Importance of Compact Random Walks for the Rheology of Transient

Letter pubs.acs.org/macroletters

Importance of Compact Random Walks for the Rheology of Transient Networks B. J. Gold,*,† C. H. Hövelmann,† N. Lühmann,† N. K. Székely,‡ W. Pyckhout-Hintzen,† A. Wischnewski,† and D. Richter† †

Jülich Centre for Neutron Science (JCNS-1) and Institute for Complex Systems (ICS-1), Forschungszentrum Jülich GmbH, 52425 Jülich, Germany ‡ Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstraße 1, 85748 Garching, Germany S Supporting Information *

ABSTRACT: Controlling the mechanical behavior of novel supramolecular materials is of the utmost importance and requires a fundamental understanding of the underlying physical processes. We present a multimethods approach to the dynamics of entangled transient polyisoprene networks. Small-angle neutron scattering (SANS) on randomly functionalized chains shows homogeneous supramolecular melts with Gaussian chain conformations. The H-bond lifetimes (dielectric α*-process) and the rheological response in terms of the loss modulus G″ differ by 2 orders of magnitude in time. Within the concept of a compact random walk (RW), where the random walker (urazole group acting as a sticker) undergoes multiple returns to its starting point and following the concept of theoretical proposed renormalized sticky bond lifetimes, we quantitatively solve this longstanding and unexplained large discrepancy: While the bond opening gives rise to the dielectric response, for rheological relaxation the association with a new partner is relevant. This takes place only after multiple returns to the original binding partner.

S

this process represents the lifetime of association as well. Nevertheless, even though both relaxations are interpreted as resulting from the same physical process, the dielectric α* and the additional rheological relaxation are found to differ by 1−3 orders of magnitude in time,14−16 a finding that up to now is not understood. In this work, we present a multimethods study on the structure and dynamics of urazole-functionalized well-entangled polyisoprene transient networks revealing a quantitative understanding of the orders of magnitude difference between rheological relaxation and transient bond lifetime in such systems. The high molecular weight, monodisperse polyisoprene (PI) that was functionalized with urazole (U) groups by an Alderene reaction17 along the backbone served as a model system for our investigations. U-groups interact via a bifunctional hydrogen-bonding motif. Hydrogenous (H) PI with a molecular weight of Mw = 84 kg/mol and deuterated (D) PI with a molecular weight of Mw = 100 kg/mol were synthesized via living anionic polymerization18 (Mw/Mn = 1.02, revealed by SEC). Functionalization degrees of ΦU = 1 mol% (∼13 groups/

upramolecular polymers functionalized with reversible linkers form an emerging class of novel materials that offer enhanced and tunable properties compared to their nonassociating counterparts. Noncovalent interactions such as hydrogen bonding,1 ionic,2 and/or metal−ligand complexes3 may be controlled by temperature, pH, light irradiation, or mechanical forces and introduce novel features such as selfhealing4−6 and/or shape memory7−9 properties in commodity materials. They are found in diverse applications where precise control of molecular properties is required, as e.g. controlled release drug delivery,10 sensors,11 coatings,12 and shock absorbers.13 The rational design of such tailor-made products presupposes a fundamental understanding of their physical properties. In particular the relaxation mechanisms that are introduced by the reassociating groups determine the rheological behavior that underlies the mechanical properties. The relaxation behavior of supramolecular polymers is commonly studied by broadband dielectric spectroscopy and small amplitude shear rheology. Both techniques reveal an additional relaxation process caused by the implemented associating groups. The characteristic relaxation time of the second dielectric process (τα*) is interpreted as the mean lifetime of the associated state. For entangled supramolecular systems the additional rheological relaxation is commonly described in terms of the sticky reptation model, predicting that © XXXX American Chemical Society

Received: November 17, 2016 Accepted: December 27, 2016

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ACS Macro Letters chain) and ΦU = 2 mol% (∼25 groups/chain) were achieved and confirmed by FTIR spectroscopy measurements (Figure S1). Functionalized supramolecular polymers often display clustering of the H-bonding groups or phase separation. In order to investigate the homogeneity of our melts SANS measurements (KWS-2, MLZ, Munich, Germany) were undertaken over a broad temperature range (T = 278−413 K). Thereby, in order to provide the necessary scattering contrast, 5 vol% of functionalized hydrogenous PI was replaced by its deuterated analogue. Figure 1 compares the SANS results for the base PI and the two functionalized samples.

Figure 2. Imaginary part of the dielectric (ε″( f), circles) and viscoelastic (G″( f), squares) response obtained at an isofrictional reference temperature of 243 K for the unfunctionalized PI (blue symbols) and the corresponding supramolecular networks bearing 1% (red symbols) and 2% (green symbols) of functional urazole groups. For an explanation of the lines see text. The verification of the thermorheological simple behavior of the supramolecular system with 2% Ugroups is shown in the inset, where we compare a direct measurement (black squares) with the results of TTS (gray lines) at a reference temperature Tref = 258 K.

For the supramolecular melts the imaginary part ε″( f) shows two well-defined relaxation processes. If considered isofrictional, the monomer reorientation follows the characteristic relaxation time of the α-process τα. No further influence of the reassociating groups compared to the linear reference system is observed. At lower frequencies a second process (α*relaxation) is observed and related to the mean lifetime of the associated state τα*. Since the associated U-group complex is centrosymmetric (see inset Figure 1) its dipole moment cancels out. The dissociation of a complex at time τα*(T) results in a time-dependent variation of the absolute value of the dipole moment, giving rise to the α*-process.25,26 Applying the time−temperature superposition (TTS) principle the viscoelastic loss modulus G″(f) was obtained over a broad frequency domain, resulting in a smooth mastercurve. The horizontal shift factors aT follow a WLF behavior (Figure S6), implying approximately thermorheological simplicity. Similar observations are also reported in the literature for systems with a bifunctional hydrogen-bonding motif.14,27 A rationalization of this counterintuitive behavior is given later. To verify this behavior experimentally, frequency sweeps (inset Figure 2, black squares) covering the whole range of the additional rheological process were carried out at T = 258 K for a functionalization degree of ΦU = 2 mol% and compared to the corresponding mastercurves obtained by TTS (inset Figure 2, gray lines), using the same reference temperature (Tref = 258 K). Because of the additional elastomeric properties resulting from associated groups, aside from a prolonged terminal behavior and an increase of the slope in the flow regime, the most prominent feature observed in the rheological relaxation spectra is the occurrence of an additional rheological relaxation process with a characteristic relaxation time τrheo. As Figure 2 shows, τrheo exceeds τα* by about 2 orders of magnitude in time. Obviously the time it needs for a mesh in the transient network

Figure 1. SANS form factors obtained at T = 298 K for the unfunctionalized PI (blue symbols) and the corresponding supramolecular networks bearing 1% (red symbols) and 2% (green symbols) of functional urazole groups. The data are shifted by factors of 2 with respect to each other. For an explanation of the fitting curves see text. Inset: H-bonded centrosymmetric urazole complex.

In all samples the PI chains display an unperturbed Gaussian random walk conformation and are well fitted by Debye functions19 including a small additive Porod-like forward scattering contribution19 resulting most likely from microscopic air bubbles inside the viscous transparent sample material. For the different samples and temperatures a radius of gyration of Rg = 105 ± 7 Å was found, well in the range of literature values.20−22 The SANS studies unambiguously demonstrate the sample homogeneity. The dipole relaxation behavior was studied using a Broadband Dielectric Converter equipped with an Alpha-AHigh-Resolution Analyzer (Novocontrol). An ARES rheometer (Rheometric Scientific Inc.) with a plate−plate geometry (plate diameter 8 mm) was used to detect the linear viscoelastic response in small amplitude shear rheology, using a strain of γ = 0.1%. Figure 2 compares the resulting imaginary parts of the dielectric and viscoelastic response. All curves are shifted to an isofrictional reference temperature Tref [K] = 243 + ΔTg(ΦU) as the glass transition temperature Tg increases linearly with the amount of reassociating supramolecular groups. This finding is commonly reported in the literature for comparable systems17,23,24 and related to a reduced segmental mobility due to group association. For the current system we found Tg(ΦU) [K] = 211 + Δ Tg(ΦU) = 211 + 2.2ΦU, where ΦU is the functionalization degree with urazole groups in mol%. 74

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Although originally developed for unentangled Rouse chains, we will show the applicability of the ansatz to entangled polymer systems. In the following we briefly explain the main features of Rubinstein’s approach. Consider a sticker, bonded for τbond ≕ τα*. To free itself, it has to break the H-bonds at an energy cost of Ea, and in order not to immediately rebind again, it needs to diffuse over a monomer distance b ≃ Vmono1/3 = Mmono/(ρNA1/3) (Vmono monomer volume, ρ polymer density, NA Avogadro constant, and Mmono monomer molar mass), which will take a time τ0 ∼ τα. τ0 and τα* are related by

to become rheologically active cannot simply be explained by the lifetime of association seen in ε″(f). Now we quantify our observations by fitting the loss part ε″( f) and G″( f) of the complex dielectric and rheological response functions by a set of Havriliak−Negami functions (Figures S4 and S5) that are commonly used for extracting characteristic times of relaxation processes. Figure 3 compares the relaxation times τα, τα*, and τrheo, obtained by this procedure, in an isofrictional representation.

τα ≃ τ0 exp(Ea /(RT )) ≡ τbond (1) * The process of group separation and therefore the activation energy of the whole dissociation process not only depend on Ea but also are coupled to the strongly temperature-dependent αprocess. Plotting τα*/τα in an Arrhenius representation (inset in Figure 3) reveals the activation energy of the dissociation process: 1 mol% U-groups Ea = 29.3 ± 4.0 kJ/mol and Ea = 28.1 ± 3.7 kJ/mol for 2 mol% U-groups. Ea is about twice as high as the enthalpy change ΔH ≈ −15.9 kJ/mol obtained by FTIR spectroscopy measurements (Figure S3). Apparently, an additional barrier of ≃12−13 kJ/mol must be overcome to separate two associated groups. Thus, Ea of τα* is much smaller than the activation energy Eαa ≃ 110 kJ/mol of τα in this temperature range. τα strongly dominates the total temperature dependence and leads to an approximate validity of TTS. After a sticker separates, it starts to explore its surrounding volume. With τopen, the time it takes to move to the next open sticker, the corresponding mean squared displacement (MSD) amounts to

Figure 3. Isofrictional representation of the relaxation times for the αprocess (circles), α*-process (stars), and the additional rheological relaxation process (squares), for the unfunctionalized PI (blue symbols) and the corresponding transient networks bearing functional urazole groups (1 mol% red symbols, 2 mol% green symbols). The temperature dependence of τα follows an identical WLF behavior for all samples (black line). The dashed lines present the prediction of the compact random walk model in the temperature range where τα* was measured (see text). The Arrhenius representation of τα*/τα is shown in the inset.

⎛ τopen ⎞x 2 ⟨Δropen ⟩ = b2 ⎜ ⎟ ⎝ τ0 ⎠

(2)

Thereby we generalize by taking the exponent x as a free and adjustable parameter. For Rouse chains x = 1/2 would be valid.31 The current system consists of entangled chains, which might lead to a smaller exponent (e.g. local reptation x = 1/4). Because the random walk of the sticker is compact for small x, the sticker undergoes an average number of returns Jopen to its former binding partner during τopen. There, each time it stays bonded for a duration of τbond, before it finally recombines with a new partner. It is this dissociation process and exchange of partners that makes the process rheologically active, implying

As may be seen, about a thousand segmental relaxation processes τα take place during the mean lifetime of the associated state τα*. Furthermore, the additional rheological relaxation process τrheo exceeds τα* by another 2 orders of magnitude, an observation that agrees with similar earlier findings.14−16 In the sticky reptation theory28 the rheological time τrheo is identified with the lifetime τlife of the temporary association of a pair of stickers. For τ < τlife the associating groups behave like permanent cross-links, causing an additional elastic contribution to the plateau entanglement modulus. Vice versa, as soon as the experimental time scale exceeds the lifetime of the temporary sticker τ > τlife the plateau modulus drops back to the entanglement plateau modulus, and simple reptation takes over. If the sticky reptation model would be valid, τrheo should approximately agree with τα*. Up to now the discrepancy between the sticker lifetime τα* and the mechanical relaxation time τrheo, which governs industrial applications, is unexplained. As shown in Section S4, the idea of bond time renormalization that was developed in the context of “sticky reptation”29,30 does not capture our experimental results. Recently, in order to describe the self-healing properties of end-functionalized Rouse chains and based on the idea of bond time renormalization, Rubinstein31 investigated the conditions for a polymer strand to become rheologically active. His considerations hint at a solution for the longstanding problem.

τrheo = Jopen τbond + τopen

(3)

Without any further calculations Equation 3 shows that τrheo will be significantly longer than τbond, exactly what is observed experimentally. For a quantitative description Jopen and τopen still have to be calculated. Equation 2 imposes Δropen2 = b2nx with n steps of size b. With the number of sites ⟨Δropen2 ⟩3/2b−3 in the explored volume, the average number of returns is given by Jopen

⎛ Δropen ⎞(2/ x) − 3 n = =⎜ ⎟ 2 ⎝ b ⎠ ⟨Δropen ⟩3/2 b−3

(4)

The root mean squared distance between two open stickers Δropen follows from the total concentration of stickers in the system ct = Ngroups/chain/(VmonoN) and the equilibrium constant K obtained from FTIR spectroscopy measurements (Figure S3), Δropen = copen−1/3 with copen ≈ (ct/(2K))1/2 from the definition of K. With these considerations all parameters are 75

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ACS Macro Letters related to experimentally accessed measures. Only the random walk exponent x for the MSD remains as an unknown parameter that can thus be evaluated on the basis of the measured data. For both functionalization degrees and temperatures shown in Table 1, we determined the random walk exponents x, resulting in an average value of x = 0.365 ± 0.01. With this exponent the values for Jopen, τopen, and Jopen·τbond were calculated. Table 1 presents the results including also ropen, τbond, and τrheo for 248 and 253 Kat these temperatures all parameters were accessible simultaneously.

experimentally accessed. Relevant parameters were the segmental relaxation time τα, the bonding time of a sticker τα*, the total concentration of stickers ct, the amount of open stickers copen, and the rheological relaxation time τrheo. SANS experiments assured the homogeneity of the samples. With these data a RW-exponent x = 0.365 ± 0.01 for the sticker MSD was revealed, well between the classical Rouse (1/2) and local reptation (1/4) behavior. In this frame the modified Rubinstein model provides a consistent description of the so far unexplained relation between the bond lifetime and the time needed to become rheologically active.

Table 1. Experimentally Accessed Quantities ropen, τbond, and τrheo and Theoretical Parameters Jopen, τopen, and Jopen·τbond

S Supporting Information *

ropen (Å) Jopen τbond (s) τopen (s) Jopen·τbond (s) τrheo (s)

1 mol%

1 mol%

2 mol%

2 mol%

248 K

253 K

248 K

253 K

41.7 192 0.0134 1.66 2.57 4.23

40.6 180 0.00403 0.569 0.725 1.30

34.2 118 0.0492 0.704 5.81 6.51

33.5 111 0.0126 0.231 1.40 1.63

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ASSOCIATED CONTENT

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.6b00880. Fourier transform infrared spectroscopy, Havriliak−Negami functions, thermorheological simple behavior: time−temperature superposition, and calculation of the bond lifetime renormalization in the framework of sticky reptation (PDF)

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Let us now discuss the outcome: The key parameter that determines the large difference between τrheo and τbond is the compactness of the sticker random walk that is governed by the exponent x. For all cases we find values scattering around x = 0.365 in between Rouse scaling (x = 1/2) and local reptation (x = 1/4). For a strongly entangled system featuring additional constraints as a consequence of the transient cross-links, this is a very reasonable result. This compact RW leads to a high number of returns to the original binding site until a different open sticker is found. In Figure 3 we have included the prediction of Equation 3 for τrheo using the average exponent x = 0.365. Based on all independently determined parameters, for x = 0.365 the theoretical prediction describes very well the Tdependence of τrheo. Within the errors, for each concentration the number of returns Jopen stays constant with T but significantly decreases with concentration. With a smaller exploration volume at higher copen the chance for return is diminishedaccording to Equation 3 Jopen increases with a power law of the distance to the next open sticker. As can be seen from Table 1, the impact of τopen together with the return contribution Jopen·τbond on τrheo is concentration dependent. For a functionalization degree of 1 mol% both contributions are in the same time range. With an increasing amount of functional groups the time Jopen·τbond spent for multiple associations with the old partner dominates more and more. This effect is determined more by the increasing association time τbond = τα*, while the number of returns is diminished. If a sticker is finally able to dissociate entirely from the old partner after Jopen·τbond, it needs only a shorter time τopen to cover the distance between the open groups ropen. In conclusion, using the concept of compact random walks together with the notion that breaking a transient bond leads to rheological activation only, if, instead of reassociation with the old partner, a new partner is found,31,32 we relate the additional rheological relaxation process, observed very frequently in the spectra of transient supramolecular systems, to its real physical origin. For this purpose in a multimethods approach all model parameters necessary to characterize the system were

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49 (0)2461 61 4714. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We declare that this work has no specific funding support. REFERENCES

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