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Imran Hussain Syed , Patrick Stratmann , Günter Hempel , Manfred Klüppel , and Kay Saalwächter. Macromolecules 2016 49 (23), 9004-9016. Abstract | ...
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Macro- and Microrheology of Heterogeneous Microgel Packings Fany Di Lorenzo† and Sebastian Seiffert*,†,‡ †

Institute Soft Matter and Functional Materials, Helmholtz-Zentrum Berlin, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany Institute of Chemistry and Biochemistry, Freie Universität Berlin, Takustr. 3, D-14195 Berlin, Germany

‡

S Supporting Information *

ABSTRACT: Microgels are soft deformable colloids that can be packed by external compression. Such packing transforms a suspension of loose microgel particles into an arrested state with properties similar to that of a macroscopic gel. This effect provides a way to purposely impart micrometer or submicrometer scale spatial inhomogeneities into these assemblies, allowing their effect to be studied. We follow this idea and prepare microgel packings that consist of major (50−99.7 No.%) fractions of soft, loosely cross-linked particles doped with defined minor (0.3−50 No.%) fractions of stiff, densely cross-linked particles. This approach creates soft microgel packings that contain defined submicrometer scale domains with very high degree of cross-linking, resembling the structure of inhomogeneous macroscopic gels. We study these inhomogeneous composites from macro- and microscopic perspectives by oscillatory shear rheology and fluorescence recovery after photobleaching to probe their macroscopic mechanics and the microscopic mobility of flexible linear tracer polymers that diffuse through them. These studies reveal an ambiguous behavior: whereas the presence of densely cross-linked domains does not exhibit any systematic effect on the bulk compressibility and microscopic tracer-chain diffusivity in the heterogeneous packings, it increases their macroscopic shear elastic modulus in a linear additive fashion. These results indicate that the impact of spatial inhomogeneities in polymer gels depends on whether the gels are probed in equilibrium or deformed states.

1. INTRODUCTION Microgels are micrometer or submicrometer sized particles that consist of a rubbery polymer network swollen by solvent.1 In contrast to colloids that are composed of incompressible and undeformable solids, microgels can be both compressed and deformed to a degree that is determined by the cross-link density and the polymer−solvent interactions of their constituent polymer network.2,3 This allows them to be packed to effective volume fractions much greater than the close packing limit for hard spheres.3,4 At such dense packing, the particles are immobilized and deformed by their neighbors; as a result, dense packed microgel suspensions exhibit properties similar to that of macroscopic, space-filling polymer gels. For example, the concentration-dependent osmotic compressibility of dense microgel packings is represented by the Flory−Rehner theory, reminiscent of macroscopic polymer gels.5 This is because if microgel particles become so densely packed that they fill the space and deform one another, the only way for the system to accommodate further external compression is to expel solvent and deswell, resembling the compressibility of macroscopic polymer gels. In addition, packed microgel suspensions can store elastic energy if they are sheared to a degree that does not yet lead to particle rearrangement and flow, again resembling macroscopic polymer gels.5 These findings suggest that dense packings of microgels can serve as model systems for macroscopic gels, allowing their physicalchemical properties to be studied. This can be conducted with good consistency and versatility because microgel packings allow macroscopic gels to be modeled simply by assembling © 2013 American Chemical Society

microgel building blocks of determined composition, elasticity, and size in a modular principle. A characteristic feature of many macroscopic polymer networks and gels, particularly of those made by free-radical cross-linking copolymerization of bi- and multifunctional monomers, is that they exhibit marked degrees of spatial inhomogeneity of their cross-linking density.6 This is a consequence of the gelation mechanism: during the early phase of a cross-linking copolymerization, extensive cyclization and multiple cross-linking reactions occur, leading to an assembly of spatially localized nanogels. As the reaction proceeds, macroscopic gelation occurs by interconnecting these clusters to a continuous, space-filling polymer network. As a result, the final gel is an irregular assembly of loosely interconnected nanogel clusters and displays pronounced concentration fluctuations on length scales of several tens of nanometers.7−11 Despite this existing picture on the origin of spatial inhomogeneities in polymer gels, much less is known about their impact. With respect to the macroscopic gel elasticity, it is often argued that the presence of densely cross-linked clusters does not play an important role because these clusters solely act as an additional small fraction of supernodes with very high internal cross-linking density that are superimposed to the much less densely cross-linked soft gel background. Upon Received: October 31, 2012 Revised: February 18, 2013 Published: March 4, 2013 1962

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Figure 1. Concept of the present work: a suspension of submicrometer sized microgel particles is packed by external osmotic compression, strongly densifying the particles. The resultant microgel packing is then probed macroscopically by oscillatory shear rheology and microscopically by monitoring the diffusive mobility of a small portion of fluorescent linear tracer polymers within the microgel matrix. To systematically study the impact of spatial heterogeneities, we prepare composite packings that consist of major (50−99.7 No.%, corresponding to 25−99 wt %) fractions of soft microgel particles doped with defined minor (0.3−50 No.%, corresponding to 1−75 wt %) fractions of stiff microgel particles, indicated by different blue shading in the schematics. The left picture is a DIC micrograph of a compressed microgel suspension adopted from ref 5 by permission of The Royal Society of Chemistry.

spacing of cross-linkable units along them, giving excellent means to prepare networks with defined, customized nanometer scale architectures. This approach can also be used to prepare networks with purposely built-in irregularities and inhomogeneities, for example, networks with bi-14,17,18 or multimodal19 network chain length distribution, networks with controlled amount and length of dangling chains,33 and networks with purposely created nanometer-scale clusters of high internal cross-linking density.15,16 However, this strategy requires sophisticated ways of polymer synthesis and modification. On top of that, even the cross-linking of highly defined precursors may be impaired by statistics, causing network irregularities that cannot be predicted and controlled.34 A simpler way to form polymer gels with controlled inhomogeneity is to assemble them from microgel building blocks with determined composition and size. This can be achieved by making use of the fact that close-packed assemblies of microgels act like macroscopic gels, both qualitatively and quantitatively.5 On this basis, the assembly of heterogeneous mixtures of different types of microgel building blocks is a simple way of preparing macrogel-type systems with determined heterogeneous compositions. In this paper, we use submicrometer scale microgel building blocks with different degrees of cross-linking and pack them together to obtain macrogel-type samples with defined heterogeneous compositions. We prepare packings that consist of a major (50−99.7 No.%, corresponding to 25−99 wt %) soft background, constituted by soft microgel particles that are synthesized with 1 mol % of cross-linker. This soft background is doped with defined minor fractions (0.3−50 No.%, corresponding to 1−75 wt %) of stiff microgel particles that are synthesized with much more, 10 mol % of cross-linker. The resulting composite packings resemble macroscopic polymer gels that contain small percentages of submicrometer scale spatial inhomogeneities of their cross-link density, just as they are typically obtained by free-radical solution copolymerization

deformation, it is this soft background that is assumed to store most of the elastic energy.12−19 As a result, the simplistic picture of affine deformation of the network strands breaks down in inhomogeneous gels.14,20,21 Instead, nonaffine response is presumed, with an extent that is proportional to the variance in the gel local elasticity.22 Within this concept, the overall gel elasticity is not an average of that of the stiff and that of the soft domains but can be expected to reflect that of the soft domains alone; this causes deviations from the simplistic theory of rubber elasticity. By contrast, a marked effect is expected if inhomogeneous gels are probed from microscopic perspectives. For example, if the diffusive mobility of nanoscopic tracers such as flexible macromolecules or rigid nanoparticles through the polymer gel matrix is considered, densely cross-linked domains in the gel can act as traps or obstacles for these tracers, largely restricting their motion.23,24 This is a crucial aspect to be looked upon when gels shall serve for the encapsulation and controlled release of nanoscopic additives or as membranes in separation techniques. To check for the applicability of these conceptual pictures, it is necessary to prepare and probe polymer gels that are complexed with defined degrees of spatial inhomogeneities. This is difficult to achieve because it must rely on a sound understanding of the impact of the experimental parameters at the moment of the gel formation on the nano- and micrometer scale structure of the resulting gels. In principle, one may utilize the circumstance that gels made by fast and uncontrolled polymerization, initiated by large amounts of initiator and/or polymerized at high temperatures, are strongly inhomogeneous, whereas those formed by gentle reactions at lower polymerization temperature are more homogeneous.9,25−29 However, there are no general relations capturing this interplay, which does in fact vary from system to system. An alternative, more sophisticated way to form gels with determined polymer network topology is to selectively interconnect prefabricated macromolecular precursor chains.11,30−32 The cross-linking pattern of these precursors can be tuned by the degree and 1963

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experiments. This particle size is chosen to keep the ratio of the size of the microgel building blocks to the characteristic length scales of the different experiments constant. The characteristic length scale of macrorheology experiments is given by the plate−plate gap in the rheometer, which is 1 mm in the context of this work. By contrast, the characteristic length scale of microscopic FRAP experiments is about 200 μm, given by the diffusion path on the time scale of the experiments.40−42 Both corresponds to about 1000× the respective particle size in their state of equilibrium swelling. Moreover, in rheology, we work with a shear strain of γ = 1%, resulting in a horizontal displacement of 10 μm at a sample height of 1 mm, corresponding to about 10 swollen-particle diameters. In FRAP, we subject the samples to a Gaussian-shaped bleaching pattern with an initial width of about 2 μm,40−42 again corresponding to about 10 swollen-particle diameters. Thus, the particle diameters of 200 nm and 1 μm consistently ensure a constant ratio of the particle size to the characteristic length in the respective experiments. The different degrees of internal particle cross-linking and the resulting different degrees of particle swelling entail marked differences of the inner segmental densities of the soft and stiff microgels. To determine the polymer concentration inside the particles, cp, in the limit of full particle swelling, we measure the viscosities of the four different microgel suspensions at high dilution and match these values to the Batchelor−Einstein equation43 to obtain their particle volume fractions. In addition, we determine the polymer weight fractions of the same suspensions by gravimetry. Comparison of these values, together with knowledge of the particle radii and particle volumes, yields the polymer content within the particles, which is cp(1 μm, 1% BIS) = 3.2 wt % (∼32 g L−1), cp(1 μm, 10% BIS) = 8.3 wt % (∼83 g L−1), cp(200 nm, 1% BIS) = 2.1 wt % (∼21 g L−1), and cp(200 nm, 10% BIS) = 7.1 wt % (∼71 g L−1). With these values, we can estimate the individual particle elastic shear moduli at swelling equilibrium; this is done with a simple formula derived from the theory of rubber elasticity, assuming each individual microgel particle to reflect an individual swollen polymer network:44 G0 = νkBT = c[NA/(MmN)] kBT,45 with ν the number density of elastically active polymer network strands, N their average number of monomer units, Mm the monomer molecular weight, c the polymer concentration, and NA the Avogadro number. We calculate GParticle(1 μm, 1% BIS) = 6.75 kPa, GParticle(1 μm, 10% BIS) = 170 kPa, GParticle(200 nm, 1% BIS) = 4.43 kPa, and GParticle(200 nm, 1% BIS) = 145 kPa (T = 15 °C). These values are in good agreement with experimental estimates on pNIPAAm microgels with similar compositions;46,47 these estimates, however, are obtained by indentation of single substrate-supported microgel particles, which can exhibit marked differences to the elasticity of microgels suspended in a solvent.48 A drawback of the upper calculation is that it is based on the assumption of fully efficient incorporation of the BIS cross-linker to form elastically active network junctions (νx‑link = cBIS), that is, N = 10 in the stiff and N = 100 in the soft microgels. It is known that crosslinking is not that efficient in the case of free-radical cross-linking copolymerization but typically occurs with a low efficiency of only ∼0.1−10%.9,10 Thus, the individual particle shear moduli can be expected to be lower than the above estimates. This is in agreement to other experimental estimates in the literature, for example, GParticle = 100−1500 Pa for acrylate-based polyelectrolyte microgels.44 2.2. Preparation of Heterogeneous Microgel Packings. To obtain heterogeneous microgel suspensions that exhibit micrometer or submicrometer scale domains with different internal cross-link densities, we mix the soft (1%-BIS cross-linked) and stiff (10%-BIS cross-linked) microgels within both the 200 nm and the 1 μm set of particles. We prepare suspensions that contain major fractions of 99, 95, 75, 50, or 25 wt % of soft microgels, mixed with minor fractions of 1, 5, 25, 50, or 75 wt % of stiff microgels within both sets of particles. These mixtures contain particle number ratios of 300:1, 60:1, 9:1, 3:1, or 1:1 of the soft and stiff microgels, corresponding to particle number percentages of 99.7, 98.3, 90, 75, or 50 No.% of soft microgels doped with 0.3, 1.7, 10, 25, or 50 No.% of stiff microgels. These compositions resemble the typical percentages of densely cross-linked domains in macroscopic polymer gels that shall be modeled in this work.

of monomers and cross-linkers. We use these model systems and probe them from macroscopic and microscopic perspectives. We study their equilibrium mechanics by macroscopic oscillatory shear rheology, as illustrated schematically in Figure 1. We also study the diffusion of fluorescently tagged linear macromolecules that are entrapped within these systems by fluorescence recovery after photobleaching (FRAP), as also illustrated in Figure 1.

2. EXPERIMENTAL CONCEPT AND EXPERIMENTAL PROCEDURES 2.1. Microgel Synthesis by Precipitation Polymerization. We focus on aqueous suspensions of cross-linked poly(N-isopropylacrylamide) (pNIPAAm) microgel particles. pNIPAAm is well-known for its thermosensitivity in aqueous media;35 this feature allows us to synthesize monodisperse microgel particles with sizes in the range of several hundred nanometers through the established technique of precipitation polymerization.36,37 We follow this approach and synthesize pNIPAAm microgels with either 1 or 10 mol % (relative to the total monomer content) of N,N-methylenebis(acrylamide) (BIS). We polymerize all these microgels at a total monomer concentration of 15 g L−1. The polymerization is triggered by addition of 5 wt % (relative to the monomer content) of ammonium persulfate (APS) at a temperature of 70 °C and occurs in the presence of sodium dodecyl sulfate (SDS). We work at four different concentrations of SDS: 0.12, 0.3, 0.6, and 0.8 g L−1. With this approach, we prepare two sets of particles: one set consists of particles with sizes of 200 nm, whereas the other set consists of particles with sizes of 1 μm. Both sets embrace two subsets of particles, one comprising particles prepared with 1 mol % of BIS and the other comprising particles prepared with 10 mol % of BIS, respectively. We polymerize small and soft (200 nm sized and 1%-BIS crosslinked) particles with 0.8 g L−1 SDS, small and stiff (200 nm sized and 10%-BIS cross-linked) particles with 0.6 g L−1 SDS, large and soft (1 μm sized and 1%-BIS cross-linked) particles with 0.3 g L−1 SDS, and large and stiff (1 μm sized and 10%-BIS cross-linked) particles with 0.12 g L−1 SDS. To promote the formation of the large, 1 μm sized particles, they are polymerized in the presence of a 0.1 mol L−1 acetate/acetic acid buffer at pH = 4.3, whereas the small, 200 mn sized particles are polymerized in water. The presence of the acetate/acetic acid buffer serves to increase the ionic strength in the reaction medium; this destabilizes the microgel nuclei, so that less of them are formed and the dimension of the final particles is larger. Together, these experimental parameters ensure equality of size of the soft (1%BIS cross-linked) and stiff (10%-BIS cross-linked) particles within both the set of 200 nm particles and the set of 1 μm particles. This is because the soft particles of both sets are formed at a higher content of SDS than the stiff particles, limiting their growth more efficiently. As a result, the stiff particles of each set are grown to a larger size than the soft particles. However, as the soft particles can swell to a higher degree after their fabrication, both types of microgels end up having very similar sizes in the equilibrium swelling state, both within the set of 200 nm particles and within the set of 1 μm particles. The final particle sizes are confirmed by dynamic light scattering conducted after purification of the particles by 1 week dialyses against water. These experiments reveal hydrodynamic particle diameters of d200 nm,1% BIS = 198 ± 4 nm, d200 nm, 10% BIS = 204 ± 4 nm, d1 μm, 1% BIS = 1046 ± 24 nm, and d1 μm, 10% BIS = 1080 ± 20 nm. The particle sizes of 200 nm and 1 μm are chosen for the following reason: spatial inhomogeneities in polymer networks span length scales of 10−100 nm.6−11 Hence, we prepare one set of particles to exhibit sizes in this range. This is given by the 200 nm particles, which exhibit sizes in the range of 10−100 nm when they are deswollen by external osmotic compression.5,38,39 In our present work, these small particles are used for the microscopic probing of the tracer mobility in dense-packed heterogeneous microgel suspensions. By contrast, we use bigger, 1 μm microgels for macroscopic oscillatory shear 1964

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formate (0.375 mol L−1). The reaction is triggered by adding APS (0.5 mmol L−1) and N,N,N′,N′-tetramethylethylenediamine (1.25 mmol L−1) and allowed to proceed for 1 h at room temperature, before the polymer is isolated by precipitation in methanol/water (1:1). To distinguish the tracer chains from the matrix polymer, we tag their primary amine groups introduced by the comonomer with fluorescein labels following established protocols.50 We use a low content of label (0.15 mol % relative to NIPAAm repeat units) that does not notably alter the polymer properties. The tracer polymers are characterized by size exclusion chromatography, dynamic light scattering, and FRAP, revealing a number-average molecular weight of MN = 130 kg mol−1 (polystyrene analogue), a polydispersity of MW/MN = 1.8, and a hydrodynamic radius of rH(DLS) = 17.7 ± 1.8 nm and rH(FRAP) = 17.7 ± 1.6 nm. 2.4. Shear Rheology. Rheology measurements are performed on a strain-controlled rheometer (Rheometrics Fluids Spectrometer II) using a parallel plate−plate geometry with a plate radius of 25 mm, separated by a gap of 1 mm. Both shear rate and strain have a maximum at the edge of the plates, and the reported data are related to this position. The surface of the plates is roughened by sandblasting to prevent wall slip, creating an average surface roughness of 34 μm; to prove the absence of wall slip, we conduct control experiments as described by Seth, Meeker, Cloitre, and Bonnecaze,51,52 as shown in Figure S1 of the Supporting Information. All measurements are carried out at 15 °C, and a solvent trap is used to avoid solvent evaporation. The microgel suspensions are presheared for 120 s at dγ/dt = 1−5 s−1, left to relax for 30 min, sheared again, and left to relax for another 15 min to eliminate any history dependence of the results. Oscillatory shear experiments are then performed at a constant strain of γ = 1% within an angular frequency range of ω = 0.001−10 rad s−1. Yielding and flow of the jammed microgel suspensions do not occur at this low strain, as assured by independent strain sweeps that denote yield strains of >10% in all samples, as shown in Figure S2. These values of the yield strain are higher than those typically found for microgel suspensions53,54 because the microgel packing is very dense in our work. In addition to the exclusion of macroscopic yielding, the strain sweeps also support the absence of microscopic interparticle slip because both the elastic and viscous moduli do not exhibit any marked dependence on strain up to γ ≈ 3%, as also seen in Figure S2 and detailed in the Supporting Information. In addition to the experiments with the plate−plate geometry, the shear viscosity of dilute microgel suspensions (c = 0.1−0.6 wt %) is measured at 20 °C using a double-wall Couette geometry (Rbob = 16 mm, Rcup = 17 mm) at shear rates of dγ/dt = 0.02−1000 s−1. 2.5. Fluorescence Recovery after Photobleaching (FRAP). To quantify the mobility of flexible tracer polymers inside the heterogeneous microgel packings, 0.6 g L−1 of the fluorescein-tagged tracer polymer is dissolved in the pNIPAAm microgel suspensions (c = 15 g L−1 in the as-obtained state) prior to their osmotic densification. Incorporating the tracer polymer at this low concentration assures that the tracer chains do not overlap and entangle with each other but just do so with the compressed microgel polymer networks. This low tracer-polymer content also precludes entropic depletion interactions between the microgel particles.55 To ensure good fluorescence quantum yield of the fluorescein tags, 0.1 mmol L−1 of Na2CO3 is added to each sample prior to and during the osmotic compression by supplementing it to the compressing PEG solutions. After the osmotic compression, the tracer-doped samples are subjected to fluorescence recovery after photobleaching (FRAP)56 experiments at T = 25 °C. For this purpose, aliquots are withdrawn from the compressed microgel pastes and sandwiched between two glass coverslips that are sealed with nail polish. FRAP experiments are performed on a Leica TCS SP2 confocal laser scanning microscope equipped with a 10× DRY objective of NA = 0.3. The low NA ensures that bleaching does not create any appreciable gradient in the zdirection; thus, we have to consider two-dimensional lateral diffusion only. In the scanning mode, the fluorophores are excited with the 488 nm line of an Ar ion laser at 20% of its maximum power, whereas fullpower irradiation with 6.2 mW at the object level is applied to bleach the fluorophore. Before bleaching, a stack of 10 images is scanned to

Densification of the heterogeneous microgel suspensions is achieved by transferring them into dialysis membrane tubes (SpectraPor regenerated cellulose membrane, MWCO 6−8 kDa) and immersing these in aqueous solutions of poly(ethylene glycol) (PEG, MN = 20 kg mol−1) with concentrations between 2.5 and 17.5 wt % until the osmotic pressures of the PEG solutions and the osmotic pressures of the microgel suspensions are equilibrated. The relationship between the osmotic pressure of PEG solutions and their PEG concentration is obtained by fitting literature data.49 We compress the suspensions of the 200 nm microgels within a range of Π = 10−500 kPa at 25 °C, whereas the suspensions of the 1 μm microgels are compressed within the same range at 4 °C; the difference in temperature is due to the different operating temperatures in the FRAP and rheology experiments: FRAP is conducted at a standard temperature of 25 °C, whereas rheology is measured at a lower temperature, 15 °C, to prevent solvent evaporation during the measurement. In all cases, the extent of compression is high enough to transform the initial lowviscous dilute microgel suspension into an elastic solid of dense-packed particles. The increase of the polymer content achieved by this method is determined by gravimetric analysis of the wet and dry weight of the dense-packed microgel suspensions. For comparison, we also compress a solution of un-cross-linked pNiPAAm chains (M = 65 000 g mol−1) and a 1%-BIS cross-linked macroscopic pNiPAAm hydrogel within the same range of osmotic pressure as applied to the microgel suspensions. The un-cross-linked polymer solution has the same polymer content as the microgel suspensions before their compression (c = 15 g L−1); the macroscopic gel has approximately the same polymer (25 g L−1) and cross-link (1 mol %) density as the 1%-BIS cross-linked microgel particles in their swollen state. To ascertain that the stiff doped-in microgels are randomly distributed within the soft-microgel background in the compressed mixtures, a control experiment is conducted with fluorescently labeled stiff microgels. Fluorescence imaging of the resulting packings reveals no indication for clusterization of the labeled stiff microgel particles upon osmotic compression of the mixed suspensions. Particulate systems are characterized by their particle volume fraction. However, this parameter does not account for particle compression, which is specifically relevant for soft microgels. To account for this effect, we consider a different parameter: the effective microgel packing fraction. This parameter is defined as ζ = nVd,5 with n the number of particles per volume and Vd the volume of a single swollen particle in dilute suspension. ζ is related to the polymer concentration in the microgel packings, c, via ζ = nVd = (c/mP)Vd = c/ (mp/Vd) = c/cP, with mP the polymer weight inside each particle and cP the polymer concentration inside each particle in the swollen state, as reported in section 2.1. We calculate Vd from the hydrodynamic radii of the particles. mP is derived from estimates of the viscosity of dilute particle suspensions and application of the Batchelor−Einstein equation43 to obtain the effective particle volume fraction in the dilute limit, as discussed above. We find ζ = 0.31c (wt %) for the 1 μm, 1%-BIS cross-linked microgels, ζ = 0.12c (wt %) for the 1 μm, 10%BIS cross-linked microgels, ζ = 0.47c (wt %) for the 200 nm, 1%-BIS cross-linked microgels, and ζ = 0.14c (wt %) for the 200 nm, 10%-BIS cross-linked microgels. These relations show that in suspensions with the same polymer content the soft, 1%-BIS cross-linked microgels are almost 3 times more densely packed than the stiff, 10%-BIS crosslinked particles. 2.3. Synthesis of Linear Flexible Tracer Polymers. For microscopic FRAP studies, the compressed microgel matrixes must be loaded with a linear tracer polymer. To avoid thermodynamic incompatibility, we choose this polymer to be the same material as the microgels: we use linear pNIPAAm chains that are long enough to entangle with the microgel polymer network. The synthesis of these tracer chains is performed in the presence of a chain-transfer agent, sodium formate, to suppress chain branching and to keep the polydispersity of the tracer-chain length low.50 We polymerize the tracer polymers from aqueous solutions of NIPAAm (0.25 mol L−1), an amine-functionalized comonomer, N-(3-aminopropyl)methacrylamide (0.15 mol % relative to NIPAAm), and sodium 1965

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record the prebleach situation. To bleach a point pattern into the confocal plane, a chosen spot is irradiated for 0.1 s with the laser settings mentioned above. After bleaching, three series with 10 images each are recorded to document the fluorescence recovery process. The temporal spacing between the single images is 1 s during the first series, 5 s during the second, and 30 s during the third. The FRAP data are analyzed with a multicomponent diffusion model.40−42 In short, we record spatially (r) and temporally (t) resolved fluorescence intensity profiles, I(r,t), that are attenuated by a Gaussian sink in the bleached region around r = 0. This pattern smears out with time due to the diffusive exchange of bleached and unbleached fluorophores, characterized by an ensemble of translational diffusion coefficients Di in respective amounts Mi, with one D;M-pair for each diffusing species i. Quantitative analysis of these data sets yields distributions of diffusion coefficients.42 For all experiments discussed in this paper, these distributions spread across about one decade on the diffusivity scale. For simplicity, we average them on a logarithmic scale to obtain only one characteristic, average tracer-chain diffusivity for each sample. FRAP probes diffusion processes on length scales of 10−100 μm.40,56 This is orders of magnitude larger than spatial inhomogeneities in polymer networks and gels, which span typical length scales of 10−100 nm.6−11 This is also larger than the submicrometer scale inhomogeneities that we impart into our microgel packings to model and study these features. As a result, the FRAP method averages over these spatial inhomogeneities. Thus, we do not need to consider anomalous diffusion; instead, we probe normal, above-micrometer scale diffusivities that reflect the net effect of all these submicrometer scale inhomogeneities. Another series of FRAP experiments are performed on compressed suspensions of fluorescently labeled microgels, both soft and stiff. These experiments ensure that the microgel particles are fully immobilized at all levels of compression discussed in this work. Redispersion of the same samples is accompanied by full reversion of the original particle mobilities in the precompressed state, assuring reversibility of the microgel packing and compression. 2.6. Microgel Density Profiles. Microgel particles that are synthesized by precipitation polymerization exhibit radialIy inhomogeneous density profiles.57,58 The specific profile of a given particle depends on many parameters, including the particle average degree of cross-linking, the synthesis and measuring temperatures, and the electrolyte content in the particle suspension. Nevertheless, a typical radial profile has the following appearance: About one-third of the particle consists of a stiff core with a box profile of the radial density, RBox. The second third exhibits a smooth decrease of the segmental density to about one-half of that in the core; this is the particle radius, R = RBox + 2σSurf. The last third exhibits further decrease of the segmental density to zero; this is another measure of the particle radius, RSANS = R + 2σSurf.57 Upon deswelling, the particle densifies from the rim to the core, leading to a progressively less diffuse profile. As the particles used in this work are strongly deswollen, their radial density profiles are homogenized by this effect.

all these experiments, we observe a considerable increase of the polymer concentration with increasing osmotic pressure at external compression stronger than Π = 10 kPa. When plain soft-microgel suspensions are compressed, the relation of Π(c) follows a power law with an exponent close to 9/4, as shown in Figure 2A (large and small + symbols). This scaling law resembles the prediction for the concentration dependence of the osmotic pressure in un-cross-linked semidilute polymer solutions.59,60 Hence, at high level of densification, the soft microgels appear to be so compressed that the chains in their constituent polymer network behave as if they were fully relaxed in a semidilute solution. As a result, the osmotic pressure arises from mixing of the solvent and polymer alone,5 whereas the additional presence of chemical cross-links is not noticed in this limit. This hypothesis is confirmed by good agreement of these Π(c) data to that of uncross-linked pNIPAAm solutions, as denoted by the small star symbols in Figure 2A. There is also good agreement to macroscopic 1%-BIS cross-linked pNIPAAm gels, as denoted by the large star symbols in Figure 2A. All these findings tie in with similar observations on other macroscopic gels;61 they also show quantitative agreement to equivalent results on suspensions of much larger, 100−400 μm noncolloidal pNIPAAm gel particles.5 The preceding rationale is supported when the polymer concentration that is achieved by the microgel compression is considered: the large and small + symbols in Figure 2A reflect polymer concentrations in the compressed soft-microgel suspensions of c = 7−25 wt % (∼70−250 g L−1). These concentrations are considerably higher than the polymer concentration inside each particle, which is cp = 3.2 wt % (∼32 g L−1) for the soft 1 μm microgels and cp = 2.1 wt % (∼21 g L−1) for the soft 200 nm microgels. An alternative way to assess the extent of microgel densification is to consider the effective packing fraction of the compressed microgel suspensions. This quantity is defined as ζ = nVd,5 with n the number of particles per volume and Vd the volume of a single swollen and unperturbed particle in a dilute microgel suspension, as detailed in section 2.2. At the verge of closepacking, ζ = 1, whereas ζ > 1 if further microgel compression and microgel particle deswelling occurs beyond this point. The effective packing fractions of the compressed soft microgel suspensions are ζ ≈ 2−8 for the 1 μm microgels and ζ ≈ 3−12 for the 200 nm microgels. Thus, external osmotic compression leads to strong densification and deswelling of these particles, yielding a state with heavily destretched network chains inside each particle. These unconstrained and flexible chains resemble a semidilute polymer solution with Π ∼ c9/4 scaling of the osmotic pressure. Compressing the plain suspensions of stiff microgel particles leads to a different result. First, the Π(c) curve of the stiff particles is shifted toward higher concentrations, as shown in Figure 2A (× symbols). This is because each stiff particle contains about 3 times more polymer than each soft particle, since the particle syntheses were conducted in a way to obtain soft and stiff microgels with similar sizes in the swollen state. If the microgels are soft, they gain their final size by swelling after the synthesis, so their inner polymer density is low (cp = 2.1 and 3.2 wt %). This is different for the stiff microgels that cannot swell as much. These particles have to be grown to a desired, comparable size by incorporating more monomer into each particle during their synthesis. As a result, their final inner polymer density is higher (cp = 7.1 and 8.3 wt %) than that of

3. RESULTS AND DISCUSSION 3.1. Bulk Osmotic Compression. Dense packing of microgels in a suspension can be achieved by bulk isotropic compression. We follow this approach and densify aqueous microgel suspensions that consist of major fractions of 99, 95, 75, 50, or 25 wt % of soft, 1%-BIS cross-linked pNIPAAm microgels mixed with minor fractions of 1, 5, 25, 50, or 75 wt % of stiff, 10%-BIS cross-linked particles by applying different external osmotic pressures. These mixing ratios correspond to particle-number fractions of 300:1, 60:1, 9:1, 3:1, or 1:1 of the soft and stiff microgels, which can also be expressed in terms of particle-number percentages of 99.7, 98.3, 90, 75, or 50 No.% of soft microgels doped with 0.3, 1.7, 10, 25, or 50 No.% of stiff microgels. The same osmotic compression is also applied to the plain soft-microgel and the plain stiff-microgel suspensions. In 1966

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Figure 2. continued diameter; additional large and small star symbols in panel A denote data obtained from macroscopic 1%-BIS polymer gels and from semidilute solutions of un-cross-linked pNIPAAm. Data in panel B are obtained from 1 μm microgels only, whereas data in panel C are obtained from 200 nm microgels only. Additional star symbols in panel C denote the tracer-chain diffusivity in un-cross-linked semidilute solutions of pNIPAAm at concentrations similar to those in the microgel packings. In all panels, full lines represent power-law trendlines with slopes as expected for macroscopic, space-filling polymer gels or polymer solutions. Dotted lines denote deviations from these trends without specific theoretical fundament and are drawn manually to guide the eye. The dashed part of the −1 trend line in panel C is an extension to the lower c-range.

the soft microgels, and so is the content of polymer in a packing with ζ > 1. Second, the Π(c) scaling of the stiff microgels exhibits a steeper course than that of the soft microgels, as also shown in Figure 2A (× symbols). This finding is addressable to the shorter network strands within their constituent polymer networks, which are more constrained and less flexible than those in the soft microgels. This constraint causes deviation from semidilute-solution-like Π ∼ c9/4 scaling. The latter argument is again supported by consideration of the concentration of polymer after the osmotic compression: the × symbols in Figure 2A reflect polymer concentrations of c = 15−25 wt % (∼150−250 g L−1) in the compressed stiffmicrogel suspensions. This is not considerably higher than the concentration of polymer per individual stiff microgel particle, which is cp = 8.3 wt % (∼83 g L−1) for the 1 μm microgels and cp = 7.1 wt % (∼71 g L−1) for the 200 nm microgels. The effective packing fractions that correspond to this less pronounced increase in concentration are ζ ≈ 1.8−3.5 only. Thus, external osmotic compression does not lead to very strong densification of these stiff particles. It does yield a state with close-packed particles, ζ > 1, but this is not accompanied by heavy destretching of the network chains inside each particle. This behavior persists up to very strong external compression of 500 kPa, where the Π(c) curves of the stiff and soft systems finally converge at a resulting polymer concentration close to c = 25 wt % (∼250 g L−1). The osmotic bulk compressibility of heterogeneous microgel mixtures with particle-number fractions of 300:1, 60:1, 9:1, 3:1, and 1:1 of soft:stiff microgels largely resembles that of plain soft systems. Independent of the content of stiff particles, the Π(c) data of all these heterogeneous mixtures scatter around the same master curve, represented by semidilute-solution-like Π ∼ c9/4 scaling, as shown in Figure 2A (large and small circles, triangles, squares, and diamonds with different grayscale coloring). Data that correspond to the same external osmotic compression (20, 50, 100, 200, or 500 kPa) resemble similar polymer concentrations in all the packed mixtures, varying between c = 7 wt % (∼70 g L−1) for 20 kPa and c = 25 wt % (∼250 g L−1) for 500 kPa external compression. Hence, the presence of micrometer and submicrometer sized heterogeneities has no systematic effect on the bulk compressibility of these composite systems within the limits of data scattering and experimental certainty. In particular, we do not observe any systematic shift of the Π(c) data from the plain-soft to the plain-stiff trend line with increasing content of stiff particles in the mixed systems. This noneffect is even encountered in packings that consist of 1:1 soft and stiff microgels, which are still dominated by the soft-microgel compressibility. To further

Figure 2. Macrogel-type scaling of (A) the osmotic pressure Π, (B) the frequency-independent part of the elastic shear modulus, G0′, and (C) the tracer polymer diffusivity, DTracer, in compressed pNIPAAm microgel suspensions as a function of the degree of microgel packing, expressed in terms of the polymer concentration, c. In the lower plot in panel B, the abscissa is normalized by the average length of the network chains in the microgel polymer networks, Nexp, as compiled in Table 1, times the effective monomer molecular weight, Mm. The inset in the upper plot shows selected frequency sweeps of the elastic and viscous part of the shear moduli, G′ (large symbols, upper data set) and G″ (small symbols, lower data set), of 50 kPa compressed microgel packings. Data symbols are the same in all panels, assigned to the different microgel mixtures in the table at the footer of the figure. In panel A, large symbols denote microgel particles with 1 μm diameter, whereas small symbols denote microgels with 200 nm 1967

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heterogeneous systems. From a fit of the high-packing data to the preceding equation, we determine the values of N for the two homogeneous and the five heterogeneous systems, as compiled in Table 1 (denoted Nexp). With this estimation, we

elucidate this finding, we probe the resulting microgel packings by macro- and microrheology. 3.2. Shear Rheology. To explore the macroscopic mechanical characteristics of the heterogeneous microgel packings under shear, we subject them to oscillatory rheology at low deformation (1%). Frequency-dependent measurements (ω = 0.001−10 rad s−1) reveal that each sample exhibits a frequency-independent plateau in its storage modulus, G0′, as shown exemplarily for 50 kPa compressed microgel packings in the upper inset plot in Figure 2B. These plateau moduli increase with increasing polymer concentration. For macroscopic, space-filling polymer gels, the simplistic theory of rubber elasticity predicts G0 = νkBT = c[NA/(MmN)] kBT,45 with ν the number density of elastically active network strands, N their average number of monomer units, Mm the monomer molecular weight, c the polymer concentration, and NA the Avogadro number. We find that the plateau moduli of the microgel suspensions investigated in this work can be fitted with this prediction at high degree of microgel packing, showing linear scaling of G0′ ≈ G0 with c, as shown in Figure 2B. By contrast, the variation of G0′ with c at lower degree of microgel packing is dominated by a very steep increase, as also shown in Figure 2B. These two different dependencies of G0 on c can be explained by the following rationale: A simple approach to describe the modulus of a microgel suspension is G0(c) ≈ f(c)GParticle, where f(c) is a function that accounts for particle−particle interactions and pair correlations, whereas GParticle is the modulus of the individual microgels. The particle−particle interaction potential has been estimated to be a power law of the interparticle distance, with an exponent that varies with the particle softness;43,44,62−64 this modeling entails a steep power-law dependence for G0 as a function of c in the region above close packing (ζ ≈ 0.5−1), typically G0 ∼ c.4−8 In this regime, particle densification can be viewed to cause progressive particle contact and polyhedral deformation, with Hertzian repulsive forces at interparticle facets.44,65 At higher packing fractions (ζ > 1), however, the suspension microstructure and contact geometry are assumed to be fixed, causing f(c) to be constant. Further densification of the packings is then only possible because the constituent microgel particles can deswell, causing GParticle to increase.44 We estimate GParticle ∼ kBT/Vx, with Vx the volume between polymer network crosslinks. Since Vx ∼ cP−1, we obtain GParticle ∼ cP, and hence, G0 ∼ cP ∼ c. This argument stems from the statistical theory of rubber elasticity;45 thus, compressed microgel packings can be treated very similarly to macroscopic gels, for which this theory predicts G0 = νkBT = c[NA/(MmN)]kBT. It has been shown in a previous work that this equivalence of micro- and macrogels is not only qualitative but also quantitative: comparison of the plateau elastic shear moduli of dense-packed suspensions of small, 1 μm colloidal microgels with that of large, 100−400 μm noncolloidal gel particles shows good agreement, both qualitatively and quantitatively.5 As the large gel particles are not subject to Brownian motion and particle−particle collision, they can be regarded as macroscopic gels, which are therefore shown to be equivalent to densely compressed packings of colloidal microgels. On the basis of the preceding rationale, we focus on the G0′(c) data in the high packing regime and discuss them in view of the simple concept of rubber elasticity. We do this to show that despite its simplicity, this established model can serve as an illustrative tool to explain the elasticity of microgel suspensions in the dense packing limit, including both homogeneous and

Table 1. Average Number of Monomers between CrossLinks in Packings of Microgel Particles Prepared with Different Concentrations of Cross-Linker, 1% and 10%a

Nexp Ncalc

1%-BIS x-linked

mix 300:1

mix 60:1

mix 9:1

mix 3:1

mix 1:1

10%-BIS x-linked

3538

3332 3520

3233 3448

2674 3088

2868 2638

2190 2189

1739

Nexp is obtained by applying the theory of rubber elasticity, G0′ ≈ G0 = c[NA/(MmN)]kBT,45 to experimental estimates of the frequencyindependent part of the elastic modulus of the microgel packings, G0′. Ncalc is calculated by linear combination of Nexp of the homogeneous suspensions, weighted by their relative amount in the heterogeneous suspensions. Nexp is smaller than what can be expected from the monomer-to-cross-linker stoichiometry used to prepare the microgel particles. This discrepancy can be addressed to the low efficieny of cross-linking in free-radical cross-linking copolymerization, which is typically only ∼0.1−10%.9,10 a

can calculate the elastic moduli of the individual microgel particles, again using the formula G = νkBT = c[NA/(MmN)] kBT, with c = cP = 32 g L−1 for the soft and c = cP = 83 g L−1 for the stiff microgels. We obtain GParticle(1 μm, 1% BIS) = 190 Pa and GParticle(1 μm, 10% BIS) = 975 Pa (T = 15 °C). This is in good agreement with literature data on micro-5,44 and macrogels9,10 with similar compositions. It is also in good agreement with G0 of the plain-soft and plain-stiff microgel suspensions at the point where they are compressed to about twice the concentration as the polymer content per individual microgel particle: c = 2cP for the soft and c = 1.5cP for the stiff microgels, as illustrated by the gray boxes in Figure 2B. These points denote the transition from steep, power-law increase of G0(c) to shallow G0 = νkBT = c[NA/(MmN)]kBT, as visualized by the steep dotted lines and the shallow full line in Figure 2B. Thus, we reason that packing fractions of ζ > 2 are needed to cause rubberlike elasticity in microgel packings. This threshold is higher than what has been discussed on the basis of the previous simple picture (ζ ≈ 1). A reason for this discrepancy might be the radially decaying density profile of microgel particles in their unperturbed swollen state, as detailed in section 2.6.57,58 The onset of particle densification causes their loosely cross-linked rims to deswell first,57 thereby delaying the onset of true dense particle contact and rubber-like elasticity of the resulting microgel pastes to higher packing fractions. As the radial dacay of the polymer and cross-linking density is more pronounced in soft than in stiff microgels,57 this delay is more pronounced in the soft (ζRubber‑Elasticity‑Onset ≈ 2) than in the stiff systems (ζRubber‑Elasticity‑Onset ≈ 1.5). The values of N for the heterogeneous microgel packings can be estimated both experimentally (Nexp) and by calculation from those in the homogeneous systems through simple linear combination of the weight fractions of the two different types of microgels within each mixture, as also compiled in Table 1 (denoted Ncalc). These calculated values of N correspond to Nexp within ∼10% error. Rescaling G0′ with Nexp allows us to represent all data on a single master curve with a power-law slope of 1 in the limit of dense microgel packing, as shown in the lower plot in Figure 2B. In this plot, the data from the upper unscaled representation are spread along the abscissa, 1968

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by DTracer ∼ c−1, as shown in Figure 2C (× symbols). In addition, the values of DTracer in the second, stiff-microgel scenario are larger than those in the first, soft-microgel scenario. To understand the difference in the concentration dependence of the tracer-chain diffusivity in soft vs stiff microgel matrixes, it is necessary to consider the length scales that are relevant for the tracer mobility. Modeling how these length scales are affected by the degree of matrix-microgel packing and densification is the key to predicting the tracer-chain diffusivity. Within the concept of reptation, the relaxation of a long linear tracer polymer that diffuses through a network matrix that acts as an array of fixed obstacles66 is constrained by the crosssectional area of the network meshes, ξ2, yielding the tracer chain relaxation time τTracer ∼ ξ−2.75 This formula serves as a basis to derive the dependence of the tracer-chain diffusivity on the polymer matrix concentration. For this purpose, it is necessary to distinguish whether chemical cross-linking or mechanical chain entanglement dominates ξ. If the spatial density of chemical network junctions is much higher than that of mere mechanical chain entanglements or if entanglements are totally absent, then chemical cross-linking sets the scale. In this limit, which has been referred to as “strangulation regime”,76 ξ resembles the spatial distance between the cross-linking junctions, which scales with their concentration as ξx‑link ∼ cx‑link−1/3. For simplicity, we assume cx‑link to be proportional to the molar concentration of crosslinker used to prepare the network. At a fixed ratio of crosslinker to monomer, cx‑link/c = constant, this entails τTracer ∼ ξx‑link−2 ∼ c2/3. This relation can be used to estimate the translational diffusion coefficient:67 DTracer ≈ RTracer2τTracer−1 ∼ (c−1/8)2(c2/3)−1 ∼ c−11/12 in the limit of a good solvent. This prediction is very close to DTracer ∼ c−1, which has been confirmed by experiments on different systems, including polystyrene tracers in poly(methyl methacrylate),75 poly(vinyl methyl ether),77 and polystyrene networks78 and dextran tracers in guar matrixes.79 At low degree of chemical cross-linking, the average spatial distance between mechanical network chain entanglements, ξent, is shorter than that between chemical cross-links. In this case, these entanglements impose the most dominant constraint on the tracer-chain diffusion through the network. In the extreme of total absence of chemical cross-links, this distance translates to the concentration of the network polymer by ξent ∼ c−3/4, which entails τTracer ∼ ξent−2 ∼ c1.5, and thus DTracer ≈ RTracer2τTracer−1 ∼ (c−1/8)2(c1.5)−1 ∼ c−1.75, again in the limit of a good solvent.67,77 In this scenario, the addition of a small fraction of chemical cross-links drastically affects the time scale of relaxation of the network chains, but it has no effect on the limiting length scale ξ, which is still dominated by chain entanglement. Hence, moderate cross-linking does not affect the mobility of flexible tracers that diffuse through this network matrix. This persists up to the point where the spatial density of chemical cross-links exceeds that of matrix chain entanglements, which then leads to the other, cross-link-dominated scenario as discussed above. This idea is supported by earlier measurements of the diffusivity of linear polyacrylamide tracers reptating through semidilute polyacrylamide matrixes that are gradually cross-linked, starting from an un-cross-linked semidilute solution with correlation length ξent and leading to a chemical network with very similar network mesh size, ξx‑link ≈ ξent.40 In the course of this process, the tracer-chain diffusivity was shown to be unaffected by the gradual matrix cross-linking

thereby emphasizing the elasticity of the heterogeneous suspensions to be systematically intermediate between that of the two homogeneous suspensions, with an elastic modulus that linearly increases with the content of stiff particles in the mixtures. In the context of rubber elasticity, this result is explainable by a simple picture of affine deformation of both the soft and stiff parts of these packings. This idea, however, is contradictive to common assumptions on the role of heterogeneities in polymer networks, which are often modeled such that their soft parts store most of the elastic energy upon deformation, whereas their stiff parts are assumed to be largely unaffected by stretching or shearing.12−19 On the other hand, this picture partly ties in with recent investigations by the Yodh group.20,21 These researchers measured the degree of nonaffine response to shear and the elastic moduli of macroscopic gels prepared at different conditions. Their studies reveal that both quantities decrease with increasing concentration of initiator and catalyst used to trigger the polymerization. Other previous investigations show that gels made by fast and uncontrolled polymerization, initiated by large amounts of initiator, are strongly inhomogeneous, whereas those formed by gentle reactions at a lower polymerization rate are more homogeneous.9,25−29 Thus, on the one hand, Yodh’s finding of reduced gel elasticity at higher initiator content during the gel polymerization supports the assumption of less efficient crosslinking in inhomogeneous gels, presuming that most of the cross-links in inhomogeneous gels are lost inside nanogel clusters that do not contribute to the gel overall elasticity. On the other hand, this picture is contradictive to Yodh’s other finding of reduced nonaffinity when gels are made at such less gentle reaction conditions, which was commented to be not understood by the authors.20 This result, however, ties in with our finding of defined reinforcement of microgel packings that are doped with determined fractions of stiff particles. These contradictive results suggest a nontrivial interplay between the microstructural topology of heterogeneous gels and their macroscopic properties. To get deeper insight, we add a microscopic perspective to our discussion by probing the diffusivity of tracer polymers within microgel packings that model inhomogeneous gels. 3.3. Tracer-Chain Diffusivity. In addition to macroscopic rheology, we perform microrheology by monitoring the motion of linear, flexible tracer polymers that diffuse through the heterogeneous microgel packings. These experiments aim to check whether the tracer polymers move according to mechanisms suggested for similar situations in macroscopic polymer gels, which is often discussed in terms of the reptation concept.66−74 We focus on probing the concentration-dependent translational diffusion coefficients of the tracer chains, DTracer(c), and discuss them in view of this theory. The concentration-dependent diffusivity of linear tracer polymers that diffuse through plain soft-microgel packings can be fitted with a power law DTracer ∼ c−1.75 in the entire range studied, as shown in Figure 2C (+ symbols). This quantitative relation is often discussed for situations in which linear tracer polymers diffuse through un-cross-linked semidilute and entangled polymer solutions;68−74 its theoretical basis is the reptation concept that models a tracer chain to be confined within a tube-like array of other chains, in combination with scaling arguments.67 By contrast, a different type of DTracer(c) scaling is observed for the diffusivity of the same tracers through plain stiff-microgel packings. In this case, the concentration-dependent tracer-chain diffusivity is represented 1969

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because ξent is always smaller than the gradually decreasing ξx‑link. In the present work, the tracer-chain diffusion in packed microgel matrixes that consist of soft, 1%-BIS cross-linked particles exhibits DTracer ∼ c−1.75 scaling, as shown in Figure 2C (+ symbols). This result indicates that these soft microgel packings are in the regime where network chain entanglement or contact and not cross-linking dominates the tracer-chain diffusivity. This argument is in agreement with the corresponding Π ∼ c9/4 scaling of the same systems under bulk osmotic compression, which also suggests dominance of a semidilutesolution-like entanglement network rather than dominance of chemical cross-linking. To substantiate this hypothesis, we probe the diffusivity of the same tracer polymer in solutions of an un-cross-linked matrix of linear pNIPAAm chains that have a similar molecular weight as the tracer, covering matrix polymer concentrations of 5, 10, and 15 wt % (about 50, 100, and 150 g L−1). The tracer-chain diffusivities in these semidilute solutions are very similar to that in the soft microgel packings at comparable concentrations and scatter around the same master curve, as shown in Figure 2C (star symbols). This finding supports the argument that chemical cross-links do indeed not impart a major hindrance on the tracer mobility in these systems; instead, topological chain contact and entanglement set the scale. By contrast, the tracer-chain mobility in packed microgel matrixes that consist of stiff, 10%-BIS cross-linked particles exhibits scaling close to DTracer ∼ c−1, as predicted for the regime in which cross-linking dominates, as also shown in Figure 2C (× symbols). This result indicates that the network topology of these stiff systems is dominated by chemical crosslinking rather than by chain entanglement. Again, this rationale is in agreement with the behavior of these systems under bulk osmotic compression, where they do not show semidilutesolution-like scaling, but a compressibility that is dominated by the much higher cross-linking density of their constituent stiff microgels. A particularly puzzling outcome of the above studies is that the tracer-chain diffusivities are faster in the stiff-microgel packings than in the soft-microgel packings. This finding is contradictive to the intuitive expectation of strong confinement of the tracer chains in the cross-link-dominated “strangulation regime”.76 This counterintuitive result can be addressed to the different extent of network chain destretching in the compressed microgel packings. In the soft-microgel packings, the polymer network strands are so destretched that they behave similar to un-cross-linked, fully relaxed, and flexible chain segments in a semidilute solution, as supported by the semidilute-solution-like scaling of the osmotic pressure in these systems, Π ∼ c9/4. By contrast, the network strands in the densely cross-linked stiff-microgel packings are not relaxed to such a degree, which is a consequence of their shorter length. These network chains are therefore constrained and less flexible, causing a steeper scaling of Π(c). In further consequence, these stiff network chains cannot fluctuate as freely as the unconstrained network strands in the soft microgels, thereby occupying less volume. This causes less pronounced obstruction of the diffusing tracer chains in the stiff-microgel packings than in the soft-microgel packings. An alternative view on this matter is to consider the tracer-chain diffusion to be subject to thermodynamic and excluded-volume interactions with the surrounding network matrix, causing D to be a mutual diffusion coefficient, Dm = (1/f m(c))(dΠ/dc), with

f m the concentration-dependent mutual friction coefficient and Π the osmotic pressure.80 The magnitude of Dm is determined by both thermodynamic (dΠ/dc) and hydrodynamic (1/f m(c)) effects, and since dΠ/dc is higher in the stiff-microgel packings than in the soft-microgel packings (cf. Figure 2A), Dm might be higher in the stiff packings, too. The stiff-microgel packings cover a limited range in the higher concentration domain only (∼15−25 wt %). This is a consequence of the low compressibility and the high polymer content of their constituent microgel particles. As a result, the DTracer ∼ c−1 scaling that corresponds to these stiff-microgel systems can be fitted in this limited high-c range only. If the −1 power-law trend line is extended to the lower concentration range, it intersects single low-c data points of other systems that do not fall into their respective dense-packed regime, as represented by the dashed part of the line in Figure 2C. In the plot of the corresponding rheology data (Figure 2B), the same samples contribute data points that do not collapse on the +1 power master curve; instead, they belong to the lower-laying regime indicated by the dotted very steep trend lines. In this limit, no pronounced microgel compression is achieved yet. Thus, the constraining length scale ξ in these systems cannot be ξent but must be ξx‑link, in agreement with the collapse of these data on the extension of the DTracer ∼ c−1 scaling in Figure 2C. In the very low concentration limit, the −1 and the −1.75 power-law trend lines converge; this occurs at a polymer concentration of about 5 wt %, which denotes the verge of dense packing and the onset of network strain destretching in the soft-microgel packings upon further compression. From this point on, when further compression and microgel densification is applied, the DTracer ∼ c−1.75 scaling in the soft systems leads to a more pronounced decrease of DTracer than the DTracer ∼ c−1 scaling in the stiff systems. As a result, the tracer diffusivity in the soft compressed systems is slower than that in the corresponding stiff systems. The preceding results show that the two microgel systems used in this work are very different. Thus, mixing these two types of microgel yields interesting heterogeneous systems composed of a loosely cross-linked background that contains domains that are cross-linked much heavier. The tracer-chain diffusivity inside these two types of domains is expected to be affected very differently by microgel compression and accompanying increase of the matrix polymer concentration: whereas it decreases with DTracer ∼ c−1.75 in the soft-microgel domains, it decreases with only DTracer ∼ c−1 in the stiffmicrogel domains if both are probed in nonmixed, plain states. Thus, a simplistic expectation is that in mixtures the soft, strongly compressed and ξent-dominated domains exert a more pronounced hindrance on the tracer mobility than the stiff ξx‑link-dominated domains. This argument suggests that dopingin fractions of stiff microgels should entail faster tracer-chain diffusion. To check on whether the latter expectation is true, we subject the heterogeneous mixtures of soft and stiff packed microgels with mixing ratios from 300:1 up to 1:1 to the same type of FRAP experiments. We observe that the tracer-chain mobility within all these mixtures largely resembles that in plain softmicrogel suspensions. All DTracer(c) data of the heterogeneous systems scatter around the same master curve, together with the data of the plain soft systems, and all these data can be represented by semidilute-solution-like D ∼ c−1.75 scaling, as shown in Figure 2C (circles, triangles, squares, and diamonds with different grayscale coloring). Hence, the presence of 1970

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densely cross-linked heterogeneities does not systematically affect the tracer-chain diffusivity in these systems within the margins of data scattering and experimental certainty. Just like in osmometry, there is certainly no systematic trend of the intermediate data to gradually transition between the plain-soft and the plain-stiff data sets. The latter finding is in contrast to the simplistic expectation that adding inhomogeneities to the compressed microgel packings should create domains of less pronounced obstruction and therefore accelerate the tracer-chain diffusion. However, this result is again in agreement with the osmometry data of these composite systems, which all scatter around a master curve represented by semidilute-solution-like Π ∼ c9/4 scaling. The common feature of these two different types of experimental situations is that they probe the microgel packings at rest. By contrast, shear rheology probes the same samples under anisotropic deformation, wherein which the presence of built-in heterogeneities has a measurable effect. Thus, the impact of spatial inhomogeneities in soft polymer gel systems appears to be dependent on the type and extent of stress exerted to the system. These ambiguous findings tie in with earlier studies on the impact of spatial polymer network inhomogeneities on the extent of light and neutron scattering. It has been shown that inhomogeneous networks do not exhibit pronounced excess scattering if they are deswollen. This is because in such states the built-in heterogeneities are buried within the deswollen network. By contrast, if the network chains unfold and stretch by swelling, network domains of high local cross-link density are uncovered and cause excess neutron and light scattering.6,8 A similar situation is encountered in this present work: the strong osmotic compression causes marked microgel deswelling, as assessed by the effective microgel packing fractions, which are all ζ > 1. As a result, the strong topological folding of the constituent network strands diminishes the impact of builtin network inhomogeneities if the gels are probed in compressed states at rest. By contrast, shear deformation leads to partial network chain unfolding and stretching, thereby pronouncing the built-in gel heterogeneities.

In conclusion, the impact of spatial inhomogeneities in soft polymer gel systems appears to be dependent on the type and extent of external stress. This rationale ties in with earlier findings on the extent of light and neutron scattering by gels, which strongly differs depending on the degree of swellinginduced network-chain unfolding. Thus, polymer network inhomogeneities may or may not be disregarded, depending on whether equilibrium or deformed states considered. The present study focuses on heterogeneous gel compositions of up to 1:1 of soft and stiff building blocks. A future point of investigation can be to go beyond this ratio and investigate whether a percolation effect occurs at the composition whereupon composite gel systems contain more stiff than soft domains. For such future work, the present concept of microgel building-block assembly provides a simple and straightforward basis. The same strategy can also serve to create other types of custom heterogeneities, not only with respect to their local elasticity but also with respect to their size, chemical composition, and spatial distribution. In addition, it is another advantage of the present approach that it provides expedient means to incorporate tracer objects such as linear polymer chains or tagged nanoparticles into the resulting systems without complications caused by unwanted chemical linking of these tracers to the gel matrixes.81 All these benefits render this strategy promising for further systematic work.

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ASSOCIATED CONTENT

S Supporting Information *

Experimental proof of absence of wall slip and interparticle slip in shear rheology. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone +49 30 8062 42294; e-mail sebastian.seiffert@ helmholtz-berlin.de. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS FRAP experiments were performed at Clausthal University of Technology, kindly hosted by Prof. W. Oppermann (Institute of Physical Chemistry). Tracer polymers were synthesized by S. Hackelbusch (FU Berlin). We thank Joris Sprakel and Jasper van der Gucht (Wageningen University) for inspiring discussions. We also thank the referees of this manuscript for their detailed, valuable, and stimulating criticism. This project was funded by the Focus Area NanoScale at FU Berlin, which is gratefully acknowledged. S. Seiffert is a Liebig Fellow of the Fund of the Chemical Industry. F. Di Lorenzo is a doctoral student of the Helmholtz Virtual Institute “Multifunctional Biomaterials for Medicine”.

4. CONCLUSIONS AND OUTLOOK The role of spatial inhomogeneties in polymer gels that are modeled by dense-packed, heterogeneous microgel assemblies depends on the viewpoint. When these model systems are probed at rest with respect to their bulk compressibility and to the diffusivity of linear tracer polymers inside them, the presence of strongly cross-linked local domains does not show any considerable effect within the limits of data scattering and experimental certainty. Instead, both quantities resemble those in plain loosely cross-linked systems. This effect persists even up to mixing ratios of 1:1 of soft and stiff microgels. By contrast, the same heterogeneities show a measurable effect if the microgel packings are probed in deformed states by shear rheology: here, the frequency-independent elastic moduli of the heterogeneous microgel packings are reflected by linear combination of the moduli of their constituent plain soft and stiff components within ∼10% error, in agreement with the simplistic concept of rubber elasticity. This finding is inconsistent with common assumptions on the impact of spatial network inhomogeneities on the macro-12−19 and microscopic23,24 properties of polymer gels, but it partly ties in with other recent work.20,21

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REFERENCES

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