Elastic and Flow Properties of Densely Packed Binary Microgel

Nov 29, 2018 - Densely packed soft microgels (pastes) behave as a yield-stress fluid. They act as soft elastic solids with finite equilibrium shear mo...
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Elastic and Flow Properties of Densely Packed Binary Microgel Mixtures with Size and Stiffness Disparities Ayaki Nakaishi,† Saori Minami,† Shun Oura,‡ Takumi Watanabe,‡ Daisuke Suzuki,*,‡,§ and Kenji Urayama*,† †

Department of Macromolecular Science and Engineering, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan Graduate School of Textile Science & Technology and §Division of Smart Textile, Institute for Fiber Engineering, Interdisciplinary Cluster for Cutting Edge Research, Shinshu University, Ueda 386-8567, Japan

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‡

S Supporting Information *

ABSTRACT: Densely packed soft microgels (pastes) behave as a yieldstress fluid. They act as soft elastic solids with finite equilibrium shear modulus (G0) below a critical stress (σc) whereas they flow like liquids above σc. The effects of size and stiffness heterogeneities in the constituent microgels on the rheological properties of the pastes are revealed using the binary microgel mixtures by an oscillatory rheometer and diffusive wave spectroscopy (DWS). The binary blends with various degrees of size and stiffness disparities are made by mixing the pastes with the same apparent particle-volume fraction (ϕeff) at various values of relative weight fraction of soft microgels (fsoft). The G0−fsoft relations for the soft/hard microgel mixtures are significantly influenced by size disparity: The relations for small size disparities well obey the logarithmic mixing rule, while those for large size disparities have a wide fsoft regime in which G0 are almost equal to those of the single small-microgel pastes (G0,small), regardless of whether the small microgels are soft or hard. The characteristic fsoft region with G0 ≈ G0,small for the mixtures with large size disparities is attributed to the developed continuous phase of small microgels (overwhelming in number) where the large microgels are discretely dispersed. The steady-state flow behavior of the binary pastes above σc obey the classical Hershel−Bulkley (H−B) equation. In each binary paste, the characteristic time (τcage) of the fast local dynamics of microgels trapped in the densely packed structures evaluated from DWS is close to the characteristic time (τHB) obtained from the parameters in the H−B equation and G0. This agreement shows that the dynamics of the positional rearrangement of microgels in the steady-state flow is closely related to the fast local dynamics in the quiescent state of the pastes, independently of the size and stiffness disparities in the constituent microgels. are allowed to move locally only in the “cage” made by the surrounding microgels. The pastes flow like liquids above σc, accompanied by the positional rearrangement of the individual microgels. These rheological properties of the pastes have potential applications as paints, coating, textured foods, and cosmetics. The size and stiffness of microgels are tunable by synthetic modifications. The stiffness of microgels is primarily governed by cross-link density. As cross-link density increases, the microgels become stiffer and the degree of swelling decreases. The effects of the particle stiffness and size on the rheological properties of pastes have been extensively investigated using a single microgel component.9,13,17−25 The stiffness of the constituent microgel significantly influences the equilibrium shear modulus (G0) of the pastes. When compared at the same ϕ, G0 and σc of the pastes increase with an increase in cross-link density whereas they are insensitive to the particle size.18,25 The modulus G0 increases with ϕ by a power law with a large

1. INTRODUCTION Microgel particles are cross-linked polymer networks with micrometer or submicrometer diameter that are highly swollen by solvents.1−7 Unlike rigid particles, they are mechanically soft and deformable, and they have a fuzzy surface with dangling chains. In addition, their volume and molecule binding are fast responsive to a change in environment due to their small dimensions. Microgels can be stabilized in a dispersion medium, and the suspensions with sufficiently low particle concentrations are low viscosity fluids. Characteristically, the soft-microgel suspensions can be concentrated far beyond the threshold volume fraction of random close packing which is estimated from the dimension in the isolated state (ϕc ≈ 0.64).8−10 High concentrations of ϕ exceeding unity can be attained because microgels can undergo elastic deformation, interpenetration, and deswelling when they contact with each other.11,12 The dense packing with ϕ ≫ ϕc prohibits the free diffusion of microgels, which changes dramatically the rheological properties of the suspensions. Such dense microgel suspensions, which are often called “pastes” or “microgel glasses”, are non-Newtonian fluids with yield stress (σc).13−16 The pastes behave as soft elastic solids at sufficiently small stresses below σc where the microgels © XXXX American Chemical Society

Received: July 30, 2018 Revised: November 15, 2018

A

DOI: 10.1021/acs.macromol.8b01625 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 1. Characteristics of Microgels code

NIPAM (mol %)

BIS (mol %)

SDS (mM)

KPS (mM)

polymerization temp (°C)

Dh (nm)

CV (%)

[η] (mL/g)

170-44 180-46 280-28 360-63 560-30 690-47 840-17 880-16 960-16 1100-18 810-29 1200-72

95 95 95 99 98 98 95 95 95 95 98 99.9

5 5 5 1 2 2 5 5 5 5 2 0.1

2 2 0.5 0.5 0.1 0 0 0 0 0 0 0

2 2 2 2 2 2 1 1 1 1 2 2

70 70 70 70 70 70 45→70 45→70 45→70 45→70 45→70 45→70

174 175 284 355 562 685 838 877 956 1052 806 1204

0.30 2.6 0.31 0.14 1.2 0.70 0.19 3.6 2.6 4.4 3.5 2.4

44.3 45.8 28.1 62.5 29.9 46.6 17.4 15.6 16.2 17.8 29.2 72.2

exponent, and the increase becomes gentler in very high ϕ regime.26−28 The steep rise of G0 with ϕ has been mainly attributed to an increase in repulsive interparticle force caused by an increase in the extent and number of flat faceting via Hertzian-like contact. The gentle increase of G0 with ϕ at high ϕ has been explained by the osmotic deswelling (densification). Recent studies using confocal fluorescence microscopy11 and superresolution microscopy12 have observed directly that the dominant event changes from the facet formation to the deswelling as ϕ increases. It has been known that the steady-state flow behavior above ϕc for various types of paste is well described by the empirical Herschel−Bulkley equations: σ = σc + kγ̇ where k and γ̇ are the constant and shear rate, respectively.29,30 Seth et al.30 proposed a constitutive model to describe the flow behavior of the pastes by considering the elastohydrodynamic slippage between the deformed microgels. This constitutive model provides a similar σ−γ̇ relation to the Herschel−Bulkley equations, which shows the importance of the elastic stress associated with the positional rearrangements of microgels under flow. In this model, the characteristic time for the elastohydrodynamic lubrication is given by the ratio of solvent viscosity and particle stiffness. These earlier studies demonstrate that the rheological properties of the pastes are controllable by ϕ and particle stiffness, but most of the results are limited to the pastes made of a single microgel component. In the case of hard-particle suspensions, the size distribution has pronounced effects on the rheological response, and the size-distribution effects have often been investigated using binary mixtures of spherical particles experimentally,31−37 theoretically,38−40 and by simulations.41 The viscosity of binary hard-sphere suspensions is not a function of only the total particle volume fraction (ϕtotal), unlike that of monodisperse suspensions. When ϕtotal is kept constant, the rheological properties of binary hard-sphere suspensions considerably depend on the size and mixing ratios. If the size disparity is sufficiently large, the viscosity of the binary mixtures shows a minimum at a relative volume fraction of small particles, which is known as the Farris effect.42 In such binary suspensions, the small particles can fill the space between the large particles, resulting in an increase in maximum packing fraction. A considerable decrease in the reduced elastic modulus, the yield strain, and stress as well as even melting of the glass to a fluid was reported for the concentrated suspensions of binary hard-sphere mixtures with large size disparities.36 In the case of binary suspensions of soft microgels, the disparity can be varied in stiffness as well as size. This unique feature further extends the controllability of the rheological

properties of pastes. When the microgels with considerably different stiffness contact with each other, the softer microgels are largely deformed whereas the stiffer ones undergo almost no deformation. The modulus G0 of the binary mixture pastes at constant ϕtotal is expected to depend on not only the relative volume fraction of soft component (ϕsoft) but also the stiffness ratio of the soft and hard ones (RG). The size ratio in hard and soft microgels (RDH/S = Dhard/Dsoft where Dhard and Dsoft are the diameters of the hard and soft ones, respectively) is also expected to have pronounced effects on G0 because the difference in ϕsoft and the relative number fraction of soft microgels (Nsmall) becomes larger with an increase in size disparity. The number fraction Nsmall is directly related to the number of the contacts between soft and hard microgels, and it also influences the dispersion of the spatial volumes occupied by large microgels. Thus, RG and RDH/S can be governing parameters for rheological response of the binary microgel pastes, in addition to ϕtotal and ϕsoft. The binary mixtures with the size and stiffness disparities are qualitatively classified into two types of combination, i.e., large-soft/small-hard and largehard/small-soft microgels. Little research has been performed to study the effects of RG and RDH/S on rheological properties of binary mixture pastes. Seiffert et al.27,43 investigated G0 for the binary mixture pastes with soft and hard microgels with similar size. They reported that the values of G0 remain almost unchanged in the range of ϕsoft > 0.5 when compared at the same total gel volume fraction. They argued that the dense pastes store elastic energy only at soft-microgel interfaces when subjected to shear deformation because only the soft microgels are deformed by faceting whereas the hard ones sandwiched between the soft ones are almost undeformed. However, much remains unclear and unknown about how R G and R D H/S influence the ϕ soft dependence of G0 of the pastes. Our previous study44 investigated the yielding behavior of the pastes including the binary pastes, but with only a single set of RG and RDH/S. Sprakel et al.45 reported the steady-state flow behavior of the binary dense microgel suspensions with various ϕtotal, but only at a fixed mixing ratio and with a single set of RG and RDH/S, because they mainly focused on the heterogeneous flow through narrow microchannels. Thus, the effects of size and stiffness disparities on the yielding and steady-state flow behavior still remain to be elucidated. The present study surveys the elastic and flow properties of the binary pastes with different degrees of RG and RDH/S which include various types of combination (large-soft/small-hard, large-hard/small-soft, soft/hard with similar size, and large/soft B

DOI: 10.1021/acs.macromol.8b01625 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules with similar stiffness). The binary pastes with randomly disordered structure are made by mixing the two components with the same ϕ. The linear viscoelasticity of the binary pastes is investigated using concertedly the oscillatory rheometry and diffusive wave spectroscopy (DWS) which cover a broad range of dynamics from the equilibrium to the fast local dynamics of microgels trapped in densely packed structures. The present results provide a definite basis to discuss how the size and stiffness disparities affect various rheological aspects of the microgel pastes including elasticity, yielding, and flow properties. They will also give more in-depth information about the tuning of the rheological properties of the pastes on the basis of molecular design of microgels.

varied from 0 to 1. The types of combination and the sample codes for the mixture pastes are listed in Table 2, and the corresponding

2. EXPERIMENTS

small-hard/ large-soft

Table 2. Characteristics of Binary Microgel Mixtures combination large/small: almost same stiffness hard/soft: almost similar size

2.1. Materials. N-Isopropylacrylamide (denoted as NIPAM, purity 98%), N,N′-methylenebis(acrylamide) (BIS, 97%), sodium dodecyl sulfate (SDS, 95%), and potassium peroxodisulfate (KPS, 95%) were purchased from Wako Pure Chemical Industries (Osaka, Japan) and used as received. The water used for the synthesis and purification was first distilled and then ion-exchanged (EYELA, SA-2100E1, Tokyo, Japan) before use. 2.2. Synthesis of PNIPAM Microgels. The chemically crosslinked PNIPAM microgels were synthesized by conventional aqueous free radical precipitation polymerization.46,47 The polymerization condition and monomer composition are shown in Table 1. First, a mixture of NIPAM, the cross-linker (BIS), and water was poured into a four-neck round-bottom flask equipped with a mechanical stirrer, a condenser, and a nitrogen gas inlet. The total monomer concentration was fixed at 150 mM in each case. Next, the monomer solutions were heated at 70 °C, stirred at 250 rpm, and sparged with nitrogen gas for 30 min to remove any oxygen dissolved in water. The appropriate amount of SDS was added as required. Then, 5 mL of initiator solution (KPS) was added to start the polymerization. The polymerization was allowed to proceed for 4 h; the obtained microgel dispersion was then cooled to stop the polymerization using an ice bath. To remove unreacted chemicals and impurities, the resultant microgel suspension was purified by two centrifugations (Avanti J-26SXP, Beckman Coulter Inc., or Himac CS 100 GX, Hitachi Koki Co., Ltd.). Furthermore, the suspensions were purified by dialysis for at least 1 week with daily changes of water. Moreover, the larger PNIPAM microgels (Dh > 800 nm) were prepared by temperature-controlled precipitation polymerization.48 Typically, the PNIPAm monomer and cross-linker (BIS) were dissolved in water. The monomer solution was heated to 45 °C and sparged with nitrogen gas for 30 min. Subsequently, the polymerization was started by addition of KPS solution (5 mL). After that, the temperature was increased from 45 to 70 °C using a temperature gradient of 1 °C/3 min. The polymerization was allowed to proceed for 4 h after addition of KPS solution and then cooled with an ice bath to stop the reaction. The obtained microgel suspension was purified by two centrifugations (Avanti J-26SXP, Beckman Coulter Inc.). Table 1 summarizes the characteristics of each microgel. In the sample code X-Y, X and Y denote the hydrodynamic diameter (Dh) and intrinsic viscosity ([η]) at 25 °C for the microgel, respectively. According to its definition, [η] of the microgels is proportional to the degree of swelling, with the result that [η] tends to decrease with increasing the cross-link density. 2.3. Preparation of Binary Mixture Pastes. Each microgel suspension was concentrated to obtain the paste by centrifugation (Himac CS120GX, HITACHI) at 25 °C. The concentration of the samples was controlled by adding deionized water. Before mixing, the particle concentration (c) of each microgel paste was adjusted so that the effective volume fraction of the microgels (ϕeff), which is defined in the later section, could be equal to be 1.1 at 25 °C unless otherwise stated. The binary blends were made by mixing the pastes with the same ϕeff (= 1.1) at various mixing ratios. The relative weight fraction of the softer microgel paste in the mixtures, which is denoted by fsoft, was

large-hard/ small-soft

a

RDH/S

RGH/S

components (hard/soft)

RDsmall/large

L/S-I

4.3

1.4

1200-72/280-28

0.23

H/S-I

0.81

560-30/690-47

0.81

H/S-II H/S-III SH/LS-I

0.68 0.68 0.23

5.3 4.0 3.3

810-29/1200-72a 810-29/1200-72 280-28/1200-72

0.68 0.68 0.23

SH/LS-II LH/SS-I

0.47 2.7

9.6 22

560-30/1200-72 960-16/360-63

0.47 0.37

LH/SS-II LH/SS-III LH/SS-IV LH/SS-V

3.1 4.9 4.9 6.1

5.6 55 35 22

1100-18/360-63 880-16/180-46 840-17/170-44b 1100-18/180-46

0.32 0.20 0.20 0.16

code

20

ϕ = 1.0. bϕ = 1.4.

Figure 1. Schematics of the binary pastes with four types of combination: (i) large/small microgels with same stiffness (L/S); (ii) hard/soft microgels with same size (H/S); (iii) small-hard/large-soft microgels (SH/LS); (iv) large-hard/small-soft microgels (LH/SS). schematics are shown in Figure 1. The mixture pastes are characterized by the two parameters: the diameter ratio RDH/S = DH/DS and the S stiffness ratio RGH/S = GH 0 /G0. The sub- and superscripts H and S S indicate hard and soft components, respectively; GH 0 and G0 denote the equilibrium shear modulus of the corresponding single-microgel pastes at ϕeff = 1.1. In most of the discussion of the results, we employ RDH/S rather than RDsmall/large = Dsmall/Dlarge where Dsmall and Dlarge are the diameters of the small and large microgels, respectively. The value of RDsmall/large for each mixture paste is also listed in Table 2. 2.4. Measurements. The hydrodynamic diameter D h for submicrometer-scale microgels was evaluated at 25 °C by dynamic light scattering (DLS, Malvern Instruments; Zetasizer NanoS, Malvern, C

DOI: 10.1021/acs.macromol.8b01625 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules UK). The DLS measurements were conducted for dilute suspensions with particle concentration range between 1.4 × 10−5 and 6.8 × 10−5 g mL−1. The ionic strength of samples was adjusted to 1 mM using NaCl solutions. We averaged the results of 15 independent measurements of the intensity autocorrelation acquired over 30 s. The Dh values were calculated on the basis of the measured diffusion coefficients using the Stokes−Einstein equation (Malvern, Zetasizer software v. 6.12). For all measurements, the sample was equilibrated for 600 s at a set temperature before the measurements. The experiment was performed in triplicate, and the average of the results was used for further analysis. For micrometer-scale microgels, D was estimated by optical microscopy (BX51, Olympus). The morphologies of each microgel were evaluated by transmission electron microscopy (TEM, JEOL JEM-2100, operated at 200 kV). The each microgel dispersion (2 μL), which was purified by centrifugation with pure water, was dried on a carbon-coated copper grid at room temperature. The intrinsic viscosity ([η]) of each microgel at 25 °C was evaluated from the viscosities of the sufficiently diluted suspensions with an Ubbelohde viscometer using Huggins and Mead−Fuoss plots. Based on the particle volume in the isolated state, the effective volume fraction of the particles (ϕeff) in the pastes was evaluated by ϕeff = c[η]/2.5, where c is a particle concentration.49 For the pastes with high concentrations, ϕeff deviates from the real volume fraction due to deformation and deswelling of microgels. As commonly done for the concentrated microgel suspensions,50−53 this study uses ϕeff as a measure of the degree of packing. The macroscopic viscoelasticity of microgel pastes was measured at 25 °C using a stress-controlled AR-G2 rheometer (TA Instruments, New Castle, DE) with a cone−plate geometry with a diameter of 40 mm and a cone angle of 1°. A solvent trap was placed around the sample to minimize water evaporation during the measurements. No appreciable slip between the samples and plates was confirmed by the agreement of the results in the tests with and without waterproof sandpaper on the plate surface. Diffusive wave spectroscopy (DWS RheoLab, LS Instruments) in the transmission geometry was employed to investigate the fast local dynamics of the pastes. TiO2 particles with a diameter 360 nm which strongly scatter light were used as tracer particles. The measurements were made at 25 °C after the specimens in the cells were equilibrated. The concentration of tracer particles (typically 1 wt %) was adjusted to satisfy the condition of multiple scattering, i.e., 7 < L/l* < 30 where l* is transport mean free paths and L (= 2 mm) is the cell thickness. The value of l*, which reflects the turbidity of the specimen, was calculated by comparing the transmitted light intensity to a calibration standard. As shown below, the DWS results obtained reflect the macroscopic dynamic viscoelasticity of the pastes.

Figure 2. TEM images of the microgels in the dry state: (a) 280-28, (b) 560-30 (c) 690-47, and (d) 1100-18.

Figure 3. Microphotographs of the binary mixture pastes of large-hard/ small-soft microgels (1100-18/180-46) of ϕeff = 1.1 with various values of relative weight fraction of large microgels (f large): (a) 1.0, (b) 0.75, (c) 0.50, and (d) 0.25. Disordered structure but with no appreciable clustering of each component is observed at every value of f large. The micrograph of the single small-microgel paste (f large = 0) is not displayed here because the structure cannot be characterized due to the small size below the resolution limit.

3. RESULTS 3.1. Size Uniformity of Microgels. The TEM images of the microgels 280-28, 560-30, 690-47, and 1100-18 in the dried state are shown in Figure 2. The values of CV for the diameter obtained from the DLS measurements, which are listed in Table 1, are 0.7 for LH/SS-III. These results indicate that the f soft dependence of G0 is significantly influenced by RDH/S when the two components have considerably different stiffness (RGH/S ≫ 1). 3.5. Steady-State Flow. The steady-state flow behavior of the mixture pastes is investigated as a function of shear stress (σ) at σ > σc. Figure 6 shows the σ dependence of shear rate (γ̇) for LH/SS-I, LH/SS-II, and SH/LS-I with various values of fsoft. The yield stress σc of each paste, which is shown by the arrow in the figure, is defined as the lowest stress reaching the equilibrium in the flow measurements. The values of σc decrease with increasing fsoft, and they are close to the values of σc obtained from the large-amplitude oscillatory measurements (Figure 4a). When σ is larger than σc, the pastes flow and reach a steady shear state. The flow behavior in the steady shear state (σ > σc) is considerably nonlinear. The flow curves are well described by the Herschel−Bulkley equation:29 σ = σc + kγ ṁ

Figure 4. (a) σa dependence of G′, G′′, and γa and (b) ω dependence of G′ and G′′ for H/S-I with ϕeff = 1.1 and fsoft = 0.5.

(1)

where m is a flow behavior index and k is a consistency coefficient with a unit of Pa·sm. The solid lines in the figures represent the fitted results, and the fitted values of σc, k, and m for each specimen are listed in Table 4. The flow curves and the fitted results for other specimens (H/S-I, SH/LS-II, and LH/SSIII) are shown in Figure S1 of the Supporting Information and Table 3. The exponent m (m = 0.48), which is constant for all mixture pastes, is close to the values of m ≈ 1/2 reported for the single-microgel pastes.29,57 In each mixture paste, k decreases with increasing fsoft. Figure 7 illustrates the double-logarithmic plots of σc versus G0 for all mixture pastes. All data collapse into a single relation which is represented by a straight line with a slope of unity. This result with the relation of σc = G0γc shows that the value of γc is almost constant (γc ≈ 4%) regardless of fsoft, RDH/S, and RGH/S, which can be seen in the inset of Figure 7 as well as in Tables 3 and 4. Our previous paper44 reported that the values of γc of the microgel pastes are primarily governed by the chemical component of microgel surface and that they are almost insensitive to the diameter and its distribution and stiffness. The present results further confirm the almost constancy of γc for the binary pastes with large size and stiffness disparities. Equation 1 is transformed into the following form with the dimensionless shear rate [(k/G0)1/mγ̇] and shear stress (σ/σc):

pastes as functions of stress amplitude (σa) and angular frequency (ω). In the linear region where σa is sufficiently small, the storage modulus (G′) is much larger than the loss modulus (G′′), and G′ is independent of σa and ω. In this regime, the strain amplitude (γa) is proportional to σa, which corresponds to the Hookean behavior of elastic solids. The plateau value of G′ in Figure 4b allows us to evaluate the equilibrium shear modulus (G0), and G0 is determined from the value of G′ at ω = 0.1 s−1. When σa increases, the log γa−log σa relationship shows a definite upward inflection at certain values of σa (σc) and γa (γc) as a result of yielding (Figure 4a), which is qualitatively similar to the earlier observations for various types of pastes.8,44,55 The modulus G′ steeply drops at around σc, and the magnitudes of G′ and G′′ are reversed when σa is sufficiently larger than σc. The values of σc and γc are regarded as yield stress and strain, respectively. The similar features are observed for all specimens. The loss modulus appears to exhibit a minimum at ω ≈ 1 s−1, although the data at low frequencies are considerably scattered. The similar relaxation was observed in several studies,27,56,57 and it was attributed to the relaxation processes associated with very slow structural rearrangements of the microgels. The values of G0, σc, and γc are evaluated for the mixture pastes with various values of RDH/S and RGH/S, and they are tabulated in Tables 3 and 4. 3.4. Equilibrium Shear Modulus. Figure 5a shows the semilogarithmic plots of G0 versus fsoft for the two simple types of

m li y1/ m | o o ojj k zz σ 1o o o =1+ o mjj zz γ o }̇ o σc γc o ojk G0 z{ o n ~

E

(2) DOI: 10.1021/acs.macromol.8b01625 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 3. Viscoelastic Parameters of Binary Microgel Pastes flow curve

LAOS H/S-I

SH/LS-II

LH/SS-III

fsoft

σc (Pa)

γc (%)

G0 (102 Pa)

σc (Pa)

m

k (Pa·sm)

τHB (10−4 s)

0 0.20 0.33 0.50 0.67 0.80 1 0 0.20 0.30 0.50 0.60 0.80 1 0 0.10 0.15 0.25 0.33 0.50 0.67 0.80 1

60 47 34 20 11 6.9 3.8 41 39 32 26 16 10 4.5 130 120 75 24 16 5.7 3.3 3.2 2.4

3.5 4.7 4.7 4.6 4.8 4.4 4.5 4.0 4.5 4.2 4.9 4.8 5.2 4.2 4.1 4.5 4.2 4.0 3.7 3.6 3.7 3.9 4.2

17 10 7.3 4.3 2.2 1.6 0.84 10 8.6 7.5 5.3 3.4 1.9 1.1 32 26 18 6.0 4.3 1.6 0.90 0.82 0.58

76 59 43 23 13 8.9 4.2 50 40 34 34 23 13 5.5 120 110 80 31 20 5.5 3.1 2.8 1.8

0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48

18 15 11 7.5 4.5 3.7 2.1 12 11 10 9.7 7.3 4.9 2.8 85 60 42 12 6.4 2.8 1.7 1.5 1.2

0.75 1.6 1.6 2.2 2.9 4.1 4.5 0.87 1.1 1.2 2.4 3.3 4.7 4.7 5.2 3.9 4.1 2.9 1.6 2.3 2.6 2.2 3.1

−4

−4

Table 4. Viscoelastic Parameters of Binary Microgel Pastes flow curve

LAOS LH/SS-I

LH/SS-II

SH/LS-I

fsoft

σc (Pa)

γc (%)

G0 (10 Pa)

σc (Pa)

m

k (Pa·s )

250 61 38 21 9.7 60 26 21 16 10 24 23 18 9.5 7.1

5.0 4.8 4.8 5.2 4.3 5.0 4.5 5.2 4.9 4.6 4.3 5.3 5.0 4.9 4.3

49 13 8.0 4.0 2.3 13 5.5 4.8 3.4 2.3 5.4 4.4 3.5 1.9 1.7

320 78 45 35 11 64 25 25 17 11 24 25 19 10 7.0

0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48

78 26 14 7.8 4.3 33 11 9.0 5.5 4.0 6.0 6.0 5.5 4.3 4.2

1/ m

τHB

m

τHB (10

1.8 3.0 2.0 2.7 2.6 4.9 2.6 2.5 1.9 2.2 0.84 1.4 1.7 3.6 4.3

s)

τcage (10 2.0 3.0 3.5 3.0 2.6 5.1 4.0 3.1 2.2 2.9 1.1 1.5 2.2 3.7 6.0

s)

δ (nm)

β

1.6 3.0 2.7 4.2 4.3 2.3 2.9 3.6 4.5 4.2 3.0 3.6 3.9 5.0 5.2

0.42 0.43 0.44 0.42 0.42 0.41 0.43 0.44 0.43 0.42 0.48 0.49 0.47 0.45 0.41

the mechanism of nonlinear steady-state flow, although they affect the values of σc, G0, and k. The similar type of universal flow curves using a constitutive model was reported for the single microgel pastes.30 3.6. Microrheology. As described above, the characteristic times for the steady-state flow (τHB) are on the order of 10−4 s. The short time scale of τHB implies that τHB is related to the fast local dynamics of the pastes. The corresponding angular frequency ω, which is given by τHB−1, is far beyond the accessible ω range in conventional oscillatory rheometer. Cloitre et al.29 showed that the local dynamics of the single-microgel pastes could be characterized by DWS. The fast local dynamics of the mixture pastes is investigated by DWS. The inset of Figure 9a shows a typical ensemble average autocorrelation function g2(t). A nonzero plateau of g2(t) at long times indicates that the

where the relation σc = G0γc is used. The quantity (k/G0)1/m has a unit of time, and it is regarded as a characteristic time for the steady-state flow (τHB):

ij k yz = jjj zzz j G0 z k {

DWS

0 0.30 0.50 0.8 1 0 0.20 0.50 0.70 1 0 0.30 0.50 0.80 1

2

(3)

The value of τHB for each specimen is listed in Tables 3 and 4. The values of τHB for most of the samples are on the order of 10−4 s. Figure 8 illustrates the reduced flow curves using the dimensionless shear rate and shear stress for the data in Figure 6. All the reduced curves collapse into a universal curve. The solid line in the figure corresponds to eq 2 with γc = 0.038 and m = 0.48. This result indicates that the parameters of the mixture pastes such as fsoft, RDH/S, and RGH/S have no significant effect on F

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Macromolecules

Figure 5. Semilogarithmic plots of G0 versus fsoft for (a) L/S-I and H/SI and (b) SH/LS-II and LH/SS-III. The lines are guides for the eyes.

tracers cannot diffuse freely whereas they are capable of Brownian motion only within cages made by densely packed microgels. A decay of g2(t) at short times represents the local motions of tracers in the cages. The mean-square displacement (MSD) of tracers (⟨Δr2⟩) is obtained from g2(t) using the formalism for the transmission geometry.58 Figure 9 displays the t dependence of ⟨Δr2⟩ for LH/SS-I, LH/SS-II, and SH/LS-I with various values of fsoft. The time profiles of ⟨Δr2⟩ for all specimens are well fitted by the following stretched exponential function:29 ⟨Δr 2⟩ = δ 2{1 − exp[−(t /τcage)β ]}

Figure 6. σ dependence of γ̇ for (a) LH/SS-I, (b) LH/SS-II, and (c) SH/LS-I with various values of fsoft. The solid lines represent the fitted results by eq 1.

(4)

where τcage is the characteristic time for local relaxation of tracers in the cage. The fitted value of each parameter is summarized in Table 4. The parameter δ represents the maximum displacement of the tracers, reflecting the cage size. The values of δ, which correspond to the maximum MSD, are on the order of nanometers, indicating that the local motion is limited in a range much smaller than the microgel diameters. The exponent β is a measure of the distribution of relaxation times.

between log G0 and fsoft (Figure 5a), indicating that G0 obeys the logarithmic mixing rule as log G0 = fsoft log G0S + fhard log G0H

GS0

(5)

GH 0

where and denote G0 of the pure pastes of soft or hard microgels, respectively, and f hard = 1 − fsoft. The G0 data of H/S-II and H/S-III with the small size disparities of RDH/S = 0.68 also obey eq 5, which is shown in the Supporting Information. The logarithmic mixing rule is a well-known empirical equation,59 and it describes some physical properties of mixture and composite systems.60,61 The serial mixing rule corresponding to the isostress condition for the two components is expressed as

4. DISCUSSION 4.1. Equilibrium Shear Modulus. The size disparities pronouncedly affect the fsoft dependence of G0 for the mixtures with finite stiffness disparities (Figure 5). The mixtures H/S-I with a small size disparity of RDH/S = 0.81 exhibit a linear relation G

DOI: 10.1021/acs.macromol.8b01625 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 7. G0 dependence of σc and for mixture pastes. The inset shows the fsoft dependence of γc.

Figure 8. σ/σc dependence of γ̇τHB for SH/LS-II, LH/SS-III, and H/S-I with various values of fsoft. The solid line depicts eq 2 with γc = 0.038 and m = 0.48.

f f 1 = softS + hard G0 G0H G0

Figure 9. t dependence of ⟨Δr2⟩ for (a) LH/SS-I, (b) LH/SS-II, and (c) SH/LS-I with various values of fsoft. The solid lines represent the fitted results by eq 4. The inset in (a) shows an ensemble-averaged autocorrelation function for fsoft = 0.8.

(6)

The parallel mixing rule satisfying the isostrain condition is given by G0 = fsoft G0S + fhard G0H

results using the following general Lichtenecker mixing rule with a flexible parameter α:62

(7)

Figure 10a shows the comparison of the data with the expectations by these three mixture rules. In the entire fsoft range, the parallel mixing rule overestimates G0 whereas the serial one underestimates. The logarithmic mixing rule, which well describes the data, is intermediate between these two rules. For SH/LS-II and LH/SS-III with large size disparities (RDH/S = 0.47 and 4.9, respectively), the log G0−fsoft relations are significantly nonlinear (Figure 5b). Figure 10b displays the fitted

G0α = fsoft (G0S)α + fhard (G0H)α

(8)

The best-fit results for the data of SH/LS-II and LH/SS-III are obtained at α = 0.60 and −0.36, respectively. In eq 8, the parallel and serial mixing rules correspond to α = 1 and 1, respectively, while α → 0 results in the logarithmic mixing rule. The parameter α characterizes the direction and degree of the curvature in the log G0−fsoft relations: The positive and negative H

DOI: 10.1021/acs.macromol.8b01625 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

(logarithmic mixing rule) for RDH/S ≈ 1 and α > 0 for RDH/S < 1 (where soft microgels are larger), while α < 0 at RDH/S > 1 (where soft microgels are smaller). In each regime, the absolute values |α| tend to increase as the size disparity increases. The sign of α depends on whether the quasi-plateau regions in the G0−fsoft relations are located in high or low fsoft region: G0 ≈ GS0 at high D fsoft for RDH/S > 1, and G0 ≈ GH 0 at low fsoft for R H/S < 1. Both quasi-plateau moduli are the moduli of the corresponding single small-microgel pastes (G0small), regardless of whether the small microgels are soft or hard. This feature is more evident in Figure 12a where the G0 data for the mixtures with the large size disparities of RDH/S = 0.47 and 4.9 are replotted against the weight fraction of small microgels (fsmall). Both types of mixture in the high fsmall region of fsmall > 0.7. This exhibit G0 ≈ Gsmall 0 indicates that the small microgels form the developed continuous phase which governs totally the modulus of the mixtures. In the corresponding fsmall region, the number fractions of large microgels (Nlarge) are extremely small (