Dynamics of Entangled Linear Supramolecular Chains with Sticky

Article pubs.acs.org/Macromolecules

Dynamics of Entangled Linear Supramolecular Chains with Sticky Side Groups: Influence of Hindered Fluctuations Mostafa Ahmadi,*,† Laurence G. D. Hawke,‡ Hadi Goldansaz,‡ and Evelyne van Ruymbeke*,‡ †

Department of Polymer Engineering and Color Technology, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15875-4413, Iran ‡ Bio and Soft Matter Division (BSMA), Institut de la Matière Condensée et des Nanosciences (IMCN), Université catholique de Louvain, Place Croix du Sud 1, 1348 Louvain-la-Neuve, Belgium ABSTRACT: The design and effective application of supramolecular transient polymer networks based on the assembly of entangled polymer building blocks requires not only precise description of relaxation mechanisms of the chain segments but also inclusion of the kinetics of reversible formation and breakage of reversible supramolecular interactions. In this work we extend the tube-based time marching algorithm to the entangled associative polymers with sticky side groups, with special emphasis on the effect of hindered fluctuations, besides sticky Rouse and sticky reptation. Two different approaches are introduced for inclusion of hindered fluctuations comprising fluctuations with extra penalty for deeper segments and stepwise fluctuations by extra friction. It is shown that there is a transition zone in dynamic moduli between the characteristic lifetime of the stickers and emergence of the final relaxation slopes, which can be characterized by almost parallel drop of loss and storage moduli with slope of 0.5 that can be assigned to hindered fluctuations alongside the blinking of stickers. Theoretical guidelines are drawn for practical application of the model by inclusion of secondary association of sticky groups in clusters with longer lifetimes. dual stimuli-responsive characteristics.25−31 The dual associative groups like orthogonal hydrogen bonding and metal ligand coordination,25,26 different levels of hydrogen bonding,27 or metal ligand coordination with different strengths28,29 can be directly incorporated in one polymer chain precursor30 or provided by physical blending of separate chains having different reversible supramolecular interactions.31 The design and effective application of supramolecular systems requires precise description of relaxation mechanisms. This includes accounting for the influence of the kinetics32,33 of reversible formation and breakage of physical bonds as well as the uncontrolled heterogeneities of cross-linking density, in addition to the motions of the chain segments as captured by the classical theories of polymer dynamics based on tube model. Supramolecular transient networks can be classified into two main groups.34 The first group, which has associative motifs at both extremities of building polymer chain blocks can form extended chains, where the degree of supramolecular polymerization is proportional to the concentration and equilibrium number of active stickers. Utilization of nonlinear building blocks35,36 or of associative motifs with functionality larger than

1. INTRODUCTION Supramolecular transient polymer networks formed by reversible interactions besides the usual covalent bonds between monomeric units have introduced new features in material science.1 The secondary bonds such as hydrogen bonding,2−4 metal ligand coordination,5 ionic complexes,6 and amphiphilic phase separation7 have finite lifetimes and can be broken several times during experiments and reformed again. The specific lifetimes of the associative groups can be triggered by external stimuli8 such as temperature,9 pH,10 exposure to light,11 or mechanical forces,12 which brings about stimuliresponsive characteristics, useful in many applications like sensors,13 actuators, controlled release drug delivery,14 selfhealing,15 and shape memory16 materials. Also, in many supramolecular systems the transient supramolecular interactions can lead to nanometric inhomogeneity of cross-linking density due to, for example, higher polarity of the associative groups,17−20 crystallization tendency,21,22 π−π stacking,23 and even uncontrolled interaction of sticky groups with the main chain body.24 Despite the fact that such unintended interactions may amplify the complexity of system and made the interpretation of network dynamics more difficult, many researchers have used them as the main source of improving the physical/mechanical properties of their polymeric systems. Moreover, in many cases two different reversible interactions have been simultaneously utilized to provide a network with © 2015 American Chemical Society

Received: April 9, 2015 Revised: September 5, 2015 Published: September 18, 2015 7300

DOI: 10.1021/acs.macromol.5b00733 Macromolecules 2015, 48, 7300−7310

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Figure 1. (a) Storage and loss moduli of 38% hydrolyzed PnBA measured at different temperatures. (b) Master curves of relaxation moduli at 45 °C for pure and 38% hydrolyzed PnBA (symbols) (only measurements at lower temperatures between −15 and 35 °C are used for hydrolyzed sample), and predictions with fluctuations based on extra friction (solid line) and with fluctuations based on extra penalty (dashed line).

two37,38 leads to the formation of 3D transient networks. The dynamics of this class of supramolecular systems can be followed, as described in the pioneer work of Cates,39 by assuming living polymeric assemblies that are subjected to random chain scission and combination from noncovalent physical junctions. This idea has been recently revisited and combined with modern relaxation mechanisms of linear and star chains including reptation, contour length fluctuations, and subsequent constraint release and tube dilation.40−42 The second group of supramolecular networks includes polymer chains having sticky groups along the backbone. Again the number of active stickers, the lifetime for being associated, and the lifetime of a free sticker determine, besides the dynamics of polymeric segments, the relaxation behavior of the whole system. The situation becomes more complicated when the polymeric precursor is well-entangled and all of the aforementioned relaxation mechanisms involved in dynamics of entangled polymer chains are affected by successive formation and breakage of physical bonds. Scholars like Stadler et al.43 and Rubinstein et al.44,45 in early works considered that the presence of stickers does not affect the topological constrains, and the chain will diffuse along the curvilinear axis of tube to relax, however, by hindered reptation, slowed down by repetitive association/dissociation of stickers. Such a system is characterized by two different internal locks, one on the time scale of sticker’s lifetime and the other on the time scale of sticky reptation of the main chain. This relaxation picture was simplified to the two-step sticky Rouse relaxation mechanism for nonentangled sticky chains, corresponding to Rouse motion of segments between stickers before sticker’s lifetime and beyond this scale after their dissociation.46−48 Other researchers have neglected sticky reptation as a main way of stress relaxation but assumed instantaneous stress relaxation of segments if all of the active stickers between any considered segment and one of the free ends of the chain become free at the same time.49−51 Despite significant progresses on description of relaxation mechanisms of entangled complex polymer architectures in recent years,52−59 hierarchical relaxation steps of well-entangled sticky chains are poorly updated, which can be mainly attributed to the lack of systematic experimental studies on the effects of number of stickers and their lifetime on rheological properties of such systems. The experimental difficulties in controlling uniform distribution of stickers without uncontrolled phase separation, crystallization, or

stacking, besides the thermorheological complexity of these systems, due to different temperature dependence of segmental motions and sticker’s dynamics3,60 are the main obstacles for application of experimental results in bringing prediction pieces together. Recently, the time marching algorithm (TMA) model developed by van Ruymbeke et al. based on classical tube theory has been successfully applied to a wide range of complex polymer architectures53−55 and supramolecular telechelic systems.41,42 In this work, we would like to extend TMA to entangled supramolecular chains with sticky side groups, with special emphasis on the effect of hindered fluctuations besides sticky Rouse and sticky reptation mechanisms. Thus, the paper is organized as follows. First, the main relaxation mechanisms involved in dynamics of the supramolecular transient networks proposed here are described. In particular, two different approaches are introduced for description of hindered fluctuations: either by considering extra penalty for fluctuations of deeper segments or by assuming stepwise fluctuations based on extra friction. Numerical calculation and analytical derivation of the accumulated times are described in the Appendix. Then, predictions based on both approaches are compared and similarities and differences are discussed. Finally, the large range of achievable rheological behavior in supramolecular networks is explored through a parametric study, and final remarks and conclusions are presented.

2. MODEL DEVELOPMENT In this section, we first describe the supramolecular system synthesized in our group and explain the model assumptions, which correspond to these specific samples. Then, all stress relaxation mechanisms considered in the model are presented. 2.1. Model Assumptions. A series of supramolecular chains with sticky side groups were synthesized by random hydrolysis of entangled narrowly dispersed poly(n-buthyl acrylate) (PnBA) at different levels. The carboxylic acid groups of different chains attract each other by weak hydrogen bonding and form a transient network. Linear rheological properties of the samples were measured at different temperatures. Details of the synthesis, characterization, and rheological properties of the samples are explained in a separate paper,60 but examples are described here to rationalize model assumptions. The relaxation moduli of the 38% hydrolyzed PnBA (determined by NMR) measured at different temperatures are shown in Figure 1a. 7301


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could be reasonably neglected in this work because in the melt state the probability of a sticker to have intrachain association is expected to be small compared with its probability to create interchain associations, this last one being already quite low. Indeed, it has been shown in ref 45 that the fraction of loops is large at low concentrations and reaches unity below the overlap concentration, while it reaches the minimum value in melt. When association lifetime of the stickers is met, the active stickers, which effectively contribute to the polymeric network, disassociate and new active ones emerge in random places to keep the equilibrium fraction of active stickers constant. After another lifetime step, the current active ones are replaced by randomly positioned new active stickers and so forth. In other words, active stickers start random blinking among all potential stickers. All relaxation mechanisms of the precursor chain are affected by the simultaneous blinking of the stickers and slowed down as described in the following sections. The association lifetime of the stickers can be inferred from rheological properties. Feldman et al.47 have considered the frequency at which the storage modulus drops to 90% of the plateau value to be a measure of sticker’s lifetime. Similar scaling leads to pairwise association lifetime of 0.5 s in our samples in agreement with dielectric measurement results.60 According to the observed experimental behavior, the supramolecular system studied here is characterized by an entangled linear chain with the number of potential stickers along the backbone. The stickers are defined by two characteristic lifetimes: the lifetime of being associated and the lifetime of being free before becoming engaged again. The fraction of active stickers present at each time step is dictated by the thermodynamics based on these two characteristics lifetimes. The relationship between characteristic lifetimes and the fraction of active stickers is out of the scope of the current study and is described somewhere else,41,43−46 but the important point is that they are interrelated, and fixing two of them, the third parameter can easily be determined. 2.2. Relaxation Modulus. The dynamics of the defined supramolecular system follows the basic mechanisms involved in relaxation of linear chains including high-frequency Rouse motions, contour length fluctuations, final reptation, and motion of neighboring chains, which causes further relaxation due to tube motion or constraint release; however, each of these steps is drastically affected by the presence of the stickers and their corresponding lifetimes. The relaxation modulus can be calculated by the following equations

As segmental motions of the PnBA precursor and lifetime of the stickers have different activation energies resulting in different temperature dependencies, samples with high hydrolysis levels show significant thermorheological complexity, especially in the low-frequency region, where segmental motions and association/dissociation of sticker happen in parallel. Situation is less complex at higher frequencies, before lifetime of stickers, because only segmental motions restricted to the strands between entanglements and active stickers are allowed. The corresponding master curve of 38% hydrolyzed PnBA in the high-frequency region at reference temperature of 45 °C is compared with pure PnBA in Figure 1b. The curves are model predictions developed in Sections 2.2 to 2.5 and are explained later. While the predictions for the hydrolyzed sample in Figure 1b show final relaxation slopes of 1 and 2 for loss and storage moduli at lower frequencies, in the experimental results, as shown in Figure 1a, a second plateau emerges at intermediate temperatures at low frequencies (see curves corresponding to 45, 65, and 85 °C) and disappears at temperatures above 100 °C.60 The appearance of the second plateau at lower frequencies can be attributed to phase separation of polar groups into microdomains with lifetimes longer than pairwise associations.60 In the current work, we neglect microphase separations and focus on pairwise associations. The effect of cluster formation is studied elsewhere.61 Equilibrium number of active (associated) stickers can be simply calculated based on the level of apparent plateau modulus compared with the plateau modulus of pure PnBA

GN0,app GN0

=

Meg app Meg

N 1 1 + S app = Meg Meg Mn

G0N

(1)

(2)

GN0,app

where and are plateau modulus of pure and hydrolyzed PnBA, respectively. Meg is the entanglement molecular weight of the pure sample, while Mapp eg is the apparent entanglement molecular weight of the hydrolyzed copolymer that accounts for both entanglements and stickers. NS is the number of active stickers per chain and Mn is the number-average molecular weight of the precursor. The obtained results (based on Meg = 16 500 g/mol, determined later) show that while there are more than 600 hydrolyzed units out of 1600 monomers, only about eight groups are actively associated and contribute to rise of plateau level. The weak hydrogen bonding of carboxylic acid groups leads to large dissociation and short association lifetimes, which make only a small fraction of stickers to be active at each instant. Weak hydrogen bonding of hydroxyl side groups in polyglycerols,62 acrylamidopyridine, and carboxyethylacrlate side groups in PnBA4 have also been reported to cause similar minor increases in plateau level. We consider that each sticker can be either open or closed. The closed stickers could make inter- or intrachain associations. It should be noted that the number of active stickers determined based on the level of apparent plateau only accounts for the stickers with interchain associations. Pairing of successive active stickers and formation of loops along the chains is expected to accelerate the relaxation of the molecular segments (through Rouse relaxation), which can also affect the final relaxation by sticky reptation; however, this contribution

G (t ) α = ϕ(t )ϕCR (t ) + ϕRouse(t ) GN0

(3)

where ϕ(t) is the chain survival probability (defined as the unrelaxed fraction of the initial oriented tube, not yet released by reptation or contour length fluctuations), ϕCR(t) is the global tube relaxation (determined by constraint release motion, CR), ϕRouse(t) is the contribution of high-frequency Rouse motions and α is the dilution exponent. The chain survival probability can be calculated by summing up the contribution of reptation and contour length fluctuations in relaxation of all chain segments ϕ(t ) =

∫0

1

φreptation(x , t ) ·φfluctuations(x , t ) dx

(4)

where x is the fractional distance changing from zero at chain’s free end to one at the middle of the chain. 7302


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fluctuation, the second one considers that active stickers are acting as extra friction points. Contour length fluctuations of linear and star chains start from chain extremities by unconstrained Rouse motions and become thermally activated with an effective potential U(x) for deeper segments. Transition between these two modes happens at the molecular segment, where potential barrier equals the thermal energy of the system. The fluctuation survival probability is defined as

2.3. Sticky Rouse Dynamics at High Frequencies. Relaxation of linear chains at very high frequencies or very low temperatures is restricted to the bond length scale. Increasing temperature or the exposure time the scale of relaxation increases beyond bond length, as we pass the glass-transition temperature. In this regime two distant backbone bonds can simultaneously change their angles, and the segment in between can change its conformation. Further increase in time or temperature leads to increase in relaxing segment’s length until further increase in relaxation scale is prohibited due to the entanglements with neighboring chains, which defines a hypothetical tube for the permissible lateral motions. From this point the Rouse relaxation mechanism is limited to one dimension, that is, only along the tube axis. The situation for the sticky chains is the same at highfrequency regime except that the limitation of relaxation scale is met much sooner due to the presence of active stickers, which are frozen at these frequencies (given the fact that their association lifetime is beyond the frequencies scanned in this regime) besides the normal restriction due to entanglements with other chains. The slower relaxation modes of Rouse, which correspond to subchains of span beyond this length and up to the entanglement’s length, are postponed to times longer than the characteristic lifetime of the active stickers. This two-step Rouse relaxation mechanism is defined as sticky Rouse and is previously used in many systems including unentangled ionomers6,48 and unentangled chain with strong hydrogen bonding side groups,47 except for the fact that the second step proceeds to the whole chain length in the case of unentangled chains, while it is restricted to the tube diameter in the current study. ϕRouse(t ) =

⎡ N ⎛ −j 2 t ⎞ 5 ⎢ ⎟+ exp⎜ ∑ 4Z ⎢⎣ j = Z + 1 ⎝ τR ⎠ app

Zapp

⎛ ⎞ t φfluctuations(x , t ) = exp⎜ − ⎟ ⎝ τfluc(x , t ) ⎠

The characteristic fluctuation time of segment x is defined based on the aforementioned fluctuation modes τearly(x) =

(5)

where τR/j is the Rouse relaxation time for the small segments when the presence of stickers is not felt yet and τS.R/j2 is the second-step Rouse relaxation time for longer segments after dissociation of stickers, defined as 2

(6-1)

τS.R = τstickerZapp2

(6-2)

(8)

(9)

As the characteristic fluctuation times of deeper segments are exponentially separated in time space, the deeper segments can effectively fluctuate in an entangled network dilated by the fraction of already relaxed segments. In the early tube-based models, the fraction of still-oriented segments is calculated based on the position of the segment along the chain, so the entanglement density should be updated by Ψα(x) = (1 − x)α. Recently Shchetnikava et al.59 have proposed a new method to calculate the fluctuations solvent based on time instead of x, compatible with other elements of TMA.53−55 Therefore, the number of entanglements is updated at each time step, as described in ref 59. A. Accumulated Time or Extra Penalty Approach. Fluctuations of a linear chain with sticky side groups follow the same mechanism with minor adjustments. The segments between the first active sticker and the closest chain end are free to fluctuate at time zero. (See Figure 2.) When characteristic lifetime of the stickers is met, the active stickers dissociate from their location and new active ones appear somewhere else to keep the fraction of active stickers constant. (The new active stickers are randomly selected from all active and passive stickers.) To account for these stickers, we propose

⎛ − j 2 t ⎞⎤ ⎟⎥ ⎝ τS.R ⎠⎥⎦

τR = τeZ2

9π 3 4 4 τeZ x 16

d ln τlate(x , t ) 1 d U (x ) = = 3Zx Ψ α(t ) dx kT dx

∑ exp⎜ j=Z

(7)

where τe and τsticker are the Rouse relaxation time of an entanglement and the characteristic association lifetime of stickers, respectively. Z = M n /M eg is the number of entanglements in the chain and Zapp is the apparent number of entanglements defined as Zapp = Z + NS, that is, the degree of polymerization between entanglements and/or active stickers. The longitudinal Rouse modes are expected to be suppressed in associative polymers due to presence of stickers, while they are considered for the unfunctionalized precursor.53−55 2.4. Hindered Fluctuations: Comparing Extra Penalty to Extra Friction Approaches. Because of the active stickers along the chains, the contour length fluctuations of the chains are slowed down and can only take place at the rhythm of the association/dissociation of these stickers. In this section, we propose two different approaches to account for this effect. While the first one consists in determining the effective time during which a molecular segment is able to relax by

Figure 2. Blinking of the three active stickers (filled circles) among seven potential stickers (unfilled stickers are not associated) at different time steps and calculation of accumulated time for free segments (light blue) between first active stickers and chain ends at both extremities. Blue and black open circles are both free, but the black ones placed in active segments do not help segments relaxation. 7303


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Figure 3. Stepwise fluctuations in associative polymers (a,b) compared with the corresponding fluctuations modes in comb architecture (c,d). Segments between chain ends and the first sticker (equivalent to branch in comb architecture) are mobile, while internal segments are frozen before lifetime of stickers (a,c). All segments are mobile and free to choose fluctuations with respect to the closer stickers or middle of the chain after lifetime of stickers (b,d).

is the fact that there is no solvent coming from relaxed side branches accelerating relaxation mechanisms of the backbone. Besides, we consider a simplified version of only two fluctuations modes. The outer segments between the first active sticker and chain ends can choose between fluctuations with respect to the first active sticker or with respect to the center of the chain, while the inner segments placed between the first active sticker and the middle of the chain have to fluctuate with respect to the middle of the chain, that is, according to the second fluctuations mode. In Section 3.1, we compare both approaches: hindered fluctuations based on accumulated times and stepwise fluctuations. 2.5. Hindered Reptation. The curvilinear diffusion along the tube axis is the last and the main relaxation mechanism for linear chains

(10)

where Δt(x,t) is the accumulated time during which a segment x was able to fluctuate. It is obvious that segments closer to chain extremities can more easily accumulate time, while fluctuations of deeper segments need more penalty. Increasing the number and lifetime of the stickers results in larger fluctuations penalty for deeper segments. The analytical description of the accumulated time for different segments is one of the main concerns of the current study and is described in Appendix A. B. Stepwise Fluctuations or Extra Friction Approach. Another approach for dealing with the fluctuations in supramolecular systems is to treat the stickers just like side branches of comb architecture so that the fluctuations may happen in a stepwise fashion.55 In this approach only the active stickers are considered to be uniformly placed along the backbone. Fluctuations with respect to closer stickers benefit from carrying less extra friction but have to overcome deeper contraction penalty (because x changes from zero at chain end to one at the position of the immobile sticker and penalty increases according to eq 9), while fluctuations with respect to the middle of the chain have inversely lower contraction barrier but more extra friction due to larger number of mobile stickers. Therefore, as illustrated in Figure 3, segments are free to choose the fastest fluctuations modes, that is, fluctuating with respect to either the closer sticker, or the middle of the chain. Details of this mechanism can be found elsewhere.55 The total friction imposed at each fluctuations mode includes monomeric friction as well as extra friction from mobile sticky points ξtot(i) = Niξ0 +

φreptation(x , t ) =

i ,odd

⎛ − i 2t ⎞ ⎛ iπx ⎞ 4 ⎟ exp⎜ sin⎜ ⎜ τ ⎟⎟ ⎝ 2 ⎠ iπ ⎝ rept ⎠

(12)

Supramolecular systems with stickers along the backbone may reptate with extra friction coming from the active stickers as described in refs 43−46 for supramolecular systems and refs 53−55 for complex polymer architectures. The situation is quite similar to the reptation of branched chains,55 except for the fact that there is no extra solvent coming from the relaxed side branches α τrept = 3τeZ3φactive (t ) +

2NSτsticker

2α φactive (t ) (13) π2 where φactive(t) is the fraction of the material not yet relaxed by fluctuations at earlier times that can be used by the reptation mechanism at the current time according to the Graessley’s criterion.53−55 The last relaxation mechanism is the contribution of tube motion due to relaxation of the neighboring chains according to CR. Lateral movement of the tube in supramolecular systems is treated in the same way as regular linear chains, and the relevant details can be found elsewhere.53−55

2(i − 1)kTτsticker a2

∑

(11)

where ξ0 is the monomeric friction, Ni is the number of mobile monomeric units at mode i, and a is the tube diameter. The only modification compared with the case of comb architecture 7304


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Figure 4. Predicted dynamic moduli and tan δ with (dashed lines) and without (solid lines) reptation, considering stepwise fluctuations (a) and fluctuations based on accumulated time (b).

3. RESULTS AND DISCUSSION 3.1. Comparing the Two Proposed Approaches for Inclusion of Hindered Fluctuations. We first add the effect of stickers in the entire frequency range by considering sticky Rouse, stepwise fluctuations (following the extra friction approach), and sticky reptation as described in the modeling section. Figure 4a shows the predicted behavior for a linear PnBA with Mw = 210k, including about 600 sticky side groups (ϕsticker = 0.38), out of which 10 stickers are active with characteristic association lifetime of 0.5 s. (See Section 2.1.) The tube model parameters were fixed to G0N = 0.18 MPa, Meg = 16500 g/mol, τe = 1.15 × 10−4 s, and α = 1. These parameters were determined based on the best fit for the pure PnBA sample (solid lines in Figure 1b). The same parameters are also used for the sticky PnBAs, and the apparent plateau modulus and entanglement molecular weight are automatically fixed based on eqs 1 and 2. Then, similar predictions are made based on the second approach, which considers fluctuations with extra penalty according to eq 10 and accumulated times determined by eq a.4. The obtained results are depicted in Figure 4b. To put stress on the differences between results of the aforementioned fluctuations approaches, we also include predictions without reptation. As explained in the modeling section, relaxation starts from high frequencies with sticky Rouse motions of segments between physical junctions due to the presence of both the active stickers and the entanglements and follows with fluctuations of the free chain ends. After these two mechanisms, further relaxation is prohibited until the characteristic lifetime of stickers is met and internal segments relax by second-step Rouse motion and hindered fluctuations. Rouse motions are taking place until reaching entanglement length, while hindered fluctuations still continue toward the middle of the chain. Final relaxation happens at lower frequencies, considering reptation or not, and terminal Maxwell relaxation slopes of 1 and 2 for loss and storage moduli is obtained, respectively. In Figure 4, one can observe that there is a transition zone between the characteristic lifetime of the stickers and emergence of the final relaxation slopes, which can be characterized by almost parallel drop of loss and storage moduli with slope of ∼0.5. This regime can be assigned to fluctuations alongside the blinking of stickers, which leads to extra penalty for fluctuations of deeper segments or stepwise fluctuations with extra friction due to presence of active stickers. Interestingly enough, predictions based on both mechanisms of fluctuations lead to more or less

similar behavior, as compared in Figure 5. Special emphasis has deliberately put on fluctuations behaviors by adding predictions

Figure 5. Predicted relaxation modulus considering stepwise fluctuations (gray lines) and fluctuations based on accumulated time (black lines) with reptation (dashed lines) and without reptation (solid lines).

made without reptation. The shapes of the curves are quite similar to the predictions of Indei et al. using a single-chain theory model.50,51 According to their model, after the association lifetime of the stickers, relaxation moduli start to drop in parallel with the slope of 0.5 until the time NS2τsticker is met, after which the terminal relaxation slopes start to emerge. 3.2. Control of Linear Rheological Properties of Entangled Supramolecular Transient Networks. As shown in the studied example, the loss modulus goes through a maximum as the sticker lifetime is met due to the second step of sticky Rouse motions, and both moduli drop in parallel by hindered fluctuations until final crossover and subsequent Maxwell slopes are observed. Depending on number and lifetime of the stickers, two different relaxation scenarios can be distinguished. The first one, in accordance with the studied example, takes place when the relaxation time of the (nonsticky) precursor chain is short, that is, about or less than the lifetime of the stickers. This situation is illustrated in Figure 6, which shows the effect of number of sticky groups (NS= 0, 5, and 15) with τsticker= 10 s, for a PnBA having Mn = 210k. The terminal crossover of the precursor chain (NS = 0) is located at frequencies larger than the frequency at which stickers start blinking in the sticky chains (NS = 5 or 15). In these cases the peak in loss modulus is followed by parallel drop 7305


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Figure 6. Effect of number of stickers on dynamic moduli: NS = 0 (dotted lines), 5 (dashed lines) and 15 (solid lines) with τsticker= 10 s for a PnBA with Mn = 210k, considering stepwise fluctuations (gray lines) and fluctuations based on accumulated time (black lines).

Figure 7. Effect of number of stickers on dynamic moduli: NS = 0 (dotted lines), 25 (dashed lines), and 75 (solid lines) with τsticker= 0.1 s, for a PnBA with Mn = 1050k, considering stepwise fluctuations (gray lines) and fluctuations based on accumulated time (black lines).

of moduli due to hindered fluctuations and final relaxation happens by reptation before reflection of a discernible second plateau, as confirmed by a small shoulder in relaxation modulus (Figure 6b). In this scenario, compared with the delay imposed by the lifetime of the stickers, chain dynamics has a small influence. In other words, after dissociation of the stickers, almost no resistance is left and final relaxation happens quickly. It must be noted that in this example the characteristic association lifetimes and number of stickers are treated as independent parameters. In practice, the sticker’s lifetime can be changed independently from their number, for example, by using controlled amount of metal ions with different coordination strengths. The second relaxation scenario corresponds to the case at which the relaxation time of the precursor chain is much longer than the lifetime of the stickers. As is readily seen in Figure 7, a pretty different behavior is obtained in this case. To amplify the difference, five times longer precursor and shorter lifetime of the stickers (τsticker = 0.1 s) are used. It is worth noting that to have a similar density of stickers and comparable plateau level, five times more active sticky groups are utilized. Here the peak in loss modulus at frequency corresponding to the lifetime of stickers is well-separated from the peak belonging to the final relaxation of the chain. In other words, there is a broad frequency domain between sticker’s lifetime and final relaxation that can be assigned to chain dynamics. The sticky reptation is much more hindered, and the relaxation due to second-step Rouse motions brings about a second distinct plateau modulus.

The corresponding relaxation modulus is similar to the instance explained by Leibler et al.43 for the case of sticky reptation. Hence, hindered fluctuations are much slower than reptation, and their effect is not discernible in this case. It can be concluded that in moderately entangled chains the final relaxation is sticker-dependent and happens immediately after the second step of sticky Rouse motions,47,48 while, if the precursor is well-entangled, the final relaxation is dominated by the chain disentanglement and a distinct second plateau can be observed following the sticker’s lifetime43 Hindered fluctuations are too slow to be noticed before reptation in the latter, while they are too fast to be seen in the former. Therefore, depending on the lifetime of the stickers, there are intermediate precursor lengths for which the effect of hindered fluctuations is more noticeable, and their inclusion in predictions is more important. It should be noted that as is visible in Figures 6 and 7, the first deep valley in loss modulus alongside with the higher plateau in storage modulus is due to lack of damping in the frozen network before lifetime of the stickers. Internal segments become mobile at once when stickers start blinking. The sharp transition in fluctuation time of the internal segments based on accumulated time approach is depicted in Figure a.2. This effect is less pronounced in stepwise fluctuations approach due to the rescaling of the fluctuations time between two modes, that is, for segments before and after the first active sticker, to keep continuity of the fluctuations time curve.54,55 3.3. Final Remarks on Model Application. Model predictions for the linear precursor and sticky PnBA are 7306


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of the extra polydispersity of sticker’s distribution and temperature dependence of sticker’s lifetime are essential to yield precise theoretical predictions.

compared with the experimental data in Figure 1b. While predictions for the nonsticky chain are in good agreement with the experimental data, there is considerable discrepancy for the case of sticky chain. Several factors can be responsible for such a deviation. The first reason is the effect of polydispersity in the precursor’s length that becomes more significant when combined with dispersity in random positioning of the stickers along and among chains. The irregular and time-dependent distribution of the association sites, that may lead to gradual decrease in the plateau (as noted in Figure 1b) is an important missing factor in the current model. Furthermore, as mentioned in Section 2.1, the emergence of a low-frequency plateau at intermediate temperatures that could be correlated to microphase separation of associative sites that leads to longer lifetimes18−20 is not included in the current model. This effect can be incorporated into the model by considering a network of associated chains with longer lifetimes. The experimental data at different temperatures can be captured by adjusting the fraction of this higher order aggregations.61 Finally, it should be mentioned that the transient associations are considered fixed in space; however, as Phantom chain model proposes, these junctions can fluctuate around their equilibrium position by Brownian motion. Indei et al.51 have shown, through the single-chain slip-link model, that these fluctuations can decrease plateau modulus in a wide frequency range under specific conditions. Such a decrease in plateau modulus can also decrease the discrepancy of the model explained here in the frequency range between τe and τsticker; however, the corresponding analytical equations are still needed to be developed to be considered in a consistent way with the rest of relaxation mechanisms explained in this work.

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APPENDIX A. ANALYTICAL DESCRIPTION OF THE ACCUMULATED TIME FOR FLUCTUATIONS OF SEGMENT X To derive the analytical representation of the accumulated time at distance x at each time step, we consider numbers of linear chains with N0 monomeric units and ϕsticker as the fraction of sticky units, out of which a fraction of ϕactive sticker is considered to be active. The characteristic lifetime of τsticker is considered to be the association lifetime of the stickers. The characteristic lifetime of stickers in the dissociated state is already fixed based on the considered ϕactive sticker. We consider the active stickers placed at equal distances along the backbone at time zero and proceed through time space with very small time steps. The association probability of a sticker is defined as41 ⎛ t − ti − 1 ⎞ φassociation(ti − ti − 1) = exp⎜ − i ⎟ ⎝ τsticker ⎠

(a.1)

When the association probability of an active sticker is dropped below a certain threshold in a certain time step, it is set free and a new one is randomly emerged among all stickers along the backbone. In this way, we let the stickers blink but keep the total number of active ones constant. The accumulated time is updated at each tiny time step for segments placed between the first active sticker and the chain ends at both extremities as it was illustrated in Figure 1. Varying the three independent parameters, ϕsticker, ϕactive sticker, and τsticker, we tried to derive an analytical expression for the accumulated time of fluctuations for each segment at each time step. Figure a.1 shows the

4. CONCLUSIONS We have extended the tube-based TMA to the entangled supramolecular chains with sticky side groups, with special emphasis on the effect of hindered fluctuations, besides sticky Rouse and sticky reptation mechanisms. Hindered fluctuations are included either with extra penalty for deeper segments based on their corresponding accumulated time or with stepwise fluctuations by extra friction coming from associated stickers. Relaxation beyond the apparent entanglement length is prevented for segments localized between active stickers, until their association lifetime is met and active stickers start random blinking. There is a transition zone characterized by almost parallel drop of loss and storage moduli with slope of 0.5 after characteristic lifetime of the stickers and emergence of the final relaxation slopes, which can be assigned to hindered fluctuations. Both approaches for inclusion of fluctuations lead to similar behavior in this zone. The hindered fluctuations are found to affect chain dynamics when precursor chains are moderately entangled, while at the same network density, sticky reptation dominates the relaxation behavior of well entangled chains. The two-step sticky Rouse relaxation mechanism leads to emergence of two discernible plateau levels in the second case, while damping peak at sticker’s lifetime is followed by the final relaxation without the appearance of perceptible second plateau in the first approach. Numbers of stickers directly affect network density and therefore lead to higher apparent plateau levels, while longer lifetimes postpone the final relaxation of the chain. To observe a second low-frequency plateau, inclusion of secondary association of sticky groups characterized by much longer lifetimes seems to be necessary. Furthermore, enclosure

Figure a.1. Normalized accumulated time, Δt/τsticker, versus x averaged on 2000 chains with N0 = 1000 and different number of active stickers (indicated in the plot) at constant τsticker= 0.1 s and ϕsticker = 0.25.

numerically calculated accumulated time, Δt, normalized by the τsticker, versus x for different number of active stickers, constant number of total stickers, and fixed characteristic lifetime at time far above τsticker (105 times larger). The presented data were obtained by averaging 2000 number of sticky chains. Interestingly enough, all curves show a quadratic dependence on x, with slopes increasing by the number of active stickers and the same intercept, equal to the final studied time. Conclusively, all curves can be fitted by the following equation, as depicted by dashed lines in Figure a.1 7307


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Figure a.2. (a) Accumulated time versus x (at time steps according to the intercept of each curve), (b) Δt versus t (at different x increasing from 0.018 to 0.987), (c) numerically obtained Δt surface versus t and x, and (d) Δt surface analytically calculated using eqs a.1−a.4 at constant τsticker = 1 s, N0 = 1000, NS = 10, and ϕsticker = 0.25. Symbols and dashed lines in (a) and (b) correspond to numerical and analytical predictions, respectively.

log Δt(i) = ax 2 + bx + log t(i)

evolution of accumulated time of different segments as time proceeds is shown in Figure a.2. Such a behavior could be suitably represented by the following equation: for x < Xtr:

(a.2)

where a and b are linear functions of number of stickers according to a = −0.1NS

and

b = −0.2NS

(a.3)

(a) Δt(i) = t(i)

Increase in the extra penalty with number of active stickers was expected. If we consider the slope of the curve as a representative of the penalty for deeper fluctuations (p = 2ax + b), given the parameters a and b, penalty doubles as we go from chain’s free ends (x = 0) down to the middle of the chain (x = 1) and increases linearly with number of stickers. Despite the fact that Figure a.1 is calculated for a specific τsticker = 0.1 s, the accumulated time at times far larger than τsticker is independent of sticker’s lifetime and normalized curves for stickers with different lifetimes overlap on the same graph. Such a behavior is anticipated because at times long enough all stickers can blink freely without feeling the restriction imposed by the association lifetime. Conversely, at times in the vicinity of the sticker’s lifetime, random blinking is not freely permitted. At initial time steps below τsticker, the segments deeper than the first active sticker are frozen, while outer segments can freely accumulate time. As time approaches, characteristic lifetime of the stickers and association probability drop below a certain threshold (0.1 considered in this work), stickers become mobile, and the accumulated time of formerly frozen segments increases exponentially toward the values predicted by eq a.2. The

if

t(i) < τsticker

(b) Δt(i) = max(τsticker , 10ax

2

+ bx + logt(i)

)

if

t(i) > τsticker

for x > Xtr: (a) Δt = 0

if

(b) Δt = 10(ax

2

t(i) < τsticker + bx + logt (i)) − 1/(t (i) − τsticker)β

if

t(i) > τsticker (a.4)

where Xtr = 2/(NS+1) is the dimensionless position of the first sticker at time zero and β is a parameter to fit the numerically predicted data. The best fitted curves as shown in Figure a.2 were obtained by considering β = 4. The numerically calculated accumulated times are compared with analytical derivations based on eqs a.1−a.4 in Figure a.2.

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AUTHOR INFORMATION

Corresponding Authors

*M.A.: [email protected]. *E.v.R.: E-mail: [email protected]. Notes

The authors declare no competing financial interest. 7308


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ACKNOWLEDGMENTS This work has been supported by the E.U. (Marie SklodowskaCurie ITN “Supolen”, no. 607937). E.v.R. is a Research Associate of the FRS-FNRS. H.G. thanks FRIA for financial support. We thank the anonymous referees for their very helpful comments.

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Dynamics of Entangled Linear Supramolecular Chains with Sticky

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