# Statistical physics of isotropic-genesis nematic elastomers: I. Structure and correlations at high temperatures

## Abstract

Isotropic-genesis nematic elastomers (IGNEs) are liquid crystalline polymers (LCPs) that have been randomly, permanently cross-linked in the high–temperature state so as to form an equilibrium random solid. Thus, instead of being free to diffuse throughout the entire volume, as they would be in the liquid state, the constituent LCPs in an IGNE are mobile only over a finite, segment specific, length–scale controlled by the density of cross-links. We address the effects that such network–induced localization have on the liquid–crystalline characteristics of an IGNE, as probed via 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 the quenched disorder associated with the cross-link constraints. To study IGNEs, we consider a microscopic model of dimer nematogens in which the dimers interact via orientation-dependent excluded volume forces. The dimers are, furthermore, randomly, permanently cross-linked via short Hookean springs, the statistics of which we model by means of a Deam-Edwards type of distribution. We show that at length–scales larger than the size of the nematogens this approach leads to a recently proposed, phenomenological Landau theory of IGNEs [Lu et al., *Phys. Rev. Lett.* **108**, 257803 (2012)], and hence predicts a regime of short–ranged oscillatory spatial correlations in the nematic alignment, of both thermal and glassy types. In addition, we consider two alternative microscopic models of IGNEs: (i) a wormlike chain model of IGNEs that are formed via the cross-linking of side-chain LCPs; and (ii) a jointed chain model of IGNEs that are formed via the cross-linking of main-chain LCPs. At large length–scales, both of these models give rise to liquid–crystalline characteristics that are qualitatively in line with those predicted on the basis of the dimer-and-springs model, reflecting the fact that the three models inhabit a common universality class.

## 1Introduction

### 1.1Nematic elastomeric materials

Nematic elastomers are fascinating materials in which the liquid crystalline order is strongly coupled to the elasticity of the underlying elastomeric network (see, e.g., Refs. [1]). This strong nemato-elastic coupling gives rise to novel, *emergent* properties in nematic elastomers that are found neither in liquid crystal nematics nor ordinary rubbery materials. One well-known example of such properties is the soft elasticity characteristic of monodomain nematic elastomers (see, e.g., Ref. [7]) that are formed via the Küpfer-Finkelmann procedure [8].

Not only are nematic elastomers fascinating but also they have proven to be a challenging subject for theoretical investigation. Part of the challenge originates in the dependence of the physical characteristics of nematic elastomers on the conditions under which the elastomers are prepared. For example, *isotropic-genesis nematic elastomers* (or IGNEs)—nematic elastomers cross-linked in the isotropic state—exhibit the so-called supersoft version of elastic response at sufficiently low temperatures (see, e.g., Refs. [9]), unlike their nematic-genesis nematic elastomer (or NGNE) counterparts. Furthermore, *in thermal equilibrium* the nematic alignment in IGNEs exhibits a polydomain structure (see, e.g., Refs. [16]) characterized by a length-scale determined solely via thermodynamic quantities such as the density of cross-links.

Another challenge to theorists comes from the fact that nematic elastomers possess a multi-level hierarchy of interdependent elements of randomness. First, there is quenched disorder in its conventional form, associated with the permanent chemical structure that originates in the cross-linking process. Second, as a result of sufficient cross-linking there arise the mean positions and r.m.s. displacements of the spatially localized polymers that constitute the elastomeric network, both of these elements being random. Third, there is the thermal disorder associated with the Brownian motion of the positional and orientational (i.e., nematogenic) freedoms in the state of the system *just prior to the instant of cross-linking*; in part, this thermal disorder is frozen in via the process of cross-linking. Fourth, there is also the thermal disorder associated with the Brownian motion of the nematogens in the state of the system long after the instant of cross-linking. It is not *a priori* evident how the interplay between the various types of randomness present in nematic elastomers resolve themselves, and thus determine the equilibrium structure and elastic response of such media. As we shall try to make clear in subsequent sections of the present Paper, in order to understand nematic elastomers it is valuable to go beyond the conventional notion of quenched disorder and, instead, to consider an amalgam of the second, third and fourth 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 of the network constituents).

### 1.2Overview

In the present Paper we investigate the static structure of nematic alignment in IGNEs that can be probed in the high-temperature regime. This investigation is partly motivated by the experiments on polydomain structure reported in Ref. [9]. We focus our considerations here on systems in the high-temperature regime and which are not subject to externally applied deformations.

Two types of nematic fluctuations are present in IGNEs: (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 local alignment pattern that is frozen in. In order to characterize such fluctuations, we make use of (i) the correlator of the local nematic order that was frozen in during the cross-linking process, this correlator being appropriately averaged over realizations of the quenched disorder (and termed the glassy correlator); and (ii) the correlator of the thermally driven departures of the nematic order from the mean frozen-in local alignment pattern, appropriately averaged over realizations of the quenched disorder (and termed the thermal correlator).

We have already mentioned that the physical properties of IGNEs depend on the conditions under which they were prepared. To reflect this fact, we make a careful distinction between two thermodynamic ensembles in our theoretical approach: the first, in which the IGNE was *prepared*, which we term the *preparation ensemble*; and the second, in which the system is *measured*, which we call the *measurement ensemble*. This structure enables our theory to capture the ability of IGNEs to remember, at least to some degree, the local nematic alignment pattern at the moment of preparation, and in addition enables the determination of the dependence of the strength of this memorization on (i) the temperature at which the system was cross-linked, and (ii) the average number of cross-links per polymer.

We note in passing that having both preparation and measurement ensembles places us in a family of disordered systems for which spontaneous replica symmetry breaking is expected to be irrelevant, in contrast with settings that do not feature a preparation ensemble; cf. Ref. [20]. (Technically, this expectation shows up in the need to investigate coupled replicas of the physical system in the neighborhood of not zero but one replica.) Thus, we do not expect our approach to yield glassy phenomena such as hysteresis in the stress-strain behavior of IGNEs—which is not unreasonable, given the absence, to date, of experimental observations of such phenomena.

The fact that the elastomer network is thermally fluctuating means that any prospective theory of IGNEs should feature a typical localization length-scale, below which the polymer constituents of the network are effectively delocalized. Our approach features such a length-scale, which leads to the possibility that nematic correlations undergo a novel, *oscillatory* form of decay with distance in a certain regime. Prior theoretical approaches built on conventional random-field models do not feature the thermal fluctuations of the elastomer network (see, e.g., Refs. [13]), and thus do not capture this intriguing phenomenon. A phenomenological random-field-type model that *does* take the thermal fluctuations of the elastomer network into account was presented in Ref. [22]. In the present Paper we derive the Landau-type free energy for IGNEs that was presented in Ref. [22], doing so via a microscopic model that involves dimers that are randomly and permanently connected by Hookean springs. The present Paper thus provides a microscopic justification for the ideas and results presented in Ref. [22].

The outline of the Paper is as follows. In Section 2 we present a microscopic dimer-spring model of IGNEs. In Section 3 we apply the replica technique and implement the Hubbard-Stratonovich scheme to decouple the interacting microscopic degrees of freedom. In Section 4 we derive a Landau-Wilson type of free energy for the IGNE, which involves an order parameter field for the isotropic-to-nematic phase transition as well as an order parameter field for the vulcanization/gelation transition. In Sec. ? we make an expansion of the Landau-Wilson free energy for small and , which is appropriate for exploring the physics in the vicinity of the transition to the random solid state. In Section 5 we determine the stationary states of the Landau-Wilson free energy. In Section 6 we derive an effective replica Hamiltonian describing local nematic order in IGNEs by setting to its stationary value but retaining fluctuations of to quadratic order. In Section 7 we then compare this effective Hamiltonian with that arising from the phenomenological Landau free energy considered in Ref. [22] and show that they are equivalent. In Section 8 we use the effective Hamiltonian to derive the glassy and thermal correlators. In Section 9 we describe two alternative microscopic models of the IGNE, viz. (i) a worm-like chain model of side-chain nematic polymer networks; and (ii) a jointed chain model of main-chain nematic polymer networks. As we shall see, at length-scales larger than the size of a nematogen, both of these models give rise to liquid–crystalline characteristics similar to those resulting from the dimer-and-springs model, reflecting the fact that the three models inhabit a common universality class. In Section 10 we make some concluding remarks.

## 2Ingredients of the model

We model an IGNE microscopically as a system of dimers in spatial dimensions that are randomly, permanently linked via springs (see Figure 1 and Ref. [23]). We envision the springs as mimicking the flexible constituents of liquid crystalline polymers whilst also serving as cross-links; the dimers mimic the stiff constituents of liquid crystalline polymers. Each dimer (labeled by , where ) consists of two particles at position vectors and separated by a fixed distance . The orientation of the -th dimer is specified by the unit vector

The dimers interact via three types of forces. First, there is an *orientational* interaction between dimers that promotes parallel or antiparallel alignment. We model this interaction via a potential of the Maier-Saupe type, viz.,

where is the position of the -th dimer’s center of mass. We assume that the aligning interaction is short-ranged, and model the interaction potential by the form . In Fourier space, the potential is given by , where specifies the range of the interaction between dimers and characterizes its strength. In addition to the orientational interaction, the dimers experience a *positional* excluded-volume interaction between particles belonging to any pair of dimers, which we model via an Edwards-type pseudo-potential [24]:

where is the strength of the excluded-volume interaction [26]. The presence of sufficiently strong excluded-volume forces stabilizes the system against collapse to a globule, even when well cross-linked. Third, any given two dimers that are connected by a spring are taken to interact additionally via a harmonic potential associated with the spring, which we take to have zero rest-length and native mean-square end separation , characteristic of Gaussian molecular chains. These springs and, specifically, the architectural information indicating which pairs of rod ends are connected to one another by springs constitutes the quenched randomness of any given realization of the system. This information takes the form , where is the total number of springs and spring connects end of rod to end of rod . These springs result in the following Hookean term in the Hamiltonian:

where denotes the temperature and we have adopted units in which Boltzmann’s constant is unity.

The total Hamiltonian for the dimers-and-springs model is then given by

For a given realization of the quenched disorder, the corresponding partition function takes the form

and the free energy is given by . As is well known [27], it is appropriate to average the free energy of the system over realizations of the quenched disorder. Denoting this average by the square brackets , one has

## 3Replicas and collective fields

### 3.1Statistics of quenched disorder

What statistical distribution should one use to compute the average of the free energy over the quenched disorder? In common with other elastomers such as isotropic rubbery systems, the quenched-disorder average for IGNEs can be performed via a variant of the Deam-Edwards distribution [28]. Such distributions reflect situations in which systems undergo *instantaneous* cross-linking: one begins with a melt or solution at equilibrium and—so rapidly that hardly any relaxation has time to occur—one introduces permanent bonds between some random fraction of the pairs of dimers that happen, at the instant of cross-linking, to be nearby one another. To construct the associated , one should respect the causal order of the cross-linking process (see, e.g. Ref. [30]). Formally, this amounts to the specification

where the multiple integral is defined via

the distribution describes the statistics of the liquid system at the instant of cross-linking, so that

and is the conditional probability that cross-links are formed between pairs of dimer-ends at positions , given that the constituents of the liquid at the instant of cross-linking are at . It is given by [30]

Here, is the dimensionless volume of the system, i.e., . The term arises from the requirement that be properly normalized over , i.e., ; it is given by

The probability features a dimensionless parameter that controls the likelihood that cross-links are actually formed. In App. ? we show that it is related to the average number of cross-linking springs per dimer via the formula .

Here, labels the replicas, and is defined via

Note that in Eq. ( ?) the term limits permutation invariance to replicas , with replica being excluded. This is consistent with our physical expectation that the preparation and measurement ensembles play distinct roles.

Next, we complete the computation of the disorder average in Eq. ( ?), which yields in terms of the replica partition function , i.e.,

in which , where the effective replica Hamiltonian is given by

### 3.2Collective fields and their physical meaning

We note that in , Eq. ( ?), the interacting (replicated) dimers are coupled with one another, which means that the trace over microscopic variables in [Eq. (Equation 8)] cannot be straightforwardly carried out. We decouple these dimers by defining microscopic collective fields and , and making a Hubbard-Stratonovich transformation involving fluctuating auxiliary fields conjugate to the collective fields. Thus, we arrive at a description in terms of uncoupled copies of a single replicated dimer, for which the trace over can be readily performed, order by order in an expansion in the auxiliary fields. The collective fields are given by

Here, we have used the symbol to denote the -fold replicated wave-vector , and have restricted the value of in to the higher-replica sector (HRS), viz., the set of replicated vectors each having at least two non-zero vector entries. Details of the Hubbard-Stratonovich transformation are given in App. ?. The result is that the replica partition function becomes a functional integral over the auxiliary fields and , whose expectation values are related to the expectation values of and via

as we demonstrate in App. ?. Note that indicate an expectation value taken with respect to the Landau-Wilson free energy discussed in Section 4 and defined in Eq. (Equation 9), and denotes an expectation value taken with respect to the Hamiltonian , Eq. (Equation 2).

Thus, except for a trivial constant, we can interpret as the joint probability that a given dimer end is found at position at the instant of cross-linking, and that the same dimer end would be found at subsequent widely-separated time instants at the positions [23]. vanishes if all dimers are delocalized, and has a nonzero value if a fraction of them are localized. Thus, serves as the order parameter that detects the phase transition from the liquid state to the random solid state. Similarly, is the nematic order parameter for the preparation state, whilst (for ) is the nematic order parameter for the measurement state.

We note that when defining we have chosen to exclude the components associated with the *one-replica sector* (denoted 1RS), i.e., the set of replicated wave-vectors that each have only one non-zero vector entry [i.e., ]. This is because this sector of the field corresponds to fluctuations in the macroscopic density of dimers, and is strongly stabilized as a result of the excluded volume interactions, regardless of the extent of the cross-linking. The physical content of the decomposition of fields into higher and lower replica- sectors is that if condensation occurs in the higher sector only, this implies the random localization of particles. On the other hand, if condensation occurs in the 1RS as well, this indicates the formation of a state having some kind of spatially modulated density structure. We are focusing on highly incompressible systems, for which density fluctuations are negligibly small. Incompressibility is enforced by taking to have a large value, so that fluctuations of 1RS counterpart to 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-replica sector* (denoted LRS) would then refer to the union of the one-replica sector and the zero-replica sector.

### 3.3Intermezzo on replicas

Compared with the more familiar example of spin glasses (see, e.g., Ref. [31]), the effective replica Hamiltonian ( ?) in our theory contains an extra replica. What meaning can we ascribe to this zeroth replica as well as to the other replicas? Physically, the zeroth replica, which originates in the Deam-Edwards distribution, corresponds to the state of the system at the instant of preparation, whilst the other replicas correspond to the state of the system when it is measured. The dependence of the measured properties on the state of the system at preparation corresponds, operationally, to the coupling in the effective replica Hamiltonian ( ?) between the freedoms belonging to the zeroth replica and those belonging to the other replicas. On the other hand, such a coupling does not imply that the measured properties influence the preparation state, as one can show via a careful consideration of the limit [30].

Because the replica framework accounts for both the preparation and measurement ensembles vis-à-vis the zeroth and other replicas, it is well adapted to investigations of nematic elastomers, whose measured properties are known to depend on their preparation histories; see, e.g., Ref. [9]. Such a dependence was already understood by Deam and Edwards and by de Gennes more than thirty years ago (see, e.g., Refs. [28]).

## 4Landau-Wilson free energy

As we demonstrate in App. ?, the local incompressibility of IGNEs allows us to express the effective replica theory in terms of the auxiliary fields and as

where the Landau-Wilson free energy per dimer is given by

We have introduced the notation

and have specialized to three spatial dimensions (i.e., ). The curly braces in Eq. (Equation 9) denote the trace of the product of the tensors and , i.e., . The symbols , and are defined via

Moreover, denotes the sum over replicated wave-vectors that are restricted to the higher-replica sector.

## 5Saddle-point approximation

Next, we consider the saddle-point equations for and , which follow from the stationarity of Eq. ( ?):

As we are concerned with macroscopically isotropic states, we require that the saddle-point value of vanishes: . As a result, the saddle-point equation ( ?) for is the same as it would be for ordinary (i.e., non-nematogenic) elastomers, and reads

in which we have defined the length-scale via . Next, by defining the quantities and , the saddle-point equation takes the form analyzed in Ref [37]:

and is therefore solved by mean of the the Ansatz

(with ), provided that the gel fraction obeys

and , which has the meaning of the distribution of inverse square localization lengths , obeys

The typical value of the localization length is , which diverges as the vulcanization transition is approached from the solid side. The saddle-point equation for , viz., Eq. ( ?), yields

This equation is automatically satisfied, which we can see as follows. The product involves a product of two Kronecker deltas , which reflects the macroscopic translational invariance of the random solid, and implies that . Next, we note that the restricted wave-vector sum can be replaced by the full wave-vector sum , as the 1RS and 0RS contributions to the restricted sum vanish. Lastly, is a scalar, implying that

For simplicity, instead of working with the full distribution , we replace it by one that is sharply peaked at . This approximation is valid as long as we are concerned with the broad implications for the nematogens of the very *presence* of localized network constituents. We expect that the spread of localization lengths would at most result in the quantitative but not qualitative modification of our results. If the localization is sharp, the Ansatz for the order parameter becomes

## 6Effective theory of structure and correlations in IGNEs

In App. ?, we derive an effective replica Hamiltonian for the liquid crystalline behavior of IGNEs, using an approximation in which is set to its saddle-point value in the Landau-Wilson free energy ( ?), but allowing to undergo fluctuations. By making this approximation we are neglecting the impact of fluctuations in on the nematic freedoms. We adopt this level of description because it is the least complicated one that is capable of revealing the impact of the localized network on the liquid crystallinity characteristic of IGNEs. As it remains constant, we do not need to consider the contribution to the free energy. Furthermore, at the present level of approximation, the term in proportional to vanishes, as we show in App. ?, as a result of the macroscopic translational and translational invariance of the random solid state. Moreover, as we show in the same appendix, at wavelengths long compared with the dominant contribution to comes from a term in . Thus, in forming an the effective replica Hamiltonian for the high-temperature liquid crystallinity of IGNEs, which we denote by , we need only consider the term together with a term in term:

Let us draw attention to the kernel , which is defined via

Here, , which we call the *disorder strength*, characterizes the strength with which the network influences the liquid crystallinity, and has the value . The kernel is the manifestation of the presence of the thermally fluctuating random elastomeric network and, in particular, encodes the central physical characteristic of the network, viz., that at long length-scales the localization appears perfect but at length-scales shorter than thermal fluctuations render the network effectively molten. Note that the characteristic length-scale beyond which is suppressed is .

To determine the effect that the preparation history has on the equilibrium liquid crystalline properties post-cross-linking, we integrate out the zeroth-replica element, , thus obtaining the effective Hamiltonian

A noteworthy feature of is that the replica-diagonal contribution to the quadratic term in is structurally distinct from the replica off-diagonal contribution. This structural feature, as well as the short length-scale liquidity encoded in , enables us to capture a richer range of physical behavior (such as oscillatory-decaying nematic correlations) than can be predicted via conventional random-field approaches, for which the replica-diagonal and off-diagonal terms contain identical coefficients.

## 7Phenomenological content of the microscopic replica theory

We now show that the effective replica Hamiltonian (Equation 17) that we have derived on the basis of the microscopic dimers-and-springs model can be interpreted as having arisen from a more phenomenological, continuum description of a liquid crystalline systems subject to a novel pair of interrelated random fields. This phenomenological description and its implications were explore in Ref. [22]. Our purpose here is not to revisit these issues in detail but rather to reveal the microscopic underpinnings of the phenomenological theory. In that latter theory, the free energy of an IGNE having a given realization of the quenched disorder is given by

where , and and are independent, Gaussian-distributed random fields, with zero means and non-zero variances, the latter being given by

Here, describes the impact of the configuration of the nematogens that is present at the instant of cross-linking on the post-cross-linking nematic alignment pattern, and accounts for the impact of the local anisotropic environment created by the localized network constituents post-cross-linking on this nematic alignment pattern. For a given realization of , the partition function is given by

and by suitably averaging over using the replica technique we obtain

where the effective replica Hamiltonian is given by

Applying the disorder statistics specified in Ref. ( ?), we arrive at the result that , i.e., the phenomenological continuum description originally reported in Ref. [22] contains the same physics as the microscopic dimers-and-springs model.

## 8Structure and correlations in IGNEs

### 8.1Diagnostic quantities

To describe the essential features of the pattern nematic ordering characteristic of IGNEs in the high-temperature regime we focus on the following pair of thermally- and disorder-averaged correlators: (i) the average, taken over realizations of the cross-linking, of the product of the local nematic order at two points, i.e.,

which we term the thermal correlator.

The correlator characterizes the strength of the thermal fluctuations of the nematic alignment away from the local mean value as well as the spatial range over which these fluctuations are correlated. Inter alia, through its range, is capable of signaling the occurrence of a continuous phase transition. The correlator is a diagnostic of particular value for nematic elastomers, as it detects the occurrence of randomly frozen (i.e., time-persistent) local nematic order. For the case where and are co-located, it is the nematic analog of the Edwards-Anderson order parameter, introduced long ago for spin glasses [38], in the sense that it measures the magnitude of local nematic ordering, regardless of the orientation of that ordering. Moreover, how varies with the separation of and determines the spatial extent of regions that share a roughly common nematic alignment. Two mechanisms are responsible for the existence of these aligned regions. First, the formation of a random network causes a local breaking of rotational invariance, which has the effect of creating randomly anisotropic environments that tend to align the nematogens locally. Second, although the equilibrium state of the system at the instant prior to cross-linking is, on average, isotropic, a snapshot of its microscopic configuration at that instant would reveal local nematic order of the type that we normally call thermal fluctuations. The cross-linking process can trap these fluctuations in, either partially or fully, the extent depending on the strength of the cross-linking and the temperature at the moment of cross-linking.

The correlators and have been computed in Ref. [22] via Eq. (Equation 18). In the present section, we re-derive the results via the replica approach to the microscopic dimers-and-springs model that we have developed in the foregoing sections. To proceed, we make use of the effective replica Hamiltonian (Equation 17) and the following identities (which we prove in App. ?):

where denotes an average performed with respect to the effective Hamiltonian , Eq. (Equation 17). To compute the quantities on the right hand side of Eqs. ( ?), we invoke the quadratic form of Eq. (Equation 17), invert the corresponding kernel, and use the replica diagonal and off-diagonal parts to obtain

Note the presence of the scale-dependent kernel in the denominators of the correlators, which plays an essential role in determining their behavior. This may appear surprising when we compare with the result one would obtain via a conventional random-field approach (for details, see App. ?). At length-scales larger than , the presence of in the denominator of leads to a *downward* renormalization of the bare critical temperature by an amount proportional to the disorder strength (and hence grows with the cross-linking density). This indicates that the nematogens are more strongly inhibited from aligning with one another if the density of cross-links is higher. Interestingly enough, a similar result has been obtained using the molecular level neo-classical elasticity theory of nematic elastomers [17]. This disordering effect of the random polymer network (on the nematic alignment) is not contained in the conventional random-field type of models ^{1}

### 8.2Oscillatory-decaying correlations

The scale-dependent kernel in the denominator of each of the correlators in Eq. ( ?) also gives rise to the possibility that the correlators undergo both oscillation and decay with distance. As discussed in Ref. [22], undergoes oscillatory decay with distance for , and the length-scale of the oscillation is given by , which is independent of . On the other hand, undergoes oscillatory decay for sufficiently large values of and sufficiently low values of (see Figure 3), with its oscillation scale given implicitly by the following equation:

The cross-over boundary between the oscillatory and non-oscillatory regimes for , which is the threshold at which the inverse oscillation wavelength , increases from zero to a non-zero value, is given by

The different regimes of behavior of and are shown in Figure 4.

## 9Alternative microscopic models of IGNEs

Instead of the dimers-and-springs model of IGNEs, one could have started with alternative microscopic models that may at first sight appear to be more faithful representation genuine IGNEs. For example, one could have started with either of the following two microscopic models: (i) Model A: a system comprising worm-like chains—pairs of which are permanently, randomly bonded by point-like cross-links—as well as stiff rods that dangle fromn each chain at regular arc-length intervals; and (ii) Model B: a system of chains, each constructed from stiff rods that are connected in series and then permanently bonded by point-like cross-links between randomly chosen rod ends. Models A and B are, respectively, caricatures of side- and main-chain nematic polymer networks; starting from either Model A or Model B, we can apply the Hubbard-Stratonovich decoupling procedure described in Secs. II to IV, and thus derive a corresponding Landau-Wilson free energy that is structurally equivalent (i.e., having the same symmetries and types of couplings) to that derived for the dimers-and-springs model; cf. Eq. (Equation 9). The purpose of this section is to establish this structural equivalence. We can then, in principle, compute the coefficients of the terms of the expansion of the Landau-Wilson free energy separately for Models A and B, apply a similar saddle-point analysis, and derive an effective replica Hamiltonian that would enable us to explore nematic fluctuations. However, the coefficients of the terms of such expansions are technically more difficult to compute than those for the dimers-and-springs model. Moreover, even if one were to succeed in computing such coefficients, the expansion would still lead, at sufficiently large length-scales, to predictions that are identical, up to an overall length-scale, to those made on the basis of the dimers-and-springs model, Eq. ( ?). This is why we chose to work with the dimers-and-springs model in deriving an effective Hamiltonian for the liquid crystallinity of IGNEs.

In what follows, we consider Models A and B separately. We define the microscopic Hamiltonian, introduce the corresponding collective fields, perform the Hubbard-Stratonovich decoupling procedure and, lastly, carry out the log-trace expansion. We then show that the auxiliary fields and the terms in the Landau-Wilson free energy corresponding to either model have the same symmetries and structure as those obtained from the dimers-and-springs model, and thus, give rise to effective Hamiltonians that are structurally equivalent to ; cf. Eq. (Equation 15). To streamline the presentation we display only the essential equations, as the formal procedure employed in this section is identical to the one employed in Secs. II and III.

### 9.1Model 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 with nematogens attached at equal intervals of arc-length along the chain. Each nematogen is represented by a unit vector , where labels the chains, and labels the nematogens. We denote the arc-length measured from one end of the chain by (with ). A segment at arc-length along chain has a position vector ; the -th nematogen resides at arc-length , and its base, which is attached to chain , has a position vector ; see Figure 5. We require that nematogens and tangent vectors to the chains interact via orientational forces that favor parallel or anti-parallel alignment.

The side-chain nematic polymer network is formed by randomly, permanently cross-linking pairs of such chains via point-like cross-links. The pairs of cross-linked chains and the number are fixed for a given realization of quenched disorder, but vary across such realizations of it. In the absence of cross-links the system is specified by the following Hamiltonian:

The partition function corresponding to this system, subject to cross-linking constraints, is given by

Here, , and denotes integration over the unit sphere of nematogen orientations. To proceed further, we can follow the strategy described in Secs. II and III, including making a Hubbard-Stratonovich transformation to introduce the auxiliary fields , to arrive at the corresponding Landau-Wilson free energy :

Here, , reflecting the fact that the excluded-volume parameter is not renormalized by the effects of cross-linking in the preparation state. Note that the Boltzmann average is defined via

We observe that is invariant under transformations of the auxiliary fields that correspond to independent translations and rotations of the replicas, as in the case of the Landau-Wilson free energy for the dimers-and-springs model (Equation 9). In addition, is a HRS field, is a 1RS field, and is a traceless and symmetric second-rank tensor field, all as with the corresponding auxiliary fields for the dimers-and-springs model. Furthermore, terms developed by expanding the log-trace part in Eq. ( ?) are structurally equivalent to those arising from the corresponding expansion of the Landau-Wilson free energy (Equation 9) of the dimers-and-springs model. In other words, as fields theories, Model A and the dimers-and-springs model are identical, up to elementary rescalings of their coefficients. We thus conclude that at length-scales larger than the microscopic scales at which they are defined, these models yield predictions that are structurally identical.

### 9.2Model B: main-chain nematic polymer network

We apply the procedure used in Section 3 and the previous subsection to to derive the Landau-Wilson free energy for Model B, a model for main-chain nematic polymer networks. This model consists of jointed chains (see, e.g., Ref. [39]) comprising rods each of length . In addition, adjacent pairs of rods interact via a bending energy that promotes their parallel alignment, and arbitrary pairs of rods also interact via orientational forces that favor parallel or anti-parallel alignment. The network is formed via the random, instantaneous cross-linking of pairs of jointed chains via permanent point-like cross-links, located at the ends of the rods. The chains are labeled by () and the rod-rod junctions are labeled by (); the position of rod end on chain is , as shown in Figure 6. The system is then specified by the following Hamiltonian:

The partition function corresponding to this system, subject to cross-linking constraints, is given by

Using the replica method and a Deam-Edwards type of distribution for the quenched randomness, together with a Hubbard-Stratonovich decoupling, we arrive at the following Landau-Wilson free energy in terms of the auxiliary fields :

The comments made at the end of the previous subsection concerning symmetries of the Landau-Wilson free energy under transformations of the auxiliary fields hold for Model B, too.

## 10Concluding remarks

The objective of this Paper has been to develop a microscopic approach to the liquid crystalline properties of isotropic-genesis nematic elastomers (IGNEs), in which local nematic order—both in the preparation and the measurement ensembles—and random localization—induced by the presence of an elastomer network—are naturally incorporated. This development, which takes as its starting point a system of dimers that are permanently connected at random by Hookean springs, serves as the underpinning to the phenomenological approach presented in Ref. [22] by providing a systematic derivation of the formulas on which the phenomenological description is based.

Specifically, by deriving an effective Hamiltonian of liquid crystallinity in IGNEs, we have shown that this microscopic approach leads to the phenomenological Landau theory of IGNEs proposed in Ref. [22] which, *inter alia*, predicts that at sufficiently large disorder strengths, both the thermal and glassy spatial correlations of nematic 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 influence of the preparation history of IGNEs on their subsequent equilibrium behavior. The consequences of these two ensembles were analyzed in detail in Ref. [22]. However, the appearance of the two ensembles took the form of a hypothesis in Ref. [22], whereas in the present Paper they come into play naturally. Lastly, we have argued that at sufficiently large length-scales, predictions made on the basis of a simple dimers-and-springs model are qualitatively identical to those made on the basis of two more realistic (but more complicated) microscopic models of IGNEs: one for side-chain liquid crystalline elastomers and one for main-chain liquid crystalline elastomers.

Apart from its relevance to the specific subject of liquid crystalline elastomers, the present work brings to light a more general issue, viz., that the concept of a quenched random field should be broadened to incorporate not only the conventional, frozen type, which does not fluctuate thermally, but also the type necessary for understanding media such as liquid crystalline elastomers, in which the frozen nature of the random field is present only at longer length-scales, fading out as the length-scale progresses through a characteristic localization length, owing to the thermal position fluctuations of the network’s constituents. The framework elucidated in the present work can be extended, with suitable modifications, to explore the statistical physics of other randomly cross-linked systems, such as smectic elastomers and various biological materials.

### Footnotes

- This distinction arises because our effective replica Hamiltonian (Equation 17) does not contain replica-diagonal contributions to the term proportional to , whereas the effective replica Hamiltonian that corresponds to the conventional random-field approach contains both replica-diagonal and replica off-diagonal contributions, all having the same coefficients.

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