . For simplicity, we write t α = t for α = 1 , . . . , n . Theparameter A α ( ≡ J T ∗ /T α ) characterizes the aligning tendencies of nematogens, and L α [ ≡ (2 T ∗ − T α ) a J / T α ] isthe generalized nematic stiffness in the (Landau-de Gennes equivalent of the) one-Frank-constant approximation [35].Hence, the nematic free energy f Q becomes f Q ( Q ) ≈ T (cid:88) p M ( T , p ) { Q p Q − p } + n (cid:88) α =1 (cid:88) p T ( A t + L p ) { Q α p Q α − p } , (4.11)where A and L are, respectively, A α and L α for α = 1 , . . . , n . The third term of the right hand side of Eq. (4.6), f C ,describes the coupling between the nematic freedoms and the elastomer and is given by f C (Ω , Q ) = n (cid:88) α (cid:54) = β (cid:88) p , q Φ αβd d d d ( p , q ) Ω − p ˆ (cid:15) α − q ˆ (cid:15) β Q αd d ( p ) Q βd d ( q )+ n (cid:88) α =0 (cid:88) p (cid:88) ˆ k Ψ αd d ( p , q , ˆ k ) Ω ˆ k Ω − ˆ k − p ˆ (cid:15) α Q αd d ( p )+ n (cid:88) α,β =0( α (cid:54) = β ) (cid:88) p , q (cid:88) ˆ k Γ αβd d d d ( p , q , ˆ k ) Ω ˆ k Ω − ˆ k − p ˆ (cid:15) α − q ˆ (cid:15) β Q αd d ( p ) Q βd d ( q )+ n (cid:88) α =0 (cid:88) p , q (cid:88) ˆ k (cid:16) Υ αd d d d ( p , ˆ k ) δ p + q , + Θ αd d d d ( p , q , ˆ k ) (cid:17) × Ω ˆ k Ω − ˆ k − ( p + q )ˆ (cid:15) α Q αd d ( p ) Q αd d ( q ) , (4.12)where the wave-vector-dependent coefficients Φ, Ψ, Θ, Υ and Γ are defined asΦ αβd d d d ( p , q ) ≡ − ∆ − p ˆ (cid:15) α − q ˆ (cid:15) β ˜ η J p J q T α T β p d p d q d q d ; (4.13a)Ψ αd d ( p , q , ˆ k ) ≡ − ∆ ˆ k ∆ − ˆ k − p ˆ (cid:15) α ˜ η (cid:96) J p T α (cid:32) p d p d + (cid:18) k α + 12 p (cid:19) d (cid:18) k α + 12 p (cid:19) d (cid:33) (4.13b)Υ αd d d d ( p , ˆ k ) ≡ − ∆ k η (cid:18) J p T α (cid:19) δ d d δ d d ; (4.13c)Γ αβd d d d ( p , q , ˆ k ) ≡ ˜ η (cid:96) J p J q T α T β ∆ ˆ k ∆ − ˆ k − p ˆ (cid:15) α − q ˆ (cid:15) β × (cid:18) k αd k αd k βd k βd + 14 (cid:0) p d k αd + k αd p d (cid:1) (cid:16) q d k βd + k βd q d (cid:17) + 18 p d p d q d q d (cid:19) ; (4.13d)Θ αd d d d ( p , q , ˆ k ) ≡ − ˜ η J p J q T α ) ∆ ˆ k ∆ − ˆ k − ( p + q )ˆ (cid:15) α × (cid:18)(cid:18) − (cid:96) | p + q |
56 + (cid:96) | p + q | − (cid:96) | p + q + 4 k α | (cid:96) | p + q + 4 k α | (cid:19) δ d d δ d d + (cid:18) −
114 + (cid:96) | p + q | (cid:19) (cid:96) ( p + q ) d ( p + q ) d δ d d + (cid:18) −
114 + (cid:96) | p + q + 4 k α | (cid:19) × (cid:96) ( p + q + 4 k α ) d ( p + q + 4 k α ) d δ d d + (cid:96) p + q ) d ( p + q ) d ( p + q ) d ( p + q ) d + (cid:96) p + q + 4 k α ) d ( p + q + 4 k α ) d × ( p + q + 4 k α ) d ( p + q + 4 k α ) d ) . (4.13e)Here, we have retained the lowest-order terms that couple Ω and Q , i.e., those proportional to Q Q
Ω and Q Ω Ω.We have, moreover, kept only the lowest-order wave-vector dependencies of their coefficients. We have, in addition,retained the term proportional to
Q Q
Ω Ω because we have found that it is the leading one responsible for thequalitatively new possibility of thermal and glassy correlators that not only decay but also oscillate with distance.The physical significance of f C is that it encodes into the theory the fact that the nematic degrees of freedominhabit an environment that is at the microscopic level anisotropic, inhomogeneous and thermally fluctuating, butthat no remnants of this anisotropy or inhomogeneity survive to the macroscopic level. f C thus also describes themutual effects of random localization and nematic alignment. In particular, it enables us to study the impact of thememorization of the nematic fluctuations in the preparation state on the nematic fluctuations in the measurementstate [22]. V. SADDLE-POINT APPROXIMATION
Next, we consider the saddle-point equations for Q andΩ, which follow from the stationarity of Eq. (4.6): δf n δ Ω (cid:12)(cid:12)(cid:12)(cid:12) ¯Ω , ¯ Q = 0; (5.1a) (cid:32) δf n δQ αd d − δ d d Tr δf n δ Q α (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯Ω , ¯ Q = 0 . (5.1b)As we are concerned with macroscopically isotropicstates, we require that the saddle-point value of Q van-ishes: ¯ Q = . As a result, the saddle-point equa-tion (5.1a) for ¯Ω is the same as it would be for ordinary(i.e., non-nematogenic) elastomers, and reads (cid:16) − ˜ η + 12 (cid:96) (cid:48) r | ˆ k | (cid:17) Ω ˆ k − ˜ η (cid:88) ˆ k (cid:48) Ω ˆ k (cid:48) Ω ˆ k − ˆ k (cid:48) = 0 , (5.2)in which we have defined the length-scale (cid:96) (cid:48) r via (cid:96) (cid:48) r ≡ b +( (cid:96) / (cid:15) ≡ η − / ˜ η and (cid:96) r ≡ √ (cid:96) (cid:48) r / ˜ η , the saddle-point equation takes theform analyzed in Ref [37]: (cid:16) − (cid:15) + 12 (cid:96) r | ˆ k | (cid:17) Ω ˆ k − (cid:88) ˆ k (cid:48) Ω ˆ k (cid:48) Ω ˆ k − ˆ k (cid:48) = 0 , (5.3)and is therefore solved by mean of the the Ansatz¯Ω ˆ k = G (cid:90) d z V (cid:90) dτ P ( τ ) e i (cid:80) α k α · z −| ˆ k | / τ (5.4)(with ˆ k ∈ HRS), provided that the gel fraction obeys G = 2 (cid:15)/ , (5.5)and P ( τ ), which has the meaning of the distribution ofinverse square localization lengths τ , obeys τ dP ( τ ) dτ = (cid:16) (cid:15) − τ (cid:17) P ( τ ) − (cid:15) (cid:90) τ dτ P ( τ ) P ( τ − τ ) . (5.6)The typical value ξ L of the localization length is (cid:15) − / (cid:96) r ,which diverges as the vulcanization transition is ap-proached from the solid side. The saddle-point equationfor Q α , viz., Eq. (5.1b), yields (cid:88) ˆ k ¯Ω ˆ k ¯Ω − ˆ k − p (cid:15) α (cid:16)(cid:0) p d p d + ( k α + p/ d ( k α + p/ d (cid:1) − D (cid:0) | p | + | k α + p / | (cid:1) δ d d (cid:17) = 0 . (5.7) This equation is automatically satisfied, which we can seeas follows. The product ¯Ω ˆ k ¯Ω − ˆ k − p (cid:15) α involves a productof two Kronecker deltas δ (cid:80) nα =0 k α , δ − p − (cid:80) nα =0 k α , , whichreflects the macroscopic translational invariance of therandom solid, and implies that p = . Next, we notethat the restricted wave-vector sum (cid:80) ˆ k can be replacedby the full wave-vector sum (cid:80) ˆ k , as the 1RS and 0RS con-tributions to the restricted sum vanish. Lastly, ¯Ω ˆ k ¯Ω − ˆ k isa scalar, implying that (cid:88) ˆ k ¯Ω ˆ k ¯Ω − ˆ k (cid:0) k αd k αd − | k α | δ d d (cid:1) = 0 . (5.8)For simplicity, instead of working with the full distribu-tion P ( τ ), we replace it by one that is sharply peaked at1 /ξ L . This approximation is valid as long as we are con-cerned with the broad implications for the nematogens ofthe very presence of localized network constituents. Weexpect that the spread of localization lengths would atmost result in the quantitative but not qualitative mod-ification of our results. If the localization is sharp, theAnsatz for the order parameter becomes¯Ω ˆ k ≈ G (cid:90) d z V e i (cid:80) α k α · z − | ˆ k | ξ L . (5.9) VI. EFFECTIVE THEORY OF STRUCTUREAND CORRELATIONS IN IGNES
In App. E, we derive an effective replica Hamiltonianfor the liquid crystalline behavior of IGNEs, using an ap-proximation in which Ω is set to its saddle-point value ¯Ωin the Landau-Wilson free energy (4.6), but allowing Q α to undergo fluctuations. By making this approximationwe are neglecting the impact of fluctuations in Ω on thenematic freedoms. We adopt this level of description be-cause it is the least complicated one that is capable of re-vealing the impact of the localized network on the liquidcrystallinity characteristic of IGNEs. As it remains con-stant, we do not need to consider the f Ω contribution tothe free energy. Furthermore, at the present level of ap-proximation, the term in f C proportional to ¯Ω ¯Ω Q α van-ishes, as we show in App. E, as a result of the macroscopictranslational and translational invariance of the randomsolid state. Moreover, as we show in the same appendix,at wavelengths long compared with ξ L the dominant con-tribution to f C comes from a term in ¯Ω ¯Ω Q α Q β . Thus,in forming an the effective replica Hamiltonian for the0high-temperature liquid crystallinity of IGNEs, which wedenote by H n [ { Q α ( · ) } nα =0 ], we need only consider the f Q term together with a term in ¯Ω ¯Ω Q α Q β term: H n [ { Q α } nα =0 ] ≈ (cid:88) p T ( A t + L p ) { Q p Q − p } + n (cid:88) α =1 (cid:88) p T ( A t + L p ) { Q α p Q α − p }− T n (cid:88) α =1 (cid:88) p H p { Q p Q α − p }− T n (cid:88) α,β =1( α (cid:54) = β ) (cid:88) p H p { Q α p Q β − p } . (6.1)Let us draw attention to the kernel H p , which is definedvia H p ≡ H e − p ξ L . (6.2)Here, H , which we call the disorder strength , char-acterizes the strength with which the network in-fluences the liquid crystallinity, and has the value G J ˜ η (cid:96) / (900 T ξ L ). The kernel H p is the manifesta-tion of the presence of the thermally fluctuating randomelastomeric network and, in particular, encodes the cen-tral physical characteristic of the network, viz., that atlong length-scales the localization appears perfect but atlength-scales shorter than ξ L thermal fluctuations renderthe network effectively “molten.” Note that the charac-teristic length-scale beyond which H ( r ) is suppressed is ξ L .To determine the effect that the preparation historyhas on the equilibrium liquid crystalline properties post-cross-linking, we integrate out the zeroth-replica element, Q , thus obtaining the effective Hamiltonian H eff [ { Q α } nα =1 ] ≡ n (cid:88) α =1 (cid:88) p (cid:18) A t + L p T − | H p | T ( A t + L p ) (cid:19) (cid:8) Q α p Q α − p (cid:9) − n (cid:88) α,β =1( α (cid:54) = β ) (cid:88) p (cid:18) H p T + | H p | T ( A t + L p ) (cid:19) (cid:8) Q α p Q β − p (cid:9) . (6.3)A noteworthy feature of H eff is that the replica-diagonalcontribution to the quadratic term in Q α is structurallydistinct from the replica off-diagonal contribution. Thisstructural feature, as well as the short length-scale liq-uidity encoded in H p , enables us to capture a richerrange of physical behavior (such as oscillatory-decayingnematic correlations) than can be predicted via conven-tional random-field approaches, for which the replica-diagonal and off-diagonal terms contain identical coef-ficients. VII. PHENOMENOLOGICAL CONTENT OFTHE MICROSCOPIC REPLICA THEORY
We now show that the effective replica Hamilto-nian (6.3) that we have derived on the basis of the mi-croscopic dimers-and-springs model can be interpretedas having arisen from a more phenomenological, contin-uum description of a liquid crystalline systems subjectto a novel pair of interrelated random fields. This phe-nomenological description and its implications were ex-plore in Ref. [22]. Our purpose here is not to revisitthese issues in detail but rather to reveal the microscopicunderpinnings of the phenomenological theory. In thatlatter theory, the free energy of an IGNE having a givenrealization χ (cid:48) of the quenched disorder is given by H χ (cid:48) [ Q ] = 12 (cid:88) p (cid:16)(cid:0) A t + L p + H p (cid:1)(cid:8) Q p Q − p (cid:9) − (cid:8)(cid:0) Y p + ( T /T ) H p Q p (cid:1) Q − p (cid:9)(cid:17) , (7.1)where χ (cid:48) = { Q , Y } , and Y and Q are independent,Gaussian-distributed random fields, with zero means andnon-zero variances, the latter being given by (cid:2)(cid:8) Q p Q p (cid:48) (cid:9)(cid:3) = 5 T δ p + p (cid:48) , A t p + L k , (7.2a) (cid:2)(cid:8) Y p Y p (cid:48) (cid:9)(cid:3) = T H p δ p + p (cid:48) , . (7.2b)Here, Q describes the impact of the configuration of thenematogens that is present at the instant of cross-linkingon the post-cross-linking nematic alignment pattern, and Y accounts for the impact of the local anisotropic en-vironment created by the localized network constituentspost-cross-linking on this nematic alignment pattern. Fora given realization of χ (cid:48) , the partition function is givenby Z χ (cid:48) ∝ (cid:90) D Q exp( − H χ (cid:48) [ Q ] /T ) , (7.3)and by suitably averaging over χ (cid:48) using the replica tech-nique we obtain (cid:2) Z nχ (cid:48) (cid:3) ∝ (cid:34)(cid:90) n (cid:89) α =1 D Q α exp (cid:32) − n (cid:88) α =1 H χ (cid:48) [ Q α ] /T (cid:33)(cid:35) = (cid:90) n (cid:89) α =1 D Q α exp( − H (cid:48) [ Q α ]) , (7.4)1where the effective replica Hamiltonian H phen [ Q α ] isgiven by H phen [ Q α ]= 12 T n (cid:88) α =1 (cid:88) p (cid:0) A t + L p + H p (cid:1) { Q α p Q α − p }− T n (cid:88) α,β =1 (cid:88) p [ { Y p Y − p } ] { Q α p Q β − p }− T ) n (cid:88) α,β =1 (cid:88) p | H p | [ { Q Q } ] { Q α p Q β − p } . (7.5)Applying the disorder statistics specified in Ref. (7.2),we arrive at the result that H phen [ Q α ] = H eff [ Q α ], i.e.,the phenomenological continuum description originallyreported in Ref. [22] contains the same physics as themicroscopic dimers-and-springs model. VIII. STRUCTURE AND CORRELATIONS INIGNESA. Diagnostic quantities
To describe the essential features of the pattern ne-matic ordering characteristic of IGNEs in the high-temperature regime we focus on the following pair ofthermally- and disorder-averaged correlators:(i) the average, taken over realizations of the cross-linking, of the product of the local nematic order at twopoints, i.e., C G ( r , r (cid:48) ) ≡ [ {(cid:104) q ( r ) (cid:105) (cid:104) q ( r (cid:48) ) }(cid:105) ] , (8.1a)which we term the “glassy correlator”; and(ii) the disorder average of the familiar correlator of thefluctuations in the nematic order, i.e., C T ( r , r (cid:48) ) ≡ (cid:2)(cid:10)(cid:8)(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:9)(cid:11)(cid:3) , (8.1b)which we term the “thermal correlator.”The correlator C T characterizes the strength of thethermal fluctuations of the nematic alignment awayfrom the local mean value as well as the spatial rangeover which these fluctuations are correlated. Inter alia,through its range, C T is capable of signaling the occur-rence of a continuous phase transition. The correlator C G is a diagnostic of particular value for nematic elas-tomers, as it detects the occurrence of randomly frozen(i.e., time-persistent) local nematic order. For the casewhere r and r (cid:48) are co-located, it is the nematic analogof the Edwards-Anderson order parameter, introducedlong ago for spin glasses [38], in the sense that it mea-sures the magnitude of local nematic ordering, regard-less of the orientation of that ordering. Moreover, how C G varies with the separation of r and r (cid:48) determines the a) b) FIG. 2: Schematic depictions of snapshots of nematogen lo-cations and orientations at a particular instant (blue, full),and at a much earlier instant (gray, shaded). (a) A conven-tional liquid crystal in the isotropic state. Such systems donot memorize the local pattern of nematogen alignment in-definitely. There is no correlation between the orientationsof blue and shaded nematogens that are depicted near oneanother. Nor is there any preference for blue and shaded ne-matogens that are depicted near one another to be the samenematogen. (b) A liquid cystalline elastomer in the macro-scopically isotropic state. Such systems do memorize the localpattern of nematogen alignment indefinitely. The orientationsof blue and gray nematogens depicted near one another arelikely to be correlated. For systems in which the nemato-gens are chemically bonded to an elastomer network, blueand shaded nematogens depicted near one another are likelyto be the same nematogen. However, for systems in which thenematogens are not chemically bonded to a network there isno preference for blue and shaded nematogens depicted nearone another to be the same nematogen. spatial extent of regions that share a roughly commonnematic alignment. Two mechanisms are responsible forthe existence of these aligned regions. First, the forma-tion of a random network causes a local breaking of ro-tational invariance, which has the effect of creating ran-domly anisotropic environments that tend to align thenematogens locally. Second, although the equilibriumstate of the system at the instant prior to cross-linkingis, on average, isotropic, a “snapshot” of its microscopicconfiguration at that instant would reveal local nematicorder of the type that we normally call thermal fluctu-ations. The cross-linking process can trap these fluctu-ations in, either partially or fully, the extent dependingon the strength of the cross-linking and the temperatureat the moment of cross-linking.The correlators C G and C T have been computed inRef. [22] via Eq. (7.1). In the present section, we re-derive the results via the replica approach to the micro-scopic dimers-and-springs model that we have developedin the foregoing sections. To proceed, we make use ofthe effective replica Hamiltonian (6.3) and the following2identities (which we prove in App. C):[ {(cid:104) q ( r ) (cid:105)(cid:104) q ( r (cid:48) ) (cid:105)} ] = lim n → (cid:104)(cid:104) Q α ( r ) Q β ( r (cid:48) ) (cid:105)(cid:105) ( α (cid:54) = β ) , (8.2a)[ {(cid:104) q ( r ) q ( r (cid:48) ) (cid:105)} ] = lim n → (cid:104)(cid:104) Q α ( r ) Q α ( r (cid:48) ) (cid:105)(cid:105) , (8.2b)where (cid:104)(cid:104) . . . (cid:105)(cid:105) denotes an average performed with respectto the effective Hamiltonian H eff , Eq. (6.3). To computethe quantities on the right hand side of Eqs. (8.2), weinvoke the quadratic form of Eq. (6.3), invert the cor-responding kernel, and use the replica diagonal and off-diagonal parts to obtain C T p = 5 T A t + L p + H p , (8.3a) C G p = 5 T TT ( A t + L p ) − | H p | + H p ( A t + L p + H p ) . (8.3b)Note the presence of the scale-dependent kernel H p in thedenominators of the correlators, which plays an essen-tial role in determining their behavior. This may appearsurprising when we compare with the result one wouldobtain via a conventional random-field approach (for de-tails, see App. F). At length-scales larger than ξ L , thepresence of H p in the denominator of C T p leads to a down-ward renormalization of the bare critical temperature T ∗ by an amount proportional to the disorder strength (andhence grows with the cross-linking density). This indi-cates that the nematogens are more strongly inhibitedfrom aligning with one another if the density of cross-links is higher. Interestingly enough, a similar result hasbeen obtained using the molecular level neo-classical elas-ticity theory of nematic elastomers [17]. This disorderingeffect of the random polymer network (on the nematicalignment) is not contained in the conventional random-field type of models [40]. In addition, a larger value of H leads to a larger amplitude of C G p , indicating thata stronger localization of the network results in a morestrongly trapped-in nematic pattern. Also note that thecorrelators C T p and C G p meet the physically sensible re-quirement that they revert to the forms appropriate toa nematic liquid in the absence of a network at length-scales very short compared with ξ L , for which H p be-comes very small, indicative of the molten character ofthe network at such scales. B. Oscillatory-decaying correlations
The scale-dependent kernel H p in the denominator ofeach of the correlators in Eq. (8.3) also gives rise tothe possibility that the correlators undergo both oscil-lation and decay with distance. As discussed in Ref. [22], C T ( r ) undergoes oscillatory decay with distance for H >H ( c ) (= 2 L /ξ L ), and the length-scale of the oscillation isgiven by ξ T,o = ξ L / (cid:112) H /H ( c ) ), which is indepen-dent of T . On the other hand, C G ( r ) undergoes oscil- FIG. 3: Real-space decay behavior of (a) the glassy cor-relator (rescaled) (cid:101) C G ( r ) ≡ (12 π L /T ) C G ( r ), for t (cid:29) T H /T A ; t = 0 . L / A ξ L , at (i) H /H ( c ) = 0 . H /H ( c ) = 40 (strong dis-order; blue, solid). (b) the thermal correlator (rescaled) (cid:101) C T ( r ) ≡ (2 π L / T ) C T ( r ), for the same parameters. On goingfrom weak to strong disorder, both correlators cross over fromsimple exponential decay to oscillatory decay of wavelengthof order ξ L . Thermal andglassy oscillationsThermal oscillationsNo oscillations t (cid:142) H (cid:142) FIG. 4: Crossover diagram for the glassy and thermal cor-relators, indicating the three qualitatively distinct regimes ofbehavior for a system that is cross-linked at a very high tem-perature. Here, ˜ H ≡ H /H ( c ) is a measure of the disorderstrength, and ˜ t is the rescaled reduced temperature, with thevalue ( A ξ L / L ) t . Above the blue solid line, both correlatorsoscillate and decay as a function of separation. Between theblue solid and red dashed lines, both correlators decay butonly the thermal one also oscillates. Below the red dashedline, both correlators decay but neither oscillates. latory decay for sufficiently large values of H and suf-ficiently low values of T (see Fig. 3), with its oscillationscale ξ G,o given implicitly by the following equation:1 + ( ξ N /ξ G,o ) + 4( ξ N /ξ L ) − ( H / A t ) e − ξ L / ξ G,o = 0 . The cross-over boundary between the oscillatory andnon-oscillatory regimes for C G , which is the threshold atwhich the inverse oscillation wavelength 1 /ξ G,o , increasesfrom zero to a non-zero value, is given by H = A t + 2 H ( c ) . (8.4)The different regimes of behavior of C T ( r ) and C G ( r ) areshown in Fig. 4.3 IX. ALTERNATIVE MICROSCOPIC MODELSOF IGNES
Instead of the dimers-and-springs model of IGNEs, onecould have started with alternative microscopic modelsthat may at first sight appear to be more faithful repre-sentation genuine IGNEs. For example, one could havestarted with either of the following two microscopic mod-els: (i) Model A: a system comprising worm-like chains—pairs of which are permanently, randomly bonded bypoint-like cross-links—as well as stiff rods that danglefromn each chain at regular arc-length intervals; and(ii) Model B: a system of chains, each constructed fromstiff rods that are connected in series and then perma-nently bonded by point-like cross-links between randomlychosen rod ends. Models A and B are, respectively, car-icatures of side- and main-chain nematic polymer net-works; starting from either Model A or Model B, we canapply the Hubbard-Stratonovich decoupling proceduredescribed in Secs. II to IV, and thus derive a correspond-ing Landau-Wilson free energy that is structurally equiv-alent (i.e., having the same symmetries and types of cou-plings) to that derived for the dimers-and-springs model;cf. Eq. (4.2). The purpose of this section is to establishthis structural equivalence. We can then, in principle,compute the coefficients of the terms of the expansion ofthe Landau-Wilson free energy separately for Models Aand B, apply a similar saddle-point analysis, and derivean effective replica Hamiltonian that would enable us toexplore nematic fluctuations. However, the coefficientsof the terms of such expansions are technically more dif-ficult to compute than those for the dimers-and-springsmodel. Moreover, even if one were to succeed in com-puting such coefficients, the expansion would still lead,at sufficiently large length-scales, to predictions that areidentical, up to an overall length-scale, to those madeon the basis of the dimers-and-springs model, Eq. (4.6).This is why we chose to work with the dimers-and-springsmodel in deriving an effective Hamiltonian for the liquidcrystallinity of IGNEs.In what follows, we consider Models A and B sepa-rately. We define the microscopic Hamiltonian, introducethe corresponding collective fields, perform the Hubbard-Stratonovich decoupling procedure and, lastly, carry outthe log-trace expansion. We then show that the auxiliaryfields and the terms in the Landau-Wilson free energycorresponding to either model have the same symmetriesand structure as those obtained from the dimers-and- springs model, and thus, give rise to effective Hamiltoni-ans that are structurally equivalent to H n ; cf. Eq. (6.1).To streamline the presentation we display only the essen-tial equations, as the formal procedure employed in thissection is identical to the one employed in Secs. II andIII. FIG. 5: Model of a side-chain nematic polymer consisting ofnematogens (red online) and worm-like chain backbone (blueonline). The chains are labeled by i (= 1 , . . . , N ), is of length L , and has S nematogens attached. The segment at arc-length τ of chain i has position c i ( τ ). The nematogens are attachedto the backbone at equally spaced arc-length intervals. A. Model A: side-chain nematic polymer network
We represent a side-chain nematic polymer as a worm-like chain (see, e.g., Ref. [39]) of length L with nemato-gens attached at equal intervals of arc-length L/S alongthe chain. Each nematogen is represented by a unitvector n i,s , where i = 1 , . . . , N labels the chains, and s = 1 , . . . , S labels the nematogens. We denote the arc-length measured from one end of the chain by τ (with0 < τ < L ). A segment at arc-length τ along chain i hasa position vector c i ( τ ); the s -th nematogen resides atarc-length sL/S , and its base, which is attached to chain i , has a position vector c i ( sL/S ); see Fig. 5. We requirethat nematogens and tangent vectors to the chains in-teract via orientational forces that favor parallel or anti-parallel alignment.The side-chain nematic polymer network is formed byrandomly, permanently cross-linking M pairs of suchchains via point-like cross-links. The pairs of cross-linkedchains and the number M are fixed for a given realizationof quenched disorder, but vary across such realizations ofit. In the absence of cross-links the system is specifiedby the following Hamiltonian:4 H SC = T N (cid:88) i =1 (cid:90) L dτL (cid:12)(cid:12)(cid:12)(cid:12) d c i ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) + λ N (cid:88) i,j =1 (cid:90) dτ dτ (cid:48) L δ ( c i ( τ ) − c j ( τ (cid:48) )) − V N N (cid:88) i,j =1 (cid:90) dτ dτ (cid:48) L J ( c i ( τ ) − c j ( τ (cid:48) )) (cid:18) d c i ( τ ) dτ · d c j ( τ (cid:48) ) dτ (cid:19) − V N N (cid:88) i,j =1 S (cid:88) s,t =1 J ( c i ( sL/S ) − c j ( tL/S )) ( n i,s · n j,t ) − VN N (cid:88) i,j =1 S (cid:88) s =1 (cid:90) dτL J ( c i ( τ ) − c j ( sL/S )) (cid:18) d c i ( τ ) dτ · n j,s (cid:19) . (9.1)The partition function corresponding to this system, sub-ject to M cross-linking constraints, is given by Z SC ∝ N (cid:89) i =1 (cid:90) D c i S (cid:89) s =1 (cid:90) d n i,s e − H SC /T × M (cid:89) e =1 δ (cid:0) c i e ( τ e ) − c j e ( τ (cid:48) e ) (cid:1) . (9.2) Here, (cid:82) D c ≡ (cid:81) ≤ τ ≤ L (cid:82) d c ( τ ), and (cid:82) d n denotes in-tegration over the unit sphere of nematogen orienta-tions. To proceed further, we can follow the strat-egy described in Secs. II and III, including making aHubbard-Stratonovich transformation to introduce theauxiliary fields { Ω ˆ k , Ω α p , Q α p } , to arrive at the correspond-ing Landau-Wilson free energy f SC : f SC = η ˜ V n (cid:88) ˆ k | Ω ˆ k | + N V n (cid:88) α =0 (cid:48) (cid:88) p λ α | Ω α p | + 12 n (cid:88) α =0 (cid:88) p J p T α Tr | Q α p | − ln (cid:68) exp (cid:16) η ˜ V n (cid:88) ˆ k Ω ˆ k (cid:90) L dτL e i ˆ k · ˆ c ( τ ) + iN ˜ V n (cid:88) α =0 (cid:48) (cid:88) p λ α Ω α p (cid:90) L dτL e i p · c α ( τ ) + n (cid:88) α =0 (cid:88) p J p T α Q αdd (cid:48) ( p ) (cid:16) (cid:90) dτL (cid:18) d c αd ( τ ) dτ d c αd (cid:48) ( τ ) dτ − δ dd (cid:48) (cid:19) e i p · c α ( τ ) + 1 S S (cid:88) s =1 ( n αs,d n αs,d (cid:48) − δ dd (cid:48) ) e i p · c α ( sL/S ) (cid:17)(cid:17)(cid:69) SC . (9.3)Here, λ α ≡ λ − η ( δ α, − / ( N ˜ V n − ), reflecting thefact that the excluded-volume parameter λ is not renor-malized by the effects of cross-linking in the preparation state. Note that the Boltzmann average (cid:104) . . . (cid:105) SC is definedvia (cid:104)· · · (cid:105) SC ≡ (cid:81) nα =0 (cid:82) D c α (cid:81) Ss =1 (cid:82) d n αs · · · exp (cid:18) − (cid:82) L dτL (cid:12)(cid:12)(cid:12) d c ( τ ) dτ (cid:12)(cid:12)(cid:12) (cid:19)(cid:81) nα =0 (cid:82) D c α (cid:81) Ss =1 (cid:82) d n αs exp (cid:18) − (cid:82) L dτL (cid:12)(cid:12)(cid:12) d c ( τ ) dτ (cid:12)(cid:12)(cid:12) (cid:19) . (9.4)We observe that f SC is invariant under transformations of the auxiliary fields that correspond to independent trans-5lations and rotations of the replicas, as in the case of theLandau-Wilson free energy for the dimers-and-springsmodel (4.2). In addition, Ω ˆ k is a HRS field, Ω α p is a1RS field, and Q α p is a traceless and symmetric second-rank tensor field, all as with the corresponding auxil-iary fields for the dimers-and-springs model. Further-more, terms developed by expanding the log-trace part inEq. (9.3) are structurally equivalent to those arising fromthe corresponding expansion of the Landau-Wilson freeenergy (4.2) of the dimers-and-springs model. In otherwords, as fields theories, Model A and the dimers-and-springs model are identical, up to elementary rescalings oftheir coefficients. We thus conclude that at length-scaleslarger than the microscopic scales at which they are de-fined, these models yield predictions that are structurallyidentical. B. Model B: main-chain nematic polymer network
We apply the procedure used in Sec. III and the pre-vious subsection to to derive the Landau-Wilson free en- ergy for Model B, a model for main-chain nematic poly-mer networks. This model consists of jointed chains (see,e.g., Ref. [39]) comprising S − (cid:96) . In addition, adjacent pairs of rods interact via abending energy that promotes their parallel alignment,and arbitrary pairs of rods also interact via orientationalforces that favor parallel or anti-parallel alignment. Thenetwork is formed via the random, instantaneous cross-linking of pairs of jointed chains via permanent point-likecross-links, located at the ends of the rods. The chainsare labeled by i (= 1 , . . . , N ) and the rod-rod junctionsare labeled by s (= 1 , . . . , S ); the position of rod end s on chain i is c i,s , as shown in Fig. 6. The system is thenspecified by the following Hamiltonian: H MC = − T (cid:96) N (cid:88) i =1 S − (cid:88) s =1 (cid:0) c i,s +1 − c i,s (cid:1) · (cid:0) c i,s − c i,s − (cid:1) + λ N (cid:88) i,j =1 S (cid:88) s,t =1 δ ( c i,s − c j,t ) − (cid:96) N (cid:88) i,j =1 S − (cid:88) s,t =1 J (cid:0) ( c i,s +1 + c i,s ) / − ( c j,t +1 + c j,t ) / (cid:1) (( c i,s +1 − c i,s ) · ( c j,t +1 − c j,t )) . (9.5)The partition function corresponding to this system, sub-ject to M cross-linking constraints, is given by Z MC ∝ N (cid:89) i =1 S (cid:89) s =1 (cid:90) d c i,s e − H MC /T δ ( | c i,s +1 − c i,s | − (cid:96) ) × M (cid:89) e =1 δ ( c i e ,s e − c i (cid:48) e ,s (cid:48) e ) . (9.6) Using the replica method and a Deam-Edwards type ofdistribution for the quenched randomness, together witha Hubbard-Stratonovich decoupling, we arrive at the fol-lowing Landau-Wilson free energy in terms of the auxil-iary fields { Ω ˆ k , Ω α p , Q α p } : f MC = S η ˜ V n (cid:88) ˆ k | Ω ˆ k | + N S V n (cid:88) α =0 (cid:48) (cid:88) p λ α | Ω α p | + S n (cid:88) α =0 (cid:88) p J p T α Tr | Q α p | − ln (cid:68) exp (cid:16) Sη ˜ V n (cid:88) ˆ k Ω ˆ k S (cid:88) s =1 e i ˆ k · ˆ c s + iN S ˜ V n (cid:88) α =0 (cid:48) (cid:88) p λ α Ω α p S (cid:88) s =1 e i p · c αs + n (cid:88) α =0 (cid:88) p J p T α Q αdd (cid:48) ( p ) S − (cid:88) s =1 (cid:0) (cid:96) ( c αs +1 ,d − c αs,d )( c αs +1 ,d (cid:48) − c αs,d (cid:48) ) − δ dd (cid:48) (cid:1) e i p · ( c αs + c αs +1 ) / (cid:17)(cid:69) MC , (9.7)6where λ α ≡ λ − η ( δ α, − / ( N S ˜ V n − ), and the Boltzmann average (cid:104) . . . (cid:105) MC is defined to be (cid:104)· · · (cid:105) MC ≡ (cid:81) nα =0 (cid:81) Ss =1 (cid:82) d c αs · · · exp (cid:16) (cid:96) (cid:80) nα =0 (cid:80) S − s =2 ( c αs +1 − c αs ) · ( c αs − c αs − ) (cid:17) (cid:81) nα =0 (cid:81) S − s =1 δ ( (cid:12)(cid:12) c αs +1 − c αs (cid:12)(cid:12) − (cid:96) ) (cid:81) nα =0 (cid:81) Ss =1 (cid:82) d c αs exp (cid:16) (cid:96) (cid:80) nα =0 (cid:80) S − s =2 ( c αs +1 − c αs ) · ( c αs − c αs − ) (cid:17) (cid:81) nα =0 (cid:81) S − s =1 δ ( (cid:12)(cid:12) c αs +1 − c αs (cid:12)(cid:12) − (cid:96) ) . (9.8) FIG. 6: Model of a main-chain nematic polymer consisting of S − (cid:96) . The chain is labeled i (= 1 , . . . , N )and the ends of the rods are labeled by s (where s = 1 , . . . , S ).Rod-end s on chain i is located at c i,s . The comments made at the end of the previous sub-section concerning symmetries of the Landau-Wilson freeenergy under transformations of the auxiliary fields holdfor Model B, too.
X. CONCLUDING REMARKS
The objective of this Paper has been to develop amicroscopic approach to the liquid crystalline proper-ties of isotropic-genesis nematic elastomers (IGNEs), inwhich local nematic order—both in the preparation andthe measurement ensembles—and random localization—induced by the presence of an elastomer network—arenaturally incorporated. This development, which takesas its starting point a system of dimers that are perma-nently connected at random by Hookean springs, servesas the underpinning to the phenomenological approachpresented in Ref. [22] by providing a systematic deriva-tion of the formulas on which the phenomenological de-scription is based.Specifically, by deriving an effective Hamiltonian of liq-uid crystallinity in IGNEs, we have shown that this mi-croscopic approach leads to the phenomenological Lan-dau theory of IGNEs proposed in Ref. [22] which, interalia , predicts that at sufficiently large disorder strengths,both the thermal and glassy spatial correlations of ne-matic alignments can undergo oscillation with decay.The development has as a core feature an ensemble— the preparation ensemble—that is distinct from the usual(measurement) ensemble of Gibbs statistical mechanics.This feature enables us to determine in detail the influ-ence of the preparation history of IGNEs on their subse-quent equilibrium behavior. The consequences of thesetwo ensembles were analyzed in detail in Ref. [22]. How-ever, the appearance of the two ensembles took the formof a hypothesis in Ref. [22], whereas in the present Pa-per they come into play naturally. Lastly, we have ar-gued that at sufficiently large length-scales, predictionsmade on the basis of a simple dimers-and-springs modelare qualitatively identical to those made on the basis oftwo more realistic (but more complicated) microscopicmodels of IGNEs: one for side-chain liquid crystallineelastomers and one for main-chain liquid crystalline elas-tomers.Apart from its relevance to the specific subject of liquidcrystalline elastomers, the present work brings to light amore general issue, viz., that the concept of a quenchedrandom field should be broadened to incorporate not onlythe conventional, “frozen” type, which does not fluctuatethermally, but also the type necessary for understand-ing media such as liquid crystalline elastomers, in whichthe frozen nature of the random field is present only atlonger length-scales, fading out as the length-scale pro-gresses through a characteristic localization length, owingto the thermal position fluctuations of the network’s con-stituents. The framework elucidated in the present workcan be extended, with suitable modifications, to explorethe statistical physics of other randomly cross-linked sys-tems, such as smectic elastomers and various biologicalmaterials.
Acknowledgments
We thank Tom Lubensky, Leo Radzihovsky KenjiUrayama, and Mark Warner for informative discussions.This work was supported by the U.S. National ScienceFoundation via grants DMR 09 06780 and DMR 1207026, the Institute for Complex Adaptive Matter,Shanghai Jiao Tong University, and the National ScienceFoundation of China via Grants 11174196 and 91130012.7
Appendix A: Calculation of the average linking number per dimer
In this series of Appendices we show various components of the calculations that are necessary for deriving theresults presented in the main body of the Paper. In the present Appendix we show that the average number of springsconnected to a dimer, [ M ] /N , is approximately given by 2 η . By definition,[ M ] = (cid:88) χ P ( χ ) M = (cid:88) χ (cid:90) D c P ( χ |{ c } ) P ( { c } ) M = 1 Z ∞ (cid:88) M =0 N (cid:88) i ,j =1 (cid:88) s ,t = − , . . . N (cid:88) i M ,j M =1 (cid:88) s M ,t M = − , MM ! (cid:16) η V N (2 πb ) D/ (cid:17) M × (cid:90) D c e H norm − H nem − H ev M (cid:89) e =1 e − b | c ie,se − c je,te | N (cid:89) i =1 δ ( | c i, − c i, − | − (cid:96) )= η V N (2 πb ) D/ Z (cid:90) D c (cid:16) N (cid:88) i,j =1 (cid:88) s,t = − , e − b | c i,s − c j,t | (cid:17) × e H norm − H nem − H ev ∞ (cid:88) M =1 M − (cid:16) η V N (2 πb ) D/ N (cid:88) i,j =1 (cid:88) s,t = − , e − b | c i,s − c j,t | (cid:17) M − N (cid:89) i =1 δ ( | c i, − c i, − | − (cid:96) )= η V N (2 πb ) D/ Z (cid:90) D c (cid:16) N (cid:88) i,j =1 (cid:88) s,t = − , e − b | c i,s − c j,t | (cid:17) × e H norm − H nem − H ev exp (cid:16) η V N (2 πb ) D/ N (cid:88) i,j =1 (cid:88) s,t = − , e − b | c i,s − c j,t | (cid:17) N (cid:89) i =1 δ ( | c i, − c i, − | − (cid:96) )= η V N (2 πb ) D/ Z (cid:90) D c N (cid:88) i,j =1 (cid:88) s,t = − , e − b | c i,s − c j,t | − H nem − H ev N (cid:89) i =1 δ ( | c i, − c i, − | − (cid:96) ) . (A1)To find a rough estimate for [ M ] /N , we shall approximate the system as a dilute gas of dimers, which involvesapproximating the quantity exp( −| c i − c j | / (2 b )) by (2 πb ) D/ /V . This leads to N (cid:88) i,j =1 (cid:88) s,t = − , (cid:90) D c e − b | c i,s − c j,t | − H nem − H ev N (cid:89) i =1 δ ( | c i, − c i, − | − (cid:96) ) ≈ N (2 πb ) D/ V Z . (A2)Substituting this into Eq. (A1), we obtain [ M ] /N ≈ η , (A3)which shows that η provides a quantitative measure of the average number of cross-linking springs per dimer.8 Appendix B: Hubbard-Stratonovich scheme
In this Appendix, we present the details of the Hubbard-Stratonovich scheme that leads to Eq. (4.2). We firstconsider the replica partition function Z n in Eq. (3.10), which can equivalently be expressed as follows: Z n = ∞ (cid:88) M =0 N (cid:88) i ,j =1 · · · N (cid:88) i M ,j M =1 (cid:88) s ,t =1 , − · · · (cid:88) s M ,t M =1 , − M ! (cid:18) V η N (2 πb ) D/ (cid:19) M × n (cid:89) α =0 (cid:90) N (cid:89) i =1 d c αi, d c αi, − e (cid:80) nα =0 (cid:80) Ni,j =1 JijTα (cid:0) ( n αi · n αj ) − D (cid:1) − b (cid:80) nα =0 (cid:80) Me =1 | c αie,se − c αje,te | × exp (cid:16) − V η N (2 πb ) D/ N (cid:88) i,j =1 (cid:88) s,t =1 , − e − b | c i,s − c j,t | − n (cid:88) α =0 N (cid:88) i,j =1 (cid:88) s,t = − , λT α δ ( c αi,s − c αj,t ) (cid:17) × n (cid:89) α =0 N (cid:89) j =1 δ ( | c αj, − c αj, − | − (cid:96) )= n (cid:89) α =0 (cid:90) N (cid:89) i =1 d c αi, d c αi, − e (cid:80) nα =0 (cid:80) Ni,j =1 J ij T α (cid:0) ( n αi · n αj ) − D (cid:1) − (cid:80) nα =0 (cid:80) Ni,j =1 (cid:80) s,t = − , λTα δ ( c αi,s − c αj,t ) × exp (cid:18) V η N (2 πb ) D/ N (cid:88) i,j =1 (cid:88) s,t =1 , − (cid:16) e − b (cid:80) nα =0 | c αi,s − c αj,t | − e − b | c i,s − c j,t | (cid:17)(cid:19) × n (cid:89) α =0 N (cid:89) j =1 δ ( | c αj, − c αj, − | − (cid:96) ) . (B1)Note that the two terms in the first exponential factor in the first equality are replicated versions of the Maier-Saupeinteraction H nem and Hookean spring potential H ev . Also note that the first term in the last exponential factor inthe first equality, − (cid:0) V η / N (2 πb ) D/ (cid:1) (cid:80) i,j (cid:80) s,t exp (cid:0) −| c i,s − c j,t | / b (cid:1) , is the term H norm ( { c i,s } ) /T in Eq. (3.5),whilst the second term in the same exponential factor is a replicated version of the excluded-volume interaction, H ev .In the last step we have summed over all realizations of the quenched disorder. This results in an exponentiation ofthe Hookean energy term, and leads to the following expression: Z n = n (cid:89) α =0 (cid:90) N (cid:89) i =1 d c αi, d c αi, − e V N (cid:80) nα =0 (cid:80) Ni,j =1 J ( c αi − c αj ) Tα (cid:0) n αi,d n αi,d − D δ d d (cid:1)(cid:0) n αj,d n αj,d − D δ d d (cid:1) × exp (cid:16) − n (cid:88) α =0 λ T α (cid:90) d r α N (cid:88) i =1 (cid:88) s = − , δ (cid:0) c αi,s − r α (cid:1) N (cid:88) j =1 (cid:88) t = − , δ (cid:0) c αj,t − r α (cid:1)(cid:17) × exp (cid:18) N V η (2 πb ) D/ (cid:16) (cid:90) d ˆ x d ˆ ye − | ˆ x − ˆ y | b N N (cid:88) i =1 (cid:88) s = − , δ (cid:0) ˆ x − ˆ c i,s (cid:1) N N (cid:88) j =1 (cid:88) t = − , δ (cid:0) ˆ y − ˆ c j,t (cid:1) − (cid:90) d x d y e − | x − y | b N N (cid:88) i =1 (cid:88) s = − , δ (cid:0) x − c i,s (cid:1) N N (cid:88) j =1 (cid:88) t = − , δ (cid:0) y − c j,t (cid:1)(cid:17)(cid:19) n (cid:89) α =0 N (cid:89) j =1 δ ( | c αj, − c αj, − | − (cid:96) ) . (B2)Here, we have replaced J i,j by its continuum limit J ( c αi − c αj ), viz., J i,j ≈ ( V /N ) J ( c αi − c αj ), where we approximate J ( c ) by a potential of Gaussian form, i.e., J ( c ) ≈ (cid:0) J / (2 πa ) d/ (cid:1) exp( − c / a ), suitable for describing short-range9interactions. Next, we define the following collective fields in Fourier space: q αd d ( p ) ≡ N N (cid:88) i =1 e i p · c αi ( n αi,d n αi,d − D − δ d d ) , (B3a) ω α p ≡ N N (cid:88) i =1 (cid:88) s = − , e − i p · c αi,s , (B3b) ω ˆ k ≡ N N (cid:88) i =1 (cid:88) s = − , e − i ˆ k · ˆ c i,s . (B3c)In Eq. (B3c), the argument of ω ˆ k can take any value in replicated Fourier space. However, in Eq. (B5) below, weshall decompose the field ω ˆ k into the part whose argument takes values from the HRS, the part whose argument takesvalues from the 1RS, and the part whose argument takes values from the 0RS. We shall then redefine ω ˆ k to be onlythose parts whose arguments belong to the HRS. Note that ω p ˆ (cid:15) α = ω α p and that the 0RS part of ω ˆ k is a constant. Tosimplify the notation, we define the following normalized averages: (cid:104)· · · (cid:105) N, n ≡ n (cid:89) α =0 N (cid:89) i =1 (cid:90) d c αi, d c αi, − πV (cid:96) δ ( | c αi, − c αi, − | − (cid:96) ) · · · ; (cid:104)· · · (cid:105) , n ≡ n (cid:89) α =0 (cid:90) d c α d c α − πV (cid:96) δ ( | c α − c α − | − (cid:96) ) · · · . (B4)We substitute the collective fields into Eq. (B2), taking care to separate out the 0RS, 1RS and HRS parts of the termsinvolving ω ˆ k . The replica partition function then becomes Z n = (4 πV (cid:96) ) N (cid:28) exp (cid:32) N n (cid:88) α =0 (cid:88) p J p T α | q αd d ( p ) | − N V n (cid:88) α =0 λT α | ω α | − N V n (cid:88) α =0 (cid:88) p (cid:48) λ α | ω α p | (cid:33) × exp (cid:18) N η V n | ω ˆ0 | + 2 N η V n n (cid:88) α =1 (cid:88) p (cid:48) ∆ p | ω p (cid:15) α | + 2 N η V n (cid:88) ˆ k ∆ ˆ k | ω ˆ k | (cid:19)(cid:29) N, n ∝ (cid:28) exp N n (cid:88) α =0 (cid:88) p J p T α | q αd d ( p ) | − N n (cid:88) α =0 (cid:88) p (cid:48) ˜ λ α p T α | ω α p | + N ˜ η V n (cid:88) ˆ k ∆ ˆ k | ω ˆ k | (cid:29) N, n , (B5)where (cid:80) p (cid:48) denotes a sum over all wave-vectors p excluding p = , we have made the following definitions: ∆ k ≡ exp( − b k ), ˜ η ≡ η , and ˜ λ α p /T α ≡ (4 N/ ˜ V ) λ α /T α − (4 η / ˜ V n )(1 − δ α, )∆ p , and ˜ λ α p is the effective excluded–volumeinteraction between dimers. For α (cid:54) = 0, the excluded–volume interaction is renormalized downwards; owing to thepresence of attractive interactions induced by cross-linking, whereas for α = 0, the excluded–volume interaction hasno such renormalization. In the case of replica α = 0, the downward renormalization due to cross-linking is canceledexactly by the correction from the term H norm ; physically, this is expected as the dimers cannot feel the cross-linkinduced attractive forces at the instant just prior to cross-linking.We now implement the Hubbard-Stratonovich transformation, which is based on the following set of equalities forcomplex variables q and ω : e − J | q | = Jπ (cid:90) d (Re ω ) d (Im ω ) e − J | ω | +2 iJ Re qω ∗ ; (B6a) e + J | q | = Jπ (cid:90) d (Re ω ) d (Im ω ) e − J | ω | +2 J Re qω ∗ . (B6b)Here, q may be thought of as a complex-variable analogue of a collective field and ω is the complex-variable analogue ofits conjugate auxiliary field. The Hubbard-Stratonovich procedure allows us to express the replica partition function Z n as a functional integral in terms of the auxiliary fields Ω [whose argument we restrict to the HRS via theconstraint Ω p (cid:15) α , in accordance with the replica-sector division laid out in Eq. (B5)], Ω α (which is formally Ω but with0its argument taking values in the 1RS), and Q α [conjugate to (resp.) ω , ω α , and q α ]. In terms of these auxiliaryfields, the replica partition function Z n is given by Z n ∝ (cid:90) D Ω n (cid:89) α =0 D Ω α D Q α exp (cid:0) − N f n [Ω , Ω α , Q α ] (cid:1) . (B7)Here, the Landau-Wilson free energy per dimer f n (scaled in units of T ) is given by f n (Ω , Ω α , Q α ) = ˜ η V n (cid:88) ˆ k ∆ ˆ k | Ω ˆ k | + 12 n (cid:88) α =0 (cid:88) p (cid:48) ˜ λ α p T α | Ω α p | + 12 n (cid:88) α =0 (cid:88) p J p T α | Q αd d ( p ) | − ln (cid:28) exp (cid:18) ˜ η V n (cid:88) ˆ k ∆ ˆ k Ω ˆ k (cid:88) s =1 , − e − i ˆ k · ˆ c s + i n (cid:88) α =0 (cid:88) p (cid:48) ˜ λ α p T α Ω α p (cid:88) s =1 , − e − i p · c αs + n (cid:88) α =0 (cid:88) p J p T α Q αd d ( p ) e − i p · c α ( n αd n αd − D − δ d d ) (cid:19)(cid:29) , n . (B8)By expanding f n for small values of the auxiliary fields, one obtains a Landau theory in terms of Ω, Ω α and Q α , which are, respectively, the order-parameter fields for the random solidification transition, the crystallizationtransition, and the isotropic-nematic transition. As we assume that IGNEs are incompressible, there will be nofluctuations in the density of dimers (which can be enforced by making ˜ λ to be extremely large), and thus there willbe no corresponding instability in the 1RS. We shall therefore disregard the contribution from Ω α . The free energy,Eq. (B8), then becomes f n (Ω , Q ) = ˜ η V n (cid:88) ˆ k ∆ ˆ k | Ω ˆ k | + 12 n (cid:88) α =0 (cid:88) p J p T α | Q αd d ( p ) | − ln (cid:104) exp (cid:0) G (Ω) + G ( Q ) (cid:1) (cid:105) , n . (B9)Note that we have introduced the abbreviations G (Ω) ≡ ˜ η V n (cid:88) ˆ k ∆ ˆ k Ω ˆ k (cid:88) s =1 , − e − i ˆ k · ˆ c s , (B10) G ( Q ) ≡ n (cid:88) α =0 (cid:88) p J p T α Q αd d ( p ) e − i p · c α (cid:18) n αd n αd − δ d d (cid:19) , (B11)and specialized to three spatial dimensions (i.e., D = 3).1 Appendix C: Proof that (cid:104) Ω ˆ k (cid:105) f = [ (cid:10) ω ˆ k (cid:11) χ ] and (cid:104) Q α p (cid:105) f = [ (cid:10) q α p (cid:11) χ ] We now prove Eq. (3.13), by deriving a general expression for [ (cid:104) q p (cid:105) . . . (cid:104) q p P (cid:105) ] ( P ∈ { , , , . . . } ) in terms of theconjugate fields Q p and the Landau-Wilson free energy f n [cf. Eq. (4.2)]:[ (cid:104) q p (cid:105) χ . . . (cid:104) q p P (cid:105) χ ] = (cid:20) Z χ (cid:90) N (cid:89) i =1 d c α i, d c α i, − q α p exp (cid:0) − (cid:0) H nem ( { n α i } ) + H ev ( { c α i,s } ) + H xlink ( { c α i,s } ) (cid:1) /T α (cid:1) × N (cid:89) i =1 δ ( | c α j, − c α j, − | − (cid:96) ) · · ·× Z χ (cid:90) N (cid:89) i =1 d c α P i, d c α P i, − q α P p P exp (cid:0) − (cid:0) H nem ( { n iα P } ) + H ev ( { c α P i,s } ) + H xlink ( { c α P i,s } ) (cid:1) /T α P (cid:1) × N (cid:89) j =1 δ ( | c α P j, − c α P j, − | − (cid:96) ) (cid:21) = lim n → (cid:88) χ P χ Z nχ n (cid:89) α =0 (cid:90) N (cid:89) i =1 d c αi, d c αi, − q α p . . . q α P p P × e − (cid:80) nγ =1 ( H nem ( { n γi } )+ H ev ( { c γi,s } )+ H xlink ( { c γi,s } ) ) /T γ n (cid:89) γ =0 N (cid:89) j =1 δ ( | c γj, − c γj, − | − (cid:96) ) , (C1)where in the second step, we have multiplied numerator and denominator by n − P copies of the factor Z χ = (cid:90) N (cid:89) i =1 d c i, d c i, − exp ( − ( H nem ( { n i } ) + H ev ( { c i,s } ) + H xlink ( { c i e ,s e } )) /T γ ) N (cid:89) i =1 δ ( | c i, − c i, − | − (cid:96) ) . (C2)This produces a factor of Z nχ in the denominator, which goes to unity once the replica limit is taken. We now performthe average over realizations of quenched disorder:[ (cid:104) q p (cid:105) χ . . . (cid:104) q p P (cid:105) χ ] = lim n → (cid:88) χ M ! Z (cid:18) V η πb ) D/ N (cid:19) M e H norm ( { c i,s } ) /T n (cid:89) γ =0 N (cid:89) i =1 δ ( | c γi − c γi | − (cid:96) ) × (cid:90) N (cid:89) i =1 d c α i, d c α i, − q α p . . . q α P p P exp (cid:32) − n (cid:88) γ =0 (cid:0) H nem ( { n γi } ) + H ev ( { c γi,s } ) + H xlink ( { c γi e ,s e } ) (cid:1) /T γ (cid:33) = lim n → Z n (cid:89) γ =0 (cid:90) N (cid:89) i =1 d c α i, d c α i, − q α p . . . q α P p P e H norm ( { c i,s } ) /T × exp n (cid:88) γ =0 T γ N (cid:88) i,j =1 J ij (cid:0) ( n γi · n γj ) − D (cid:1) + λ T γ N (cid:88) i,j =1 (cid:88) s,t =1 , − δ ( c γi,s − c γj,t ) × exp V η πb ) D/ N N (cid:88) i,j =1 (cid:88) s,t =1 , − e − b (cid:80) nγ =0 | c γi,s − c γj,t | N (cid:89) i =1 δ ( | c γi, − c γi, − | − (cid:96) )= lim n → B n (4 πV (cid:96) ) N Z (cid:42) q α p . . . q α P p P e H norm ( { c i,s } ) /T × exp n (cid:88) α =0 (cid:88) p (cid:48) (cid:32) N J p T α { q α p q α − p } − N ˜ λ p T α | ω α p | (cid:33) + N ˜ η (cid:88) ˆ k ∆ ˆ k | ω ˆ k | (cid:43) N, n , (C3)where B n is a constant. Here, { α , . . . , α P } ⊆ { , . . . , n } , as we are interested in nematic correlators in the measure-ment ensemble. In the second step, we have summed over M to obtain an exponential function. To proceed further,2we note that the factors of q α p can be generated by introducing, into the replica partition function Z n , a sourcefield H α p that is linearly coupled to q α p ; we denote the resulting replica partition function by the symbol Z n [ H ].By functionally differentiating Z n [ H ] with respect to H α p , one can recover the correlators of q α p . At the end of thecomputation, one has to take the source limit H →
0. Following this procedure, one obtains[ (cid:104) q p (cid:105) χ . . . (cid:104) q p P (cid:105) χ ] = lim n → lim H → ( − ) P B n (4 πV (cid:96) ) N Z e H norm ( { c i,s } ) /T (cid:28) δδ H α p . . . δδ H α P p P e N (cid:80) nα =0 (cid:80) p J p Tα { q α p q α − p } × e − (cid:80) nα =1 (cid:80) p H α p q α − p − N (cid:80) nα =0 (cid:80) p (cid:48) ˜ λ p Tα | (cid:101) Ω α p | + N ˜ η (cid:80) ˆ k ∆ ˆ k | (cid:101) Ω ˆ k | (cid:29) N, n = lim n → lim H → ( − ) P B n (4 πV (cid:96) ) N Z e H norm ( { c i,s } ) /T (cid:28) δδ H α p . . . δδ H α P p P e N (cid:80) p J p T { q α p q α − p } × e N (cid:80) nα =1 (cid:80) p J p Tα (cid:8)(cid:0) q α p − TαNJ p H α p (cid:1)(cid:0) q α − p − TαNJ − p H α − p (cid:1)(cid:9) × e − (cid:80) nα =1 (cid:80) p ( Tα )22 NJ p J − p { H α p H α − p }− N (cid:80) nα =0 (cid:80) p (cid:48) ˜ λ p Tα | ω α p | + N ˜ η (cid:80) nα =0 (cid:80) ˆ k ∆ ˆ k | ω ˆ k | (cid:29) N, n . (C4)In performing this calculation, we have completed the square in q α p , generating extra quadratic terms ( T α ) NJ α p J α − p { H α p H α − p } . The Hubbard-Stratonovich transformation can now be performed, with q α p − T α NJ p H α p , ω α p , and ω ˆ k being the auxiliary fields. Making use of the following relation (which one can prove) B n (cid:28) e N (cid:80) p J p T { q α p q α − p } + N (cid:80) nα =1 (cid:80) p J p Tα { q α p − TαNJ p H α p q α − p − TαNJ − p H α − p } × e − N (cid:80) nα =0 (cid:80) (cid:48) p ˜ λ p Tα | ω α p | + N ˜ η (cid:80) ˆ k ∆ ˆ k | ω ˆ k | (cid:29) N, n = (cid:90) D Ω n (cid:89) α =0 D Ω α D Q α e − Nf n (Ω ˆ k , Ω α , Q α ) − (cid:80) nα =1 (cid:80) p H α p Q α − p , (C5)it follows that[ (cid:104) q p (cid:105) χ · · · (cid:104) q p P (cid:105) χ ] = lim n → lim H → ( − ) P (4 πV (cid:96) ) N Z δδ H α p · · · δδ H α P p P × (cid:90) D Ω n (cid:89) α =0 D Ω α D Q α e − Nf n (Ω , Ω α , Q α ) − (cid:80) nα =1 (cid:80) p H α p Q α − p − (cid:80) nα =1 (cid:80) p ( Tα )22 NJ p J − p { H α p H α − p } = lim n → Z n Z (cid:82) D Ω (cid:81) nα =0 D Ω α D Q α (cid:0) Q α p · · · Q α P p P (cid:1) e − Nf n (Ω , Ω α , Q α ) (cid:82) D Ω (cid:81) nα =0 D Ω α D Q α e − Nf n (Ω , Ω α , Q α ) ≡ (cid:104) Q α p · · · Q α P p P (cid:105) f , (C6)as we were aiming to establish. On going from the second to the third line of Eq. (C6), we have multiplied numeratorand denominator by Z n , and we have also taken the limit n →
0, which implies that Z n → Z [as one can seefrom Eq. (3.10) by taking the limit n → P = 2, we recover Eq. (3.13).We can also derive Eq. (8.2) by using Eq. (C1) and taking the replica limit:[ (cid:104) q p (cid:105) · · · (cid:104) q p P (cid:105) ] = lim n → (cid:82) D Ω (cid:81) nα =0 D Ω α D Q α (cid:0) Q α p · · · Q α P p P (cid:1) e − Nf n (Ω , Ω α , Q α ) (cid:82) D Ω (cid:81) nα =0 D Ω α D Q α e − Nf n (Ω , Ω α , Q α ) ≈ lim n → (cid:82) (cid:81) nα =1 D Q α (cid:0) Q α p · · · Q α P p P (cid:1) e − NH eff ( Q α ) (cid:82) (cid:81) nα =1 D Q α e − NH eff ( Q α ) ≡ (cid:104)(cid:104) Q α p · · · Q α P p P (cid:105)(cid:105) , (C7)where in the second step, we have approximated the functional integrals by the values that are stationary with respectto Ω and exclude density fluctuations Ω α .3 Appendix D: Terms in the Landau-Wilson expansion
In this section we expand the log trace term in Eq. (B9) for small Ω and Q α . This gives us f n (Ω , Q ) = ˜ η V n (cid:88) ˆ k ∆ ˆ k | Ω ˆ k | − (cid:104) G (Ω) (cid:105) , n − (cid:104) G (Ω) (cid:105) , n + 12 n (cid:88) α =0 (cid:88) p J p T α | Q αab ( p ) | − (cid:104) G ( Q ) (cid:105) , n − (cid:104) G ( Q ) (cid:105) , n − (cid:104) G ( Q ) (cid:105) , n − (cid:104) G (Ω) G ( Q ) (cid:105) , n − (cid:104) G (Ω) G ( Q ) (cid:105) , n − (cid:0) (cid:104) G (Ω) G ( Q ) (cid:105) , n − (cid:104) G (Ω) (cid:105) , n (cid:104) G ( Q ) (cid:105) , n (cid:1) . (D1)
1. Terms proportional to ΩΩ First, we compute the quadratic term for the vulcanization part of the Landau theory: (cid:104) G (Ω) (cid:105) , n = ˜ η (cid:88) ˆ k , ˆ k ∆ ˆ k ∆ ˆ k Ω ˆ k Ω ˆ k (cid:88) s,t (cid:68) e − i (cid:0) ˆ k · ˆ c s +ˆ k · ˆ c t (cid:1)(cid:69) , n = ˜ η (cid:88) ˆ k ∆ k Ω ˆ k Ω − ˆ k (cid:18) n (cid:89) α =0 sin( k α (cid:96) ) k α (cid:96) (cid:19) . (D2)At length-scales much larger than (cid:96) or b , we can approximate the above result by (cid:104) G (Ω) (cid:105) , n ≈ ˜ η (cid:88) ˆ k (cid:16) − (cid:0) b + (cid:96) (cid:1) | ˆ k | (cid:17) | Ω ˆ k | . (D3)
2. Terms proportional to
ΩΩΩ
We next compute the cubic term for the vulcanization part of the Landau theory: (cid:104) G (Ω) (cid:105) , n = ˜ η (cid:88) ˆ k , ˆ k , ˆ k ∆ ˆ k ∆ ˆ k ∆ ˆ k Ω ˆ k Ω ˆ k Ω ˆ k × (cid:88) s ,s ,s (cid:68) e − i (cid:0) ˆ k · ˆ c s +ˆ k · ˆ c s +ˆ k · ˆ c s (cid:1)(cid:69) , n = ˜ η (cid:88) ˆ k , ˆ k , ˆ k δ ˆ k +ˆ k +ˆ k , ˆ0 ∆ ˆ k ∆ ˆ k ∆ ˆ k Ω ˆ k Ω ˆ k Ω ˆ k × (cid:18) n (cid:89) α =0 sin( | − k α + k α + k α | (cid:96)/ | − k α + k α + k α | (cid:96)/ n (cid:89) α =0 sin( | − k α − k α + k α | (cid:96)/ | − k α − k α + k α | (cid:96)/ (cid:19) . (D4)At length-scales much larger than (cid:96) and b , we can approximate this result by (cid:104) G (Ω) (cid:105) , n ≈ ˜ η (cid:88) ˆ k , ˆ k , ˆ k δ ˆ k +ˆ k +ˆ k , ˆ0 Ω ˆ k Ω ˆ k Ω ˆ k . (D5)
3. Terms proportional to Q α Q α Next, we compute the quadratic terms from the nematic part of the Landau theory: (cid:104) G ( Q ) (cid:105) , n = (cid:88) α,β (cid:88) p , q (cid:88) s,t =1 , − J p J q ( T α ) Q αd d ( p ) Q βd d ( q ) × (cid:68) e − i ( p · c α + q · c β ) (cid:0) n αd n αd − δ d d (cid:1)(cid:0) n βd n βd − δ d d (cid:1)(cid:69) , n . (D6)4Note that this term vanishes for α (cid:54) = β . For α = β , we find that (cid:104) G ( Q ) (cid:105) , n = 14 π (cid:88) α (cid:88) p | J p | ( T α ) Q αd d ( p ) Q αd d ( − p ) × (cid:90) d n α (cid:0) n αd n αd n αd n αd − n αd n αd δ d d − n αd n αd δ d d + 19 δ d d δ d d (cid:1) = 115 (cid:88) α (cid:88) p | J α ( p ) | ( T α ) Q αd d ( p ) Q αd d ( − p ) (cid:0) δ d d δ d d + δ d d δ d d − δ d d δ d d (cid:1) . (D7)Here, the notation (cid:82) d n denotes integration over the unit sphere, and we have used the equalities (valid for D = 3): (cid:82) d n n d n d = π δ d d and (cid:82) d n n d n d n d n d = π ( δ d d δ d d + δ d d δ d d + δ d d δ d d ).
4. Coupling terms a. Terms proportional to Ω Q α Q β and ΩΩ Q α The terms that couple the nematic order parameter to the vulcanization order parameter are the ones that give riseto the physics of nematic elastomers. At cubic order in Ω and Q (i.e., Ω Q α Q β or ΩΩ Q α ) there are two such terms.They were computed in Ref. [23] and are given by (cid:104) G (Ω) G ( Q ) (cid:105) , n = ˜ η n (cid:88) α (cid:54) = β (cid:88) p , q ∆ − p ˆ (cid:15) α − q ˆ (cid:15) β J p J q T α T β Ω − p ˆ (cid:15) α − q ˆ (cid:15) β Q αd d ( p ) Q βd d ( q ) p d p d q d q d , (D8a) (cid:104) G (Ω) G ( Q ) (cid:105) , n = ˜ η (cid:96) n (cid:88) α =0 (cid:88) p (cid:88) ˆ k ∆ ˆ k ∆ − ˆ k − p ˆ (cid:15) α J p T α Ω ˆ k Ω − ˆ k − p ˆ (cid:15) α Q αd d ( p ) × (cid:0) p d p d + ( k α + 12 p ) d ( k α + 12 p ) d (cid:1) . (D8b) b. Terms proportional to ΩΩ Q α Q β Terms proportional to ΩΩ Q α Q β arise in two forms: connected and disconnected. We first consider the disconnectedtype, which is given by (cid:104) G (Ω) (cid:105)(cid:104) G ( Q ) (cid:105) . By using the expressions obtained for (cid:104) G (Ω) (cid:105) , n in Eq. (D2) and (cid:104) G ( Q ) (cid:105) , n in Eq. (D7) we obtain (cid:104) G (Ω) (cid:105) , n (cid:104) G ( Q ) (cid:105) , n = 8˜ η n (cid:88) α =0 (cid:88) ˆ k (cid:88) p ∆ k (cid:18) J p T α (cid:19) Ω ˆ k Ω − ˆ k Q αd d ( p ) Q αd d ( − p ) . (D9)Next, we consider the connected type, (cid:104) G (Ω) G ( Q ) (cid:105) , n : (cid:104) G (Ω) G ( Q ) (cid:105) , n = ˜ η n (cid:88) α,β =0 (cid:88) ˆ k , ˆ k (cid:88) p , q (cid:88) s,t =1 , − ∆ ˆ k ∆ ˆ k Ω ˆ k Ω ˆ k J p J q T α T β Q αd d ( p ) Q βd d ( q ) × (cid:68) e − i (cid:80) nγ =0 (cid:0) k γ · c γs + k γ · c γt (cid:1) − i p · c α − i q · c β × (cid:0) n αd n αd − δ d d (cid:1)(cid:0) n βd n βd − δ d d (cid:1)(cid:69) , n . (D10)Note that in this expression we have to evaluate terms such as (cid:104) e i p · c s (cid:105) , n , (cid:104) n d n d e i p · c s (cid:105) , n and (cid:104) n d n d n d n d e i p · c s (cid:105) , n . We recall the definitions c s ≡ c + ( s(cid:96)/ n , (D11a)5 (cid:104)· · · (cid:105) , n ≡ n (cid:89) α =0 (cid:90) d c α d c α − πV (cid:96) δ ( | c α − c α − | − (cid:96) )= (cid:90) n (cid:89) α =0 (cid:90) d c α d u α πV (cid:96) δ ( | u | − (cid:96) )= (cid:90) n (cid:89) α =0 (cid:90) d c α d n α du α | u α | πV (cid:96) δ ( | u | − (cid:96) )= (cid:90) n (cid:89) α =0 (cid:90) d c α d n α πV . (D11b)Here, we have decomposed coordinates into center of mass c = ( c + c − ) / u = c − c − , and havedefined the integral over the unit sphere as (cid:90) d n π ≡ (cid:90) − d cos θ, (D12)where θ is an angle that we can define with respect to the direction of the wave-vector. The following formulas areessential for the next step in our calculation: (cid:90) d n π e i p · n = sin | p || p | , (D13a) (cid:90) d n π e i p · n (cid:0) n d n d − δ d d (cid:1) = − (cid:90) d n π (cid:18) ∂∂p d ∂∂p d + 13 δ d d (cid:19) e i p · n = − (cid:18) ∂∂p d ∂∂p d + 13 δ d d (cid:19) (cid:90) − d (cos θ ) e i | p | cos θ = − (cid:18) ∂∂p d ∂∂p d + 13 δ d d (cid:19) sin | p || p | = (cid:18)(cid:18) | p | − | p | (cid:19) sin | p | + 3 | p | cos | p | (cid:19) (cid:18) p d p d − | p | δ d d (cid:19) ≈ − (cid:0) p d | p | p d | p | − δ d d (cid:1) , (D13b) (cid:90) d n π n d n d n d n d e i p · n = (cid:16) − sin | p || p | − | p || p | + 3 sin | p || p | (cid:17)(cid:0) δ d d δ d d + δ d d δ d d + δ d d δ d d (cid:1) + (cid:16) − cos | p || p | + 6 sin | p || p | + 15 cos | p || p | −
15 sin | p || p | (cid:17)(cid:0) δ d d p d p d + δ d d p d p d + δ d d p d p d + δ d d p d p d + δ d d p d p d + δ d d p d p d (cid:1) + (cid:16) sin | p || p | + 10 cos | p || p | −
45 sin | p || p | −
105 cos | p || p | + 105 sin | p || p | (cid:17) p d p d p d p d ≈ (cid:16) − | p |
210 + | p | (cid:17)(cid:0) δ d d δ d d + δ d d δ d d + δ d d δ d d (cid:1) + (cid:16) − | p | − | p | (cid:17)(cid:0) δ d d p d p d + δ d d p d p d + δ d d p d p d + δ d d p d p d + δ d d p d p d + δ d d p d p d (cid:1) + (cid:16) − | p | | p | (cid:17) p d p d p d p d . (D13c)6Using Eqs. (D11) and (D13), we can now compute (cid:104) G (Ω) G ( Q ) (cid:105) , n : (cid:104) G (Ω) G ( Q ) (cid:105) , n ≈ η n (cid:88) α =0 (cid:88) p , q (cid:88) ˆ k J p J q ( T α ) ∆ ˆ k ∆ − ˆ k − ( p + q )ˆ (cid:15) α Ω ˆ k Ω − ˆ k − ( p + q )ˆ (cid:15) α Q αd d ( p ) Q αd d ( q ) × (cid:16)(cid:0) − (cid:96) | p + q |
56 + (cid:96) | p + q | − (cid:96) | p + q + 4 k α |
56 + (cid:96) | p + q + 4 k α | (cid:1) δ d d δ d d + (cid:0) −
114 + (cid:96) | p + q | (cid:1) (cid:96) ( p + q ) d ( p + q ) d δ d d + (cid:0) −
114 + (cid:96) | p + q + 4 k α | (cid:1) (cid:96) ( p + q + 4 k α ) d ( p + q + 4 k α ) d δ d d + (cid:96) p + q ) d ( p + q ) d ( p + q ) d ( p + q ) d + (cid:96) p + q + 4 k α ) d ( p + q + 4 k α ) d ( p + q + 4 k α ) d ( p + q + 4 k α ) d (cid:17) + 2˜ η (cid:96) n (cid:88) α,β =0( α (cid:54) = β ) (cid:88) p , q (cid:88) ˆ k J p J q T α T β ∆ ˆ k ∆ − ˆ k − p ˆ (cid:15) α − q ˆ (cid:15) β Ω ˆ k Ω − ˆ k − p ˆ (cid:15) α − q ˆ (cid:15) β Q αd d ( p ) Q βd d ( q ) × (cid:16) k αd k αd k βd k βd + 14 ( p d k αd + k αd p d )( q d k βd + k βd q d ) + 18 p d p d q d q d (cid:17) , (D14)in which the Q α matrices are constrained to be symmetric and traceless. Appendix E: Deriving the effective Hamiltonian for liquid crystallinity
In this Appendix we derive the effective Hamiltonian for liquid crystallinity, Eq. (6.1). The first step in the derivationis to set Ω to its saddle-point value ¯Ω in f n of Eq. (D1) whilst retaining fluctuations of Q , and evaluate the terms (cid:104) G (Ω) G ( Q ) (cid:105) , n , (cid:104) G (Ω) G ( Q ) (cid:105) , n , (cid:104) G (Ω) G ( Q ) (cid:105) , n and (cid:104) G (Ω) (cid:105) , n (cid:104) G ( Q ) (cid:105) , n with Ω = ¯Ω. Aswe are considering length-scales large compared with b or (cid:96) , we shall make the approximations ∆ ˆ k ≈ J p ≈ J .(The wave-vector dependent parts of ∆ ˆ k and J p would generate corrections of higher order in the wave-vector.) Byfurther arguing that fluctuation modes with p > ξ − L can be neglected in f C ( ¯Ω , Q ) (see App. E 5), we shall arrive atthe effective Hamiltonian (6.1). (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n This term is given by (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n ≈ ˜ η (cid:96) n (cid:88) α =0 (cid:88) p (cid:88) ˆ k J T α ¯Ω ˆ k ¯Ω − ˆ k − p ˆ (cid:15) α Q αd d ( p ) (cid:0) p d p d + ( k α + 12 p ) d ( k α + 12 p ) d (cid:1) . (E1)The product ¯Ω ˆ k ¯Ω − ˆ k − p ˆ (cid:15) α involves a product of two Kronecker deltas, δ (cid:80) nα =0 k α , δ − p − (cid:80) nα =0 k α , , that enforces macro-scopic translational invariance, and implies that p = . Equation (E1) then becomes (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n ≈ n (cid:88) α =0 (cid:88) ˆ k ˜ η (cid:96) J T α ¯Ω ˆ k ¯Ω − ˆ k k αd k αd . (E2)By decomposing the full wave-vector sum (cid:80) ˆ k into three separate parts corresponding to contributions from the HRS,1RS and 0RS, the right hand side of Eq. (E2) becomes proportional to n (cid:88) α =0 (cid:88) ˆ k − (cid:88) ˆ k ∈ RS − (cid:88) ˆ k ∈ RS ¯Ω ˆ k ¯Ω − ˆ k k αd k αd . (E3)7Here, the 0RS part vanishes, and the 1RS part also vanishes, because ¯Ω ˆ k is proportional to δ (cid:80) nγ =0 k γ , , which is zeroif ˆ k belongs to the 1RS. We are thus left with the full wave-vector sum; it is proportional to (cid:88) ˆ k (cid:90) d z d z V k αd k αd e − ξ L | ˆ k | + i (cid:80) nγ =0 k γ · ( z − z ) = (cid:88) ˆ k (cid:90) d c d u V k αd k αd e − ξ L (cid:80) γ | k γ − i u ξ L | − ( n +1) | u | ξ L ∝ δ d d ξ L (cid:18) πξ L (cid:19) ( n +1) D/ (cid:18) πξ L n + 1 (cid:19) D/ (cid:0) − n + 1 (cid:1) = 0 , (E4)where in the last step we have taken the replica limit. On going from the first to the second step, we have changed tocenter-of-mass coordinate c and relative coordinate u , and on going from the second to the third step we haveshifted the wave-vector k γ → k γ − i u / (2 ξ L ) and integrated over k γ ( γ = 0 , , . . . , n ) and r . Thus, the term (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n vanishes. (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n To lowest order in wave-vector, this term is given by (cid:104) G (Ω) G ( Q ) (cid:105) , n ≈ ˜ η (cid:96) (cid:88) α (cid:54) = β (cid:88) p , q J T α T β p d p d q d q d ¯Ω − p (cid:15) α − q (cid:15) β Q αd d ( p ) Q βd d ( q )= ˜ η (cid:96) (cid:88) α (cid:54) = β (cid:88) p , q (cid:90) d z V J GT α T β p d p d q d q d e − i ( p + q ) · z − ξ L ( p + q ) Q αd d ( p ) Q βd d ( q )= G ˜ η (cid:96) (cid:88) α (cid:54) = β (cid:88) p J T α T β p d p d p d p d e − p ξ L Q αd d ( p ) Q βd d ( − p ) (E5) (cid:104) G ( ¯Ω) (cid:105) , n (cid:104) G ( Q ) (cid:105) , n This term vanishes. To show this, it is sufficient to consider the value of (cid:80) ˆ k ¯Ω ˆ k ¯Ω − ˆ k . Using the formula (5.9) for ¯Ω ˆ k and the decomposition (cid:80) ˆ k ≡ (cid:80) ˆ k − (cid:80) ˆ k ∈ RS − (cid:80) ˆ k ∈ RS , we have that (cid:88) ˆ k ¯Ω ˆ k ¯Ω − ˆ k = (cid:88) ˆ k − (cid:88) ˆ k ∈ RS − (cid:88) ˆ k ∈ RS G (cid:90) d z d z V e − ξ L | ˆ k | − i (cid:80) nγ =0 k γ · ( z − z ) = G (cid:32) − V n (cid:90) n (cid:89) γ =0 d k γ (2 π ) D (cid:90) d c d r V e − (cid:80) nγ =0 (cid:16) ξ L | k γ + i r ξ L | (cid:17) − (1+ n ) r ξ L (cid:33) = G (cid:18) − V n · VV π ) (1+ n ) D (cid:16) πξ L n (cid:17) D/ (cid:16) πξ L (cid:17) (1+ n ) D/ (cid:19) = n ln (cid:18) V (2 π ) D ξ DL (cid:19) G + O ( n ) . (E6)After taking the replica limit n →
0, the above result vanishes. Here, we have used the definition (cid:80) ˆ k ≡ V n (2 π ) (1+ n ) D (cid:82) d ˆ k when performing the integral over ˆ k . In the last step, we have integrated over ˆ k , r , and c . The term (cid:104) G ( ¯Ω) (cid:105) , n (cid:104) G ( Q ) (cid:105) , n is therefore zero in the replica limit. (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n We now consider the contribution (cid:104) G ( ¯Ω) G ( Q ) (cid:105) . Despite its being of higher order in ¯Ω, this term has long wave-length properties that are sufficiently hard, and it is therefore important for the purpose of studying the polydomain8state of IGNEs. It contains a part that is diagonal in replica space (i.e., proportional to Q α Q α ) and a part that isoff-diagonal in replica space (i.e., proportional to Q α Q β for α (cid:54) = β ).First, let us consider the replica-diagonal part. From Eq. (D14), with Ω set to its saddle-point value ¯Ω, andby making use of the fact that ¯Ω ˆ k ¯Ω − ˆ k − ( p + q ) (cid:15) α ∝ δ (cid:80) γ k γ , δ − (cid:80) γ k γ − p − q , , which implies that p + q = , we seethat replica-diagonal terms with prefactors in powers of | p + q | vanish. Thus the replica-diagonal contribution to (cid:104) G (Ω) G ( Q ) (cid:105) (with Ω set to ¯Ω) becomes proportional to the quantity, n (cid:88) α =0 (cid:88) p (cid:88) ˆ k ¯Ω ˆ k ¯Ω − ˆ k Q αd d ( p ) Q αd d ( − p ) (cid:16)(cid:0) − (cid:96) | k α | + 263 (cid:96) | k α | (cid:1) δ d d δ d d +16 (cid:0) −
114 + (cid:96) | k α | (cid:1) (cid:96) k αd k αd δ d d + 863 (cid:96) k αd k αd k αd k αd (cid:17) . (E7)The term proportional to δ d d δ d d (without the wave-vector prefactors) vanishes, as we showed in Eq. (E6). On theother hand, the remaining terms should also vanish, as ¯Ω ˆ k ¯Ω − ˆ k is isotropic in wave-vector space. The vanishing ofthe term in (cid:80) nα =0 (cid:80) ˆ k k αd k αd ¯Ω ˆ k ¯Ω − ˆ k comes about by the same argument as was presented in App. E 1. By means ofa calculation similar to that performed in App. E 1, one can also show that the following result holds in the replicalimit: (cid:88) ˆ k k αd k αd k αd k αd ¯Ω ˆ k ¯Ω − ˆ k ≡ (cid:88) ˆ k − (cid:88) ˆ k ∈ RS − (cid:88) ˆ k ∈ RS k αd k αd k αd k αd ¯Ω ˆ k ¯Ω − ˆ k = 0 . (E8)Next, we compute the replica off-diagonal contribution to (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , whose terms we divide into the followingthree classes: (i) a term with prefactor proportional to kkkk , (ii) terms with prefactors proportional to kk but not kkkk , and (iii) terms with prefactors that do not depend on k [see Eq. (D14)].First, we compute the replica-off-diagonal term proportional to kkkk : n (cid:88) α (cid:54) = β (cid:88) p , q (cid:88) ˆ k ¯Ω ˆ k ¯Ω − ˆ k − p ˆ (cid:15) α − q ˆ (cid:15) β Q αd d ( p ) Q βd d ( q ) k αd k αd k βd k βd = G V n +3 (cid:90) n (cid:89) γ =0 d k γ (2 π ) d p (2 π ) d q (2 π ) (cid:90) d z V d z V n (cid:88) α,β =0( α (cid:54) = β ) Q αd d ( p ) Q βd d ( q ) k αd k αd k βd k βd × exp (cid:18) − (cid:88) γ (cid:54) = α,β ξ L (cid:12)(cid:12)(cid:12)(cid:12) k γ + i z − z ξ L (cid:12)(cid:12)(cid:12)(cid:12) − ( n − | z − z | ξ L − ξ L ( p + q ) − i ( p + q ) · z − ξ L (cid:12)(cid:12)(cid:12)(cid:12) k α + i z − z ξ L + 12 p (cid:12)(cid:12)(cid:12)(cid:12) + ξ L (cid:12)(cid:12)(cid:12)(cid:12) i z − z ξ L + 12 p (cid:12)(cid:12)(cid:12)(cid:12) − ξ L (cid:12)(cid:12)(cid:12)(cid:12) k β + i z − z ξ L + 12 q (cid:12)(cid:12)(cid:12)(cid:12) + ξ L (cid:12)(cid:12)(cid:12)(cid:12) i z − z ξ L + 12 q (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . (E9)We replace the coordinates z and z by the relative coordinate R and center-of-mass coordinate C , defined respec-tively by R = z − z and C = ( z + z ) /
2, and integrate over C , which results in a factor of (2 π ) d V δ p + q , . Next,we integrate over q to enforce the equality q = − p . We then integrate out k γ (for γ (cid:54) = α, β ). The right-hand side ofEq. (E9) then becomes G V n +2 (2 π ) ( n − D (cid:18) πξ L (cid:19) ( n − D/ n (cid:88) α,β =0( α (cid:54) = β ) (cid:90) d k α (2 π ) D d k β (2 π ) D d p (2 π ) D (cid:90) d R V Q αd d ( p ) Q βd d ( − p ) × exp (cid:16) − p ξ L − ( n + 1) | R | ξ L − (cid:0) | k α | + | k β | (cid:1) ξ L (cid:17) × (cid:0) k α − i R ξ L − p (cid:1) d (cid:0) k α − i R ξ L − p (cid:1) d (cid:0) k β − i R ξ L + p (cid:1) d (cid:0) k β − i R ξ L + p (cid:1) d . (E10)Because the exponent is invariant under rotations of R , k α , k β and p , any terms having prefactors proportional to R d R d , k αd k αd , k βd k βd or p d p d will, respectively, become proportional to δ d d on integrating over R , k α , k β and9 p . However, on contracting with Q d d , these terms will yield no contributions, owing to the tracelessness of Q d d .Thus, the term (E10) effectively becomes G V n +2 (2 π ) ( n − D (cid:18) πξ L (cid:19) ( n − D/ n (cid:88) α,β =0( α (cid:54) = β ) (cid:90) d k α (2 π ) D d k β (2 π ) D d p (2 π ) D (cid:90) d R V Q αd d ( p ) Q βd d ( − p ) × exp (cid:16) − p ξ L − ( n + 1) | R | ξ L − (cid:0) | k α | + | k β | (cid:1) ξ L (cid:17) × (cid:16) ξ L (cid:0) R d R d p d p d + R d R d p d p d + R d R d p d p d + R d R d p d p d (cid:1) + 116 ξ L R d R d R d R d + 116 p d p d p d p d (cid:17) (E11)Then, by integrating over R , k α and k β , we obtain the following result: n (cid:88) α,β =0 α (cid:54) = β (cid:88) p , q (cid:88) ˆ k ¯Ω ˆ k ¯Ω − ˆ k − p ˆ (cid:15) α − q ˆ (cid:15) β Q αd d ( p ) Q βd d ( q ) k αd k αd k βd k βd = G ξ L n (cid:88) α,β =0 α (cid:54) = β (cid:88) p (cid:0) δ d d δ d d + δ d d p d p d ξ L + 18 p d p d p d p d ξ L (cid:1) e − p ξ L Q αd d ( p ) Q βd d ( − p ) . (E12)By means of similar calculations, we obtain the following results: n (cid:88) α,β =0 α (cid:54) = β (cid:88) p , q (cid:88) ˆ k ¯Ω ˆ k ¯Ω − ˆ k − p ˆ (cid:15) α − q ˆ (cid:15) β Q αd d ( p ) Q βd d ( q ) (cid:0) p d k αd + k αd p d (cid:1)(cid:0) q d k βd + k βd q d (cid:1) = G ξ L n (cid:88) α,β =0 α (cid:54) = β (cid:88) p (cid:0) δ d d p d p d ξ L + p d p d p d p d ξ L (cid:1) e − p ξ L Q αd d ( p ) Q βd d ( − p ) , (E13) n (cid:88) α,β =0 α (cid:54) = β (cid:88) p , q (cid:88) ˆ k ¯Ω ˆ k ¯Ω − ˆ k − p ˆ (cid:15) α − q ˆ (cid:15) β Q αd d ( p ) Q βd d ( q ) p d p d q d q d = G n (cid:88) α,β =0 α (cid:54) = β (cid:88) p p d p d p d p d e − p ξ L Q αd d ( p ) Q βd d ( − p ) . (E14)Putting the pieces together, we then obtain (cid:104) G (Ω) G ( Q ) (cid:105) , n ≈ G ˜ η (cid:96) ξ L n (cid:88) α,β =0 α (cid:54) = β (cid:88) p J T α T β e − p ξ L × (cid:0) δ d d δ d d + 5 δ d d p d p d ξ L + 338 p d p d p d p d (cid:1) Q αd d ( p ) Q βd d ( − p ) . (E15)
5. Effective Hamiltonian for liquid crystallinity
The effective Hamiltonian for liquid crystallinity, H n [ { Q α } nα =0 ] in Eq. (6.1), can now be derived using thecoefficients obtained in App. D. Defining H n [ { Q α } nα =0 ] by H n [ { Q α } nα =0 ] ≡ f Q ( Q ) + f C ( ¯Ω , Q ) , (E16)0we have H n [ { Q α } nα =0 ] ≈ n (cid:88) α =0 (cid:88) p J p T α | Q αab ( p ) | − (cid:104) G ( Q ) (cid:105) , n − (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n − (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n − (cid:0) (cid:104) G ( ¯Ω) G ( Q ) (cid:105) , n − (cid:104) G ( ¯Ω) (cid:105) , n (cid:104) G ( Q ) (cid:105) , n (cid:1) ≈ (cid:88) p (cid:32) T ( A t + L p ) { Q p Q − p } + n (cid:88) α =1 T ( A t + L p ) { Q α p Q α − p } (cid:33) − (cid:88) p − T n (cid:88) α =1 (cid:88) p H p { Q p Q α − p } + 12 T n (cid:88) α,β =1( α (cid:54) = β ) H p { Q α p Q β − p } , (E17)where H p is defined as in Eq. (6.2). Eq. (E17) is Eq. (6.1). In performing this derivation we have made twoapproximations. Our first approximation consists in neglecting terms of cubic or quartic order in Q , as we areconsidering an IGNE that is both prepared and measured at high temperatures. Our second approximation consistsin neglecting those terms in Eq. ( ?? ) that vanish with pξ L whilst retaining those terms that do not vanish with pξ L .It may appear that terms with factors of larger powers in pξ L would dominate over those with smaller powers of pξ L for large pξ L ; however, the same terms also come equipped with exponential-damping factors of exp( − p ξ L / pξ L contributions can effectively be neglected. We are thus left with the small pξ L modes, andfor these, the terms in f C ( ¯Ω , Q ) that vanish with pξ L are obviously smaller than those that do not vanish with pξ L .The effective Hamiltonian thus obtained allows us to predict a novel regime in which correlators C T and C G undergospatial decay and oscillation (see Sec. VIII B). As we shall see in the following subsection, the terms neglected by oursecond approximation would not result in a qualitative change to this prediction.
6. Corrections arising from neglected terms
In the previous subsection the effective Hamiltonian H n [ { Q α } nα =0 ] was derived under the approximation that weneglect terms originating in f C ( ¯Ω , Q ) that contain factors of pξ L . In this subsection we consider the modifications tothe thermal and glassy correlators C T and C G arising from these neglected terms. For simplicity, we consider the caseof a system prepared at a high temperature, so that the nematic order parameter in the zeroth replica, Q , may beset to zero. Taking the neglected terms into account, one has the effective Hamiltonian H eff [ { Q α } nα =1 ] = 12 T (cid:88) α (cid:88) p (cid:0) A t + L p (cid:1) { W α p W α p } − ν T (cid:88) α (cid:54) = β (cid:88) p e − p ξ L { W α p W β p } + 12 T (cid:88) α (cid:88) p (cid:0) A t + L p (cid:1) X α p · X α p − ν T (cid:88) α (cid:54) = β (cid:88) p (cid:0) p ξ L (cid:1) e − p ξ L X α p · X β p + 12 T (cid:88) α (cid:88) p (cid:0) A t + L p (cid:1) φ α p φ α p − T (cid:88) α (cid:54) = β (cid:88) p (cid:16) ν (cid:0) p ξ L (cid:1) e − p ξ L + ν p ξ L e − p ξ L (cid:17) φ α p · φ β p , (E18)where we have introduced ν ≡ ( G ˜ η J / (cid:96)/ξ L ) and ν ≡ ( G ˜ η J / (cid:96)/ξ L ) . We have also made use of thefact that any symmetric, traceless tensor Q p can be decomposed (in a wave-vector-dependent manner) into its fiveindependent modes φ p , X p and W p : viz., Q d d ( p ) = p d | p | p d | p | φ p + P Td d p d | p | X d ( p ) + P Td d p d | p | X d ( p ) + W d d ( p ) , (E19)1where the modes φ p , X p and W p are given by φ p ≡ p d | p | p d | p | Q d d ( p ) , (E20a) W d d ( p ) ≡ P d d P d d Q d d ( p ) − (cid:0) P d d P d d Q d d + φ p (cid:1) P d d , (E20b) X d ( p ) ≡ P d d p d | p | Q d d ( p ) , (E20c)and the projection operator P Td d is defined via P Td d ≡ δ d d − p d | p | p d | p | . (E21)By using Eq. (E18) we can compute the thermal and glassy correlators for the five independent modes in Fourierspace. The thermal fluctuation correlators are given by[ {(cid:104) W p W − p (cid:105) − (cid:104) W p (cid:105)(cid:104) W − p (cid:105)} ] = 2 T A t + L p + ν T − e − p ξ L , (E22a)[ (cid:104) X p · X − p (cid:105) − (cid:104) X p (cid:105) · (cid:104) X − p (cid:105) ] = 2 T A t + L p + ν T − (cid:0) p ξ L (cid:1) e − p ξ L , (E22b)[ (cid:104) φ p φ − p (cid:105) − (cid:104) φ p (cid:105)(cid:104) φ − p (cid:105) ] = T A t + L p + T − (cid:16) ν (cid:0) p ξ L (cid:1) e − p ξ L + ν p ξ L e − p ξ L (cid:17) . (E22c)We note that a given mode’s correlator undergoes oscillatory decay in real space if the least stable mode has a nonzerowave-vector. Equivalently, if we Taylor expand the denominator of that correlator in powers of the wave-vector aboutzero and find that the effective stiffness (i.e., the coefficient of the quadratic term) is negative then the correlatorundergoes oscillatory decay in real space. We can check that the effective stiffness does indeed become negative atsufficiently large values of the disorder strength (which is proportional to ν ) for the modes W p and φ p , although itdoes not for X p . The full thermal fluctuation correlator C T ( r ) is the sum of the inverse Fourier transforms of thecorrelators for the modes in Eq. (E22). Thus, at sufficiently large disorder strength, C T ( r ) will undergo oscillatorydecay.To determine whether the glassy correlator C G ( r ) undergoes oscillatory decay at sufficiently large values of thedisorder strength we apply a similar analysis, computing the glassy correlator for each mode:[ {(cid:104) W p (cid:105)(cid:104) W − p (cid:105)} ] = 2 ν e − p ξ L (cid:0) A t + L p + ν T − e − p ξ L (cid:1) ; (E23a)[ (cid:104) X p (cid:105) · (cid:104) X − p (cid:105) ] = 2 ν (1 + 10 p ξ L ) e − p ξ L (cid:16) A t + L p + ν T − (cid:0) p ξ L (cid:1) e − p ξ L (cid:17) ; (E23b)[ (cid:104) φ p (cid:105)(cid:104) φ − p (cid:105) ] = ν (cid:0) p ξ L (cid:1) e − p ξ L + ν p ξ L e − p ξ L (cid:16) A t + L p + T − (cid:0) ν (cid:0) p ξ L (cid:1) e − p ξ L + ν p ξ L e − p ξ L (cid:1)(cid:17) . (E23c)The full thermal fluctuation correlator C G ( r ) is the sum of the inverse Fourier transforms of the correlators for themodes in Eq. (E23). As for the thermal correlator, the denominators of the glassy correlators for the modes W p and φ p (but not X p ) can acquire minima at non-zero wave-vectors for sufficiently large disorder strengths. In real space,such nonzero-wave-vector minima imply that the correlator C G ( r ) undergoes oscillation as it decays. Appendix F: A conventional random field approach
In this Appendix we provide the steps leading to the thermal and glassy correlators that one would obtain from aconventional random field approach. In this approach, an IGNE subject to a given realization of quenched disorder H can be described by the Landau free energy H rf = 12 (cid:88) p (cid:0) A t + L p (cid:1) { Q p Q − p } − (cid:88) p { H p Q − p } . (F1)2Here, H is a quenched random field having a Gaussian distribution defined by mean and variance[ H d d ( p )] = 0 , (F2a)[ { H d d ( p ) H d d ( − p ) } ] = ∆( δ d d δ d d + δ d d δ d d + δ d d δ d d ) . (F2b)By using the replica technique to eliminate the quenched disorder, one obtains the effective Hamiltonian H ef = 12 n (cid:88) α =1 (cid:88) p (cid:0) A t + L p (cid:1) { Q α p Q α − p } + 12 n (cid:88) α,β =1 (cid:88) p ∆ { Q α p Q β − p } . (F3)The inverse of the Hessian matrix corresponding to H ef is given by (for α, β = 1 , . . . , n ) (cid:32) δ H ef δQ αd d ( p ) δQ βd d ( − p ) (cid:33) − = 1 A t + L p δ αβ + ∆ (cid:0) A t + L p (cid:1) αβ , (F4)where repeated Cartesian indices d , d mean that they are to be summed over, and αβ is an n × n matrix with eachentry having the value of unity. Equation (F4) yields for the thermal and glassy correlators the results C T p = 5 T A t + L p , (F5a) C G p = 5 T ∆ (cid:0) A t + L p (cid:1) . (F5b)Note that the denominators of C T p and C G p , calculated via a conventional random field approach developed in thisappendix, do not feature the length-scale-dependent function H p that appears in the denominators of (and plays anessential role in determining the behavior of) the correlators (8.3) derived from the microscopic dimer-and-springsmodel. [1] J. K¨upfer and H. Finkelmann, Macromol. Chem. Rap.Commun. 12, 717 (1991).[2] M. Warner, E. M. Terentjev, Liquid Crystal Elastomers (Oxford University Press, Oxford, 2003).[3] K. Urayama,
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