Revisiting Segmental Order: A Simplified Approach ... - ACS Publications

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Revisiting Segmental Order: A Simplified Approach for Sulfur-Cured Rubbers Considering Junction Fluctuations and Entanglements Beatriz Basterra-Beroiz,† Robert Rommel,† François Kayser,† Juan L. Valentín,*,‡ Stephan Westermann,*,† and Gert Heinrich§ †

Goodyear Innovation Center Luxembourg, Avenue Gordon Smith, L-7750 Colmar-Berg, Luxembourg Institute of Polymer Science and Technology (CSIC), c/Juan de la Cierva 3, 28006 Madrid, Spain § Leibniz-Institut für Polymerforschung Dresden e.V., Hohe Straβe 6, 01069 Dresden, Germany ‡

S Supporting Information *

ABSTRACT: The use of modern multiple-quantum proton NMR experiments for the determination of cross-link density requires precise knowledge of several model-dependent physical and structural quantities, like the dipolar static frequency or the definition of vector segmental order. In this paper, different models for describing segmental order of the polymer backbone, based on different assumptions about the contribution of crosslinks and entanglements, are critically reviewed and applied for the analysis of unfilled natural rubber samples with average mass between cross-links and entanglements determined from network tube model fittings of stress−strain data. A recent theoretical model developed by Lang and Sommer, which allows for the consideration of junction fluctuations, is adapted for the analysis of NMR experimental results. After having verified the correlation between the calculations with this enhanced theoretical treatment and the residual dipolar coupling from multiple-quantum NMR experiments carried out in a low-field spectrometer, a new simplified approach to determine the segmental order parameter is proposed for sulfur-cured rubbers. broad range of elastomers.6,7,16,18−21 The DQ-NMR method is based on the residual second moment of the dipolar interaction, for which the relationship with cross-link density had been previously revealed through transversal relaxation experiments (e.g., refs 4 and 22). In spite of the advances done during the past years, a detailed quantitative discussion of the calculation of cross-link density from DQ-NMR experiments, which is influenced by several model factors as well as by the contribution of entanglements,16,23−25 is still missing. In a recent study,7 it was shown that the use of the tube model of rubber elasticity26 for the analysis of quasi-static uniaxial stress−strain measurements provided high consistency at the quantitative level with equilibrium swelling experiments using the Helmis−Heinrich−Straube model,27 which considers real networks with entanglements. This study constituted a step toward the convergence of the approaches for the data analysis of different experimental methods (mechanics and swelling) using the physical picture provided by the tube model as solid basis. Furthermore, the average mass between cross-links (Mc) resulting from the tube model analysis was compared to the NMR measurable, i.e., the residual dipolar coupling (Dres), showing a directly proportional relationship. The obtained

I. INTRODUCTION The characterization of rubber network structure, and in particular of cross-link density, is a fundamental question of high importance in rubber science and technology due to the partial dependence of physical properties on these parameters.1−3 Compound optimization for each application requires the adaptation and refinement of rubber formulations. In the case of complex systems, like passenger and truck tires, several types of rubbers can appear in different structural parts (e.g., natural rubber, styrene butadiene rubber, butadiene rubber, butyl rubber, ethylene propylene diene monomer rubber, and their blends). In this context, the evaluation of the required properties for industrial applications by means of versatile, inexpensive, and quick experimental methods becomes a key aspect. The characterization of cross-link density by different experimental methods is a recurring subject in scientific literature (e.g., refs 4−7). In particular, NMR spectroscopy has shown to be a useful tool for the study of the structure of polymer networks through the characterization of chain mobility (see for example examples in refs 8−14). During the past decade, the progress in the double-quantum (DQ) or more generally multiple-quantum (MQ) low-field NMR experiments and data treatment13,15−17 made this method increasingly popular, in part because the correlations with the results of other experimental methods showed to be satisfactory for a © XXXX American Chemical Society

Received: January 15, 2018 Revised: February 15, 2018

A

DOI: 10.1021/acs.macromol.8b00099 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules slope (1307 Hz·kg/mol)7 was around 2 times the result derived from a simulation by Saalwächter et al. (617 Hz·kg/mol).16 Similarly, in a recent publication28 a difference in a factor 2 was obtained with the simulations for PDMS, which was considered to be reasonable when considering a possible dependence of the prefactor on the network functionality. In addition, the expected contribution of entanglements to the NMR signal as the intercept of the linear fitting 1/Mc (tube model) − Dres was not observed, which required further investigations that could provide molecular arguments to explain this observation. In this context, the fundamentals of the relationship between residual dipolar coupling and molecular parameters (cross-link and entanglement densities), with special interest in the molecular models, underlying assumptions, and required model factors, are critically revisited in this work. Finally, a proposal for the analysis of polymer networks is presented and applied to the study of unfilled natural rubber, after which a pragmatic approach for moderately cross-linked rubbers (which is the most usual scenario in technical applications) is proposed.

between the residual dipolar coupling Dres (related to the residual dipolar coupling second moment M2,res) and the effective static dipolar constant Dstat/k (related to the effective second moment M2,stat eff), and it is therefore accessible through experiments: S b = ⟨P2(cos θ )⟩t =

3 m 5

M 2,res M 2,stat eff

(2)

where Dres results as a time average over the fluctuations of the dipolar tensor, representing an average over multiple inter- and intrasegmental dipolar couplings, as it will be discussed in following sections. Therefore, a link between the vector order parameter, which is dependent on the network structure, and the DQ-NMR measurable Dres can be made:

Dres 3 m= 5 Dstat /k

(3)

DQ-NMR experimental data consist of the so-called reference (Iref) and double-quantum (IDQ) intensities as a function of the DQ-evolution time (τDQ). These experiments are carried out far above the glass transition temperature Tg in order to access the plateau of the autocorrelation function32 C(t) for polymer networks, which defines Sb.13 The dynamic contribution from structural units with long characteristic relaxation times (like polymeric sol content, dangling chain ends, or plasticizers) is then identified and eliminated, obtaining the so-called normalized double-quantum intensity (InDQ) via data treatment procedures available in the literature.13 The residual dipolar coupling can be calculated approximately by fitting InDQ buildup curve to an expression that defines a Gaussian distribution of Dres13 or through an improved data treatment procedure consisting of an adapted Fast Tikhonov Regularization ( f tikreg) with a kernel function calibrated for homogeneous networks, like natural rubber (NR), polybutadiene rubber (BR), or polydimethylsiloxane (PDMS),33 and which allows for additionally visualizing the distribution of Dres in the frequency range. An illustrative example of application of this approach is the study of Valentiń et al.,34 who successfully used it to analyze differences in network structure of polybutadiene and natural rubber compounds for both sulfur and peroxide cure. In the following two subsections, the static limit dipolar constant Dstat and the vector segmental order m will be discussed in more detail. 1. Definition of the Static Limit Dipolar Constant, Dstat. The definition of Dstat as a function of the proton gyromagnetic ratio (γH) and the distance between two protons (rij) is given by the following expression:15

II. THEORETICAL BACKGROUND II.A. Relationship between DQ-NMR Measurements and Rubber Network Structure. The fundamental expression for establishing a quantitative correlation between the NMR measurable Dres and the network constraints (cross-links and entanglements) is based on a relationship between the tensor segmental order, i.e., the local dynamic order parameter of the polymer backbone (Sb) related to the residual dipolar coupling obtained from DQ-NMR experiments and the residual bond orientation also called the vector segmental order parameter (m). For dense systems, like dry networks, Gaussian statistics of the end-to-end vector are assumed and the equation reads Sb =

Dres = Dstat /k

(1)

The numerical prefactor in eq 1 is not universal and depends on the statistics of bond orientations (for example, 3/2 for a lattice model),29 which can be influenced, among other factors, by the conformational restrictions imposed by entanglements and topological constraints, e.g., deviations from Gaussian endto-end vector statistics in the range of short ends separations of longer chains.30 Details on the origin of Sb and m as well as investigations about the relationship between them in the swollen state can be found in ref 31. The dynamic order parameter obtained by NMR reflects the orientation dependence of the dipolar coupling according to the second Legendre polynomial P2(cos θ), where θ is the orientation of the internuclear axis with respect to the magnetic field. A preaveraged dipolar tensor is defined to simplify the treatment, and then θ represents the orientation of the local symmetry axis of motion (assuming to be along the backbone) rather than an individual bond or internuclear vector. According to this statement, the time dependence of θ(t) will reflect the orientation fluctuation of the polymer backbone. This preaveraging (within a Kuhn segment) embodies an effective static limit dipolar constant Dstat/k, with a modeldependent factor16 k = 1/P2 cos(α) where α is the angle between the internuclear coupling vector and the segment orientation.15 In consequence, the order parameter of the polymer backbone in rubber networks is defined as the quotient

Dstat =

μ0 γH 2ℏ 4π rij 3

(4)

The static dipolar frequency is inverse to the cube of the distance between dipoles, and therefore the dipolar frequency is higher the closer the spins. The value of Dstat is reduced by the fast local motions that occur within the Kuhn segment, and the residual dipolar orientation tensor is projected over the polymer backbone.16 The latter is captured by the factor k. The resulting effective static dipolar constant Dstat/k corresponds to the time average over the fluctuations of the dipolar tensor covering the time until τs is reached, which is defined as the characteristic time corresponding to the glass transition temperature.32 B


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Macromolecules The determination of the effective Dstat/k is a major shortcoming for the use of double-quantum experiments in a stand-alone manner in order to derive quantitative information on network structures. Gotlib et al.35 calculated the value of k36 for different geometrical directions, which was also discussed in depth by Fry and Lind37 in their study of highly cross-linked resins. In the case of NR, the result k = 2 (for the case in which chain and dipolar vector are parallel) was used by some authors as projection factor for predominant methylene groups (e.g., refs 22 and 38). Given that this description considers neither the effect of multiple spins but just a pair nor the existence of other motions (more complex than simple rotations) within a Kuhn segment, a matter of modeling (with the associated ambiguities) should be assumed in order to take these effects into consideration. In these terms, k could be considered a calibration factor. In order to minimize this problem, rotationalisomeric-state simulations have been applied to determine the rescaled dipolar constant for polyethylene.39 More recently, Saalwächter et al.16 applied atomistic spin dynamics simulations (which were verified using NMR experiments) to obtain the effective static limit dipolar constant for natural rubber and cisbutadiene rubber. In the particular case of NR they obtained (Dstat /k)NR = 6300 Hz 2π

that the value of the static frequency would vary for blends with different fractions of the composing single polymers. 2. Definition of the Vector Segmental Order, m. In addition to the value of Dstat/k, the relationship between Dres and cross-link density strongly depends on the model describing the vector segmental order m. The anisotropy in the relaxation of the polymer chains after the excitation caused by a radiofrequency pulse sequence is due to the presence of topological constraints that limit this movement, namely crosslinks and entanglements, whose contributions are part of the model for the calculation of m. In order to account for this effect, several models have been proposed in the literature: a. Affine Model. The use of the affine model in the context of NMR research is based on the relationship between refraction indices in birefringence,5 for which Kuhn and Grün45 formulated the stress optical rule: 3 r2 5N

Sb =

(6)

In eq 6, r indicates the ratio of the end-to-end vector to its average, unperturbed melt state (considered to be effectively equal to the unity for undeformed samples) and N is the number of statistical chain segments between structural constraints. The replacement of N in terms of the number of statistical segments between cross-links Nc and entanglements Ne can be done in two different manners: a.i. Single Term, Function of Cross-Link Density. The affine model considers that cross-links are the dominant contribution and predicts for the segmental order:

(5) 22,40−42

Considering that Dstat/2π is in the range of 20−24 kHz, the above value has a related k ≈ 3.2, which is in between the predominant methylene (k = 2) and methyl values (k = 4) calculated using more simplistic geometrical factors for a system of two spins.22,35 However, this approach based on atomistic molecular dynamics simulation is not applicable to more complex copolymers, limiting its use. In conclusion, the value of the factor k, necessary to establish a correlation between the measured Dres and cross-link density, is both a material- and model-dependent16,41 constant that is consequently subject to the uncertainty in the method chosen for its determination. Some publications mention estimations of the expected errors from the correlation between Mc and Dres due to systematic errors in the equations (that include the assumed Dstat/k). For instance, Schlögl et al.21 estimate that the 1/Mc,NMR values have uncertainties of at least 50%; Chassé et al.43 give a value of 40% for PDMS and discuss that the average mass between cross-links by DQ-NMR in dry samples is underestimated by 30% (which is attributed to the suitability of the chosen model for the estimation of the calibration factor); Vaca Chávez et al.32 consider that the values of the autocorrelation function C(t) in their study have an error of around 40% due to the model dependence of the factor Dstat/k. Alternative approaches consist of the calculation of a calibration factor between the DQ-NMR result and an effective cross-link density by other experimental methods: For styrene− butadiene rubber (SBR), Mujtaba et al.20 used a calibration with the moduli from dynamic mechanical experiments, whereas Syed et al.25 obtained the calibration factor for nitrile butadiene rubbers (NBR) by combining the NMR measurements with the molecular parameters obtained from uniaxial stress−strain experiments. In the case of our previous study on unfilled NR,7 it was found that a linear fitting without intercept relates Dres and cross-link density calculated from tube model fittings of uniaxial stress−strain curves. Finally, the application of DQ-NMR experiments and generalization of the method for the quantitative analysis of cross-link density in blends is still an open challenge,44 given

m=

1 Nc

(7)

where Nc is the average number of segments between crosslinks. This model was used to interpret NMR data in several publications (e.g., refs 13, 19, and 34) and corresponds to an affine model as described by Lang46 in his extensive study on segmental order. a.ii. Two Additive Terms, Accounting for Cross-Links and Entanglements. In 2005, Saalwächter et al.16 assumed in a simplistic ad hoc approach that the NMR observable consisted of additive contributions of equal weights from cross-links and entanglements in order to account for the observed intercept in the fitting of 1/Mc values from equilibrium swelling and DQNMR: 1 NNMR

≈

1 1 + Nc Ne

(8)

Equation 8 has also been recently used for the study of EPDM by Saleesung et al.47 This approach is equivalent to the classical data treatment used for the analysis of T2 relaxation experiments, in which the additive contribution from crosslinks and entanglements is assumed (e.g., refs 4, 12, 22, and 48). However, the variation of the entanglement density upon curing7,49 as well as the need of an additional measurement on the un-cross-linked counterpart (which presents experimental complexities21) represents a practical shortcoming for the application of this strategy. In addition, for lowly cross-linked samples a square root behavior as proposed by Lang and Sommer50 (whose model will be presented below) is expected. These authors also showed that the additivity of the contributions of cross-links and entanglements does not agree with the tube model of Edwards.51 C


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Macromolecules b. Phantom Model. Analogously to the affine model, two cases are distinguished among the phantom models considering free junction fluctuations: b.I. Single Term, Function of Cross-Link Density. The effective number of statistical segments in a phantom network, which depends on the network functionality f,52 was applied first to DQ-NMR calculations by Chassé et al.:43

m=

f−2 1 f Nc

segmental order concept to entangled rubbers by considering a chain of length Nc connected to an elastic background at the cross-link points, which fluctuate in a range n. The total chain length is thus Nc′ = Nc + 2n. Each monomer h56 of the chain can be connected to the elastic background through a virtual chain of length N′h and diffuses along the path of the confining tube formed by the neighboring entanglements. Basing their arguments on the correlation of monomer fluctuations and segmental order, Lang and Sommer46,50 derived the following expression for the monomer h

(9) 6,21

Subsequent publications have often used this model; however, it should be noted that while it is an appropriate description for model samples where the contribution of crosslinks is dominant, the extension of its use to other types of samples has to be validated after the consideration of entanglement effects. b.ii. Two Additive Terms, Accounting for Cross-Links and Entanglements. An alternative approach considering both the phantom model prefactor and additive contributions of crosslinks and entanglements was used in a very recent article.25 The picture would correspond to a network cross-linked and entangled where junctions can fluctuate freely. This ad hoc approach is denoted in the following as phantom phenomenological and is described through the equation m=

f− f

2⎛ 1 1⎞ + ⎟ ⎜ Ne ⎠ ⎝ Nc

m(Nc , h) ≈

d 2⎞ d2 ⎞ 1⎛ 1⎛ ⎜1 − 0 ⎟ = ⎜1 − 20 ⎟ Nc ⎝ bL ⎠ Nc ⎝ b Nc ⎠

(12)

with

y2 =

2Nh′ 3Np

(13)

and assigning erfc to the complementary error function as well as Ne and Np to the average number of segments between entanglements inside the tube and of the primitive path, respectively. The length of the virtual chain can be calculated according to the following equation: Nh′ =

(n + h)[Nc′ − (n + h)] N n + n2 + hNc − h2 = c Nc′ Nc + 2n (14)

(10)

where h varies between 0 and Nc along the chain. Equation 12 needs additional corrections (written in terms of the rootmean-square length of a bond l, the effective bond length le, and a coefficient associated with chain repulsion ce57 as well as the fraction of elastically active material ϕac) in order to account for the increased size of the average square chain extension and the monomer fluctuations along the tube.46 The final expression to be used for calculations is

A similar proposal based on the consideration of free junction fluctuations and the sum of cross-links and a term of trapped entanglements was proposed by Campise et al.28 c. Sommer−Heinrich−Straube. A tube model based26 prediction of segmental order was formulated by Sommer et al.53,54 The starting point was the continuous chain model of Edwards51 using harmonic-like configurational constraints. The vanishing constraint case led to the classical Kuhn and Grün result of freely rotating chains. Two cases were distinguished: (i) weak constraints (valid for dominating cross-links and weak entanglement restrictions; this expression was recently recovered in the study of Ott et al.55) and (ii) strong constraints (corresponding to the behavior of entanglementdominated networks). The second case is of higher interest for this study and, by extension, for most of the real and technically relevant rubber compounds, where moderately cross-linked networks are considered. In this case, the obtained segmental order parameter is m=

1 y2 e erfc(y) Ne

m(Nc , h) ≈

2

1 le 2

( )(1 − ) l2

cele l Nc

1 y′ 1 e erfc(y′) 2/3 Ne ϕac

(15)

with ⎛ l 2 ⎞⎛ cl y′2 ≈ ⎜ e2 ⎟⎜⎜1 − e e ⎝ l ⎠⎝ l Nh′

⎞ 2N′ ⎟ h ⎟ 3N ⎠ p

(16)

for Nh′ and Nc ≫ 1. Note that these correction factors as written in ref 46 would lead to negative results; instead, the equations should be consistent with eq 15 in the same reference. The values l = 2.636, le ≈ 1.23l, and ce ≈ 0.41 used for the bond fluctuation model (BFM)46 are taken from ref 57. As shown in ref 46, the segmental order expression introduced in eq 15 in the present work provides good agreement with simulation data in the center of the chains while deviations are obtained for the monomers close to the cross-linking points. An alternative expression which predicts the same result for the monomers in the inner part of the chain and fits better simulation data for the monomers close to the cross-link positions was found by Lang:46 z mh ≈ NpNh′ (17)

(11)

The lateral tube dimension d0 represents in this case the confining potential of the chain segments,26 which are oriented along the tube, L (L = Ncb) is the contour length, and b is the length of a Kuhn statistical segment. It should be noted that the application of this model presents limitations in the strong constraints case since the main equations result from the averaging over all the segments, while the NMR measurable is based on the averaging over the measurement of each segment (and therefore the local orientation needs to be considered). Consequently, a good representation is only expected in the case of weak constraints. d. Lang−Sommer Model.46,50 Lang and Sommer proposed recently a theoretical treatment for the extension of the

where z is approximately constant for all the simulated networks. D


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Macromolecules

(free junction fluctuations, Ac = 0.5) the fluctuations are maximum (n = Nc/2) while the result n = 0 is obtained for the affine model (Ac = 1, fixed junctions). Inserting the result of eq 23 into eq 22 gives

The relationship between Np and Ne and their correlation with the average number of segments between entanglements obtained from the entanglement modulus Ge in the tube model fittings58 needs to be taken into account in order to perform quantitative calculations within this model. The following correspondence will be used between Np and Ne in the framework of the tube model of Heinrich et al. in consistency with the definition of the modulus:26 Np Ne

=

4 5

m=

=

1 Nc

(18)

∫0

Nc

z Np

z Nc + 2n Nc Np

∫0

Nc + 2n Ncn + n2 + hNc − h2 Nc

m∝

(19)

which gives m=

z Nc + 2n Nc Np

⎛ Nc 2 arctan⎜⎜ 2 ⎝ 2 Ncn + n

⎞ ⎟ ⎟ ⎠

(20)

Considering the relationship arctan(x) =

⎛1⎞ π − arctan⎜ ⎟ ⎝x⎠ 2

Table 1. Summary of Segmental Order Models and Their Main Assumptions

(22)

59

The microstructure factor Ac is a prefactor that appears within the definition of the cross-link modulus Gc60 and which accounts for the fluctuations of cross-links (more details are provided in the Experimental Section, where the basics of the tube model of rubber elasticity and its main parameters for fittings of stress−strain data are summarized). For a combined chain, Ac is defined as the ratio between the number of segments in the real (Nc) and in the combined (Nc + 2n) chain,61 and consequently, n can be written according to the following expression:

n=

Nc(1 − Ac) 2Ac

(25)

(21)

and the power series expression of arctan(1/x), for which the approximation arctan(1/x) ≈ 1/x holds for x ≫ 1, i.e., n ≪ Nc, a simplified expression for segmental order can be derived: ⎡ ⎤ z Nc + 2n ⎢ π 2 Ncn + n2 ⎥ − 2 m= ⎢2 ⎥ Nc Nc Np ⎣ ⎦

1 NpNc

This expression shows the theoretically predicted dependence50 with the inverse of NpNc . In this work, eq 20 and its approximation (eq 22) in terms of n or eq 24 in terms of Ac are used for a quantitative analysis of m. Initially, differences between eqs 20 and 22 will be evaluated in the Results and Discussion section for systems of known m from simulations.46 The most precise analytical expression eq 20, which includes the explicit consideration of entanglements and junction fluctuations, will be used for the analysis of the segmental order for unfilled NR upon knowledge of Nc, Ne, and Ac, which are obtained from the analysis of stress−strain curves7 with the nonaffine tube model of rubber elasticity.26,62,63 Finally, the segmental order obtained by application of Lang−Sommer model (which includes the constraints due to entanglements and junctions fluctuations that appear in the common description of elasticity of real rubber networks) will be compared with those reached by using other models previously presented (and applied) in the literature. In this sense, an overview on the segmental order models for data analysis is presented in Table 1, where a short identifying name is given to each model in order to clarify and simplify the referencing to them during the analysis and discussion.

dh

1 dh Ncn + n + hNc − h2 2

(24)

The factor between brackets in eq 24 is only dependent on the Ac factor so that the dependence of m on Nc and Np can be easily identified:

The results of the different segmental order approaches will be compared for unfilled NR samples in dry state in the Results and Discussion section. II.B. New Analytical Expression for the Consideration of Junction Fluctuations. In order to use the Lang−Sommer model for calculation, it is necessary to derive an analytical expression where the segmental order m is written as a function of the average number of segments between cross-links, Nc, and entanglements, Ne. To achieve this, eq 17 needs to be integrated (considering eq 14) for the real chain (between h = 0 and h = Nc): m=

⎡ 2⎤ 2z ⎢ π − 2 1 − Ac + ⎛ 1 − Ac ⎞ ⎥ ⎜ ⎟ 2Ac AcNpNc ⎢ 2 ⎝ 2Ac ⎠ ⎥⎦ ⎣

model

consideration of entanglements

junction fluctuations

equation

affine affine phenomenological phantom phantom phenomenological Sommer−Heinrich−Straube integrated Lang−Sommer

no yes no yes yes yes

fixed fixed free free fixed constrained

7 8 9 10 11 20

II.C. Analytical Expression for the Calculation of the Parameter z. The use of the equations derived in the previous section requires a quantitative value of the constant z, which in ref 46 showed to be almost constant for the different investigated chain sizes. A simple approach for its estimation is proposed in this section based on the observation that eqs 15 and 17 provide approximately the same results in the middle of the chain, while differing for the positions near the cross-linking points. This assumption about the agreement of both functions is an approximation since, in reality, eq 15 shows higher m values at the middle of the chain than the simulation results, which agrees very well with eq 17 (see Figures 10 and 13 in ref 46). Considering the middle point in eqs 15 and 17, for which h

(23)

which allows for considering cross-link fluctuations in segmental order in consistency with the tube model approach for mechanics and swelling. Note that for the phantom model E


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Macromolecules = Nc/2 and N′h = Nc/4 + n/2 and inserting the BFM parameters from ref 57, results in

(

Np z≈

Nc 4

+

(

Ne 1.513 −

n 2

Table 2. Molecular Parameters Obtained from the Tube Model Fitting Results of Uniaxial Stress−Strain Curves for Unfilled NR Compounds Reported in Ref 7

)

0.763 Nc

)

⎞ ⎛⎛⎛ N ⎛N n⎞ n⎞⎞ 1 exp⎜⎜⎜⎜⎜ c + ⎟ − 0.5043 ⎜ c + ⎟ ⎟⎟ ⎟⎟ ⎝4 2⎠ 2 ⎠ ⎠ Np ⎠ ⎝⎝⎝ 4 ⎛ erfc⎜ ⎜ ⎝

⎛⎛ N ⎛N n⎞ n⎞⎞ 1 ⎜⎜⎜ c + ⎟ − 0.5043 ⎜ c + ⎟ ⎟⎟ ⎝4 2⎠ 2 ⎠ ⎠ Np ⎝⎝ 4

⎞ ⎟ 1 ⎟ ϕ 2/3 ⎠ ac (26)

sample

Gc (MPa)

Ge (MPa)

Ac

d0 (nm)

Nc

Ne

160 163 166 169 172 175 178 181 184 187 190

0.345 0.409 0.341 0.430 0.376 0.443 0.244 0.473 0.385 0.320 0.320

0.369 0.275 0.233 0.334 0.338 0.339 0.223 0.372 0.337 0.335 0.207

0.802 0.695 0.699 0.732 0.768 0.729 0.772 0.736 0.761 0.801 0.684

1.93 2.23 2.42 2.02 2.01 2.01 2.48 1.92 2.01 2.02 2.57

40.4 29.4 35.5 29.4 35.3 28.5 54.7 26.9 34.2 43.4 37.0

19.3 25.7 30.4 21.1 20.9 20.8 31.7 19.0 21.0 21.1 34.2

Replacing n by the use of eq 23: z≈

0.447 Nc

(1.513 − ) 0.763 Nc

⎛ erfc⎜ ⎜ ⎝

⎛N ⎜⎜ c − ⎝ Ac

⎛⎛ N exp⎜⎜⎜⎜ c − ⎝⎝ A c AcNe

Nc ⎞ 1 ⎟⎟ Ac ⎠ 16 Ne 5

⎞ Nc ⎞ 1 ⎟ ⎟⎟ 16 Ac ⎠ Ne ⎟⎠

define the macroscopic stress (σ) as a function of the applied extension ratio (λ) in the case of uniaxial extension:

⎡ ⎤ ⎢ (1 − δ 2)(λ − λ−2) ⎥ δ 2(λ − λ−2) σ = Gc⎢ − ⎥ 2 2 ⎡1 − δ 2 λ 2 + − 3 ⎤ ⎥ ⎢ ⎡1 − δ 2 λ 2 + 2 − 3 ⎤ ⎣ ⎦ λ ⎦ ⎣⎣ ⎦ λ 2Ge + (− λ−β − 1 + λ β /2 − 1) β (29)

5

⎞ ⎟ 1 ⎟ ϕ 2/3 ⎠ ac

(

(27)

Assuming Nc ≫ n, the following equation becomes a reasonable approximation of eq 26: z≈

NpNc

(

3 1−

1 2 Nc

)

⎛ Nc ⎛ ⎜1 − erfc⎜ ⎜ 4Np ⎜⎝ ⎝

⎛ N ⎛ ⎛1⎞ ⎜ ⎟ exp⎜⎜ c ⎜⎜1 − ⎝ Ne ⎠ ⎝ 4Np ⎝ 1 Nc

⎞⎞ 1 ⎟⎟ ⎟ 2/3 ⎠ ⎟⎠ ϕac

1 Nc

(

)

)

In eq 29, Gc is the cross-link modulus, Ge is the entanglement modulus, δ is a chain finite extensibility parameter related to the trapping of entanglements, and β is a parameter accounting for the release of topological constraints upon deformation. For more details and a list of references on each of the involved parameters, the reader is referred to our previous publication7 where this topic was covered in more detail. Although the description and discussion of the theoretical background of this model are out of the scope of this work, it is important to mention that this approach has been broadly applied for the understanding of the elastic behavior of rubbers26,62,63 (in some cases, also in conjunction with DQ-NMR experiments),21,25 and it is considered a powerful constitutive model for engineering applications.64 Therefore, it constitutes a solid, physically based model for the determination of average molecular parameters that can be used as reference for the segmental order calculations using the different physical models presented in the previous section. For this purpose, the fit parameters of interest are two: the crosslink modulus Gc and the entanglement modulus Ge. The latter contains molecular information related to entanglements, which is represented by the radius of the undeformed tube d0 and the average mass between entanglements Me:

⎞⎞ ⎟⎟⎟⎟ ⎠⎠

(28)

The preceding expressions allow for an analytical approximated calculation of the parameter z. It should be noted that the values of z using eqs 26, 27, and 28 are underestimated due to the chosen comparison point for both expressions eqs 15 and 17. In addition, the consideration of the neglected terms in the approximated expression will lead to higher values of z. It is also important to note that the requirement Np ≤ Ne46 was not fulfilled for all the simulation results in ref 46.

III. EXPERIMENTAL SECTION Ge =

The set of analyzed samples consisted of unfilled natural rubber vulcanized with different contents of sulfur and accelerator (Ncyclohexyl-2-benzothiazolsulfenamide, CBS), with amounts of each varying in a systematic manner and constant concentrations of the rest of the ingredients (N-(1,2-dimethylbutyl)-N′-phenyl-p-phenylenediamine 6-PPD, ZnO, stearic acid). The samples were mixed in a tangential laboratory mixer and vulcanized in 2 mm sheets at 150 °C up to t90. Each of the compounds is identified by a three-number denomination. Three compounds (163, 172, and 184) were mixed with the same formulation and were used as repeatability check. More details about formulations can be found in a previous publication7 and the Supporting Information (Table S1). Mechanical Testing. Uniaxial quasi-static (10 mm/min) stress− strain tests of S2 samples of 2 mm thickness were carried out in a Zwick tensile tester at 23 °C until break. In this work, the nonaffine tube model of rubber elasticity developed by Heinrich, Straube, and Helmis26 including non-Gaussian chain extensibility corrections62,63 has been applied to obtain the molecular parameters (Table 2) that

1 4 6

kBTnsb2 d0

2

=

1 ρRT 6 Me

(30)

where kB is the Boltzmann constant, R the gas constant, T the temperature, ρ the density of rubber, b the length of the Kuhn segment, and ns the polymer segment number density, calculated as ns = ρNA/Ms, where Ms is the mass of the statistical Kuhn segment and NA is Avogadro’s constant. The value of Ne is then directly obtained from the relationship with the tube radius d0:58 Ne =

Me 4d 2 = 20 Ms b

(31)

In the case of the average number of statistical segments between cross-links Nc, it is calculated from the cross-link modulus Gc:60

Gc = Ac

ρRT ρRT = Ac Mc NcMs

(32) 7,60

where Ac is the microstructure factor defined as F


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Macromolecules Table 3. Results of the Segmental Order Calculations Using the Analytical Integral Solution Eq 20 and the Proposed Approximated Expression 22 for the Simulation Data in Ref 46a calculations using simulation results in ref 46 m analytical integral eq 20

relative difference eq 20 (%)

m approximated expression eq 22


0.073 0.045 0.032 0.022 0.017

7 3 1 3 1

0.062 0.041 0.030 0.022 0.017

9 6 6 5 2

a

The relative difference columns indicate the deviation between the calculations in this paper taking as reference the values from simulations in ref 46.

Table 4. Results of the Calculations of the Constant z Using an Analytical Expression Eq 26 and the Proposed Approximated Expression 28 for the Simulation Data in Ref 46a calculations using simulation results in ref 46 z analytical integral eq 26


z approximated expression eq 28


0.434 0.425 0.397 0.386 0.372

5 6 2 5 1

0.414 0.413 0.393 0.386 0.374

0 3 1 5 2

a

The relative difference columns indicate the deviation between the calculations in this paper taking as reference the values from simulations in ref 46. −K 1 1 Ke c Ac = + 1/2 c 2 erf Kc π

with Kc =

3Ne 2Nc

2

the parameter z. It should be noted that the difference between eqs 20 and 22 is higher for the shortest Nc value (15%, taking as reference the analytical result) and minimized for the longest Nc (1%). Table 3 shows the estimated z values making use of the proposed approach based on the agreement of analytical expressions in the middle of the chain eq 26 and its approximation eq 28. Both calculations show low relative differences taking the simulation results as reference. In the case of the approximation eq 28 it is expected to be better for longer chains taking into account the assumptions taken for its calculation; however, in all the cases the obtained accuracy is satisfactory and sometimes, surprisingly, even better than for the complete eq 26. The expressions for the calculation of z are dependent on n and therefore on Ac. The results in ref 46 varied slightly over several Nc lengths, while the related values of Ac varied only in a small range. It should be noted that z is calculated here considering a single point (the one at the middle of the chain) while in ref 46 the reported values were obtained from a fitting versus simulation results. IV.B. Comparison of Segmental Order Models for Unfilled NR Compounds. The segmental order models reviewed in the section, as well as the analytical expression for the application of the Lang-Sommer model calculated at the beginning of this section (eq 20), are applied for unfilled NR compounds using the average number of statistical segments between cross-links Nc and entanglements Ne as obtained by fittings of stress−strain curves using the tube model (Table 2). The Nc values obtained from the tube model fits were shown to be in very good agreement with the results of the analysis of equilibrium swelling experiments with the Helmis−Heinrich− Straube swelling model.7 The calculated microstructure factor Ac from mechanics (Table 2) is used to determine the value of n for each sample, which is required for applying the integrated Lang−Sommer model. For all the selected segmental order models, the resulting correlation with the residual dipolar

(33)

. The above equation implies that the Ac value, which

reflects the amount of fluctuations of the cross-links in the rubber network, depends on the sample structural characteristics, i.e., the Ne/ Nc ratio. For that reason, a self-consistent numerical code was used for the calculation of the microstructure factor Ac for each unfilled NR compound separately. An example of the application of the tube model for the fitting of the stress−strain curves of the studied unfilled NR samples is shown in the Supporting Information (Figure S1). DQ-NMR Spectroscopy. Double-quantum proton NMR experiments were performed by using an improved Baum and Pines pulse sequence15 at 80 °C (high above the glass transition temperature of NR Tg ∼ −57.5 °C) in a Bruker low-field NMR spectrometer operating at 0.5 T equipped with a 10 mm variable temperature probe with a 90° pulse length of 3.2 μs. Two rubber disks of 8 mm diameter punched out from the 2 mm sheets were used as specimens. The estimation of the nonelastic fraction of the rubber network43 was done from experimental data of DQ experiments at 40 °C in deuterated toluene. The sample size was adapted so that swelling was not restricted by the tube walls.

IV. RESULTS AND DISCUSSION IV.A. Validation of Integrated Lang−Sommer Model Using Simulation Results. The results of the segmental order calculations using the exact eq 20 and its approximation (eq 22) are compared in Table 3 with the simulation results of Lang et al., whose detailed data (Np, Ne, n, z) for the different chain sizes (Nc from 16 to 256) can be found in ref 46. The relative differences for the calculations with the exact and approximated expressions are below 10%, indicating a very high agreement between simulation results and the predictions from the new analytical expressions. The sample with Nc = 16 is the one presenting the highest difference between them, which could be related to the higher influence of the segments in the vicinity of cross-link points or to the uncertainties in the determination of G


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Macromolecules coupling Dres from DQ-NMR measurements is displayed in Figure 1.

In terms of absolute values, the integrated Lang−Sommer model as applied in this paper is close to the Sommer− Heinrich−Straube and the phantom model results. Note that the observed differences in the estimation of m are directly influenced by the value of z and the relative influence of entanglements, which in this case is similar for all the samples. As explained in the Theoretical Background, the Sommer− Heinrich−Straube model is a good approach only for weak constraints, and therefore the good agreement with the integrated Lang−Sommer model is not expected to hold universally. It is also important to note that the good agreement between the integrated Lang−Sommer and the phantom model results is not valid in general. A detailed mathematical analysis of the dependence of the integrated Lang−Sommer model equation on Nc and Ne shows that this model is close to the phantom model approximately in the parameter range Nc ≈ Ne. The dependence of the integrated Lang−Sommer m on the Nc/ Ne ratio is more evident at low Nc values, for which a higher value of Nc/Ne is translated into a higher difference with the phantom model. On the other hand, for Nc/Ne < 1, the integrated Lang−Sommer model provides values below the predictions of the phantom model. It is also obvious that the models excluding the influence of entanglements (affine and phantom) represent an oversimplified picture of real network structures, where also entanglements are known to play an important role. In addition, the correlation of m resulting from the phantom and/or the affine model with Dres has a complex interpretation since the NMR observable depends on all network constraints (cross-links and entanglements). Models where entanglements have been taken into consideration are expected to be more realistic and better suited to predict in a quantitative way the segmental order vector of a rubber network according to the NMR measurements. However, the phenomenological affine and phantom models with entanglements overestimate the predicted value of m (by factors from ∼1.5 to 2 for the phantom and from ∼2.5 to 4 for the affine) with respect to the two models with a more robust and consistent physical background to define the molecular parameters of entangled real networks, i.e., the Sommer−Heinrich−Straube and Lang− Sommer models, respectively. Note that the phenomenological phantom and affine models are based on ad hoc arguments which assume the same effect of cross-links and entanglements, which might result too simplistic to describe the behavior of complex cross-linked and entangled polymer networks. In summary, Figure 1 indicates that the residual dipolar coupling, i.e., the DQ-NMR observable, reflects the segmental order of rubber backbone and its variation with the cross-link density (independently of the adopted model), but the quantitative estimation of m from NMR measurements is strongly influenced by the assumed model. IV.C. Interpretation of the Slope of the Linear Fitting between the NMR Observable (Dres) and the Vector Segmental Order (m). After having evidenced the consistency of the segmental order trends predicted by the different models versus the residual dipolar coupling, a deeper analysis on the quantitative aspects of the relationship Dres−m is done in this subsection using the integrated Lang−Sommer equation (eq 20) as this approach could be considered the most general and complete model to date based on a theoretical robust and realistic picture incorporating cross-links, entanglements, and junction fluctuations.

Figure 1. Predicted segmental order (m) resulting from the different segmental order models (calculated using the tube model results in Table 2) presented in the introduction (affine, eq 7; affine phenomenological, eq 8; phantom, eq 9; phantom phenomenological, eq 10; Sommer−Heinrich−Straube, eq 11; integrated Lang−Sommer, eq 20) versus the residual dipolar coupling (Dres) from low-field DQNMR experiments.

For the calculation with the Lang−Sommer model, the calculation with eq 27 gives an average z = 0.249. In order to obtain this result, the presence of elastically inactive fractions (like sol and dangling chains) was taken into account by measuring the nonelastic rubber fraction ωnonelastic through DQNMR experiments of samples swollen in deuterated toluene. Then, the elastically active fraction was calculated as ϕac = 1 − ωnonelastic. The range of nonelastic rubber fractions (in swollen state) for the analyzed set of sulfur cured unfilled NR samples was between 7 and 12%. It is noted that the small fraction of nonelastic network fractions (compared to other reported NR samples34) does not have a large influence in the result, since z = 0.235 is obtained if all the chains are assumed to be active (ϕac = 1). The average value of z = 0.249 is used in the subsequent analysis in order to avoid that the scattering of the individual z estimations for each sample influences the analysis of the dependence of m on cross-links and entanglements. Neglecting the elastically inactive fractions in the tube model analysis from mechanics will also slightly influence the m values for all the models; however, in view of the very low nonelastic rubber fraction, this effect is not expected to be significant. In the following discussion, the main aim is to compare the variation of the order parameter according to the adopted model, taking as reference the integrated Lang−Sommer model as it represents the most complete and recent theoretically based model among the ones available, and it includes the contribution to rubber elasticity from entanglements and junction fluctuations through a physically based theoretical approach. A deeper discussion about the validation of each individual model is out of the scope of this work. Figure 1 displays the systematic increase of the m values from all the models with increasing Dres, as expected from eq 3, which can be reasonably well fitted to a line without intercept within the experimental scatter of the data. H


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Table 5. Analysis of the Factors Influencing the Relationship between Dres and m and Their Effect on the Obtained Dstat/k linear fitting without intercept integrated L−S eq with z = 0.249 no correction of interchain effects slope (Hz)

Dstat/k (Hz)

15100 prefactor (Gaussian stat) = 3/5 25200

prefactor (BFM) = 1/1.213 18300

integrated L−S eq with zavg = 0.387

correcting the interchain effect (factor = 1.466) 10800 prefactor (Gaussian stat) = 3/5 18000

prefactor (BFM) = 1/1.213 13100

The linear fittings (with and without intercept) of the residual dipolar coupling data as a function of the segmental order (predicted by using the analytical eq 20) show high correlation coefficients (R2 > 0.85). The obtained slope value of the studied samples is reduced by ∼15% when allowing for the fitting of an intercept. The y-intercept of the linear fitting has no physical meaning according to the theory background (see eq 3), and it is subject to important uncertainties since it has an associated relative error around 60%. For these reasons, the fitting with two degrees of freedom (slope and intercept) will be not used in the subsequent discussion about the interpretation of the slope. Nevertheless, it is important to consider that the possible presence of an intercept needs to be carefully discussed for different types of rubber compounds, since this term could be present as a consequence of experimental uncertainties and systematic errors (e.g., related to limitations of the models applied to extract the correlated parameters) as well as due to sample-related factors, such as the heterogeneous distributions of molecular structural parameters or the presence of significant nonelastic chain fractions. The slope of the linear fitting of Dres versus m without yintercept is ∼15 100 Hz, with a relative error around 2%. Considering the relationship between Sb and m in eq 3, a value of (Dstat/k)/2π ∼ 25 200 Hz is obtained, which is between 3 and 4 times higher than the one obtained from atomistic simulations16 and almost equal to the static frequency Dstat/2π when preaveraging of intrasegmental motions are not considered. This unexpected result, which in principle could seem in conflict with the NMR background explained in previous sections (e.g., the concept of NMR submolecule), can be due to several factors that significantly modify the factor Dstat/k. In what follows, this initial result will be discussed in comparison to analogous frequencies which result from using different assumptions in the chain model, the fitting approach or interchain effects, which are summarized in Table 5. The first aspect of discussion relates to the chain model used to calculate Dstat/k. The initial calculation above was performed using the Gaussian chain model for establishing the numerical factor relating Sb and m. The predictions of segmental order from Lang and Sommer,46,50 however, are based on simulations using the bond fluctuation model, which has a direct impact on the coefficients in the expression of m in comparison to the Gaussian prediction. In this case, a different relation between Sb and m holds according to Sommer and Saalwächter:29 S b(BFM) ≅ S b· 1.374 =

3 m m ·1.374 = 5 1.213

no correction of interchain effects 9700 prefactor (Gaussian stat) = 3/5 16200

prefactor (BFM) = 1/1.213 11800

correcting the interchain effect (factor = 1.466) 6900 prefactor (Gaussian stat) = 3/5 11500

prefactor (BFM) = 1/1.213 8400

selecting the middle segment of the chain as reference, while being directly influenced by the assumptions considered in eq 26 and the values of ref 57. A reasonable alternative is to use the average value zL̅ ang ≈ 0.387 from the simulation results fittings of Lang,46 which showed only slight variations upon large differences in chain size. The use of this value for calculation with eq 20 would provide a quite different slope of 9700 Hz, related to a different (Dstat/k)/2π value (see Table 5). These observations show that the precise parametrization of Dstat/k is an important issue to take into account and cannot be easily overcome by simulations, which evaluate the vector order parameter for a backbone bond that is parallel to the chain tangent vector. In consequence, they are not sensitive to any rotational motions around the chain axis; a factor that is effectively encoded in the NMR measurements.16 In addition to these aspects, it is known that the measured residual dipolar coupling represents an average over multiple intra- and intersegmental couplings.13,65 Especially the latter is a factor difficult to determine, and it might be, at this moment, the main source of uncertainty in order to obtain quantitative estimations of average cross-link density from DQ-NMR measurements. In a recent article,66 multiple-quantum (MQ) NMR experiments of high-molecular-weight polybutadiene using an isotope dilution strategy were used to quantify the interchain dipole−dipole coupling factor which is not taken into account in the theoretical models for data interpretation presented in previous sections. Furtado et al.66 concluded that a systematic deviation appeared in the local segmental order parameter by a factor of around 1.4, which would mean that the measured Dres overestimate the intrachain residual dipolar coupling by this factor. The consideration of this effect for quantitative data evaluation of DQ proton NMR experiments reduces the estimated (Dstat/k)/2π prefactor that could reach values of 8400 Hz when the average z from Lang’s simulations46 is applied (see Table 5). In this sense, Fatkullin and co-workers65,67 have evaluated different models that seem to indicate an increasing relative intermolecular effect for polymer melts when the evolution time increases in the constrained Rouse regime (regime II). This frequency-dependent effect could have some influence for weakly cross-linked samples. Lang46 also discussed the influence of polydispersity on segmental order, which had been previously pointed out also by other authors (e.g., ref 68). Higgs and Ball69 had earlier demonstrated that the second Legendre polynomial characteristic of chain orientation in networks follows the same expression in both the monodisperse and polydisperse cases for phantom networks. Saalwächter and Sommer30 discussed the lack of agreement between experimental results and the predictions considering polydispersity of randomly cross-linked NR and concluded that the influence of polydispersity on Dres is negligible due to spatial averaging of chain dynamics. The lack

(34)

for which values of (Dstat/k)/2π ∼ 18 300 Hz would be obtained. As discussed in the previous section, the proposed approach for the calculation of z was shown to underestimate its value by I


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experimental technique (like mechanical testing) and calibrate it versus the NMR data, which avoids the need to assume an arbitrary Dstat/k value. IV.D. Proposal of a Simplified and Pragmatic Approach To Determine Segmental Order Parameters for Sulfur-Cured Rubbers. The full equation for data analysis proposed in this paper, i.e., eq 20, is based on a state-of-the-art model which is nowadays the only available approach for accounting for entanglements, cross-links, and junction fluctuations in the segmental order expression based on the solid physical background given by the tube model of rubber elasticity. The trend predicted by eq 20 has been successfully validated versus NMR experimental data; however, the complexity of this expression complicates its application for the determination of the segmental order parameter in a practical context. For this reason, a simplified and more pragmatic approach is proposed in this section in order to ease the analysis of real and technically relevant sulfur-cured rubber compounds. The first simplification in this regard relates to the microstructure factor Ac. Along this work, the variability of Ac as a result of the changes in the molecular parameters (Nc and Ne) of each sample has been considered. However, the application of an average value of Ac (e.g., Ac ≈ 0.6760 for sulfur-cured rubbers) is often used in the literature for data analysis of mechanical data.25,60 The widely used approximated value of Ac ≈ 0.67 corresponds to a value of Kc ≈ 1.22, and it is equivalent to assume Nc ≈ Ne. The applicability of this simplification is justified in the case of the unfilled NR samples discussed previously in this work, since all the studied compounds are in the range of moderately cross-linked networks. This is also the most usual scenario in technical applications and the most relevant range for the industry and academia, since it is intermediate between the limit of the entanglement-dominated range (Nc ≫ Ne) and the limit of weak entanglement constraints (Nc ≪ Ne). Therefore, replacing n in terms of Ac (eq 23) into eq 20 gives

of appearance of the polydispersity effect was also identified in the study of Valentin et al. on NR.18 In the case of the (random) sulfur-cured NR samples analyzed in this paper, the measured distributions of Dres are narrow (variance to average ratio between 0.07 and 0.15); therefore, the effect of polydispersity is not expected to be significant in the calculated values. In order to complete the discussion, it should be mentioned that some authors discuss a possible overestimation of Dres (which would have as consequent effect an overestimation of Dstat/k) if the NMR measurement time is shorter than the characteristic chain relaxation time.70 According to Schlögl et al.,21 the measured Dres reflects time scales in the range 10−100 μs, which gives limited insights into large scale relaxation motions. This influence of this effect is expected to be small for networks, since in this case the measurement far above Tg should allow to observe the dynamics in the plateau of the autocorrelation function, being possible to make a link between NMR measurements and network structure parameters like cross-link and entanglement densities. Finally, when comparing to literature values, it should be taken into account that different authors give different references for the average mass of the statistical segment; for instance, Saalwächter uses Ms = 168 g/mol to calculate the factor of 617 Hz in the correlation between Dres and 1/Mc (which in this case would lead to higher frequencies), while in this paper the value Ms = 131 g/mol is used.7 This value corresponds to the calculation using the equations in ref 71 considering the structure of the cis-1,4-polyisoprene monomer for which a bond length of 1.49 Å and a monomer mass of 68 g/mol are obtained and the following data: characteristic ratio C∞ ≈ 5.1;72 number of main chain bonds in one monomer nb ≈ 3.8;73 length of the statistical Kuhn segment b = 0.88 nm.74 Importantly, it should also be noted that the value of the effective static frequency Dstat/k is arbitrary, given that it will depend on the method and assumed motional model. This problem has been clearly pointed in the past, as explained by Saalwächter and Heuer:41 “Dstat/k is of course a model-dependent quantity, and in particular the absolute value results for the time scale of fast segmental modes depend on this choice.” In addition, the relative errors derived from the application of simulation factors for quantitative calculations expected to be between 20% and 50% according to the literature.16,21,32,43 On the basis of these arguments, it is concluded that the obtained estimations for Dstat/k between 8400 and 13 100 Hz by using the BFM, and intersegmental prefactors are realistic and within error bars of the atomistic simulation from Saalwächter et al.16 The value of Dstat/k obtained through this approach is subjected to uncertainties derived from different factors such as value of z, the error in the determined fitting factors or the estimated factor for intermolecular interactions or the experimental errors (in the determination of Mc, Me, Ac, and fraction of elastically inactive fractions). To sum up, the uncertainties and material-dependent parameters in the relationship between m and Sb make the accurate calculation of cross-link density through their individual determination and combination very difficult. This was evidenced by the results reported in Table 5 for the unfilled NR samples. Consequently, the final proposal from the present work for a theoretically and conceptually better justified approach to estimate cross-link density from NMR in a semiquantitative manner is to calculate the vector segmental order through eq 20 obtaining data from an additional

m≈

2z Nc Ne

(35)

which simplifies further when considering Nc ≈ Ne: m≈

2z Nc

(36)

While the simplified eq 36 is useful to understand the trend of experimental data, it is important to remark that the full equation (eq 20) is preferred for a quantitative analysis of m since it considers the actual value of Ac for each sample. Nevertheless, in the case of no availability of independent information about Nc, Ne, and Ac, the simplified eq 36 enables a simple and pragmatic approach to obtain a reasonable estimate for the segmental parameter m based on DQ-NMR measurements only. If Ac is higher than the assumed value of 0.67, eq 36 underestimates segmental order. Assuming ideal networks composed of long chains without nonelastic rubber fractions, ϕac ≈ 1, and using Ac ≈ 0.67, an approximated value z ≈ 0.174 is obtained. The obtained value is smaller than the results reported in ref 46, for which the corresponding Ac values (using n as starting point) are estimated to be between 0.78 and 0.95. The estimated value of 0.174 for z, which is underestimated according to the discussion in subsection C, would lead to m J


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molecular dynamics simulations are not available. It should be noted that the determination of the (calibrated) Dstat/kdependent factor needs to be done for each material in order to obtain quantitatively meaningful values of m from NMR. In addition, the calculation of cross-link density from DQ-NMR can only be done accurately upon characterization of the entanglement density, since m depends on both the constraints of cross-links and entanglements. Considering the successful application of the adapted Lang− Sommer model, a pragmatic and simplified approach for real and technically relevant sulfur cured rubber compounds (i.e., moderately cross-linked networks with Nc ∼ Ne) was proposed. Assuming typical values of the constant z and the microstructure factor Ac, the determination of the density of entanglements with the tube model prior to NMR experiments is not required, being possible to make approximated estimations of cross-link density for industrial rubbers using the DQ-NMR technique. The present study has shown a way to use the DQ-NMR technique performed in low-field NMR spectrometers as a tool to access cross-link density in a quick and inexpensive manner. This has been demonstrated not only through the multiple existing experimental evidence correlating their results with the outcome of other experimental methods but also independently through the adaptation of the state-of-the-art theoretical model of Lang and Sommer46,50 (which is the most complete and realistic segmental order model nowadays). Given the listed uncertainties related to the numerical factors that appear in the conversion from residual dipolar coupling to cross-link density (e.g., effect of polydispersity and intersegmental interactions, assumption of a model-dependent Dstat/k value, characterization of the average mass of the statistical segment of each polymeric system of interest, etc.), the application of calibration with well-known experimental techniques, like mechanical testing using the tube model, is strongly recommended. In this way, the application of the DQ-NMR technique could be further generalized to other polymeric systems, even when the relevant parameters are not available in the literature, taking as basis the robust relationship of the residual dipolar coupling and network structure parameters which was discussed in this paper. In this sense, the dependence of the cross-link density value on the used model and prefactor highlights the importance of clearly specifying the used approach for each reported result, as should be common practice for every experimental method.

values below those of the phantom model revealing the limitation of this type of approximations. Using instead the average value of z = 0.249 calculated with the mechanical data values for unfilled NR, eq 36 would only be a good approximation (up to 10% error) for samples 163, 166, and 190, which have Ac values close to the one assumed (0.695, 0.699, and 0.684, respectively). Finally, the dependence on cross-link density predicted by the simplified expression of m in eq 36 explains the observed linear trend between 1/Nc and Dres through a fitting without intercept in our previous study (m ∝ 1/Nc).7 In addition, it also partially explains the higher scattering obtained for correlations between Dres/2π and m by neglecting the effect of varying junction fluctuations (i.e., samples with different Ac).

V. CONCLUSIONS In this study, several models and approaches for the calculation of segmental order in terms of the number of statistical segments between cross-links and entanglements have been reviewed, and their use has been exemplified for a set of unfilled natural rubber samples with different contents of sulfur and accelerator. The main aim of our work was (i) to point out and try to quantify the different factors involved in this calculation, influencing the final results according to the different approaches, and (ii) to define a data analysis procedure that can be used for the determination of cross-link density of technical elastomers from DQ-NMR data. The state-of-the-art theoretical model from Lang and Sommer,46,50 which considers a phantom network with cross-links attached to an elastic background, has been used for the first time for the analysis of experimental data in order to provide an analytical expression to calculate the segmental order of networks considering junction fluctuations and the number of segments between cross-links Nc and between elastically active entanglements Ne that can be used for the analysis of DQ-NMR measurements. The obtained expression has been validated versus the simulation data in ref 46. It should be noted that the use of the Lang−Sommer approach not only improves the understanding of DQ-NMR data but also allows for aligning the fundamentals for the calculations of mechanics, swelling, and NMR data by using a common physical picture and background. In order to establish a direct quantitative correlation between the data obtained by tube model fittings of uniaxial quasi-static stress−strain curves and experimental NMR results, the length of the chain connecting the cross-link points to the elastic point n is expressed in terms of the microstructure factor Ac through the relationship of both magnitudes with the network modulus Gc. The results of the application of the different models to define m show that, in general, all of them correlate qualitatively well with the NMR experimental data Dres through linear fittings without intercept in the case of the studied unfilled NR samples. The calculations with the new analytical expression proposed in this paper present a very good correlation with the NMR data. The quantitative analysis of the obtained slope shows that there are several factors affecting the value of the effective static dipolar constant Dstat/k. Among others, interchain couplings and the bond fluctuation model prefactor between Sb and m, are two elements with a direct and critical influence on the obtained model-dependent result. In this sense, the progress in the modeling of segmental order sets a basis for the further studies on the value of Dstat/k for different (and more complex) polymeric materials where atomistic

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00099. Brief information about recipe of samples and details about the tube model fitting procedure of stress−strain curve (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (J.L.V.). *E-mail: [email protected] (S.W.). ORCID

Juan L. Valentín: 0000-0002-3916-9060 K


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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Goodyear S. A. for the permission to publish this paper and Prof. J. U. Sommer and Dr. M. Lang (IPF Dresden) for fruitful discussions. The present project was supported by the National Research Fund Luxembourg (project reference 6916525), whose financial support is acknowledged. J.L.V. thanks the Ministerio de Ciencia e Innovación (Spain) for his Ramon y Cajal contract.

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