Brownian motion in a viscoelastic medium modelled by a Jeffreys fluid

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This article can be cited before page numbers have been issued, to do this please use: Y. L. Raikher, V. V. Rusakov and R.Perzynski, Soft Matter, 2013, DOI: 10.1039/C3SM51956B.

http://dx.doi.org/10.1039/c3sm51956b

http://pubs.rsc.org/en/journals/journal/SM

Brownian motion in a viscoelastic medium

modelled by the Jeffreys fluid

Yuriy L. Raikher ∗ Victor V. Rusakov † Régine Perzynski ‡

September 11, 2013

Abstract

Theory of Brownian motion of a particle in a viscoelastic Jeffreys fluid is ex-

tended for the case of rotational motion. The employed rheological model com-

bines two viscous mechanisms (instantaneous and retarded) and, in contrast to the

Maxwell model, does not produce artifacts and works robustly when applied to dif-

fusion of tracer particles in real complex fluids. With the aid of it, specific features

of the dynamic susceptibility of a magnetic Jeffreys suspension and the viscous

power losses induced by an ac field are analyzed deriving the conclusions valid for

active microrheology and magnetic hyperthermia. In general aspect, it is shown

that the developed phenomenology provides an archetypal “frame” for a number

of mesoscopic models used to describe confined random walk transport processes

in a variety of systems of both biological and inorganic origin.

1 Introduction

Brownian motion in complex fluids is a subject of intense and yet increasing interest.

This ubiquitous fluctuational process is very important from the fundamental as well∗Institute of Continuous Media Mechanics, Ural Branch of RAS, Perm, 614013 and Ural Federal Uni-

versity, Ekaterinburg, 620083, Russia; E-mail: [email protected]†Institute of Continuous Media Mechanics, Ural Branch of RAS, Perm, 614013, Russia and Perm Na-

tional Research Polytechnical University, Perm, 614990, Russia; E-mail: [email protected]‡Université Pierre et Marie Curie, Laboratoire PECSA, UMR 7195, case 51, 4 place Jussieu, 75005 Paris,

France; E-mail: [email protected]

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View Article OnlineDOI: 10.1039/C3SM51956B


from practical viewpoints. Diffusion of micro- and/or nanosize particles plays essential

role in the physical chemistry, chemical technology, and material science of inorganic

composites and polymers. Diffusion-mediated transport is an integral part of cell and

tissue biology and biotechnology, while diffusion-aided pharmacokinetics is a vital

subject in medicine. All these applications are in need for the theory of Brownian

motion beyond the classical limit of a linearly viscous (Newtonian) fluid. This need

is two-fold. From one hand, such a theory can provide a more detailed description

of the molecule and nanoparticle penetration through complex systems of inorganic,

organic and biological origin. On the other hand, if to use the Brownian particles

as microprobes, the theory shows the way to extract reliable rheology info from the

recorded data.

The issue of prime interest delivered by the theory of Brownian motion is the mean-

square (MS) displacement of a particle in a given medium. As soon as it is evaluated,

its Laplace (one-side Fourier) transform yields the complex modulus G = G′ + iG′′

of the analyzed substance. Such an approach, termed microrheology, is widely used

to study complex media, including biological substances, cell parts (e.g. membranes),

and even cells in vivo, see1 for the state-of-art review. Tracking the motion of fine

solid particles enables one to probe small-scale intrinsic deformations in these me-

dia2–8 that is unavailable in the conventional macrorheology. Besides that, these tests

can be performed on the samples of tiny volume and reach to much higher frequency

range. Another great advantage of microrheology is a possibility to use it in both pas-

sive (the particles driven solely by fluctuations) and active (the particles excited by

external fields) variants.

Hereby we present a theory of Brownian motion in a viscoelastic fluid, whose non-

Newtonian behavior is modeled by the Jeffreys rheological scheme,9 see Fig. 1. The

latter structure has one extra element in comparison with the well-known Maxwell

model. Namely, besides the retarded viscosity (chained elastic and viscous parts), it

contains a purely viscous element. With this simple modification, a consistent phe-

nomenological description of free and forced diffusion in a Jeffreys fluid (JF) is devel-

oped, which works well for living polymer solutions. Considering dipole particles as

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an example, we show that JF model is easily extendable to include additional forces

and frictions working on the Brownian particle. Besides that, when used for microrhe-

ology description, JF model proves to be robust and does not produce the non-physical

effects inherent to Maxwell model. Finally, we demonstrate that the theory of Brown-

ian motion in JF has a much wider significance because it makes a uniting “frame” that

takes in a variety of mesoscopic models of confined random walk (CRW) processes.

This specific type of non-standard diffusion is an important issue in cell biophysics10

as well as in the material science of inorganic nanopore and membrane materials.11 In

particular, our phenomenology shows that the existence of the “cage”, which is a key

hypothetical object in any CRW model, is a natural property of a viscoelastic medium

of the Jeffreys-type.

The paper is organized in the following way. In Sec. 2 we remind the dynamic

and Langevin equations for the translational motion of a particle in a JF. The general

expression for the MS displacement is derived, reduced to the inertia-free form, and

proven to work well for living polymers. In Sec. 3 the theory is re-formulated for the

rotational Brownian motion, and the occurrence of two diffusion regimes—fast and

slow—is shown. The crossover between this modes, manifesting itself as the parti-

cle confinement, is commented on, and the advantages of JF in comparison with the

Maxwell one are emphasized. On the basis of the worked out rotational model, in Sec.

4 the linear response of a dilute viscoelastic suspension of dipolar particles to an ac

field is obtained. It is shown that the presence of two diffusion mechanisms yields

a double-peak shape of the low-frequency absorption spectrum. Sec. 5 sums up the

essence of the obtained results and outlines their theoretical significance and possible

applicational aspects.

2 Langevin equations and mean-square displacement

We begin with a one-dimensional translation motion of a particle with mass m embed-

ded in a viscoelastic JF and subjected to a regular force f = −∇U with the potential

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N

M m

x

G

Figure 1: Schematic representation of the Jeffreys rheological model.

U . The pertinent set of equations was obtained in our preceding work:12

dxdt

= v,dvdt

=−ζN

mv+

1m

(Q− ∂U

∂x

)+

1m

yN(t),

dQdt

=− 1τM

[Q−ζMv+ yM(t)] ,⟨yα(t)yβ (t

′)⟩= 2Dα δαβ δ (t− t ′), (1)

where the dynamic variable Q has the meaning of a time-dependent dissipative force. In

equations (1) the friction the coefficients ζα = 6πηα a are taken for a spherical particle

in the Stokes approximation. As seen from Fig. 1 and the second of equations (1),

the dissipative resistance to the particle motion has two sources. One of them (index

α = N) is a viscous (Newtonian) friction, while the other (index α = M) is the retarded

(Maxwellian) friction that imparts viscoelasticity to the medium. The same indices

mark the random forces yα(t) and the diffusion coefficients Dα . The Maxwell stress

relaxation time τM = ζM/K in (1) is defined by the spring rigidity parameter K, related

to the elastic modulus G (see Fig. 1), as K = 6πGa. In other words, τM is taken in the

standard microrheology approximation: assuming coincidence of form factors for the

viscous and elastic forces.6,7

As well known, see,13,14 for example, the set of the Langevin equations (1) cor-

responds to a Fokker-Planck equation with several diffusion coefficients, one per each

phase variable. The particular forms of those coefficients follow from the equilibrium

condition requiring that the state of a particle in a thermostat is entirely determined by

its energy and does not depend on relaxation mechanisms. Therefore, the equilibrium

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distribution function has a generalized Maxwell-Boltzmann form:12

W0(x,v,Q) ∝ exp{− 1

T

[mv2

2+U(x)+

Q2

2K

]}. (2)

The term ∝ Q2 is associated with the dynamic elasticity of the system, since the particle

motion induces some amount of elastic energy in the viscoelastic environment. As

follows from (2), in thermal equilibrium the phase variables (x,v,Q) are statistically

independent. Moreover, since v(t) and Q(t) are Gaussian random processes, they are

defined by their second statistical moments: 〈v2〉 = T/m, 〈Q2〉 = T K, 〈vQ〉 = 0;

henceforth the angular brackets denote statistical averaging over the equilibrium state,

i.e., with the distribution function (2).

From the first of equations (1), one finds the MS displacement in the form

〈(∆x)2〉= 2t∫

0

dt ′t ′∫

0

dt ′′〈v(0)v(t ′− t ′′)〉. (3)

The Cauchy problem for the velocity autocorrelation function also follow from (1) after

multiplying it by the initial value v(0) and performing the ensemble averaging. Using

the fact that, due to the causality principle, the correlations of v(0) with random forces

are zeroes, one arrives at

[(ddt

+ γm

)(ddt

+ γM

)+

Km

]〈v(t)v(0)〉= 0, (4)

〈v(0)v(0)〉= 〈v2〉= T/m,ddt〈v(t)v(0)〉

∣∣t=0 =−γm〈v2〉

with γm = 1/τm = ζN/m being the velocity relaxation rate and γM = 1/τM . The solution

of equations (4) is

〈v(t)v(0)〉= 〈v2〉λ2−λ1

[(λ2 + γM)eλ2t − (λ1 + γM)eλ1t

], (5)

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where λ ’s are the roots of the characteristic equation, which we present in the form

(λ + γM)(λ + γMM/q)+ γ2MM = 0, (6)

introducing the dimensionless parameters

M = ζMτM/m = ζ2M/(mK) and q = ζM/ζN = τM/τN . (7)

The first of them, the Maxwell number, characterizes the relation between the elastic,

viscous and inertial forces experienced by the particle. Note that M comprises only

the parameters of Maxwell elements of the Jeffreys model. The second parameter, q,

relates the two viscosities of the JF. In the representation (7) one has γM = (q/M)γm.

Solution of (7) yields the roots

λ1,2 =−12

γM

[1+M/q±

√(M/q−1)2−4M

]. (8)

Substituting this in the autocorrelation function (5), one obtains via equation (3) the

MS displacement of a particle in the JF:

〈(∆x)2〉= 2〈v2〉λ2−λ1

[λ2 + γM

λ2

(eλ2t −1

λ2− t

)− λ1 + γM

λ1

(eλ1t −1

λ1− t

)]. (9)

As seen from (5), at t < τm the particle motion is inertia-dominated (ballistic),

so that expression (9) reduces to 〈(∆x)2〉 ∼= 〈v2〉t2 [1− t/3τm]. For microrheology,

however, actual is the inertia-free regime (m → 0), i.e., τm � τM or, equivalently,

M� (1+q)/q. For that case the roots of the characteristic equation (6) are

λ1 ∼=−γMM/q, λ2 ∼=−γM(1+q), (10)

and the one-dimensional MS displacement takes the form

〈(∆x)2〉= 2TτNK(1+q)

{t +

τMq1+q

[1− e−(1+q) t

τM

]}, (11)

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with an obvious extension for three-dimensional case: 〈(∆r)2〉= 3〈(∆x)2〉.

We note that formula (11) was obtained in Ref.12 from the inertia-free Langevin

equations. Here we, first, derive expression (9) valid for a particle of a finite mass

and then confirm (11) passing to the limit m→ 0. In the terms of Jeffreys material

parameters, general expression (9) takes the form

〈(∆r)2〉= 6TτNK(1+q)

{t + τM

q(M−1)(1+q)M

[1− exp

(−q+M

2qτMt)

sinh(β t +ψ)

sinhψ

]},

(12)

with the phase given by

cothψ =3M−1+M(M+1)/q

(M−1)√(M/q−1)2−4M

, and β =1

2τM

√(M/q−1)2−4M. (13)

In Ref.12 an example was given demonstrating that formula (11) is well capable of

describing the motion of probing microparticles in a “living polymer”, i.e., a concen-

trated solution of wormlike micelles. Fig. 2 extends this illustration. Pane (a), taken

from,12 refers to the particles in cetyltrimethylammonium bromide / potassium bro-

mide solution.15 In pane (b) equation (11) is applied to the Brownian motion in the so-

lution of hexa-ethylene glycol mono-n-hexadecyl ether (giant micelles).16 We remark

that for either of these viscoelastic solutions the estimates for the ratios τM = ηM/G

and T/aG are given in the reported data. Therefore, when fitting the measurements,

only one of the three Jeffreys constants, viz. ηN , is treated as an adjustable parameter.

As seen, the achieved agreement spans over about eight time decades.

3 Mean-square displacement in the rotational motion

The active microrheology as well as the magnetic hyperthermia applications require

the theory of rotation Brownian motion in a viscoelastic medium. The passage from

the translation case is done as follows. To introduce the angular variables, we con-

sider a vector of constant length “frozen” in the particle body. As such, one can use,

for example, the major axis of an axisymmetrical particle, the direction of maximal

electric polarizability, or the particle permanent electric / magnetic dipolar moment,

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Figure 2: Application of formula (11) to diffusion measurements with: (a) particles ofdiameter 1 µm in a micellar solution CTAB/KBr15 with parameters τM = 1.05 s andq = τM/τN = 1.1×104; (b) particles of diameter 0.7 µm in a giant-micelle solution16

with parameters τM = 80 ms and q = 690.

if any. As long as the free rotational diffusion is considered, the particular choice of

this orientation marker does not matter. The origin of the marker becomes important,

however, in a situation, where the free and forced particle motions combine. Being in-

terested in “active” magnetic microrheology, in below we consider a viscoelastic fluid

with embedded single-domain (i.e., nanoscale) ferromagnetic particles possessing high

magnetic anisotropy. In this case the particle orientation is rendered by that of its mag-

netic moment µµµ . The rotational diffusion of µµµ is free in the absence of magnetic field,

but is directly influenced by it as soon as the field is switched on.

It is worthwhile to remark the difference between the translational and rotational

Brownian motion from the microrheological viewpoint. The translational mode is well

fit for the “passive” measurements: those, where the probing particles are driven just

by thermofluctuational forces. The resulting irreversible displacements (diffusion) are

registered with some optical method, e.g. DWS.15 The situation of rotational motion,

although admitting the “passive” variant,17 is rather well suited for “active” microrhe-

ology. Indeed, by subjecting the sample to a uniform ac field, one can easily excite

rotational oscillations of the dipolar particles embedded in the tested substance. This

periodic motion induces the alternating electric/magnetic polarization signal recordable

with radio-frequency technique. Analyzing this response, one can obtain the microrhe-

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ology data even in the cases, where the experimental conditions exclude the use of

optical methods, e.g. the fluid is opaque or the particles are too small. Another case of

forced rotational motion, evolving on thermofluctuational background, is the magnetic

hyperthermia. There the embedded magnetic nanoparticles excited by an applied ac

field are used as the source of local heating.

To simplify the description, we consider planar rotations of the particle. This is

feasible because the treatment of spatial rotations, albeit being much more cumber-

some, does not yield any essentially different results, just changes some numerical

coefficients. A set of good examples to that effect related to Brownian orientational

motion could be found in Ref.,18 see chapters 3 and 7 there. The advantage of the

planar problem, similarly to the case of Maxwell fluid,19 is that it admits an analytical

description.

We consider, first, the free Brownian rotations, i.e., the case of zero external field.

Then the orientation of the model particle (dipolar rotator) is determined by a single

angle ϑ , which the dipolar moment µµµ makes with a chosen axis of the coordinate

frame. The elements of the Jeffreys scheme are now transformed to their rotational

analogues, namely, the elasticity and friction coefficients are introduced as K = 6GV

and ζα = 6ηαV , where V = (4π/3)a3 is the particle volume. The pertinent equations

of motion follow from those of Sec. 2 upon replacements x⇒ ϑ and m⇒ I with I

being the moment of inertia of the rotator. Proceeding directly to the inertia-free limit,

from the transformed equation (11) one gets

⟨(∆ϑ)2⟩= 2T

ζN(1+q)

[t +

qλM

(1− e−λMt

)], λM =

1+qτM

. (14)

In the JF (Fig. 1), friction comes from two sources. The Newtonian element has the

response time τN = ζN/K, while that of the Maxwellian element is τM = ζM/K, their

ratio being q = τM/τN = ζM/ζN , as defined by (6). Complete domination of one of the

relaxation modes turns JF in a Newtonian (q= 0) or a Maxwell (q=∞) fluid. There are

two types of situations, where contributions from both relaxation mechanisms might

matter. The case of weak viscoelasticity (q� 1) is of minor interest since there the

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Brownian motion closely resembles that in a Newtonian fluid anyway. Non-trivial,

however, is the case of developed viscoelasticity (q� 1). There the timescales are

significantly different (τN � τM , i.e., q� 1), and the crossover region between the

diffusion modes is well pronounced. Expanding formula (14) under assumption q� 1

and for short times (t ≤ τN), one obtains the mode of fast rotational diffusion:

⟨(∆ϑ)2⟩≈ 2T

ζNt(

1− t2τN

). (15)

At the long timescale (t� τN), where the retarded viscosity is important, formula (14)

yields ⟨(∆ϑ)2⟩≈ 2T

ζM[t + τM] , (16)

indicating that at t ≥ τM the diffusion becomes stationary and is slowed down (q+1)≈

q times in comparison with the initial (fast) process.

We remark important consequences following from relations (15) and (16). First,

no matter how small the Newtonian viscosity is, it solely defines the initial stage of

diffusion. This is the essential distinction of the Jeffreys model from the Maxwell one.

In the Maxwell model equation (16) applies to any time interval, including t→ 0, where

it yields a finite value of the MS displacement:⟨(∆ϑ)2

⟩= 2T/K meaning complete

rotational localization of the particle. In other words, the particle surrounding behaves

as a perfect spring (no friction). Under such conditions the quality factor of the system

tends to infinity, so that any high-frequency excitation induces huge oscillations.19,20

Contradicting both the experimental evidence and physical expectations, such an effect

is an apparent artifact, which proves inappropriateness of the Maxwell model for the

Brownian motion. As far as we know, this issue remains not very much known, and

these non-physical oscillations are still being predicted in complex fluid systems, see,21

for example. The JF model is free of this flaw. There the Newtonian (non-retarded)

viscosity ensures the high-frequency friction, so that the short-time (t < τN) behavior

of the MS displacement is “regularized” and turns to zero at t→ 0, see (15).

Another notable fact following from the existence of two diffusion regimes (at short

and long times) is their crossover in the intermediate range τN < t < τM . In a JF with

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pronounced viscoelasticity (q� 1), the duration of this transient process is about qτN ,

i.e., exceeds τN many times. At this interval the value of MS fluctuation, which is

attained in result of the fast diffusion, remains practically time-independent:

⟨(∆ϑ)2⟩'Θ

2, Θ =q

1+q

√2TK

. (17)

Accordingly, in the plots⟨(∆ϑ)2

⟩(t) this region should yield a plateau similar to those

occurring in the translational motion, see Fig. 2. The temporal constancy of the angular

fluctuation may be termed “dynamic confinement” due to the fact that in this regime

the Maxwell part of the Jeffreys scheme works mostly as an elastic link and keeps the

particle within a region of the size Θ. The dynamic confinement is the tighter the higher

the JF elasticity K. Moreover, this confined fluctuation does not explicitly depend on

temperature. Assuming that the elastic modulus of JF is of the high-elasticity origin,

we set G' nT , where n is the cross-link density. Introducing the reference scale of the

matrix as `' n−1/3, e.g. the mesh size of a polymer network, one obtains the effective

location size in the form

Θ' q√π(1+q)

(`

2a

)3/2

. (18)

Revisiting the translational Brownian motion in a JF (Sec. 2), we find that there the

confined fluctuation and the effective size of the cage are given by the relations

⟨(∆rrr)2⟩' L2, L' q`

1+q

(`

πa

)1/2

, (19)

which point out that the translational “cage effect” has a stronger dependence on the

intrinsic scale of the matrix and a weaker dependence on the probe size than it is in the

rotational case. Note that at high viscoelasticity (q = ηM/ηN � 1) the effective size of

either cage but slightly depends on the ratio of the Jeffreys viscosities.

Summing up, we remark that the dynamic confinement is an inherent feature of

Brownian motion of a particle in a Jeffreys fluid. Its signature is virtual constancy of

the particle MS displacement, either positional or angular, along some intermediate

time interval. Such a situation in a natural way leads to a concept of a particle locked

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inside a cage. Evidently, this model cage is not a real physical entity but a helpful

representation for the occurring viscoelastic effect. This follows from the fact that Θ

as well as L depend on both the inner matrix scale (mesh size) and the size of the

Brownian probe.

4 Dynamic magnetic susceptibility

In “active” magnetic microrheological probing, the free Brownian motion of the sus-

pended particles is combined with the forced one induced by an ac field. Consider

a viscoelastic fluid, where the particle concentration is low. This, on the one hand,

prevents the properties of the matrix from modification by the dopant (particles); on

the other hand, this ensures a possibility to neglect the interparticle interactions of any

kind. Therefore, one deals with an ensemble of non-interacting particles, which is sub-

jected to the probing (low-amplitude) linearly polarized magnetic field HHH cosωt. The

coordinate ϑ introduced in Sec. 3 is re-defined, and now denotes the angle, which the

particle magnetic moment µµµ makes with the direction of HHH.

The response characteristic of the system, needed for the microrheology analysis

or for characterizing the efficiency of magnetic hyperthermia, is the Fourier component

of the dynamic magnetic susceptibility in the direction of the probing field: χ(ω) =

Mω/Hω with M being the system magnetization. From the general linear response

theory, see,18 for example, one finds

χ(ω) = χ0

[1+ iω

∫∞

0dteiωt G (t)

], G (t) = 〈cosϑ(t)cosϑ(0)〉/

⟨cos2

ϑ⟩

; (20)

here χ0 = χ(0) = nµ2⟨cos2 ϑ

⟩/T = nµ2/2T is the static (ω = 0) susceptibility, n the

particle number concentration, and G (t) the dipolar correlation function. Note that, as

before, the statistical averaging is performed over the equilibrium state at zero external

magnetic field.

Since the set of stochastic equations for the rotational motion of a dipolar particle is

linear, the angular displacement in the presence of the thermal white noise is a Gaussian

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random process. This enables one to accomplish averaging and transform the dipolar

correlation function (20) in

G (t) = exp(− 1

2

⟨(∆ϑ)2⟩) , ∆ϑ = ϑ(t)−ϑ(0); (21)

as seen, G is defined by the equilibrium MS angular fluctuation of the particle magnetic

moment.

Substituting phase fluctuation (14) in the correlation function (21) and then in the

linear response formula (20), one finds

χ(ω)

χ0= 1+ iω

∞∫0

dt exp[

iωt− t(1+q)τD

]exp[−Θ

(1− e−λMt

)], (22)

where τD = ζN/T is the standard rotational thermal relaxation (Debye) time defined

with respect to the Newtonian viscosity and parameter Θ is introduced in (17). Ex-

panding the integrand in (22) in powers of Θ, one gets the representation

χ(ω)

χ0= e−Θ

∞

∑k=0

Θk

k! [1− iωτD(1+q)/(1+qk/Θ)], (23)

which is quite feasible for numeric calculations.

For a nearly Newtonian fluid (q→ 0) expression (23) tends to the standard Debye

formula

χ(ω)/χ0 ∼= χD(ω)/χ0 = [1− iωτD]−1 . (24)

For a fluid with a pronounced viscoelasticity (q� 1), several situations may occur. If

the elastic component is so weak that it ensures the condition Θ� 1, from expression

(21) it follows that the MS fluctuation evolves mostly via the fast diffusion process. In

this limit⟨(∆ϑ)2

⟩' 2Tt/ζN , see (15), and the result of integration in (22) is expectedly

close to the Debye susceptibility (24). At strong elasticity (Θ� 1), when the slow

diffusion (16) dominates, integral (22) again renders the Debye-type formula but now

with a different reference time. Namely, instead of τD it is τS = (1+q)τD� τD. Thus,

one sees that in both limits, q→ 0 and q→ ∞, the system behaves as if possessing

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just one relaxation mechanism and, accordingly, is described by a single-peak dynamic

susceptibility. More complex shapes emerge at moderate Θ’s, where the contributions

of both diffusion mechanisms are comparable.

Figure 3: Dynamic susceptibility curves for a suspension of dipolar particles in vis-coelastic medium with q = 13 (a) and q = 690 (b); unitless temperature Θ = 0.1 (1),0.2 (2), 0.5 (3), 1.0 (4).

Fig. 3 displays the frequency dependencies of the imaginary (out-of-phase) compo-

nent of the magnetic susceptibility obtained with the aid of expression (16) for several

values of the viscoelastic parameters Θ and q = ηM/ηN . With the chosen frequency

units, one peak is always centered close to ωτD ∼ 1, and the other one is located in

the vicinity of ωτD ≈ 1/(1+q). The case q = 13 refers to amebae cytosol;5 while the

value q = 690 is drawn from application of JF model to experiment16 interpreted in

Fig. 2b. Both plots of Fig. 3 evidence domination of the single-mode regime at low and

high Θ and demonstrate the double-mode relaxation in the intermediate range. The

obtained splitting seems strong enough to ensure identification of the viscoelasticity

of the suspensions matrices from their magnetic spectra. Transforming the reference

double-mode condition Θ' 1 to the dimensional form, one gets the appropriate radius

of the probe for that case q > 1:

aref ' 12 (T/G)1/3. (25)

In a high-elasticity material the reference mesh size is `∼ (T/G)1/3, one sees that the

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double-peak regime of χ ′′ emerges when the particle diameter 2a is of the order of `.

This ensures that in the regime Θ∼ 1 the probe in full interacts with the mesh.

5 Discussion

5.1 Jeffreys vs Maxwell models

An important fact following from our considerations is the failure of the Maxwell

model with a single relaxation time to be used for the Brownian motion theory in a

viscoelastic fluid. To prove that, let us compare JF and its limit q→ ∞ corresponding

to the case of Maxwell fluid. Consider the velocity autocorrelation function (5). It

shows that the character of the particle motion is defined by the sign of the determinant

in equation (6). Accordingly, in the material parameter space {M,q} there are the lines

(neutral curves), which separate the regions of monotonic and oscillatory relaxation. In

unitless form, the neutral curves are determined by equation (1−M/q)2 = 4M, where

the Maxwell number is introduced in (7). Sorting out the roots of the quadratic equa-

tion, one finds that for a particle embedded in a JF the oscillatory relaxation conditions

are

M1+2

√M

< q <M

1−2√

Mfor M < 1/4;

M1+2

√M

< q for M > 1/4. (26)

The illustration is given in Fig. 4, where the oscillatory regime takes place inside the

“sector” in between the two neutral curves.

In the Maxwell limit (q→ ∞) conditions (26) reduce to a single bound M > 1/4.

Converting it in the dimensional form, one arrives at the relation

4ζ2M > mK ⇒ η

2M > mG/(24πa) , (27)

which shows that the oscillatory regime in a Maxwell fluid emerges, when the retarded

viscous friction dominates the inertia and elastic forces. In other words, in a Maxwell

fluid the oscillatory relaxation is the stronger the greater is friction. This conclusion,

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Figure 4: Regions of oscillatory (white) and monotonic (gray) relaxations in the planeof viscoelasticity parameters separated by the branches of the neutral curve; dashesmark the line M = 1/4; the region M > 1/4 whatever q corresponds to the oscillatorydiffusion of a particle in a Maxwell fluid.

being in contradiction with the physical understanding of friction in dynamic processes,

might be valid only in a purely formal sense. Looking closer, one finds that this non-

physical result stems from the absence of the “fast” (non-retarded) viscosity in the

Maxwell model. Due to that, in the Maxwell fluid any particle excitation takes the

form of oscillatory motion, whose quality factor Q' 2√

M = 2ζM/√

mK is the greater

the higher is the long-time (“slow”) viscosity ηM and holds the easier the smaller the

particle mass. Consequently, tending m to zero (the inertia-free limit), one gets the

oscillatory regime for any Maxwell medium. All this evidences that a meaningful

description of Brownian motion in a Maxwell viscoelastic fluid is impossible. This is

noteworthy since the oscillatory motions of Brownian particles had been predicted in a

number of theoretical works done in the Maxwell approximation, see15,19,20 and21 as

a fresh example.

The fundamental advantage of the JF model is that it “regularizes” the Brownian

motion theory in a viscoelasic medium, thus enabling one to construct a description

free of artifacts. To see that, consider a typical viscoelastic system, where M,q� 1.

According to (26), the oscillatory relaxation occurs at q>√

M/2 or, in the dimensional

form√

mK > 12 ζN . Using the definitions of the material parameters and substituting in

(28) their reference values τM ≈ 0.1 s, ηM ∼ 10 P, and ρ ∼ 1 g/cm3, one arrives at an

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estimate for the lower bound for the particle size:

a∗ = 32

√τMηM/2ρ ∼ 1cm, (28)

implying that oscillatory diffusion becomes formally possible only at a > a∗. However,

the obtained value of a∗ is so high that for such particles the diffusion process is com-

pletely insignificant. This explains, why the developed theory of Brownian motion in

a JF does not entail any artifacts and is robust either in its simplest form (with two vis-

cosities) or with any modifications implying a distribution of the viscosity coefficients.

5.2 Viscoelasic losses in hyperthermia

The above-presented results have one more important aspect. It concerns magnetic hy-

perthermia: the induction of local heating by working on suspended magnetic nanopar-

ticles with an ac field. This method is an issue of high interest for theranostics,22–26

and oncology27–31 as a means of thermally-mediated treatment of malignant cells. As

far as we know, until now both the theoretical studies and experiment interpretations

are based on the “dichotomy” that the carrier media, where the particles are embedded

to, either have Newtonian rheology or are solid. This assumption is used to evaluate

the viscous losses (i.e., specific heating) produced by the particle rotations. However,

when considering magnetic hyperthermia of biological objects, especially when it is

performed inside single cells,1,32 it is but natural to expect that the carrier fluid has

non-Newtonian rheology. The dynamic susceptibility of a Jeffreys magnetic suspen-

sion provides an example enabling one to assess, in what way do the viscous losses

change in comparison with the Newtonian case. As here the rigid dipole model is

used, this comparison applies for the cases, where highly anisotropic (cobalt or barium

ferrite) particles are used.

The principal characteristic of heat generation in the magnetic hyperthermia prob-

lems is the specific loss power (SLP), i.e., the rate of the ac field energy absorption. In

the linear response limit this quantity is defined as SLP = ωImχp/ρ , where χp is the

magnetic susceptibility per unit volume of the particle and ρ the specific density of the

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particle material. The expression for SLP in a suspension with the Jeffreys matrix is

easily derived from equation (23). The frequency dependencies of this SLP under vari-

ation of viscoelasticity for two values of the elasticity parameter Θ are shown in Fig.

5. In a standard Newtonian suspension (q = 0) with viscosity ηN , function SLP(ω)—

see curves 1—starts with quadratic ascend and then gradually saturates. For a Jeffreys

suspension (q 6= 0), where two viscosities (ηN and ηM) are present, each of the curves

(2 or 3) virtually combines of two Debye-type absorption lines. Due to that, such func-

tions SLP(ω) acquire intermediate flattened parts (plateaux). The ultimate cause of this

effect is the “dynamic confinement” of the embedded particles caused by interplay of

the two relaxation (diffusion) mechanisms accessible for them in the JF.

Therefore, in a JF in the low-frequency range ωτD� 1 the particles, when moving,

perturb the structure of the viscoelastic matrix at the mesoscopic scale `. Accordingly,

the energy is absorbed as if in a highly viscous system: ηM = GτM , and the obtained

SLP values turn out to be orders of magnitude greater than those for a Newtonian

matrix with the same viscosity ηN . As the frequency grows, the system enters the

domain, where the particle confinement occurs. Upon that, the situation inverts, and

the expected viscoelastic SLP (the plateau) falls much below the Newtonian level. As

Fig. 5 shows, the frequency, at which the lines (e.g. 1 and 2) cross, is the lower the

greater is the viscosity ratio parameter q. With the further frequency increase, the

system leaves the interval of “dynamic confinement”, and the viscoelastic SLP begins

to grow fast approaching the Newtonian limit and joining it at the saturation level.

For completeness, we note that the saturation behavior of our SLP dependencies at

ω → ∞ is inherent to any Debye-type dispersion curve and is a direct consequence of

neglecting the particle inertia, which we adopt here.

Therefore, taking the JF as a model example, the problem of the Brownian particle-

mediated viscous losses can be solved to a detail. The analysis of this solution reveals

some important features, formerly not accounted for, which might affect heat genera-

tion by ac field-driven magnetic particles embedded in a viscoelastic matrix.

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Figure 5: Specific loss power in a suspension of dipolar particles in a Jeffreys fluidwith elasticities Θ = 0.2 (a) and Θ = 1 (b); the viscous parameter q = 0 (1, Newtonianfluid), 13 (2), 690 (3).

5.3 Mesoscopic aspect

It turns out that, as any sound phenomenology, the theory of Brownian motion in a

JF provides an archetypal “frame” for a number of mesoscopic models developed for

physically rather different systems. We refer to the type of behavior known as confined

random walk (CRW) motion.33 CRW processes are encountered when tracking pro-

tein transport on the cell surfaces,34 performing microrheology tests on the semi-dilute

polymer solutions,35 monitoring penetration of nanoscale objects and ions through in-

organic sponges,11 etc. The individual features of the considered systems put aside, the

mesoscopic basis of all CRW models is universal. It implies that each Brownian parti-

cle resides in a “cage” filled with a low-viscous fluid. The particle can diffuse within

the cage, but the cage walls are impenetrable. For example, in a polymer solution, this

in-cage fluid is identified with the pure solvent.35 The cage, in its turn, also undergoes

diffusion, but in a much less compliant medium with the viscosity close to that of the

solution as a whole and, thus, measurable in macroscopic experiments. Both diffusion

modes are assumed to be normal (standard random walks), and in accordance with the

reference scales their rates are termed Dmicro and Dmacro. The overall particle motion

is taken to be a superposition of the two basic ones. This yields a time dependent

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diffusion coefficient in the form

D(t) = Dmacro +(Dmicro−Dmacro) exp(−t/τ), (29)

with τ being a reference relaxation time. Evaluation of the MS displacement with

equation (29) yields for the 3D case:

⟨r2(t)

⟩= 6Dmacro

[t + τ

Dmicro−Dmacro

Dmacro

(1− e−t/τ

)]. (30)

From dimensional considerations it follows that combination (τDmicro)1/2 determines

the reference size of the cage, where the Brownian particle is locked. Assigning the

role of the cage to some real element/object in the matrix structure is a central point for

any CRW model.

Comparing (30) with equation (11) for 3D case, one finds that, apart for the nota-

tions, they are identical, and, thus, the “material” parameters of any CRW model could

be directly related to the rheological coefficients of the Jeffrey model. This coinci-

dence entails important conclusions. As shown in Sec. 2–4, the theory of Brownian

motion in JF can be developed consistently all the way up from a simple rheologi-

cal scheme. Therefore, besides being useful for particular needs, e.g. to account for

nanoparticle motion in living polymers, this theory provides a general “frame” for the

CRW approach and transforms into any particular CRW model as soon as the coeffi-

cients are specified. Vise versa, any process resembling walking confined diffusion can

be readily interpreted as the Brownian motion in an effective JF. Being based on phe-

nomenology, our model provides valid general description even for the cases, where

the precise meaning of coefficients is not clear. This could help in dealing with CRW

models. For example, theory35 sets the cage size equal to Rd , the thickness of depletion

layer in a polymer solution, and takes the pertinent formula from,36 where Rd is shown

to be a monotonically increasing function of the probing particle size. Given that, the

model35 predicts that the value of MS displacement (the squared cage size L2) should

grow with the particle radius a. However, the experimental data, see Fig. 10 in Ref.,35

evidence the opposite dependence. Taken rigorously, such a contradiction could have

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fatally compromised the CRW model despite that it accounts quite well for other ex-

perimental facts. Adoption of the JF approach enables one to separate the essence of

the model from the mesoscopic details needing revision. Looking at equation (19), one

sees that the JF approach yields the correct type of the L(a) behavior.

The last but not least notable issue is that the proposed theory of Brownian motion

in JF is deduced from a fundamental set of dynamic equations. This set can be in a

clear way extended for more complicated problems: additional external forces, dissi-

pation mechanisms, etc. For example, modeling of the hyperthermic effect (Sec. 4)

in a nonlinear case, i.e., strong ac fields, does not pose any difficulties. This is much

unlike CRW descriptions which typically are ad hoc constructions or numerical mod-

els inspired by experiments. On the other hand, the mesoscopic CRW theories, being

specified and proven valid in a number of particular problems, fill the general scheme

with physical content and by that stimulate its further development and modifications.

Conclusions

The theory of Brownian motion in a viscoelastic fluid, which obeys the linear Jeffreys

model, is developed. For the cases of micro- and nanoparticles used as the probes

in microrheology, the theory is reduced assuming that the particle inertia is negligi-

ble. The expressions for the mean-square displacement are found for both translational

and rotational thermal motions. It is shown that the model is robust against the non-

physical effect—oscillations increasing with the growth of friction—that necessarily

turn out when the fluid rheology is described by the Maxwell viscoelasticity. In gen-

eral case, the time dependence of the particle mean-square displacement comprises the

stages of: fast diffusion, dynamical confinement, and slow diffusion. Under appropri-

ate conditions, all these three types of the particle thermal motion are experimentally

resolvable.

For the case of a dipolar (e.g. magnetic) particles embedded in a Jeffreys fluid, the

linear magnetic response to an ac probing field is evaluated. It is found that the low-

frequency part of the dynamic magnetic susceptibility has a double-peak shape. The

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heights of the peaks and inter-peak distance are determined by the phenomenological

material parameters of the rheological model. Therefore, these parameters could be

found from magnetic measurements. Aimed at the theory of magnetic hyperthermia,

an example is given, how the viscoelasticity of the matrix affects the energy losses

when an embedded dipole particle is excited by an ac field.

The presented description is based on a clear and simple rheological scheme. In re-

sult of its consistent treatment the statistical characteristics of Brownian motion are ob-

tained. Being simple and robust, the initial model could be easily extended to study the

particle free and forced diffusion in the systems with more complicated non-Newtonian

rheology. Another important advantage of our phenomenological approach is that

it makes a general “frame” whereto fits any of the variety of mesoscopic confined-

random-walk models of diverse origins. Due to that, the model can be used as a reliable

theoretical tool even in the cases, where the meanings of its material parameters are not

precisely specified.

The work was done under auspices of projects RFBR 11-02-96000, RAS Program

12-P-1-1018 and grant MIG-08 from Perm Regional Administration.

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