Two step micro-rheological behavior in a viscoelastic fluid
Rohit Jain, Félix Ginot, Johannes Berner, Clemens Bechinger, Matthias Krüger
TTwo step micro-rheological behavior in a viscoelastic fluid
Rohit Jain, ∗ F´elix Ginot, ∗ Johannes Berner, Clemens Bechinger, and Matthias Kr¨uger Institute for Theoretical Physics, Georg-August Universit¨at G¨ottingen, 37073 G¨ottingen, Germany Fachbereich Physik, Universit¨at Konstanz, 78457 Konstanz, Germany (Dated: February 23, 2021)We perform micro-rheological experiments with a colloidal bead driven through a viscoelasticworm-like micellar fluid and observe two distinctive shear thinning regimes, each of them displayinga Newtonian-like plateau. The shear thinning behavior at larger velocities is in qualitative agreementwith macroscopic rheological experiments. The second process, observed at Weissenberg numbersas small as a few percent, appears to have no analog in macro rheological findings. A simple modelintroduced earlier captures the observed behavior, and implies that the two shear thinning processescorrespond to two different length scales in the fluid. This model also reproduces oscillations whichhave been observed in this system previously. While the system under macro-shear seems to be nearequilibrium for shear rates in the regime of the intermediate Newtonian-like plateau, the one undermicro-shear is thus still far from it. The analysis suggests the existence of a length scale of a fewmicrometres, the nature of which remains elusive.
I. INTRODUCTION
Stochastic processes are of general importance, from afundamental point of view but also regarding technicaland biological applications. As a result, they have beenthe subject of intense research over the past years. Asimple example is Brownian motion which has been in-tensively studied, both in equilibrium and in presence ofvarious types of external driving, in experiments and the-oretically [1–3]. In particular in case of entirely viscous,i.e. Newtonian solvents the treatment of such processes israther straightforward because it can be described withinthe framework of a Markovian theory. The correspond-ing Langevin equation is then also linear in the sense thatthe dissipative parts (i.e., friction), remain linear in theparticle’s velocity. To go beyond these model systems,a variety of nonlinear properties of complex fluids havebeen studied [4]. An important example concerns shear-ing of complex fluids [5–13], which, experimentally, in-volves macroscopic rheometers [14–17]. In such investiga-tions, many nonlinear features have been observed, e.g.,shear-thinning or thickening, the phenomenon of yield-ing, or the multifaceted nonlinear response to oscillatoryshear. Following these macroscopic studies, the develop-ment of microrheology, where a single microscopic probeparticle is driven through a (nonlinear) medium, allowedto investigate the fluids’ behavior at much smaller lengthscales [5–13]. In related experimental and theoreticalstudies, several consequences of the nonlinearity of thebath have been reported. For example, it has been ob-served that the driven probe experiences an effectivetemperature which differs from the true bath tempera-ture [6, 18], that it shows superdiffusive behavior [9, 13],or shear thinning [6, 12]. More recently, experiments re-ported the occurrence of oscillatory modes [19], whichare seen in a regime of linear, e.g., Newtonian-like be- ∗ Both authors contributed equally to this work. havior. The oscillatory dynamics in Ref. [19] was repro-duced using a generalized Langevin equation with neg-ative memory at long times, which can induce persis-tent motion [20, 21] and stress overshoots [22]. Further-more, it was noted that the non-linear properties of abath can already be detected in equilibrium [23]. Forexample, in a nonlinear bath, the effective friction mem-ory kernel may depend on the external potential [23–28].These findings illustrate the difficulty to match microand macro-rheology measurement, and make the studyof non-Markovian baths all the more important.Here we study the motion of a particle trapped in aviscoelastic fluid, for different driving velocities, where,compared to previous work [19], we extend the exper-imentally accessible regime towards smaller velocities.We observe the previously reported linear regime, wherethe flow curve shows a plateau. For even smaller veloc-ities, however, the viscosity is seen to increase in a pro-nounced manner, so that two distinct plateaus and twoshear thinning processes are found. Theoretically, we re-produce this behavior by use of a previously introducedstochastic Prandtl Tomlinson Model (SPT) [23, 27]. Thismodel also reproduces the previously observed oscilla-tions [19, 27] for the given shear rates. The model im-plies that each shear thinning process corresponds to ade-equilibration of an important set of bath degrees offreedom, and it allows to estimate the important lengthscales involved in these degrees.
II. EXPERIMENTAL DETAILS
Our experiments were performed in an equimolar solu-tion of the surfactant cetylpyridinium chloride monohy-drate (CPyCl) and sodium salicylate (NaSal) in deionizedwater at a concentration of 7 m m and at constant tem-peratures of 25 ° C and 30 ° C, respectively. Under suchconditions, the mixture forms an entangled viscoelasticnetwork of worm-like micelles [29], with a structural re-laxation time on the order τ s = 2 . a r X i v : . [ c ond - m a t . s o f t ] F e b ■■■■■■■ ■■■■ ■■■■■■■ ●●● ● ●●●●●●●● ●●●●●●●●●●●●● ● ●●● ● ■■ ■ ° C, exp. ● ° C, exp. 25 ° C, sim.30 ° C, sim. η [ P a - s ] v ( nm / s ) FIG. 1. Microviscosity as a function of the driving speed(flowcurves). Symbols correspond to experimental data for25 ° C (blue squares) and 30 ° C (orange circles). Solid col-ored lines correspond to simulations using the stochastic PTmodel. Both temperatures exhibit the same two plateaushapes. For v >
200 nm s − , we recover the shear thinningbehavior which is expected from macro-rheological measure-ments at 25 ° C, shown as a black dashed line. However, for v <
200 nm s − , the micro-viscosity further increases towardsanother plateau reaching values more than twice as high asthe intermediate plateau. Dashed lines and data points onthe y -axis give linear response values from simulation and ex-periment, respectively. microscopic recoil experiments [30]. Typical length scalesof worm-like micelles are between 100 and 1000 nm [31],and their typical mesh size is on the order of 30 nm [32].A small amount of silica particles with 2 R = 2 . µ m di-ameter has been added to the fluid and a single particlehas been optically trapped by a focused laser beam ofwavelength of 1064 nm. This creates a harmonic poten-tial with trap stiffness κ . The latter is determined fromthe equilibrium probability distribution of the particlein equilibrium, see e.g. Ref. [19]. To probe the micro-rheological properties, the sample is translated with con-stant velocity − v relative to the static optical trap bya piezo-driven stage with v between 4 and 400 nm s − .Note that this driving is equivalent to a trap which ismoving at velocity v with respect to a fluid at rest. Over-all it leads to a local shear rate of v / R close to the par-ticle, and a Weissenberg number of W i = v τ s / R . Theso-obtained Weissenberg numbers range between a fewpermille to almost unity in our experiments. We therebyextend the measurements by one decade towards smallerWeissenberg numbers compared to Ref. [19].When driving the particle with constant velocitythrough the fluid, it experiences a drag force which leads to a displacement relative to the trap center. UsingStokes law and the trap stiffness, we can measure thevelocity-dependent microviscosity, given by η = κ πRv (cid:104) x ( t ) − v t (cid:105) . (1)Here, (cid:104) x ( t ) − v t (cid:105) directly corresponds to the average po-sition of the particle, relative to the center of the trap.The resulting flow curves are shown in Fig. 1, for twotemperatures 25 ° C (blue squares), and 30 ° C (orange cir-cles). Both curves exhibit the same trend. For drivingspeeds approaching Weissenberg numbers of order unity,the viscosity shows shear thinning, as expected. This isalso confirmed by the macro-viscosity which is also shownin Fig. 1 for the temperature of 25 ° C for comparison(black dashed line, obtained by a plane-plane rheome-ter). Indeed, for that temperature, the data correspond-ing to micro and macro viscosities agree reasonably wellfor v >
50 nm s − . We note that this data range wasavailable in Ref. [19]. From this close resemblance, to-gether with the fact that the Weissenberg number is ofthe order of only 5% for v = 50 nm s − , at first glanceone might conclude that these velocities are in the regimeof linear response.In contrast to this interpretation, however, the obser-vation of oscillations in the so called mean conditionaldisplacements (see details below), suggests that the sys-tem is far from equilibrium even at the small drivingspeed [19] of v = 50 nm s − . When extending the rangeof velocities to smaller values, an astonishing observationis made: The viscosity shows a pronounced second shearthinning transition, connected to another plateau in theflow curve, of more than twice the viscosity value com-pared to the second plateau. The corresponding shearthinning process sets in at a Weissenberg number ofroughly 2%, implying that this dimensionless quantity isnot useful to understand that process. This conjecture isunderpinned by the observation that the macro-viscosityseems to not display the shear thinning at the mentionedsmall velocities. III. THEORY AND MODEL
To rationalize these but also previous experimental ob-servations [23], we consider a simple model where thetracer particle is coupled to a small number of bath par-ticles which mimic the fluid environment (see sketch ofFig. 2). Each bath particle is coupled to the tracer viaa nonlinear interaction potential V int (specified below).As argued in Ref. [23], the potential V int is taken to beunbounded in the sense that it is finite for any particleseparation. This allows the bath and tracer particles tobe arbitrarily far away from each other, and to travelwith different (mean) velocities. Within this model, e.g.the effect of shear thinning is reproduced, because tracerand bath particles are able to move at different (average)speeds. For simplicity, in the following we assume V int to FIG. 2. a. Sketch of the experimental system, highlighting the probe suspended in a worm-like micellar fluid. The particle istrapped with an optical tweezer, using a highly focused laser beam to form a harmonic potential for the particle. b. Sketchof the Stochastic Prandtl-Tomlinson model. The tracer particle (bare friction coefficient γ ) is subject to a harmonic potential V ext with stiffness κ , moving at constant speed v . The tracer is also linked to several (here two) bath particles with frictioncoefficients γ and γ , respectively, through periodic interaction potentials V int ,i . follow a sinusoidal function V int ,i = V ,i cos (cid:18) πd i ( x − q i ) (cid:19) (2)where V ,i is the amplitude and d i is a length scale cor-responding to bath particle i . x and q i are the positioncoordinates of tracer and i -th bath particle, respectively.The parameter d i in Eq. (2) mimics an important lengthscale in the micellar bath, see above for typical numbers.In addition, the tracer particle is subjected to a harmonicpotential of stiffness which results from the laser tweezer.As the tweezer is moving with velocity v relative to thefluid, the trap potential V ext reads, V ext = 12 κ ( x − v t ) . (3)Accordingly, the equations of motion (with N bath par-ticles) are given by γ ˙ x ( t ) = − κ ( x − v t ) − N (cid:88) i =1 πd i V ,i sin (cid:18) πd i ( x − q i ) (cid:19) + ξ ( t ) , (4) γ i ˙ q i ( t ) = 2 πd i V ,i sin (cid:18) πd i ( x − q i ) (cid:19) + ξ i ( t ) . (5)Here, γ and γ i are the bare friction coefficients of tracerand bath particle i , respectively. These coefficients arelinked to the corresponding random forces ξ and ξ i viathe following standard properties, (cid:104) ξ i ( t ) (cid:105) = 0 and (cid:104) ξ i ( t ) ξ j ( t (cid:48) ) (cid:105) = 2 k B T γ i δ ij δ ( t − t (cid:48) ) . (6)We assume in the following that the driving started atan infinite time in the past, so that the system is in asteady state for t >
0. Practically, this means that from experimental as well as simulated trajectories, the initialparts, corresponding to equilibration, are removed.The interaction potential of Eq. (2) makes this modelreminiscent of the so-called Prandtl-Tomlinson (PT)model, which is used to study dry friction [33, 34].Eqs. (4) and (5) extend this model to allow the back-ground (our bath particles) to be stochastic and dynamic,and to contain several bath particles. This StochasticPrandtl Tomlinson (SPT) model [23, 27] thus reducesto the original PT model when setting N = 1 and let-ting γ approach infinity, so that the bath particle be-comes a stationary background potential. This differ-ence is quite intuitive, as our micellar background is dy-namic, while the potentials considered in dry friction arerather static. Note that the notion ’Stochastic Prandtl-Tomlinson model’ has been used for other models whichare different from the one used here [35, 36].The solid curves shown in Fig. 1 give the outcomes ofthe model of Eqs. (4) and (5) (see Table I for parametersused), using Eq. (1). For small velocities v , we observea linear response regime, where the micro-viscosity is in-dependent of v . In this regime, the viscosity can be ob-tained also via linear response (see Eq. (35) in Ref. [23]),shown as the horizontal dashed lines in the graph. Itis insightful to discuss the case of high potential barriers V ,i /k B T , which appear appropriate to fit the experimen-tal data (see Table I). Then the regime of small drivingspeeds corresponds to the case where all particles move(approximately) with the same average velocity v . Be-cause of this, the microviscosity in Eq. (1) can be foundto a good approximation from η ≈ πR ( γ + (cid:80) i γ i ). Inthe graph, the resulting value is not shown, as it is indis-tinguishable from the dashed lines. Experimentally, thiscorresponds to the case where the slow colloidal parti-cle drags its surrounding with it, and thus feels a largefriction.The interaction potential V int ,i in Eq. (2) supports amaximal force of πd i V ,i . If the force between the tracerand the bath particle exceeds that value, the bond rup-tures, and the velocity of the bath particle is (on average)smaller than v : As a result, the system shows a shearthinning behavior. The critical velocity where this hap-pens can be estimated by balancing the mentioned max-imal force with the drag force γ i v of particle i , yielding(in absence of noise) v i,cr ≈ πγ i d i V ,i . For the curvesin Fig. 1, we use two bath particles, with distinct criti-cal velocities, resulting in the two-plateau structure seenin the graph. As mentioned, for very small velocities,both bath particles move (approximately) with the samevelocity as the tracer. The smaller critical velocity cor-responds to the “larger” bath particle, i.e., the one witha larger value of d i . Using the values of Table I, we es-timate this velocity to be 27 and 36 nm s − for the twotemperatures, respectively, which fits well to the numer-ical curves. Beyond this velocity, the viscosity decreasestowards the second plateau. On the regime of the secondplateau, the larger particle is thus far from equilibrium,while the second (“smaller”) bath particle is still close toequilibrium. Once the second critical velocity (estimatedto 321 and 655 nm s − , respectively) is reached, also thesecond particle starts shear thinning. The model thus im-plies the interpretation that two distinct sets of degreesof freedom of the bath display very different critical ve-locities, yielding an intermediate state with half of themout of equilibrium. Notably, the often employed notionof fast and slow degrees of freedom [37] is demonstratedhere explicitly, at least in our model.The flow curve concerns the average position (cid:104) x − v t (cid:105) of the particle, as seen from Eq. (1), and we now aim toaddress the particle’s fluctuations. Subtracting the par-ticle’s mean position, i.e. using X ( t ) = x ( t ) − (cid:104) x − v t (cid:105) ,yields a stochastic variable X with zero mean. Its fluc-tuations can be quantified using the so-called Mean Con-ditional Displacement (MCD) [19]. It is defined by (cid:104) X (cid:105) X = (cid:82) ∞−∞ dX X P ( X, t | X , P ( X, t | X , X . Inprevious work, we observed that MCDs show oscillationsin this system [19], which were observed at a temperatureof 25 ° C, for Weissenberg numbers in the range of 0 .
04 to0 .
34. Can the SPT model also explain the occurrence ofoscillations? To address this question, we show in Fig. 3experimental and simulated MCD curves, this time fora temperature of 30 ° C, to also emphasize the same phe-nomenology at the two temperatures. The MCD curvesgenerally are found to a good approximation linear in X (both in experimental as well as in simulated data),so that division by X yields X -independent curves asshown. The figure distinguishes between positive andnegative values of X , yielding the positive and nega-tive flanks shown. The observed mirror symmetry of thedata with positive and negative X further emphasizesthe linearity in X . Fig. 3a) shows a small velocity of v = 16 nm s − , which corresponds to the regime of lin-ear response in Fig. 1. For this velocity, the MCD curve decays monotonically to zero, as expected near equilib-rium [19], and also in agreement with the SPT model.Fig. 3b) shows a larger velocity of v = 200 nm s − , avelocity placed on the intermediate plateau in the flowcurve in Fig. 1, thus in the regime corresponding to thecurves shown in Ref. [19, 27]. This curve shows pro-nounced oscillations, which are reproduced by the SPTmodel. How can these be understood? As describedabove, the intermediate plateau is beyond the critical ve-locity of the larger bath particle, which is thus far fromequilibrium. It thus moves with an average speed muchsmaller than v . It can for the sake of argument be as-sumed to stand still, so that the tracer is moving in astationary periodic potential. It is thus subject to a pe-riodic force, which results in the seen oscillations. Theperiod of oscillations is in this approximation given by d/v , which matches well the one observed in Fig. 3b). InRef. [19], the frequency of oscillations was indeed foundto scale linearly with v , an observation which can nowbe understood. This analysis thus identifies an importantlength scale in the system, of the order of 15 µ m. Passingover spatial variations on that scale (which are almoststationary as seen from the colloidal particle) seems tocause the observed oscillations.The amplitude of oscillations is larger in the SPTmodel as compared to experiments, which we attributeto a number of idealizations of the model. For example,the background potential of the SPT model is perfectlyperiodic with a sharp length scale d i . A real micellar solu-tion, however, will exhibit a range of length scales, whichnaturally leads to decoherence. Indeed adding more bathparticles, with slightly different parameters (see Table II)leads to a loss of coherence, and the amplitude of theresulting oscillations is reduced. Notably, adding morebath particles with appropriate parameters keeps the flowcurve unaltered, but changes the MCD. This confirms theexpectation that flow curve and MCD are not one to onerelated. The flow curve concerns the mean motion, andthe MCD quantifies fluctuations.In Ref. [23], a single bath particle was found to be suf-ficient to capture experimental observations in the STPmodel. It is natural that a system close to equilibrium(as in Ref. [23]) is easier to model compared to a systemfar from equilibrium, as addressed here. Indeed, an openquestion is whether the larger length scale of ∼ µ mcan be detected in equilibrium.Finally, we note that, while the MCD curves and theflow curve each can easily be modelled quantitatively, wehad problems fitting both curves with the same param-eters. We attribute this to the different values of trapstiffness used for flow curve and MCDs of 1 . µ N m − and0 . µ N m − , respectively, for experimental reasons. Thismakes fitting both curves simultaneously even more chal-lenging. We therefore allowed the amplitude of flowcurveto be a free parameter in our simulations, which turnedout to be 4 . (a) (b) ExperimentsPT Model - - - - t ( s ) 〈 X ( t ) 〉 X / X v =
16 nm / s ExperimentsPT Model - - - - t ( s ) v =
200 nm / s FIG. 3. Mean conditional displacements (a) in the linear response regime, 16 nm s − and (b) in the regime of intermediateplateau, 200 nm s − . Circles correspond to experiments (at 30 ° C), and lines correspond to the Prandtl-Tomlinson simulations(solid and dotted lines for 2 and 3 bath particles, respectively). While we recover previously observed oscillations [19] for200 nm s − , the linear regime (left, 16 nm s − ) exhibits none. Adding more bath particles (solid line for 3 bath particles), leadsto decoherence, and thus decreases the amplitude of the oscillations. IV. CONCLUSION
In this work we describe micro-rheological experimentswhere a single colloid is driven by an optical tweezerwithin a viscoelastic fluid. The presented fluid, a micellarsolution, shows a flow curve with two distinct shear thin-ning regimes, with an intermediate plateau in between.Theoretical modeling via a stochastic Prandtl Tomlin-son model captures the observed behavior, and impliesthat, on the intermediate plateau, one set of bath de-grees of freedom is far from equilibrium, while anotherset is still in equilibrium. The intermediate plateau cor-responds to the linear response regime of macro-rheology,so that the shear thinning process at even smaller drivingvelocities is a purely microscopic effect, and can thus eas-ily be overlooked. The mean conditional displacementsshow oscillations on the intermediate plateau, which arealso reproduced in the theoretical model. The oscilla-tions allow extraction of a length scale, which is as largeas 15 µ m, and whose nature and origin have to be inves-tigated in future work. V. ACKNOWLEDGEMENTS
The authors thank Boris M¨uller for discussions at theearly stages of this work. The theoretical parts of thiswork build on the initial findings provided in his the-sis [27].
Funding : FG acknowledges the support bythe Alexander von Humboldt foundation. This project is funded by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) – SFB 1432 – Project-ID 425217212. RJ acknowledges the support by theG¨ottingen Campus QPlus program.
Competing inter-est : The authors declare no competing interest.
VI. DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
Appendix A: Simulation parameters
In this appendix, we provide the simulation parametersused for generating the flow-curves in Fig. 1 and MCDcurves in Fig. 3 given in the main text.
T κ ( γ , γ , γ ) ( V , , V , ) ( d , d )[ ° C] [ µ N m − ] [ µ N s m − ] [ k B T ] [nm]25 1 . . , . , .
5) (7 . , .
0) (400 , . . , . , .
0) (4 . , .
6) (320 , For the MCD curves (dotted lines) in Fig. 3, we usea different trap-stiffness, κ = 0 . µ N m − . The addi-tional MCD curves with three bath particles (solid lines)in Fig. 3 are generated using the parameters in Table II. ( γ , γ , γ , γ ) ( V , , V , , V , ) ( d , d , d )[ µ N s m − ] [ k B T ] [nm](0 . , . , . , .
5) (4 . , . , .
8) (320 , , ° C with κ = 0 . µ N m − . As discussed in the main text, including more bath par-ticles into the model leads to de-coherence, resulting intothe reduction in amplitude of oscillations in the MCD.We choose the parameters for the third bath particle (seeTable II) in such a way that the flow curves remain un-affected. In Fig. 4, we have compared the flowcurve gen-erated with the parameters of Table I, i.e with two bathparticles with that of Table II, i.e. with three bath par-ticles. ● ●● ● ●●●●●●●● ●● ●● ●●●●●●●●● ● ● ●● ●● ° C, exp.30 ° C, sim. [ ] ° C, sim. [ ] η [ P a - s ] v ( nm / s ) FIG. 4. Comparison of flowcurves simulated at 30 ° C with twoand three bath particles.[1] U. Seifert, Rep. Prog. Phys. , 126001 (2012).[2] K. Sekimoto, Progr. Theoret. Phys. Suppl. , 17(1998).[3] J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996).[4] R. G. Larson,