How are mobility and friction related in viscoelastic uids Juliana Caspers1aNikolas Ditz2Karthika Krishna Kumar2F elix Ginot2Clemens Bechinger2Matthias Fuchs2 and Matthias Kr uger1

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How are mobility and friction related in viscoelastic ﬂuids?

Juliana Caspers,1, a) Nikolas Ditz,2Karthika Krishna Kumar,2F´elix Ginot,2Clemens Bechinger,2Matthias Fuchs,2

and Matthias Kr¨uger1

1)Institute for Theoretical Physics, Georg-August Universit¨at G¨ottingen, 37073 G¨ottingen,

Germany

2)Fachbereich Physik, Universit¨at Konstanz, 78457 Konstanz, Germany

(Dated: 7 October 2022)

The motion of a colloidal probe in a viscoelastic ﬂuid is described by friction or mobility, depending on

whether the probe is moving with a velocity or feeling a force. While the Einstein relation describes an

inverse relationship valid for Newtonian solvents, both concepts are generalized to time-dependent memory

kernels in viscoelastic ﬂuids. We theoretically and experimentally investigate their relation by considering

two observables: the recoil after releasing a probe that was moved through the ﬂuid and the equilibrium mean

squared displacement (MSD). Applying concepts of linear response theory, we generalize Einstein’s relation

and thereby relate recoil and MSD, which both provide access to the mobility kernel. With increasing

concentration, however, MSD and recoil show distinct behaviors, rooted in diﬀerent behaviors of the two

kernels. Using two theoretical models, a linear two-bath particle model and hard spheres treated by mode-

coupling theory, we ﬁnd a Volterra relation between the two kernels, explaining diﬀering timescales in friction

and mobility kernels under variation of concentration.

I. INTRODUCTION

Observing the Brownian motion of colloidal probe par-

ticles can be used to investigate complex ﬂuids, soft mate-

rials or biological tissues1–4. The technique of ’microrhe-

ology’ provides insight into local material properties and

thus extends macrorheological investigations. Recently,

such investigations tracking colloidal probe particles were

performed in model complex ﬂuids, where the macro-

scopic rheological response is rather well characterized.

An example are wormlike micellar solutions, for which

local ﬂow curves, i.e. nonlinear force-velocity relations5,6,

particle oscillations during shearing6–9 and transient par-

ticle motion10,11 were investigated. The existence of a

number of diﬀerent relaxation channels for the probe mo-

tion was recorded, which could be considered an intrin-

sic property of the system of viscoelastic ﬂuid plus im-

mersed colloidal particle. In contrast, in dense colloidal

suspensions12–15, a delocalization transition at ﬁnite forc-

ing strength was discovered16. At a critical force, the

probe decouples from the surroundings and its motion

records very atypical bath particle ﬂuctuations.

The friction force experienced by an individual Brown-

ian particle and the velocity by which it moves relative to

a Newtonian solvent are related by a friction coeﬃcient

γ. Alternatively, the velocity the particle attains when

subject to a force can be written in terms of a mobility

µ. For the mentioned case of Newtonian solvent, mobil-

ity and friction coeﬃcients are each other’s inverses and

are also connected to the diﬀusion coeﬃcient via temper-

ature in the famous Stokes-Einstein-Sutherland relation

D0=kBT µ =kBT

γ; here kBis Boltzmann’s constant17.

In viscoelastic ﬂuids, where memory matters, it is

well known that both coeﬃcients generalize to time-

a)Electronic mail: j.caspers@theorie.physik.uni-goettingen.de

dependent (retarded) kernels whose time-dependence en-

codes the temporal correlations of the ﬂuid18,19. Rear-

rangements of the ﬂuid take longer with e.g. increasing

concentration and thus forces at earlier times still inﬂu-

ence the velocity at the present time. A similar memory

of motion at earlier times also aﬀects the friction force

at present. Regarding the Einstein relation, the kernels

are then in general no longer related15,20, not even at

zero frequency, i.e., large times. For example, the fric-

tion memory kernel may depend on the conﬁnement of

a probe particle21–24. There is so far no complete un-

derstanding how the time-dependencies of mobility and

friction kernels are related.

Recent microrheology experiments of colloidal probes

in worm-like micellar solutions indicated that recoil spec-

tra could provide crucial insights10,11. It is known

that such micellar solutions in the semi-dilute regime

are well described by a Maxwell model with a sin-

gle timescale25–28; at higher concentrations more com-

plex relaxation behavior can occur29,30. The Maxwell

model captures the memory on macroscopic scales such

as macrorheological measurements. Moreover, previous

works could systematically ﬁt the equilibrium Brownian

motion, for example the mean squared displacement, of

immersed colloids to Maxwell’s model with a single relax-

ation time10,11,31–34. Yet, ’recoil’ measurements, which

test the back-motion of the colloidal probe when released

after it was moved with optical tweezers, recorded a very

diﬀerent temporal evolution. Most notably, the recoil dy-

namics exhibit at least two timescales, with the faster one

being much shorter than Maxwell’s macroscopic relax-

ation time10. Experiments of partial loading and partial

relaxation (close to equilibrium) showed the same two

timescales, which generates the question how recoil and

equilibrium mean squared displacement are related.

In the present work, we build on these ﬁndings and

study equilibrium mean squared displacements of a probe

and its recoils after weak drivings. Applying concepts

arXiv:2210.02801v1 [cond-mat.soft] 6 Oct 2022

FIG. 1. Left: Sketch of transient recoil experiment (gray line)

after a colloidal probe particle in a complex ﬂuid was per-

turbed with force F(red line), applied e.g. with optical tweez-

ers (inset), for a time tsh. Right: Equilibrium mean squared

displacement (MSD) of a colloidal particle in a wormlike mi-

cellar solution (CPyCl/NaSal 7 mm, open blue symbols). The

two experiments are related via a linear response relation.

from linear response theory enables us to determine the

relation between the two experiments. We will ﬁnd that

one (the equilibrium mean squared displacement) is dom-

inated by the retarded friction kernel, whose integral

grows strongly with density, while the other (recoils) pro-

vides direct access to the mobility kernel. Analysing two

theoretical models, this explains the diﬀering behaviors

recorded in either experiments.

The manuscript is organized as follows: we start with

the derivation of a linear response relation between recoil

and MSD in section II. In section III A we introduce the

experimental system and in section III B compare exper-

imental MSD and experimental recoil, i.e. test the linear

response relation. The analysis of two theoretical models,

a linear two-bath particle model and mode coupling the-

ory, is presented in section IV. In section V we discuss

the relation between these two models and explain the

observed concentration-dependence of recoil and MSD.

II. LINEAR RESPONSE RELATION BETWEEN RECOIL

AND MSD

In this work we theoretically and experimentally study

the motion of a colloidal probe particle in a viscoelastic

ﬂuid, namely a wormlike micellar solution, keeping in

mind that the theoretical analysis is valid for other vis-

coelastic ﬂuids, such as dense colloidal suspensions, as

well. Speciﬁcally, we are interested in the experimental

setups depicted in Fig. 1. Its left panel shows so called re-

coils, where the particle is dragged by a force for a certain

amount of time, and is then released at time t= 0. Due

to the deformed microstructure and accumulated strain

in the viscoelastic ﬂuid, after releasing, the particle per-

forms a backward motion opposite to the direction of the

forced motion. We deﬁne the (positive) recoil δx(t) as

δx(t)≡ − [hx(t)i−hx(0)i],(1)

with h. . . idenoting averaged quantities. The right hand

side of Fig. 1 shows the mean squared displacement

δx2(t)≡[x(t)−x(0)]2eq

18, obtained in absence of any

driving.

Here, we derive a relation between these quantities.

This derivation is explicit, based on a microscopic system

of identical spherical particles, obeying overdamped dy-

namics17. However, the ﬁnal equation, Eq. (14), encodes

the general form of the ﬂuctuation-dissipation-theorem

(FDT)18, and is thus valid in more general settings.

One of the particles is considered as probe particle.

The free diﬀusion coeﬃcient is D0=kBT/γ and the

particles interact via forces Fi=−∂iVwhere quantities

without index, such as F=−∂V , refer to the probe

particle, and ∂is a spatial gradient. Perturbing the probe

with a small time-dependent force Fex(t), leads to the

Smoluchowski operator17

Ω(t)=Ωeq +D0

kBTFex(t)·(−∂).(2)

The solution for the probability density is split in a sim-

ilar fashion

Ψ(t) = Ψeq +δΨ(t),(3)

where Ψeq ∝exp(−V/kBT) and δΨ arises from Fex. This

results in the linearized Smoluchowski equation for the

deviation of the probability distribution

∂tδΨ(t) = ΩeqδΨ(t)−D0

kBTFex(t)·∂Ψeq +O(F2

ex).(4)

The solution is (with β= 1/kBT)

δΨ(t) = −β

γZt

−∞

dt0Fex(t0)·e(t−t0)Ωeq FΨeq .(5)

Using this to calculate the linear response of an observ-

able Ato the external force we obtain a form, which may

be familiar from the ﬂuctuation dissipation theorem,

hA(t)ilr =hAieq −β

γZt

−∞

dt0Fex(t0)·DFe(t−t0)Ω†

eq AEeq .

(6)

In this isotropic system, we assume, without loss of gen-

erality, the external force to point in x-direction. We

are interested in the displacement of the particle hx(t)ilr,

linear in applied force. For symmetry reasons, only its

xcomponent is ﬁnite. Taking its time derivative results

in the application of the Hermitian conjugate Ω†of the

Smoluchowski operator on the variable x. We obtain

∂thx(t)ilr =γ−1Fex(t)−Zt

−∞

dt0γ−1Fex(t0)M(t−t0),

(7)

where

M(t−t0) = D0DβFxe(t−t0)Ω†

eq βFxEeq (8)

is the mobility-kernel, the correlation of interaction force

felt by the probe particle, evaluated in equilibrium.

The desired connection to the equilibrium mean

squared displacement (MSD) starts from 18,35

∂tδx2(t)=2D0−2D0Zt

ds0M(s0).(9)

Comparing Eq. (9) with (7) lets us conclude

hx(t)ilr =1

2kBTZt

−∞

dt0∂tδx2(t−t0)Fex(t0) (10)

which can be checked by reinserting. This result is the

central linear response relation generalizing Einstein’s re-

lation between friction and diﬀusion coeﬃcients to arbi-

trary time dependencies of the mean drift and the quies-

cent mean squared diﬀusion.

Upon inserting this into the deﬁnition of a recoil,

Eq. (1), we obtain for the recoil in linear response

δx(t) = 1

2kBTZ0

−∞

dt0Fex(t0)∂t0δx2(t−t0)−δx2(−t0)

−1

2kBTZt

dt0Fex(t0)∂tδx2(t−t0).

(11)

In the case of a time independent force acting from t=

−tsh to t= 0, as used in our experiments, the recoil

simpliﬁes to (t > 0)

δx(t) = Fex

2kBTδx2(t) + δx2(tsh)−δx2(t+tsh).(12)

For ergodic ﬂuids, where the MSD is diﬀusive for large

correlation times, we deﬁne the long-time diﬀusion coef-

ﬁcient D∞as

lim

t→∞

dtδx2(t) =: 2D∞.(13)

With this, the recoil takes an even simpler relation in

the limit of long shear times, tsh → ∞ (corresponding

to the limit where the probe attained its stationary drift

velocity before release),

δx(t) = Fex

2kBTδx2(t)−2D∞t.(14)

Eq. (14) shows a relation between recoil and MSD, and

interestingly it is the MSD minus its long time asymp-

tote that appears. Eq. (14) also already hints that recoil

experiments, probing an interesting non-equilibrium as-

pect of the system, may give important insights into the

equilibrium dynamics as well.

III. TEST IN MICELLAR SOLUTIONS

Having established a relation between MSD and the

recoil, we aim to apply it to experimental data obtained

in wormlike micellar solutions.

We note right away that our experiments use optical

tweezers, and are not performed by controlling the driv-

ing force, so that the requirements for Eq. (14) to be valid

are not strictly given.

A. Experimental setup

In the experiments we analyzed viscoelastic equimo-

lar cetylpyridinium chloride monohydrate (CPyCl) and

sodium salicylate (NaSal) solutions with concentrations

ranging from 5 mmto 9 mm(recoil measurements are lim-

ited to 7 mm), which are contained in a 100 µm thick

sealed sample cell. The cell was kept at a constant tem-

perature of 25 °C, leading to the formation of an entan-

gled network of giant worm-like micelles28. To probe the

ﬂuid’s microrheology, we suspended a small amount of

silica probe particles with diameters 2.73 µm in the solu-

tion.

For the measurement of recoils the colloidal probe is

trapped in an optical tweezer, built of a Gaussian laser

(λ= 1064 nm) and a high magniﬁcation microscope ob-

jective (100×, NA = 1.45). Using relatively large trap-

ping strengths (see Table I in Appendix A), the probe is

positioned in the center of the trap and we can apply a

constant-velocity perturbation via a relative motion be-

tween the probe/trap and the ﬂuid. This is achieved

by a computer controlled piezo-driven stage that trans-

lates the sample cell with constant velocity vand that is

synchronized with the laser intensity. The time of trans-

lation or shear time tsh is chosen suﬃciently long, such

that the probe reached a non-equilibrium steady state

before release. To avoid interactions with the sample

walls, the trap was located at least 30 µm away from any

surface. We extract the probe’s trajectory by video mi-

croscopy with a frame rate of 100 Hz, which, using a cus-

tom Matlab algorithm36 yields an accuracy of ±6 nm of

the particle position. For further details regarding the

experimental recoil setup, we refer to the recent work of

Ginot et. al11 and its Supplementary Material.

To test the linear response relation, Eq. (14), we also

measured the mean squared displacement of freely diﬀus-

ing colloidal particles suspended in the micellar solution,

and at the surface of the sample cell. The 2-dimensional

trajectories r(t)=(x(t), y(t))Tof the colloidal particles

were extracted using video microscopy and above men-

tioned Matlab algorithm. Due to the system’s isotropy,

we calculated the (one-dimensional) MSD, after subtract-

ing drift from the trajectories, and by averaging over both

dimensions,

δx2(t)≡1

2D|r(t0+t)−r(t0)|2Et0

.(15)

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Howaremobilityandfrictionrelatedinviscoelasticuids?JulianaCaspers,1,a)NikolasDitz,2KarthikaKrishnaKumar,2FelixGinot,2ClemensBechinger,2MatthiasFuchs,2andMatthiasKruger11)InstituteforTheoreticalPhysics,Georg-AugustUniversitatGottingen,37073Gottingen,Germany2)FachbereichPhysik,UniversitatKonstan...

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How are mobility and friction related in viscoelastic uids Juliana Caspers1aNikolas Ditz2Karthika Krishna Kumar2F elix Ginot2Clemens Bechinger2Matthias Fuchs2 and Matthias Kr uger1.pdf

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How are mobility and friction related in viscoelastic uids Juliana Caspers1aNikolas Ditz2Karthika Krishna Kumar2F elix Ginot2Clemens Bechinger2Matthias Fuchs2 and Matthias Kr uger1

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