New Interpretation of Shear Thickening in Telechelic Associating

Article pubs.acs.org/Macromolecules

New Interpretation of Shear Thickening in Telechelic Associating Polymers Giovanni Ianniruberto* and Giuseppe Marrucci Department of Chemical, Materials, and Industrial Production Engineering, University Federico II, Piazzale Tecchio 80, 80125 Naples, Italy ABSTRACT: Shear thickening of telechelic associating polymers, particularly of hydrophobically modified ethoxylated urethanes (HEURs), is classically attributed to either nonGaussian behavior of the chains, also called finite extensible nonlinear elasticity (FENE), or to flow-enhanced formation of network strands; however, in a recent paper [Macromolecules, 45, 888 (2012)], Suzuki et al. show that neither of the previously mentioned interpretations can hold true, as they find shear thickening while at the same time the network structure has essentially remained the equilibrium one, except for the fact that reassociation to the network of the strands that continuously detach from it is anisotropically enhanced in the shear gradient direction. In the present paper, we propose a different mechanism that might explain shear thickening, namely, the stress contribution arising from the repulsive interaction between the flowerlike micelles that are forced to interpenetrate one another by the shear flow. In support of this idea is the fact that shear thickening is found only at low concentrations when the flowerlike micelles are nearly intact and essentially wellseparated at equilibrium because relatively few bridging chains exist that percolate the network. Shear thickening is found to disappear with increasing concentration, possibly because micelles already interpenetrate at equilibrium, so that the flowerlike structure is reduced in favor of an increasing number of bridging chains. To develop the mathematical model in a simple way, the network of the bridging chains is here described by using a suitable dumbbell model. The dumbbell dynamics also regulates the repulsive micelle interactions. Predictions of the model qualitatively compare with the existing data, although for a better, more quantitative comparison, we suggest that Brownian simulations of the network of flowerlike micelles be developed. network, both at equilibrium and in flow. Their theory, however, only predicts shear thinning in fast flows, whereas in some cases associating polymers exhibit shear thickening in a range of shear rates, followed by shear thinning in even faster flows. Hence, a number of proposals were made over the years to explain shear thickening, ranging from non-Gaussian behavior of the chains8−16 to shear-induced increased association,17−23 but no general agreement has emerged. HEUR polymers may also show another peculiar phenomenon, that is, strain hardening in large step shear deformation,10 or in large amplitude oscillatory shear (LAOS),24,25 or in shear startup at high shear rates.15,26 In the linear regime of slow shear flows, during startup the ratio η+0 (t) = σ(t)/γ̇ of the timedependent shear stress σ to the shear rate γ̇ is a monotonically increasing function approaching the zero-shear viscosity η0. In fast shear startup, ordinary polymers typically show nonmonotonic curves for η+(t), which go through a maximum before reaching the steady-state viscosity η < η0; however, at short times the response remains linear; then, with increasing t, the curve deviates downward from the linear one; that is, η+(t)

1. INTRODUCTION It has long been known that above some critical concentration c* associating polymers form temporary networks.1−3 In particular, HEUR telechelic polymers form flowerlike micelles at low concentrations in water, with the hydrophobic ends clustered together in the micellar core, while the hydrophilic polymers, usually poly(ethylene-oxide) or PEO, form closed loops (resembling the petals of a flower) that swim in water. Because of thermal motion that periodically extracts chain ends from the micellar core, occasionally two or more micelles become connected by bridging chains, that is, chains whose two ends are imbedded in the cores of two different neighboring micelles. Above c*, the bridging chains percolate the system, and their number further increases with increasing the concentration. The network that is thus formed is temporary, however, meaning that thermal motion detaches and reattaches chain ends continuously. Hence, the system remains liquid-like, albeit with a viscosity much larger than if the chains were not associated. Many theories for the dynamics of temporary networks were developed over the years, the first of them going far back in time.4,5 What is often referred to as the classical theory for associating polymers was developed by Tanaka and Edwards,6,7 who derived probability distributions for chains attached to the © XXXX American Chemical Society

Received: May 15, 2015 Revised: July 21, 2015

A

DOI: 10.1021/acs.macromol.5b01048 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules ≤ η+0 (t) at all times. On the contrary, HEUR polymers may generate η+(t) curves that deviate upward from the linear one (strain hardening), go through a maximum, and then fall below the linear curve to finally approach a shear-thinning viscosity. Koga et al.15 suggest that also this peculiar phenomenon is due to the non-Gaussian behavior of the chains; however, they also note that for a quantitative match with the data, unrealistically large deviations from the Gaussian behavior would be needed or else a dominance of short chains in the polydisperse sample, a possibility of uncertain validity. Very recently, new important data were provided by Watanabe and coworkers,26 showing thickening for the shear viscosity in a range of shear rates where the first normal stress coefficient essentially remains at the value of linear viscoelasticity. Moreover, relaxation following shear flow in such a range of shear rates coincides with relaxation in the linear limit, showing that the equilibrium structure has been preserved. These new data contradict both of the two previously mentioned interpretations for the shear-thickening phenomenon (non-Gaussian behavior or shear-induced increased association) and demand that some new physics be considered. Watanabe and coworkers suggest that the difference between first normal stress coefficient and viscosity “... results f rom reassociation of the HEUR strands (to the network) being in balance with dissociation (hence relaxation is the same as in the linear range) but anisotropically enhanced in the shear gradient direction.” They also present new data of shear startup showing strain hardening at high shear rates, both for η+(t) and (more weakly) for the first normal stress coefficient, but do not comment on this phenomenon. The present paper is organized as follows. We first discuss in Section 2 an important property of linear viscoelasticity for these systems, namely, the dominant relaxation time, and its dependence on concentration and molar mass. Such a discussion opens the way to a simple model of the nonlinear behavior of the temporary network, developed in Section 3, that, however, (similarly to the classical theory of Tanaka and Edwards6,7) does not predict shear thickening and hence needs to be augmented with an additional contribution. This is provided in Section 4, where we propose that shear thickening and strain hardening arise from the shear-induced forced collisions of the mutually repelling flowers. Indeed, shear thickening is only observed at low concentrations, just above c*, when there are many quasi-intact flowers and relatively few percolating bridging chains. Several data11,23,27 consistently show that shear thickening disappears at large c, when in fact most chains are bridges and flowers have become irrelevant. Predictions of the proposed theory are then compared qualitatively with existing data and discussed in Section 5. Final comments conclude the paper.

conclusion is confirmed by the Arrhenius-type dependence of τ0 on temperature as well as by the exponential dependence of τ0 on the length of the hydrophobic chain end.3 Naively one might then be tempted to assume the identity: τ0 = β−1 0 ; however, this identity is contradicted by the fact that τ0 is found to also depend on concentration c and molar mass M of the polymer,3,23,28 whereas β0 is a chain-end property, certainly independent of c and M. Notice that the identity τ0 = β−1 0 would imply that relaxation of a chain of the temporary network occurs in a single detachment event. On the contrary, to explain the observed dependence of τ0 on c and M, it may be useful to recall the so-called free path (FP) model, which was proposed long ago.9 The FP model postulates that relaxation of the chains of the temporary network generally occurs through a sequence of detachments because when a chain end detaches from a micellar core it is soon recaptured by a neighboring one. If one indicates with a the mean free path between consecutive captures, a chain end diffuses with a diffusion coefficient given by D0 = β0a2. On the contrary, relaxation requires that chain ends diffuse over a distance on the order of the size of the polymeric coil. Hence, if R20 is the mean square end-to-end distance of the polymer at equilibrium, one obtains τ0 =

R 02 R2 = β0−1 20 ∝ β0−1(cM1/2)2/3 D0 a

(1)

where numerical coefficients are omitted, and the last proportionality is obtained by assuming, as done by Tripathi et al.,23 that R20 is proportional to M and that a is inversely proportional to the cubic root of the micelle number density and hence (because micellar cores contain a nearly fixed number of chain ends) to the cubic root of the molecule number density c/M. The power law with c and M predicted by eq 1 is tested in Figure 1 against data from different authors.3,23,28 All data refer

2. RELAXATION TIME 2.1. Equilibrium State. A peculiar feature of telechelic associating polymers is the fact that they often show a Maxwelltype response in linear viscoelasticity at low frequencies, with a single well-defined relaxation time τ0.3,28 (Here and in the following the index 0 refers to equilibrium properties.) This feature contradicts the behavior of ordinary polymers, which always exhibit a spectrum of relaxation times and speaks in favor of the fact that relaxation of associating polymers (rather than by friction, as is the case for ordinary polymers) is controlled by the frequency β0 of chain-end detachments from the micellar cores (a thermally activated process). This

Figure 1. Relaxation time data from different authors plotted versus the group cM1/2. The straight line shows the 2/3 slope predicted by eq 1. The C20 data point and the data of Uneyama et al. have been divided by 60 and 10, respectively, as explained in the text.

to C16-HEUR’s (i.e., with n-hexadecane in the hydrophobic chain end), with the exception of a single C20 data point, which has been reduced to an equivalent C16 through the factor 1/60, as obtained from Figure 6 of Annable et al.,3 which gives the exponential dependence of τ0 on the number of chain-end carbon atoms. Although the data of Uneyama et al.28 also refer to a C16, the urethane group of their HEUR is different from B


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Macromolecules that of Annable et al.3 and Tripathi et al.23 and seemingly generates larger values of the relaxation time. The τ0 data of Uneyama et al.28 in Figure 1 were therefore divided by 10 for a better comparison with the other data. Overall, Figure 1 shows that the scaling predicted by eq 1 is in good agreement with the observations. More in detail, however, the data of Annable et al.3 denounce deviations at very low concentrations, where indeed the FP model is expected to fail because micelles are sufficiently apart that a single detachment event may become sufficient for relaxation. Annable et al.3 even postulate that at low concentrations τ0 falls below β−1 0 because superbridges formed by a series of bridging chains are probably present, and detachment of any bridging chain in the series relaxes them all. The deviations found in the data of Uneyama et al.28 at high concentrations seem to indicate a decrease in the mean free path a with increasing concentration stronger than c−1/3, possibly due to a micellar core volume larger than that of the other HEURs. 2.2. Nonequilibrium Relaxation Time. During flow or deformation the relaxation time can be modified because the detachment frequency β departs from the equilibrium value β0. At equilibrium the detachment frequency is given by the formula β0 = Ω exp(−E/kBT), where Ω is a large basic thermal frequency, kBT is Boltzmann constant times absolute temperature, and E is the depth of the energy well that keeps chain ends together in the micellar core. The larger E is (e.g., by increasing the length of the hydrophobic chain end), the less frequent detachment is.3 During flow or deformation chains get stretched, and the extra tension F that is generated helps extracting chain ends from their micellar core more frequently than at equilibrium. As for all activated processes, the augmented frequency becomes ⎛ E − Fl ⎞ ⎛ Fl ⎞ β = Ω exp⎜ − ⎟ ⎟ = β0 exp⎜ ⎝ kBT ⎠ ⎝ kBT ⎠

situation is only the temporary status of a chain between two consecutive attachments to the network, and because friction is negligibly small such status lasts much less than the bridging one. Moreover, the contribution to the stress tensor of a chain only depends on its end-to-end vector R, irrespective of whether the chain is in the bridging or pendant status. The important distinction is between bridging chains and those forming loops within micelles, the latter not contributing to the stress tensor of the impermanent network within the dumbbell model. Similarly, in a recent paper by Sing et al.29 that accounts with great accuracy for all type of chains (bridging, pendant, and loop ones), loop chains do not contribute to the stress, differently from what we will develop in Section 4 of the present paper, where additional physics enters the picture. We adopt the simplifying assumption that the concentrationdependent proportion between bridging and loop chains remains fixed at the equilibrium value also during flow. In other words, the number density ν of elastically active chains of the network, differently from Sing et al.,29 is taken to be constant (but see Section 5 for further discussion on this point). The number density ν is linked to the modulus G of linear viscoelasticity through the classical equation G = νkBT, valid to within a numerical coefficient somewhat less than unity due to fluctuations of the network junctions. With the previously mentioned assumptions the impermanent network is fully described by the time-dependent distribution function of the end-to-end vector R of the active chains, Ψ(R;t), obeying the well-known Smoluchowski equation for the elastic dumbbell (e.g., see the book of Larson30) ⎤ 3fR ⎞ ∂Ψ ∂ ⎡ ⎛ ∂Ψ = · ⎢D ⎜ + Ψ 2 ⎟ − Ψk·R ⎥ ⎥⎦ ∂t ∂R ⎢⎣ ⎝ ∂R R0 ⎠

In eq 4, k = k(t) is the velocity gradient tensor and f = f(R/ Rmax) accounts for deviations of the entropic elastic force from the Gaussian behavior when R approaches the fully extended chain length Rmax, the Gaussian behavior corresponding to f = 1. At equilibrium, f = 1 is expected, and the distribution function reduces to the Gaussian form Ψ0(R) = (2πR20/3)−3/2 exp(−3R2/2R20). At any time t during a flow with an assigned velocity gradient k, the stress tensor σ is obtained from the equation σ = ν⟨FR⟩, where F is the elastic force F = 3f kBTR/R20, and calculation of the ensemble average ⟨FR⟩ requires solving eq 4 for the distribution function Ψ(R;t). Furthermore, it should be recalled that the diffusion coefficient D in eq 4 is not a constant, as it is related to the magnitude of the force F through the exponential form of eq 3. The mathematical difficulties of this nonlinear problem are overcome by a few classical decoupling and preaveraging approximations.30 Indeed, if eq 4 is multiplied throughout to the dyadic product RR and integrated over the whole R space, while D and f are replaced by their preaveraged values D̅ and f ̅, respectively, the following ODE is obtained for the average ⟨RR⟩ (a similar equation could be derived by using the Tanaka and Edwards theory6,7)

(2)

where l is related to the length of the hydrophobic chain end. As a consequence of the change in detachment frequency, both the chain-end diffusion coefficient D and the relaxation time τ depart from their equilibrium value. From eq 2 we may expect ⎛ Fl ⎞ ⎛ Fl ⎞ D = D0 exp⎜ ⎟ ⎟ , τ = τ0 exp⎜ − ⎝ kBT ⎠ ⎝ kBT ⎠

(4)

(3)

3. DUMBBELL MODEL FOR TELECHELIC ASSOCIATING POLYMERS The FP model used in the previous section to derive eq 1 leads quite naturally to the idea that the impermanent network of associating telechelic polymers might be described through the use of a dumbbell model. Indeed, the elastic dumbbell model assumes that the resistance to motion be concentrated at the ends of the chain, which is exactly the case of the bridging chains of the impermanent network formed by telechelic associating polymers. The resistance to motion is not due to friction (which would be distributed all along the chain and is in fact negligible in this case); rather, it is determined at the chain ends by the detachment frequency β. The chain-end mobility is then described by the diffusion coefficient D previously defined. Notice that the dumbbell model does not distinguish between bridging and pendant chains because the pendant

6Df̅ ̅ d⟨RR ⟩ = 2D̅ δ − ⟨RR ⟩ + k·⟨RR ⟩ + ⟨RR ⟩·kT dt R 02

(5)

Here δ is the unit tensor and kT is the transpose of k. eq 5 is coupled to the equations for D̅ and f ̅ that are C


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Figure 2. Predictions of the dumbbell model for η, η+ (solid lines) and Ψ1, Ψ+1 (dashed lines), with parameter values l/R0 = 0.12, Rmax/R0 = 11 in (a) and (c) and l/R0 = 0.05, Rmax/R0 = 5 in (b) and (d). Panels (a) and (b) report shear viscosity and first normal stress coefficient in steady shear. Panels (c) and (d) show the corresponding startup curves for shear rates γ̇τ0 = 0.1, 10, and 100 from top to bottom.

⎛ f ̅ R̅ − R ⎞ 0 ⎟ D̅ = D0 exp⎜3l R 02 ⎠ ⎝

f̅ =

νkBT. The curves in Figure 2a were obtained for the parameter values l/R0 = 0.12, and Rmax/R0 = 11 (choice dictated by comparison with data in Section 4), but similar curves are obtained for other values of the parameters, as shown in Figure 2b. In all cases, both the viscosity and the first normal stress coefficient come out shear thinning. In Figure 2c,d shear startup curves are reported for the same parameter values of Figure 2a,b, respectively. It is apparent that either by increasing the shear rate to very large values, as in Figure 2c, or by decreasing either l/R0 or Rmax/R0 or both as in Figure 2d, strain hardening appears due to non-Gaussian effects. The cusps in the curves of Figure 2d coincide with the beginning of the conformational transition of PEO, which starts abruptly in eq 7′ because of the Heaviside function. The strain-hardening behavior in Figure 2c,d is in line with similar predictions by Koga et al.15 It is suggested that the simple dumbbell model developed in this section describes, at least qualitatively, the behavior of associating telechelic polymers at not too low concentrations, that is, when shear thickening is not found. The dumbbell model will be further discussed in Section 5, with particular attention to the limitations arising from the preaveraging approximations and from the assumption of a shear-rate independent value of ν.

(6)

1 R max −1⎛ R̅ ⎞ 3 ⎜ ⎟ 3 R̅ ⎝ R max ⎠

(7)

where R̅ , R0 are short notations for (⟨R ⟩) , respectively. For ordinary polymers 3−1 would be the inverse Langevin function or some approximation to it. For PEO in water, however, deviations from the Gaussian behavior occur differently because of a conformational transition due to loss of hydrogen bonds, as revealed by the AFM data of Oesterhelt et al.,31 so that f ̅ is better described by the approximate form 2

f̅ =

20(R̃ − 0.6)H(R̃ − 0.6) 1 + 2 R̃ 1 − R̃

1/2

(R02)1/2,

(7′)

where R̃ = R̅ /Rmax and H(...) is the Heaviside step function. The first term on the r.h.s. of eq 7′ comes from the simplest approximation to the inverse Langevin function, frequently used in the so-called FENE-dumbbell model. The second term approximates the contribution arising from the conformational transition described by Oesterhelt et al.31 Equations 5−7 are readily solved numerically for all Cartesian components of ⟨RR⟩ in any given flow, and the corresponding stress is obtained as σ = 3νkBTf ̅⟨RR⟩/R20. By way of example we plot in Figure 2a the steady-state shear viscosity and first normal stress coefficient predicted by eqs 5−7. The shear rate is normalized to τ0 = 6R02/D0 and the stress to G =

4. SHEAR THICKENING 4.1. Interacting Micelles. As mentioned in the Introduction, we here propose that the shear thickening (and even the strain hardening) observed in associating telechelic polymers at low concentrations is due to the additional stress arising from D


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Macromolecules the shear-induced collisions among the flowerlike micelles. Indeed, below c* micelles mutually repel because of excluded volume, and such a repulsion is expected to also survive for concentrations just above c* when bridging chains percolate the system. Although the just formed impermanent network will somehow “lock in place” the micelles, these will remain as far apart as possible one from the other. We will assume that the flowerlike micelles are spherical objects of size R0 (i.e., of a size similar to that of the single chain) and that the mean distance between the centers of neighboring micelles, called d0, is close to R0 so that micelles barely touch one another or not at all. As shown in the following, particularly important to possibly explain shear thickening and strain hardening is the case d0 > R0. Of course, because d0 scales with c−1/3, with increasing c neighboring micelles interpenetrate more and more, yet under those conditions bridging chains are expected to become much more numerous, and the very existence of the “flowers” will gradually be lost. Figure 3 schematically shows what happens to neighboring micelles in a shear flow. In Figure 3a two micelles are depicted

where F was obtained as the derivative of U with respect to d. The power 0.2 affecting the monomer number N results from assuming that because of excluded volume the polymer size (and the flowerlike micelle size as well) is linked to b and N as R0 ≈ bN0.6 If we now introduce the distance vector d, the force vector F is written as Fd/d, and the dyadic product Fd = Fdd/d is expected to contribute to the stress tensor whenever d is less than R0. Within the affinity assumption, the vector distance d is readily linked to the initial coordinates (x0, y0, z0) of the moving sphere and to the current value of the shear deformation γ. Indeed the current coordinates of the moving sphere coincide with the Cartesian components of d, and are given by x = x0 + γy0, y = y0, and z = z0. The problem is how to average the dyadic product Fd over the initial coordinates of the moving sphere and how to account for the fact that the network is impermanent and therefore that during a collision process the two spheres may be instantly pushed apart because one of the spheres detaches itself from the network. We will approach this problem in two steps. We first ignore that the network is impermanent and calculate the average ⟨Fd⟩ as a function of the shear deformation γ by integrating over the initial coordinates. More specifically we will calculate the functions Sxy(γ; R0/d0) = −⟨(R20 − d2)xy/R30d⟩ and N1(γ; R0/d0) = −⟨(R20 − d2)(x2 − y2)/R30d⟩, parametric in the ratio R0/d0. Next, we will use such functions in a suitable integration over time (to be shown in Section 4.2) that mimics the startup of a shear flow of the impermanent network, all the way up to the steady state. The integration over the initial location of the moving micelles is best written by switching to spherical coordinates, that is, by writing: x0 = r cos θ, y0 = r sin θ cos φ, and z0 = r sin θ sin φ. Then, Sxy(γ; R0/d0) is calculated as

Figure 3. Schematic of micelle interaction. (a) Noninteracting micelles at equilibrium. (b) Shear deformation forces micelles to interpenetrate.

Sxy(γ ; R 0/d0) = −

at equilibrium, at a distance r larger than R0. Then, a shear flow is started along the x direction with the gradient along y. As a consequence, the sphere to the left is brought to collide with the sphere at the origin. We will temporarily assume that the motion of the micelle centers is affine (since they belong to the network), and consequently the spheres will interpenetrate, as depicted in Figure 3b, where the volume V of the spherical caps of the interpenetrating micelles is given by V = π (R0 − d)2(2R0 + d)/12, where d is the current distance between the centers (d < R0). The energy U of the interacting micelles is then obtained from (see, e.g., equation 2.89 in Doi−Edwards book32) U = 0.5kBTvVν2m, where v is the excluded volume of a monomer of the polymer chains, and νm is the number density of the monomers (assumed to be uniform within the volume V). Finally, we take v ≈ b3, where b is the Kuhn length of the polymer and νm ≈ 2npN/(πR30/6), where np is the number of polymers per micelle, N the number of Kuhn segments (monomers) per chain, and the factor 2 accounts for the fact that within the volume V the monomer density is twice that in an isolated micelle. The energy U and the intensity F of the repulsive force are then estimated to be

∫0

∞

dr ̃ 4πr 2̃ g (r )̃

2π

dφ

xy R 03d

∫0

π

dθ sin θ

max(0, R 02 − d 2)

(9)

In eq 9, r̃ = r/d0, g(r̃) is the radial distribution function of the micelles at equilibrium, and the function Max(0,R20 − d2) equals the largest of the two arguments; that is, no contribution to the integral is given by the moving micelles whose initial coordinates are such that the current value of the distance d is larger than R0 and therefore are not touching the test micelle fixed at the origin. In eq 9, although the upper limit of the outermost integral is formally infinity, at any given γ there is a limiting distance beyond which no sphere will reach that at the origin. Such a distance is given by rmax(γ) = R0(2/(2 + γ2 − γ(γ2 + 4)1/2))1/2 that, for large γ, reduces to rmax(γ) = γR0. The equilibrium radial distribution function of the micelles is not known, except for the previously mentioned assumption that, by considering the repulsive nature of the interaction, g(r̃) is taken to be zero for r̃ < 1. Actually, because the repulsive interaction is soft, g(r̃) will decay to zero (more or less rapidly) as r̃ decreases below unity, but we take g(r̃) = 0 for simplicity. For r̃ ≥ 1, a plausible form for such a function is depicted in Figure 4 and is described through the equation

(R − d)2 (2R 0 + d) U 3 = np2N 0.2 0 kBT π R 03 R 2 − d2 R 2 − d2 F 9 = − np2N 0.2 0 3 = −C 0 3 kBT π R0 R0

∫0

g (r )̃ = 1 +

⎤ ⎛ 5π 2⎡ 3 ⎞ ⎟ cos(2.5πr ) sin(2.5πr )̃ − ⎜ − ̃⎥ 2⎢ ⎝ 12 ⎦ 5π ⎠ r̃ ⎣

exp(3 − 3r )̃

(8) E

(10) DOI: 10.1021/acs.macromol.5b01048 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

However, it may be appropriate to relax this assumption by considering that, because of the compressive force between the interacting spheres, the spheres are (elastically) somewhat displaced from the affine configuration, the elasticity being provided by the network to which the spheres are attached. As schematically depicted in Figure 5, the spheres would be

Figure 5. Displacement from the affine configuration (a) of the interacting micelles (see Figure 3b) to a more separate one (b) with d > daff, as dictated by the force balance of eq 12.

Figure 4. Assumed radial distribution function of the micelles at equilibrium.

displaced along the line of the centers (instantly, since friction is negligible) from the affine distance daff to a somewhat larger distance d obeying the following algebraic equation

that satisfies the condition whereby the integral of g(r̃) − 1 to the whole space must be zero. Of course our detailed choice of g(r̃) is arbitrary, but we have verified that different choices (like, e.g., the overdamped form g = 1 + ((exp(3 − 3r̃))/r̃2) for r̃ ≥ 1, or else the inclusion of an exponential decay for r̃ < 1) do not modify the results in any significant way. An equation similar to eq 9 is written for N1(γ, R0/d0), where in the innermost integral the difference x2 − y2 replaces the product xy. It is further noted that for d0 > R0, in view of the “hole” of size d0 in the radial distribution function, micelles do not overlap at equilibrium, and the integral for Sxy in eq 9 (as well as that for N1) remains zero up to a minimum value of γ that depends on the ratio R0/d0. Indeed, it can be shown that the first moving sphere touching that in the origin will do so only at γmin = d0/R0 − R0/d0. Similarly, because with increasing γ spheres move from one side to the other of that in the origin, therefore contributing to Sxy with the opposite sign, it is expected that Sxy goes through a maximum, and that, beyond some γmax, Sxy will effectively drop to zero. Although x2 − y2 does not change sign from left to right, N1 also decreases with increasing γ beyond some γmax. This is due to the fact that, while at low γ the colliding spheres have y larger than x, at high γ colliding spheres with x larger than y are also present, thus providing contributions with the opposite sign. In the case of d0 < R0 micelles overlap also at equilibrium; however, Sxy and N1 are zero at equilibrium because of the isotropic micelle distribution. It is finally noted that the symmetry of the shear deformation requires that Sxy and N1 are odd and even functions of γ, respectively; moreover, because the response is that of a rubberlike solid (albeit a very special one), we expect that the relationship N1(γ) = γSxy(γ), proven by Rivlin (see Larson30), is valid in this case. Such an equality is also known as Lodge−Meissner relationship,33 holding true for step strains of polymeric liquids. Up to this point, it was assumed that the current coordinates of the moving sphere x, y, z and d = (x2 + y2 + z2)1/2 appearing in eq 9 were linked affinely to the initial coordinates, that is, that (x, y, z) = (xaff, yaff, zaff), where

kBT (R 02 − d 2) R 03

kBT (d − daff ) R 02

(12)

where the pushing force of the interacting spheres (on the l.h.s.) is balanced by the elastic one (on the r.h.s.), assumed to be proportional to the displacement. The numerical factor χ accounts for the constant C of the compressive force in eq 8 as well as for the connectivity of the network. One may note that χ = 0 would imply d = R0; that is, the spheres do not actually interact. At the opposite extreme, χ → ∞ implies d → daff. Once d is obtained from daff by fulfilling eq 12, the coordinates x and y to be used in eq 9 are given by x = xaff

d d , y = yaff daff daff

(13)

Figure 6 shows typical results for the functions Sxy(γ) and N1(γ) obtained from eqs 9−13 for some values of the parameters R0/d0 and χ. It is noted that in all cases it is N1(γ) = γSxy(γ), as expected, and as explicitly shown in Figure 6a. (In the other panels, N1(γ) is omitted for better clarity.) In Figure 6c, the crucial difference between the cases where R0/d0 is smaller, or else larger, than unity is to be emphasized. If d0 is larger than R0, there is no linear range, that is, Sxy remains zero up to γmin = d0/R0 − R0/d0, whereupon it abruptly goes up to reach a maximum value, after which it goes down again, asymptotically approaching zero as γ goes to infinity. On the contrary, when d0 is smaller than R0, that is, when concentration is large enough that micelles are already forced to interpenetrate somewhat at equilibrium, the function Sxy(γ) is linear in γ, at small deformations. For future reference, it should also be noted that in the latter case Sxy(γ) at its maximum value is larger by orders of magnitude with respect to the case when micelles are well separated at equilibrium. It should finally be noted that the Sxy(γ) curves so obtained overlook the possibility of multiple interactions; that is, they ignore the occurrence, whereby some monomers belonging to more than two micelles are brought by the shear deformation to occupy the same volume. We are also ignoring screening of the excluded volume potential that takes place when concentration is large. We will

xaff = −r cos θ + γr sin θ cos φ , yaff = r sin θ cos φ , zaff = r sin θ sin φ

=χ

(11) F


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Figure 6. Characteristic micellar interaction functions Sxy(γ) (solid lines) and N1(γ) (dashed line). (a) Parameter values are R0/d0 = 0.48 and χ = 0.1 (later to be used in Figures 7 and 8 for a comparison with data). (b) Effect of the χ parameter at constant R0/d0 = 0.48: χ = 1, 0.1, 0.01, from top to bottom. (c) Effect of the ratio R0/d0 at constant χ = 0.1: R0/d0 = 1.1, 1.01, 1, 0.99, 0.9, 0.8, 0.48, from top to bottom.

further comment on the case d0 < R0 when discussing Figure 9 in Section 5. 4.2. Startup of a Shear Flow. During startup of a shear flow with a shear rate γ̇, at any time t from the start the shear deformation is γ = γ̇t, but the contribution of the micelles to the shear stress is not proportional to Sxy(γ) because during time t some of the interacting micelles will have detached from the network and reattached to it in an equilibrium-like configuration. In other words, we have to account for the relaxation time τ of the network discussed in Section 2, which (in the nonlinear range) is itself a function of time according to eq 3. There follows that the shear stress contributed by the interacting micelles during shear startup will be proportional to the integral over past deformation history ⎛

t

Ixy(t ) =

∫−∞ τd(tt′′) exp⎜⎝−∫t′

t

dt ″ ⎞ ⎟Sxy[γ(t , t ′)] τ (t ″ ) ⎠

Calculation of Ixy(t) then requires knowing the evolution of the relaxation time τ during flow. Such information is obtained by solving the dumbbell model equations of Section 3 that provide (among other things) the time dependence of the diffusion coefficient (see eq 6), from which τ(t) is obtained as τ/τ0 = D0/ D̅ . Needless to say, the contribution of the micelles to the first normal stress difference is obtained from an equation similar to eq 15 with N1(γ) in place of Sxy(γ). There remains to combine the shear stress contributed by the bridging chains, approximately calculated with the dumbbell model, [σxy(t)]dumbbell, to that of the colliding micelles, proportional to Ixy(t). To this end, one should know the relative proportion of micelles to bridging chains. In the concentration range we are considering, with c just above c*, bridging chains are relatively few, and we may assume that the number ratio of micelles to bridging chains is of order unity, that is, νmicelle ≈ νdumbbell = ν. We then write for the overall shear stress σxy(t)

(14)

σxy(t ) ≈ [σxy(t )]dumbbell + νkBTCIxy(t )

where γ(t,t′) is the shear deformation between t′and t. Because the flow starts at t = 0 the integral in eq 14 is conveniently broken up in one from − ∞ to 0, plus one from 0 to current time t, so that eq 14 is rewritten as ⎛ Ixy(t ) = Sxy[γ(t )] exp⎜ − ⎝ +

∫0

t

∫0

t

⎛ dt ′ exp⎜ − τ(t ′) ⎝

where C is the numerical constant in the expression for the repulsive micellar force, eq 8. An equation similar to eq 16 will be used for the first normal stress difference N1(t). Notice that when c is just above the percolating concentration c*, using the FP and the dumbbell models is a particularly crude approximation. Leaving better alternatives to future developments, we will here examine the predictions of eq 16. To compare, at least qualitatively, the predictions of eq 16 with the recent data of Watanabe and coworkers,26 we need to estimate the values of the relevant parameters. We start with

dt ′ ⎞ ⎟ τ(t ′) ⎠

∫t′

t

(16)

dt ″ ⎞ ⎟Sxy[γ(̇ t − t ′)] τ (t ″ ) ⎠ (15) G


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Figure 7. Startup of shear flows predicted by the interacting micelle model, eq 16. Solid lines refer to γ̇τ0 = 0.1, 10, 100, from top to bottom. Dashed lines, for γ̇τ0 = 1, show shear thickening. Dotted lines give the micelle contribution for γ̇τ0 = 1, 10, 100, from right to left. The curves for γ̇τ0 = 10 and γ̇τ0 = 100 show strain hardening, mainly due to micelle interaction. The second bump appearing for γ̇τ0 = 100 is due to nonlinear stretching of the bridging chains, as shown in Figure 2c for the dumbbell model.

to τ0 = 6R20/D0 and stress to G = νkBT. Figure 7 also shows the stress contribution arising from micelle interactions (dotted curves), while the separate contribution of the bridging chains was previously reported for the same values of parameters in Figure 2c. For γ̇τ0 = 100, the solid curves in Figure 7 exhibit a shoulder and a maximum, the former arising from the maximum in the micelle contribution, while the latter is contributed by the non-Gaussian response of the bridging chains. (See Figure 2c.) Figure 8 reports the steady-state values of viscosity and first normal stress coefficient versus the shear rate, as obtained from

the number np of polymers per micelle, appearing in the constant C. From the value of the shear modulus G = νkBT, Watanabe and coworkers deduce that the number density ν of elastically active chains is 2.1% of the total number of chains. Accounting for the neglected numerical coefficient (somewhat less than unity) in the relationship G = νkBT, this number is in fact somewhat higher, perhaps somewhere between 3 and 4%, with the rest of the chains (∼96%) forming loops in the micelles. Then, because in this range of concentrations the number of micelles is of the same order of that of active chains, there follows that the number np of polymers per micelle is several tens, say np ≈ 96/4 ≈ 25. Next, from the PEO Kuhn segment length b = 0.7 nm31 and the length of the all-trans configuration of the monomer equal to 0.358 nm,31 we derive that a PEO Kuhn segment is made of 1.96 monomers and hence has a molar mass of 86. Because the polymer used by Suzuki et al.26 has Mn = 3.4 × 104, we get for the number of Kuhn segments N = 395. Hence, C = 9n2pN0.2/π ≈ 5900 and Rmax/R0 ≈ bN/bN0.6 = N0.4 ≈ 11, the latter being needed in the dumbbell calculations. Also needed for the dumbbell equations is the ratio l/R0 appearing in eq 6. Now, R0 ≈ bN0.6 ≈ 25 nm, while from the chain-end chemistry of the HEUR, by assuming that both the alkane sequence and the urethane group are within the micellar core (as suggested by the large values of relaxation time reported in Figure 1), we estimate l ≈ 3 nm, and hence l/R0 ≈ 0.12. To calculate Sxy(γ) we need to assign values to R0/d0 and χ. Consistently with the previous estimate that np ≈ 25 and knowing that the polymer concentration c is 0.01 g/ cm3,26 we estimate d0 ≈ (c5 /Mnnp)−1/3 = 52 nm. (5 is the Avogadro number.) Hence it is R0/d0= 0.48, smaller than unity, implying that the micelles do not interpenetrate at equilibrium, that is, that the shape of Sxy(γ) is that reported in Figure 6a, rather than that of the uppermost curves in Figure 6c. Finally, the choice of χ is not a simple task, although we expect it to be less than unity because C is a large number, yet χ cannot be too small because interacting spheres that want to displace themselves too much are probably stopped also by jumping into other spheres. Not knowing better, we have arbitrarily assigned χ = 0.1. Figure 7 shows the predicted behavior for η+(t) and for the first normal stress coefficient Ψ+1 (t) in shear startup at different shear rates, as obtained by using the parameter values just determined. As in Figure 2, time and shear rate are normalized

Figure 8. Shear viscosity (full line) and first normal stress coefficient (dashed line) from the interacting micelle model, showing shear thickening in a range of shear rates.

eq 16. Comparison of Figure 8 with Figure 2a for the dumbbell contribution (both obtained with the same parameter values) clearly shows the shear thickening arising from the flowerlike micelle interaction.

5. DISCUSSION The important message conveyed by Figures 7 and 8 is that both strain hardening and shear thickening are predicted by the proposed model, that is, by somehow accounting for the repulsive interaction between the flowerlike micelles; however, a closer comparison with the data of Suzuki et al.26 shows both H


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Figure 9. Predictions of the interacting micelle model for R0/d0 = 1.1 and χ = 0.1. (a,b) Shear startup curves for γ̇τ0 = 0.1, 0.5, 1, 2, 5, 10, and 20 from top to bottom. (c) Steady-state shear viscosity and first normal stress coefficient versus the shear rate.

equations of the dumbbell model. Specifically, we expect that large errors arise from preaveraging the chain-end diffusion coefficient D, which is exponentially sensitive to the chain endto-end distance R. (See eq 6.) By preaveraging we certainly lose directionality in the sense that because during flow chains elongate only in the x direction, D should grow larger than the equilibrium value specifically for those longer chains, not for other chains that have a shorter length, closer to equilibrium. Preaveraging mixes them all, assigning the same enhanced diffusivity D̅ to all chains without distinction. We further recall that the same diffusivity also controls the relaxation time and hence the detachment events of interacting micelles. Now, should directionality be accounted for, the enhanced diffusivity preferentially decreases the dimension of the chains in the x direction, which would reduce the difference x2 − y2 more than the product xy, thus possibly explaining the observed different behavior of the first normal stress coefficient from that of the viscosity. In other words, we agree with the analysis of Watanabe and coworkers,26 whereby “reassociation...(is) anisotropically enhanced in the shear gradient direction”. Unfortunately preaveraging destroys the anisotropy. From the preaveraged value D̅ of the diffusivity obtained from the dumbbell model (eqs 5−7) we can also calculate the relaxation time as τ/τ0 = D0/D̅ . It is then noted that, at the beginning of shear thickening, that is, at γ̇τ0 ≈ 0.3 in Figure 8, we find τ/τ0 = 0.99, implying that relaxation from such steady state would be undistinguishable from relaxation in the linear range. Even at γ̇τ0 = 1, that is, well within the shear thickening region, it is τ/τ0 = 0.88. Also such a feature compares favorably with the results of Watanabe and coworkers and further

positive and negative aspects of the prediction that need to be discussed in some detail. In Figure 7 strain hardening appears to start at values of time that are inversely proportional to the shear rate γ̇, that is, at a fixed value of the shear deformation γ. Such a feature favorably compares with the data reported by Watanabe and coworkers,26 which indicate a value of γ ≈ 2. From theory, it is to be expected that the value of γ at which strain hardening first appears is determined by γmin = d0/R0 − R0/d0, previously defined in Section 4 as the shear deformation at which the closest micelles first come in contact. From the previously estimated value of R0/d0 ≈ 0.48, we then find γmin ≈ 1.6 that (nicely) nearly coincides with the experimental value of γ at which strain hardening shows up. Less favorable is the comparison between the breadth of the shear startup maximum in fast flows, which is smaller in the data of Suzuki et al.26 than in the predicted curves in Figure 7. In other words, data reveal that after the maximum the stress decreases toward the steady-state value faster than predicted by the proposed theory. Even more dramatic is the discrepancy between data and predictions at smaller shear rates, that is, when shear thickening first appears. Indeed, as shown in Figure 8, shear thickening of the first normal stress coefficient takes place together with that of the viscosity, similarly to the predictions of Koga and Tanaka16 but contrary to what was shown by the recent data of Watanabe and coworkers26 and also by the older data of Pellens et al.25 We tend to believe, however, that the just mentioned discrepancies are not to be attributed to wrong physics but rather to the mathematical approximations used to simplify the I


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6. CONCLUSIONS In the first part of this paper we have explicitly shown that at not too low concentrations the dominant, single relaxation time of HEUR telechelic polymers scales with the concentration and the molar mass of the polymer in agreement with the prediction of the FP model,9 a fact just briefly mentioned by Tripathi et al.23 Within the FP model, the distinction between bridging and pendant chains becomes irrelevant because a pendant chain is but a bridging one in the act of diffusing. Moreover, because friction is negligibly small, the time spent in the pendant state is negligible with respect to that in the bridging state. Next, we have shown that, as a consequence of the FP model, and provided the role played by the chains forming loops within the flowerlike micelles can be neglected, the dynamics of the network in the nonlinear range can be described by a FENE-dumbbell model, endowed with a variable, chaintension-dependent diffusion coefficient. The dumbbell model systematically predicts shear thinning; that is, qualitatively at least, it describes the behavior observed in HEUR-associating polymers at not too low concentrations. At low concentrations, HEUR telechelic polymers show, in a range of shear rates, shear thickening at steady state and at larger shear rates, strain hardening during shear startup, followed by a stress drop toward a shear-thinning viscosity. As a main contribution of the present paper, we have then suggested, and shown by suitable calculations, that both of these phenomena can be explained by accounting for the repulsive interaction between the micelles being brought to interpenetrate one another by the shear flow. Such an effect disappears with increasing concentration both because the micelles already interpenetrate at equilibrium and because under those conditions many loop chains become bridging chains. The model here developed does not explain all of the observations. Most prominently, it does not predict that thickening of the shear viscosity can occur when the first normal stress coefficient has essentially remained at the linear value, a fact recently reported by Watanabe and coworkers.26 Following their analysis of this strange result, we argue that the discrepancy is probably due to the mathematical approximations that are typically used, in the present paper as well, to solve the dumbbell model. We therefore propose that Brownian simulations of the impermanent network might resolve the issue. We are fully aware of the many assumptions explicitly or implicitly made in the course of the analysis presented in this paper and also of the crudeness of the analysis itself to be considered as a first approximation, but we believe in the important conclusion that non-Gaussian behavior of the bridging chains (or FENE effect) or shear-induced chain association is not needed to explain shear thickening. The repulsive micellar interaction, here proposed, might represent the solution, at least in some cases. For different systems, other mechanisms are better candidates to explain shear thickening. For example, shearinduced chain association appears to be the correct explanation in the case of polysaccharides.34 Conversely, the mechanism proposed in this paper might prove significant in systems where repulsive and attractive interactions coexist. For example, we suggest to make experiments on solutions containing both telechelic and semitelechelic polymers, like those used by van

confirms that shear thickening is not due to non-Gaussian behavior, which arises at much larger shear rates, nor to increased association, because the structure of the network has essentially remained that of equilibrium, except for the fact that flow has brought the micelles to overlap. A slight decrease in the relaxation time associated with shear thickening is also in line with the indication arising from parallel superposition data.24 We have previously observed that the breadth of the shear startup maximum in fast flows comes out larger than shown by the data. In fast flows another discrepancy is found in the slopes of the shear-thinning viscosity and of the first normal stress coefficient, both somewhat less steep in Figure 8 than shown by the data.26 A possible source of these discrepancies is the assumption we have made in Section 3 that, during flow, the number density ν of active chains remains fixed at the equilibrium value. Now, because the frequency β of detachments of the active chains grows exponentially with increasing R (and on average R certainly increases in fast flows), while the frequency of detachments of the loop chains remains at the equilibrium value β0, we should expect that some of the active chains convert themselves into loop ones, that is, that ν decreases in fast flows. A decreasing ν with increasing γ̇ would make the slopes in Figure 8 steeper. Also, a decreasing ν with increasing time (at large shear rates) would make the stress to drop more quickly after the maximum; that is, the breadth of the shear startup maximum would be smaller, in better agreement with data. Again, however, accounting for the change of ν due to change in the frequency of detachments in fast flows cannot be dealt with in a preaveraged way. From the previous discussion emerges the necessity of dealing with the dumbbell model without the use of the preaveraging approximations adopted in the present paper, perhaps by developing a suitable Brownian simulation of the network dynamics, inclusive of the repulsive interaction between the spherical micelles. We plan to do so in the near future. We conclude this section by showing what comes out from the model for a case where it is R0/d0 > 1, that is, when micelles already interpenetrate at equilibrium because of a larger concentration. Figure 9 shows the predictions of the model for R0/d0 = 1.1, with the other parameters being the same of the case previously considered. In Figure 9 shear thickening and strain hardening have disappeared, although the contribution to the stress from the interacting micelles is now dominant with respect to that from the bridging chains. The disappearance of both shear thickening and strain hardening is due to the qualitative difference of the function Sxy(γ) with respect to the case R0/d0 < 1, as previously shown in Figure 6c, because Sxy(γ) now includes a linear range. The very large values of viscosity and first normal stress coefficient in Figure 9 are very probably not to be taken as real, however. Indeed, when micelles interpenetrate at equilibrium, most chains switch from the loop conformation to the bridging one, and the excluded volume interaction of the flowerlike micelles (that now miss many petals) loses force. Also, the excluded volume interaction gets partly screened because of the increased concentration. To better appreciate the change, notice that going from R0/d0 = 0.48 to 1.1 implies that the concentration has been increased by the factor (1.1/0.48)3 = 12, that is, more than 10-fold. It is no wonder that shear thickening has disappeared, as many data confirm. J


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Macromolecules Ruymbeke et al.35 Indeed, because the semitelechelic molecules cannot become bridging chains and only contribute to the “petals” of the flower-like micelles, the structure of the network can be tuned by controlling the ratio between the two species. As a consequence, one can better control the ratio between repulsive and attractive interactions and their contribution to the rheological behavior.

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(29) Sing, M. K.; Wang, Z. G.; McKinley, G. H.; Olsen, B. D. Soft Matter 2015, 11, 2085. (30) Larson, R. G. Constitutive Equation for Polymer Melts and Solutions; Butterworth Publishers: Boston, 1988. (31) Oesterhelt, F.; Rief, M.; Gaub, H. E. New J. Phys. 1999, 1, 6.1− 6.11. (32) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, U.K., 1986. (33) Lodge, A. S.; Meissner, J. Rheol. Acta 1972, 11, 351. (34) Jaishankar, A.; Wee, M.; Matia-Merino, L.; Goh, K. K. T.; McKinley, G. H. Carbohydr. Polym. 2015, 123, 136. (35) van Ruymbeke, E.; Vlassopoulos, D.; Mierzwa, M.; Pakula, T.; Charalabidis, D.; Pitsikalis, M.; Hadjichristidis, N. Macromolecules 2010, 43, 4401.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge financial support from EU through the ITN SUPOLEN project (grant no. 607937) on supramolecular assembly of polymeric structures.

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REFERENCES

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New Interpretation of Shear Thickening in Telechelic Associating

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