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Microrheology and the fluctuation theorem in dense colloids
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Microrheology and the fluctuation theorem in dense colloids
L. G. Wilson1(a),A.W.Harrison1,W.C.K.Poon1 and A. M. Puertas2(b)
1 COSMIC and School of Physics & Astronomy, The University of Edinburgh, Kings Buildings Mayfield Road, Edinburgh EH9 3JZ, UK, EU 2 Group of Complex Fluids Physics, Department of Applied Physics, University of Almeria 04120 Almer´ıa, Andaluc´ıa, Spain, EU
received 12 November 2010; accepted in final form 14 February 2011 published online 14 March 2011
PACS 82.70.Dd – Colloids PACS 83.80.Hj – Suspensions, dispersions, pastes, slurries, colloids PACS 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
Abstract – We present experiments and computer simulations of a “tracer” (or “probe”) particle trapped with optical tweezers and dragged at constant speed through a bath of effectively hard colloids with approximately the same size as the probe. The results are analyzed taking the single-particle case and assuming effective parameters for the bath. The effective microscopic friction coefficient and effective temperature of the tracer are obtained. At high probe velocities, the experimental microviscosity compares well with the viscosity from bulk rheology, whereas a correction due to hydrodynamic interactions (absent in the simulations) is necessary to collapse the simulation data. Surprisingly, agreement is found without any need of hydrodynamic corrections at small probe velocities. The dynamics of the tracer inside the trap shows, both in the simulations and experiments, a fast relaxation due to solvent friction and a slow one caused by the collisions with other particles. The latter is less effective in dissipating the energy introduced by the moving trap and causes increasing fluctuations in the tracer motion, reflected as higher effective temperature.
Copyright c EPLA, 2011
Using tools such as optical tweezers, it is now possible absence of hydrodynamic interactions (HI). Two limiting to manipulate and measure the response of soft matter situations are possible: dragging a probe particle by a on the sub-micron scale. In active microrheology [1], a constant F, or translating it at constant U. When the known external force is applied to a bead in a complex probe is held in a trapping potential, e.g. using laser environment, and the response of the bead (and, less tweezers, and translated through the sample, a mixed frequently, of the environment) is recorded. Experimen- mode appears which reduces to constant U in a stiff tally, one of the attractions is that such microrheological trap. Simulations have shown that these two limits are techniques require much smaller samples than bulk indeed different [5]. Experimentally, Habdas et al. [6] rheology. Theoretically, by studying fluctuating quantities have dragged magnetic beads through a hard sphere at mesoscopic length scales, microrheology raises issues of (HS) colloidal suspension with a constant F, and their fundamental statistical mechanics related to fluctuations results have been compared with theory [4]. Earlier, Meyer and irreversibility [2]. et al. [7] studied the forces acting on a probe particle In microrheology, effective friction coefficients, γ,are held by optical tweezers in suspensions of teflon particles defined through the stationary relation F = γ U , where being translated at constant U, where the probes were U and F are the mean velocity of and force on the significantly larger (∼×10) than the bath colloids. Their tracer, respectively. In this letter, we are concerned with results do not agree with bulk rheology, but more recent the microrheology of colloidal suspensions. Here, theories experiments by the same group [8] compare well with the have been developed for low particle concentrations [3] theoretical work of Squires and Brady [3]. and up to vitrification [4], relating U and F in the Other microrheological experiments have confirmed the fluctuation theorem (FT), which states that the ratio of − (a)Present address: The Rowland Institute at Harvard - 100 Edwin the probabilities of producing an entropy σ and σ is H. Land Boulevard, Cambridge, MA 02142, USA. equal to exp(στ), where τ is the length of the interval over (b)E-mail: [email protected] which σ is measured, for large τ [9–11]. The single-particle
58007-p1 L. G. Wilson et al. problem can be described by the Langevin equation [12]. allows direct comparison with experimental data. Density 4 3 2 For dense systems, simulations in the constant F limit is reported as volume fraction, φ = 3 πa [1 + (δ/a) ]nc, have shown that the transient FT is applicable up to with nc the particle number density. The equations of vitrification, though a diverging waiting time is necessary motion were integrated with a time step δt =5· 10−4 [14]. for the stationary state FT to apply [13]. In contrast to Newtonian dynamics or pure Brownian We have performed experiments and Brownian dynam- dynamics, this model allows both momentum transfer to ics simulations to study the dynamics of a tracer in a other particles and dissipation to the fluid. harmonic trap and dragged at constant speed through a Our experimental system has been described in detail system of (effective) HS (the “bath”) up to high volume elsewhere [15]. Briefly, we used sterically-stabilized fractions, φ ∼ 0.5. Both in experiments and simulations, PMMA spheres (average radius ahost = 870 nm, poly- the tracer bead is similar in size to the bath particles. We dispersity σ ≈ 10%) suspended in an index-matching find that the steady-state distribution of the tracer posi- mixture of cis-decalin and cycloheptyl bromide (viscosity tion inside the trap can be rationalized by considering the η =2.6 mPa s), where they behave as nearly-perfect bath as a continuum with effective parameters; the excep- HS [16]. Index-mismatched sterically stabilized melamine tion occurs at high velocities and low density in the simu- particles, radius aprobe = 950 nm, were used as probes, lations, where hydrodynamic interactions (absent in our trapped using a standard implementation of optical tweez- simulations) become important. The effective friction coef- ers with a typical trap stiffness of k ≈ 4 × 10−6 Nm−1, ficient agrees in some limits with the shear viscosity from calibrated using the power spectrum method [17] in bulk rheology of HS and with theoretical results at low bare solvent. We translated the sample stage at constant φ [3]. The effective temperature increases both in exper- velocity using a computer-controlled DC motor and iments and simulations as the trap speed increases. We monitored the time-dependent position of the trapped then study the tracer position correlation function, where probe particle using a quadrant photodiode. A typical a slow relaxation due to the collisions with bath particles experimental run lasted around half an hour. Both in appears for large velocities or volume fractions, which is the experiment and simulations the tracer and bath incompatible with a Langevin model. Thus, any quantity particles are of similar size, making the contribution of dependent on the dynamics, such as the work distribution, tracer-particle and particle-particle collisions comparable, which we study in the final part of this letter, cannot be but also the applicability of an effective model for the properly described within a simple effective model. bath more problematic [18]. We simulated a system of 1000 polydisperse quasi-hard The problem of an isolated particle trapped in a particles. These undergo damped Newtonian dynamics to harmonic potential moving at constant speed (Fij =0 mimic colloidal dynamics. The position of particle j, rj, in eq. (1)) through a viscous medium involves two obeys a Langevin equation: time scales, the Brownian time, τB, and the time scale set by the dragging velocity, τtrap. The compe- r F − r F m¨j = ij γ0 ˙ j + ηj(t)(+ trap ), (1) tition between them is measured by a P´eclet number, i Pe = τB/τtrap = vtrapa/(6D0), where D0 = kBT/γ0 is where Fij is the force exerted by particle i due to the the diffusion coefficient of the particle. Onsager and −36 soft-core potential V (rij )=kBT (r/σij ) , with kBT the Machlup [19,20] showed that in the steady state, and thermal energy and σij the centre-to-centre distance, neglecting the inertial term, the distribution of displace- γ0 is the friction coefficient of the solvent, and ηj is a ments was random force obeying the fluctuation dissipation theorem, k 2 ηi(t)ηj(t ) =6kBTγ0δ(t − t )δij . A particle is selected p(α)= exp −kα /2kBT , (2) randomly and trapped in a harmonic potential. This 2πkBT tracer (or probe) particle thus also experiences the trap where α is x − xst, y − ytrap or z − ztrap, with xst the force, Ftrap = −k(r − rtrap), with k the spring constant solution of the stationary-state equation, i.e. γ0vtrap = and rtrap =(xtrap,ytrap,ztrap) the position of the trap −k(xst − xtrap). This expression reproduces the distri- minimum. The trap moves at constant velocity vtrap along bution of displacements obtained for very low φ, both the x-axis of the simulation box of dimensions Lx,L,L, experimentally and in simulations (data not shown). with Lx =8L; all particles are subject to Brownian For concentrated systems, fig. 1, the distributions are motion (eq. (1) without Ftrap). The friction coefficient fitted well by Gaussian functions in most cases. The non- is set to γ0 = 50, in conventional colloidal simulation Gaussian shape of the data in the simulations at high vtrap units (mean radius a =1, kBT =1 and m = 1), and the and lower φ is an inherent characteristic of the simulations. spring constant k = 1000. Particle radii are distributed In the absence of HI, the forces on the probe due to the according to a flat distribution of width 2δ =0.2a to solvent and due to direct inter-particle collisions become avoid crystallization. Converting these data to physical separately resolvable in the limit of high probe speed. The units using a particle radius a = 900 nm (see below), simulations at lower speed and higher volume fractions, yields the time unit 2.39 × 10−5 s, the medium viscosity and the experiments (where HI play a role) conform more −6 −1 is η0 =0.036 mPa s and k =5.47 × 10 Nm ,which closely to a mean-field description. However, the mean
58007-p2 Microrheology and the fluctuation theorem in dense colloids
15 φ=0.10 15 φ 100 =0.19 φ φ=0.30 =0.20 10 φ=0.45 φ=0.56 10 5 0 γ
0 /
γ 10 -1 -0.5 0 P(x*)
5
φ=0.40 1
0 0.001 0.01 0.1 1 10 100 -1.5 -1 -0.5 0 Pe
15 Fig. 2: (Colour on-line) Relative effective friction coefficient as 15 φ=0.19 a function of Pe; the volume fractions indicated refer to the 10 experimental data (points with error bars). The lines represent simulation data, for φ = 0, 0.10, 0.20, 0.30, 0.40, 0.50 and 0.55 5 10 from bottom to top. The horizontal lines at low Pe mark the value of γ/γ0 at Pe → 0 determined from simulations. 0 P(x*) -0.5 -0.25 0
5 φ=0.45 with absolute values increasing with φ, in agreement with theoretical expectations [3,4]. The experimental data is restricted to a smaller range of Pe where only the final 0 part of the decay and the high-Pe plateau are visible, again -0.5 -0.25 0 with absolute values increasing with φ. The absence of HI x* = (x-xtrap)/a in the simulations gives rise to a reduced value for γ/γ0 in the high-Pe limit, showing the decreasing importance Fig. 1: (Colour on-line) Position distribution functions for the dragged tracer particle, relative to the centre of the optical of the Brownian contribution and direct interactions to trap, with distances along the x-axis scaled by the particle hard-sphere viscosity at high shear rates [21,22]. radius. The upper panel shows simulation data for φ =0.40 and Figure 4 shows a comparison of microrheological data different values of Pe (from right to left: Pe = 1.67, 8.34, 16.7, with bulk rheology measurements. The lower panel shows 41.6, 83.4 and 166.6); the inset shows data for φ =0.20 at the experimental low-shear viscosity data [23,24] and simula- same values of Pe. The lower panel shows experimental data tions [25] compared with the low-Pe plateau of γ/γ0 from comparable to the simulations, for φ =0.45 and φ =0.19 (inset) our simulations (the latter is obtained as the average of —the Pe values, from right to left are Pe = 12.3, 24.6, 49.2, the data at the lowest Pe, and shown in fig. 2 by the hori- 98.4 and 147. In each panel, the continuous (red) curves are zontal lines). In this regime, the bath particles’ Brownian Gaussian fits to the data, and the vertical (green) line indicates motion is the dominant contributor to the viscosity, and the trap position. the probe’s environment closely conforms to the mean-field picture. The reverse is true in the high-Pe case shown in the upper panel: at high probe speeds, the surrounding and variance of the distributions are not correctly given medium does not have time to rearrange to accommo- by eq. (2). Instead, there is now an effective friction date the probe, and individual particle interactions emerge coefficient, γ,andaneffective temperature Teff ,which from the relatively smooth, Brownian regime. Whereas the characterize the mean and variance of the measured probe experimental microviscosity agrees with macrorheological position distribution, respectively. We determine these data at high shear rates [15,24], the values from simula- parameters by fitting Gaussians to the data, solving for tions are significantly lower, presumably due to the neglect γ via γvtrap = −k(xst − xtrap) and taking the variance of of HI. Neither the experiments nor the simulations for the the distribution to be kBTeff /k. microviscosity show shear thickening, observed in conven- Figures 2 and 3 show the fitted values of γ and Teff tional rheology of HS at high Pe. from fitting simulations and experiments as a function Interestingly, we can obtain agreement on the high- of Pe. In the case of the friction coefficient, fig. 2, the shear plateau by estimating the effect of hydrodynamics shape of all simulation curves is very similar, with plateaux using data for the short-time self-diffusion coefficient as s at low and high Pe and a decreasing trend in between, a function of volume fraction, Ds(φ). This quantity is
58007-p3 L. G. Wilson et al.
Bulk high-shear viscosity φ γ γ =0.10 high / 0 from simulations 10 φ=0.19 s γ / γ x D (φ) / D φ=0.30 high 0 s 0 φ=0.45 γ / γ from experiments 0 high 0 φ=0.56 γ 1 10 Theory /
eff,x Scaled theory T high γ
1 0 10 0 0.1 0.2 0.3 0.4 0.5 0.6 10 100
2 Experimental bulk low-shear viscosity 10 Fit to simulation bulk viscosity γ γ low / 0 from simulations Theory
0 Scaled theory γ / 1 eff,y low
T 10 γ
1 0 10 0 0.1 0.2 0.3 0.4 0.5 0.6 10 100 φ Pe Fig. 4: (Colour on-line) Upper panel: experimental high- Fig. 3: (Colour on-line) Upper panel: effective temperature in shear viscosity from buk rheology [24] is compared with our the axis parallel to the dragging direction, as a function of Pe, experimental and simulation microrheological data without with the same colour and symbol cod as fig. 2. Lower panel: and with the “hydrodynamic correction factor” (see text). effective temperature in the direction perpendicular to the drag Lower panel: comparison of bulk low-shear viscosity [24] with direction as a function of Pe, with same colour and symbol effective drag coefficients obtained from fits to molecular codes again. dynamics data (based on the Krieger-Dougherty form) [25] and the low-Pe plateau from our own simulations, fig. 2. The dashed black lines represent the theoretical predictions from straightforward to measure using optical tweezers [15] Squires and Brady [3], for the constant-velocity (upper line) or dynamic light scattering [26], and we take the ratio and constant-force (lower one) cases. s Ds(φ)/D0 as a hydrodynamic correction factor. Multi- plying the “structural” contribution to the microviscosity from simulations by this correction factor brings the On the other hand, our microviscosity results should be former into agreement with micro/macrorheological data comparable to the theoretical predictions of Squires and for the high-shear rate viscosity, fig. 4 (top panel). The Brady [3], who also neglected HI. In their theory, a bead structural and hydrodynamic contributions are therefore trapped in a harmonic potential and dragged appears as “decoupled” in this sense. an intermediate case between the constant-velocity and More puzzling is the finding that the simulated micro- constant-force limits. Our simulation results agree with viscosities in the low-Pe regime agree with experimental their predictions at low φ both at low and high Pe, but macro-rheology [24] without the need for hydrodynamic deviate above φ ∼ 10% for low Pe and φ ∼ 40% for high corrections, fig. 4 (lower panel). This is unexpected, Pe, as shown in fig. 4 (continuous lines). Obviously, the s because HI is expected to play an important role in the theoretical predictions can be rescaled with Ds(φ)/D0 to viscosity at all shear rates [21,27]. If this agreement is not agree with the experimental values. Alternatively, Squires fortituous, it indicates that the relative weight of HI in the and Brady proposed to scale their low density theory to viscosity of the HS increases with Pe, and is negligible at higher densities with the contact value of the pair distri- the single-particle level. We do not currently understand bution function, ∆η ∼ φgeq(2; φ), which was confirmed by this finding. simulations [5,8]. Using the Carnahan-Starling equation to
58007-p4 Microrheology and the fluctuation theorem in dense colloids estimate the pair distribution function at contact, 0.004 − eq 1 1/2φ 2 Pe = 0.0833 g (2; φ)= 3 (3) st 0.003 Pe = 0.833 (1 − φ) Pe = 4.167 Pe = 8.333 the theoretical results have been scaled and plotted as Pe = 20.833 dashed lines in fig. 4. The agreement with simulations at 0.002 the low-Pe limit improves, in accordance with previous simulations, but worsens the high-Pe comparison [8]. Returning to the probe position distributions measured 0.001
in both experiments and simulations, fig. 1, we find
2 tion, whereas for large vtrap it must force the surround- ∗ ∗ kBT γ0 x (t + τ)x (t) = exp (−kτ/γ0)+ vtrap . (4) ing particles out of the way. These (elastic) collisions pass k k momentum to other particles, serving as an indirect and Note that both the long-time plateau and the decay time slower dissipation mechanism (thus raising the temper- scale are controlled by the friction coefficient, and the ature). The splitting into two processes of the decay of amplitude depends only on temperature. The correlation the correlation function is characteristic of a concentrated functions from simulations and experiments are presented system and reminiscent of approaching the glass transi- ∗2 in fig. 5 (with the long-time plateaux subtracted: xst = tion, and therefore cannot be described by the simple 2 (vtrapγeff /k) ) for constant density and increasing Pe. The exponential in eq. (4), as shown by the thin broken lines overall relaxation is slower as Pe (or the colloid density) in fig. 5. increases, and the amplitude raises, implying an increase Therefore, the dynamics of the particle inside the in both the friction coefficient and the temperature —this trap is incorrectly described by the mean-field model, shows that the trends in figs. 2 and 3 are insensitive to i.e. the bath of colloidal particles cannot be properly transient events. described as an effective fluid, although the tracer position Additionally, the correlation functions show two relax- distributions are indeed Gaussian. It is interesting to show ation mechanisms: a fast one, almost independent of the that this failure of the description of the microscopic trap velocity, due to the friction with the solvent in a time dynamics shows up when the distribution of the work τ v F scale γ0/k; and a second, slower, one, which is absent at performed by the trap, Wτ = 0 dt trap trap is studied.
58007-p5 L. G. Wilson et al.
0.04 ∗∗∗
Pe = 0.0833 AMP acknowledges the financial support from the Junta 0.03 Pe = 0.833 de Andaluc´ıa, project P09-FQM-4938. LGW and AWH Pe = 4.167 Pe = 8.333 held EPSRC studentships, and WCKP was funded by trap Pe = 20.833 EPSRC (EP/D071070/1). 0.02 ) * v τ REFERENCES P(W 0.01 [1] Waigh T. A., Rep. Prog. Phys., 68 (2004) 685. [2] Evans D. J. and Searles D. J., Adv. Phys., 51 (2002) 15929. 0 Squires T. M. Brady J. F. 17 -100 0 100 200 300 400 [3] and , Phys. Fluids, (2005) W / v 073101. τ trap [4] Gazuz I., Puertas A. M., Voigtmann Th. and Fuchs M., Phys. Rev. Lett., 102 (2009) 248302. Fig. 6: (Colour on-line) Distributions of work from simulations [5] Carpen I. C. and Brady J. F., J. Rheol., 49 (2005) at constant density φ =0.40 for different Pe as labeled. The 1483. lines represent the theoretical predictions from the Langevin [6] Habdas P., Schaar D., Levitt A. C. and Weeks equation with the effective parameters from figs. 2 and 3. E. R., Europhys. Lett., 67 (2004) 477. [7] Meyer A. et al., J. Rheol., 50 (2006) 77. [8] Sriram I., Meyer A. and Furst E. M., Phys. Fluids, (Any other quantity that depends on the details of the 22 (2010) 062003. dynamics of the particle inside the trap will show a [9] Evans D. J., Cohen E. G. D. and Morriss G. P., Phys. similar breakdown.) The distribution of work, P (Wτ ), has Rev. Lett., 71 (1993) 2401. been calculated for an isolated probe particle, and in the [10] Wang G. M. et al., Phys. Rev. Lett., 89 (2002) 050601. stationary state it is a Gaussian with a mean and width [11] Carberry D. M. et al., Phys. Rev. Lett., 92 (2004) determined by γ0 and bath temperature [12]. In fig. 6, 140601. [12] van Zon R. and Cohen E. G. D., Phys. Rev. E, 67 P (Wτ ), with τ = 1, is presented for constant φ =0.40 and increasing Pe from simulations. All of the distributions are (2003) 046102. [13] Williams S. R. and Evans D. J., Phys. Rev. Lett., 96 almost Gaussian, but cannot be described with the same (2006) 015701. effective parameters as the position distributions, as shown [14] Paul W. and Yoon D. Y., Phys. Rev. E, 52 (1995) 2076. in the figure. The maximum of the distribution (mean [15] Wilson L. G., Harrison A. W., Schofield A. B., work), which depends only on the equilibrium position of Arlt J. and Poon W. C. K., J. Phys. Chem. B, 113 the tracer inside the trap (and therefore on the effective (2009) 3806. friction coefficient) is correctly captured by the mean- [16] Underwood S. M., Taylor J. R. and van Megen W., field model. But the width, which depends also on the Langmuir, 10 (1994) 3550. dynamics of the tracer in the trap (the temperature), is [17] Gittes F. and Schmidt C. F., Signals and Noise not correctly predicted. Interestingly, because the work in Micromechanical Measurements, Vol. 55 (Academic distributions are approximately Gaussian and the mean Press) 1998, Chapt. 8. Squires T. M. 24 is correct, the qualitative predictions of the FT are valid, [18] , Langmuir, (2008) 1147. [19] Onsager L. and Machlup S., Phys. Rev., 91 (1953) such as the existence of negative work, and the ratio of 1505. positive to negative work. [20] Dean Astumian R., J. Chem. Phys., 126 (2007) 111102. In conclusion, we have used simulations and experi- [21] Brady J. F., Curr. Opin. Colloid Interface Sci., 1 (1996) ments in conjunction to reveal more about the dynamic 472. processes involved in active microrheology. A simple mean- [22] Khair A. S. and Brady J. F., J. Fluid Mech., 557 (2006) field theory of the response of a HS colloidal host is valid in 73. the limit of some averaged quantities and low Pe, but this [23] Segre` P. N., Meeker S. P., Pusey P. N. and Poon picture breaks down when the transient dynamics of the W. C. K., Phys. Rev. Lett., 75 (1995) 958. host are considered. We have also shown that it is possi- [24] Phan S.-E., Russel W. B., Cheng Z., Zhu J., Chaikin P. M., Dunsmuir J. H. Ottewill R. H. ble to decouple the structural and hydrodynamic compo- and , Phys. Rev. 54 nents of the microviscosity; this has implications for the E, (1996) 6633. [25] Sigurgeirsson H. and Heyes D. M., Mol. Phys., 101 development of simulation techniques where complicated (2003) 469. Simulation results for the shear viscosity are hydrodynamic interactions may be approximated using collected from the literature and an “empirical” fit based mean-field experimental data, facilitated by the separation loosely on the Krieger-Dougherty equation is given. in timescales between the structural and hydrodynamic [26] van Megen W., Phys. Rev. E, 73 (2006) 011401. processes. [27] Brady J. F., J. Chem. Phys., 99 (1993) 567.
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