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What Controls the Structure and the Linear and Nonlinear Rheological Properties of Dense, Dynamic Supramolecular Polymer Networks? Tingzi Yan,† Klaus Schröter,† Florian Herbst,‡ Wolfgang H. Binder,‡ and Thomas Thurn-Albrecht*,† †

Experimental Polymer Physics, Institute of Physics, and ‡Chair of Macromolecular Chemistry, Institute of Chemistry, Martin Luther University Halle-Wittenberg, Halle 06120, Germany S Supporting Information *

ABSTRACT: We investigated a series of telechelic polyisobutylenes, previously shown to exhibit self-healing, by means of small-angle X-ray scattering and rheology. All samples form a dense, dynamic network of interconnected micelles resulting from aggregation of the functional groups and leading to viscoelastic behavior. The dynamic character of this network manifests itself in the appearance of terminal flow at long time scales. While the elastic properties are distinctly molecular weight dependent, the terminal relaxation time is controlled by the functional end groups. The yielding properties under large deformation during startup shear experiments can be understood by a model of stress activation of the dynamic bonds. Stress relaxation experiments help to separate the nonlinear response into two contributions: a fast collapse of the network and a slow relaxation, happening on the time scale of the terminal relaxation. The latter is also known to control self-healing of the collapsed structure.

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INTRODUCTION The incorporation of dynamic (supramolecular) bonds in polymer systems makes these materials responsive to external stimuli enabling many fascinating applications.1−3 One important application is the development of self-healing materials.4−6 Especially appealing are materials forming dynamic networks which are capable of autonomous healing, not requiring an external trigger. Dynamic polymer networks (dynamic gels) were studied extensively in the past decades.1,2,7−12 Gels were either formed by different dynamic bonds (hydrogen bonds,13−17 metal− ligand coordination,18,19 ionic interactions,20−22 and so on) or demixing of triblock copolymers.11,23 They may contain solvent or not. A prominent example for gels including solvent are hydrogels.24 Here, we are interested in dense networks without solvent to achieve higher mechanical robustness for applications as structural parts and long-term stability. Dynamic bonds in such networks behave like permanent bonds on time scales shorter than their lifetime. Hence, dynamic networks also show an elastic plateau in the dynamic modulus similar to conventional rubbery networks, which is the basis of their application as load carrying materials. The dependence of the elastic modulus on molecular weight of the network chains has been studied for conventional rubbery networks for a long time. For dynamic networks, cross-linked by dynamic bonds at the chain ends, detailed systematic investigations were taken up only recently.15,25 Because of the dynamic character of the bonds, dynamic gels can rearrange on long time scales.26 This feature enables the healing of cracks in the network structure. The application of © XXXX American Chemical Society

different dynamic gels as self-healing materials has been studied over the past few years.27−32 Leibler30,33 produced a gel-like rubbery material by mixing low-molecular-weight fatty acids and urea and could show that the sample heals after cutting into two parts. Rowan31 synthesized a metallosupramolecular polymer and found that the material is self-healable after exposure to ultraviolet light. A telechelic polypeptide functionalized with collagen-like triple helices has been synthesized by Skrzeszewska et al.,34 which self-healed after breaking the sample by strong shear. Schmidt and colleagues studied ionomeric elastomer model systems which also show selfhealing at elevated temperatures.21,22,35 The network is crosslinked by several ionic clusters per chain. The authors found a correlation between the relaxation time of the dynamic network and the self-healing ability of the material. But the structural information from small-angle X-ray scattering (SAXS) is not very detailed. Generally, the combination of structural and rheological studies on well-controlled and characterized dynamic gels is scarce. It requires a combination of sophisticated synthetic methods and characterization techniques. The usual method to test the self-healing properties of materials is to cut the sample into two parts, put them together for different times, and then perform tensile measurements.22,30,36,37 With such an experimental approach it is possible to decide whether a sample is self-healable or not. But Received: November 18, 2016 Revised: February 15, 2017

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DOI: 10.1021/acs.macromol.6b02507 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Scheme 1. Chemical Structure of the Samples

although there have been theoretical attempts,38 it is difficult to relate the observations to the fundamental viscoelastic properties of the material. Using a combination of linear and nonlinear rheological experiments on a well-defined supramolecular network, we could recently show that the terminal relaxation process is responsible for the self-healing kinetics in a dynamic network of bifunctionalized PIB chains.39 This network had been previously shown to exhibit self-healing.16 The chemical structure is designed to enable a clear distinction between chain dynamics and network relaxation by their different temperature dependencies. We here extend this work and present a study of a series of these samples with different molecular weights. As will be shown below, molecular weight is an additional control parameter to adjust the viscoelastic properties of the network. Small-angle X-ray scattering and rheology have been used to confirm the structure of the samples as a densely connected network of aggregates. The dynamic character of the network shows up as terminal flow at long time scales. The elastic properties in the rubbery plateau region of this network can be controlled by the molecular weight, while the terminal relaxation time is molecular weight independent. The nonlinear network properties were studied in start-up shear experiments. Yielding at high strain indicates network failure. The shear rate dependence of the yield point can be described by a model from the literature.34 Stress relaxation in the nonlinear region identifies two contributions, the slower one happening on the time scale of the terminal relaxation in the linear response.

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an aluminum disk and attached onto a Linkam hot stage TMS 94 by thermal conducting paste. The experiments were done at a series of temperatures in the range of −40 to 120 °C during heating and cooling. Rheological Measurements. An Anton Paar MCR 501 rheometer was used for all rheological measurements. The sample fixture consisted of parallel plates with a diameter of 8 mm. Sample thickness was in most cases between 0.3 and 0.5 mm. The lower plate is connected to a Peltier element and controls the sample temperature together with the streaming nitrogen gas inside the sample chamber. Although the sample deformation is not homogeneous in a parallel plate geometry, we use this compromise also for the nonlinear rheological measurements. This allows us to cope with the very limited amount of sample available. The limit for linear response behavior was checked first. Afterward, rheological measurements in the linear range were done in a wide range of temperatures and angular frequencies. Measurements at each temperature were performed after an equilibration time of at least 10 min. Repetition of measurements in cooling or heating confirmed the reproducibility of the results. Nonlinear rheological measurements were performed as startup shear experiments with different shear rates or stress relaxation experiments at large strains. To erase the history and reshape the disturbed sample geometry after each nonlinear experiment, the sample was heated up to 120 °C, cooled to the new measurement temperature, and equilibrated there for at least 15 min.

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RESULTS AND DISCUSSION Structural Characterization by Temperature-Dependent SAXS. We expect for these telechelic polymers demixing between the unpolar PIB chains and the polar chain-end groups similar to former results for thymine/triazine functionalized PIB chains.25 Hence, it is important to characterize these aggregation phenomena first. Figure 1a shows the small-angle X-ray scattering (SAXS) patterns of sample PIB8K-BA2 at different temperatures. A peak at a scattering vector value q of around 0.1 Å−1 and a shoulder around a q value of 0.17 Å−1 can be observed. This is a clear indication for the formation of aggregates. The peak position is nearly temperature independent. This indicates that the mean distance between the aggregates or their number density stays the same over a broad range of temperatures. But with increasing temperature, the intensity decreases for peak and shoulder. The positional correlations between the aggregates decrease or their fluctuations increase accordingly. We expect that this is caused by a diminishing size of the aggregates with increasing temperature and, correspondingly, an increase of the fraction of free chains, not bound to the aggregates. The scattering results for samples PIB14K-BA2 and PIB4KBA2 are similar to those of PIB8K-BA2, and they have been shown in our former article39 or are included in the Supporting Information as Figure S1, respectively. However, for the sample with the highest molecular weight PIB28K-BA2 the SAXS results differ a bit. As shown in Figure 1b, at 20 and 40 °C, there is only a weak peak at q ≈ 0.075 Å−1. With increasing temperature, again the intensity decreases for peak and shoulder, and they disappear at around 100 °C. The SAXS

SAMPLES AND EXPERIMENTAL METHODS

Samples. A series of polyisobutylenes (PIB) were synthesized, functionalized at both chain ends with barbituric acid groups (see Scheme 1). The molecular weights vary from below to above critical entanglement molecular weight (Mc = 13.1 kg/mol).40 A homomolecular PIB with a molecular weight of 30 kg/mol was investigated for comparison. Table 1 gives relevant informations about the

Table 1. Sample Characteristics samples telechelic polymers

homopolymer a b

PIB4K-BA2 PIB8K-BA2 PIB14K-BA2 PIB28K-BA2 PIB30K

Mna [kg mol−1]

PDIb

3.9/4 8.7/7.9 14/13.8 28.4/26.7 30

1.3 1.2 1.2 1.1

Molecular weights obtained by NMR/GPC and MALDI-MS. Polydispersity index PDI = Mw/Mn as obtained by GPC.

molecular weights and defines the abbreviations for the sample names. Detailed information about the synthesis and characterization of the samples can be found in ref 16. SAXS Measurements. Details of the experimental setups have already been published.39 In brief, small-angle X-ray scattering (SAXS) experiments were performed with a setup consisting of a Rigaku rotating anode, a focusing X-ray optics from Osmic, and a Bruker 2Ddetector. Cu Kα radiation was used, covering a range of scattering vectors q from 0.04 to 0.4 Å−1. The sample was placed into the hole of B


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the same one because the distance between the aggregates is comparable to or even below the calculated end-to-end distance of a Gaussian chain (see below). Hence, we expect a large amount of bridging chains in the system connecting the aggregates. This is consistent with experimental and simulation results on triblock copolymers forming spherical micelles,44,45 where the fraction of bridging chains is around 50%−80%. Beside bridging chains, also free chains, dangling chains, and loop chains are expected, which are also indicated in Figure 2. The tentative picture of the transient network corresponds fully to the model of a supramolecular network formed by telechelic chains with sticky end monomers in ref 26 and is similar to our previous work on a mixture of telechelic chains with complementary end groups.25 Assuming that one can interpret the peak position q* in the scattering data as a measure of an interparticle distance, one can estimate the average distance between the centers of the scattering objects, the aggregates, as d = 2π/q*. This corresponds also to the end-to-end distance for the bridging telechelic polymers. The peak position q* does not change much with temperature (see Figure 1) but distinctly with molecular weight. We plot 2π/q* and the calculated end-to-end distance of a Gaussian chain REE vs the molecular weight Mw in Figure 3 for the four samples at 20 °C. REE was calculated by

Figure 1. (a) Scattering intensity of PIB8K-BA2 at a series of temperatures (120, 100, 80, 60, 40, 20, and 0 °C; from bottom to the top). The data are shifted vertically. (b) Scattering intensity of PIB28K-BA2 at a series of temperatures (120, 100, 80, 60, 40, and 25 °C; from bottom to the top). The curves are shifted vertically.

patterns for all samples are reversible; i.e., the peaks reappear after cooling from 120 °C. These phenomena can be understood by considering the telechelic polymer as asymmetric triblock copolymers, with a long nonpolar PIB middle block and polar barbituric acid end blocks. The end blocks will tend to assemble and form spherical aggregates because of their low volume fraction. The PIB chains will emanate in a brushlike configuration from the surface of these aggregates. The peak in the scattering data of sample PIB8K-BA2 highlights the existence of such aggregates because in a disordered melt without aggregates normally only a correlation hole would be observed. The spatial arrangement of the aggregates is not an ordered state because higher order peaks are missing. Hence, the scattering data of sample PIB8KBA2 point to a disordered spatial arrangement of spherical aggregates. Similar results are known for covalently bound, asymmetric block copolymers41−43 and from our recent work on monofunctional supramolecular polymers.25 Figure 2 shows a tentative picture of the aggregate structure formed by telechelic PIB. The two end groups of each PIB chain are more likely located in two different aggregates than in

Figure 3. Measured distance between aggregates (2π/q*) (■) and calculated end-to-end distance REE for a Gaussian chain (△) in dependence on molecular weight for four telechelic polymers. The line is a linear fit with slope 1/3.

the equation REE = √Nb for a homopolymer chain with the number of Kuhn segments N = Mw/MKuhn (molecular weight of Kuhn segments MKuhn = 273 g/mol) and the Kuhn length b = 1.3 nm.46 For the lowest molecular weight sample, PIB4K-BA2, the calculated end-to-end distance of a Gaussian chain is comparable to the distance between the aggregates. For higher molecular weight samples, the distance is smaller than the calculated end-to-end distance of a Gaussian chain. A linear curve fit log(2π/q*) ∼ log M gives a slope of 1/3, smaller than the value of 1/2 for a Gaussian chain. A similar slope of 1/3 was found in ref 15. A slope of 1/3 fits also to a picture with a molecular weight independent aggregation number (see below). In this case, the volume occupied by all chains belonging to one aggregate will scale linearly with the molecular weight M of the single chain. The linear dimensions will grow accordingly with the power of 1/3. The conformation of the bridging chains is determined by the conditions of constant aggregation number, uniform density, and space filling. Although the chains are in a coiled conformation as the value for the lowest molecular weight in Figure 3 shows, the distance between the aggregates and

Figure 2. Tentative picture of aggregates formed by the telechelic PIB. The red spheres are the aggregates formed by barbituric acid groups, and the blue lines are the connecting PIB chains. The gray lines represent free telechelic PIB chains. d is the distance between nearby aggregates. R1 is the radius of the aggregates. C


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Macromolecules therefore also the end-to-end-distance of the bridging chains increase less strongly than for a Gaussian chain. However, we have no direct information about the conformation of free chains. We expect that the interaction between nearby aggregates in Figure 2 contains two contributions. One is an attractive, longrange interaction because of the existence of bridging chains. In addition, there will be a repulsive interaction which prevents interpenetration between neighboring, brushlike coronas.47,48 To enable a more quantitative analysis of the scattering data, we replace the real interaction potential by a much simpler effective hard sphere potential.49 As we will see, this potential is able to reproduce the most important features of the scattering data. The only parameter of the hard sphere potential is the radius of the hard spheres, here called R2. It represents the range of the repulsive interaction. R2 is an effective parameter, there is no distinct structural boundary between the effective hard spheres and the surrounding matrix, and no scattering contrast arises there. Such an effective hard sphere radius can be defined for instance by the Barker−Henderson theory. In our notation it is given by48 R 2 = RHS = R1 +

1 2

Figure 4. Fit of SAXS intensity with Perkus−Yevick model for PIB14K-BA2 from −40 to 120 °C in 20 K steps. Open symbols represent the original data, and the solid lines are the fit curves. The data are shifted vertically for different temperatures.

∞

∫2R

1

[1 − exp( −U (r )/kT )] dr

(1)

where U(r) is the repulsive part of the interaction potential of two particles of distance r between their midpoints. This is analogous to the definition of an excluded volume. As mentioned earlier, we neglect the attractive part of the interaction in this respect. Because the scattering data in Figure 1 suggest liquid-like correlations between the positions of the scattering objects, we use the structure factor of the Perkus−Yevick hard sphere model to fit the SAXS data.25 The scattering intensity is then given by the following equation:

(2)

Figure 5. Fit parameters (volume fraction of hard spheres f, radius of aggregates R1, and radius of hard sphere interaction R2) for four samples of different molecular weight (PIB4K-BA2, ■; PIB8K-BA2, ●; PIB14K-BA2, ▲; PIB28K-BA2, ▼) in dependence on temperature.

The structure factor of interacting objects S(q,R2,f) is based on the Perkus−Yevick model.42 It depends on the volume fraction f of the effective hard spheres with radius R2. Because there is no structural boundary at R2, the only scattering contrast arises from the aggregated chain ends as the core of the hard spheres. Hence, the form factor Φ(qR1) in the scattering function depends on the radius R1 of the aggregates. We allow for a distribution in the size of the aggregates and use a lognormal distribution f p with a width σ and a mean value of R1. q is the scattering vector. Both K (prefactor) and C (the scattering background) are constants. The detailed fitting equation can be found in our previous work.25 Since the interaction between the aggregates is not purely repulsive, as discussed above, the results for f and R2 from the curve fit will be only effective values. As an example, the scattering intensity (symbols) for PIB14K-BA2 at different temperatures and the corresponding fit curves (lines) are shown in Figure 4. Obviously, the model is able to represent all important features of the experimental data. The fit parameters for the four samples with different molecular weight in dependence on temperature are shown in Figure 5. The error bars are based on the statistical error from the fit procedure. Volume fraction f and R2 have errors smaller than the symbol size.

As already mentioned, the number density of aggregates does not change much within the investigated temperature range. In comparison, the data for the size of the aggregates R1 are relatively noisy. This results from the missing well-defined minima of the form factor in the scattering data. Hence, the trend with temperature is also not very obvious. There seems to be a slight decrease with increasing temperature above room temperature. Obviously, the aggregates are relatively stable over this temperature range. The size of the aggregates R1 is nearly the same for all samples; only the sample PIB4K-BA2 shows a 20% smaller radius. This means the aggregation number is also nearly independent of molecular weight. It seems that the size of the polar end groups itself together with the requirement of spacefilling limits aggregation. This result holds as long as the demixing between the polar end groups and the main chain is strong enough. Possibly, the somewhat reduced size of the aggregates in the case of PIB4K-BA2 is related to a reduced thermodynamic driving force for demixing due to the low molecular weight. Now we discuss the results for the parameters of the repulsive potential. The effective interaction range R2 grows with molecular weight. This result is compatible with the

∞

I(q) = KS(q , R 2 , f )

∫R =0 Φ(qR1)fp (R1, R1̅ , σ ) dR1 + C 1

D


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Figure 6. Complex modulus (G′, G″) vs angular frequency ω for (a) sample PIB8K-BA2 and (b) sample PIB28K-BA2 at different temperatures [−30 (|), −20 (−), −10 (∗), 0 (×), 10 (+), 20 (⊙), 30 (⬠), 40 (★), 50 (⬡), 60 (▷), 70 (◁), 80 (◇), 90 (▽), 100 (△), 110 (○), and 120 °C (□)].

the temperature and molecular weight dependence of R1 given above applies also here. Next we want to compare this aggregation number to the total number of chain ends available. To simplify the estimation, we assume that the aggregates arrange in an ordered structure, where one can do the calculation based on the unit cell. Since a body centered cubic (bcc) ordered structure was found in our former supramolecular PIB system25 and the peak position shifted only slightly around the order− disorder transition temperature, we also assume the bcc packing here. Hence we identify q* from the SAXS data with q110. The lattice parameter l (the size of the unit cell) is then √2·2π/q*. Because the unit cell contains two aggregates and each chain has two ends, one ends up with a number of chain ends g′, available per aggregate

picture of molecular brush chains emanating from the aggregates, for which the size and interaction range will grow with molecular weight. Equation 1 for the effective hard sphere radius R2 together with theoretical estimates for the interaction potential between spherical particles with a polymer brush layer of thickness L48 results in values for R2 which increase with molecular weight, in agreement with our results. But a quantitative comparison is not possible without assumptions about the molecular weight dependence of L. Like R2, also the distance between the aggregates increases with molecular weight (see Figure 3). But because of the stronger increase of the distance compared to R2, the volume fraction f influenced by this repulsion decreases (see Figure 5). A larger part of every bridging chain is outside the strong repulsive range for higher molecular weights. To come back to the structure of the interconnected network, we now want to estimate the aggregation number. The aggregation number g is defined as the number of end groups in each aggregate. It can be estimated from the ratio of the volume of the whole aggregate divided by the volume of a single end group g=

4 πR13 3 Mend ρNA

g′ =

( 2 ·2π /q*)3 M ρNA

(4)

where ρ is the density of polyisobutylene (0.92 g/cm3 at 20 °C50), q* is the peak position in the SAXS pattern, and M is the molecular weight of the whole supramolecular polymer. If we take PIB14K-BA2 at 20 °C as an example, q* = 0.078 Å−1, and the calculated g′ is around 60. This means 60 chain ends are in the volume belonging to one aggregate. Hence, the fraction of aggregated end groups is estimated for this sample to be around 22/60 = 0.36. Other end groups belong to free chains or dangling chains, consistent with the disordered micellar fluid structure from the SAXS analysis. In fact, the fraction of aggregated end groups is similar to 0.35 for all samples, except for PIB4K-BA2 where it is a bit smaller at around 0.2. Both the aggregation number and the fraction of free chains are higher than typical values from the model in ref 26, although the latter can be adjusted by a smaller value of the bonding energy ε in the model. Hence, a direct quantitative comparison of experimental results and model predictions is not possible.

(3)

where ρ is the mass density for the end groups (we assume ρ ≈ 1 g/cm3), NA is Avogadro’s number, R1 is the radius of the aggregate from the SAXS curve fit, and Mend = 264 g/mol is the molecular weight of the barbituric acid group. For sample PIB14K-BA2 at 20 °C, R1 from the fit is 13.3 Å, and the calculated aggregation number is around 22. The aggregation number is similar for all samples, except for PIB4K-BA2 where it is slightly smaller, around 13. Because the estimated aggregation numbers are solely based on R1, the discussion of E


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on molecular weight. The value for the plateau modulus was taken at room temperature at that frequency where the loss modulus (respectively the damping factor for sample PIB28KBA2) shows a minimum. The result is shown in Figure 8. With

In conclusion, the SAXS analysis shows that the telechelic supramolecular PIB samples form aggregates. A fit with the Perkus−Yevick model gives informations about the size and number of the aggregates. An aggregation number can be estimated, which is mostly around 20 and does not change much with temperature or molecular weight. Identification of Micellar Network by Linear Rheology. Small-angle X-ray scattering identified the aggregation of the chain ends, but gives no information about the topology of the chains. To obtain further information here, one needs rheological investigations. Figure 6 shows the complex shear modulus for samples PIB8K-BA2 and PIB28K-BA2 from 120 to −30 °C. The results for samples PIB14K-BA2 and PIB4K-BA2 are very similar to those of sample PIB8K-BA2, and they have been shown in our former article39 or are included in the Supporting Information as Figures S2 and S3, respectively. One notices terminal flow at low frequencies or high temperatures and a broad plateau in the real part of the modulus for sample PIB8K-BA2. At the highest frequencies, the transition to a Rouse relaxation process can be seen. If one considers the low molecular weight of the single chains (just below Mc for PIB), the only possibility to explain the plateau in the modulus is the existence of a dynamic network through physical cross-links. Hence, the aggregates of chain ends, identified by SAXS, must be connected by bridging chains. This result is analogous to the cross-linked micelles in a mixture of thymine and triazine functionalized telechelic chains from our previous paper.25 Figure 6 demonstrates the three relaxation regimes, initial Rouse regime, intermediate rubbery regime, and terminal relaxation regime, in agreement with a model of the dynamics in supramolecular polymer networks formed by associating telechelic chains.26 To demonstrate the influence of molecular weight on the rheological properties directly, Figure 7 compares the frequency dependence of the real part of the modulus for the different samples at the same temperature of 20 °C. The data clearly demonstrate that besides the existence of this rubber plateau in a system of low molecular weight chains, there are two other properties fundamentally different from simple polymeric melts. The height of the rubber plateau is molecular weight dependent, while the terminal relaxation time is not. First, we analyze the dependence of the plateau modulus

Figure 8. Dependence of the plateau modulus at room temperature (■) on molecular weight. The calculated values (eq 6) for a fully endcross-linked network, including entanglements, are shown for comparison as dashed line. The horizontal blue dashed line gives the plateau modulus for a highly entangled homopolymer PIB.

increasing molecular weight the plateau modulus first decreases steeply. Above the critical molecular weight Mc = 13.1 kg/mol for a corresponding homopolymer (vertical line), it levels off and becomes nearly constant at around the value of a wellentangled homopolymer PIB melt, 0.32 MPa (horizontal dashed line).46 A simple estimate of the plateau modulus can be made by rubber theory. If one assumes that all chains are active network strands, i.e., all telechelic polymer chains are bridging chains between aggregates, the plateau modulus is given in the affine model by15

ρRT (5) M 50 3 where ρ is the density of the polymer (0.92 g/cm at 20 °C ), R is the gas constant, T is absolute temperature, and M is the molecular weight between two cross-link points, i.e., in our case the molecular weight of the polymer. This equation considers only cross-links between chain ends. If one wants to include also physical cross-links by entanglements, which become relevant for molecular weights above Mc, one can in a first step simply add up the number densities of end links and entanglements51 G=

⎛1 1 ⎞ G = ρRT ⎜ + ⎟ Me ⎠ ⎝M

(6)

The calculated values for the plateau modulus are included in Figure 8 (molecular weight between entanglements Me = 6.9 kg/mol40). One notices that the calculation predicts the correct trend of decreasing plateau modulus with increasing molecular weight and leveling off at high molecular weights. Although the calculated values nearly match the experimental values for small molecular weights, one has to keep in mind that for the calculation all the telechelic polymer chains were assumed to be bridging chains. This contradicts our earlier assumption of the existence of free chains, not participating in the network. Hence, the experimental values for the plateau modulus are astonishingly high. Similar unexpectedly high values were also reported in the literature for a PnBA melt, cross-linked by a metal−ligand binding system,52 and a dynamic network of PEA chains.15

Figure 7. Real part of the modulus G′ vs angular frequency ω for the samples of different molecular weight at the same temperature of 20 °C (PIB4K-BA2, ■; PIB8K-BA2, ●; PIB14K-BA2, ▲; PIB28K-BA2, ▼). F


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Figure 9. Master curves of (a) sample PIB8K-BA2 and (b) sample PIB28K-BA2 with a reference temperature of 25 °C, constructed with shift factors from homopolymer PIB. The symbols are used in the same way as in Figure 6.

modulus and η the zero shear rate viscosity. All methods give comparable results (see Supporting Information Figure S5). In the following we will use the simple estimate from the reciprocal of the angular frequency ω at the crossover between G′ and G″ as the terminal relaxation time. The temperature dependence of the terminal relaxation time is shown in an Arrhenius representation in Figure 10a for the four telechelic polymers. One notices that the relaxation times are rather similar and not strongly molecular weight dependent. This is different from simple polymeric melts, for which the terminal relaxation time is always strongly molecular weight dependent.

Master curves for sample PIB8K-BA2 and PIB28K-BA2 were constructed with shift factors from the PIB homopolymer. The same strategy to separate the temperature dependence of relaxation times for the chain modes and the aggregates was used before by us.25 The result is shown in Figure 9. The Rouse relaxation part overlaps well while the flow part does not. The Rouse relaxation is related to chain dynamics of PIB while the terminal flow is controlled by dynamic bonds of the chain ends. Beside these general similarities, there are small differences visible between Figures 9a and 9b. The master curve for the sample with the highest molecular weight PIB28K-BA2 looks somewhat different in the plateau region. There is not the usual, distinct minimum in G″ between α relaxation at high frequencies and terminal relaxation at low frequencies. Instead, one finds a relatively flat frequency dependence there. This finding points to extra relaxation processes with a broader distribution of relaxation times. In addition, the master curve construction fails not only for the position but also for the intensity of the terminal relaxation. The height of the G″ peak for the terminal relaxation increases with temperature (a zoom into the plateau region can be found in the Supporting Information Figure S4). The corresponding extra relaxation process lies in the same frequency range as the terminal relaxation process of a homopolymer of about the same molecular weight. Hence, one could speculate about reptation processes in the slightly entangled corona of the supramolecular aggregates as a possible reason for this observation. In fact, similar effects were found for entangled star polymers.53,54 Next, we analyze the terminal flow part and discuss the terminal relaxation time. The relaxation time spectrum was calculated by the rheometer software. One example is shown in the Supporting Information Figure S6. The relaxation time spectra show a peak with a width of about 1 decade. Alternatively, the characteristic relaxation time can be calculated as the inverse of the angular frequency where G′ and G″ cross. The third possibility is to calculate the relaxation time from the simple Maxwell equation τ = η/G, where G is the plateau

Figure 10. (a) Logarithm of terminal relaxation time τ vs 1000/T for the four supramolecular samples (PIB4K-BA2, ▽; PIB8K-BA2, △; PIB14K-BA2, ○; PIB28K-BA2, □) and a homopolymer PIB with molecular weight of 30 kg/mol (◇) and the corresponding linear fit. (b) Activation energy vs molecular weight (same symbols as in (a)). G


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Figure 11. (a) Stress−strain curves for PIB14K-BA2 at 25 °C for different shear rates (Wi = 0.33, black; Wi = 1.04, red; Wi = 3.3, blue; Wi = 10.4, light blue; Wi = 33, magenta; Wi = 100, green; Wi = 330, dark blue; Wi = 1000, brown). (b) Stress−strain curves for PIB14K-BA2 at 10 °C for different shear rates (Wi = 9, black; Wi = 28, red; Wi = 90, blue; Wi = 280 , light blue; Wi = 900, magenta; Wi = 2800, green; Wi = 9000, dark blue). (c) Stress−strain curves for PIB4K-BA2 at 20 °C for different shear rates (Wi = 1.1, black; Wi = 3.5, red; Wi = 11, blue; Wi = 35, light blue; Wi = 110, magenta. (d) Stress−strain curves for PIB4K-BA2 at 15 °C for different shear rates (Wi = 0.5, black; Wi = 5, red; Wi = 50, blue).

This result is an additional indication that the terminal flow is not controlled by chain dynamics but by dynamic bonds which are localized. The results are consistent with predictions by Tanaka et al.55 for a transient network model with reversibly associating end groups. The model gives modulus curves which also depend only weakly on the molecular weight of the polymer chains. For comparison, Figure 10a includes also the terminal relaxation time of a homopolymer PIB with a molecular weight of 30 kg/mol. Obviously, the temperature dependence of the relaxation time of the homopolymer is considerably weaker than that of the telechelic polymers. This is again an indication that in the case of the functionalized polymers the terminal flow is controlled by the dynamic bonds, not by chain dynamics. Over a broad temperature range one would expect a WLF type temperature dependence. But in our range of about 4 decades for the relaxation time this simplifies to an approximately Arrhenius type behavior. Correspondingly, a fit with a linear function in Figure 10a gives the flow activation energy. The results for the different samples are shown in Figure 10b in dependence on molecular weight. As already stated, the activation energy for the homopolymer is much smaller compared to the supramolecular samples. For the four supramolecular samples, the activation energy decreases slightly with increasing molecular weight. In conclusion, the analysis of the linear rheological properties confirms the existence of a dynamic network of supramolecular aggregates, connected by telechelic polymer chains. The flow properties of this network are controlled by the dynamics of the hydrogen bonds and not by the chain dynamics. This dynamic process controls also the kinetics of self-healing in ruptured dynamic networks, as we have demonstrated in a previous publication.39 Nonlinear Rheology: Fracture of the Dynamic Network. Nonlinear mechanical properties are important for many

applications. Additionally, they give also fundamental informations not accessible by linear response experiments. One could expect, for instance, an influence of mechanical stress on the lifetime of the dynamic bonds and hence on the dynamics of the whole network. We used startup shear experiments to investigate fracture and nonlinear stress relaxation. Beginning at time zero, the sample is deformed with a constant shear rate. Shear stress and strain are recorded over time up to strain values, much larger than during linear response experiments. Whether one observes typical nonlinear behavior depends on the value of the Weissenberg number Wi, the product of the shear rate and the longest relaxation time in the system, Wi = τγ̇. For values of Wi < 1 the sample has enough time to relax during deformation and linear behavior is found. Only for Wi > 1 typical signatures of nonlinear behavior are observed. We selected samples PIB14K-BA2 and PIB4K-BA2 for nonlinear experiments to detect possible molecular weight dependencies. The higher molecular weight sample PIB28KBA2 will not be discussed because of the additional influence of chain entanglements. Figure 11a shows the results for a series of startup shear experiments for sample PIB14K-BA2 at 25 °C.39 The terminal relaxation time from linear response experiments at this temperature is 33 s. As expected, for Wi < 1 the dynamic network shows normal viscoelastic flow. The shear stress first increases with shear strain and becomes constant at high strain values, indicating terminal flow. Figure 12a demonstrates that one can successfully fit the data from the startup shear experiments at small Weissenberg numbers with a simple Maxwell model ⎡ ⎛ G ⎞⎤ σ = ηγ ⎢̇ 1 − exp⎜ − t ⎟⎥ ⎝ η ⎠⎦ ⎣ H

(7) DOI: 10.1021/acs.macromol.6b02507 Macromolecules XXXX, XXX, XXX−XXX

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startup shear measurements on sample PIB4K-BA2 at the two temperatures 20 and 15 °C. They are again similar to those of sample PIB14K-BA2. The most obvious difference is that the yield strain is smaller than for sample PIB14K-BA2 at the same Weissenberg number (see discussion below). The results for the yield stress and strain values for both samples PIB14K-BA2 and PIB4K-BA2 for different temperatures are collected in Figures 13a,b as a function of

Figure 12. (a) Time dependence of shear stress (□) and fit curve () for the startup shear measurement on sample PIB14K-BA2 with Wi = 0.33 at 25 °C. (b) Enlarged representation of the initial part of the startup shear curve for different shear rates (Wi = 0.33, blue; Wi = 3.3, magenta; Wi = 33, light blue; Wi = 330, black). A linear curve with a slope corresponding to the plateau modulus of 3 ·105 Pa from the dynamic measurements is shown as a dashed line.

where σ is shear stress, η viscosity, γ̇ shear rate, G plateau modulus, and t time. The results of the fit G = 1.9 × 105 Pa and η = 7.25 × 106 Pa·s are comparable to the independently measured data G = 3 × 105 Pa and η = 9 × 106 Pa·s from dynamic measurements in the linear range. At Weissenberg numbers greater than one, the dynamic network has no time to reconstruct. In this case, the dynamic bonds act as permanent cross-links. This leads to an elastic response and a linear increase of stress over strain for small strain values. Figure 12b demonstrates that the slope of the stress strain curves at small deformations for high shear rates approaches the value of the plateau modulus from linear response measurements. At large deformations the weak dynamic bonds fail, leading to macroscopic yielding. The shear stress goes through a maximum with a sharp drop at high deformations. The maximum is taken as the yield point.56,57 Of course, this behavior cannot be modeled by linear response parameters anymore. Please note that yield stress for such a dynamic network is different from a yield stress for brittle solids or glasses.58 There the yield stress is defined for vanishing shear rate. Here one needs a shear rate larger than the inverse terminal relaxation time. Figure 11a shows that yield stress and strain both increase with shear rate. Experimental results from literature on associative polymer networks find that yield strain is either constant,59,60 increases,34 or decreases61 with increasing shear rate. The reason for these differences is unclear. Figure 11b shows the results of the startup shear measurements of sample PIB14K-BA2 at the lower temperature of 10 °C. The results are similar to those at 25 °C. Higher values of the Weissenberg number can be reached for the lower temperature. At such high shear rates there are some indications of strain hardening before yielding, i.e., an increase in slope with increasing deformation. Figures 11c and 11d show the results of the

Figure 13. Dependence of (a) yield strain γc and (b) yield stress σc on Weissenberg number for the two samples at different temperatures (PIB14K-BA2: 10 °C (□) and 25 °C (○); PIB4K-BA2: 15 °C (△) and 20 °C (▽)). The dashed line in (a) shows exemplarily the expected behavior according to the model of ref 62.

Weissenberg number. Beside some scatter, yield strain and stress are nearly temperature independent for constant Weissenberg number for each sample. Hence, the temperature dependence of the nonlinear behavior is already captured by the temperature dependence of the terminal relaxation time from linear rheology. Yield stress and strain are approximately linear functions of the logarithm of shear rate. This does not fit to the model prediction from ref 62, where the yield strain in the associative dynamic network is directly proportional to the shear rate. The yield strain increases with increasing molecular weight while the yield stress decreases. Yielding in a self-assembled polymer network during startup shear deformation has been investigated and modeled in the literature.34 The model assumes that yielding happens if the duration of the experiment reaches the average lifetime of the dynamic bonds. Please notice that the latter changes during deformation. The average lifetime is calculated from the survival probability of a bond, which in turn is related to the disassociation rate of the dynamic bonds. The key assumption is that the disassociation rate of a single bond increases exponentially with the applied stress. Obviously, this assumption is different from the model,26 where network rearrangement happens through cluster exchange processes, not single bond exchange. Together with a linear stress strain relation before yielding, the assumptions of ref 34 result in a yield stress σc of I


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σc =

Gξ ⎛ Kδτ ⎞ ln⎜ γ ⎟̇ Kδ ⎝ ξ ⎠

Figure 15 shows the result for sample PIB14K-BA2 at two different temperatures. Each part of the figure shows the result

(8)

This model explains the logarithmic dependence of yield stress on shear rate. A quantitative comparison still contains several parameters which we tried to estimate independently. G is the elastic modulus, taken as the plateau modulus from the linear response experiments. ξ is the typical distance between network junction points, i.e., the distance corresponding to the peak position in the SAXS scattering data. The undisturbed lifetime of the bonds τ is taken as the terminal relaxation time from the linear response data. δ is a width that characterizes the bond potential. It determines how strongly the disassociation rate of a dynamic bond depends on the locally applied force. δ was taken to be the diameter of the aggregates, estimated by the previous SAXS fit. Table 2 lists the parameters used. Table 2. List of Parameters Used for Scaling Yield Stress vs Strain Rate sample temp τ/s δ/nm ξ/nm

PIB4K-BA2 15 °C 20 °C 517 113 2.25 5.4

PIB14K-BA2 10 °C 25 °C 900 33 2.7 7.6

Figure 15. Startup shear (black squares) and the following stress relaxation with strain of 150% (green symbols) and 500% (cyan symbols) for sample PIB14K-BA2 at 25 °C (a) and 10 °C (b). The corresponding fit curves are included (red line for 150% and blue line for 500% deformation).

The only remaining free parameter is K, a stress concentration factor. This value was optimized to give a maximal overlap between the data of different samples, measured at different temperatures. Figure 14 shows the result for a common value of K = 3.82. The axes were scaled according to eq 8 to show the linear relationship σc ∼ ln γ̇.

for two measurements: one with a final deformation of 150% (smaller than the yield strain) and one with a deformation of 500% (larger than the yield strain). In case of a startup shear strain smaller than the yield strain, the following stress relaxation after cessation of shear shows only a slow process which corresponds to a viscoelastic fluid. When the strain is larger than the yield strain, the following stress relaxation shows two processes: a faster one in addition to the already known slower one. The time scale for the slower process seems similar to the terminal relaxation time. We interpret the other, faster process as the collapse of the hydrogen bonding network, caused by yielding. To make a more quantitative comparison, we fit the stress relaxation data with a stretched exponential function

Figure 14. Scaled yield stress σc vs scaled shear rate γ̇ for the samples PIB4K-BA2 and PIB14K-BA2 at different temperatures (same symbols as in Figure 13).

σ(t ) = σ0 exp( − (t /τ )β ) = γG0 exp(− (t /τ )β )

(9)

where σ(t) is the stress measured at time t after cessation of shear, σ0 is the shear stress at the beginning of stress relaxation, τ is the relaxation time, β is the stretch exponent, γ is the shear amplitude, and G0 is an elastic modulus. The corresponding fit curves are included in Figure 15. We can see that the model fits the experimental data at long times very well. The fit parameters are collected in Table 3. For comparison, the table includes also plateau modulus values and terminal relaxation times from the dynamic measurements shown before in ref 39. No quantitative analysis of the width of the relaxation time distribution was attempted there, and hence no width parameter β is given in these cases. In all cases the values for the relaxation time from the stress relaxation experiments are comparable to the terminal relaxation time from linear response experiments. This proves that terminal relaxation of the dynamic network is indeed the reason for the observed slow process in stress relaxation. For

Obviously, the model is able to scale measurements of yield stress for different samples and different temperatures to a common master curve. It satisfactorily explains the logarithmic dependence of yield stress on shear rate. In addition, the parameters of the model correlate well to physical parameters of the samples, determined independently by SAXS investigations. Stress Relaxation in the Nonlinear Region. In order to learn more about the rupture of bonds, stress relaxation measurements were conducted at different strain levels. All measurements started with startup shear with the same shear rate of 1 rad s−1. This increasing deformation was stopped at different levels of strain, before or after passing the yield point. The corresponding final strain was held constant afterward and the time evolution of stress was recorded. J


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nevertheless both buildup dynamic networks with similar elastic properties. Sample Tr-PIB-Tr + Th-PIB-Th showed a gelation transition in dependence on temperature, accompanied by strong changes in the shape of the rheological curves. This makes a direct estimation of the terminal relaxation time at the common temperature of 40 °C in Figure 16 impossible. But it is clear from Figure 16 that the terminal relaxation time would be much longer than the one for the sample PIB4K-BA2. The bond dynamics in a system with triazin−thymine dynamic bonds is different from a system with barbituric acid groups. The former are probably stronger, which leads to longer lifetimes of the bonds and longer terminal relaxation times. Because of the gelation transition, a direct comparison of activation energies for the terminal relaxation time in both samples is not possible. Nevertheless, Figure 16 demonstrates that elastic and flow properties can be independently tuned by the molecular weight of the backbone chains and the functional groups.

Table 3. Fit Parameters for Linear and Nonlinear Stress Relaxation on Sample PIB14K-BA2 temp (°C)

deformation

G0/MPa

τ/s

25 25 25 10 10 10

linear 150% 500% linear 150% 500%

0.314 0.263 0.138 0.314 0.245 0.067

33 19.96 21.4 900 620 533

β 0.58 0.56 0.59 0.53

deformations below the yield point also the magnitude of the elastic behavior, characterized as elastic modulus G0, is comparable to the plateau modulus value from linear response. For larger deformations beyond yielding, the fast collapse of the dynamic network relaxes already a large fraction of the stress. This is represented as strongly reduced values of the elastic modulus G0 for the slow process. The stretch exponent around β = 0.55 signals a broad distribution of relaxation times. The value is compatible with the width of the relaxation time spectrum of about 1 decade, observed in linear response experiments.63 The similar values for the stretch exponent β and time constant τ for small and larger deformations at each temperature mean that one could superimpose both curves in the long time range of Figure 15 by a vertical shift. This corresponds to a separation of time and deformation dependence for the stress relaxation function. These stress relaxation experiments show that instability in the network sets in at the yield point. Beyond yielding, stress is reduced considerably by a relatively fast process, and the following slow relaxation is controlled by the same time scale as the usual terminal relaxation from linear response. Influence of Different Functional Groups. The molecular weight and temperature dependence of the mechanical properties has been demonstrated in this article for a single set of functional end groups. To show the orthogonal influence of different functional groups on the mechanical properties, we compare in Figure 16 the rheological data of sample PIB4KBA2 with those of a bifunctional mixture Tr-PIB-Tr + Th-PIBTh of similar molecular weight, but different end groups, from our former study.25 Both samples have a similar molecular weight of about 4000 g/mol, which leads to a similar network density and hence also a similar height of the elastic plateau. Although sample Tr-PIBTr + Th-PIB-Th as a mixture of two telechelic polymers is more complex than sample PIB4K-BA2 from this study,

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CONCLUSIONS

A series of telechelic PIB samples of varying molecular weight, functionalized with barbituric acid groups at the chain ends, have been thoroughly investigated by SAXS and rheology. All samples show micellar aggregation of the functional groups in the melt state. The size and number density of these aggregates were characterized by SAXS. From the size we estimated an aggregation number of about 20 chain ends, which is not strongly molecular weight or temperature dependent in the investigated range. Rheology in the linear response regime proofs that these aggregates form a densely connected network. In addition, this network shows terminal flow because of the dynamic character of the hydrogen bonds for the functional groups. The temperature dependence of the terminal relaxation time is controlled by the functional groups, not by chain dynamics. At the same time, the terminal relaxation time shows only a weak molecular weight dependence. Startup shear deformation with high shear rates leads to shear yielding followed by fast stress relaxation caused by network failure. The dependence of yielding on shear rate can be explained by a model, where the lifetime of the dynamic bonds depends exponentially on the mechanical stress. Stress relaxation experiments in the nonlinear region help to separate the behavior into two contributions: a fast collapse of the network and a slow relaxation, corresponding to the normal terminal relaxation. The latter controls then also self-healing of the collapsed structure.39 Figures 7 and 16 demonstrate that it is possible to independently control the elastic properties and the terminal flow properties of dynamic telechelic networks by the molecular weight of the chains and the functional groups at the chain ends, respectively. Different from polymer melts the terminal relaxation time is not strongly molecular weight dependent. But the elastic properties are molecular weight dependent similar to permanent polymer networks, while dynamic networks retain their ability to flow on long time scales. These general guidelines may help to improve such self-assembled, functional materials.

Figure 16. Complex modulus (G′ (squares), G″ (circles)) vs angular frequency ω for samples PIB4K-BA2 (filled symbols, this work) and Tr-PIB-Tr + Th-PIB-Th (open symbols, ref 25) at a common temperature of 40 °C. K


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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.6b02507. Additional material with further scattering data and rheological measurements (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (T.T.-A.). ORCID

Tingzi Yan: 0000-0001-7159-6938 Wolfgang H. Binder: 0000-0003-3834-5445 Thomas Thurn-Albrecht: 0000-0002-7618-0218 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the German Science Foundation DFG (Projects TH 1281/5-1, BI 1337/7-1, and SFB TRR 102), the projects BI 1337/8-1 and BI 1337/8-2 within the SPP 1568, and the state of Saxony-Anhalt (Research Network Nanostructured Materials) for financial support.

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REFERENCES

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