Orientational order and glassy states in networks of semiflexible polymers
aa r X i v : . [ c ond - m a t . s o f t ] M a r Orientational order and glassy states in networks of semiflexible polymers
Martin Kiemes,
1, 2
Panayotis Benetatos,
1, 3 and Annette Zippelius
1, 2 Institut f¨ur Theoretische Physik, Georg-August-Universit¨at G¨ottingen,Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany Max-Planck-Institut f¨ur Dynamik und Selbstorganisations, Bunsenstraße 10, D-37073 G¨ottingen, Germany Theory of Condensed Matter Group, Cavendish Laboratory, University of Cambridge,19 J. J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom (Dated: November 7, 2018)Motivated by the structure of networks of cross-linked cytoskeletal biopolymers, we study orienta-tionally ordered phases in two-dimensional networks of randomly cross-linked semiflexible polymers.We consider permanent cross-links which prescribe a finite angle and treat them as quenched disor-der in a semi-microscopic replica field theory. Starting from a fluid of un-cross-linked polymers andsmall polymer clusters (sol) and increasing the cross-link density, a continuous gelation transitionoccurs. In the resulting gel, the semiflexible chains either display long range orientational order orare frozen in random directions depending on the value of the crossing angle, the crosslink concen-tration, and the stiffness of the polymers. A crossing angle θ ∼ π/M leads to long range M -foldorientational order, e.g., “hexatic” or “tetratic” for θ = 60 ◦ or 90 ◦ , respectively. The transition tothe orientationally ordered state is discontinuous and the critical cross-link density, which is higherthan that of the gelation transition, depends on the bending stiffness of the polymers and the cross-link angle: the higher the stiffness and the lower M , the lower the critical number of cross-links.In between the sol and the long range ordered state, we always expect a gel which is a statisticallyisotropic amorphous solid (SIAS) with random positional and random orientational localization ofthe participating polymers. PACS numbers: 87.16Ka,82.70Gg,61.43Er
I. INTRODUCTION
The cytoskeleton is a network of linked protein fiberswhich plays an important role in several functions of eu-karyotic cells such as maintenance of morphology, me-chanics and intracellular transport [1]. Cytoskeletalfibers, such as F-actin can be described as semiflexiblepolymers with a behavior intermediate between the twoextreme cases of rigid rods and random coils. The func-tion of the actin cytoskeleton is modulated by a largenumber of actin-binding proteins (ABPs) [2, 3]. The or-ganization of actin filaments into networks is regulated byABPs which can be classified into two broad categories; cross-linkers promote binding of the filaments at finitecrossing angles whereas bundlers promote formation ofbundles consisting of parallel or antiparallel filaments.F-actin is a polar semiflexible polymer and some ABPsbind filaments with a specific polarity whereas others arenot affected by the filament’s polarity. In order to un-ravel the physics of these complex aggregates, in vitro solutions of actin filaments with controlled cross-linkershave been studied [4].The stiffness of semiflexible filaments gives rise toorientational correlations and promotes the formationof structures with long range orientational order. Theisotropic-nematic transition in solutions of partially flex-ible macromolecules has been studied theoretically us-ing the Onsager approach [5] or the Maier-Saupe ap-proach [6, 7]. In Ref. [7], the role of the solvent istaken into account and a very rich phase transition ki-netics is predicted. Aggregation and orientational or- dering of Lennard-Jones macromolecules with bendingand torsional rigidity, with or without solvent, has re-cently been investigated in Ref. [8] using molecular dy-namics simulations. Experimental investigations of theisotropic-nematic transition in lyotropic F-actin solutionshave been carried out in Ref. [9] and measurements ofthe associated order parameter are presented in [10].The excluded volume effect is not the only mechanismwhich drives an isotropic-nematic transition in solutionsof semiflexible polymers. Assemblies of cytoskeletal fil-aments exhibit a structural polymorphism due to inter-actions mediated by a wide range of ABPs, as shown inRef. [11]. Theoretical attempts to study the formation ofordered structures in this kind of systems involve a gener-alized Onsager approach [12–14], a Flory-Huggins theory[15], and a semi-microscopic replica field theory [16]. InRefs. [12–15] the filaments are modeled as perfectly rigidrods whereas in Ref. [16] they are considered to be semi-flexible polymers and the thermal bending fluctuations(finite persistence length) are fully taken into account.
In vitro structural studies by Wong et al. [17] of F-actin in the presence of counterions have shown the for-mation of sheets (“rafts”) with tetratic order without anycross-linking proteins. It appears that electrostatic in-teractions are the main mechanism behind the effective“ π/
2” cross-linking of the actin filaments in this experi-ment. A theory for tetratic raft formation by rigid rodsand reversible sliding “ π/
2” cross-linkers has been pro-posed by Borukhov and Bruinsma [13].In Ref. [16], a three-dimensional melt of identical,fixed-contour-length semiflexible chains is considered andrandom permanent cross-links are introduced. The cross-links are such that they fix the relative positions of thecorresponding polymer segments and constrain their ori-entations to be parallel or antiparallel to each other. Theaim of the present study is to examine the effect of cross-links which prescribe a finite crossing angle on a two-dimensional version of the previous model. We distin-guish the case of sensitive cross-links that are perceptiveof the polymers’ polarity and unsensitive cross-links thatare not. In contrast to previous studies where the rigidrods have a finite width which allows for nematic order-ing `a la Onsager, our semiflexible polymers are consideredone-dimensional objects and the sole cause for the emer-gence of orientationally ordered phases is the interplayof the finite persistence length of the polymers and thecross-link geometry.Using the polymer stiffness and the cross-link densityas control parameters, we obtain a phase diagram whichinvolves a sol and various types of orientationally orderedgels. For appropriate values of the control parametersand crossing angle θ ∼ π/M ; M ∈ Z , we predict theemergence of an exotic gel with random positional lo-calization and M -fold orientational order (e.g., hexaticfor M = 6 or tetratic for M = 4). Similar phases havebeen predicted [14] for a different system in three spacedimensions: a semi-dilute solution of charged rods withfinite diameter in the presence of polyvalent ions thathave the function of (non-permanent) cross-linkers andmay favor various crossing angles. Besides the orien-tationally ordered phase, we also predict a statisticallyisotropic amorphous solid (SIAS) with random positionaland orientational localization of its constituent polymersappearing right at the gelation transition.This article is organized as follows. In Section II, weintroduce our model and the Deam-Edwards distribu-tion which parametrizes the quenched disorder associatedwith the cross-links. In Section III, the disorder-averagedfree energy is presented as a functional of a coarse-grained field which plays the role of an order parame-ter. Then, in Section IV, we discuss different variationalAns¨atze that express positional and orientational local-ization. The symmetries imposed by the cross-linkingconstraints allow the emergence of specific orientation-ally ordered phases. The corresponding free energies arecalculated variationally in the saddle-point approxima-tion and a phase diagram is obtained. We summarizeand give an outlook in Section V. Finally, details of thecalculations are given in the appendices. II. MODEL
We consider a large rectangular two-dimensional vol-ume (area) V which contains N identical semiflexiblepolymers. A single polymer of total contour length L is represented by a curve in 2 d space with r ( s ) denotingthe position vector at arc-length s . Bending a polymercosts energy according to the effective free energy func-tional (“Hamiltonian”) for the wormlike chain (WLC) model [18] H W LC ( { r ( s ) } ) = 12 κ Z L d s (cid:18) d t ( s )d s (cid:19) . (1)Here we have introduced the tangent vector t ( s ) =d r ( s ) / d s and chosen a parametrization of the curve, inaccord with the local inextensibility constraint of theWLC, such that | t ( s ) | = 1. The position vector is re-covered from t ( s ) as r ( s ) = r (0) + R s t ( τ )d τ . Hencethe conformations of a single polymer, that can be bentbut not be stretched, are alternatively characterized by t ( τ ) , ≤ τ ≤ L , and r (0). The bending stiffness isdenoted by κ ; it determines the persistence length L p according to κ = L p k B T /
2. Throughout the rest of thepaper we set k B T = 1. The WLC model can describe stiffrods, obtained in the limit L/L p → L p /L → H ev = N V X k =0 X m ∈ Z λ | k | ,m | ρ k ,m | (2)with ρ k ,m := 1 N N X i =1 L Z L d s e i kr i ( s ) e imψ i ( s ) (3)being the Fourier transformation of the positional-orientational density. t i ( s ) = (cos ψ i ( s ) , sin ψ i ( s )) de-notes the orientation of monomer s on polymer i . Thecoefficients λ | k | ,m depend only on the absolute value ofthe vector k in order to preserve the rotational symme-try of the system. They are later chosen large enoughto provide stability with respect to density modulations.For details, see Appendix C.The Hamiltonian H W LC is invariant with respect to in-terchanging head and tail of the filaments, i.e., the energyis unchanged under reparametrizations of the contour ofone polymer i by { r i ( s ) → r i ( L − s ) , ∀ s ∈ [0 , L ] } . How-ever, in the following, we want to consider filaments witha definite polarity which could for example arise due tothe helicity of F-actin. The WLC Hamiltonian is not sen-sitive to such a polarity, but the cross-links may or maynot differentiate between the two states of the filament,as discussed below.We now introduce M permanent (chemical) cross-linksbetween pairs of randomly chosen monomers. A singlecross-link, say between segments s and s ′ on two chains i and j , constrains the two polymer segments to be atthe same position, i.e. r i ( s ) = r j ( s ′ ), and fixes their rela-tive orientation by a constraint expressed by the function∆ ( t i ( s ) , t j ( s ′ ) , θ ). The constraint of relative orientationis most easily formulated in polar coordinates where ∆just fixes the crossing angle ψ − ψ ′ to some prescribedvalue θ . In the case of cross-links sensitive to polarity wesimply have ∆( ψ − ψ ′ , θ ) = δ (cid:0) ψ − ψ ′ − θ (cid:1) , (4)where the weight of the delta function is such that R π ψ π δ ( ψ ) = 1. In the unsensitive case on the otherhand, changing head and tail of either filament leads tofour equivalent situations that correspond to two differ-ent crossing angles (see Fig. 1). In this case the cross-link(4) thus appears with the two equally likely cross-linkingangles θ and θ + π . We point out that in our model thepolarity is only recognized by the cross-links and doesnot enter into the Hamiltonian. θ + π θ + π θ θ FIG. 1: 4 possibilities for two filaments to be joined by anunsensitive cross-link.
The delta function in (4) is the simplest way to modelthe angular constraint of the cross-links. It is howevermuch more realistic to introduce an effective angularcross-linking potential that allows for fluctuations aroundthe preferred direction. A simple model for these “soft”cross-links is given by∆( ψ − ψ ′ , θ ) = 1 I ( γ ) e γ cos( ψ − ψ ′ − θ ) . (5)The cross-link stiffness parameter γ controls the varianceof the fluctuations of the angle and it is clear that in thelimit γ → ∞ we recover the simple delta function modelof “hard” cross-links. In the following we will first explorethe case of hard cross-links, because it is mathematicallysimpler. It gives rise, however, to artifacts which dis-appear when considering the more physical model whichfavors certain angles but does not enforce them strictly.The cross-links are permanent and do not break up orrebuild. Hence we are led to study the equilibration ofthe thermal degrees of freedom { r i ( s ) } in the presence of quenched disorder represented by a given cross-link con-figuration C = { i e , j e ; s e , s ′ e } Me =1 which is characterizedby the set of pairs of polymer segments that are involvedin a cross-link. The central quantity of interest is thecanonical partition function Z ( C ) = * M Y e =1 δ ( r i e ( s e ) − r j e ( s ′ e ))∆ ( t i e ( s e ) − t j e ( s ′ e ) , θ ) + H . (6) Here the thermal average < ... > H is taken with respectto the Hamiltonian H := H W LC + H ev and the partitionfunction depends on the cross-link realization C . Thefree energy F is expected to be self-averaging so that weare allowed to compute the disorder averaged free energy[ F ] = − [ln Z ], where [ ... ] denotes the “quenched” averageover all cross-link configurations according to some dis-tribution P ( C ). We assume that the different realizationsobey a Deam-Edwards-like [19] distribution P ( C M ) ∝ M ! (cid:18) µ VN (cid:19) M * M Y e =1 δ ( r i e ( s e ) − r j e ( s ′ e )) + H , (7)implying that polymer segments that are likely to be closeto each other in the uncross-linked melt also have a highprobability of getting cross-linked. Note that the proba-bility of cross-linking is independent of the relative orien-tation of the two segments, so that the cross-link actuallyreorients the two participating segments. The parameter µ controls the mean number of cross-links in the system:[ M ] /N ∼ O ( µ ) .Given the Hamiltonian, the constraints due to cross-linking and the distribution of cross-link realizations, thespecification of the model is complete and we proceed tocalculate the disorder averaged free energy [ F ]. III. REPLICA FREE ENERGY
The standard method to treat the quenched disorderaverage is the replica method, representing the disorderaveraged free energy as [ F ] = − [ln Z ] = − lim n → ([ Z n ] − /n , i.e., we have to calculate the simultaneous disor-der average of the partition sums of n non-interactingcopies of our system. In the end, we extract the disorderaveraged free energy [ F ] from [ Z n ] as the linear ordercoefficient of its expansion in the replica number n .In this section, we present only the essential formulasand present the details of the calculation in Appendix B.The disorder average, [ Z n ], gives rise to an effectively uni-form theory, however, with a coupling of different repli-cas:[ Z n ] ∝ * exp µ V N N X i,j =1 Z s,s ′ δ (ˆ r i ( s ) − ˆ r j ( s ′ )) ∆( ˇ ψ i ( s ) − ˇ ψ j ( s ′ ) , θ ) !+ H n +1 , (8)where the average h ... i H n +1 is over the ( n + 1)-fold repli- cated Hamiltonian H . To simplify the notation wehave introduced the abbreviations ˆ r ≡ (cid:0) r , r , . . . , r n (cid:1) ,ˇ ψ ≡ (cid:0) ψ , . . . , ψ n (cid:1) , and the shorthand notation R s ≡ (1 /L ) R L d s . Furthermore, δ (ˆ r ) ≡ Q nα =0 δ ( r α ) and ∆denotes for sensitive cross-links∆ s ( ˇ ψ, θ ) ≡ n Y α =1 δ ( ψ α − θ ) (9)and∆ u ( ˇ ψ, θ ) ≡ ( n Y α =1 δ ( ψ α − θ ) + n Y α =1 δ ( ψ α − ( θ + π )) ) (10)if they are unsensitive to polarity of the filaments. Forsoft cross-links we introduce the corresponding defini-tions.Different polymers are decoupled via a Hubbard-Stratonovich transformation, introducing collectivefields, Ω(ˆ x , ˇ ϕ ), whose expectation values are given by (cid:10) Ω(ˆ x , ˇ ϕ ) (cid:11) = 1 N N X i =1 Z s (cid:10) δ (ˆ x − ˆ r i ( s )) δ ( ˇ ϕ − ˇ ψ i ( s )) (cid:11) . (11)The field h Ω i quantifies the probability to find monomer s on chain i at position x in replica 0, at position x inreplica 1,... and at position x n in replica n and similarlyto find it oriented in the direction e = (cos( ϕ ) , sin( ϕ ))in replica 1,... and oriented in the direction e n =(cos( ϕ n ) , sin( ϕ n )) in replica n . Sometimes it is conve-nient to use its equivalent representation in Fourier space (cid:10) Ω(ˆ k , ˇ m ) (cid:11) = 1 N N X i =1 Z s (cid:10) e i ˆ k ˆ r i ( s ) e i ˇ m ˇ ψ i ( s ) (cid:11) . (12)In terms of these collective fields the effective replicatheory is given by[ Z n ] ∼ Z D{ Ω } exp ( − N F ( { Ω } )) . (13)The replica free energy per polymer reads F = µ V n X ˆ k X ˇ m ∆ | Ω | − ln z ( { Ω } ) , (14)with the effective single chain partition function z ( { Ω } ) = * exp (cid:18) µ V n X ˆ k X ˇ m ∆ Ω Z s exp n − i ˆ k ˆ r ( s ) o exp n − i ˇ m ˇ ψ ( s ) o(cid:19)+ H WLC n +1 . (15)The average h ... i W LCn +1 refers to the ( n + 1)-fold replicatedHamiltonian of a single wormlike chain. The sum P ˆ k over replicated wave vectors is restricted to the com-bination of the so-called “0-replica sector” (0RS) thatcontains only the point ˆ k = (0 , . . . ,
0) and the “higher-replica sector” (HRS) where at least two wave vectors indifferent replicas are non-zero: k α = 0 and k β = 0 with α = β .Note that in the above overview we left out the “1-replica sector” (1RS) that consists of ( n + 1)-fold repli-cated vectors ˆ k α where only one entry (the α th) is non-zero, i.e. ˆ k α = (0 , ..., k α , ..., m being arbi-trary. The corresponding fields ˜Ω α ( k , ˇ m ) describe regularmacroscopic density fluctuations (modulated states). Inthis work we focus on the properties of the macroscopi-cally translationally invariant amorphous solid state andassume that the inter-polymer interactions (2) are suchthat periodic density fluctuations are suppressed. SeeAppendices B and C for more details on that issue. IV. VARIATIONAL APPROACH
We shall only discuss the saddle-point approximationto the field theory on the right hand side of (13) replacingthe field Ω by its saddle point Ω sp which has to be calcu-lated from δ F /δ Ω | Ω sp = 0. In fact, even the saddle pointequation is too hard to solve, because the conformationaldistribution of the WLC is very complex. Similarly, it isnot possible to perform a complete stability analysis ofthe Gaussian theory. Hence we restrict ourselves to avariational approach and in the following we are goingto construct Ans¨atze which capture the symmetry of thedifferent physical states which may emerge.What behavior do we expect? Upon increasing thenumber of cross-links up to about one per polymer, thereshould be a gelation transition from a liquid to an amor-phous solid state. A finite fraction of the polymers formthe percolating cluster and are localized at random po-sitions. The other polymers belong to finite clusters orremain un-cross-linked and are still free to move aroundin the volume. This scenario has been found to be validfor a variety of different models [20–24]. A similar replicafield-theoretic approach has beeen used by Panyukov and FIG. 2: Sketch of the tetratic phase (M=4).
Rabin to study the properties of well cross-linked macro-molecular networks [25, 26].For semiflexible polymers the positional localization inthe macroscopic cluster is accompanied by orientationallocalization. The cross-links create locally an orienta-tional structure. Supposing that the semiflexible poly-mers are rather stiff, a long range orientationally orderedstate can be established. Otherwise we may find an ori-entational glass, i.e., the directions of the polymer seg-ments are frozen in random directions in analogy to thelow temperature phase of a spin glass[27]. We call sucha phase statistically isotropic amorphous solid (SIAS).Can we expect long range orientational order for ar-bitrary crossing angles θ ? We first consider a specialcase, namely that the crossing angle is such that aninteger multiple of the crossing angle, M θ , equals 2 π ,e.g. θ = 120 ◦ and M = 3. Such a choice of crossingangle allows for orientationally ordered states providedthe chains are sufficiently stiff. In the case of sensitivecross-links we expect M -fold discrete rotational symme-try, in the case of unsensitive cross-links and odd M weexpect 2 M -fold symmetry because cross-links are estab-lished including both the angles θ and θ + π (see Fig. 1).Sketches of gels with long range four-fold order ( θ = 90 ◦ )or long range three-fold or six-fold order respectively( θ = 120 ◦ / ◦ ) are shown in Fig. 2 and Fig. 3. Thecase θ = k πM with k = 1 , . . . , M − { r i ( s ) } . The disorder averaged parti-tion function of Eq.(8) is invariant under spatially uni-form translations and spatially uniform rotations of eachreplica separately .Only the fluid state is expected to have the full sym-metry of the partition function Eq.(8) because here, thepolymers and finite clusters of polymers are free to sam-ple the complete volume and can take any orientation. FIG. 3: Sketch of the triangular phase (M=3) for sensitive orthe hexatic phase (M=6) for unsensitive cross-links.
This implies for the order parameter (cid:10)
Ω(ˆ k, ˇ m ) (cid:11) = δ ˆ k, ˆ0 δ ˇ m, ˇ0 , (16)i.e., each replica is separately invariant under translationsand rotations.We will only investigate gels with random localizationof a fraction of the particles. In other words, we re-strict ourselves to amorphous solids and do not considerperiodic density fluctuations. It might be of interest tostudy one-dimensional periodic density modulations withan overall orientation of the WLCs perpendicular to thewave vector,– reminiscent of bundles. However this isnot the topic of the present paper, where we stick to theincompressible limit.If the translational and rotational symmetry is brokenspontaneously, then we expect a non-trivial expectationvalue of the local density ρ i,s ( x , ϕ ) := δ ( x − r i ( s )) δ ( ϕ − ψ i ( s )) (17)in a particular equilibrium state. For example, in the amorphous solid phase a finite fraction of particles shouldbe spatially localized at preferred positions and possi-bly oriented along preferred directions. A simple Ansatzquantifying such a scenario is the following: h ρ i,s ( x , ϕ ) i ∝ e − ξ ( x − a ) e η cos( ϕ − ϕ ) . (18)The vector a is the preferred mean position of monomer s and the thermal fluctuations around the preferredposition are controlled by the localization length ξ . Thepreferred orientation is given by ϕ and η parametrizesthe variance of the orientational distribution.An amorphous solid is macroscopically translationalinvariant , so that the spontaneous symmetry breakingof translational invariance is local . All macroscopic ob-servables should display translational symmetry and inparticular all moments of the local density˜Ω J ( k , ... k J , m , ...m J ) (19):= N P Ni =1 R s h ρ i,s ( k , m ) i ... h ρ i,s ( k J , m J ) i∼ δ k + ... k J , are non-zero only, if the wave vectors add up to zero (seereference [21] for a more detailed discussion).As far as the rotational symmetry is concerned, we willconsider two possibilities: a statistically isotropic state(SIAS) as well as states with true long range orienta-tional order. In the former case the spontaneous sym-metry breaking of rotational invariance is local and re-stored globally analogously to what happens in the caseof translational symmetry. All macroscopic properties,such as the moments, are non-zero only, if the angularmomentum “quantum” numbers add up to zero:˜Ω J ( k , ... k J , m , ...m J ) ∼ δ k + ... k J , δ m + ...m J , (20)For the long range ordered case the simplest Ansatzconsists in assuming a local M -fold symmetry. This im-plies for the moments˜Ω J ( k , ... k J , m , ...m J ) ∼ δ k + ... k J , J Y α =1 δ m α , Z M , (21)i.e. each m α has to equal an integer multiple of M .Note that a rotation affects both r i ( s ) and t i ( s ) andthat consequently, a system with e.g. M -fold symmetryshould be described by an Ansatz Ω(ˆ k , ˇ ϕ ) which isonly invariant under common rotations of the k α and ϕ α . We have chosen our approach for simplicity andincorporated the symmetry with respect to the orien-tational and position vector separately. Consequently,the Ansatz is symmetric under individual rotationsof the spatial and angular argument. An interestingextension of the present work would involve the studyof Ans¨atze which couple positional and orientationaldegrees of freedom. Some preliminary work in thisdirection has been done in the context of random net-works of covalently connected atoms or molecules[28, 29]. Rephrasing our results in replica language, frozenfluctuations in a single equilibrium state correspond toa non-zero expectation value of the density, ρ i,s ( k , m ),within one replica. The full statistics of the local staticfluctuations that we need for the characterization ofthe macroscopic symmetries in the gel is specified bythe order parameter field Ω(ˆ k , ˇ m ), encoding momentsof arbitrary order. In particular macroscopic transla-tional invariance requires P nα =0 k α = 0; macroscopicrotational invariance requires P nα =1 m α = 0; long rangeorientational order for crossing angle θ = πM requires m α = lM with l ∈ Z .Let us now encode these physical expectations in avariational Ansatz: The liquid state is characterized byΩ sp (ˆ k , ˇ m ) = δ ˆ k , ˆ0 δ ˇ m, ˇ0 . It becomes unstable at a criti-cal cross-link density, µ c , where a percolation transitiontakes place and a macroscopic cluster is formed contain-ing a finite fraction of the polymers. To account for thefraction of localized particles Q in the percolating clustercoexisting with mobile particles (fraction 1 − Q ) in finiteclusters, we make the following general Ansatz for theexpectation value of the order parameter:Ω sp (ˆ k , ˇ ϕ ) = (1 − Q ) δ ˆ k , ˆ + Q ω (ˆ k , ˇ ϕ ) δ , P nα =0 k α (22)The first term describes the sol phase which is charac-terized by perfect spatial homogeneity and orientationalisotropy. The second part describes the amorphous solidwhich is macroscopically translational invariant as re-flected in δ , P nα =0 k α . The details of the amorphous solidphase under consideration have to be implemented in theorder parameter ω (ˆ k , ˇ ϕ ).Plugging the generic form of the order parameter (22)into the replica free energy (14) we get F sp = µ V n n − Q + Q X ˆ k Z ˇ ϕ ˇ ϕ ′ ω (ˆ k , ˇ ϕ ) ∆( ˇ ϕ − ˇ ϕ ′ , θ ) ω (ˆ k , ˇ ϕ ′ ) δ , P nα =0 k α o (23) − µ V n (1 − Q ) − ln * exp n µ QV n X ˆ k Z ˇ ϕ ˇ ϕ ′ δ , P nα =0 k α ∆( ˇ ϕ − ˇ ϕ ′ , θ ) ω (ˆ k , ˇ ϕ ) Z s e i ˆ k ˆ r ( s ) δ ( ˇ ϕ ′ − ˇ ϕ ( s )) o+ H WLC n +1 We expect the gel fraction to be small close to the gela-tion transition and thus expand the log-trace contribu-tion of the free energy in Q . The terms linear in Q cancel,as they should for the expansion around Q = 0 to be jus-tified. A. M -fold orientationally ordered amorphous solid
1. Hard cross-links
Assuming that the thermal fluctuations of the poly-mers are to a certain degree suppressed by their stiffnessand a sufficient number of cross-links has been formed,long range orientational order may be present as sketchedin Fig. 2 and Fig. 3. The Ansatz for the M -fold orien-tationally ordered amorphous solid favors M preferredorientational axes separated by angles πM . A simple wayto incorporate this symmetry is the following Ansatz forthe replica order parameter ω (ˆ k , ˇ ϕ ) = e − ξ ˆ k e η P nα =1 cos( Mϕ α ) I n ( η ) . (24)Here I denotes the modified Bessel function of the firstkind; it ensures the proper normalization.For ˆ k = ˆ , the above order parameter is a probabilitydistribution and thus specifies the local orientational or-der completely. In experiment on the other hand one hasaccess to low order moments only. The simplest physi-cal order parameter being sensitive to the degree of longrange M -fold orientational order is given by S M := 1 N N X i =1 Z s D cos( M ψ i ( s )) E (25) ∼ hD cos( M ψ i ( s )) Ei (26)= η O ( η )where in the second line, we replaced the average overall monomers of the system by the disorder average ofone arbitrary monomer. This expression allows to relate S M to our effective one-particle theory. h . . . i denotes thethermal expectation value of the system for a given in-stance of disorder C . In order to establish the connection between the variational parameter η and the physical or-der parameter S M we evaluate (26) using (24) and findthat S M is in leading order proportional to η .We plug the Ansatz (24) in (23) and perform the ˆ k summations and ˇ ϕ integrations. If we have chosen M ac-cording to the symmetry considerations of the previoussection the constraint ∆ drops out. We are left with thefree energy as a function of three variational parameters:the gel fraction Q , the spatial localization length ξ andthe degree of orientational order as measured by η . Todetermine these parameters we minimize the free energy.The resulting equation for the gel fraction is universaland has been derived previously [21]. There are two so-lutions Q = 0 and Q ∼ µ ( µ −
1) which implies that thephase transition from sol to gel takes place at µ c = 1.In order to determine the remaining parameters ξ and η we assume them to be small near the transition and doa Taylor expansion of the free energy where we leave outterms which do not depend on ξ or η because these termsare irrelevant for the variation of F . More precisely, itwill turn out that ξ ∝ Q , being thus small close to thetransition, and we will keep contributions up to order1 /ξ in our expansion. As for the variational parameter η , we will find that it actually jumps from 0 to a finitevalue, i.e that there is a first order orientational transi-tion. We expand all the same in η and obtain at least aqualitative picture. Details on the calculations are givenin Appendix D.Since our Ansatz is replica-symmetric it is straightfor-ward to extract the part of F linear in the replica index n and we find F n = µ Q ( − µ Q L ξ + µ L ξ g ( LL p ) + δ M, µ L ξ n η l ( LL p ) − η
16 ˜ l ( LL p ) + η
16 ˜˜ l ( LL p ) o (27)+ η (cid:16) − µ h ( M LL p ) (cid:17) − η (cid:16) − µ ˜ h ( M LL p ) (cid:17) + 31288 η (cid:16) − µ ˜˜ h ( M LL p ) (cid:17) ) Note that for
M > η and the positional lo-calization specified by ξ . Such terms appear in higherorder but will not be considered here because we restrictourselves to the vicinity of the gel point. The persistancelength L p and the localization length ξ are both rescaledby the contour length L of the polymers.The functions g , h , ˜ h , ˜˜ h and l , ˜ l , ˜˜ l go to zero for large ar-gument and are for small argument approximately given by g ( x ) ∼ − x , h ( x ) ∼ − x ˜ h ( x ) ∼ − x , ˜˜ h ( x ) ∼ − xl ( x ) ∼ − x , ˜ l ( x ) ∼ − x ˜˜ l ( x ) ∼ − x (28)For the definitions of these functions see Appendix F.Minimizing F for M > ξ yields1 ξ = 23 L µ Qg ( LL p ) = µ Q R g . (29)Hence the filaments are localized as soon as a percolat-ing cluster of cross-linked chains has formed and the gelfraction is finite. The localization length is independentof M and its scale is set by the radius of gyration R g ofthe filament which is ∼ L for stiff chains and ∼ L forrandom coils.For finite polymer flexibility LL p the incipient gel hasno long range orientational order. Increasing the numberof cross-links beyond µ = 1 we find a first-order tran-sition that corresponds to a minimum of the free energyat nonzero η , which is first metastable and eventuallybecomes the global minimum (see Fig. 4). The criti-cal values of µ for the first appearance of a metastableordered state ( µ ), the first order phase transition ( µ )and the disappearance of the metastable disordered state( µ ) are shown in Fig. 5. In the limit LL p → ∞ the phasetransition coincides with the sol-gel transition, whereasfor more flexible filaments higher cross-link densities arerequired. For rather stiff polymers we find that the tran-sition takes place at µ ∼ . M L/L p . (30)In addition, Fig. 5 shows the dependence of η c , whichis the value of the variational parameter η at the phasetransition, on the polymer flexibility. There is a ten-dency to a lower degree of orientational localization forhigher values of the polymer stiffness. This behavior isqualitatively different from the behavior predicted for thelyotropic nematic ordering of partially flexible rods byKhokhlov and Semenov in the corresponding range offlexibility [5]. However, a direct comparison of the twomodels is not feasible. In our case, orientational order-ing is due to the cross-links and the critical density ofcross-links µ of the transition decreases for increasingpolymer stiffness. There are thus two competing effects:stiffer polymers should lead to a stronger orientationallocalization of the polymers whereas the smaller numberof cross-links should have the opposite effect. It seemsthat the lower number of cross-links plays the dominantrole in the dependence of η c ( L/L p ).We point out that the cross-linking angle θ = 2 π/M enters the free energy as a rescaling of the polymer flex-ibility L/L p through the parameter M . This impliesthat the higher the value of M (the smaller the angle),the higher the polymer stiffness required for the transi-tion at a given cross-link density. The reason for thisscaling is that within our mean-field description we aredealing with one single chain in an effective medium. M-fold order has to be propagated along the chain fromone cross-link to the next and the M scaling reflects theproperties of the WLC in the calculation of the corre-sponding correlator. -2024681012 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 f o r (cid:1) FIG. 4: Plot of the orientational free energy of the 3-fold sym-metric case for
L/L p = 0 .
05 and µ = 1 . , . , . µ µ µ µ η crit M LL p η FIG. 5: Variational parameter of the incipient ordered state η c and cross-link densities µ , µ and µ where a metastableordered minimum appears, where it becomes the global mini-mum (phase transition) and where the minimum at zero cor-responding to the disordered state becomes unstable (plot for M > µ < µ < µ for finite stiffness. For a first-order transition the expansion in η is not re-ally justified and can only give qualitative results. Evenworse, the expansion of F in η exhibits an oscillatingbehavior: the orders 2 , , , . . . provide a stable orienta-tional free energy (for large η ) in a region around µ = 1and LL p = 0, but the orders 4 , , , . . . are always unsta-ble in the above region because they diverge asymptoti-cally to minus infinity. However, considering the purelyorientational free energy f or,hard := µ Q ( ln (cid:16) ∞ X q =1 I q ( η ) I ( η ) (cid:17) (31) − µ Z s ,s ln (cid:16) ∞ X q =1 I q ( η ) I ( η ) e − q M κ | s − s | (cid:17)) it is obvious that for any finite κ the Gaussian part islarger than the log-trace contribution as long as the cross-link density µ does not become too large. So, as longas we restrict ourselves to a region close enough to thesol-gel transition asymptotic stability is garantied anda qualitative picture can be obtained by truncating theexpansion at order 6, 10 or even higher.For the polar case, i.e M = 1, there are additionalterms ins the free energy (27) which couple spatial andorientational part. At first sight it might be tempting toargue that they can be neglected close to the transitionbecause they are proportional to ξ ∼ Q . As it turnsout this is not correct: the orientational transition for agiven polymer stiffness is shifted to significantly highervalues of µ . But all the same, the qualitative picture ofthe transition is still valid.In order to analyze the orientational transition we firstcalculate the stationarity equation with respect to 1 /ξ L ξ = 23 µ Qg ( LL p ) + η l ( LL p ) − η ˜ l ( LL p ) + η ˜˜ l ( LL p ) (32)and plug it again into the free energy. Keeping contri-butions up to order η we obtain again a power series in η . The transition scenario that we found for M > µ . For rather stiff polymers we find approximately µ ∼ . L/L p (33)and the critical µ is considerably larger than what wewould obtain from (30) without the coupling terms.At first sight, it seems a bit curious that the case M =1 is set apart by its coupling term. This is, however,only due to the low order expansion and the couplingterms for M > yet . Because ofthe higher symmetry of the orientational contributions,only higher order spatial contributions may lead to non-vanishing coupling terms.If long range orientational order could exist only forrational values of θ/ π , then it would be inaccessible inexperiment. We show in the next section that the longrange ordered states discussed above are also present ifwe introduce crosslinks that favor given crossing angleson average only.
2. Soft cross-links
Which changes do we expect when using soft cross-links (5) that do not rigidly fix the intersection angle to one particular value, but allow for fluctuations about agiven mean direction? Soft cross-links surely will make itharder to establish long range order, so the first expecta-tion is that the orientational transition will need a highercross-link density µ to take place. In particular, the be-haviour in the limit of stiff rods will change qualitatively:In the case of hard cross-links the macroscopic network isa completely rigid object, even at cross-link densities justabove the critical value of µ = 1, whereas in the case ofsoft cross-links there are still the degrees of freedom offluctuations around the preferred directions of the cross-links left. Instead of approaching a combined spatial and orientational transition right at µ = 1 upon increasingthe polymers stiffness, it appears possible that for softcross-links the long range order transition will take placenot at µ = 1 but at a higher cross-link densitiy becausethe networks needs to be stabilized. However, in the limitof stiff rods and hard cross-links we should recover thecombined transition right at µ = 1.We now discuss the modifications of the free energydue to the soft cross-links: Keeping terms only up toorder Q as before, we find that in the case M > f or,soft := (34) µ Q ( ln (cid:16) ∞ X q =1 I q ( η ) I ( η ) I Mq ( γ ) I ( γ ) (cid:17) − µ Z s ,s ln (cid:16) ∞ X q =1 I q ( η ) I ( η ) I Mq ( γ ) I ( γ ) e − q M κ | s − s | (cid:17)) . The soft cross-links give rise to additional factors I Mq ( γ ) I ( γ ) that equal 1 in the limit of hard cross-links γ → ∞ .They appear linearly in the Gaussian and quadraticallyin the contribution of the log-trace term. Their rangelying between 0 and 1 it is clear that the log-trace contri-bution is smaller with respect to the Gaussian so that thetransition occurs at a higher cross-link concentration (ascompared to the case of hard cross-links). This is con-firmed by a numerical analysis of the Taylor expansionup to 10th order in η .The effect of soft cross-links on the orientational phasetransition is illustrated in Fig. 6 for M = 3: First of all,it shows that for finite γ the phase transition takes placeat a finite distance from µ = 1, even in the limit ofstiff rods. Moreover, comparing the curves correspond-ing to increasing values of γ , i.e. to harder and harderorientational cross-link constraints, the curves convergetowards the solid curve at the bottom that was drawn forperfectly hard cross-links as they should.The case M = 1 involves additional coupling termsbetween spatial and orientational parameters that needto be calculated. But as before, they don’t lead to abehavior that differs qualitatively from what we foundfor M > μ L/L P μ μ μ μ μ ∞ FIG. 6: Plot of the critical µ ( L/L p ) where the phase tran-sition takes place for M = 3 and stiffness parameters γ =80 , , ,
320 and hard cross-links.
In the preceding section we found that the higher M the more cross-links are needed to get into the long rangeordered phase. This holds true for soft cross-links, too,as we checked numerically. The related calculations arepresented in Appendices D and F. B. Statistically isotropic amorphous solid (SIAS)
In the preceding two sections, we found that (given asuitable cross-linking angle θ ) there is a phase boundary µ ( LL p ) above which long range orientational order be-comes possible. For lower cross-link density, long rangeorientational order vanishes. But the positional local-ization of the polymer segments that takes place in themacroscopic cluster is always accompanied by orienta-tional localization. The corresponding alternative to longrange order are glassy states, where the average orienta-tions of the polymer segments are frozen in random direc-tions, so that isotropy is restored on a macroscopic level.We expect such states for WLCs with a small persistencelength, such that the order induced by a cross-link can-not be sustained along the contour length up to the nextcross-link. This will be particularly severe, if M is large.Frozen orientations for polar filaments are describedby the distribution (18) for a single site. However the di-rection of localization varies from chain to chain, so thataveraging over the whole sample implies an average overthe locally preferred orientation ϕ assuming all direc-tions to be equally likely. For convenience we introducethe unit vectors u α = (cos ϕ α , sin ϕ α ) and denoting the local preferential axes by e we get Z d e π exp (cid:16) η e · ( n X α =1 u α ) (cid:17) = I (cid:16) η | n X α =1 u α | (cid:17) . (35)Altogether the order parameter for polar filaments in theglassy state then reads ω (ˆ k , ˇ ϕ ) = e − ξ ˆ k I n ( η ) I (cid:16) η | n X α =1 u α | (cid:17) . (36)What is the physical order parameter and how does itscale with the parameter η ?Locally, we have for each localized polymer segmenta polar moment h t i ( s ) i ∼ η = 0, but because of theorientational disorder it vanishes globally. We thus needan Edwards-Anderson like order parameter and choose q = 1 N N X i =1 Z s h t i ( s ) i · h t i ( s ) i . (37)As for the M -fold order parameter in the preceding sec-tion, we relate the variational parameter η to q and findthat in lowest order q = η O ( η ) . (38)We plug the Ansatz (36) for the replicated order pa-rameter into the saddle point free energy and keep onlyterms which depend on ξ or η . We obtain the simplestnon-trivial free energy by including terms up to secondorder in η . F n = µ Q ( − µ Q L ξ + µ L ξ g ( LL p ) (39) − η
16 Λ( γ ) (cid:18) − µ Λ( γ ) h ( 2 LL p ) (cid:19) + η L ξ µ Λ( γ ) l ( LL p ) ) where we have introduced the shorthand notationΛ( γ ) := I ( γ ) I ( γ ) . The functions g, h, l are given by (28)in the M-fold section. Minimizing the above free energywith respect to ξ , yields qualitatively the same result asfor the long range ordered state, namely L ξ = 23 µ Qg ( LL p ) . (40)The orientational part shows a behavior different fromthe M-fold case: The stationarity equation with respectto η gives rise to η = L ξ µ l ( LL p )1 − µ Λ( γ ) h ( LL p ) (41)The coupling terms implies a non-zero value η , i.e. ori-entational localization, as soon as positional localization1 µ Sol L/L p M−foldSIAS
FIG. 7: Phase diagram in the plane of cross-link concentra-tion µ and polymer flexibility L/L p . To the right of thevertical dotted line, SIAS order becomes stable, wheras M-fold order appears to the right of the continuous tilted linewhich depends on M. sets in. Hence glassy orientational order is enslaved topositional localization and the orientational order param-eter grows continuously at the gelation transition. Vary-ing γ we see that the softer the cross-links are, the smalleris also the variational parameter η , i.e. the degree of ori-entational order.Note that, in contrast to the M -fold ordered case, thelimit n → η instead of minimizingit as it is well known from spin glasses [27]. C. Phase diagram
The results of the previous sections can be summarizedin a phase diagram, presented in Fig. 7. The controlparameters are the cross-link density measured by µ ,the polymer flexibility measured by L/L p , and the cross-linking angle θ . Irrespective of the cross-linking angle,independent of the stiffness of the filaments and of thesoftness of the cross-links there is a continuous gelationtransition accompanied by random local orientationalordering (SIAS phase) at the critical cross-link density µ c = 1. This glassy ordering has been encountered pre-viously for randomly linked molecules with many legs( p -Beine) [20, 22]. The free energy of the SIAS is abovethe free energy of the sol, which however is unstable be-yond µ = 1 and hence, is not available in this region ofthe phase diagram.What is new, is the appearance of a state with longrange orientational order, if the crossing angle θ = kM π where k, M ∈ Z . This phase is characterized by a spon-taneous breaking of the rotational symmetry. For sensi-tive cross-links the symmetry of the orientational orderis M -fold, for unsensitive cross-links and odd M the re-sulting phase has 2 M -fold symmetry. The free energy ofthe long ranged ordered state is below the free energy of the isotropic sol as well as below the free energy of theSIAS. Hence, we expect it to win as soon as it appears,even though we cannot do a complete stability analysisbeyond the variational Ansatz.We find that the appearance of long range order ispushed to higher values of µ as the constraint for thecrossing angle is softened. The same effect is observedfor increasing flexibility of the polymers, – because itbecomes more difficult to sustain the orientation of thepolymers –, and increasing M .When reading the phase diagram, we should keep inmind that the parameters µ and L/L p are not thermo-dynamic ones like a temperature or a chemical potentialbecause changing either of them changes the disorder en-semble, too. This means that two points in the phasediagram correspond effectively to two different systems. V. CONCLUSIONS - DISCUSSION
In this paper, we studied the role of the cross-linkingangle in the formation of orientational order in randomnetworks of semiflexible polymers in two dimensions. Wehave used a variational Ansatz to map out a phase di-agram. Besides a statistically isotropic amorphous solid(SIAS) we find more exotic gels with random positionalorder coexisting with long ranged orientational order,whose symmetry is dictated by the crossing angle. Inanalogy to liquid crystals with long range orientationalorder and thermal centre of mass motion like in a fluid,these gel phases might be termed “glassy crystals”– withlong range orientational order and frozen in random po-sitions like in a glass. It is interesting to note that thetetratic ordering, which corresponds to M = 4 in oursystem, has been predicted and/or observed in a varietyof physical systems with quite different constituents andunderlying physical mechanisms [14, 30–32].Because of the peculiarities of two dimensions, we ex-pect fluctuations to affect positional localization [33, 34].It is known, in the context of 2d defect-mediated melt-ing, that at finite temperature positional order can onlybe quasi-long ranged whereas orientational order can betruely long ranged [35]. A study of the correspondingphenomena for our positionally amorphous and orienta-tionally ordered system is a very interesting direction forfurther investigation.We point out that the finite bending rigidity is an es-sential ingredient of our model. It allows the effectivedecoupling of positional and orientational degrees of free-dom close to the gelation transition. In the case of in-finitely stiff polymers on the other hand, a rigid cross-link would automatically fix both position and orienta-tion. The nature of the emerging network is an interest-ing problem which goes beyond the scope of this paper.Another possible extension of our work concerns moreelaborate variational Ans¨atze probing the appearance ofcombined M-fold and glassy orientational order. Here,the chains in the gel fraction are assumeed to be orien-2tationally localized in preferential directions which varyfrom chain to chain but macroscopically average in anM-fold pattern.The three-dimensional generalization of our model hasto deal with the fact that a finite cross-linking angle be-tween two wormlike chains prescribes a cone and not aplane. If we want to describe cross-links with torsionalrigidity, we need to go beyond the simple WLC modeland use the helical WLC [36].In this work, we focused on orientational and glassyorder in networks of semiflexible polymers mediatedthrough appropriate cross-links. We assume that theexcluded-volume interaction is such that it supports amacroscopically translationally and rotationally invari-ant liquid which, upon cross-linking, may give rise to or-dered networks. Although this assumption is mathemat-ically consistent and facilitates the analytical treatmentof our model, it may be challenged in experimental real-izations where the excluded volume may lead to lyotropicalignment in dense systems. In a future extension of ourmodel, one may envisage adding a Maier-Saupe aligningpseudopotential as in Refs. [6, 7] and studying its inter-play with the cross-link induced interaction. Acknowledgments
P.B. acknowledges support during the later part of thiswork by EPSRC via the University of Cambridge TCMProgramme Grant and the Project of Knowledge Inno-vation Program (PKIP) of the Chinese Academy of Sci-ences, Grant No. KJCX2.YW.W10.
Appendix A: Evaluating expectation values - theWLC propagator
The wormlike chain propagator G ( ϕ, s ; ϕ ′ , s ) quanti-fies the probability that the tangential vector of monomer s points into the direction ϕ provided that the tangen-tial vector of monomer s points into the direction ϕ ′ : G ( ϕ, s ; ϕ ′ , s ) (A.1):= D δ ( ϕ − ψ ( s )) δ ( ϕ ′ − ψ ( s )) E H WLC = 1 N Z s s D{ t } e − κ R s s d τ ( d t d τ ) Here, N denotes the normalization and ψ ( s ) the anglecorresponding to the unity vector t ( s ). In principle, thepath integral includes all the monomers from 0 to L , butthe monomers that lie not between s and s do not affectthe result and can be integrated out. In order to performthe remaining path integral we write down a discretizedversion of the above expression replacing the continuousdegrees of freedom by a finite number l of them suchthat t corresponds to t ( s ) and that t l corresponds to t ( s ). The distance ǫ between neighbors is determined by lǫ = | s − s | . Expressing the integral and the derivativesof the wormlike chain Hamiltonian by their discretizedversions and calling the normalization constant for thediscretized path integral N ǫ we arrive at [37] G ǫ = 1 N ǫ l − Y i =2 (cid:18)Z d ϕ i π (cid:19) exp − κ ǫ l − X i =1 (cid:18) t i − t i +1 ǫ (cid:19) ! = 1 N ǫ l − Y i =2 (cid:18)Z d ϕ i π (cid:19) exp − κǫ l − X i =1 (1 − cos( ϕ i − ϕ i +1 )) ! We want now to perform the integrations over the ϕ i . Forthat purpose it is convenient to decouple ϕ i and ϕ i +1 bymeans of e a cos ϕ = ∞ X q = −∞ I q ( a ) e iqϕ where the I q ( a ) denote modified Bessel functions. Per-forming the integrations we arrive at G ǫ = 1 N ǫ ∞ X q = −∞ (cid:16) e − κǫ I q ( κǫ ) (cid:17) l − e iq ( ϕ − ϕ l ) and integrating over ϕ π and ϕ l π we find that the nor-malization is given by N ǫ = (cid:0) exp( − κ/ǫ ) I ( κ/ǫ ) (cid:1) l − .Last, we take the limit ǫ → lǫ = | s − s | constant. It is thus possible to express the modifiedBessel functions by the asymptotic expansion I q ( a ) ∼ exp( a ) √ πa (cid:0) − q − a + . . . (cid:1) and the propagator converges to G ( ϕ, s ; ϕ ′ , s ) = ∞ X q = −∞ e − κ q | s − s | e iq ( ϕ − ϕ ′ ) (A.2)Calculating a general 2-point correlation function of the(real valued) observables O and O at positions s and s respectively by means of the above propagator we find D O ( ψ ( s )) O ( ψ ( s )) E H WLC (A.3)= ∞ X q = −∞ e − κ q | s − s | ˆ O ( q ) ˆ O ∗ ( q ) , where ˆ O and ˆ O denote the Fourier transformation ofthe observables. We learn from this expression that onlythe Fourier components to the same q couple to eachother and that the rate of the exponential decay of cor-relations along the filament scales with q . Appendix B: Mean-field replica free energy and thesaddle point equations
In order to obtain the disorder averaged free energy[ F ] by means of the replica method we need to calculate3the disorder averaged n -fold replicated partition function[ Z n ] ∝ ∞ X M =0 M Y e =1 N X i e ,j e =1 Z s e ,s ′ e X σ e M ! (cid:18) µ V N y (cid:19) M D M Y e =1 ( δ (ˆ r i e ( s e ) − ˆ r j e ( s ′ e )) × (B.1) × e γ P nα =1 cos( ψ αie ( s e ) − ψ αje ( s ′ e ) − θ σe ) I n ( γ ) )E H n +1 where r αi ( s ) and ψ αi ( s ) denote the position vector andangle of orientation of segment s belonging to polymer i inside the α th replica. For sensitive cross-links θ σ equalsalways the single crossing-angle θ , i.e. the normalization y = 1, and we can omit the summation over σ , but inthe unsensitive case θ σ takes the two values θ = θ and θ = θ + π and so, y = 2.Observing that the formula factorizes in the cross-linkindex e it is possible to perform the sum over the numberof cross-links M that leads to an exponential function[ Z n ] ∝ * exp µ V N N X i,j =1 Z s,s ′ δ (ˆ r i ( s ) − ˆ r j ( s ′ )) ×× ∆( ˇ ψ i ( s ) − ˇ ψ j ( s ′ ) , θ ) !+ H n +1 where for sensitive cross-links the function ∆ is definedas ∆ s ( ˇ ψ, θ ) ≡ e γ P nα =1 cos( ψ α − θ ) I n ( γ ) (B.2)and in the unsensitive case on the other hand as∆ u ( ˇ ψ, θ ) ≡ (cid:26) e γ P α cos( ψ α − θ ) I n ( γ ) + e γ P α cos( ψ α − ( θ + π )) I n ( γ ) (cid:27) . (B.3)After the disorder average, all the sites, i.e. all thepolymer segments, are equivalent but still appear explic-itly, coupled by the cross-linking constraint ∆ and thedelta functions. Expressing the delta functions in Fourierspace we can rewrite our formula in terms of the quantity Q (ˆ k , ˇ m ) = 1 N N X i =1 Z s e i ˆ k ˆ r i ( s ) e i ˇ m ˇ ψ i ( s ) Using this definition and writing H ev explicitly the repli-cated partition function reads[ Z n ] ∝ (cid:28) exp (cid:16) µ N V n X ˆ k , ˇ m ∆ ˇ m | Q (ˆ k , ˇ m ) | (cid:17) exp (cid:16) − λ N V n X α =0 X k =0 X m | ρ α ( k , m ) | (cid:29) H WLC n +1 . In section II the parameter λ was introduced in its mostgeneral form depending on | k | and m , but as it turnsout a constant is sufficient for our purpose and allows inthe following for a more compact notation. Because ofthe symmetry | Q (ˆ k , ˇ m ) | = | Q ( − ˆ k , − ˇ m ) | it is only thereal part of ∆ ˇ m that contributes and we thus redefine thesensitive kernel ∆ s accordingly as∆ s, ˇ m = Q nα =1 I m α ( γ ) I n ( γ ) cos (cid:16) X α m α θ (cid:17) . (B.4)For unsensitive cross-links ∆ u, ˇ m equals zero if the sum P nα =1 m α is not even, but takes otherwise the values ofthe sensitive kernel in (B.4).We are now going to rearrange the contributions ofintra-polymer repulsion and Deam-Edwards distributioninto contributions belonging to the following subsets ofthe space of ( n + 1)-fold replicated vectors ˆ k : • The 0-replica sector (0RS) consisting only of ˆ0 • The 1-replica sector (1RS) including all vectors ofthe form ˆ k = ( ~ , . . . , k α , . . . ,~
0) where k α = 0 • The higher-replica sector (HRS) containing all theˆ k where wave vectors in at least to replicas are non-zero, i.e. there are α = β ∈ { , , . . . , n } with k α =0 and k β = 0In the following we denote the sum over 0RS and HRSby P ˆ k and the sum over the 1RS as ˜ P k . For the 1RSwe obtain the new kernel˜∆ α ˇ m = λ N V n Y β = α =1 δ m β , − µ V n ∆ ˇ m (B.5)Using ˜ Q α ( k , ˇ m ) as shorthand notation for Q (ˆ k , ˇ m ) whenonly the wave vector k α in replica α is non-zero the ex-pression reads[ Z n ] ∝ (cid:28) exp (cid:16) − N n X α =0 ˜ X k , ˇ m ˜∆ α ˇ m | ˜ Q α ( k , ˇ m ) | (cid:17) exp (cid:16) µ N V n X ˆ k , ˇ m ∆ ˇ m | Q (ˆ k , ˇ m ) | (cid:17)(cid:29) H WLC n +1 . In order to decouple the sites we are now going to applya Hubbard-Stratonovich transformation. The symmetry Q (ˆ k , ˇ m ) = Q ∗ ( − ˆ k , − ˇ m ) and the analogous relation for˜ Q α ( k , ˇ m ) have to be reflected by their correspondingfields Ω(ˆ k , ˇ m ) and ˜Ω α ( k , ˇ m ). After the transformation[ Z n ] can be written as[ Z n ] ∝ Z D{ ˜Ω α , Ω } exp (cid:16) − N F ( { ˜Ω α , Ω } ) (cid:17) (B.6)where the integrals over the complex fields are meant tobe integrations over real and imaginary parts separately.4The replica free energy F is given by F = n X α =0 ˜ X k X ˇ m | ˜Ω α | + µ V n X ˆ k X ˇ m | Ω | (B.7) − ln * exp (cid:18) i n X α =0 ˜ X k X ˇ m p ˜∆ ℜ (cid:16) ˜Ω α Z s exp n − i k α r α ( s ) o exp n − i ˇ m ˇ ψ ( s ) o(cid:17) + µ V n X ˆ k X ˇ m √ ∆ ℜ (cid:16) Ω Z s exp n − i ˆ k ˆ r ( s ) o exp n − i ˇ m ˇ ψ ( s ) o(cid:17)(cid:19)+ H WLC n +1 Using the saddle point approximation we replace the inte-gral (B.6) by its maximal contribution. The correspond-ing values of the order parameter fields have to fulfill theself-consistency equations that arise from ∂ F ∂ ℜ ˜Ω α ( k , ˇ m ) = 0 , ∂ F ∂ ℑ ˜Ω α ( k , ˇ m ) = 0 ,∂ F ∂ ℜ Ω(ˆ k , ˇ m ) = 0 and ∂ F ∂ ℑ Ω(ˆ k , ˇ m ) = 0 . Note that there are many ways to perform the HS trans-formation, each of them leading to a different replicafree energy F . The resulting saddle-point replica freeenergy on the other hand is always the same as it canbe checked easily. As a consequence, we are free to rede-fine the fields for later convenience by doing the replace-ments ℜ Ω(ˆ k , ˇ m ) → √ ∆ ˇ m ℜ Ω(ˆ k , ˇ m ) and for the imagi-nary part accordingly. The advantage of this transfor-mation is that the saddle-point fields Ω SP (ˆ k , ˇ m ) are nowdirectly related to an expectation value of Q (ˆ k , ˇ m ) as pre-sented in the main part, equation (12). This connectionbetween Q and its corresponding field Ω is derived bymeans of an external field that couples to Q . Performingthen the Hubbard-Stratonovich transformation and tak-ing the logarithmic derivatives with respect to real andimaginary part of the field on both sides of the equation,i.e. for both the microscopic and the field theoretic rep-resentation of our theory, results in the desired relationbetween Q and Ω.The 0RS/HRS part of the free energy in terms of thenew fields reads then F = µ V n X ˆ k X ˇ m ∆ | Ω | (B.8) − ln * exp (cid:18) µ V n X ˆ k X ˇ m ∆ ℜ (cid:16) Ω Z s exp n − i ˆ k ˆ r ( s ) o exp n − i ˇ m ˇ ψ ( s ) o(cid:17)(cid:19)+ H WLC n +1 The 1RS part is treated in the next section.
Appendix C: Stability of the 1-replica sector
The fields of the 1-replica sector, ˜Ω α ( k , ˇ m ), can be sub-divided into the fields where ˇ m is such that only the en-try m α is non-vanishing and those corresponding to theother possible values of ˇ m . In the former case the fieldsdescribe M -fold orientationally symmetric density fluc-tuations with wave vector k in replica α and have thusa clear physical meaning. The other fields break replicasymmetry in the sense that they describe density fluc-tuations e.g. in replica α accompanied by purely orien-tational fluctuations in other replicas. These fields areunphysical and need thus to equal zero.In order to study the stability of the 1RS with respectto fluctuations in the physical fields ρ α ( k , m ) it is suffi-cient to study stability for one replica only. The expan-sion of the corresponding free energy reads up to secondorder F RS = ˜ X k X m | ˜ ρ ( k , m ) | (C.1)+2 ˜ X k , k X m ,m ˜ ρ ( k , m )˜ ρ ( k , m ) p ˜∆ m p ˜∆ m Z s ,s D e i ( k r + k r ) e i ( m ψ + m ψ ) E| {z } C ( k , k ; m ,m ) with r x := r ( s x ) and ψ x := ψ ( s x ). Choosing λ > µ thekernel ˜∆ m = λ − µ I m ( γ ) I ( γ ) cos( mθ ) is positive because I m ( γ ) I ( γ ) ≤
1. It is thus possible to redefine the fields byintroducing ρ ( k , m ) := √ ∆ m ˜ ρ ( k , m ) without changingthe stability. The corresponding free energy is given by F RS = ˜ X k X m λ − µ I m ( γ ) I ( γ ) cos( mθ ) | ρ ( k , m ) | (C.2)+2 ˜ X k , k X m ,m ρ ( k , m ) ρ ( k , m ) C ( k , k ; m , m )Let us now consider the two contributions to F RS sep-arately: It is evident that the first term is a positivequadratic form provided that λ is large enough. If weconsider stiff rods as the limiting case of very unflexi-ble wormlike chains the matrix of the second quadraticform can be diagonalized by switching back from theFourier modes m to the real space variables ϕ . Writ-ing r ( s ) = r + s t and integrating over r and t we findfor C Z s ,s D e i ( k r + k r ) δ ( ϕ − ψ ) δ ( ϕ − ψ ) E = δ k , − k δ ( ϕ − ϕ ) sin ( k t / k t / (C.3)and know thus that the second quadratic form is positivesemi-definite. Altogether we have thus proved stability5in the case of stiff rods, but we assume that this resultwill at least hold for wormlike chains with large L/L p ,too. Appendix D: Variational free energy: M-foldsymmetric long ranged order
In this Appendix, we present some details on thederivation of the variational free energy (27) of the M-fold orientationally ordered amorphous solid. We insertthe corresponding replica order parameter Ansatz (24)into the general replica free energy (23) and expand it inpowers the variational parameters ξ and η .
1. Gaussian part
The Gaussian part reads f G = µ Q V n X ˆ k δ P nα =0 k α , e − ξ ˆ k Z ˇ ϕ , ˇ ϕ ∆( ˇ ϕ , ˇ ϕ )1 I n ( η ) e η P nα =1 (cos( Mϕ α )+cos( Mϕ α )) . The spatial part is easily computed by replacing the sumover the replicated Fourier variables ˆ k by an integral andrepresenting the delta function in Fourier space. Per-forming the resulting integrations we arrive at f G = 1 n + 1 (cid:18) πξ (cid:19) n f . For the orientational contribution f o we find f o (D.1)= Z ˇ ϕ , ˇ ϕ ∆( ˇ ϕ − ˇ ϕ , θ ) e η P nα =1 (cos( Mϕ α )+cos( Mϕ α )) I n ( η ) ∼ n ln (cid:18)Z ϕ ,ϕ ∆( ϕ − ϕ , θ ) e η (cos( Mϕ )+cos( Mϕ )) I ( η ) (cid:19) = 1 + n ln (cid:16) ∞ X q =1 I q ( η ) I ( η ) I Mq ( γ ) I ( γ ) cos( M qθ ) (cid:17) . For θ = kM π the cosine equals 1 and vanishes. In orderto obtain the last line we replaced the three functions de-pending on the ϕ ’s by their Fourier representations. Per-forming then the integrations lead to Kronecker Deltasthat cancel two of the three sums over Fourier modes.Altogether we find for the Gaussian contribution inlinear order in the replica index nf G = µ Q n − − ln(4 πξ )+ln (cid:16) ∞ X q =1 I q ( η ) I ( η ) I Mq ( γ ) I ( γ ) (cid:17)o . Considering unsensitive crosslinker the correspond-ing expression reads f o = 1+ n ln (cid:16) ∞ X q =1 δ Mq, Z I q ( η ) I ( η ) I Mq ( γ ) I ( γ ) cos( M qθ ) (cid:17) , (D.2)i.e. for odd M the contributions corresponding to odd q are projected out. For M even on the other hand, weobtain the same result as for the sensitive cross-links.Note that for hard crosslinks ( γ = ∞ ) there is an al-ternative expression for the f o . For sensitive cross-linksit is given by f o (D.3) ∼ n ln (cid:18)Z ϕ ,ϕ ∆( ϕ − ϕ , θ ) e η (cos( Mϕ )+cos( Mϕ )) I ( η ) (cid:19) ∼ n ln (cid:18) I ( η ) Z ϕ e η (cos( M ( ϕ + θ ))+cos( Mϕ )) (cid:19) ∼ n ln (cid:18) I ( η ) Z ϕ e η (cos( Mϕ + Mθ )) cos( Mθ )) (cid:19) ∼ n ln I (2 η cos( Mθ )) I ( η ) ! and in the unsensitive case we have f o ∼ n ln I ( η ) 12 (cid:26) I (cid:0) η cos( M θ (cid:1) (D.4)+ I (cid:0) η cos( M ( θ + π )2 ) (cid:1)(cid:27)! .
2. Log-trace contributions
Let us now turn to the log-trace contribution that wewill expand up to the third order in the gel fraction Q : f lt = µ Qf lt, + µ Q (cid:16) f lt, − f lt, (cid:17) + µ Q (cid:16) f lt, + 2 f lt, − f lt, f lt, (cid:17) + O ( Q ) . f lt, (D.5)= * V n X ˆ k Z ˇ ϕ, ˇ ϕ ′ δ , P nα =0 k α e − ξ ˆ k ×× e γ P nα =1 cos( ϕ α − ϕ ′ α − θ ) I n ( γ ) e η P α cos( mϕ ′ α ) I n ( η ) ×× Z s e i ˆ k ˆ r ( s ) δ ( ˇ ϕ ′ − ˇ ψ ( s )) + = 1 V n Z s * Z ˇ ϕ e γ P nα =1 cos( ϕ α − ψ ( s ) α − θ ) I n ( γ ) e η P α cos( mϕ α ) I n ( η ) + = 1 V n . In the second step we use ˆ r ( s ) = ˆ r (0) + R s d τ ˆ t ( τ ) andtransform the original path integral D{ ˆ r ( s ) } into anintegral dˆ r (0) V n +1 and a path integral D{ ˆ ψ ( s ) } over angularvariables. Performing the ˆ r (0) integration we get aKronecker delta setting ˆ k to zero. The last line followsfrom the fact that the functions are normalized and thetwo integrations with respect to ˇ ϕ and ˇ ψ ( s ) simply leadto 1.The second-order contribution reads f lt, = 1 V n Z s ,s *X ˆ k X ˆ k δ , P nα =0 k α δ , P nα =0 k α e − ξ (cid:16) ˆ k + ˆ k (cid:17) / e i ( ˆ k ˆ r ( s )+ ˆ k ˆ r ( s ) ) (D.6) Z ˇ ϕ ˇ ϕ ′ Z ˇ ϕ ˇ ϕ ′ ∆ ( ˇ ϕ − ˇ ϕ ′ , θ ) ∆ ( ˇ ϕ − ˇ ϕ ′ , θ ) δ ( ˇ ϕ ′ − ˇ ψ ( s )) δ ( ˇ ϕ ′ − ˇ ψ ( s )) e η P nα =1 { cos( Mϕ α )+cos( Mϕ α ) } I n ( η ) + . Before Taylor expanding the expression in the variationalparameters, we need to perform the summations over ˆ k and ˆ k . We integrate over dˆ r (0) as before and get aKronecker Delta imposing ˆ k = − ˆ k . There is thus onlyone summation left. We replace this sum by an integraland represent the Kronecker delta in Fourier space. Afterthe two Gaussian integrations we find f lt, = 1 V n (cid:18) πξ (cid:19) n n + 1 Z s ,s (D.7) * e ξ ( n +1 P nαβ =0 f α f β − P nα =0 ( f α ) ) ×× Z ˇ∆ ϕ , ˇ∆ ϕ e γ P nα =1 (cid:0) cos(∆ ϕ α )+cos(∆ ϕ α ) (cid:1) I n ( γ ) ×× e η P nα =1 { cos( M ( ψ α +∆ ϕ α ))+cos( M ( ψ α +∆ ϕ α )) } I n ( η ) + using the shorthand notations f α ≡ R s s d s t α ( s ) and ψ αi := ψ α ( s i ). This expression can be expanded up tothe desired order and the remaining task then consists incalculating the corresponding correlation functions. De-tails are given in Appendix F.For the calculation of the third-order term, we proceedthe same way as before and need thus to perform threesums over wave vectors. With the abbreviation f αl ≡ R s l s d τ t α ( τ ) the result reads7 f lt, = 1 V n (cid:18) πξ (cid:19) n (cid:18) πξ (cid:19) n (cid:18) n + 1 (cid:19) Z s ,s ,s (D.8) * exp ( ξ (cid:18) n + 1 n X α,β =0 (cid:0) f α f β + f α f β + f α f β (cid:1) − n X α =0 (cid:0) ( f α ) + ( f α ) + f α f α (cid:1)(cid:19)) ×× Z ˇ∆ ϕ , ˇ∆ ϕ , ˇ∆ ϕ e γ P nα =1 (cid:0) cos(∆ ϕ α )+cos(∆ ϕ α )+cos(∆ ϕ α ) (cid:1) I n ( γ ) e η P nα =1 { cos( M ( ψ α +∆ ϕ α ))+cos( M ( ψ α +∆ ϕ α ))+cos( M ( ψ α +∆ ϕ α )) } I n ( η ) + . Appendix E: Variational free energy: SIAS1. Gaussian part
In the following, we present the derivation of the SIASvariational free energy. Let us first turn to the Gaussiancontribution. It is given by f g = µ Q V n X ˆ k δ , P nα =0 k α e − ξ ˆ k × (E.1) × Z ˇ ϕ, ˇ ϕ ′ ∆( ˇ ϕ − ˇ ϕ ′ , θ ) I ( η | P nα =1 u α | )I ( η | P nα =1 u ′ α | )I n ( η )where u α = (cos ϕ α , sin ϕ α ). The spatial part is treatedthe same way as for the M-fold case, but the orienta-tional contribution is slightly more complicated becauseit does not factorize in the replica index. By means ofthe integral representation of the Bessel function, I ( η ) = 12 π Z π d ϑ e η cos ϑ , (E.2)we get a factorizing expression and find in linear order in n for the orientational part of the free energy f o = 1 + n Z ϑ ,ϑ ln (cid:16) Z ϕ ,ϕ ∆( ϕ − ϕ , θ ) × (E.3) × I ( η ) e η (cos( ϕ + ϑ )+cos( ϕ + ϑ ) (cid:17) = 1 + n Z ϑ ln X q ∈ N I q ( η ) I ( η ) I q ( γ ) I ( γ ) cos( ϑ ) ! . In order to obtain the last line, we simply switch fromthe angular variables to Fourier space. The expression isthen easily expanded up to the desired order.
2. Log-trace contributions
The first-order contribution of the log-trace part givesthe same (trivial) result as for the long range orderedcase. As for the second and third order term, we again have to perform the ˆ k summations first, but this calcu-lation is done exactly the same way as before.The corre-sponding results are obtained from those of the M-foldcase by simply replacing the M-fold orientational distri-butions in (D.7) and (D.8) by the corresponding SIASdistributions.Details on the calculation of the individual terms of thesecond order log-trace contribution are given in AppendixG. Appendix F: Evaluation of the expectation values:M-fold
In our expansion we include all the terms up to firstorder in ξ and up to sixth order in η .
1. Spatial part
Let us consider the calculation of the ξ -term: Z s ,s n + 1 n X α,β =0 (cid:10) f α f β (cid:11) − n X α =0 (cid:10) ( f α ) (cid:11) = − n Z s ,s Z s s d τ Z s s d τ ′ h t ( τ ) t ( τ ′ ) i = − n Z s ,s Z s s d τ Z s s d τ ′ e − κ | τ − τ ′ | =: − nL g ( LL p ) (F.1)The first expectation value gives a contribution for α = β only. Combining the two terms and noticing that eachreplica gives the same contribution we obtain the prefac-tor of − n . It is interesting to note that the function g isclosely related to the radius of gyration R g . R g := 12 Z s ,s D ( r ( s ) − r ( s )) E (F.2)= 12 Z s ,s Z s s d τ d τ ′ h t ( τ ) t ( τ ′ ) i = L g ( LL p )8
2. Coupling terms
In order to calculate the lowest order coupling termswe expand the spatial part in first order. For θ = kM π we find14 ξ *n n + 1 n X α = β =0 f α f β − nn + 1 n X α =0 ( f α ) o × (F.3) × Z ˇ∆ ϕ , ˇ∆ ϕ e γ P nα =1 (cos ∆ ϕ α +cos ∆ ϕ α ) I n ( γ ) ×× e η P nγ =1 { cos( M ( ψ γ +∆ ϕ γ ))+cos( M ( ψ γ +∆ ϕ γ )) } I n ( η ) + . We can omit the second term in the first line of the aboveequation because it is already proportional to n and willin the end give rise to contributions proportional to n .Writing for the moment only the terms that are directlyinvolved in the thermal expectation value, we have n X α = β =1 *n cos ψ ατ cos ψ βτ ′ + sin ψ ατ sin ψ βτ ′ o ×× e η P nγ =1 { cos( M ( ψ γ +∆ ϕ γ ))+cos( M ( ψ γ +∆ ϕ γ )) } + = − n δ M, ( D cos ψ τ e η ( c + c ) E D cos ψ τ ′ e η ( c + c ) E| {z } ( ∗ ) + D sin ψ τ e η ( c + c ) E D sin ψ τ ′ e η ( c + c ) E ) × n Y γ =3 D e η ( c γ + c γ ) E| {z } ( ∗∗ ) (F.4)where we used the abbreviations c αi := cos( ψ α ( s i ) +∆ ϕ αi ). This expectation value is only non-vanishing for M = 1 because of the relation (A.3): the Fourier trans-formed of cos( ψ τ ) has contributions for the modes q = ±
1, but the Fourier transformed of exp( η cos( M ψ ( s ))only for q = ± Z M . The prefactor of − n ∼ n ( n − α and β from the spatial contribution with the γ ’s fromthe orientational contribution.We need thus to compute three types of expectation value. We find1 I ( γ ) Z ∆ ϕ , ∆ ϕ e γ cos ∆ ϕ e γ cos ∆ ϕ × (F.5) × D cos( ψ τ ) e η { cos( ψ +∆ ϕ )+cos( ψ +∆ ϕ ) } E = X q ∈ N I q ( η ) I q +1 ( η ) I q ( γ ) I q +1 ( γ ) I ( γ ) ×× n e − κ ( q | s − τ | +( q +1) | s − τ | ) +e − κ (( q +1) | s − τ | + q | s − τ | ) o , D sin( ψ τ ) e η { cos( ψ +∆ ϕ )+cos( ψ +∆ ϕ ) } E = 0 (F.6)and 1 I ( γ ) Z ∆ ϕ , ∆ ϕ e γ cos ∆ ϕ e γ cos ∆ ϕ × (F.7) × D e η { cos( ψ ( s )+∆ ϕ )+cos( ψ ( s )+∆ ϕ ) } E = I ( η ) + 2 X q ∈ N I q ( η ) I q ( γ ) I ( γ ) e − κ q | s − s | . From (F.4) we obtain the coefficients of the couplingterms proportional to η ξ , η ξ and η ξ by sampling all theways to collect the corresponding power in η from thedifferent factors. For e.g. η ξ the only way to collect anfactor η is to expand the two averages in ( ∗ ) in firstorder and leave out ( ∗∗ ). The result is − µ ξ ( η I q I l ( α ) (F.8)+ η − I I l ( α ) + I I I l ( α ) − I I l ( α ) ! + η I I l ( α ) − I I I l ( α )+ 116 I I I l ( α ) + 124 I I I I l ( α ) − I I I l ( α ) − I I I l ( α ) + 8 I I l ( α ) !) The appearing correlation functions are defined as fol-9lows: l ( x ) := 7 − − x + e − x − x + 2 x x (F.9) l ( x ) := − − − x + 2e − x + 18e − x + 6 x x (F.10) l ( x ) := 1600 x (cid:16) − − − x + 400e − x (F.11)+16e − x − − x + 180 x (cid:17) l ( x ) := 181000 x (cid:16) − − − x (F.12)+2025e − x + 324e − x − − x + 1260 x (cid:17) l ( x ) := 17200 x (cid:16) −
297 + 400e − x − − x (F.13)+25e − x + 360 x (cid:17) l ( x ) := 11800 x (cid:16) −
37 + 100e − x − − x (F.14)+225e − x + 60 x (cid:17) l ( x ) := 11350 x (cid:16) − − − x + 225e − x (F.15)+25e − x − − x + 120 x (cid:17) In the case of hard cross-links it is convenient to intro-duce the following functions:˜ l ( x ) := 16 (cid:16) l ( x ) + 4 l ( x ) − l ( x ) (cid:17) (F.16)and ˜˜ l ( x ) := (cid:16) l ( x ) + 128 l ( x ) − l ( x ) (F.17)+ 23 l ( x ) + l ( x ) − l ( x ) − l ( x ) (cid:17)
3. Orientational contributions
As for the calculation of the purely orientational partwe notice first that it is factorizing in the replica index α . It is thus possible to take the replica limit directly bymeans of the expansion h . . . i n = 1 + n ln h . . . i + O ( n )and we have f or = 1 + n Z s ,s ln D Z ∆ ϕ , ∆ ϕ e γ (cos ∆ ϕ +cos ∆ ϕ ) I ( γ ) × (F.18) × e η (cos( M ( ψ +∆ ϕ ))+cos( M ( ψ +∆ ϕ )) I ( η ) E! The thermal average can be performed using formula(A.3). Keeping only the contribution linear in the replica index we obtain f or = Z s ,s ln D Z ∆ ϕ , ∆ ϕ e γ (cos ∆ ϕ +cos ∆ ϕ ) I ( γ ) × (F.19) × X q ∈ Z e − q κ | s − s | I q ( η ) I ( η ) e iq (∆ ϕ − ∆ ϕ ) E! . The ∆ ϕ -integrations lead to a Fourier transformation ofthe soft cross-link contributions and noticing that theresulting expression is symmetric in q we find f or = Z s ,s ln ∞ X q =1 I q ( η ) I ( η ) I Mq ( γ ) I ( γ ) e − q M κ | s − s | ! (F.20)Expanding the orientational free energy for hard cross-links in η and performing the s - and s -integrations itis convenient to introduce the function h ( x ) h ( LL p ) ≡ Z s ,s e − κ | s − s | (F.21)and to build of h ( x ) the functions˜ h ( x ) ≡ h ( x ) + h (2 x ) − h (4 x ) (F.22)and ˜˜ h ( x ) ≡ h ( x ) + 3 h (2 x ) + h (3 x ) (F.23) − h (4 x ) − h (5 x ) + 148 h (9 x ) . Appendix G: Evaluation of the expectation values:SIAS
The calculation of the expectation values in the sec-ond order contributions of the SIAS log-trace part canbe done along the same lines as above using again theintegral representation of the Bessel function (E.2). Thecalculation of the lowest order spatial contributions is thesame as for the M-fold case. For the purely orienta-tional we obtain f lt ,or = Z s ,s Z ϑ ,ϑ ln (cid:16) ∞ X q =1 I q ( η ) I ( η ) I q ( γ ) I ( γ ) × (G.1) × e − q κ | s − s | cos( q ( ϑ − ϑ )) (cid:17)) . coupling term is given by14 ξ * n + 1 n X α = β =0 f α f β − nn + 1 n X α =0 ( f α ) × (G.2) × Z ϑ ,ϑ Z ˇ∆ ϕ , ˇ∆ ϕ e γ P nα =1 (cos ∆ ϕ α +cos ∆ ϕ α ) I n ( γ ) ×× η n X γ,δ =1 (cid:16) cos( ψ γ − ϑ +∆ ϕ γ )+cos( ψ γ − ϑ +∆ ϕ γ ) (cid:17) × (cid:16) cos( ψ δ − ϑ + ∆ ϕ δ ) + cos( ψ δ − ϑ + ∆ ϕ δ ) (cid:17)+ As in (F.3) the second term in the first line of the ex-pression is already proportional to n and will thus finallylead to a contribution of O ( n ) that is not of interest forus. After integrating with respect to the ϑ -variables andthen with respect to the ϕ -variables we obtain in the end − n η ξ I ( γ ) I ( γ ) l ( α ) . (G.3)The results for the SIAS can be expressed in terms of thefunctions g , h and l that we already introduced in thepreceding section. [1] J. Howard, Mechanics of Motor Proteins and the Cy-toskeleton (Sinauer Associates, Sunderland, MA, 2001).[2] S. J. Winder and K. R. Ayscough, J. Cell Sci. , 651(2005).[3] P. A. Janmey, Proc. Natl. Acad. Sci. USA , 14745(2001).[4] A. R. Bausch and K. Kroy, Nature Phys. , 231 (2006),and references therein.[5] A. R. Khokhlov and A. N. Semenov, Physica , 605(1982).[6] M. Warner, J. M. F. Gunn, and A. B. Baumgartner, J.Phys. A , 3007 (1985).[7] A. J. Spakowitz and Z.-G. Wang, J. Chem. Phys. ,13113 (2003).[8] W.-J. Ma and C.-K. Hu, J. Phys. Soc. Jpn. , 054001(2010).[9] J. Viamontes and J. X. Tang, Phys. Rev. E. ,040701(R) (2003).[10] J. Viamontes, S. Narayanan, A. R. Sandy, and J. X.Tang, Phys. Rev. E. , 061901 (2006).[11] O. Lieleg, M. M. A. E. Claessens, and A. R. Bausch, SoftMatter , 218 (2010).[12] I. Borukhov, R. F. Bruinsma, W. M. Gelbart, and A. J.Liu, Proc. Natl. Acad. Sci. USA , 3673 (2005).[13] I. Borukhov and R. F. Bruinsma, Phys. Rev. Lett. ,158101 (2001).[14] R. F. Bruinsma, Phys. Rev. E , 061705 (2001).[15] A. G. Zilman and A. S. Safran, Europhys. Lett. , 139(2003).[16] P. Benetatos and A. Zippelius, Phys. Rev. Lett. ,198301 (2007).[17] G. C. L. Wong, A. Lin, J. X. Tang, Y. Li, P. A. Janmey,and C. R. Safinya, Phys. Rev. Lett. , 075501 (2003).[18] N. Saitˆo, K. Takahashi, and Y. Yunoki, J. Phys. Soc.Jpn. , 219 (1967).[19] R. T. Deam and S. F. Edwards, Proc. Trans. R. Soc. Lon.Ser. A , 317 (1976).[20] P. M. Goldbart and A. Zippelius, Europhys. Lett , 599 (1994).[21] P. M. Goldbart, H. E. Castillo, and A. Zippelius, Adv.Phys. , 393 (1996).[22] O. Theissen, A. Zippelius, and P. M. Goldbart, Int. J.Mod. Phys. B , 1945 (1996).[23] S. Ulrich, X. Mao, P. M. Goldbart, and A. Zippelius,Europhys. Lett. , 677 (2006).[24] S. Ulrich, A. Zippelius, and P. Benetatos, Phys. Rev. E , 021802 (2010).[25] S. V. Panyukov and Y. Rabin, Phys. Rep. , 1 (1996).[26] S. V. Panyukov and Y. Rabin, in Theoretical and Math-ematical Methods in Polymer Research , edited by A. Y.Grosberg (Academic, 1998).[27] M. M´ezard, G. Parisi, and M. A. Virasoro,
Spin GlassTheory and Beyond (World Scientific, Singapore, 1987).[28] K. A. Shaknovich and P. M. Goldbart, Phys. Rev. B ,3862 (1999).[29] P. M. Goldbart, in Rigidity Theory and Applications ,edited by M. F. Thorpe and P. M. Duxbury (SpringerUS, 2002), Fundamental Materials Research, pp. 95–124.[30] A. Donev, J. Burton, F. H. Stillinger, and S. Torquato,Phys. Rev. B , 54109 (2006).[31] K. Zhao, C. Harrison, D. Huse, W. B. Russel, and P. M.Chaikin, Phys. Rev. E , 40401(R) (2007).[32] V. Narayan, N. Menon, and S. Ramaswamy, J. Stat.Mech. P01005 , 1742 (2006).[33] N. D. Mermin and H. Wagner, Phys. Rev. Lett. , 1133(1966).[34] P. M. Goldbart, S. Mukhopadhyay, and A. Zippelius,Phys. Rev. B , 184201 (2004).[35] D. R. Nelson, Defects and Geometry in Condensed MatterPhysics (Cambridge University Press, Cambridge, 2002).[36] H. Yamakawa,
Helical Wormlike Chains in Polymer So-lutions (Springer-Verlag, New York, 1997).[37] H. Kleinert,