Inhomogeneities and local chain stretching in partially swollen networks

According to classic single-chain theories of rubber elasticity such as the affine or phantom models, the length N and the state of stretching R/R0 of network chains are directly reflected in the magnitude of segmental orientation correlations, as quantified by a dynamic order parameter S f R/N (this relation holds for chains in the bulk and q solvent). S can be determined by suitable NMR techniques, and is a viable molecular probe of the network structure, as it is directly proportional to the elasticity modulus. Furthermore, previous studies (see Saalwächter et al., Soft Matter, 2013, XX, XXX, in this issue for a review) have convincingly demonstrated the validity of the phantom model for the prediction of S in equilibrium-swollen networks. We here investigate changes in the degree of local chain stretching reflected in S as a function of the degree of (partial) swelling Q 1⁄4 V/V0 in different networks prepared in bulk and in states of increasing dilution. Previous work has already revealed a non-monotonic dependence of S on Q, indicating strongly subaffine local deformation in the early stages of swelling. The width of the distribution in S is also accessible, and increases significantly during the early stage of swelling, indicating the well documented presence of swelling heterogeneities. We find that beyond this early stage, the network chains deform affinely. A back-extrapolation of the affine deformation range to zero swelling allows for conclusions on the actual crosslink density of the networks corrected for the effect of topological or packing constraints, which we refer to as the “phantom reference state” of a bulk network. Deviations of the network elasticity from this state constitute non-classical contributions, possible origins of which are discussed.


Introduction
An important property of exible polymer chains is their elasticity, which is based on the interplay of the coupling of their conformation with external stress and thermal motions.Various molecular theories of rubber elasticity have been developed in order to describe their elastic response to deformation, i.e., uniaxial stretching and isotropic swelling, with respect to the molecular structure of the network. 1,2The most prominent are the affine 3,4 and the phantom models. 5,6s opposed to the affine model assuming xed junctions, the phantom model allows for uctuations of crosslinks around their average position, constrained only by connectivity but not by next-neighbor packing.Thus, the uctuations strongly depend on the crosslink functionality.A variety of mechanical studies demonstrate that the elastic properties of real polymer networks are between these two limiting extremes. 7,8The interpenetration of individual network strands whose Gaussian chain conformation is not affected by crosslinking 9 causes topologically neighboring crosslinks to be more remote than spatial neighbors, 10 and topological constraints such as nextneighbor packing or entanglements should affect the spatial range of crosslink uctuations.In real networks the magnitude of the latter is thus smaller than that predicted by the phantom model and the topological interactions contribute directly to the modulus. 11,12The concept of restricted crosslink uctuations is supported by the results of mechanical studies 7,13 and the experimentally determined range of crosslink uctuations, which is smaller than theoretical predictions. 14n swelling experiments the solvent uptake leads to an isotropic dilation of the network.The classical network models mentioned above assume an affine deformation of the individual network strands (note that the phantom model also predicts an affine deformation of the average crosslink positions).6][17][18][19][20] The overall swelling process is found to be sub-affine especially at low degrees of swelling, meaning that on a microscopic scale the network does not deform as the whole sample.Particularly the change of the radius of gyration of network strands upon swelling was found to be signicantly smaller than that expected from affine deformation. 21,22These observations were explained in terms of a desinterspersion process, 19,23,24 which can be interpreted as a topological unfolding. 25During the progressive swelling, solvent molecules replace monomer units of network chains separating the individual network strands and thus the mutual interpenetration.Thereby it is assumed that the network can swell up to a certain degree without change of the chain conformation, i.e., the crosslink separation (end-toend distance) and the radius of gyration. 23,26Only at higher swelling degrees the swelling process is dominated by local chain stretching. 24he desinterspersion process is highly dependent on the network structure and thus on the preparation conditions.Network inhomogeneities created during the crosslink reaction are frozen-in by the permanent chemical interconnection of the polymer chains and cannot be released.This leads to large-scale network heterogeneities in swollen networks 27 which are observed as frozen-in concentration uctuations. 28,29n the present work, we investigate the progressive swelling of PDMS networks prepared by random crosslinking and endlinking in solution as well as radiation crosslinking in bulk, and of natural (NR) and isoprene (IR) rubber samples by 1 H doublequantum (DQ) NMR 30 and equilibrium swelling experiments.The NMR experiments are performed at well-dened swelling degrees and evaluated in terms of residual dipolar coupling constant distributions, which directly reect the magnitude and distribution of segmental orientation correlations, as quanti-ed by a segmental average dynamic order parameter S.This quantity is directly related to the state of stretching of the network chains, and its distribution over the ensemble of segments (monomers) reports on the sample heterogeneity.
The results are discussed in terms of a two-stage swelling process rst observed by Cohen-Addad, 18,19 who however used a less quantitative NMR technique.Our own previous work on swollen networks 20 was concerned with the NMR detection of swelling heterogeneities, but our results were also subject to systematic errors related to a non-optimal data analysis.Relying on improved procedures, 31 we are now in a position to study quantitatively the state of chain stretching and its distribution over the sample, in particular considering the role of network defects, i.e., elastically inactive chains that do not carry load.Previous applications of the improved method lead us to conclude on the validity of the phantom model for equilibriumswollen networks, 32 and on subtle but important effects of the swelling thermodynamics via excluded-volume (solvent quality) effects on the NMR observables, 33,34 which will also be discussed in the present context.
We demonstrate that above a swelling threshold Q 0 , all investigated samples independent of the rather different network structure feature a universal dependence of the local chain stretching on the degree of swelling, which in fact corresponds to affine deformation.The initial, sub-affine stage is characterized by a signicant increase of width of the segmental order parameter distribution, and is studied depending on the preparation conditions.We show that the desinterspersion of the network chains upon swelling must be accompanied by the release of entanglement or packing constraints, which in the bulk state also affect a large part of the network defects, rendering them elastically active on rather long timescales unless the network is swollen.Order parameters measured in bulk samples are corrected for packing/topological contributions by a back-extrapolation from the affine deformation range to an unswollen reference state, establishing what we refer to as the "phantom reference network."The soobtained results are compared to results of equilibrium swelling experiments and the observations are discussed in the context of the possible non-classical contributions to network elasticity.

Network preparation
We compare different poly(dimethylsiloxane) (PDMS) networks made by end-linking or random crosslinking of accordingly functionalized precursor polymers or by radiation crosslinking of linear PDMS, as well as elastomers based upon natural and synthetic poly(isoprene) rubber.
Commercially available randomly vinyl-functionalized (r) and di-vinyl end-functionalized (e) PDMS precursor polymers and appropriate two-and four-functional crosslinker molecules, respectively, were used as-received from ABCR Company for the preparation of PDMS networks.The reaction is performed in the presence of a Pt-based catalyst.The precursor polymers were characterized by gel permeation chromatography (GPC) and liquid-state NMR in order to determine the molecular weight, the polydispersity (PD) and the stoichiometry of the crosslink reaction, respectively.The rPDMS samples were crosslinked in the presence of 20 wt% toluene using different amounts of the two-functional crosslinker to realize different crosslink densities. 32The end-functionalized PDMS precursor polymers, ePDMS-21 (M n ¼ 4.8 kDa, PD ¼ 2.0) and ePDMS-25R (M n ¼ 10.6 kDa, PD ¼ 2.1), were crosslinked in solution at different polymer volume fractions, 4 p,c , using a stoichiometric amount of the four-functional crosslinker and toluene as the solvent.Details on the crosslinking mechanism were published recently. 35The relative weight of the toluene in % as compared to the weight of the polymer is indicated by tXXX in the sample name.The samples crosslinked in the presence of up to 300 wt % toluene were cured for 7 days at room temperature.All samples crosslinked at higher solvent content were cured for 14 days.At the end of the crosslinking time the whole sample volume was gelled and the solvent was evaporated carefully.Completely dry networks were obtained within 2 days up to 14 days, depending on the weight fraction of the solvent.
Radiation-crosslinked PDMS networks were prepared by 10 MeV high-energy electron irradiation (Beta-Gamma-Service, Saal, Germany).The xPDMS-t23 and -t61 networks are based upon linear precursors of M n z 5 and 50 kDa, respectively, and the applied radiation dose (200, 350 or 500 kGy) is indicated as the last part of the sample name.
Natural (NR) and isoprene rubber (IR) samples were prepared from natural or synthetic latex suspensions, coagulant-dipped and dried to form thin (<1 mm) lms and then crosslinked by an innovative UV curing method based upon the thiol-ene reaction using poly-functional thiol and a photoinitiator. 36,37Unlike thermally activated peroxide-crosslinked rubbers, 38 which are substantially inhomogeneous, these networks are rather homogeneous and compare well with conventional sulfur-cured NR.Details of the curing conditions, recipes, and a detailed account of the sample properties will soon be published elsewhere.The UV power, irradiation time, and the amount of crosslinker were varied to realize different crosslink densities.The samples are simply numbered indicating increasing relative crosslink density.We note that results from conventionally prepared (masticated, roll-milled and sulfur-cured) natural rubber were found to be qualitatively the same, stressing the model character of these samples.

Experimental procedures
1 H double-quantum (DQ) NMR experiments were carried out on a Bruker minispec mq20 spectrometer operating at a 1 H resonance frequency of 20 MHz with a 90 pulse length of 2.2 ms and a dead time of 13 ms.The experiments and raw data analysis were performed following previously published procedures. 30,31,38,39he experiments on PDMS networks, partially swollen in styrene and (deuterated) toluene, were performed at 308 K, which was shown to be the q-temperature of PDMS/styrene systems. 40Analogous experiments on NR/IR samples swollen in deuterated toluene were conducted at 343 K.For preparation, about 30 mg of the sol-extracted sample was cut into small pieces and swollen with a well-dened amount of a solvent in a tightly sealed NMR tube.The corresponding swelling degrees, Q, were calculated assuming additive volumes of polymer and solvent. 19The PDMS samples were stored for 1 day in order to equilibrate in agreement with results of swelling kinetics. 19or NR/IR, it is important to consider light-and oxygeninduced degradation reactions upon swelling in toluene. 41A complete exclusion of light and air proved unfeasible; so for each swelling degree a new sample was prepared and measured aer about 10 min aer adding the solvent, which proved sufficient for equilibration of the small sample pieces. 41quilibrium swelling experiments were preformed and evaluated as described in our previous work. 32,41

Background of DQ NMR and relation to swelling experiments
In double-quantum (DQ) or more generally, multiple-quantum (MQ) NMR two different signal functions are obtained experimentally as a function of the pulse sequence duration, and analyzed in a combined fashion to obtain (i) the amount of isotropically mobile, thus non-elastic components referred to as defect fraction u def , which comprises potential sol (if not extracted), short dangling ends and loop structures, and (ii) the so-called normalized DQ intensity build-up curve.This curve encodes the residual dipolar coupling constant D res (unit: rad s À1 ) arising from the non-isotropic orientation uctuations of actual network chains as well as its potential distribution in an inhomogeneous sample.
The through-space 1 H-1 H dipole-dipole coupling within monomer units is a weak rst-order perturbation of the Larmor frequency, and depends on the xed internuclear distance and on the instantaneous orientation of a given proton pair.This spin interaction is averaged to zero in the case of fast isotropic reorientations, but a nite and well-dened average value D res remains for monomers in network chains uctuating between two xed ends (crosslinks or other topological constraints).In elastomers with chemically simple monomers, D res is an effective value characteristic of all protons in a given monomer, and is proportional to the dynamic chain order parameter S, where h.i denotes the thermal average over all conformations of a network chain and [.] denotes the structural average over all segments in the sample.The angle q refers to the instantaneous segmental orientation with respect to the end-to-end vector of the given chain.The proportionality factor in eqn ( 1) can be derived on the basis of model considerations, 39 but as we want to avoid discussions of their validity, we will only discuss the objective experimental result D res in the following.Importantly, with our method we can not only reliably determine from the DQ build-up curve the structure-averaged D res described by eqn ( 1), but also its distribution in an inhomogeneous sample.In this study, it was essential to use a previously published, optimized numerical procedure 31 to obtain the precise distribution function of D res .Below, we will mostly discuss the numerically calculated standard deviation s of the distribution in terms of its dimensionless width parameter s rel ¼ s/D res with respect to the arithmetic distribution average.
Based upon the mentioned model considerations, 39 and realizing that R 0 2 ¼ Nl 2 , D res can be related to the (apparent) network chain molecular weight M c $ N, e.g., For NR, the reference coupling in the numerator is 617 Hz.The functionality of the crosslinks f enters through application of the phantom model, which was recently shown to provide an improved correlation even in the bulk. 32The phantom model also predicts the following relation for M c obtained from equilibrium swelling experiments, where r p is the polymer density, V s the solvent molar volume, c the Flory-Huggins interaction parameter, f * p the polymer volume fraction at swelling equilibrium (generally, ) the volume fraction of the elastically active polymer, i.e., it is corrected for elastically inactive network defects as determined by DQ NMR.Note that u def,sw to be used for f * p,el in eqn (3) must be determined in the swollen state.It will be shown below that u def,sw can be substantially bigger than u def determined in the bulk, in particular in rather lowly crosslinked networks.Finally, a comparison of eqn ( 2) and (3) shows that D res is proportional to the inverse (1/M c,app ) of eqn (3) irrespective of the crosslink functionality f, as demonstrated previously. 32For more details on these issues and equilibrium swelling experiments in general, we refer to this reference.

Network characterization results
All samples were investigated by equilibrium swelling experiments and by DQ NMR measurements on dry and equilibrium swollen networks in order to obtain information about the network structure and in particular the amount of non-elastic defects.For the solution-crosslinked ePDMS sample series, the sol fraction u sol extracted during the swelling experiments is on the order of a few percent (2-4%) for networks crosslinked in the bulk (4 p,c ¼ 1) up to polymer fractions of 4 p,c z 0.15.Only at lower polymer concentration considerably larger sol fractions (10-40%) are observed for the ePDMS-25R samples.The rather low extracted sol fractions demonstrate that both ePDMS series are characterized by a very good conversion of the precursor chains over a broad range of dilutions during the crosslink reaction.This is also conrmed by the nearly complete consumption of the crosslinker and the terminal vinyl-groups observed during investigations of reaction kinetics. 35he defect fraction, u def , determined by DQ NMR on dry samples, reveals a similar dependence on the polymer fraction during crosslinking, 4 p,c , as shown in Fig. 1a.The defect fraction of roughly 2% is observed over a broad range of dilutions (4 p,c ¼ 1-0.15) for both sample series, and a signicant increase of the estimated defect fraction is only observed for samples crosslinked at lower 4 p,c , especially for the ePDMS-25R networks.
The defect fractions determined in the equilibrium swollen networks differ from those determined in the corresponding dry networks by about one order of magnitude, as demonstrated in Fig. 1a.The defect fractions decrease with increasing polymer volume fractions during crosslinking 4 p,c for both series of endlinked networks, and appear to follow an inverse proportionality.The discrepancy between defect fractions found in the swollen vs. the dry state is particularly large for intermediate polymer volume fractions (4 p,c ¼ 0.7-0.15),which corresponds to previous observations in randomly crosslinked PDMS networks made from short precursor chains. 32he apparent relative distribution width s rel (Fig. 1b) determined in dry samples, reporting on the spatial distribution of crosslinks, features a non-monotonic dependence on the amount of diluent during crosslinking, going through a minimum.In the high-concentration region, ranging from bulk down to polymer fractions of 4 p,c ¼ 0.15 the distribution width narrows continuously.The narrowest observed distribution widths s rel are about 0.15 and 0.13 for ePDMS-21 and ePDMS-25R, respectively, which are less than half of the distribution width of the respective bulk crosslinked samples.Such rather small values are usually observed for natural rubber prepared of high molecular weight precursor chains, 39 and also for the NR and IR samples studied herein.
We attribute the higher inhomogeneities at high preparation concentration to insufficient mixing of the reactants when the viscosity is high.The higher relative distribution width at low concentration may indicate inhomogeneities arising from the formation of micro-gels and local sub-networks before a percolated structure is formed in a more dilute system.Also in this range, the network chains are partially swollen in their own defects, and we note that the dry-state defect content (Fig. 1a) and the corresponding inhomogeneity (Fig. 1b) exhibit similar trends in the different samples.The even higher s rel values measured in the equilibrium swollen state of all networks support this point.We earlier attributed this presumably universal feature to swelling inhomogeneities. 20In this work, making use of the new and more reliable numerical procedure to characterize such rather wide D res distributions and thus their average, 31 we are in a better position to draw quantitative conclusions from the average D res (Q) values.
Finally, Fig. 1c reports defect fractions for the NR and IR samples as a function of their crosslink density.In agreement with earlier ndings for conventional sulfur-vulcanized NR, 38 the apparent defect fractions in the dry state are always rather low, but increase substantially upon swelling, in particular at lower crosslink density.We stress that the sample series investigated herein covers the lower range of application-relevant crosslink densities.

Results and discussion
The main focus of the present work is the study of the local chain stretching of polymer networks reected in the segmental orientation S f D res upon swelling over a broad range of swelling degrees ranging from the dry state to the equilibrium swollen state.First, the results are discussed in terms of Cohen-Addad's two-stage scenario of the swelling process, 18,19 and a universal affine relation between the determined D res and the degree of swelling beyond a certain swelling threshold is demonstrated.In the second part, the inuence of the solvent quality and the preparation conditions on the progressive network swelling is discussed.We nd that previous conclusions on a general dependence of the phenomenon on the polymer concentration at preparation 19 need to be rened, in particular in light of the large defect fractions of typical samples.
3.1 Two-stage swelling and affine stretching at high swelling Fig. 2a shows the apparent average D res of ePDMS-21 networks, end-crosslinked in solution at different polymer volume fractions, 4 p,c , at various degrees of swelling in a good solvent (toluene).Analogous results were obtained for the second series of networks end-linked in solution, ePDMS-25R.Varying dependencies of D res on Q are observed, which correspond qualitatively to results in the literature. 19,20,42Deviations from the literature data, in particular in the initial part, are however apparent, and we stress the better reliability of the present data analysis.This concerns the precise consideration (subtraction) of non-elastic defects and the robust distribution analysis. 31We note that the increase of the apparent defect fraction to the swollen-state plateau value shown in Fig. 1 roughly occurs during the rst stage.Specically for the more defect-rich lowly crosslinked (lower D res (Q ¼ 1)) samples, the initial decrease of D res would be even more pronounced if the growing D res ¼ 0 defect fraction were not subtracted and considered in the plotted average value.In the second regime, the plotted D res safely represents a constant fraction of elastically active network chains, while in the rst regime, a varying amount of defects with slow chain relaxation time contributes to the average.PDMS networks crosslinked at low polymer concentrations as well as all NR and IR samples show a distinct decrease of the coupling constant upon swelling.As 4 p,c increases, the decrease of D res is less pronounced and for the samples crosslinked at 4 p,c > 0.6 even a slightly positive slope is observed, in contrast to Cohen-Addad's observations. 18,19For an interpretation, the rst stage at low swelling is oen discussed in terms of a strongly subaffine desinterspersion process, 19,23,24 which can be interpreted as a topological unfolding of the network strands induced by the swelling solvent. 25This will be addressed in the second part.
Beyond a certain swelling degree Q 0 represented by the dotted line in Fig. 2 all networks feature a similar, monotonically increasing dependence of the coupling constant on the swelling degree, suggesting actual chain stretching in the second stage.For the end-linked PDMS series, a systematic change of Q 0 is observed, indicating a dependence of D res (Q) at low swelling degrees on preparation conditions of the networks which will be discussed below.For all IR and NR samples a qualitatively similar dependence of D res on Q is observed, as shown in Fig. 2b.Starting from the dry samples the coupling constant decreases upon swelling up to a certain swelling degree threshold Q 0 .The latter has a constant value for all investigated rubber samples as indicated by the dotted line in Fig. 2b.
A log-log representation of the data (see Fig. 6) reveals that the monotonic increase can be described by a power-law for all investigated samples, with a scaling exponent n def describing the local chain stretching of elastically active network strands.Simple geometric considerations assuming an affine deformation of junction distances R, using D res f R 2 and Q f R 3 , predict a strict monotonic increase of D res $ Q 2/3 , thus n def ¼ 2/3.This is obviously violated in the rst stage to different degrees by most samples.For an objective test of possibly uniform deformation behavior, the second-stage data from all were tted to a power-law, with the deformation exponent n def and a back-extrapolated D res,n as free parameters.Sample ts are shown in Fig. 3a.As highlighted by Fig. 3b, the tted range beyond Q 0 is characterized by an almost constant relative distribution width s rel , indicating no serious structural changes and supporting the assumption of uniform deformation behavior.Interestingly, the obtained deformation exponents n def are very close to the affine prediction.The tted values range between 0.59 and 0.67.Discrepancies may be attributed to experimental inaccuracies, e.g.some loss of the solvent before sealing of the vessels.
Of course in the rst stage, a distinct broadening of the D res distribution upon swelling, attributed to the appearance of swelling heterogeneities, 20 is observed for all samples.Fig. 3a also shows that the tted D res,n deviates considerably from the value determined in the corresponding dry sample, D res (Q ¼ 1).The difference characterizes the rst-stage affinity; a quantication of the associated release of topological constraints will be presented in the second part.
The nding of affinity in the second stage is universal, as is demonstrated by a log-log plot of a normalized D res (Q)/D res,n vs. Q for the second stage of all investigated samples; see Fig. 4. Again, only weak deviations towards slightly subaffine behavior are evidenced.This also suggests a specic physical meaning of the coupling constant, D res,n , from the power law t to the latestage swelling, back-extrapolated to a hypothetical network state at Q ¼ 1 for all samples, which were prepared in rather different ways.This will be addressed in the second part.
For a more detailed picture, the full D res distribution functions were examined directly depending on Q. Fig. 5a shows the   distributions of a solution end-linked PDMS network and a synthetic rubber, rPDMS-25R t020 and respectively, determined at different degrees of swelling, The dry samples (Q ¼ 1) feature a rather narrow distribution, indicating a homogeneous network structure and thus distribution of crosslinks.Owing to swelling, the distributions are broadened, involving higher and lower coupling constants with respect to the dry sample.Especially the high-coupling contribution rises, which directly illustrates the increased residual order of the load-carrying network strands due to the chain stretching upon swelling.At low and intermediate swelling degrees the maxima of the distributions, representing the most probable chain order, are shied slightly towards lower coupling constants whereas at higher swelling degrees above Q 0 the opposite is observed.
In order to compare the D res distributions at different Q, they were normalized with respect to their maximum amplitude and plotted depending on a reduced D res scaled by Q À2/3 of the corresponding swelling degree implying the assumption of an affine network deformation.The so-obtained distributions differ signicantly at low and intermediate swelling degrees as depicted in Fig. 5b.This emphasizes again the non-affine increase or even decrease of D res and the broadening of the distribution parameterized by s rel .However, at higher swelling degrees beyond Q 0 , the distributions are in very good agreement over their entire width, which is especially highlighted by the nearly perfect coincidence of the maxima.Thus, the whole inhomogeneous structure of a network, irrespective of its structure or preparation condition, is deformed affinely beyond the threshold swelling degree Q 0 .

Low and intermediate degrees of swelling
The observed behavior of the apparent chain stretching as reected in D res in the rst stage of the swelling process at low swelling degrees is usually discussed in terms of chain desinterspersion, as mentioned above.During swelling, the solvent molecules replace monomer units of the network chains which represent their own solvent in dry polymer networks, 19 leading to a separation of the individual network strands.As indicated by scattering studies, this can proceed through topological unfolding without any change of the conformation of the network chains, 23,25,26 meaning that the end-to-end distance of the individual network strands does not necessarily change.The changes in D res can therefore be complex and the question arises whether single-chain approaches are applicable to describe the rst stage.
Cohen-Addad et al., 19 based upon his sample series, has concluded a universal D res (Q) behavior that scaled vertically from sample to sample with the respective polymer volume fraction 4 p,c during crosslinking.We now nd that the initial decay of D res (Q) is not universal and that its dependence on preparation conditions is more complex.A log-log plot of representative data, see Fig. 6, shows that a power law appears applicable in the rst stage, but that it is not only the prefactor but also the exponent that varies with the concentration at preparation, 4 p,c .In agreement with Cohen-Addad's work, we nd that the crossover swelling degree Q 0 is the same for different precursor polymers and depends only on 4 p,c .However, we nd a power-law dependence Q 0 $ 4 p,c À0.25AE0.02, in contrast to an exponent of À5/8 reported in the earlier work, which was motivated by a scaling idea based on the c* theorem. 19trikingly, the latex-based UV-crosslinked NR and IR samples investigated herein, as well as conventional (roll-milled) sulfur-vulcanized NR (data not shown), exhibit more or less the same negative initial slopes of D res (Q), although these samples were crosslinked in the bulk and had a widely varying defect content.Another option for network formation in the bulk used herein is radiation crosslinking using an electron beam.Corresponding xPDMS sample data are compared in Fig. 7 with data of ePDMS samples that have almost matching trends in the high-Q region.The xPDMS samples exhibit affine behavior over the whole Q range, apparently complementing the trend seen for the ePDMS series towards high 4 p,c (see Fig. 2a).The big difference between the NR/IR and xPDMS samples is probably due to the rather inhomogeneous structure of the latter, as inferred from the large values of s rel ¼ 0.6 and 0.8, and the comparably high defect fractions (about 0.1-0.4) in the dry state.As mentioned, these networks are to a degree swollen in their own defects, unfolding the swelling inhomogeneities and masking the rst stage.Nevertheless, the widely different trends observed in the NR/IR samples and the PDMS samples suggest that there is no simple and universal dependence of the rststage behavior on just the network chain dimensions.Another possible issue in interpreting the initial behavior or D res (Q) is thermodynamic nature.If a topological desinterspersion process occurred without changes in the chain dimension R, 23,26 D res $ R 2 would a priori be constant in the rst stage.A decrease could then be explained by excluded-volume effects arising upon swelling in a good solvent (removing the excluded-volume screening in the melt state), as shown in one of our earlier publications. 33There, we have for the rst time compared D res (Q) for swelling in good vs. q solvent, supporting the assumption.However, the data were subject to systematic errors related to the then not yet sufficiently robust distribution analysis; so we show improved data in Fig. 8.
Clear differences between good (toluene) and q (styrene at 308 K (ref.40)) solvents are indeed observed; yet these are restricted to the high-Q stage, where D res (Q eq ) at swelling equilibrium Q eq follows our expanded theoretical treatment covering the good as well as solvent ranges. 33,34Importantly, virtually no differences are observed in the rst stage, which rules out excluded-volume effects as the origin of the negative slope, and leaves us with a possible desinterspersion or a release of "packing constraints", as initially assumed.
In quantifying the "non-classical" contribution to the segmental orientation correlations D res and thus to the linearregime elasticity measured in bulk samples, we turn to a closer analysis of the back-extrapolated values D res,n , obtained by ts to the affine second stage using eqn (4).In Fig. 9, where both quantities are plotted vs. the apparent crosslink density from swelling experiments, we nd non-zero intercepts for the former, but near-zero intercepts for the latter.In previous works, 32,38,41 the non-zero intercept has been interpreted as entanglement contribution, and a very recent study 43 conrms that this contribution to the elasticity modulus can be successfully extracted using the non-linear Mooney-Rivlin analysis.Strikingly, the D res,n values are proportional to the crosslink density estimated by swelling virtually without intercept.For the few samples investigated in q solvent, the data indicates an intercept even closer to zero, conrming that the slightly lower good-solvent values arise from the excludedvolume effect on D res .Obviously, this effect on the absolute value of D res,n is rather small.Both quantities, D res and D res,n , may well be inuenced by topologically trapped entanglements,   which are physical but elastically active crosslinks contributing also in the high-deformation range.In way, the correlation shows that D res,n has a well-dened physical meaning, reecting the true crosslink density.Since our previous work has demonstrated the good applicability of the phantom model for swollen samples, 32,34,41,44 we refer to D res,n as describing the "phantom reference state" of a network.The deviation from the actual bulk value, DD res ¼ D res À D res,n , thus provides a measure for the non-classical elasticity contributions.For all observed samples, the data in Fig. 9 suggest this contribution to be almost independent of crosslink density, but dependent on the sample nature.
We end this paper with a discussion of the possible contributing factors to DD res .One possible scenario is that the crosslink uctuations around their mean position that enter the phantom model could be constrained by surrounding chains in the bulk network, 11 and be released upon swelling.Considering that the uctuations are a priori not dependent on the degree of deformation, 45 this means that the phantom-model correction factor for the crosslink density (more precisely: network chain density $1/M c ) could be different from the phantom prediction (f À 2)/2 in eqn ( 2) and (3).In the rst stage, a continuous decrease could be expected from a starting value close to its affine xed-junction limit of 1 down to a value of 0.5 (for a crosslink functionality f ¼ 4), or even less when the average f is lower, as is the case for the lower crosslinked defect-rich PDMS samples based on short precursors. 32SANS studies have in fact shown that uctuations of crosslinks around their average position are of smaller magnitude in dry networks than those predicted by the phantom model, 14 while diffusion NMR results have conrmed enhanced segmental mobility in the swollen state. 46Furthermore, mechanical studies of solution-crosslinked samples have indicated an elasticity modulus that is larger than the phantom prediction. 8s to the origin of constraints, one could intuitively assume that topologically remote but spatially close crosslinks play an important role.Thus, the crosslink density itself could be a measure of the constraints on crosslink uctuations. 47Thus, for samples with lower crosslink density in the dry state the decrease of D res at low swelling degrees should be less pronounced.From Fig. 2, we however take that this is not the case; for all samples of this study, the initial decrease is most pronounced for the samples with lowest crosslink density.
We also stress that such prefactor-based arguments would mainly change the slope of D res in Fig. 9, but not necessarily lead to an intercept, as can be inferred from eqn (2).Note that the phantom prefactor cancels from a comparison of D res measured in the bulk and 1/M c,app from equilibrium swelling, eqn (3) only when the phantom model is applicable in both states.Otherwise, the slope would change.We remind again that most of the discussed D res,n values are specic for good solvent, and that (ultimately better comparable) values obtained for q solvent are slightly higher, which adds a weak bias to these arguments.
In trying to interpret the data in Fig. 9, we stress that the relative contribution of DD res to the total D res is considerably smaller for PDMS than for NR/IR at comparable crosslink density.This is in fact in line with a simple dependence on the entanglement molecular weight, which is usually taken to contribute as D res f G $ 1/M c + 1/M e to both NMR and the elasticity modulus G. Since M e is about 2-3 times higher for PDMS than for NR/IR, the observed trend could be explained.We further note that DD res shows some dependence on precursor molecular weight, but does not depend on the concentration at preparation, which varies widely.This is not a contradiction to the entanglement argument, as the entangled structure could well depend somewhat on the conformation of the network chains, which are known to be in a "supercoiled" state in deswollen networks prepared in solution. 48,49Also, it is not clear whether the entanglement effect represents the above discussed constraint to mere junction uctuations, or acts as an additional constraint also in a xed-junction scenario.
However, the usual entanglement interpretation does not resolve one outstanding problem: the fact that lowly crosslinked dry samples always feature rather low apparent defect fractions even when they have rather high actual defect fractions identi-ed in the swollen state (see Fig. 1).In other words, D res of weakly crosslinked samples reects to a signicant part anisotropically mobile defects.This is one of the central outstanding problems of the NMR methodology and network physics in general, as it shows that parts of the defect structures have reduced mobility and thus contribute to the elastically active fraction, and thus also to the linear-regime elasticity, 43 when embedded in the bulk network matrix.This is primarily a matter of timescale, as the given NMR approach probes the anisotropy of segmental orientation uctuations roughly on the ms range.Similarly, a polymer chain behaves elastically on a given time or frequency scale only when it cannot relax the internal stress faster than the applied external stimulus.We expect this to be the case only for entangled defects with correspondingly long relaxation times.However, most of the investigated PDMS networks were prepared of rather short prepolymers, in which the expected defect structures are short dangling chains or short loops 32,50 which should not be subject to a signicant tube constraint.Similar arguments hold for the defects in NR/IR, where also short loops are expected when a crosslink forms within a given chain.
We suggest this problem to be related to previous deuterium NMR experiments of strained samples, in which even lowmolecular deuterated probe chains show a line splitting, thus a motional anisotropy. 51The phenomenon has been explained in terms of a nematic-like mean eld, which is in different theoretical approaches hypothesized to represent a second contribution to network elasticity on top of the regular entropy term. 24,52,53Related arguments concerning local ordering of polymer chains have already been a subject in the famous 1979 Faraday Discussion; see for instance the introductory lecture of Flory. 54Now for the given problem, a strained sample exhibits anisotropic excluded-volume interactions, as directly demonstrated by an anisotropic amorphous halo in X-ray scattering, 43 and this means that even a large-scale diffusive average of a small probe molecule would not render its orientation uctuations isotropic.In transferring this argument to the unstrained state and to short pending defect structures, one would have to assume that excluded-volume interactions are also anisotropic in the unstrained state.As isotropy must restored at large length scales, such a local anisotropy would have to exhibit a correlation length in the nm range, at least corresponding to the length scale probed by the constrained diffusion of pending defects.We anticipate that further experimental and theoretical work along these lines may ultimately help to resolve some of the open questions.

Conclusions
We have presented an in-depth NMR study of local orientation correlations of segments in partially swollen networks prepared under different conditions, i.e., end-linking of short PDMS precursors, radiation-crosslinking of different linear PDMS, and random crosslinking of natural and synthetic poly(isoprene) rubber.The method probes the local degree of network chain stretching, and is known to reect the dry-state crosslink density relevant for the mechanical behavior in linear response.In agreement with earlier work, the local chain stretching has been found to be rather inhomogeneous, and follow a two-stage process upon step-wise swelling.Based upon a reliable data analysis, and the observations of qualitatively different trends in the different sample series, it is concluded that previous observations on the rst stage are not universal, but that the second stage is characterized by an affine deformation of average crosslink positions.Based on previous evidence of the applicability of the phantom model in this range, we have established a back-extrapolation procedure of this affine range to a hypothetical "phantom reference state" of the bulk network, which by way of a comparison with results from equilibrium swelling was shown to reect the true crosslink density of the networks.
The deviation of this reference state from the actual dry state observables, e.g. the NMR-based crosslink density or elasticity modulus in the linear regime reects non-classical contributions to the network elasticity.Our data appear roughly consistent with a constant offset contribution that does not depend on crosslink density, ruling out simple arguments on crosslink-induced constraints to junction uctuations.Rather, the data appear consistent with the common interpretation as an apparently additive entanglement constraint (noting that deviations are expected for very low crosslink densities if the NMR experiments were conducted at very high temperatures 55 ).A big outstanding problem is however the role of network defects.The lowly crosslinked samples investigated in this study have been shown to exhibit a rather high (oen >20%) content of elastically inactive dangling structures when investigated by NMR in the swollen state; however the larger part of these defects contribute to the elastic response reected in the NMR response of the dry state and also in the mechanical response, at least on timescales up to the ms range.While such defects in many of our samples are not expected to exhibit molecular weights beyond the entanglement threshold, the question as to why they relax so slowly is still to be resolved.We envisage a possible explanation in terms of locally anisotropic excluded-volume constraints that may exhibit a sufficiently high correlation length even in unstrained bulk networks.These issues will be further investigated in future work, focusing in particular on the correlation of the NMR-determined apparent and true defect fractions with time-or frequency-dependent rheology results.We anticipate that the new concept of the "phantom reference network" established in this study may be of help to resolve relevant open questions in the science of elastomers.

Fig. 1
Fig. 1 (a) Weight fraction of network defects, u def , determined by DQ NMR in dry and equilibrium swollen ePDMS-21 and -25R networks, along with (b) the variation of the relative width of the coupling constant distribution, s rel , depending on the polymer volume fraction during crosslinking, 4 p,c .Both plots are double-logarithmic.(c) Defect fraction found in dry and swollen NR and IR as a function of NMR crosslink density $D res .All lines just guide the eye; the solid lines in (a) suggest a u def $ 4 p,c À1 dependence.

Fig. 2
Fig. 2 Average residual dipolar coupling constant, D res , as a function of the swelling degree, Q, for (a) PDMS networks end-crosslinked in solution at different polymer volume fractions and (b) natural and synthetic poly(isoprene) rubbers.

Fig. 3
Fig. 3 (a) Comparison of D res (Q) dependencies for different samples, along with fits of the second-stage data to eqn (4).(b) The relative distribution width s rel grows with Q in the first stage but is nearly constant in the second stage, indicating uniform deformation of the whole inhomogeneous network structure.

Fig. 4
Fig. 4 Log-log plot of second-stage D res data normalized with respect to the back-extrapolated coupling constant, D res,n , as a function of Q for various samples swollen in good (solid symbols) and q solvents (open symbols).The dashed line has a slope of 2/3 and indicates the affine prediction.

Fig. 5
Fig. 5 (a) Coupling constant distributions of ePDMS-25R t020 and IR-10 determined at different swelling degrees, Q, and (b) the same distributions normalized to their maximum amplitude plotted vs. a coupling constant reduced with respect to affine deformation.

Fig. 6
Fig. 6 Comparison of D res (Q) for networks prepared from different molecular weight pre-polymers, ePDMS-21 and ePDMS-25R, end-crosslinked in solution at different polymer concentrations, 4 p,c , in log-log representation.The dotted lines are power-law fits, and the vertical solid lines represent the crossover swelling degrees Q 0 .

Fig. 7 D
Fig. 7 D res (Q) for two radiation-crosslinked xPDMS samples compared to two solution end-linked ePDMS samples chosen to approximately match in the second stage of swelling.The lines are fits of eqn (4) with constant exponent 2/3 to the xPDMS sample data over the whole data range.

Fig. 8 D
Fig.8D res (Q) for three ePDMS-21 samples prepared at different concentrations, comparing swelling in good (toluene) and q solvents (styrene at 308 K).

Fig. 9
Fig. 9 Average residual coupling D res determined in dry networks (open symbols), and the respective back-extrapolated values, D res,n , obtained by evaluation of the late-stage swelling data according to eqn (4) (filled symbols), both as a function of the apparent crosslink density from swelling experiments given by eqn (3) for the (a) PDMS sample series and (b) the IR/NR series.The lines are linear fits.The log-log inset in (a) further supports the fact that the difference between the two quantities is mainly due to an almost constant contribution DD res that becomes negligible for the relations at high crosslink density.