We analyze the effects of polymer concentration, *C*_{p}, and physical crosslinking ratio, *r*_{c}, on the linear viscoelasticity of lightly entangled, linear, associating polymers with multiple crosslinkable groups along the chain backbone. To accomplish this goal, we utilize three novel tools: a robust tunable chemistry based on electrophilic methacryl-succinimidyl modified poly(*N*-isopropylacrylamide); a modified state-of-the-art entanglement theory, called the discrete slip-link molecular model; and a novel experimental technique, surface fluctuation specular reflection. This experimental technique allows the extraction of the complex viscoelastic modulus covering up to six decades of frequency, from the sample's surface fluctuations at constant temperature. We demonstrate that our theoretical model with a consistent set of parameters can qualitatively reproduce the observed features of the complex viscoelastic modulus over the entire probed frequency range. Furthermore, the effects of *C*_{p} and *r*_{c} on the viscoelastic properties agree with scaling laws predicted by the “sticky reptation model” and other published experimental data. Finally, we discuss nonadditivity of the effects of entanglements and transient associations.

## I. INTRODUCTION

The attachment of transient crosslinks on a polymer chain backbone allows for a significant modification of its viscoelastic properties. This modification is achieved by constraining the chain dynamics in a way somewhat similar to what entanglements do, but by suppressing reptation. Variation of the type and binding energy of the association, as well as the interaction between solvent and polymer, results in an even more versatile variation of mechanical properties. This flexibility of properties manifests itself in a variety of physical phenomena, including self-healing, [1,2] extension hardening, [3] and shear thinning and dilatancy [4,5], which are of great importance in several industrial and biomedical applications.

The rich rheological behavior of such materials is determined by the interplay of polymer thermodynamics and kinetics of reversible crosslinking on the one hand, and entanglement dynamics on the other. A fundamental understanding of the underlying molecular mechanisms, which might allow designing systems with desirable viscoelastic properties, is currently problematic due to the lack of systematic experimental rheological data covering a broad frequency range. This limitation arises in part from differing temperature dependencies of the characteristic friction coefficient of the bare chains, and the stickers' association dynamics (AD), which forbids application of the time-temperature superposition principle [6]. In addition to the difficulties of taking rheology measurements, molecular modeling requires independent determination of additional association parameters, such as the number of crosslinkable groups per chain and their association/dissociation rates. These quantities can also be affected by polymer concentration, and independent determination of their values is not easy, but is necessary if something better than curve fitting is sought.

The third limitation is related to the difficulties of preparing model supramolecular materials that would allow for careful control of the relevant parameter space without substantially altering the precursor polymer. In a recent publication by Shabbir *et al.* [3], some of the authors studied such model systems. By analyzing the effects of hydrogen bonding on the linear rheology of entangled polymer melts, they concluded that the chain dynamics dominated by association and dissociation of stickers is slower than that dominated by entanglement dynamics. In this paper, we employ a generalization of our detailed molecular model to further examine this conclusion.

The goal of this study is to obtain and interpret the linear rheology of lightly entangled multisticker associating polymer systems over a broad frequency range. To accomplish this goal, we utilize three novel tools: a robust tunable chemistry, a detailed molecular model, and a rheometer able to cover many decades of frequency of small samples. To collect data over such a frequency span, we measure thermal fluctuations on the materials' surfaces, from which the complex viscoelastic moduli are then extracted with the help of a fluctuation-dissipation relation. The samples are prepared at various polymer concentrations (mass fraction), *C*_{p}, and transient crosslinking ratios, *r*_{c}. The obtained data are simultaneously analyzed by applying a generalization to the single-chain discrete slip-link model (DSM) with a consistent set of parameters and confronted with previously observed power-law dependencies predicted by the sticky reptation model of Rubinstein and Semenov [7]. This work is the first attempt to generalize the DSM, originally developed for homopolymer melts, to semidilute solutions of associating polymers. Ultimately, we seek to test whether our novel experimental technique in combination with our detailed molecular model can be used for measuring and predicting the viscoelastic properties of a highly tunable multisticker supramolecular polymer system.

## II. EXPERIMENTAL SECTION

### A. Materials

Model supramolecular-associative (“sticky”) polymers are made from methacryl-succinimidyl (MASI) modified supramolecular poly(*N*-isopropylacrylamide) (pNIPAAm). It has been shown [8,9] that this material can be used to prepare supramolecular polymer gels of broadly tunable mechanical characteristics by varying the chains' degree of polymerization as well as the number, strength, and type of crosslinkable groups, while keeping the same precursor polymer. Materials used in this paper are prepared by replacing the MASI moieties by terpyridine (TPy) sticky groups and by adding Mn(NO_{3})_{2} ions to interlink the TPy stickers. Characterization by size exclusion chromatography is conducted using PSS GRAM-1000/100-7 *μ*m columns, 70 °C, *N*-methylpyrrolidone eluent with 0.05 mol l^{–1} of lithium bromide, and benzoic acid methylester as internal standard. Molecular weight calibration with polystyrene standards reveals the number-average molecular weight of *M _{n}*= 61 kDa and polydispersity index of PDI = 1.7. The amount of MASI in the copolymer, determined by 1H NMR (Bruker AC 700 spectrometer), is 4.8 mol. % relative to the amount of

*N*-isopropylacrylamide (NIPAAm) repeat units [8,9]. This corresponds to ≈22 crosslinkable groups per chain with number-average molar mass. The samples are prepared by disolving pNIPAAm in DMF, which is a good solvent. Obtained polymer solutions in concentrations of

*C*

_{p}= 20, 30, and 40 wt. % are then mixed under a continuous stirring with Mn(NO

_{3})

_{2}in the crosslinking ratios defined as

*r*

_{c}= 2⋅

*C*(Mn

^{2+})/

*C*(TPy)= 0.25, 0.5, and 1. The overlap concentration of our basis polymer is $Cp*\u22484.5\u2009wt.\u2009%$ [8], whereas the entanglement threshold is at $2\u22125Cp*$. We can, therefore, consider the probed systems, covering the range of $4.5\u22129Cp*$, to be in the lightly entangled regime. Assuming pairwise associations between crosslinkers and sticky polymer side groups according to $Mn2++2TPy\u2192Mn(TPy)22+$, there is a complete conversion corresponding to

*r*

_{c}= 1.0 [8]. Every step of the sample preparation takes between 6 and 12 h. Homogeneity of the obtained mixtures is validated by visual inspection and reproducibility of all measured rheological data.

In Table I, we compile several details of all measured samples. The samples are named in accordance with their respective polymer concentrations “C” and crosslinking ratios “r.”

. | Materials . | DSM parameters . | ||
---|---|---|---|---|

Sample . | C_{p} (wt. %)
. | r_{c}
. | β
. | β_{st}
. |

20C | 20 | 0.0 | 14 | — |

20C0.25r | 20 | 0.25 | 14 | 16 |

20C0.5r | 20 | 0.5 | 14 | 8 |

20C1r | 20 | 1.0 | 14 | 4 |

20C2r | 20 | 2.0 | — | — |

30C | 30 | 0.0 | 10 | — |

30C0.25r | 30 | 0.25 | 10 | 16 |

30C0.5r | 30 | 0.5 | 10 | 8 |

30C1r | — | — | 10 | 4 |

40C | 40 | 0.0 | 6.5 | — |

40C25r | 40 | 0.25 | 6.5 | 16 |

40C0.5r | — | — | 6.5 | 8 |

40C1r | — | — | 6.5 | 4 |

. | Materials . | DSM parameters . | ||
---|---|---|---|---|

Sample . | C_{p} (wt. %)
. | r_{c}
. | β
. | β_{st}
. |

20C | 20 | 0.0 | 14 | — |

20C0.25r | 20 | 0.25 | 14 | 16 |

20C0.5r | 20 | 0.5 | 14 | 8 |

20C1r | 20 | 1.0 | 14 | 4 |

20C2r | 20 | 2.0 | — | — |

30C | 30 | 0.0 | 10 | — |

30C0.25r | 30 | 0.25 | 10 | 16 |

30C0.5r | 30 | 0.5 | 10 | 8 |

30C1r | — | — | 10 | 4 |

40C | 40 | 0.0 | 6.5 | — |

40C25r | 40 | 0.25 | 6.5 | 16 |

40C0.5r | — | — | 6.5 | 8 |

40C1r | — | — | 6.5 | 4 |

### B. Surface fluctuation measurements

For analyzing the bulk viscoelastic properties of these supramolecular materials, we employ a surface fluctuation specular reflection technique (SFSR) [10], and complement it by classical rheometry measurements of small-angle oscillatory shear deformations conducted at the same temperature. The SFSR allows noninvasive probing of extremely small thermal fluctuations on the material's free surface by measuring the power spectral density of thermal noise (PSD), *S*(*ω*).

The viscoelastic complex modulus in the frequency range consistent with that of *S*(*ω*), which covers up to six decades of frequency, is obtained by fitting the equation [11]

where Φ(*k*) is the weighting function of surface spatial mode *k*, determined by the laser wavelength and size. The power spectral density of the surface height, *P*(*k*, *ω*), is related to the frequency dependent mechanical susceptibility of spatial mode *k*, *χ*(*k*, *ω*), through a fluctuation-dissipation relation. Under conditions where bulk viscoelasticity is expected, *χ*(*k*, *ω*) is determined by the viscoelastic complex modulus, *G*^{*}(*ω*), the surface tension, *σ*, and the density, *ρ* (=1000 kg m^{−3}), of the material. The fitted PSD data are computed by averaging over at least 100 measurements to reduce noise.

The value of the surface tension for all tested samples (*σ* = 36 ± 1 mN m^{−1}) is obtained by matching *G*^{*}(*ω*) measured by the SFSR and classical shear rheometry at constant temperature, 25 °C. The same values have been obtained by conducting SFSR measurements using two different laser beam sizes (2*R _{l}* = 55

*μ*m and 2

*R*= 83

_{l}*μ*m). The fitting of the two PSD spectra obtained from the same sample allows for the determination of both surface tension and viscoelastic modulus [12]. We also analyzed the surface tension of the samples using the classical method of contact angle measurement. The angles were measured geometrically by evaluating pictures of small drops of the solutions sitting on a silicon wafer. The results so measured varied in the range of 40 and 50 mN m

^{−1}. We note that the precision of the manual geometrical determination of the contact angle is rather low, and therefore, we conclude that the utilized

*σ*= 36 ± 1 mN m

^{–1}is a reasonable estimate.

The complete theoretical background and technical details of the SFSR experimental technique, together with full validation of the measurements of *G*^{*}(*ω*) of materials ranging from soft gels to permanently crosslinked rubbers, can be found elsewhere [10–13].

### C. Small-amplitude oscillatory shear measurements

All small-amplitude oscillatory shear measurements are conducted following standard procedures and using TA Instruments advanced rheometric expansion system (ARES) equipped with a small-cone-angle cone-and-plate and parallel-plate fixtures of radii 25 and 8 mm, respectively. All samples were equilibrated at 25 °C for approximately 30 min prior to the frequency sweep measurement.

## III. THEORETICAL SECTION

### A. The discrete slip-link model

For analyzing the effects of *C*_{p} and *r*_{c}, we utilize a generalization of the single-chain DSM, with an added associating/dissociating dynamics of stickers distributed along the chain's backbone. The DSM is a mathematical model for a single-chain in mean-field of entangled polymers. It is compliant with the nonequilibrium thermodynamics formalism called GENERIC [14] and is a single mathematical object for arbitrary chain architecture in arbitrary deformation [15–17]. DSM was originally established for entangled melts of homopolymers, where quantitative rheological prediction is possible. Without stickers, DSM requires two adjustable parameters, one for friction per Kuhn-step length and another for entanglement density, whose values should be independent of the chain architecture, crosslinking, or blending. In the present analysis, we assume that the Kuhn-step friction is independent of *C*_{p} and *r*_{c}. [18] Implementation details of the DSM model can be found in Refs. [15]–[17] and [19]–[21].

In Fig. 1, we schematically demonstrate the construction of our generalization. The effect of entanglements is modeled via pointlike slip-links distributed along the chain. The stress of the chain relaxes via a combination of different types of chain dynamics. Sliding dynamics (SD) is implemented identically to prior work via the chain's diffusive motion along its primitive path in a manner similar to reptation dynamics in the tube picture. Constraint dynamics (CD) is controlled by creation and destruction of slip-links. Slip-links are not permanent, but can be destroyed either by SD or CD. The lifetime of sliplinks is either assigned from self-consistent determination [20] or, as in the present case, is determined via Doi–Takimoto algorithm [22], which is also identical to our prior work.

In this paper, we apply the DSM to semidilute solutions of multisticker associating polymers by adding chain AD through stickers distributed along the chain. The stickers' positions on a chain are fixed and therefore the number of Kuhn-segments between neighboring stickers is preserved at all times. Also, shuffling of the chain segments is prohibited through associated stickers representing crosslinks. As one might expect, by adding AD we inhibit SD of the chain and, consequently, the CD as well.

We do not consider any microphase separation. Affine motion is assumed for both physical crosslinking points of stickers and entanglements. Also, chain strands are treated as Gaussian. Loop formation due to intrachain association is prohibited in the present model.

### B. Parameters of the model

The construction of our model is shown schematically in Fig. 1. Besides the number of Kuhn segments per chain, *N*_{K}, the model uses two parameters to represent chain entanglement dynamics: the entanglement activity, *β* (approximately the average number of Kuhn steps per entanglement strand), and the Kuhn segment relaxation time, *τ*_{K} (related to the Kuhn-segment friction). Furthermore, there are three parameters for chain-chain associating dynamics: the crosslinking “activity,” *β*_{st} (approximately the average number of Kuhn segments per strand in between two active or associated adjacent stickers along the chain), the lifetime of associated stickers, *τ*_{L} (i.e., time duration during which stickers are forming crosslinking points), and their free time *τ*_{f}. The model elementary time is *τ*_{K}, and hence, in numerical simulations, we represent all time-related quantities ($\tau L,\u2009\tau f,\u2009t,\u2009\omega ,\u2009\eta 0$) in units of *τ*_{K}.

In the model predictions, we adjust two parameters: the entanglement activity, *β*, and the ratio between the lifetime of an associated sticker and its free time, *τ*_{L}/*τ*_{f}. These parameters are determined so that we can reproduce features of the experimental *G*^{*}(*ω*) data qualitatively. Quantitative agreement is not yet possible because the samples have polydispersity, which is ignored in the model implementation here for simplicity. On the other hand, *τ*_{K} is roughly estimated from the zero-shear viscosity as shown in the next section [Fig. 6(B)]. The other parameters are determined from the polymer chemistry and solution conditions.

In Fig. 2, we validate the implementation of AD in the DSM simulations. In the limit of no entanglements, we compare the predictions of the transient network model of Indei and Takimoto [22] with our predictions. Solid curves are predictions from the Indei–Takimoto model for an unentangled associating polymer network, where the number of cross-linkable cites per association polymer *N _{a}* is 5, 10, and 20 from bottom to top, and the association/dissociation ratio

*τ*/

_{L}*τ*is 100 (bottom figure), and 1 (top figure). In the Indei–Takimoto model, the curves are smooth because the numerical calculation of

_{f}*G*

^{*}does not rely on computer-generated random numbers. We observe a quantitative agreement between both models, as expected.

### C. Determination of the parameter values

#### 1. Chain length

Our model is capable of accurately accounting for arbitrary molecular weight distributions [20,23]. However, for the sake of simplicity, all theoretical predictions are made for monodisperse chain length, *N*_{K}. In this paper, we determine *N*_{K} from the number-average molecular weight *M _{n}*. Since the number-average degree of polymerization is 465 [8], we put

*N*

_{K}= 47 assuming 10 monomers per Kuhn segment [24].

#### 2. Entanglement activity β

The entanglement activity *β* is related to the average number of Kuhn steps per entanglement strand. The average number of entanglements per chain is given by $\u27e8Z\u27e9=(NK\u22121)/(\beta +1)$ [16]. We determine *β* by comparing the features of the experimental *G*^{*}(*ω*) [Fig. 5(A)] and DSM predictions for each polymer concentration. The result is shown in Table I and plotted in Fig. 3. We see that *β* is a decreasing function of the polymer concentration. Also, the estimated *β* values are larger than 6.5 for all conditions studied. The number of Kuhn steps per entanglement strand estimated from these *β* values is large enough to assume Gaussian statistics for the strands.

Figure 3 indicates that the entanglement activity of our system obeys the power-law

A good agreement is achieved when *β*_{melt} = 2. This value is lower than expected for melts probably due to the error from our monodisperse simplification. Also, note that all our experiments are made only for semidilute solutions up to *C*_{p} ≤ 40 wt. %, and hence, there is no guarantee that Eq. (2) is satisfied for high concentration regimes or melts. Therefore, we do *not* claim that entanglement activity for melts is 2. On the other hand, we see below that the exponent −5/4 is consistent with the scaling prediction for semidilute polymer solutions in a good-solvent condition (Fig. 6) [25].

Now, we dare to estimate an approximate entanglement molecular weight of the melts as *M _{e}* =

*M*

_{K}

*β*

_{melt}≈ 2.6 kDa, where the Kuhn segment molar mass

*M*≈ 1.3 kDa and

_{K}*β*

_{melt}= 2. This value is consistent with

*M*of other common polymers (Polyethylene, Polybutadiene, Polyisoprene, etc.). Since NIPAAm monomer has a large side group and molar mass of ≈131 Da, one might expect a value of

_{e}*M*higher than reported here. It is, however, expensive to estimate the exact value of

_{e}*M*from the viscoelastic data of our fairly polydisperse samples in the semidilute regime. Nevertheless, our experimental data certainly do indicate the presence of entanglements and we propose consistent values of

_{e}*β*, which are in a qualitative agreement with the data in the semidilute regime. A more detailed analysis allowing for precise determination of

*M*would require direct

_{e}*G*

^{*}(

*ω*) measurements for melts of better-defined samples with a narrow distribution of molecular weights.

#### 3. Crosslinking activity β_{st}

The parameter *β*_{st} corresponds to the average number of Kuhn segments between two neighboring associated stickers (or crosslinks) along the chain. That is, $\beta st\u2248NK/\u27e8Nstb\u27e9$, where $\u27e8Nstb\u27e9$ is the average number of crosslinks per chain. For a fixed *C _{p}*, $\u27e8Nstb\u27e9$ is controlled by the product of the average number of crosslinkable points per chain (≈22), and the crosslinking ratio. Therefore, in this paper, we assume a relation

*β*

_{st}= 4/

*r*

_{c}. The values of

*β*

_{st}used in this study are listed in Table I.

#### 4. Lifetimes of association τ_{L} and dissociation τ_{f}

We assume $\tau f=\tau K$ and $\tau L=100\tau K$ for all samples. Thus, the ratio between association and dissociation rates is always fixed to $\tau L/\tau f=100$ for all polymer concentrations and crosslinking ratios. According to [7] a higher degree of association may lead to an increase in the lifetime of effective association. However, taking an accurate account of this complex correlation is beyond the scope of this paper.

The ratio $\tau L/\tau f$ may be estimated as [*ML*_{2}]/[*M*], where [*ML*_{2}] and [*M*] represent the concentrations of associated supramolecular complexes and free metal ions, respectively. We can roughly estimate these concentrations from the equilibrium constant of the association $K=[ML2]/[M][L]2$, where [*L*] is the concentration of free TPy. In our case, the equilibrium constant is *K* ≈ 3.1 × 10^{2 }M^{−2} [9]; thus, we find that [*ML*_{2}]/[*M*] is roughly ≈5 depending on *r _{c}* and

*C*

_{p}. This value is 1 order of magnitude smaller than that we assumed (

*τ*

_{L}/

*τ*

_{f}= 100). However, note that according to Fig. 4(a) of [22],

*τ*

_{L}/

*τ*

_{f}dependence on

*G*

^{*}(

*ω*) is quite small when [26] 10 <

*τ*

_{L}/

*τ*

_{f}< 100. Upon decreasing

*τ*

_{L}/

*τ*

_{f}from 100 to 10, the plateau modulus decreases only about 10%. Considering this fact together with all the assumptions in the estimation of

*τ*

_{L}/

*τ*

_{f}and accuracy of the isothermal titration calorimetry experiment to measure

*K*(±10%), [9] our value of

*τ*

_{L}/

*τ*

_{f}= 100 is reasonable.

On the other hand, we are not aware of any experimental techniques to directly measure *τ*_{f} itself. We assumed that it is equal to the elementary time of the system *τ*_{K}, which is physically unrealistic. However, this simplification can be justified by the following. We estimated the effect of larger values of *τ*_{f}, keeping *τ*_{L}/*τ*_{f} constant, but did not observe any qualitative change in the results (see Figs. 8 and 9). This is because of the “perfect relaxation” treatment of polymer strands in DSM (also see discussion in Sec. III D). Under this treatment, small *τ*_{f} comparable to *τ*_{K} does *not* produce a major change in *G*^{*}(*ω*) from imperfect-relaxation effects.

### D. Calculation of the dynamic modulus

Stress predictions in DSM can be found from chain conformations by the average

where *n* is the number density of chains. In brackets, *F*(Ω) is the Helmholtz free energy of the chain with conformation $\Omega :=(Z,Nstb,{Ni},{Qi})$, where *Z* is the number of entanglement strands, $Nstb$ is the number of stickers engaged in physical crosslinks, *N _{i}* and $Qi$ are, respectively, the number of Kuhn steps and the connector vector between two neighboring slip-links or crosslinks (

*i*− 1 and

*i*) (see Fig. 1).

In this paper, we assume a Gaussian free energy, which is a good approximation for equilibrium dynamics of a random walk chain with a sufficient number of steps

where *T* is the temperature and *k*_{B} is the Boltzmann constant. The Green–Kubo formula is used to calculate the stress relaxation modulus *G*_{DSM} (*t*) [27].

Note that DSM does not predict the fast relaxation modes within elastically effective strands, since these dynamics have been coarse-grained out of the model. Therefore, we add back fast relaxation modes using a more detailed model, which introduces no new parameters [17]. We do not aim to reproduce exactly the high-frequency dynamics in a semidilute solution, which are expected to include Zimm modes within blobs of concentration-dependent size, and Rouse dynamics of blobs within a strand [28,29]. We approximate the fast segmental relaxation modes by Rouse dynamics. The number of modes was adjusted so as to approximate experimental data at the highest frequencies. However, the friction was determined consistently with *τ*_{K}.

Assuming sufficient time separation between relaxation mechanisms of SD, CD, and AD determined by entanglement and crosslinking densities on the one hand and stress equilibration due to fast Kuhn segments motion on the other hand, the total stress relaxation modulus can be computed as

where *G*_{R}(*t*) is the contribution due to fast segmental Rouse relaxation modes, computed as shown in [17].

The first term of the relaxation modulus in Eq. (5) assumes that the detached sticker attaches again at a distant position after it relaxes. In our model, this perfect relaxation is realized irrespective of the value of *τ*_{f}. Fast relaxation modes of the subchain are added by the second term in Eq. (5). Even though we assumed small *τ*_{f}, it does not mean in the perfect relaxation assumption that the detached sticker attaches almost immediately after the detachment. And, actually, in our experimental system, it should not. Fortunately, in the perfect relaxation assumption, the value of *τ*_{f} itself is not important, but the result depends only on *τ*_{L}/*τ*_{f}. This is why our results are not affected by *τ*_{f}, if *τ*_{L}/*τ*_{f} is fixed.

The frequency-dependent complex modulus is obtained by fitting *G*(*t*) with a number of Maxwell modes for analytic conversion to the frequency domain.

In Fig. 4, we demonstrate an example of *G*_{DSM}, predicted with model parameters: *N*_{K} = 47, *β* = 10, and crosslinking density, *β*_{st}, varying between the “no stickers” case, and the highly crosslinked system with *β*_{st} = 4. The data demonstrate that with increasing number of stickers (decrease of *β*_{st}), the modulus at small *t* also increases, indicating an increase in the total number of elastically active segments, but at the same time, the second plateau develops, which indicates a slowdown of the entanglement dynamics due to the stickers.

## IV. RESULTS AND DISCUSSION

We now confront measured experimental data with our theoretical predictions by DSM. We also extract *C*_{p} and *r*_{c} dependencies of the zero-shear-rate viscosity and elastic plateau modulus to compare them with the power laws predicted by the “sticky reptation” model [7,30]. In Fig. 5(A), we show *G*^{*} data obtained by SFSR and classical shear rheometry data for variations of the probed polymer concentrations (*C*_{p} = 20, 30, and 40 wt. %). In every subfigure, we vary the crosslinking ratios of the systems in the range between *r*_{c} = 0, no crosslinks to, *r*_{c} = 1, which represents a complete conversion of the crosslinkable groups. Samples 30C1r, 40C0.5r, and 40C1r were found to be too rigid for SFSR measurements. The upper limit for the elastic modulus is reached when thermal fluctuations of the sample approach the same magnitude as the noise from the experimental setup. Also, we are not able to obtain reliable data close to the high-frequency crossover at *C*_{p} = 20 and 40 due to high noise level of the experiment, mainly coming from the preamplifier of the photodiode. In Fig. 5(B), we present dimensionless DSM predictions of *G*^{*}(*ω*), in which the change of polymer concentration and crosslinking ratio are modeled by variations of the parameters *β* and *β*_{st}, respectively, in accordance with the experiments (see Table I).

Comparison of the experiments to the DSM predictions in Fig. 5 demonstrates that all features of the measured data can be predicted with a consistent set of parameters *β* and *β*_{st} that qualitatively captures the *C*_{p} and *r*_{c} dependence of the analyzed systems.

Unfortunately, the broad distribution of molar masses in the experimental data [Fig. 5(A)] smears out some of the richer features originating from the interaction between entanglements and stickers. For example, the high- and low-frequency plateaus overlap, making the extraction of *τ*_{L} difficult for these data.

On the other hand, the different relaxation regimes are clearer in the model prediction of our monodisperse chains [Fig. 5(B)]. Following the high-frequency regime (indicated by “a” in the figure) where the segmental dynamics is observed, the model predicts the onset of a high-frequency plateau (b) of $G\u2032(\omega )\u221d(\u27e8Z\u27e9+\u27e8Nstb\u27e9)$, where $\u27e8Nstb\u27e9\u2248NK/\beta st$ is the average number of associated stickers at any given time per chain (assuming that *τ*_{L} ≫ *τ*_{f}). At times shorter than *τ*_{L} (i.e., $\omega \tau K>\tau K/\tau L=10\u22122$), the system behaves like a network of elastic strands between both associated stickers and entanglements. This is well pronounced for the most crosslinked samples (*β*_{st} = 4 or *r _{c}* = 1), whereas, for the less associated samples (

*β*

_{st}= 8, 16 or

*r*= 0.5, 0.25), this plateau is partially blurred due to a larger contribution from segmental motion.

_{c}At the intermediate frequency zone (c), the chain motion is dominated by Rouse-like dynamics due to repeated association and dissociation of stickers along the chain, and thus, its respective frequency range is estimated to be $\u221d\u27e8Nstb\u27e92$. The terminal relaxation zone (e) is dominated by disentanglement, although their time scales are increased by the presence of stickers. Indeed, from comparing samples without stickers (20C, 30C, and 40C), one sees the sign of an appearance of a low-frequency plateau (d) with increasing *C*_{p}, which suggests the formation of an entanglement network. However, the same effect is observed upon increasing the degree of association at constant concentration. This indicates that the effect of entanglements in the terminal zone is strongly influenced by the stickers.

We note that by increasing *r*_{c} above 1 results in no further increase in viscosity or modulus, as suggested by comparing 20C2r and 20C1r [Fig. 5(B)]. In fact, something opposite happens. It can be speculated that an excess of crosslinkers causes modification of the associating reactions, which might now be described as Mn^{2+} + TPy → Mn(TPy)^{2+}. This results in a reduction of the number of elastically effective strands. One could also argue that the slight decrease in the modulus as compared to 20C1r might be attributed to experimental uncertainty. However, the near-perfect match of the SFSR and rheometry data of the same sample suggests consistency in the experimental data.

Given the good predictions demonstrated by the model in Fig. 5, we seek to further validate our model and experimental data by analyzing the *C*_{p} and *r*_{c} dependencies of the elastic plateau modulus and zero-shear-rate viscosity. For this purpose, we confront it with scaling laws predicted by the sticky reptation model [7,30]. That model utilizes a tube picture and is developed for predicting the dynamics of entangled chains with many stickers along the backbone. Accordingly, at time scales shorter than *τ*_{L}, the chain dynamics is essentially that of a chemically crosslinked network. However, following the successive dissociations of several stickers eventually allows the chain to relax by reptating along its tube. The rate of chain motion along the tube is thus controlled by the number of stickers per chain and their binding strength [30].

In Fig. 6, we demonstrate how *C*_{p} and *r*_{c} influence the lower-frequency modulus $GN,20$ associated with the longest relaxation time, *τ*_{max}, and zero-shear viscosity *η*_{0}, given by

and

Our predictions for $GN,20$ and *η*_{0} are given in dimensionless forms as $GN,20/(nkBT)$ and $\eta 0/(nkBT\tau K)$, respectively. In order to find $GN,20$ having the right dimension, we multiplied the dimensionless predictions by $nkBT=\rho CpRT/Mn$ for each polymer concentration *C*_{p} (*R* is the gas constant and the solution density is assumed to be *ρ* = 1000 kg m^{−3} for all samples). We found that the thus obtained $GN,20$ values are almost 1-order of magnitude larger than the experimental results for all polymer concentrations studied. It should be noted that this is not surprising because we are using *β* values that are determined from the feature of the experimental *G*^{*}(*ω*) rather than the plateau height, whose exact value here is unknown from polydispersity. In other words, it is possible that our estimate for *β* is too low (see above) and that the height of the experimental plateau is further reduced by polydispersity.

In Fig. 6(A), predicted $GN,20$ values are multiplied by a factor of 0.17 (filled symbols) for all polymer concentrations and crosslinking ratios to compare with the experimental results (empty symbols). On the other hand, we need *τ*_{K} to get the zero-shear viscosity with the right dimension from the nondimensional prediction results. We found that if *τ*_{K} is on the order of 1 *μ*s, then the predicted *η*_{0} values become comparable to the experimental results (if the same factor 0.17 as in the plateau modulus is applied to the predicted *η*_{0} values for all polymer concentrations and crosslinking ratios). In Fig. 6(B), we assumed *τ*_{K} = 0.7 *μ*s.

The data plotted in Fig. 6(A) are consistent with the postulation that entanglements dominate chain dynamics in the terminal relaxation zone. Indeed, upon stress relaxation due to multiple association/dissociation of stickers, the modulus is expected to drop to the level of an analogous system without stickers. In this case, the *C*_{p} dependence of $GN,20$ can be estimated as $GN,20\u221d(NK/\beta )n\u221dCp9/4$, which is in agreement with our experimental data and observations [31,32].

The variation of zero-shear viscosity of all the investigated samples with *C*_{p} is shown in Fig. 6(B). The data show that *η*_{0} is sensitive to both *C*_{p} and *r*_{c}. All data points follow approximately the same power-law dependence of *C*_{p} with an exponent close to 4 at concentrations $\u22655Cp*$. According to the predictions by the sticky reptation theory of Rubinstein and Semenov [7], a scaling of $\u2248Cp3.75$ falls in the concentration range corresponding to unentangled strands between associated stickers. Our theoretical predictions may be consistent with this scaling. Small deviations at lower *C*_{p} toward shallower power-law dependence are consistent with the experimental data of Candau and coworkers [31], who attributed this decrease in slope to the reduction of entanglement density below its network percolation limit, where chains exhibit Rouse dynamics.

Variation in the crosslinking density manifests itself by a vertical shift of the *η*_{0}(*C*_{p}) curves. It can be noted that both the DSM and the experimental data show similar shift factors. Unfortunately, our experimental data are limited, because many measurements have not reached a well-established terminal zone. Therefore, it is impossible to extract the exact dependence of *η*_{0} on *r*_{c} and validate the existing theoretical predictions. The DSM predictions, however, fall on the same line by normalizing over $\beta st\u22122.5$ (see Fig. 7), which means $\eta 0\u221d\beta st\u22122.5\u221drc2.5$. This result is comparable with that obtained by Rubinstein and Semenov [7], who observed a change of *r*_{c} scaling from 2 to 3, by introducing renormalization of the bond lifetime. But the agreement is coincidence because our single-chain model does not include such an effect.

As demonstrated in Figs. 5 and 6, the dependence of *β* on *C*_{p} (see Fig. 3) is fairly consistent with the *C*_{p} dependence of our experimental data and previously published works in the semidilute regime. We note that in order to verify the predictions of the model in a broader possible range of *C*_{p}, the theoretical predictions of the systems with *β* = 3 and *β* = 20, corresponding to *C*_{p} = 0.7 and *C*_{p} = 0.16, respectively, have been added to Figs. 6 and 7.

Finally, we demonstrate that the effects from the physical crosslinks formed by stickers and the entanglements are not additive, according to our theory. In Fig. 8, we show DSM modeling results of the system (without fast segmental Rouse modes), where the total number of cross-links and slip-links remains the same (≈11), whereas their combination is varied. In the first system, we have only physical crosslinks, the second has only entanglements, and the third with both. The average number of associated stickers (or crosslinks) per chain, $\u27e8Nstb\u27e9$, in the first system is approximately equal to the average number of entanglements, $\u27e8Z\u27e9$, in the second. Also, these two systems have approximately the same longest relaxation time, indicated as $1/\tau max\u27e8Nstb\u27e9$ and $1/\tau max\u27e8Z\u27e9$, respectively. In the third system, we have an equal number of physical crosslinks and entanglements, but their total is still 11.

Figure 8 shows the loss moduli predicted for these three systems. The system with physical crosslinks shows a peak in $G\u2033$ only at *ω* = 1/*τ*_{L} followed by Rouse-like dynamics at lower frequencies, covering a frequency range of width $\u221d\u27e8Nstb\u27e92$ until its terminal relaxation zone. The system with entanglements only shows a single peak of $G\u2033(\omega )$ at $\omega \u2009\u2248\u2009\tau e\u22121\u27e8Z\u27e9\u22123.5$, where *τ _{e}* is the Rouse relaxation time of a single entanglement strand [33]. Finally, the system with both entanglements and physical crosslinks shows one shoulder at the low-frequency regime ($\omega \u22481/\tau max\u27e8Z\u27e9,\u27e8Nstb\u27e9$) and one peak at higher frequency (

*ω*≈ 1/

*τ*

_{L}) in $G\u2033(\omega )$. The shoulder characterizes the relaxation dynamics due to disentanglement, and the high-frequency peak is due to physical crosslinks. Location of the high-frequency peak of the mixed system

*ω*≈ 1/

*τ*

_{L}is the same as that of the system with the crosslinks only. In other words, the evidence of sticker dynamics in

*G*

^{*}is entirely unaffected by the presence of entanglements.

On the other hand, the peak in $G\u2033$ associated with entanglement dynamics moves to a frequency lower by 1 order of magnitude. Hence, the presence of stickers *does* strongly affect the entanglement relaxation evident in *G*^{*}. The presence of stickers strongly suppresses the relaxation from SD, but the entanglement presence does not change the relaxation from stickers. This coupling is an indirect result of the model. In the formulation of our mathematical model, we assumed that neither *local* sticker dynamics nor *local* Kuhn-step shuffling was affected by the presence of the other object. Not surprisingly, *global* reptation is strongly suppressed by the stickers, resulting in an increased longest relaxation time.

## V. CONCLUSIONS

We have illustrated complex structure-property relationships in entangled associating polymers by analyzing the effects of various molecular parameters on their linear viscoelastic properties. In particular, we have demonstrated that the terminal relaxation zone is dominated by the entanglements' dynamics, which contradicts earlier conclusions [3], and that effects from entanglements and stickers are synergistic and nonadditive. We have utilized a novel experimental technique, SFSR, and a detailed theoretical model, DSM, to study the linear viscoelastic properties of a semidilute multisticker supramolecular network. Specifically, we have demonstrated that DSM with a consistent set of parameters can reproduce the effects of *C*_{p} and *r*_{c} on the viscoelastic properties of these systems. The theoretical predictions and experimental data were also demonstrated to be in agreement with scaling laws predicted by the sticky reptation model and published experimental data. These conclusions have been made under several simplifying assumptions including monodispersity, constant associating/dissociating rates of stickers at different *C*_{p} and *r*_{c}, network percolation. The SFSR technique can be used to obtain reliable and reproducible complex dynamic viscoelastic modulus data at a broad frequency range (≈6 decades) at constant temperature and vanishingly small strains. Together, these can serve as a robust toolbox for the systematic study of the effects of entanglements, associating dynamics and their interplay on viscoelastic and flowing properties of multisticker associating polymers.

## ACKNOWLEDGMENT

J.D.S. acknowledges the Army Research Office, Grant No. W911NF-11-2-0018 for financial support.

### APPENDIX: ADDITIONAL MODEL VALIDATION

##### 1. Effect of *τ*_{L} and *τ*_{f}

We here demonstrate the effects of various values of *τ*_{L} and *τ*_{f} on the shape of the dynamic modulus predicted by our model.

In Fig. 9, we present the same system as presented in Fig. 8, but, this time we use *τ*_{L} = 1000, *τ*_{f} = 10, and *τ*_{L} = 1000, *τ*_{f} = 1, while other model parameters are kept identical to those used in Fig. 8.

The position of the high-frequency peak in the red and purple curves is determined by *τ _{L}*, and therefore, we do not observe any change. Also, in the “no entanglement” case, both systems have identical average number of associated stickers $\u27e8Nstb\u27e9\u224811$, and therefore, the shape of the curves does not change significantly. The effect of different

*τ*

_{f}is apparent in the purple curve, where, as expected, extended free time of stickers leads to the faster stress relaxation of chain segments.

##### 2. Comparison of model predictions and experimental data

In Fig. 10, we demonstrate overlay of the DSM predictions with the respective experimental data of the 20 and 30 wt. % solutions also shown in the left and middle panels in Fig. 5. The shape of the curves is well captured with the slight discrepancies attributed to the simplifications (e.g., monodispersity) made to the model. For the 20 wt. % dataset, we also demonstrate fast relaxation modes (not shown in Fig. 5), predicted by the model with *β* = 14.

## References

*C*

_{p}, we attempted to extract the time-temperature superposition shift factors for nonassociating polymers (

*r*

_{c}= 0) using classical rheometry with stress-controlled Anton Paar Physica MCR 302 rheometer equipped with cone-plate geometry (cone angle 1°, cone diameter 50 mm). The obtained results (not shown here) appeared very noisy and hence extraction of the reliable shift-factors was somewhat difficult. For the quantitative analysis of the shift factors the data must be collected at lower temperatures to have a stronger signal and cover high frequency range of

*G*

^{*}. Unfortunately, our current experimental capabilities do not allow to probe rheological signal at the temperatures lower than 10 °C.