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Effect of Connectivity on the Structure and the Liquid−Solid Transition of Dense Suspensions of Soft Colloids René-Ponce Nzé,† Taco Nicolai,† Christophe Chassenieux,† Erwan Nicol,*,† Susanne Boye,‡,§ and Albena Lederer‡,§ †

LUNAM Université, Université du Maine, Institut des Molécules et Matériaux du Mans UMR-CNRS 6283, Avenue Olivier Messiaen, 72085 Le Mans, Cedex, France ‡ Polymer Separation Group, Leibniz-Institut für Polymerforschung Dresden e.V., Hohe Str. 6, D-01069 Dresden, Germany § Technische Universität Dresden, D-01062 Dresden, Germany S Supporting Information *

ABSTRACT: Aqueous solutions of multiarm flower-like poly(ethylene oxide) (PEO) were formed and connected to various degrees by self-assembly. The structure was rendered permanent by in situ UV-irradiation. Dense suspensions of these single and connected soft colloids were studied by static and dynamic light scattering and viscosity measurements. The concentration dependence of the osmotic compressibility, the dynamic correlation length, and the viscosity of single flowers was shown to be close to that of equivalent PEO star-like polymers demonstrating that the effect of forming loops on the interaction is small. It was found that the osmotic compressibility and the dynamic correlation length of dense suspensions are not influenced by the bridging. However, when flower polymers are connected into clusters, motion in dense suspensions needs to be collective over larger length scales. This causes a much stronger increase of the viscosity for dense suspensions of interpenetrated clusters compared to single-flower polymers.

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INTRODUCTION Multiarm star polymers may be considered as soft polymeric colloids, and their behavior in dense suspension has attracted much attention from an academic point of view1−3 and for their potential in designing new materials such as hydrogels.4 The osmotic compressibility of suspensions of spherical colloids decreases sharply with increasing volume fraction (φ) when it approaches dense packing, whereas the viscosity increases sharply.5,6 The excluded volume interaction between star polymers increases with increasing number of arms, and as a consequence the viscosity increases more sharply with increasing concentration.2,7−10 In dense suspensions, the colloids are constrained by the surrounding colloids, which sometimes is expressed as caging. Movement is possible only by collective motion of the colloids that leaves a sufficiently large space for a colloid to move, i.e., to jump out of its cage. The effect of the colloids softness on the dynamics has been investigated by comparing the behavior of particles with various architectures (star polymers, dendrimers, grafted colloidal particles, microgels) in dense suspensions.2,3,11 The behavior of soft colloids ranges between that of linear polymer chains and that of hard spheres. The increase of viscosity with increasing effective volume fraction is steeper when the colloids are harder. It has been demonstrated that rheological properties and the liquid−solid transition (glass © XXXX American Chemical Society

transition or crystallization) in soft colloidal systems can be manipulated by changing the chemical and physical parameters of the colloids.3 The effect of attractive interaction between close packed colloids has also been studied in detail.12,13 Weak attraction causes an increase of the mobility of the colloids, but strong attraction leads to a long-lived percolating network, which reduces the mobility. When the bonds are permanent, no large scale movement is possible. The question arises as to the influence of limited connectivity on the mobility in dense suspensions. When finite size permanently bound clusters are formed, large scale movement will still be possible but the mobility will be reduced depending on the cluster size. Finite size clusters can only be formed at low concentrations so that in order to obtain dense suspensions of the clusters, a large fraction of the solvent needs to be removed. This has been done for rigid clusters of hard spheres, and the viscosity of dense suspensions of such clusters has been investigated.5,14−17 It was found that rigid clusters behaved as colloidal particles with a larger effective volume. Therefore, for the same concentration of colloids the viscosity was higher when they were in the form of clusters. However, rigid clusters cannot Received: June 17, 2015 Revised: October 5, 2015

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DOI: 10.1021/acs.macromol.5b01317 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Scheme 1. Schematic Representation of Flower Formation and Bridging with Increasing Polymer Concentration

Scheme 2. Dense Suspensions of Individual Flowers and Clusters of Bridged Flowers

Scheme 3. Synthesis Route of PMEA7-b-PEO270-b-PMEA7 Triblock Copolymer

concentration the average size of the clusters increases until at a critical concentration a system spanning network is formed. In order to obtain permanently bound flexible clusters, the selfassembled dynamic clusters were fixed by in situ cross-linking of the cores. Using this methodology, we were able to study the effect of connecting close packed soft colloids with flexible bonds in dense suspensions. Scheme 2 shows a schematic representation of dense suspensions of individual flower polymers and of clusters of bridged flowers. Here we report on the structure and the viscosity of aqueous solutions of flower-like poly(ethylene oxide) (PEO) as a function of the concentration. We will first show that the properties of dense suspensions of unconnected flower-like PEO and equivalent star-like PEO are close. We will then discuss the effect of connectivity on the structure and the viscosity of suspensions of close packed flower polymers. We

interpenetrate, and the individual colloids cannot become close packed, which means that the issue of the influence of permanent bonds on the mobility of densely packed colloids cannot be addressed with these systems. In order to properly investigate this effect, one needs to use clusters of soft colloids with flexible bonds that can interpenetrate. As far as we are aware, the influence of connectivity on the dynamics of dense suspensions of close-packed soft colloids has not yet been investigated. One way to make connected soft colloids is to exploit selfassembly of triblock copolymers into flower-like polymers, which has been studied in detail.18,19 Flower polymers resemble star polymers, but both ends of the arms are connected to the core so that they form loops.20 When self-assembly takes place at higher polymer concentrations, flower polymers connect randomly by forming bridges (see Scheme 1). With increasing B


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Figure 1. (a) Intensity autocorrelation functions for solutions of R8 measured at 90° at different concentrations. (b) Corresponding relaxation time distributions. After UV cross-linking micellar solutions at a given concentration, solutions were either diluted or concentrated (first by rotary evaporation and then pumping under vacuum) in order to cover a broad range of concentrations. Asymmetrical Flow Field-Flow Fractionation. AF4 measurements were performed to determine the molecular weights and radii of different fractions with an Eclipse 3 system (Wyatt Technology Europe, Germany). The channel spacer made of poly(tetrafluoroethylene) (PTFE) had thicknesses of 350 μm, and the channel dimensions were 26.5 cm in length and from 2.1 to 0.6 cm in width (LC350 μm). The membranes used as accumulation wall composed of regenerated cellulose with a molecular weight cutoff of 10 kDa (Superon GmbH, DE). Flows were controlled with an Agilent Technologies 1200 series isocratic pump equipped with vacuum degasser. The detection system consists of a multiangle laser light scattering detector (DAWN EOS from Wyatt Technology Europe, Germany) operating at a wavelength of 690 nm and a refractive index detector (Optilab T-rEX from Wyatt Technologies Europe, Germany) operating at a wavelength of 620 nm. All injections were performed with an autosampler (1200 series, Agilent Technologies Deutschland GmbH). The channel flow rate was maintained at 1.0 mL min−1 for all AF4 operations. Samples were injected during the focusing/relaxation step with 0.2 mL min−1 within 2 min. The focus flow (Ff) was set at 3 mL min−1 for 3 min. For all samples the injection volume was 100 μL with a sample concentration between 1 and 1.5 g L−1. Starting crossflow for elution step with a duration of 30 min with linear decreasing gradient to 0 mL min−1 varies for all fractions (R1: Fx,start = 1.5 mL min−1; R4: Fx,start = 1.0 mL min−1; R8: Fx,start = 0.3 mL min−1; and R12: Fx,start = 0.3 mL min−1). The refractive index increment (dn/ dC) of the copolymer in water was assumed to be that of pure PEO in water, i.e., (dn/dC) = 0.13 mL g−1. Collecting and processing of detector data were made by the Astra software, version 5.3.4.20 (Wyatt Technology, USA). Light Scattering. LS measurements were done using commercial static (SLS) and dynamic light scattering (DLS) equipment (ALVLangen, Germany) equipped with an He−Ne laser emitting vertically polarized light at λ = 632 nm. The temperature was set at 20 °C and controlled by a thermostat bath to within ±0.1 °C. Measurements were made at scattering angles (θ) between 30° and 150°. The refractive index increment (dn/dC) of the copolymer in water was assumed to be that of pure PEO in water, i.e., (dn/dC) = 0.13 mL g−1. The normalized electric field autocorrelation function (g1(t)) was calculated from the intensity correlation function (g2(t)) determined by DLS using the so-called Siegert relation.23 g1(t) was analyzed in terms of a relaxation time distribution:

emphasize that the aim of this study was not to investigate the behavior of a different type of soft colloid in dense suspensions. We will show that the behavior of the clusters of randomly branched flowers in dense suspensions cannot be explained by considering them simply as another kind of soft colloid. Rather we need to consider these suspensions as densely packed flower-like polymers that are cross-linked in various extents (see Scheme 2).

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EXPERIMENTAL SECTION

Block Copolymer Synthesis. The poly(2-methacryloyloxyethyl acrylate)-b-poly(ethylene oxide)-b-poly(2-methacryloyloxyethyl acrylate) (PMEA-b-PEO-b-PMEA) triblock copolymer was synthesized according to procedures described elsewhere19,21 (Scheme 3). Briefly, a triple hydrophilic poly(2-hydroxyethyl acrylate)-b-poly(ethylene oxide)-b-poly(2-hydroxyethyl acrylate) (PHEA-b-PEO-b-PHEA) copolymer was synthesized by polymerizing 2-hydroxyethyl acrylate from a brominated-PEO macroinitiator using copper(0)-mediated reversible deactivation radical polymerization (RDRP) as described previously.21 The following [HEA]/[initiator]/[Me6-TREN]/[CuBr2] = [7]/[1]/ [0.2]/[0.1] ratio was used with copper wire as a catalyst in DMSO. The polymerization was stopped at 96% monomer conversion and exhibited first-order kinetics up to this conversion. An average degree of polymerization of 7 was estimated by 1H NMR. Hydrophobic and cross-linkable methacrylate moieties were then introduced by esterification of the pendant hydroxyl groups with methacryloyl chloride22 in order to obtain an amphiphilic triblock copolymer bearing polymerizable groups on the hydrophobic blocks. The quantitative functionalization of the hydroxyl groups was checked by 1 H NMR in DMSO-d6. The final triblock PMEA7-b-PEO270-b-PMEA7 had a number-average molar mass of 13 500 g mol−1. Photo-Cross-Linking of the Triblock Copolymer Solutions. Amphiphilic triblock copolymer solutions were photo-cross-linked at C = 1, 4, 8, and 12 g L−1 in water, giving samples R1, R4, R8, and R12 respectively. For photo-cross-linking experiments a solution of DMPA photoinitiator (0.01 M) was prepared in THF. The molar ratio of DMPA to polymer was fixed to about 20 molecules per micelle considering that the average number of polymer chains per micelle is 30. The required amount of solution of DMPA was placed on the walls of a glass vial, and THF was evaporated under a gentle flow of argon, after which the polymer solution was introduced into the vial that was sealed with a rubber septum. The solution was stirred overnight. Just before the photo-cross-linking experiment, the solution was bubbled by purging argon for 15 min to remove traces of dissolved oxygen. The samples were photo-cross-linked in a glass vial by irradiating for 90 s with UV light at 365 nm (Dymax BlueWave 200 UV lamp).

g1(t ) = C

∫ A(log τ) exp(−t /τ) d log τ

(1) DOI: 10.1021/acs.macromol.5b01317 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules All correlation functions could be well-described using a log-normal distribution for the fast mode and a generalized exponential for the slow mode:

A(log τ ) = kτ p exp[− (τ /τ*)s ]

Scheme 4. Synthesis of Flowers and Bridged Flowers

(2)

As an example, g2(t) and the corresponding relaxation time distributions are shown in Figure 1 for different concentrations of R8. Similar results were obtained for the other samples. DLS showed that all solutions were ergodic. Dilute solutions showed a single fast relaxation mode after spurious scatterers were removed by filtration with 0.2 μm pore size Anatope filters, but more concentrated suspensions showed an additional slow relaxation mode. This mode was at least partially caused by spurious scatterers that could not be removed completely by filtration. The relative contribution of spurious scatterers increased strongly with increasing concentration because the scattering due to cooperative concentration fluctuation decreased strongly. We do not exclude that slow relaxation processes of the concentration fluctuations in the dense suspensions also contributed, but it is not possible to distinguish them from the spurious scattering. The average relaxation rate (Γ) of both modes was found to be q2dependent, but here we have considered only the fast mode which reflected cooperative diffusion. The cooperative diffusion coefficient (Dc) was calculated as Dc = Γ/q2. At sufficiently low concentrations, where interaction is negligible, the z-average hydrodynamic radius (Rh) of the solute can be calculated from the diffusion coefficient (D0) using the Stokes−Einstein relation:

Rh =

kT 6πηD0

multidetection asymmetrical flow-field flow fractionation (Figure 2a). For sample R1 one can clearly distinguish between the main peak representing single flowers (80 wt %) and smaller peaks at longer times. A distinct relatively large fraction of single flowers could still be observed for R4 (42 wt %), but separation into discrete populations was not possible for R8 and R12. Multidetection analysis allowed determination of the molar mass (M) of each population of sample R1 (Figure 2b). The molar mass of the main peak corresponding to single flowers was M ≈ 5.0 × 102 kg mol−1. The molar masses of the second and third peaks were about 2 and 3 times larger, respectively, and corresponded to dimers and trimers of flowers. For a more detailed investigation of the cluster size distribution as a function of the polymer concentration and a comparison with theoretical predictions and numerical simulations, see refs 18 and 19. Individual flower polymers contained 37 PEO chains in the corona, i.e., 74 end-blocks in the core. The structure of the clusters was also determined using light scattering. Weight-average molar masses (Mw), radii of gyration (Rg), and hydrodynamic radii (Rh) of cross-linked samples obtained from AF4, SLS, and DLS are summarized in Table 1. The weight-average number of flowers per cluster was calculated by dividing the molar mass of the clusters by that of the individual flowers. It was 1.2, 2.5, 6.5, and 21 for R1, R4, R8, and R12, respectively. Comparison of Flower- and Star-like Polymers. Sample R1 consisted mainly of individual flower polymers, and its behavior was compared with that of equivalent star polymers reported elsewhere.26 The star polymers contained almost the double amount of PEO chains (63) with half the length of the chains in the flower polymers. The structure of the flower polymers was therefore close to that of the star polymers, except that pairs of arms were connected at their extremities so that they formed loops. Considering that the hydrophobic core of the flowers contained 74 PMEA groups and had a density of about 1 g/mL, we find that its radius is 3.4 nm, which is much smaller than the overall size of the stars or the flowers. Figure 3a compares the dependence of Ir/(MwKC) on the effective volume fraction (φe) of the star and flower polymers with the Carnahan−Starling equation27 that describes the behavior of hard spheres. The effective volume fraction was calculated as φe = C/C*, where the overlap concentration C* was chosen in such a way that the initial dependence of Ir/ (MwKC) on the volume fraction corresponded to that of hard

(3)

with η the solvent viscosity, k Boltzmann’s constant, and T the absolute temperature. At higher concentrations cooperative diffusion is influenced by interaction. In static light scattering experiments the relative excess scattering intensity (Ir) was determined as the total intensity minus the solvent scattering divided by the scattering of toluene at 20 °C. When an additional slow mode was present, Ir was multiplied by the relative amplitude of the fast mode. After this correction Ir is related to the osmotic compressibility ((dπ/dC)−1) and the z-average structure factor (S(q)):24,25 Ir = KCRT (dπ /dC)−1S(q)

(4)

with R the gas constant and T the absolute temperature. In dilute solutions Ir is related to the weight-average molar mass (Mw) and the z-average structure factor (S(q)):24

Ir = KCM w S(q)

(5)

with C the solute concentration and K an optical constant that depends on the refractive index increment. S(q) describes the dependence of Ir on the scattering wave vector: q = (4πn/λ) sin(θ/ 2). The number of chains per flower-like polymer was calculated by dividing Mw with the molar mass of the block copolymer. At higher concentrations, interactions influence the scattering intensity, and the result obtained by extrapolation to q = 0 represents an apparent molar mass (Mapp). Rheology. Continuous flow and oscillatory shear measurements were done using a stress imposed rheometer (AR2000 and DHR3, TA Instruments) with a cone and plate geometry (diameters: 20 and 40 mm). Paraffin oil was added around the samples to avoid solvent evaporation.

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RESULTS AND DISCUSSION Clusters containing different amounts of flowers were synthesized by photo-cross-linking the core of self-assembled amphiphilic PMEA7-b-PEO270-b-PMEA7 triblock copolymers at different concentrations using a procedure described by Kadam et al.18,19 (see Scheme 4). The mass distribution of the polymers formed at 1, 4, 8, and 12 g L−1 (R1, R4, R8, and R12, respectively) was analyzed by D


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Figure 2. (a) AF4 fractograms and concentration signals as a function of elution time of the samples used in this study. (b) AF4 fractogram and evolution of Mw with elution time for sample R1.

interact directly as was reported for systems with a much smaller number of arms.29−31 The viscosity of the flower and star polymer solutions normalized by the water viscosity (ηr) is plotted as a function of φe in Figure 4. The dependence of ηr on φe was close for the

Table 1. Characteristics of Samples R1, R4, R8, and R12 AF4

light scattering

sample

Mw (kg mol−1)

Đ

Rg (nm)

R1 R4 R8 R12

0.60 × 103 1.3 × 103 3.3 × 103 10.5 × 103

1.1 1.7 3.0 5.8

17 24 48 85

Mw (kg mol−1)

Rg (nm)

Rh (nm)

C* (g/L)

× × × ×

34 54 83

14 22 29 42

115 80 65 50

0.55 1.0 3.1 10.5

103 103 103 103

Figure 4. Dependence of the relative viscosity on the effective volume fraction for aqueous solutions of flower (○) and star (△) PEO. The solid line represents the behavior of hard spheres, and the dashed line represents the behavior of linear PEO chains. Figure 3. (a) Dependence of Ir/KCMw on the effective volume fraction for flower (○) and star (△) PEO. The solid line shows the behavior of hard spheres and the dashed line that of linear PEO chains taken from ref 28. The inset shows a close-up of the initial concentration dependence. (b) Dependence of Dc/D0 on the effective volume fraction for flower- and star-like PEO.

star and flower polymers, but there could be small effects due to differences in the friction between overlapping coronas of arms and loops. A description of such subtle differences requires a more detailed investigation, which was outside the scope of the present article where we focus on the effect of connectivity. For hard spheres, the viscosity diverges at a critical volume fraction, and the dependence on φe is well described by the Krieger− Dougherty equation32 (see solid line in Figure 4). The increase of ηr for the star and flower polymers was weaker for φe > 0.3 but still much stronger than that of linear PEO. This intermediate behavior, which is characteristic for soft interpenetrable colloids, was already discussed elsewhere for star polymers.1,2,10 Effect of Connectivity. The effect of connecting the flower polymers on the concentration dependence of Ir/KC at q → 0 is shown in Figure 5a. The average molar mass of the clusters increased with increasing concentration at which they were prepared, which explains the increase of Ir/KC at low concentrations. It was previously reported that large clusters of randomly bridged flowers have a self-similar structure with a fractal dimension df ≈ 2,18 implying that the density of the clusters decreased linearly with increasing size. Therefore, larger clusters start to interact at lower concentrations causing Ir/KC

spheres: C* = 115 g L−1 for both the star and flower polymers. The value of C* obtained is this way is equivalent to that obtained using the second virial coefficient (A2) that describes the initial concentration dependence of Mw: C* = A2/4. The results for star and flower polymers superimposed very well, implying that their excluded volume interactions were the same. The dependence of the osmotic compressibility on the volume fraction of the star and flower polymers was close to that of hard spheres up to φe ≈ 0.4 and much stronger than that of linear PEO reported in ref 28 (see Figure 3a). However, Ir/ (MwKC) decreased less steeply than hard spheres at higher φe because the star and flower polymers are soft. The dependence of the cooperative diffusion coefficient on φe determined by DLS was also the same for the star and flower polymers (Figure 3b), confirming the static light scattering result that interactions between stars and flowers are the same within the experimental errors. There is no indication that the hydrophobic cores E


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Figure 5. Concentration dependence of Ir/KC (a) and cooperative diffusion coefficient (b) for samples R1, R4, R8, and R12.

Figure 7. (a) Angular frequency dependence of the storage (filled symbols) and loss (open symbols) in the linear response regime for dense suspensions of R8. (b) Superposition of the results obtained at different concentrations with as reference the results at C = 200 g/L using horizontal shift factors ac = 0.1 and 0.04 for C = 140 and 120 g/ L, respectively. It is assumed that the vertical shift factors are proportional to C. The solid lines have slope 1.

to decrease at lower concentrations. However, for C > 30 g L−1, Ir/KC was the same as for the individual flower polymers, which means that at higher concentrations the osmotic compressibility was dominated by the interaction between the individual flower polymers. Such behavior is not expected for stiff clusters and is possible only because the clusters consist of soft spheres with flexible bonds. The same conclusion can be drawn from the concentration dependence of Dc that was also approximately the same for C > 30 g/L (see Figure 5b). Figure 6 shows that good superposition with the Carnahan− Starling equation is obtained up to φe = 0.3 for all systems

concentration) (see Figure 7b), showing that the increase of the viscosity with increasing concentration was principally caused by a slowing down of the dynamics. The frequency dependence of G′ expected for purely viscous liquids (G′ ∝ ω2) was not reached even for the lowest concentration investigated. This dependence should be reached at lower frequencies or concentrations, but when G′ is much smaller than G″, it is difficult to obtain accurate values. The viscosity was independent of the shear rate up to at least 100 s−1 for η < 1 Pa·s and showed shear thinning for more viscous suspensions (see Supporting Information, Figure S1). The effect of connectivity on the concentration dependence of the low shear rate viscosity normalized by that of the solvent (ηr) is shown in Figure 8a. As expected, results obtained from

Figure 6. Dependence of Ir/KCMw on the effective volume fraction. Symbols are as in Figure 2a. The results for the equivalent star polymers are shown as crosses. The inset is a close-up of the initial concentration dependence.

assuming that C* increased with respect to R1 by a factor of 1.4, 1.8, and 2.3 for R2, R4, and R8, respectively (see Table 1). However, for φe > 0.3 the results do not superimpose in this representation. This means that the clusters of flower polymers cannot be considered as colloids with a lower density as is generally assumed to be the case for rigid clusters because they can interpenetrate. Figure 7a shows the angular frequency (ω) dependence of the storage (G′) and loss (G″) shear moduli for dense suspensions of sample R8 at different concentrations. Oscillatory shear measurements in the linear response regime showed G″ ∝ ω for low-viscosity solutions, which is characteristic for purely viscous liquids. A weaker frequency dependence was found for more viscous solutions at high frequencies, but in all cases the loss modulus remained larger than the storage modulus at least up to ω = 100 rad/s. It is expected that elasticity of the suspensions increases with increasing polymers concentration, but this effect is very small compared to the large increase of the relaxation time. The results could be superimposed by horizontal and vertical shifts (assuming that the moduli were proportional to the polymer

Figure 8. (a) Concentration dependence of the relative viscosity for samples R1, R4, R8, and R12. Symbols are as in Figure 2a. Filled symbols represent data obtained from the oscillatory shear measurements at low frequencies. The inset shows the same data in a log-linear representation. The solid lines represent fits to an exponential increase. (b) Dependence of ηr on the effective volume fraction. The results for the equivalent star polymers are shown as crosses. The inset shows a close-up of the initial concentration dependence. The solid and dashed lines represent the behavior of hard spheres and flexible linear chains, respectively.

oscillatory shear at low frequencies are the same as those obtained from flow measurements at low shear rates; however, the Cox−Merz rule does not apply as the results do not superimpose over the whole range of frequencies and shear rates (see Figure S2). The concentration dependence of zero shear viscosity could be well approximated by an exponential increase: ηr = exp(aC) with a = 3.3 × 10−2 L/g for R1 and R4, a F


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Macromolecules = 6.2 × 10−2 L/g for R8, and a = 8.0 × 10−2 L/g for R12. It appears that the relatively low degree of bridging for R4 did not have a significant influence on the viscosity, contrary to the more extensively connected flowers. One approach for understanding the effect of cross-linking on the viscosity is to consider the clusters of flowers as soft colloids with radius Rh and with a density that decreases with increasing size. In this case a better comparison of the effect on the viscosity is obtained by plotting the viscosity as a function of φe. Vlassopoulos et al.3,11 compared the dependence of ηr on φe for noninteracting spherical colloids with different degrees of softness and showed that ηr increases more steeply if the colloids are harder. The behavior of all particles is situated between the extremes of hard spheres and flexible linear chains. In the present case, the average size of the clusters increases, but their density decreases with increasing degree of crosslinking. Therefore, one would expect that the dependence of ηr on φe becomes less steep with increasing degree of crosslinking. The dependence of ηr on φe for the single and cross-linked flowers is shown in Figure 8b. For comparison, we also show the limiting behavior of hard spheres and flexible linear polymers. The initial increase of ηr when the clusters do not overlap superimposes reasonably well in this representation, showing that one may consider the clusters as soft colloids as long as they do not overlap significantly. However, it is clear that the dependence of ηr on φe at larger φe is not what would be expected if the clusters of flowers could simply be considered as soft colloids. For R4 we do indeed observe a weaker increase of ηr than for the individual flowers (R1), but larger clusters R8 and R12 show a steeper dependence, while they are much less dense and therefore much softer. This comparison shows that a different approach is needed to understand the behavior of the viscosity of these suspensions. What has not been considered is that the clusters can interpenetrate so that in dense suspensions the individual flowers of different clusters can come into close contact as was depicted in Scheme 2. Dense suspensions of the clusters of flowers are in our view best regarded as dense suspensions of spherical colloids (individual flowers) that are connected to a limited extent. In such a situation the volume fraction of the clusters is no longer the relevant parameter, but rather the volume fraction of the individual flowers. The latter does not change with the degree of cross-linking and is proportional to C. Therefore, the representation Figure 8a is the more useful representation to evaluate the effect of cross-linking on the viscosity in dense suspensions. As was mentioned above, the steep increase of ηr in dense suspensions of spherical colloids can be qualitatively explained by the increasing extent to which the movement of a colloid is constrained by the surrounding colloids and requires to be cooperative. When flowers are connected into clusters, the cooperative motion will need to take place over longer distances as flowers belonging to the same cluster have to move together. Therefore, it is expected that spontaneous motion slows down with increasing cluster size. However, it is important to also consider the large dispersity of cluster sizes (see Figure 2a). In particular, R4 still contains a large fraction of single flowers (42%), which means that the relatively small clusters of this sample are not interpenetrated. This may explain why the viscosity of R4 is close to that of single flower and star polymers. Another important consideration is the softness of the flowers and the flexibility of the connections. This allows

the connected flowers to move even for φe > 1. A more quantitative explanation of the effect of connectivity is difficult because one needs to consider the detailed structure of the polydisperse clusters and the cooperative mobility on larger length scales. Numerical simulation may help to gain a deeper understanding of the effect of bridging on the mobility of the flowers.

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CONCLUSION The osmotic compressibility and the viscosity of equivalent multiarm star and flower polymers in dense suspensions are nearly the same. This shows that excluded volume interaction between star polymers is not significantly modified if pairs of arms in the corona are connected at their extremity and form loops. Bridging of the flower polymers leads to the formation of clusters of flowers, but it does not influence the homogeneous distribution of the flowers in dense suspensions and therefore the osmotic compressibility is not influenced. An increasing degree of cross-linking causes a strong rise of the viscosity. This effect cannot be understood by considering the clusters with larger size as softer colloids. Rather, the increase of viscosity with increasing connectivity is caused by the restriction of movements of connected flowers which need to be cooperative on larger length scales.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01317. Flow curves of sample R8 at various concentration; Cox−Merz representation of sample R8 at 160 g/L (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (E.N.). Funding

This work was funded by the Région Pays de la Loire (project ‘Nanofleurs’). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Lazhar Benyahia for his help with the rheological measurements. REFERENCES

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