INTRODUCTION. Magnetic fluids (ferrofluids) are the dispersions of single domain ferromagnetic particles in a nonmag netic liquid. The diameter of particles in ...
ISSN 1061933X, Colloid Journal, 2011, Vol. 73, No. 3, pp. 327–339. © Pleiades Publishing, Ltd., 2011. Original Russian Text © A.Yu. Zubarev, L.Yu. Iskakova, D.N. Chirikov, 2011, published in Kolloidnyi Zhurnal, 2011, Vol. 73, No. 3, pp. 320–333.
On the Nonlinear Rheology of Magnetic Fluids A. Yu. Zubarev, L. Yu. Iskakova, and D. N. Chirikov Ural State University, pr. Lenina 51, Yekaterinburg, 620083 Russia Received June 16, 2010
Abstract—The influence of chain aggregates on the rheological properties of magnetic fluids is studied the oretically. The dependence of effective viscosity on the shear rate of stationary flow is investigated. The char acter and the relaxation time of the viscosity of fluids after the jumpwise changes in the shear rate are deter mined. DOI: 10.1134/S1061933X11030203
INTRODUCTION Magnetic fluids (ferrofluids) are the dispersions of singledomain ferromagnetic particles in a nonmag netic liquid. The diameter of particles in typical mag netic fluids ranges from 7 to 20 nm. In order to avoid the irreversible coagulation of particles, they are coated with stabilizing layers 2–3 nm thick. Depend ing on the type of ferrofluid, these layers can be formed from surfactant molecules or have an ionic nature. Immediately after the first publications on the syn thesis of stable ferrofluids in the 1960s, these systems began to attract a great deal of attention from researchers and practitioners due to their broad range of unique properties, which are promising for applica tion in different technologies. Physical fundamentals and procedures for synthesizing ferrofluids are described in [1]. The current state in the study of fer rofluids and their practical applications is analyzed in [2]. One of the main features of ferrofluids is their abil ity to change rheological properties under the action of an external magnetic field. Extremely dilute disper sions in which interactions between particles were ignored were originally considered in the theory of magnetoviscous effect in ferrofluids [3, 4]. Maximal rise in the viscosity of ferrofluids under the action of field predicted by these models does not exceed a few percents. Meanwhile, experiments carried out in recent years demonstrated that many ferrofluids exhibit strong magnetoviscous effect. The rise in their viscosity achieves one or two decimal orders of magni tude [5, 6]. A particularly strong effect is observed upon the orientation of the field along the gradient of the flow rate of ferrofluid. According to experiments, these strong magneto viscous effects can be associated with only the pres ence of large aggregates of ferroparticles connected by the forces of magnetic interaction [6–9]. Two types of aggregates formed in ferrofluids, i.e., linear chains and bulky dense droplets formed by a large number of fer
roparticles, are known [6–15]. Note that, at present, there is no unambiguous answer on the conditions of the formation of chain or droplet aggregates. At the same time, according to the analysis, strong magne torheological effects can be caused by both types of aggregates [8]. It is also worth mentioning that experi ments carried out in [16] demonstrated rheological effects, which, in combination with strong magneto viscous effect, in principle, cannot be explained using the models of weakly interacting single particles. In general, current theories of magnetorheological properties of ferrofluids are developed for stationary flows for which the size and orientation of aggregates are unambiguously determined by the composition of the ferrofluid, the strength of the applied magnetic field, and the gradient of the macroscopic flow rate of a medium. Along with stationary flows, significant interest is generated by flows of ferrofluids under conditions when the velocity gradient rapidly changes with time. According to experiments [17], typical ferrofluids pos sess pronounced viscoelastic properties, while the time of their rheological relaxation changes with the variation in the velocity gradient from the hundredths to dozens of seconds. These values of the relaxation time are approximately four to five decimal orders of magnitude larger than the values predicted by classical theory [3, 4] for the case of noninteracting particles. It was attempted to construct the model of vis coelastic properties of ferrofluids with chain aggre gates [18]. In this model, it was taken into account that each value of the flow velocity gradient of ferrofluids corresponds to the specific deflection angle of the chain axis from the direction of the magnetic field. The value of this angle determines the macroscopic viscous stress and, hence, the viscosity of the ferrof luid. As the velocity gradient changes, chains are reori ented, not instantaneously, but rather in some finite period of time. This period of time of chain rotation in the model [18] is responsible for the relaxation relation between the stress and the gradient of flow velocity,
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i.e., for the viscoelasticity of the ferrofluid. Note that, although the time of hydrodynamic relaxation pre dicted by this model is substantially closer to the experimental data [17], it turned out that this value is one to two decimal orders of magnitude larger than the experimental value. Thus, the physical nature of viscoelastic effects in ferrofluids has not yet been elucidated. It seems prob able that the relaxation time of viscous stress in ferrof luid measured in [17] is determined by the kinetics of the formation–disintegration of aggregates during the changes in the flow velocity gradient. As far as we know, the influence of the rearrangement of aggregates in magnetic fluids on their rheological properties has not been studied before. Below, we present the simple model of the evolu tion of chain aggregates in ferrofluids involved in the shear flow. Based on this model, we determined vis coelastic properties of ferrofluid, the dependence of the characteristic time of hydrodynamic relaxation on the strength of magnetic field, as well as the value and character of the variations in the gradient of flow velocity. KINETIC OF GROWTH OF CHAIN AGGREGATES The analysis of the kinetics of the growth (destruc tion) of chains and the influence of these processes on the macroscopic rheological properties of ferrofluid, will be based on the data reported in [19] and approx imations used previously in [7, 8, 18, 20]. Regardless of the strong simplification of ferrofluid model used in these approximations, they make it possible to esti mate the stationary viscosity of ferrofluid [5–7, 9, 17, 21], which satisfactorily corresponds to the experi mental data. First, we consider the model of monodisperse fer rofluid formed by spherical singledomain particles. Naturally, real fluids are polydisperse systems; the typ ical diagrams of particle size distribution in ferrofluids can be found in [5]. The account of polydispersity makes the problem of identifying internal microstruc ture of ferrofluid extremely complicated. It is known (see, e.g. [5]) that magnetic effects in ferrofluids are primarily associated with the presence of large parti cles capable of forming aggregates. The volume con centration of these particles in ferrofluids is usually low, about one to two percents. In our model, we con sider only large particles assuming that their volume concentration is on the order of 1.5%. It is also known [4, 5] that the magnetic moment m of large ferromag netic particle is frozen into its body and rotates together with the particle. We will assume that this condition is also fulfilled in our model. Second, we ignore the fluctuationinduced flexi bility of chains and consider them as straight rods. Magnetic moments of particles in each chain are sup posed to be arranged along its axis. In this case, the
interaction between particles in the chain is assumed to be stronger than their interaction with external magnetic field and the binding energy of particles is substantially higher than thermal energy kT . The lat ter condition is necessary to form aggregates in ferrof luid. Third, we disregard the interaction between chains that is justified at the low concentration of particles in the system. The validity of this approximation will be additionally discussed at the end of this communica tion. Stationary Chain Size Distribution Chain aggregates are specific density fluctuations. The number of particles in the chain is the stochastic parameter that is determined by the competition between the magnetic attraction of particles, their thermal motion and, in the case of macroscopic shear flow, by the hydrodynamic forces that rupture the chain. In a quiescent medium, the maximal number of particles in the chain is infinite. Upon the deforma tioninduced motion of a medium, long chains can be ruptured by hydrodynamic forces. Therefore, there is the finite maximal number of particles in the chain, nc . This number can be calculated from the balance between the forces of magnetic attraction of particles and hydrodynamic forces that rupture the chain. Upon the orientation of the magnetic field along the flow velocity gradient, the nc value was estimated in [5] as follows:
Dr , γ
2
(1) ε = 23m , Dr = kT 3 . d kT πη0d Here, d is the hydrodynamic diameter of the particle defined with allowance for the presence of stabilizing shell, m is the magnetic moment of particle, Dr is the coefficient of rotational diffusion of single ferroparti cle, η0 is the viscosity of liquid carrier, parameter ε is the dimensionless quantity referred to thermal energy kT , (the energy of magnetodipole interaction between contacting particles), and γ is the gradient of the mac roscopic flow velocity of ferrofluid. We employ the Gaussian unit system, which is the most convenient to analyze magnetic phenomena. Expression (1) was successfully used in [7, 8, 17, 21] to qualitatively esti mate stationary magnetoviscous effects in different ferrofluids. For the simplicity, we consider expression (1) for nc to be exact equality. We denote the number of nparticle chains in the volume unit of medium by g n. If the convective motion of particle in the vicinity of chains can be ignored as compared to their diffusion, the formation and rup ture of chains can be considered to occur in the quies cent medium. In this case, the distribution function for g n can be determined similarly to the equilibrium function with allowance for constraints on the maxi
nc ∝ ε
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mal length of chain. The equilibrium function can be found from the condition of the minimum free energy of the system [19]. Flow of Free Particles to Chain The kinetics of chains growth is governed by the addition and evaporation of particles to and from the chains. To find the flux of particles to the chains, we employ the model that was successfully used to describe the kinetics of the growth of the chains of polarizable particles. It is assumed that the chain growth occurs according to the chain–particle mech anism of aggregation and chain aggregation is ignored. This approximation is substantiated by the facts that, in real systems, the number of single particles exceeds that of chains and the mobility of single particles is higher than chains. We also assume that particles are only added to the poles of chains. This assumption is justified by the fact that, for the lateral addition of par ticles, they must overcome the potential barrier related to the orientation of the magnetic moments of parti cles and chains with respect to the field, the height of barrier being equal to about ε kT . It is quite improba ble to overcome this barrier by diffusion, since, under the conditions of chain formation that affect the rhe ology of ferrofluids, parameter ε is much larger than unity. Note that very long chains are thermodynami cally unstable and should be transformed into dense voluminous droplets [23]. We suggest that the proba bility of the emergence of such long chains is negligi ble. Let us first consider the flow of particles to chains; the flow of evaporating particles will be determined below. Let the chain be composed of n particles and be surrounded by free particles with concentration c∞ (the number of particles in unit volume) at an infinite distance from the chain. The concentration of parti cles in the point r of the space is denoted by c ( r ). We only take into account the interactions of the free par ticle with the nearest end particle of the chain. As was demonstrated in [22], in this approximation, the mag netic attraction of free particle to the chain is similar to the attraction to single particle. Let us consider the single particle that models the chain and place it in the origin of coordinate system. It is convenient to use a spherical coordinate system with the polar axis directed along the applied field H. Since the concentration of particles (and chains) in a ferrofluid is supposed to be low, we can confine our selves to the consideration of their interactions only with the chain considered. Then, the equation for the diffusion of free particles in the vicinity of this chain can be written in the following form:
∂c = ∇ D r ∇c + ∇ D r c∇u r . [ ( ) ] [ ( ) dd ( )] ∂t COLLOID JOURNAL
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Here, udd is the dimensionless energy of dipole–dipole interaction (in kT units) between the fixed and free particles; and D ( r ) = D0ψ ( r ) , where D0 is the diffu sion coefficient of free particles and ψ(r) = 1 − 3d 4r is the correction for the hydrodynamic interaction between particles [24]. The following boundary conditions for the problem of particle aggregation are standard:
c → c∞, r → ∞, c → 0, r → 0.
(3)
In general, the analytical solution of Eq. (2) cannot be found. As in [19, 22], we consider the quasistation ary approximation
(
)
∂ ⎡D ψ r r 2 ∂ c + c ∂ u r ⎤ = 0, 0 ( ) dd ( ) ⎦⎥ ∂ r ⎣⎢ ∂r ∂r
(4)
where
udd = −
ε 3 3x 3
(5)
x = r d is the dimensionless distance between the free and end particles of the chain, and udd is the dimen sionless energy udd, averaged over the region of attrac tion between the chain and free particle. This approx imation is substantiated elsewhere [22]. It is easy to establish that, in our case, there is a full analogy with the problem of the aggregation of colloi dal particles with spherically symmetric interaction potential. The solution of the problem is wellknown [25] and allows one to obtain the following expression for flux J of free particles to the chain: ∞
J = Ac∞,
A=
exp ( udd ) 4πD0d , W = dx. W ψ ( x) x 2 1
∫
(6)
Evolution of Ensemble of Chains Now, we describe the evolution of the ensemble of chains. Variations in the distribution function g n over time occur by virtue of the addition of particles to chains and their evaporation. We suggest that only end particles evaporate from chains (because the end par ticle is bound to the chain weaker than the internal particle, the probability of its evaporation is higher). As a result, we arrive at the following Smoluchowski kinetic equation:
⎛ B ⎞ ∂g n B = − Ag1 ( g n − g n − 1 ) + B1 ⎜ n g n + 1 − n−1 g n ⎟ , ∂t Bn ⎠ (7) ⎝ Bn + 1
n > 1. Here A is the coefficient of capture defined in Eq. (6) and Bn are the coefficients that determine the
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intensity of particle evaporation from the chain, which can be expressed as follows [19]:
sh ( κn) (8) Bn = A exp ( −ε) . v κn The first term in the righthand side of Eq. (7) defines the rate of the variations in the concentration of n particle chains due to the addition of free parti cles to these chains, as well as to n − 1 particle chains; the second term defines this rate of variations in the concentration due to the evaporation of single parti cles from chains with n + 1 and n particles. For the convenience of further calculations, the form of coef ficients at g n + 1 and g n in the desorption term of Eq. (7) is chosen in the form of fractions. Equation for g1 has the following form: nc − 1
nc
Bn − 1 B2 ∂g1 = −2 Ag12 − Ag1 g n + 2 1 g 2 + B1 g n. (9) B B ∂t n 2 n=2 n=3
∑
∑
In this equation, it is taken into account that the rela tive mobility of particles that form the doublet is approximately two times higher than that of the single particle and quiescent chain, and that two new single particles are emerged upon the decomposition of the doublet. Equations analogous to Eqs. (7) and (9) were derived in [19, 22]. For the number of chains with the maximal number of particles nc , we have B ∂ g nc = Ag nc − 1 g1 − B1 nc − 1 g nc . ∂t B nc
(10)
Equation (10) takes into account that chains can contain no more than nc particles. The nc changes with variations in shear rate γ (nc decreases with an increase in shear rate). According to the analysis, the hydrody namic forces that rupture the chain have a maximum in its middle part. Consequently, after an abrupt increase in γ , we should expect that all chains are divided in halves and that the number of particles in each of the halves is larger than new critical value nc. The rupture of chains occurs much faster than their further evolution related to the convective diffusion motion of ferroparticles. Thus, in the first approxima tion, we can consider that, after an increase in γ , the division of chains proceeds instantaneously. The divi sion of too long chains should be maintained under initial conditions for Eqs. (7), (9), and (10) that describe the further evolution of chains. Set of Eqs. (7), (9), and (10) is a closed system and automatically satisfies the following condition: nc
∑ ng
n
= C,
(11)
n =1
where C is the constant equal to the total particle con centration ϕ v in the suspension.
Set of Eqs. (7), (9), and (10) with corresponding initial conditions can be easily solved numerically. MATHEMATICAL MODEL OF VISCOSITY OF MAGNETIC FLUID As was mentioned above, the statistical model of the viscoelastic properties of ferrofluids containing chain aggregates was proposed in [18]. Moreover, the kinetic of the variations in chain size distribution after the change in shear rate was not taken into account; it was assumed that this process proceeds instanta neously. Only the reorientation of chains was taken into account. Further analysis demonstrated that, for ferrofluids used in [17], the characteristic time of hydrodynamic relaxation calculated with allowance for only this mechanism varies approximately from 10–2 and 10–1 s, whereas the experimental time was equal to about 10 s. We consider now the model of the dynamics of the effective viscosity of ferrofluids containing chain aggregates that accounts for the kinetics of their growth. In order to focus attention on the effects asso ciated with the kinetics of chain growth, we suggest that, unlike [18], after the change in the magnetic field and/or the gradient of flow rate, variations in the ori entation of chains proceed instantaneously. As in [7, 15, 18, 19, 22], we simulate the chain composed of n particles by the prolate ellipsoid of revolution with minor and major axes equal to d and nd, respectively. It is fundamentally significant that the volume of this ellipsoid is equal to the total volume π nd 3 6 of all par ticles in the chain. Taking advantage of the known results of statistical hydromechanics of the suspensions of nonspherical particles [26], the expression for the Cartesian compo nents of the mean tensor of viscous stresses σ can be written as follows: σ ik = σ iks + σ ika , ⎡( 2α γ − ρ e e δ γ ) + n ik n j s n ik js ⎣⎢ + ( ζ n + β nλ n ) ( ei e j n γ jk + ek e j n γ ji ) +
σ ik = 2η0 γ ik + η0 s
+ β n ( ωij e j ek
n
+ ωkj e j ei
n
)+
(12)
− β n d ei e k n ⎤ , ⎥⎦ dt H a σ ik = κkT ei hk − ek hi , hi = i , 2v H ⎛ ∂u ∂u ⎞ ⎛ ∂u ∂u ⎞ γ ik = 1 ⎜ i + k ⎟ , ωik = 1 ⎜ i − k ⎟ . 2 ⎝ ∂x k ∂xi ⎠ 2 ⎝ ∂x k ∂xi ⎠ Here and below, the used symbols denote the following operations: + ( χ n − 2λ nβ n ) ei ek e j e s
... =
∑ ... nv g , n
n
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...
0
∫
= ...e φ 0n ( e )d e,
i, ..., k = x, y, z,
where u is the average velocity of ferrofluid. Unit vec tor e is oriented along the chain axis, φn is the distribu tion function over e, normalized to unity, and φ 0n is the equilibrium distribution function. Parameters α n...ζ n are defined and described in the appendix. Under assumption that ε Ⰷ 1 and ε Ⰷ κ , it can be considered that the magnetic moment of all particles in the chain is oriented along its axis. Thus, the equi librium distribution function φ0n ( e) coincides with this function for rigid particles with magnetic moment m n. Using the wellknown data of the equilibrium theory of ferrofluids (see, e.g. [1, 4]), we can write
φ0n
( e) = κn exp [κn ( eh )] . 4πsh ( κn)
(13)
In order to determine the nonequilibrium function φn, it is necessary to find the solution of corresponding Fokker–Planck equation. Using the wellknown form of this equation for the ellipsoids of revolution (see, e.g. [26]), we have
∂φ n ∂φ + λ n ( es γ sl − emesel γ ms ) n + ∂t ∂el ∂φ + ωls es n − 3λ nel es γ ls ϕn − ∂el (14)
⎡ ⎤ ∂φ − Dnκn ⎢( e j emhm − h j ) n + 2e j h j φ n ⎥ = ∂e j ⎣ ⎦ 2 2 ⎛∂ φ ∂φ ∂ φn ⎞ = Dn ⎜ 2n − 2es n − e j es ⎟, ∂es ∂e j ∂e s ⎠ ⎝ ∂e j
331
integrate the obtained expression with respect to all orientations of vector e. As a result, in the linear approximation with respect to the components of the flow rate gradient, we arrive at the following equations:
d ek dt
n
= − 1 ek τ1n +
n
0 ωkj e j n
(
0
+ λ n ei n γ ik − eke j es
+ Dnκn ( hk − eke j
n
0 n
)
γ js +
(16)
hj ) ,
and
d ei ek dt
(
× ei es
0 n
n
(
= − 1 ei ek τ 2n
γ sk + ek es
0 n
)
γ si + ωij e j ek
− 2λ n ei ek es e j + Dnκn ( ek
n
)
− 1 δik + λ n × 3
n
0 n
hi − 2 e j eiek
0 n
+ ωkj e j ei
0 n
− (17)
γ sj + n
h j + ei n hk ) ,
where
τ1n = 1 , τ 2n = 1 , 2D n 6Dn and zero superscripts correspond to equilibrium moments calculated based on equilibrium distribution function φ 0n. Calculating nonequilibrium statistical moments in Eqs. (16) and (17) using trial functions (15), we arrive at the set of differential equations with respect to coef ficients ai (t ) and bik (t ) at given n. After solving this sys tem, we can use function (15) to determine nonequi librium moments. A similar approach resulted in the satisfactory agreement of analytical calculations with the data of laboratory [5, 7] and computer [27] exper iments.
where kT , Dn = 3 πη0d nδ n
δn =
(
VISCOSITY RELAXATION
)
2 n +1
(
2
3n n 2α 0 + β0
)
.
Parameter Dn is the coefficient of rotational diffusion of nparticle chain. We are not aware of the exact solution of this equa tion for arbitrary κ n value. For finding the approxi mate solution, we employ the effective field method proposed in [4] and developed in [7, 8, 15, 18]. In this method, which is the variant of trial function method, nonequilibrium distribution function should be sought in the form:
(
0 φn = φn ⎡⎣1 + ai ei − ei
0 n
) + b (e e ik
i k
− eiek
0 n
)⎤⎦ ,
(15)
where ai and bik are the components of unknown vec tor and tensor, respectively, which should be defined independently. For this purpose, we multiply both parts of Eq. (14) by the components of vector e and tensor ei em and COLLOID JOURNAL
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The problem of calculating coefficients ai and bik is not too complex, albeit cumbersome. Let us introduce the Cartesian coordinate system x, y, z, whose Oz axis is directed parallel to the external magnetic field ( H z = H = const, H x, y = 0). We assume that that the velocity u of ferrofluid is directed along the Ox axis with the gradient oriented toward the Oz axis, i.e., along the external magnetic field H . The shear rate γ xz = γ zx = ω xz = −ω zx is denoted by γ. According to direct estimates, the characteristic time of chain reorientation, τ1n, for typical commer cial ferrofluids is no longer than 10–2–10–1 s, even for fairly long chains composed of several dozen particles, which is much shorter than the time of rheological relaxation (from one to ten seconds) measured in experiments [17]. As follows from then results of our calculations, time τ1n is substantially shorter than the time of the evolution of the ensemble composed of chain aggregates that determines the time of changes
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in the viscosity of ferrofluids. Thus, further we ignore the reorientation of chains assuming that this process proceeds instantaneously that permit us to confine ourselves to the stationary approximation for Eqs. (16) and (17). In this approximation, after simple but slightly cumbersome calculations, we arrive at the fol lowing set of equations for coefficients a x and bxz (in this case, other components ai and bij turned out to be identically equal to zero): 0⎤ ⎡⎛ 1 ⎞ 2 0 2 ⎢⎜ τ ⎟ e x n + Dn κn e x e z n ⎥ a x + ⎣⎝ 1n ⎠ ⎦ 0 0⎤ ⎡⎛ ⎞ + ⎢⎜ 1 ⎟ e x2e z + Dn κn e x2e z2 ⎥ 2bxz = n n ⎣⎝ τ1n ⎠ ⎦ 0 0 0 = ⎡⎢λ n e z n − 2 e x2e z + e z n ⎤⎥ γ n ⎣ ⎦
(18)
)
(
and
)
( (
⎡⎛ 1 ⎞ 2 0 2 2 0 2 0 ⎤ ⎢⎜ τ ⎟ e x e z n + Dn κn 2 e x e z n − e x n ⎥ a x + ⎣⎝ 2n ⎠ ⎦ 0 0 0 ⎡⎛ ⎞ + ⎢⎜ 1 ⎟ e x2e z2 + Dn κn 2 e x2e z3 − e x2e z ⎤⎥ 2bxz = n n n ⎦ (19) ⎣⎝ τ 2n ⎠ 0 0 0 0 0 = ⎡⎢λ n e x2 − 4 e x2e z2 + e x2 + e z2 − e x2 ⎤⎥ γ. n n n n n⎦ ⎣ Substituting representation (15) into relation (12), we obtain
(
)
)
σ xz = 2ηγ, where the effective viscosity
η = η0 ⎡1 + ⎣⎢
(
(20)
(
α n + 1 ⎢⎡( ζ n + βnλ n ) ex 2⎣
) + 2 (χ
2 0 n
2 0
+ ez
n
)+
0 − 2λ nβ n ) e x2ez2 ⎥⎤ + n⎦ (21) 0 ⎤ 2 0 2 + 1 κkT A1 ex + B1 e x ez , ⎥ n n 2v η0 ⎦ bxz ax A1 = , B1 = . γ 2γ is introduced. Once the particle number distribution in the chain g n is determined and the dependence of effective vis cosity η on time can be found from the set of Eqs. (7), (9), and (10) using this set in expression (21).
+ βn ez2
0
n
− ex2
(
0
n
n
)
DEPENDENCE OF STATIONARY VISCOSITY ON SHEAR RATE Much of the experimental data for magnetic sus pensions is approximated by simple rheological law η η0 = 1 + ϕM −Δ [28–30], where M ∝ γ H 2 is the dimensionless Mason number equals to the ratio of hydrodynamic force that ruptures bonds between par ticles to the force of magnetic attraction between par
ticles and Δ is a certain exponent. This power law fol lows from two models. The first model considers the linear chain aggregates of magnetic particles with identical lengths equals to the maximal length of chains, which cannot be ruptured at given γ and H . This model gives Δ = 1 [28–31]. The other model deals with droplet aggregates and leads to Δ = 2 3 or 1, depending on the supposed mechanism of the disrup tion of droplets by the hydrodynamic forces [32, 33]. However, Δ = 2 3 or 1 values were not observed in experiments. Intermediate Δ values were obtained in [28–30, 34]. Moreover, in experiments carried out with different suspensions of nonBrownian magnetic particles, Δ values increase with H and ϕ from 2/3 to 1 [28, 29]. All of the aforesaid is evidence of the insuf ficient understanding of the physical nature of rheo logical phenomena in ferrofluids. Moreover, condition M Ⰶ 1 is typical for the majority of experiments. Therefore, minor errors in the values of exponent Δ lead to large errors in the prediction of viscosity. Note that the Δ = 1 value was obtained with full disregard for the effect of Brownian motion on the shape and sizes of both the chains [31] and droplets [33]. The Δ = 2 3 value was obtained in [32] based on the analysis of thermal evaporation of particles from droplet aggregates; thermal fluctuations in the shape and sizes of droplets were not taken into account in this case. It was shown [35] that changes in exponent Δ with field H and concentration ϕ can be explained using the chain model with allowance for the statistical scatter of chain sizes associated with the thermal motion of particles. The latter motion can be ignored in magnetorheological suspensions at fairly strong magnetic fields. Estimates demonstrated that, in such systems, the energy of magnetic interaction of micronsized magnetizing particles could be by several orders of magnitude higher than the energy of thermal motion kT . In ferrofluids, due to the small sizes of fer romagnetic nanoparticles, thermal motion plays important role and its disregard leads to very signifi cant errors. Nevertheless, the data of rheological experiments with ferrofluids can also be approximated by the power law η η0 = 1 + ϕM −Δ (see, e.g. [36]). Our goal is to theoretically study the dependence of the effective viscosity of ferrofluids on the shear rate and to estimate exponent Δ. Relation (21) used by us takes into account both the statistical particle distribution in chains and the effect of shear rate γ on the character istic sizes of chains and, hence, on the rheological characteristics of ferrofluids. Results of the calculations of the dimensionless sta tionary viscosity of typical ferrofluids are shown in Figs. 1 and 2. A small plateau can be seen in the left hand sides of curves in Fig. 1, which denotes that, at low shear rates, the ferrofluid behaves itself as a quasi Newtonian medium. COLLOID JOURNAL
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333
η ln ⎛ – 1⎞ ⎝η ⎠ 0
3 2
15
2
2 0
10
1
–2
5
1
–4 0
2
4
6
8
3 γ· × 10 Dr
Fig. 1. Dependences of dimensionless effective viscosity η η0 of ferrofluid on dimensionless shear rate γ Dr at dif ferent values of dimensionless strength of magnetic field κ: (1) 1, (2) 2, and (3) 3. Volume fraction of particles ϕ = 0.015; viscosity of liquid carrier η0 = 0.13 Pa s; particle diameter d = 16 nm.
The data presented in Fig. 2 demonstrate that the power dependence of reduced viscosity η η0 − 1 on flow rate gradient γ is only fulfilled at some parts of the rheological curve. However, since this power depen dence is traditionally used to interpret experimental results, it was of interest to study the dependence of effective exponent Δ determined by the average slope angle of curves in Fig. 2 on γ. This dependence is shown in Fig. 3 at different values of the strength of the magnetic field. As can be seen from Fig. 3, exponent Δ first increases with the shear rate. Then, at γ > Dr all chains are ruptured and only single particles are left. After wards, the viscosity begins to slightly depend on γ, i.e., Δ ≈ 0. Note that, for modern ferrofluids, condition γ Dr = 1 is fulfilled at γ ≈ 10 4 s–1, which is nearly inac cessible for the majority of modern rheometers. More over, at such high shear rates, one of the main condi tions of the applicability of developed model fails, namely, the condition of the smallness of Peclet shear number calculated from particle sizes. Therefore, the portions of curves that correspond to inequality γ > Dr, cannot pretend to exactly describe the behav ior of the system. At low magnetic field and low shear rate, exponent is Δ small because, in this case, the number of chains, the sizes of which are comparable with the maximal length of chain, is very small in the system. Figures 4 and 5 present the data [21] on measuring viscosity as a function of shear rate. According to these COLLOID JOURNAL
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0
–5
10
5
γ· ln Dr
Fig. 2. The same that in Fig. 1 in logarithmic coordinates: 1 ⎯ κ = 1; 2 ⎯ 3.
experiments, exponent Δ is equal to 0.67 and 0.87 at dimensionless strength of magnetic field equal to 9.8 and 26.2, respectively. As can be seen from these data, the Δ values are the larger, the higher is the magnetic field. It follows from the comparison of Figs. 1, 2, 4, and 5 that the theoretical values of viscosity coincide with experimental data by the order of magnitude, particularly at low shear rates. Experimental values of exponent Δ increase with the applied magnetic field, which also agrees with the conclusions based on our model and evidences of its adequacy at least in main physical aspects. Unfortunately, it seems impossible to perform more correct comparison of theoretical and experimental data [21] because ferrofluids used in [21] Δ 0.8 0.6 0.4
2
1
0.2 0
⎯5
0
5
γ· ln Dr
Fig. 3. Dependences of dimensionless number Δ of ferrof luid on logarithm of dimensionless shear rate γ Dr at dif ferent values of dimensionless strength of magnetic field κ: (1) 1, (2) 2, and (3) 3. Parameters of fluid are same as in Fig. 1.
334
ZUBAREV et al.
η η0
η ln η0
4
2
20
2
3
1
2
10
1 1
0
20
40
60
80
100
0 –6
–5
–4
–3
–2
3 γ· × 10 Dr
Fig. 4. Experimental dependences [21] of dimensionless effective viscosity η η0 of APG513A1 ferrofluid on dimensionless shear rate γ Dr at different values of dimen sionless strength of magnetic field κ: (1) 9.8 and (2) 26.2.
were polydisperse to great extent. In addition, experi ments in [21] were carried out at high magnetic fields for which strong inequality κ Ⰷ ε, was fulfilled, i.e., the energy of interaction between particles and the field was much higher than the energy of interaction of particles with one another. Our model was developed for the opposite case, κ < ε. DEPENDENCE OF RELAXATION TIME OF EFFECTIVE VISCOSITY ON VARIATIONS IN SHEAR RATE In this section, we consider the dynamic effects that are observed after the jumpwise change in the shear rate. Chain size distribution function g n(t) remains unchanged with a decrease in shear rate at the initial time moment; hence, initial conditions for kinetic equations have the simple form
g n ( 0) = g n0,
at
n ≤ nc,
g n ( 0) = 0,
at
n > nc.
γ· ln Dr Fig. 5. Dependences of Fig. 4 in logarithmic coordinates.
chains are formed whose sizes twice smaller than the initial size. If an increase in the shear rate is relatively small, secondary chains are not ruptured after the rup ture of primary chains. If an increase in the shear rate is large, fairly long secondary chains are also ruptured in half. As a result, four new chains are formed from one chain. Upon a very large increase in the shear rate, chains can be broken into eight or even more parts; however, we will not consider these situations. Within previous assumptions and at moderate increase in γ, when only primary chains are divided in halves, initial conditions (7), (9), and (10) for the problem can be written as follows:
g n ( 0) = g n0,
g n ( 0) = g n0c2 + g n0c2 + 1,
n
n=
nc1 , 2
nc1 . 2
where nc1 and nc2 are the maximal numbers of particles in the chain before and after an increase in the shear rate, respectively. At a large increase in the shear rate, the initial con ditions can be expressed as follows: COLLOID JOURNAL
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ON THE NONLINEAR RHEOLOGY OF MAGNETIC FLUIDS η η0
335
η η0
1
4.2
2
12 5 10 4 8 4.1
3
3
6
2
4
4.0
1 50
0
100
150 t, s
Fig. 6. Dependences of dimensionless effective viscosity η η0 of magnetic fluid on time t upon the variations in shear rate from different values of γ 1 Dr to γ 2 Dr = 0.7 × 10–3: at 1 – γ 1 Dr = 7 × 10–3, 2 –3 × 10–3, 3 – 2 × 10–3, 4 – 1.3 × 10–3, 5 – 1 × 10–3. Parameters of fluid are same as in Fig. 1; dimensionless strength of magnetic field κ = 3.
g n ( 0) = g n0,
nc2 , 2
n
2011
0
5
10
15 t, s
Fig. 7. Same as in Fig. 6 for γ 2 Dr = 7 × 10–3: 1 – γ 1 D r = 0.7 × 10–3, 2 – 2.2 × 10–3, 3 – 3 × 10–3.
was solved under initial conditions (23) and (24) cor responding to the division of chains with n > nc2 in halves. As can be seen from Fig. 6, the viscosity rises with time after a jumpwise increase in the shear rate γ . The increase in the viscosity is explained by the lengthen ing of chains with increasing nc. In the case of a jumpwise increase in the shear rate at its fixed final value (the initial values of γ are differ ent), the behavior of viscosity is ambiguous (Fig. 7). At a small increase in γ (curve 3), the viscosity of the fer rofluid rises with time. This is associated with the fact that the longest chains are ruptured and the large number of short chains is formed for this case, with the latter chains becoming longer over time. An increase in the characteristic length of chains leads to a rise in the viscosity. Conversely, after a large increase in γ (curve 1), the viscosity decreases over time due to the fast decrease in nc and the appearance of a large num ber of short debris of chains. At a moderate jump in shear rate (curve 2), competing processes of the growth and rupture of chains take place. Let us now define the characteristic relaxation time τ of the viscosity as the time during which (after jump wise change in γ ) the difference between initial and final values of viscosity decreases by a factor of e. It is worth mentioning that, as can be seen from Fig. 7 (curve 2), in some cases, the relaxation of viscosity can be characterized by at least two times, i.e., an increase in viscosity is changed for its slow fall. Figure 8 demonstrates the results of our calcula tions of τ, as well as the results of measuring this time [17]. Note that, unlike our calculations with jumpwise change in γ, viscoelastic effects were studied in [17] in
336
ZUBAREV et al.
τ, s 40
τ, s 1
1
30
30 20
2
20
3 10
10 2
0
5
10
15 H, кА/m
0
·
6 Δγ × 10 3
4
2
Dr
Fig. 8. Dependences of relaxation time τ on strength of magnetic field H : (1), (2) theoretical calculation upon the variations in shear rate γ Dr (1) from 7 × 10 ⎯3 to 7 × 10–4 and (2) from 7 × 10–4 to 7 × 10–3, (3) experimental data [17]. Parameters of fluid are same as in Fig. 1.
Fig. 9. Relaxation time τ as function of value of jumpwise decrease in dimensionless shear rate Δ γ D r at different values of dimensionless field strength κ: (1) 3 and (2) 1. Final value of dimensionless shear rate is fixed and equal to γ 2 Dr = 0.7 × 10–3.
experiments with the oscillating time dependence of γ . The flow parameters that correspond to curve 2 pro vide for a monotonic decrease in the viscosity after jumpwise increase in γ. The values of relaxation time calculated by us (Fig. 8) coincide in their order of magnitude with experimental data [17]. One should note that, in pre vious models, the obtained values of relaxation time differ from experimental data by several decimal orders of magnitude. It seems natural to suggest that the relaxation time measured in experiments with oscillating flows lies between the values calculated under the assumption of jumpwise increase and a decrease in shear the rate. Figures 9–12 demonstrate the dependences of the relaxation time of a system on the jump in shear rate. As can be seen from these figures, relaxation time depends to great extent on the final value of shear rate at its stepwise change. The longer the time of the vis coelastic relaxation of ferrofluid, the greater is the final value of this rate; this is related to the fact that, after the change in flow rate, the transition to new stationary state needs longer time for systems with long chains (low shear rates) than for systems with short chains (high shear rates). Upon a decrease in the shear rate, the relaxation time is always longer than upon its increase in the shear rate. This result can easily be understood, since, after a decrease in the shear rate, chains grow and, in the stationary state, they are characterized by fairly large sizes. However, as the shear rate increases, unnecessarily long chains are ruptured in half; then,
depending on the situation, debris grow slightly or the characteristic length of chains decreases further due to the evaporation of particles from the chains. Both of these processes require less time than it is necessary for the growth of particles after a decrease in the shear rate. The lower the magnetic field, the smaller is the relaxation time, since, in the low magnetic field, the size of chains is not large. The length of chains increases with the strength of magnetic field and the
τ, s 1
30
20
10 2 4
5
6
7
3 γ· Δ × 10 Dr
Fig. 10. Same as in Fig. 9 at fixed initial value of shear rate γ 1 Dr = 7 × 10–3. COLLOID JOURNAL
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ON THE NONLINEAR RHEOLOGY OF MAGNETIC FLUIDS τ, s
337
τ, s 8
6 6
1 4
2 1 4
2
3 2
2 4
2
6 3 γ· Δ × 10 Dr
0
2
4
6
3 γ· Δ × 10 Dr
Fig. 11. Times of relaxation to (1) stationary and (2) max imal values of viscosity at κ = 3 and (3) to stationary value at κ = 3 as function of value of jumpwise increase in dimensionless shear rate Δ γ D r . Final value of dimen sionless shear rate is fixed and equal to γ 2 Dr = 7 × 10–3.
Fig. 12. Same as in Fig. 11 at the fixed initial value of shear rate γ 1 Dr = 0.7 × 10–3 at κ: (1) 3 and (2) 1.
evolution of the ensemble of longer chains proceeds with a longer characteristic time.
account the hydrodynamic interactions of elongated particles (chains).
CONCLUSIONS The dependences of the viscosity η of a magnetic fluid on the shear rate γ and applied magnetic field H were studied. It was assumed that nonlinear rheologi cal effects in magnetic fluids are explained by the assembly of colloidal ferroparticles into linear chain aggregates. Theoretical values of relaxation times correspond to real experimental values in their order of magni tude. In contrast to traditional models and in agree ment with experiments, the proposed model predicts an increase in exponent Δ with shear rate γ and the strength of magnetic field. The dependences of Δ on shear rate γ and field strength γ were explained by the Brownian motion of particles and the rupture of aggregates due to the evaporation of particles from aggregates, which were not taken into account in pre vious models. The relaxation of the viscosity of magnetic fluid after jumpwise changes in the shear rate was studied. It was shown that the relaxation time depends on the value and the character of changes in γ in a complex manner. Note that, upon the substantial increase in the vis cosity of ferroparticles due to the formation of chains, the model of noninteracting particles seems to be fairly strong approximation. However, at present, there is no reliable theory that makes it possible to take into COLLOID JOURNAL
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At the same time, the experience of the construc tion of rheological models (see, e.g. [5, 7, 8, 17, 21, 31–34]) demonstrated that the approximation of non interacting aggregates leads to reasonable (at least by the order of magnitude) agreement with experiment. This gives us grounds to expect that, upon the descrip tion of nonstationary relaxation phenomena, such an approach will allow us to account for fundamentally significant aspects of viscoelastic phenomena in fer rofluids. The agreement between our calculations and experimental data [17] support the adequacy of the proposed model of noninteracting chains, at least for a low concentration of ferroparticles. ACKNOWLEDGMENTS This work was supported by the Ministry for Research and Education of the Russian Federation, project nos. 2.1.1/2571 and 2.1.1/1535; Federal Goal Oriented Program, grant nos. 02.740.11.0202, 02 740115172, GK330, and NK43P(4); and by the Russian Foundation for Basic Research, grant nos. 100196002Ural, 100296001Ural, and 10 0200034. APPENDIX: EQUILIBRIUM MOMENTS AND KINETIC COEFFICIENTS Equilibrium moments of the orientation vector of n particle chains have the following forms:
338
ei
ZUBAREV et al. 0 n
= hi L1, 0 ei eke j n
= 1 (1 − L2 ) δik + 1 (3L2 − 1) hi hk , 2 2
0 n
eiek
= 1 ( L1 − L3 ) ( δik h j + δij hk + δ kj hi ) + 2
+ 1 (5L3 − 3L1 ) hi h j hk , 2
ei ek el em
0 n
∞
∞
ds β'0 = , + 1 s ( ) n2 + s Q
∫
(
0
α ''0 =
)
sds
∫ (1 + s ) 0
2
, Q
∞
β''0 =
=
sds
∫ (1 + s ) ( n
2
0
Q = (1 + s ) n 2 + s,
, +s Q
)
= 1 (1 − 2L2 + L4 )( δikδlm + δimδ kl + δil δ km ) + 8 + 1 (6L2 − 5L4 − 1)( hi hkδlm + 8 + hi hmδ kl + hi hl δ km + hl hmδik + hl hkδim + hmhkδli ) +
⎡ ⎤ 2 2 α 0 = − 2 1 ⎢2 + 1 ln 2n − 1 − 2n n − 1 ⎥ 2 n − 1 ⎣n ⎦ n −1 at n > 1,
+ 1 (3 − 30L2 + 35L4 ) hi hk hl hm, 8
β 0 = 2 at n = 1, 3
0 n
ez
= L1,
ez2 0 ez5 n
e x2ez 0 e x2ez3 n
0 n
0
= L2,
n
0
ez3
= L3,
n
ez4
0 n
α 0 = 2 at n = 1, 3
(
= L4,
0 e x2 n
= L5,
= 1 (1 − L2 ) , 2 2 2 0 e x ez = 1 ( L2 − L4 ) , n 2
= 1 ( L1 − L3 ) , 2
= 1 ( L3 − L5 ) , LJ = LJ ( κn) , 2
ζn =
βn =
)
2 n −1
(
2
n n α 0 + β0 2
)
χn = − + 2 , 2 nα 0β''0 nβ'0 n + 1 nα '0
(
8
)
) (
(
)
(
(
)
β'0 = 2 at n = 1, 5 1 β' =
∫( 0
)
∫ 0
∞
α '0 =
(n
0
2
)
−1
2
)
ds
∫ (1 + s ) 0
(
)
4 n2 − 1
2
(
)
0
ds , β0 = (1 + s ) Q
)
)
β''0 = 4 at n = 1, 15 β'' = − 1
Here, ∞
2
⎡ ⎤ 3 × ⎢n 2n 2 − 5 − ln 2n 2 − 1 − 2n n 2 − 1 ⎥ 2 ⎣ ⎦ 2 n −1 at n > 1,
(
2 λ n = n2 − 1. n +1
ds , 2 n +s Q
)
4 n2 − 1
2 ⎡ ⎤ × ⎢n 2n 2 + 1 − 4n − 1 ln 2n 2 − 1 + 2n n 2 − 1 ⎥ 2 ⎣ ⎦ 2 n −1 at n > 1,
ρ n = 1 ⎡2 α''0 − β''0 + 3n α 0α''0 − β0β''0 ⎤ , ⎦ 3nα'0β''0 ⎣
∞
(
0
0
2α ''0
α0 =
α'0 = 2 at n = 1, 5 1 α' =
(
)
2
)
⎡ 2 ⎤ 3 × ⎢n + 2 − ln 2n 2 − 1 + 2n n 2 − 1 ⎥ at n > 1, 2 ⎣ n ⎦ 2 n −1 α ''0 = 4 at n = 1, 15 1 α'' =
,
− 2 , nβ'0 n + 1 nα'0
(
4
(
(
⎤ 1 ⎡n − 1 ln 2n 2 − 1 + 2n n 2 − 1 ⎥ ⎢ 2 n − 1⎣ 2 n − 1 ⎦ at n > 1, 2
J = 1, 2, 3, 4,
L1 ( x ) = cth ( x ) − 1 , L2 ( x ) = 1 − 2 L1 ( x ) , x x L3 ( x ) = 1 + L1 ( x ) − 3 L2 ( x ) , x x 4 1 L4 ( x ) = 1 − L3 ( x ) , L5 ( x ) = + L1 ( x ) − 5 L4 ( x ) . x x x The values of kinetic coefficients in expressions (12) for effective viscosity are as follows: αn = 1 , nα'0
β0 =
)
2
, Q
(n
2
)
−1
2
(
)
2 ⎡ ⎤ × ⎢3n + 2n + 1 ln 2n 2 − 1 − 2n n 2 − 1 ⎥ at n > 1. ⎣ ⎦ 2 n2 − 1
COLLOID JOURNAL
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