Multiple dynamic regimes in concentrated microgel systems
David A. Sessoms, Irmgard Bischofberger, Luca Cipelletti, Véronique Trappe
aa r X i v : . [ c ond - m a t . s o f t ] M a y Multiple dynamic regimes in concentratedmicrogel systems
By David A. Sessoms , Irmgard Bischofberger , Luca Cipelletti , andV´eronique Trappe
1. Department of Physics, University of Fribourg, Switzerland2. LCVN UMR5587, University of Montpellier 2 and CNRS, France
We investigate dynamical heterogeneities in the collective relaxation of a concen-trated microgel system, for which the packing fraction can be conveniently varied bychanging the temperature. The packing fraction dependent mechanical propertiesare characterised by a fluid-solid transition, where the system properties switch froma viscous to an elastic low-frequency behaviour. Approaching this transition frombelow, we find that the range ξ of spatial correlations in the dynamics increases.Beyond this transition, ξ reaches a maximum, extending over the entire observablesystem size of ∼ ξ on volume fraction is reminiscent of the behaviour recently observed at the jam-ming/rigidity transition in granular systems (Lechenault et al. Keywords: microgels, soft colloids, glasses, jamming, heterogeneous dynamics,photon correlation imaging
1. Introduction
Soft deformable colloids, such as foam bubbles, emulsion droplets, star polymersand microgels can be efficiently packed beyond the close packing conditions of hardspheres. At such high packing fraction, the constituents of the system touch eachother, exerting direct forces on each other, which results in their deformation. Con-sequently we expect the dynamics in such a ‘squeezed’ state to be mainly determinedby force balances. Any imbalance in the forces, i.e. stresses, will lead to motion andthus to reconfiguration of the system. Indeed, we can presume that the dynamicsof deformable spheres is predominantly determined by either stress-imbalances ortemperature depending on their packing fractions, and the goal of this work is toassess some of the differences between stress-driven and thermally-driven dynamics.
Phil. Trans. R. Soc. A 367, 5013-5032 (2009)
TEX Paper
D. A. Sessoms and others -6 -5 -4 -3 -2 -1 j / s g G p / ( / R ) Figure 1. Volume fraction dependence of the mechanical properties of deformable spheresinferred from data sets obtained for hard spheres in the lower volume fraction range(dashed line) (Meeker et al. et al. η seemingly diverges at the glasstransition volume fraction, φ g , which is also characterised by the onset of low-frequencyelasticity G p . Beyond φ g , G p increases quickly towards the jamming transition (randomclose packing) φ j . Beyond φ j , the elasticity increases nearly linearly with ( φ − φ j ). The lowshear viscosity is normalized by the solvent viscosity η s , while the modulus is normalizedby the modulus intrinsic to the emulsion droplet, σ/R , with σ the surface tension and R the radius of the drop. Our investigation is motivated by the assumption that the high packing fractionbehaviour of repulsive deformable spheres is determined by two critical conditions:a) the glass transition at which the particles become so densely packed that the dif-fusion of the particles past their nearest neighbours becomes highly improbable onan experimental time scale and b) the jamming transition at which the particles arejammed together, forming a network of direct contacts. That both transitions maybe significant can be inferred from the combined datasets obtained by respectivelyMeeker et al. (1997) and Mason et al. (1995).Meeker et al. (1997) investigated the low shear viscosity η of hard sphere col-loidal systems. According to their investigations, the low shear viscosity exhibits anapparent divergence at the experimental glass transition φ g = 0 .
58, where they findthat an unconstrained fit of the Krieger-Dougherty equation, η /η s = (1 − φ/φ c ) − β ,to their data yield the parameters φ c = 0 .
55 and β = 1 .
8, where η s is the viscosity ofthe background medium; these fit results are shown as a dashed line in fig. 1. Mason et al. (1995) investigated the elasticity of dense emulsions beyond φ c = 0 .
55; thevolume fraction dependence of the elastic modulus G p normalized by the ratio ofthe surface tension σ to droplet radius R is shown as a solid line in fig. 1. Accordingto their investigations, the volume fraction dependence is determined by both theexperimental glass transition at φ g and the jamming transition at φ j ∼ .
64. For φ g < φ < φ j , the elasticity increases quickly with increasing volume fraction, whilefor φ > φ j the elasticity increases approximately linearly with φ − φ j . Reflectingthe squeezed nature of the system at very high φ , the elasticity converges towardsthe elasticity intrinsic to emulsion droplets, i.e. σ/R , for φ approaching unity (Prin-cen 1983). Inferring that the emulsion droplets would behave like the hard spheresystem below the glass transition, we therefore expect three states for a deformable Phil. Trans. R. Soc. A 367, 5013-5032 (2009) ynamics of concentrated microgel systems φ < φ g the system behaves like a supercooled fluid, at φ g < φ < φ j thesystem is in a glassy state, at φ > φ j the system is in a squeezed state.In this paper we present our work on the dynamics of thermosensitive micro-gels, as an example of deformable spheres, and we show that these three states arecharacterised by a distinct behaviour of dynamical heterogeneity. More precisely,spatial correlations of the dynamics are relatively short-ranged and of small am-plitude in the supercooled state, grow dramatically in the glassy state, where theyspan the whole system, and finally become more localized, but of large amplitude,in the squeezed state.
2. Sample characteristics: conformational and mechanicalproperties
We use aqueous solutions of poly-N-isopropylacrylamide (PNiPAM) microgels, whichexhibit a lower critical solution temperature (LCST) at ∼ ◦ C (Senff & Richter-ing 1999). Below the LCST, the dimensions of our PNiPAM microgels exhibit aremarkable sensitivity to temperature, which enables us to vary the volume frac-tion of a given system by a factor of 1.7 by varying the temperature from 20 ◦ C to30 ◦ C. Though very convenient for the study of the high volume fraction behaviour,working with microgel solutions calls for some caution, if the aim is to compare theirbehaviour to other soft sphere systems like foams and emulsions. Indeed, microgelsolutions are solutions and not dispersions; they are in thermodynamic equilibriumwith the solvent and do not possess a well defined interface. They thus can partlyinterpenetrate and/or compress under their own osmotic pressure. Despite this en-hanced difficulty, their unique temperature sensitivity bears two main advantages:the volume fraction can be varied by changing the temperature and, more impor-tantly, the sample history can be controlled uniquely by temperature, with no needto apply a preshear to the sample, as is typically required by colloidal systems.Indeed, the study of the dynamics of highly concentrated systems, in particular ofglassy and squeezed systems, can tremendously suffer from an ill-controlled historyof the sample. The thermosensitive microgel systems offer the unique advantage toallow for the preparation of the system in a fluid-like state at low volume fraction(high temperature), which is subsequently quenched to the solid-like states by in-flating the particle (decreasing the temperature), so as to obtain a higher volumefraction. This protocol is not only convenient to use in almost any experimentalset-up, it is to a certain extent more meaningful than the application of a highshear stress or rate, which is generally used to erase the sample history (Cloitre etal. et al. ◦ Cand reduced pressure using a rotary evaporator. This stock solution is then used toproduce more dilute samples. The actual concentration of the stock is determinedby drying a defined solution volume and determining the residual amount of mi-crogels to obtain both a concentration in weight% and a concentration in g/ml,where the latter is defined relative to the solution volume at a room temperature of
Phil. Trans. R. Soc. A 367, 5013-5032 (2009)
D. A. Sessoms and others ∼ ◦ C. Due to the use of an ionic initiator, our microgels are charged. To screenthese charges, we add a solution of sodium-thiocyanate, NaSCN, to all our PNiPAMsolutions, so that the final salt concentration in our systems is 0 .
03 M.We characterise the temperature dependent conformational properties of ourmicrogels at a concentration of 4 · − g / ml by static and dynamic light scatteringusing a commercial light scattering apparatus. We characterise both salt-free andsalted microgel solutions to ascertain that the salt does not significantly changethe solubility of the microgels. Both systems exhibit essentially the same temper-ature dependence for T smaller than the lower critical solution temperature T c .Increasing the temperature leads to a gradual decrease of the particle size up to T c , where we find that the temperature dependence of the hydrodynamic radius R h and the radius of gyration R g is well approximated by a critical-like functionof form R g,h = A ( T c − T ) a . The resulting fit parameters for R h are T c = 33 . ◦ C, A = 86 . a = 0 . R g , we find T c = 33 . ◦ C, A = 46 . a = 0 . T < T c (Senff & Richtering 2000; Clara Rahola 2007). This is due to the unevendistribution of cross-linkers, the cross-linking density being larger in the centre ofthe microgels than on the edges (Mason & Lin 2005). Because of the higher densityof the core, the core scatters significantly more than the lower density edges, suchthat we essentially measure the radius of the highly cross-linked core in static lightscattering, while we capture the hydrodynamically effective radius of the particle indynamic light scattering. As the temperature approaches the critical solution tem-perature, the overall density of the microgel increases, such that R h approaches R g .Beyond the LCST, the characterisation of the microgel dimensions is only possiblefor the salt-free solution, where the unscreened charges of the particles prevent ag-gregation. In this range of temperature we find that R g /R h ∼ = 0 .
75, near the valueexpected for hard sphere colloids (Dhont 1996).Near the LCST, a change in temperature affects both intra- and inter-molecularinteractions of the microgels (Wu et al. η of microgel solutions in the range ofconcentrations of 1 . · − g / ml to 4 . · − g / ml. Our rheological measurementsare performed in the lower concentration range (1 . · − g / ml to 1 . · − g / ml)with an Ubbelohde viscometer; in the higher concentration range (8 . · − g / mlto 4 . · − g / ml) we use a commercial stress-strain rheometer equipped with acone and plate geometry (cone and plate radius R = 25 mm, cone angle α = 1 . ◦ ).To ensure the best possible temperature control and to avoid evaporation in therheometer, we use a temperature hood combined with a solvent trap (Sato & Breed-veld 2005). This guarantees that our samples will not exhibit evaporation-inducedchanges over an experimental time window of 2–3 hours. At higher concentrations,the history of our samples is carefully controlled using the following protocol: thesamples are equilibrated in a fluid state at 32 ◦ C, where we additionally apply ashear rate of ˙ γ = 1000 s − to fully erase any previous quench history. The temper-ature is then decreased to the desired temperature and equilibrated at this tem-perature for 500 s prior to any experiment, where we have tested that our systems Phil. Trans. R. Soc. A 367, 5013-5032 (2009) ynamics of concentrated microgel systems -3 -2 -1 -3 -2 -1 -1 / s c [g/ml] (a) (V h (T) / V h (20(cid:176)C)) *c [g/ml] (b) nom (c) Figure 2. (a) Concentration dependence of the relative low shear viscosity η /η s of microgelsolutions at temperatures ranging from 20 ◦ C–30 ◦ C; from left to right: T = 20, 22, 24, 26,28, 30 ◦ C. (b) Renormalizing the concentration by the degree of shrinkage of the microgelparticles with temperature results in a single master curve. (c) Assuming a voluminosityof k = 17 . / g at T = 20 ◦ C, c is converted into a nominal volume fraction φ nom . Alllines denote critical-like divergences. do not exhibit any significant changes in their rheological properties beyond thiswaiting time.The low shear viscosity is determined in shear rate experiments, where we rampthe shear rate from ˙ γ = 10 − s − – 10 s − , depending on temperature and microgelconcentration. For each shear rate, we maintain ˙ γ for a sufficiently long time to allowfor the system to reach steady state. The steady state viscosity η ( ˙ γ ) exhibits a shearrate dependence that is typical of colloidal suspensions. At low concentrations andhigh temperature, we find that η ( ˙ γ ) exhibits almost no dependence on ˙ γ ; in thesecases we determine η as the mean of η ( ˙ γ ), η = h η ( ˙ γ ) i . At higher concentrationsand lower temperatures the system exhibit shear thinning behaviour; in these caseswe obtain η by fitting η ( ˙ γ ) to the Cross-equation, η ( ˙ γ ) = η ∞ + η − η ∞ C ˙ γ ) m (2.1)where η ∞ is the high shear viscosity and 1 /C is a measure of the crossover shearrate denoting the onset of shear thinning.In fig. 2(a) we report the relative viscosity η /η s as a function of concentration c .The measurements are performed at temperatures ranging from 20 ◦ C to 30 ◦ C. Ourdata exhibit distinct variations as the temperature is changed. At any given tem-perature, we find that η /η s increases dramatically with concentration, exhibitinga critical-like divergence at some critical concentration. This critical concentrationsystematically shifts to higher values as the temperature is increased. That this be-haviour is entirely due to the change in the particle volume is shown in fig. 2(b). Asimple renormalisation of the concentration by the ratio V h ( T ) /V h (20 ◦ C) collapsesthe data onto a single master curve, where we use V h (20 ◦ C) ≈ R h as a referencevolume, such that the concentration axis refers to the one obtained at 20 ◦ C. Thisalmost perfect collapse demonstrates the equivalence between varying the volumefraction by changing the particle concentration and varying the volume fraction bychanging the particle volume via temperature (Senff & Richtering 1999, 2000). Forour microgels, this equivalence holds up to T = 30 ◦ C; for temperatures larger than
Phil. Trans. R. Soc. A 367, 5013-5032 (2009)
D. A. Sessoms and others T = 30 ◦ C, the concentration dependence of the viscosity exhibits deviations fromthe scaling behaviour, reflecting a change in the particle-particle interactions. Ourmaster-curve is well approximated by a critical-like function, η /η s = (1 − c/c c ) α ,with a critical concentration of c c · V h ( T ) /V h (20 ◦ C) = 0 .
035 g / ml and a criticalexponent of α = 2 .
25. To express concentrations in terms of volume fractions, weassume a voluminosity of k = 17 . / g for our systems at T = 20 ◦ C, which we useto define a nominal volume fraction according to φ nom = k · c · V h ( T ) /V h (20 ◦ C).This voluminosity is chosen based on different mapping techniques, including theone where the low concentration dependence of η /η s is mapped to the behaviourexpected for hard spheres (Senff & Richtering 1999, 2000). However, we find thatdifferent approaches can lead to a variation in k of almost 20%, such that the in-dicated nominal volume fraction φ nom can only be regarded as an approximategauge of the volume fraction. Nonetheless, for convenience we use φ nom insteadof c · V h ( T ) /V h (20 ◦ C) to indicate our concentrations. For the concentration andtemperature dependence of η /η s this results in the dependence shown in fig. 2(c),where the critical divergence according to η /η s = (1 − φ nom /φ nom,c ) α occurs at φ nom,c = 0 . . / ml;this system covers the nominal volume fraction range 0 . ≤ φ nom ≤ .
81 inthe temperature range 30 ◦ C ≥ T ≥ ◦ C. To gain insight in the temperature-dependent mechanical behaviour of this system, we perform an oscillatory shearexperiment at a fixed angular frequency ( ω = 10 rad / s) and fixed strain ( γ = 0 . ◦ C to 20 ◦ C at a rate of 0 . ◦ C / s, where we en-sure that the strain of γ = 0 .
002 is well within the linear viscoelastic regime at alltemperatures investigated. Converting concentration and temperature into nominalvolume fractions enables us to report the φ nom -dependence of the loss modulus G ′′ and storage modulus G ′ in fig. 3, where we indicate the critical condition obtainedfrom the divergence of the low shear viscosity as a solid vertical line. In the volumefraction range below φ nom,c , we find that G ′′ initially dominates over G ′ ; with in-creasing φ nom , G ′ increases more quickly than G ′′ and eventually dominates over G ′′ . Indeed, upon approach of the critical conditions we expect the characteristicrelaxation time to increase dramatically. Accordingly, the characteristic frequencydenoting the cross-over from the high frequency elasticity behaviour to the low fre-quency viscous behaviour systematically shifts to lower frequencies. In a test wherewe probe the mechanical response at a given frequency, we thus expect to observe atransition from a regime in which G ′′ dominates, to a regime in which G ′ dominates,consistent with the observed behaviour.That our system indeed undergoes a fluid to solid transition at φ nom,c can beseen in the development of the frequency dependent responses of this system asthe temperature is decreased. At T = 29 ◦ C ( φ nom = 0 .
53; fig. 3(a)) the materialresponse function is essentially determined by viscous losses; G ′′ increases nearlylinearly with frequency, the magnitude of G ′′ ( ω ) /ω is consistent with the low shearviscosity measured in steady shear experiments. Decreasing the temperature to T = 27 ◦ C ( φ nom = 0 .
61; fig. 3(b)), the material response function exhibits thetypical features of a viscoelastic fluid; the response is characterised by a cross-overfrequency beyond which G ′ is the dominating modulus, while G ′′ is dominatingover G ′ at lower frequencies. Decreasing the temperature further to T = 26 ◦ C Phil. Trans. R. Soc. A 367, 5013-5032 (2009) ynamics of concentrated microgel systems -1 nom,c (f)(e)(d)(c)(b) G ’ () , G " () [ P a ] nom (a) -1 -1 (b) G ’ () , G " () [ P a ] (a) -1 (d) (e) (f)(c) -1 -1 -1 G ’ () , G " () [ P a ] s (T) / s (20(cid:176)C) -1 s (T) / s (20(cid:176)C) -1 s (T) / s (20(cid:176)C) Figure 3. Development of storage G ′ (solid symbols) and loss modulus G ′′ (open symbols)during a temperature ramp from 30 ◦ C to 20 ◦ C for a microgel system with c = 0 . / ml.Additional results obtained for a microgel system with c = 0 . / ml are shown asdashed underlying lines. The measurements are performed at a constant frequency of ω = 10 rad / s and a constant strain of γ = 0 . G ′ and G ′′ for: (a) c = 0 . / ml at 29 ◦ C corresponding to φ nom = 0 .
53; (b) c = 0 . / ml at27 ◦ C corresponding to φ nom = 0.61; (c) c = 0 . / ml at 26 ◦ C corresponding to φ nom =0.65; (d) c = 0 . / ml at 24 . ◦ C corresponding to φ nom = 0.69; (e) c = 0 . / mlat 20 ◦ C corresponding to φ nom = 0.81; (f) c = 0 . / ml at 20 ◦ C corresponding to φ nom = 0.91. The solid line in each graph corresponds to the fit of the data shown in (a).To account for the temperature dependence of the background viscosity the frequenciesare normalised by the ratio of the solvent viscosity to the solvent viscosity at 20 ◦ C. ( φ nom = 0 .
65; fig. 3(c)) finally leads to a response which is typical for soft glassymaterials: G ′ is dominating over G ′′ in the entire frequency range investigated and isnearly frequency independent; G ′′ exhibits a minimum, indicative of some residualslow relaxation process.Beyond φ nom,c , G ′ increases with increasing φ nom , where we can take G ′ ( ω =10 rad / s), shown in the main graph of fig. 3, as a measure of the plateau modulus G p . Though the increase in G p does not exhibit any further characteristic feature Phil. Trans. R. Soc. A 367, 5013-5032 (2009)
D. A. Sessoms and others that may indicate a change in the samples mechanical properties, the frequencydependence of G ′ and G ′′ qualitatively changes as we increase φ nom beyond ∼ . c = 0 . / ml. The results obtained from the temperature ramp from 30 ◦ Cto 20 ◦ C are shown as dashed underlying lines in fig. 3, demonstrating the exten-sion of the concentration range to φ nom = 0 .
91. By increasing the φ nom from 0.69(fig. 3(d)) to 0.81 (fig. 3(e)) to 0.91 (fig. 3(f)), we observe that the difference be-tween G ′ and G ′′ decreases with increasing φ nom ; the minimum in G ′′ graduallydisappears; at φ nom = 0 . G ′ and G ′′ exhibit essentially the same frequency depen-dence. Moreover, while the loss modulus in the high frequency range has a similarmagnitude in the investigated range of 0 . ≤ φ nom ≤ .
69, they significantly in-crease beyond φ nom ∼ .
7. To show this, we report in fig. 3(a)-(f) the fit describingthe data at φ nom = 0 .
53, where we account for the temperature dependence ofthe background viscosity by normalising the frequencies with η s ( T ) /η s (20 ◦ C), theratio of the solvent viscosity at the experimental temperature to the one at 20 ◦ C.The change in the shape of the material response function in conjunction withthe increase of the loss modulus in the high frequency range seemingly indicatesthat for φ nom > . G p on φ nom re-sembles the one of emulsions beyond random close packing (see fig. 1). Thus, weseemingly miss the range where the mechanical behaviour is defined by a strongincrease of G p between φ g and φ j . Such behaviour is in agreement with results ob-tained for similar PNiPAM-microgels (Senff & Richtering 1999; Senff & Richtering2000) and related core-shell particles (Senff et al. et al. et al. G p between φ g and φ j has been ob-served for non-deformable hard sphere colloids (Le Grand & Petekidis 2008). Thus,while emulsions still exhibit hard sphere behaviour between φ g and φ j , the softnessof microgels and core-shell particles determines the behaviour over the entire rangeof volume fraction above φ g .
3. Spatially and temporally resolved collective dynamics
We investigate the collective dynamics of our microgel system with c = 0 . / ml,spanning the transitional range 0 . ≤ φ nom ≤ .
81. To obtain the dynamics withtemporal and spatial resolution, we use the recently introduced photon correlationimaging (PCIm) technique applied to a small angle light scattering experiment asdescribed by Duri et al. (2009). Coherent laser light with an in vacuo wavelength of
Phil. Trans. R. Soc. A 367, 5013-5032 (2009) ynamics of concentrated microgel systems λ = 633 nm illuminates the sample at normal incidence. We use a lens with a focallength of f L = 72 . r a = 7 . q ; in our case q = 1 µ m − . The magnification factor M ≃ ×
483 pixels, corresponding to a rectangular observation window ofapproximately 5 . × . within the sample.To follow the space and time-dependent fluctuations in the dynamics of oursystem, we record the space-resolved speckle images in time, where we use a cameraexposure time of 2 . .
167 Hzdepending on the sample dynamics. Any change of the sample configuration resultsin a change of the speckle pattern, which we quantify by calculating a space andtime resolved intensity correlation function. We divide the full image into regions ofinterest (ROIs) of 37 ×
37 pixels corresponding to 350 × µ m within the sample.The local degree of correlation c I ( t w , τ, r ) between two images taken at time t w and t w + τ and for a ROI centered around the position r is calculated according to: c I ( t w , τ, r ) = 1 β h I p ( t w ) I p ( t w + τ ) i ROI ( r ) h I p ( t w ) i ROI ( r ) h I p ( t w + τ ) i ROI ( r ) − I p is the intensity measured at a single pixel, h . . . i p ∈ ROI ( r ) representsan average over pixels belonging to a ROI centered in r , and the coefficient β is a coherence factor that depends on the speckle to pixel size ratio (Goodman1984); it is chosen so that c I ( t w , τ, r ) → τ →
0. The spatially averaged buttime-resolved degree of correlation, c I ( t w , τ ), is obtained by taking the averagesover a single ROI encompassing the full image. A further average over time yieldsthe usual intensity correlation function g ( τ ) − g ( τ ) − c I ( t w , τ ), where . . . denotes an average over t w . Thespatially resolved information on the change of configuration between t w and t w + τ can be conveniently visualized by constructing a “dynamic activity map (DAM)”.Representative dynamical activity maps of our sample at different temperaturesare shown in fig. 7; a physical interpretation of these maps will be given later.Each metapixel of the dynamical activity map corresponds to a ROI and its coloursymbolises the local degree of correlation, which we indicate in units of the standarddeviation of c I ( t w , τ, r ), σ (Duri et al. Phil. Trans. R. Soc. A 367, 5013-5032 (2009) D. A. Sessoms and others c I ( t w , ) t w [s] (a) t w [s] (b) Figure 4. Time evolution of the spatially-averaged degree of correlation c I ( t w , τ ) for atemperature quench from T = 32 ◦ C to (a) T = 28 . ◦ C ( φ nom = 0 .
57) and to (b) T = 24 . ◦ C ( φ nom = 0 . τ = 0 . τ = 3 s,150 s, 600 s, 1500 s, 3000 s, 5100 s, 9000 s, 15000 s, 27000 s and 42000 s. In both cases,there is a transient increase in c I ( t w , τ ), which exceeds the duration of the temperaturequench of 50–100 s. The time needed to reach a quasi-stationary behaviour is significantlylonger for φ nom > φ nom,c than for φ nom < φ nom,c . reached, the control is extremely stable; the sample temperature is recorded duringthe experiments and the fluctuations rarely exceed ± . ◦ C. All interfaces arelightly coated with a thermal grease to ensure efficient heat transfer.Prior to any experiment, our sample is equilibrated at 32 ◦ C to ensure full flu-idization as a starting condition. The moment we lower T to the desired experi-mental temperature is defined as t w = 0. The temperature is typically stabilizedat the set temperature within 50–100s, depending on the temperature differencebetween starting and set temperature. We follow the evolution of the dynamics ofour samples by recording the speckle pattern, subsequently calculating the spatiallyaveraged instantaneous degree of correlation c I ( t w , τ ) at different lag times. Repre-sentative examples of the evolution after the quench for respectively φ nom < φ nom,c (28 . ◦ C, φ nom = 0 .
57) and φ nom > φ nom,c (24 . ◦ C, φ nom = 0 .
69) are shownin fig. 4(a) and (b). In both cases c I ( t w , τ ) increases after the quench to thenreach a quasi-stationary behaviour, indicating that the system is characterised bya well-defined slow relaxation process for both φ nom < φ nom,c and φ nom > φ nom,c .Though this is not surprising for φ nom < φ nom,c , where the system rheology isdefined by a low shear viscosity (fluid-like behaviour), it is somewhat unexpectedfor φ nom > φ nom,c , where the system rheology is defined by a low frequency elas-ticity (solid-like behaviour). Indeed, beyond φ nom,c , we would in principle expectthat any residual slow dynamics would continuously evolve with t w , exhibiting thetypical characteristics of ageing solid-like systems (van Megen et al. et al. et al. φ nom,c both lead to a quasi-stationary slow dynamics, we find distinct differences in theevolutionary behaviour between the two conditions. While the quasi-stationary be-haviour is reached after ≈
500 s for φ nom < φ nom,c , we typically need to wait 20000– 30000 seconds to reach the quasi-stationary behaviour for φ nom > φ nom,c . These Phil. Trans. R. Soc. A 367, 5013-5032 (2009) ynamics of concentrated microgel systems -1 g ( ) - [s] Figure 5. Mean autocorrelation function determined by taking time averages of c I ( t w , τ ) over the time windows where the dynamics of the system is quasi-station-ary. From left to right T = 29 . , . , . , . , . , . , . , . ◦ C; accordingly φ nom = 0 . , . , . , . , . , . , . , .
81. Open symbols denote systems that ex-hibit a low shear viscosity, closed symbols denote systems that exhibit a low frequencyelastic modulus. The solid line indicates g ( τ ) − .
95, the degree of correlation cho-sen for the investigation of the temporal and spatial fluctuations in the dynamics of oursystem. differences in the waiting time seemingly reflect the differences between the sit-uation in which the system is quenched to a state where the individual particlediffusion is nearly suppressed (supercooled fluid) and the one where the system isquenched to a state where the individual particle diffusion is suppressed (glassysystem).The quasi-stationary dynamics enables us to characterise the mean dynamicsof our system at different volume fractions by determining the space and time av-eraged correlation functions g ( τ ) − c I ( t w , τ ) over the time window where weobserve the dynamics to be quasi-stationary. As shown in fig. 5, g ( τ ) − φ nom ; the characteristic decay times evolve from ≈
10 s to ≈ ( τ ) − ( τ ) − . ◦ C and speckle images taken at the set temperature. Indeed, this temper-ature jump leads to a complete reconfiguration of the system and thus guaranteesthat our procedure yields the lowest possible degree of correlation, which we useas a measure of the baseline value. As we are working at low q , we presume thatwe capture the entire short time relaxation spectrum within the time window ac-cessible with the camera, such that we normalize the intercept of our correlationfunctions to one by using the g ( τ ) − τ = 0.The mean correlation functions exhibit a not well developed two-step decayover a wide range of φ nom , which we attribute to instabilities in our experimentalset-up. Indeed, it is worth recalling that our experiments are performed at low q ,well within the q -range where we probe collective diffusion. Estimating the exper-imental observational length-scale as 2 π/q = 6 . µ m, we find that we are probing Phil. Trans. R. Soc. A 367, 5013-5032 (2009) D. A. Sessoms and others a length-scale that corresponds to ≈
30 particle diameters. The probed dynamicsthus entails fluctuations of the refractive index on length-scales large compared tothe particle size. We therefore do not expect to observe the typical features of glassydynamics probed at length scales comparable to the particle size, where the fastlocal diffusion of a particle trapped in a cage of nearest neighbours and the slowstructural rearrangement of the cage it-self lead to a two-step decay in the intensitycorrelation function (van Megen & Underwood 1993; van Megen et al. q the displacements of the particles diffusing within the cages of nearest neighboursare insufficient to lead to any significant dephasing of the light and thus are notcontributing to the decay of the intensity correlation function. Instead, we expectthe temporal evolution of the density fluctuations to be the main cause for a decayin g ( τ ) − q .To quantify these dynamics, we choose to focus on the data obtained at lagtimes where g ( τ ) − c I ( t w , τ ) = 0 .
95, as marked by the horizontal linein fig. 5. Though somewhat arbitrary, this choice is based on the following con-siderations. When analyzing the characteristic decay of a correlation function to p g ( τ ) − /e , we generally consider that the system has to reconfigure on alength scale of ∼ π/q . For a significantly smaller amount of dephasing, like theone chosen, we can think of the lengthscale over which things have to reconfigureto be reduced. As mentioned before, 2 π/q is in our experiment rather large, cor-responding to ≈
30 particle diameters; by monitoring the fluctuations in c I witha mean of c I ( t w , τ ) = 0 .
95, we expect to resolve heterogeneities in collective rear-rangements on lengthscales smaller than 30 particle diameters. Moreover, in orderto have sufficient statistics in c I ( t w , τ ), the system has to reconfigure several timesover the chosen observational lengthscale. For the largest φ nom investigated, thetime to reconfigure so that c I ( t w , τ ) = 0 .
95 is already of the order of 10 s, nearto the duration of the experiment; here, the processing of a lower degree of corre-lation and thus larger lag time is precluded because of the finite duration of ourexperiment. Finally, the choice of a lag time which is significantly smaller thanthe characteristic decay time enables us to use the direct noise correction schemedescribed by Duri et al. (2005, Sec. IVc).From our c I -data with a mean of c I ( t w , τ ) = 0 .
95, we extract three quantitiesthat respectively characterise the average dynamics, its temporal fluctuations, andthe spatial correlations in the dynamics of our system. The dependence of thesequantities on φ nom is shown in fig. 6 along with the mechanical characteristics,which we use as a gauge of the transitional behaviour. As a parameter characterisingthe average dynamics, we determine the lag time at which c I ( t w , τ ) = 0 . τ c .Though τ c significantly increases as φ nom is increased, the fluid-solid transition,as defined by the mechanical properties of our system, is not reflected by anypronounced feature in the φ nom -dependence of τ c . For the volume fraction rangebelow φ nom,c , the characteristic time strongly increases, while at the transition thisincrease is somewhat slowed down; for φ nom > . τ c reaches a constant value.To characterise the temporal fluctuations in the dynamics of our system, wecalculate χ t , the temporal variance of c I ( t w , τ ), again fixing τ = τ c such that c I ( t w , τ c ) = 0 .
95. We recall that c I ( t w , τ ) is the instantaneous degree of correlationobtained by taking the pixel average in Eq. (3.1) over the entire image. Thus, χ t quantifies the temporal fluctuations of the spatially averaged dynamics, similarlyto the dynamical susceptibility χ introduced in numerical works on glassy systems Phil. Trans. R. Soc. A 367, 5013-5032 (2009) ynamics of concentrated microgel systems -4 -3 -2 -1 / s (a) G p [ P a ] c [ s ] (b) no r m (c) IIIII nom I (d) Figure 6. Volume fraction dependence of mechanical and dynamic properties, dividedinto 3 distinct regimes ( i , ii , and iii ). (a) Low shear viscosity (open circles) and plateaumodulus G p (half-filled circles). The apparent divergence of the low shear viscosity clearlydefines the boundary between regime i and ii . (b) The decay time τ c increases steadilyup to φ nom ≈ .
73, after which it becomes approximately constant (regime iii ). (c) Thenormalised temporal variance χ norm monotonically increases with no distinct featuresover the entire investigated range of φ nom . (d) The range of spatial correlations of thedynamics, ξ (see text for definition) increases with φ nom in regime i , peaks in regime ii and finally decreases for highly squeezed states, regime iii . The lines serve as guides forthe eye. (Laˇcevi´c 2003). To gauge the significance of χ t with respect to the noise stemmingfrom the finite number of speckles recorded, we normalise our data with data ob-tained in reference measurements, where we use freely diffusing colloidal particles asa model system exhibiting homogeneous dynamics. For the Brownian particles, χ t contains only the noise contribution. Any excess of the variance with respect to thevalue obtained for the Brownian particles can thus be ascribed to temporally het-erogeneous dynamics. For all φ nom investigated, the normalised temporal variance χ norm is significantly larger than the one expected for homogeneous dynamics, asshown in fig. 6(c); this indicates that the dynamics of our system is characterised byheterogeneities at all volume fractions investigated, temporal heterogeneity becom-ing more pronounced as the volume fraction increases, as denoted by the increaseof χ norm with φ nom . Phil. Trans. R. Soc. A 367, 5013-5032 (2009) D. A. Sessoms and others
Figure 7. Representative dynamic activity maps for the PNiPAM solution quenched toregime I ( T = 29 . ◦ C , φ nom = 0 . T = 24 . ◦ C , φ nom = 0 . T = 20 . ◦ C , φ nom = 0 . c I ( t w , τ, r ) is given in terms of its temporal standard deviation, σ . As discussed for instance in Trappe et al. (2007), χ norm depends on both theamount of temporal fluctuations of the dynamics at a given location and the rangeof spatial correlations of the dynamics. To characterise the spatial fluctuations inthe dynamics of our system, we process the speckle images calculating the degree ofcorrelation with spatial resolution, c I ( t w , τ, r ). By contrast to the mean dynamicsand the temporal variation of the dynamics, the dynamic activity maps obtainedfrom a space-resolved analysis at a lag τ = τ c reveal striking differences dependingon whether φ nom is below φ nom,c , just above φ nom,c or above φ nom ≈ .
73, where weidentify the three conditions as regime I, regime II, and regime III. At φ nom = 0 . c I ( t w , τ, r ), whichcan be visually assessed from the fluctuations of the intensity in the dynamicalactivity map shown in fig. 7 (left image). These fluctuations appear to be weaklycorrelated. At φ nom = 0 .
69 (regime II), the DAMs have, at any given time t w ,essentially the same intensity level in all metapixels, as shown in fig. 7 (middleimage). This suggests that the dynamics are correlated over distances comparableto the system size. At φ nom = 0 .
81 (regime III), we find large variations in theDAM intensity, with regions of high dynamical activity coexisting with “quieter”zones, as shown in fig. 7 (right image). The boundary between the high and lowdynamical activity zones appears to be well defined; these zones typically extendover a sizeable fraction of the field of view, but do not extend over the entire fieldof view like in regime II. As a further difference between the three regimes, wenote that the temporal evolution of the spatial heterogeneities strongly depends onwhether we quench the system to regime I, II, or III. While for regime I and II thedynamical activity within one region quickly switches from low to high and viceversa, the dynamic activity within a given zone persists for a long time in regime III.To quantify the spatial correlations of the dynamics, we calculate the four-point correlation function G (∆ r, τ ) introduced by Duri et al. (2009). This functioncompares the local dynamics on a time scale τ in regions separated by a distance∆ r . Its definition is similar to that used in numerical work (Laˇcevi´c et al. G (∆ r, τ ) = B D δc I ( t w , τ, r ) δc I ( t w , τ, r ′ ) E ∆ r , (3.2)where δc I ( t w , τ, r ) = c I ( t w , τ, r ) − c I ( t w , τ, r ), B is a constant such that G (∆ r, τ ) → r →
0, and h . . . i ∆ r is the average over all pairs of r and r ′ corresponding tothe same distance, ∆ r = | r − r ′ | . As for the other dynamical quantities reported infig. 6, we analyse G for τ = τ c . The spatial correlation functions corresponding tothe three regimes shown in fig. 7 are shown in fig. 8. Phil. Trans. R. Soc. A 367, 5013-5032 (2009) ynamics of concentrated microgel systems G ( r , c ) r [mm] Figure 8. Spatial correlation of the dynamics, G (∆ r, τ c ), as defined in the text. The rangeof spatial correlations of the dynamics is the smallest in regime I (triangles), becomescomparable to the system size in regime II (squares), and finally decreases in regime III(semi-filled circles). In regime I, G decays over a few metapixels, corresponding to about 2 mm.By contrast, in regime II G ≈ G to some functional form, suchas an exponential. While this approach would be sound at the lowest and highestvolume fractions investigated, it would fail in regime II, because G is essentially flatover the full accessible range of spatial delays. Instead, we introduce a normalizedcorrelation length by defining ξ = ∆ r − max Z ∆ r max G (∆ r, τ c )d∆ r , (3.3)where ∆ r max = 5 . G in fig. 8, ξ starts at a low level at low φ nom , increases dramatically up to almost unity in regime II and finally decreasesin regime III, as shown in fig. 6(d).Consistent with the growth of ξ observed here in regime I and II, confocal mi-croscopy (Weeks et al. etal. et al. φ g , which we identify as the volume fraction at which the low shear viscosityapparently diverges, is much larger than the one reported for the self-diffusion ofhard spheres, where the dynamics are correlated only up to a few particle sizes(Weeks et al. q and agiven ROI extends over nearly 2000 particles. Our experiment thus probes the recon-figuration of large spatial density fluctuations and the correlation length measuredindicates how zones of higher or lower dynamical activity are correlated with each Phil. Trans. R. Soc. A 367, 5013-5032 (2009) D. A. Sessoms and others other; this does not necessarily correspond to the actual number of cooperatively re-arranging particles that can be measured in real space experiments. Indeed, spatialcorrelations that span the whole sample have recently been inferred in a study of thedynamical heterogeneities of xenospheres by diffusing wave spectroscopy (Ballesta et al. (2008)). In this study, Ballesta et al. showed that the correlation length growswith increasing volume fracion to eventually span the whole sample near the maxi-mal packing conditions, in agreement with the behavior observed in regime I and II.For the xenospheres, measurements beyond the maximal packing condition were notpossible as the xenospheres are essentially non-deformable. The striking decreaseof ξ at volume fractions beyond φ j observed in our system is instead reminiscentof the behavior reported for a 2-dimensional driven granular system (Lechenault etal. φ dependence in the spatial corrrelation of thedynamical heterogeneities has been observed as well.Together with these suggestive analogies, our data can be rationalized consid-ering the transition from a situation where the particles are densely packed but donot exert direct contact forces on one another (regime I and II) to a situation wherethey do (regime III). In regime I (supercooled state) and II (glassy state) the dy-namics is purely determined by thermal motion. With increasing volume fraction,the dynamics becomes increasingly cooperative, reaching in regime II a situationwhere any dynamical activity at a given zone will require the neighbouring zone tobe dynamically active as well. By contrast, a quench into regime III (squeezed state)imposes large deformations on the microgels. The dynamic activity observed in thisregime is therefore likely to be at least partly due to imbalanced stresses; these leadto rearrangements, which do not appear to equilibrate the stress imbalances. As aresult of this the dynamical activity persists in time and elastically propagates inspace, albeit remaining well localized, exhibiting sharp boundaries to dynamicallyinactive zones.Finally, we note that the almost complete lack of hallmarks in χ norm at φ j is somewhat surprising. Indeed, we generally would expect that χ norm becomesmaximal when ξ is maximal. The increase of χ norm beyond φ j indicates that theamplitude in the dynamical fluctuations of the spatially averaged signal c I ( t w , τ )dramatically increases beyond φ j , thereby compensating the effect of the decreasingcorrelation length. A full understanding of this effect in conjunction with the φ dependence of τ c is the subject of further research.
4. Conclusions
We have characterized the mechanical and dynamical properties of a PNiPAM-microgel system whose dimensions are conveniently varied by temperature. Despitethe fact that these microgels do not possess a well-defined interface, allow for partialinterpenetration and compression, a remarkable number of the mechanical featuresof hard and soft, deformable spheres with well-defined interfaces are reproduced.The low shear viscosity appears to critically diverge at some nominal critical volumefraction φ nom,c , which we define as the transition from a ‘supercooled’ to a ‘glassy’state. As the volume fraction of our microgel system is ill-defined, the indication interms of nominal volume fractions does not allow for any conclusions with respectto the exact volume fraction at this transition. Moreover, previous work has shownthat the lower density shells of our microgels partly interpenetrate near φ nom,c Phil. Trans. R. Soc. A 367, 5013-5032 (2009) ynamics of concentrated microgel systems φ nom,c . The indication ‘supercooled’ and ‘glassy’ hererefer to the inferred conditions that the microgels are able to escape out of a cageof nearest neighbours in the ‘supercooled’ state, while this is not the case in the‘glassy’ state, where we additionally presume that the microgels do not exert directcontact forces on each other in the ‘glassy’ state. Despite the evident disadvantagesin using microgels to investigate the fluid-solid transitions of repulsive systems, theadvantage of varying the volume fraction by changing the temperature enables usto precisely control the history of our sample, which we believe prevails over thedisadvantages. All our experiments are performed by first equilibrating the systemin the fluid state at high temperature, which is then quenched to a given solid stateby lowering the temperature. The residual dynamics in the solid-like states are thusnot affected by the shear history of the sample other than the one imposed by theinflation of the particles.We measure this residual dynamics at a q -vector where we probe the collec-tive diffusion of our system. Approaching the critical nominal volume fraction frombelow, we find that the extent of the spatial correlation in the collective dynamicsincreases, reflecting the increasing constraints set by the increasing volume fraction,whereby a local reconfiguration of the system can only occur when the neighbouringareas also reconfigure. Beyond φ nom,c the range of spatial correlations extends overthe entire observational window. The mechanical properties are characterised by alow frequency elasticity; the dissipative losses probed at high frequencies indicatesthat the local dynamics of the particles are still determined by the hydrodynam-ics of individual particles. Increasing the volume fraction further, we identify asecond transition, the transition to the ‘squeezed’ states. In these states we findthat the storage and dissipative contributions in the mechanical properties of thesystem become increasingly coupled and that the high frequency dissipative lossesbecome significantly larger than the ones observed at lower φ nom . We attributethis behaviour to friction between the particles resulting from the fact that theparticles exert direct forces on one another. In this state, the dynamics exhibitspatial heterogeneities that are characterised by large zones of respectively highand low dynamical activity. This behaviour appears to be a hallmark of squeezedsystems, where the quench into this state does not necessarily lead to a balanced-stress situation. This imbalance leads to rearrangements, which again may or maynot result in a balanced-stress situation, such that the dynamical activity persistsin time. The ranges of the dynamically correlated regions here are smaller than inthe ‘glassy’ state, indicating that stress-driven rearrangements are more localizedthan the thermally-driven rearrangements in the glassy state. To our knowledge,our data are the first demonstrating the intrinsic changes in the dynamical het-erogeneities of deformable colloidal systems as the origin of the dynamical processswitches from purely thermal to stress-driven. We gratefully acknowledge financial support from the Swiss National Science Foundation(grant no. 200020-117755 and 200020-120313), CNES and CNRS (PICS no. 2110). L. C.acknowledges support from the Institut Universitaire de France.
Phil. Trans. R. Soc. A 367, 5013-5032 (2009) D. A. Sessoms and others
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