- Email: [email protected]
Magnetic rotary microrheology in a Maxwell fluid
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233 www.elsevier.com/locate/jmmm
Magnetic rotary microrheology in a Maxwell fluid Yu.L. Raikher, V.V. Rusakov Institute of Continuous Media Mechanics, Ural Branch of RAS, Perm 614013, Russia Available online 16 November 2005
Abstract A ferroparticle driven by an AC field in a Maxwell fluid is studied with regard to magnetic microrheology. Orientational kinetics is simplified by neglecting the particle rotary inertia. While in a Newtonian fluid the inertialess approximation for nanoparticles is valid practically unboundedly, in a viscoelastic matrix there exists an important restriction on the stress relaxation time. Assuming weak nonequilibrium, the magneto-orientational relaxation times and low-frequency magnetic spectra are found in the presence of a constant (bias) field. Another way of particle-mediated probing of the matrix involves the field-induced birefringence of a suspension; under zero bias field the corresponding (quadratic) susceptibility is found. r 2005 Elsevier B.V. All rights reserved. PACS: 83.10.Pp; 75.50.Mm; 83.60.Bc; 82.70.Gg; 81.05.Qk Keywords: Magnetic nanoparticles; Magnetic fluids; Microrheology; Maxwell fluid
Recent years are marked by growing interest toward microrheology as a tool to explore the structure of complex fluids at the mesoscopic scale. For that, solid particles of the reference sizes 10–100 nm are embedded in a test fluid sample and the info on their dynamic behavior is collected and analyzed. If external forces are absent, the sole cause of the tracer displacements is the Brownian diffusion. Application of an external force modifies the tracer motion and puts it under control. A convenient way to do so is to set out nanoparticles in a regular motion by an AC field HðtÞ, the output signal being the dynamic magnetization MðtÞ of the tested sample. Under apparent reformulation, interpreting of the magnetic microrheology data basically coincides with the task of theoretical description of dynamic magnetization of ferrofluids at high dilution [1,2]. For ferrofluids based on complex media the theory is yet at the start. The pertinent models split into two main classes: (1) coarse systems with the grains of a micron size, where the particle inertia is essential; (2) nanoscopic suspensions, where the inertialess limit well known in the theory of linearly viscous magnetic fluids might apply. In our model, we consider the microrheology test system as a Corresponding author. Tel.: +7 342 237 8323; fax: +7 342 237 8487.
E-mail address: [email protected] (Y.L. Raikher). 0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.10.086
dilute ferrocolloid with a viscoelastic (Maxwell) matrix possessing a single stress relaxation time tM . The colloid particles are identical ‘‘single-domain’’ and magnetically hard. Their magnetic moments have constant values and are rigidly fixed in the particle bodies. Accordingly, the state of each particle is completely determined by the orientation of its unit vector e ¼ l=m. The set of equations describing under this condition the rotary dynamics of a particle has the form [1] d o ¼ JU þ QðtÞ; dt q J ¼ e , qe I
d e ¼ ðx eÞ, dt ð1Þ
d Q þ Q ¼ zx þ yðtÞ, dt hyi ðtÞyj ðt0 Þi ¼ 2zTdij dðt t0 Þ, tM
where J is the rotation operator, yðtÞ a normalized white noise, I the particle moment of inertia, U the interaction energy between the magnetic moment and the external field, Q the friction torque against the carrier fluid, and T the temperature in energy units. Adopting the Stokes flow regime, we set the particle rotary friction coefficient z ¼ 6ZV , where V is the particle volume.
ARTICLE IN PRESS e230
Y.L. Raikher, V.V. Rusakov / Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233
For a Newtonian fluid (tM ¼ 0) Eq. (1) reduces to d x þ zx ¼ JU þ yðtÞ. (2) dt From Eq. (2) one sees that tI ¼ I=z is the ‘‘inertial’’ time which determines the decay of the particle angular velocity. For a sphere of radius a10 nm in water one has tI 10211 s, so that the frequency range of the inertialess approximation in the Newtonian case is rather wide: up to 1010 Hz. To derive the inertialess approximation for a magnetic particle in a Maxwell fluid, we first eliminate from Eq. (1) the viscous torque Q and then set to zero the particle moment of inertia. This results in the stochastic equation I
zij oj ¼ J i U~ þ yi ; zij ¼ zdij þ tM J j J i U, d ~ H . U ¼ U H þ tM dt
(3)
Tilde in the last formula means that the energy function is defined with respect not to a real but to a renormalized (shown as the argument) magnetic field. Note that tensor zij incorporates the usual U. The next step is to associate Eq. (3), following known rules [3], with the kinetic Fokker–Planck equation (FPE) which determines the distribution function W ðe; tÞ of the particle magnetic moment. However, this turns out to be impossible because the so constructed FPE does not admit the equilibrium solution in the Boltzmann form. The conflict is resolved by modification of Langevin equation (3) upon adding an extra drift term and an anisotropic noise. The new equation writes oi ¼ bij J j U~ þ Ai þ zi ðtÞ
(4)
with the notations bij ¼ ðzij Þ1 ¼ ½zdij þ tM J i J j U1 , hzi ðtÞzj ðt0 Þi ¼ 2Dij ðeÞdðt t0 Þ. Constructing FPE associated with Eq. (4) and analyzing it in equilibrium, one determines the drift vector and the angle-dependent diffusivity tensor: Ai ¼ J j Dij and Dij ¼ Tbij . In result, a correct FPE is obtained that in a compact form may be written as q W J i bij WJ j ½U~ þ T ln W ¼ 0. (5) qt Eqs. (4) and (5) show that, in a Maxwell matrix, the particle angular mobility depends on the type of the orientational potential and, thus, is anisotropic. The effective friction coefficient increases at J i J j U40 and diminishes at J i J j Uo0. Therefore, a Brownian particle that ‘‘actively probes’’ the viscoelastic microrheology, lingers near the potential minima and hastens to leave the potential maxima. We solve FPE (5) by the effective-field method [4], taking the solution in the form W ðe; tÞ ¼ W 0 ðe; H 0 Þf ðe; tÞ;
(6)
where W 0 is the equilibrium solution with allowance for a constant (bias) magnetic field H 0 . The coordinate dependence of the function f is preset: f ðe; tÞ ¼ 1 þ ai ðtÞðei hei i0 Þ þ bik ðtÞðei ej hei ej i0 Þ,
(7)
thus reducing the problem to that of finding the timedependent vector a and symmetrical tensor bij ; the subscript 0 here means equilibrium averaging. In the effective-field method, a is assumed to be linear in the perturbing field H 1 , and bij quadratic in that. The afore-presented considerations have in fact a more general relevance, e.g., they could be applied to a nonconducting suspension of electrical dipoles. But here we keep to the magnetic context and specify the orientationdependent part of the particle energy for the case of a magnetically hard particle in the form: UðeÞ ¼ mH 0 e½h þ eðtÞh1 ;
eðtÞ ¼ H 1 ðtÞ=H 0
(8)
with the probing field H 1 . The equilibrium magnetic moment is given then by the Langevin function as hehi0 ¼ hcos Wi0 ¼ LðxÞ; LðxÞ ¼ coth x 1=x; x ¼ mH 0 =T. (9) As a last formal step, FPE (5) is transformed into a set of chain-linked equations for the statistical moments of W ðtÞ. In this set, formally infinite, one or two first equations are retained, closed by means of expansion (7), and then solved for a and bij . Magnetic relaxation times: To find them, one sets H 1 and b to zero. Then the problem reduces to evaluation of the relaxation times for the longitudinal and transverse projections of vector a (we mark them with respect to H 0 ). After necessary calculations one arrives at finite analytical expressions [5], which render tk and t? in units of the Debye time tD ¼ z=2T characterizing the rate of rotary diffusion of the particle in a viscous fluid. The bias-field dependence is described with the aid of parameter x, see Eq. (9): the parameter that measures the viscoelasticity is q ¼ tM =tD . It relates the Maxwell time, which is the characteristics of the matrix, to the Debye time. A novel feature of the modified relaxation times is the ability of the derivatives dt=dx to change sign in the range of small x if q is sufficiently large. A detailed analysis [5] yields that such a behavior is justifiable only for t? . The point is that the basic assumptions of the inertialess approximation, which we use all throughout, hold only for qp1. Linear dynamic susceptibilities: Here tensor b in expansion (7) is once again set to zero while the probing field H 1 is assumed to have both longitudinal and transverse components. Calculation yields 1 iot M wk ¼ 3w0 cos2 W 0 hcos Wi20 , 1 iotk 1 iotM 3 w? ¼ w0 1 cos2 W 0 , ð10Þ 2 1 iot?
ARTICLE IN PRESS Y.L. Raikher, V.V. Rusakov / Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233
e231
1 1
1
0.8
2
q=0
0.6 3 0.4
4
0.2
q=0
2
0.4 Im ||/ 0
Re ||/ 0
0.5
3
0.3 0.2
4
0.1
5
0
5
0 0
1
1
2
D
3
4
5
1
0
0.1
2
D
3
1
2
0.8
1
4
5
q = 0.8 2
3 0.6 0.4
3
q = 0.8
0.2 0
0 0
1
2
D
3
4
0
5
1
2
D
3
4
5
Fig. 1. Components of the longitudinal dynamic magnetic susceptibility as functions of frequency at the magnetizing field x ¼ 0:1 (1), 0.5 (2), 1.5 (3), 3 (3), and 5 (4).
where w0 is the static initial susceptibility of the suspension. In Eq. (10), the stress relaxation effect in the matrix is reflected by the presence of the time tM and the relaxation times tk;? . Basic differences with the conventional case are deduced from Figs. 1 and 2 where, as an example, the frequency dependencies of the real and imaginary parts of the susceptibilities are plotted. The upper panes correspond to a Newtonian fluid while the lower panes render the calculation by formulas (10) at the value of q close to the largest allowed in the inertialess limit. One sees that both figures have much in common: with the growth of q, crossings of the lines in the real part of the spectrum disappear, the lines themselves become more flat and well separated. Flattening of the imaginary parts with growth of x can be expected: enhancement of the bias field ‘‘freezes’’ orientational motion of the particles, this tendency holds for any q. Notably, for the curves w00 ðoÞ the direction of the peak shift with x growth depends on q. At small q (not shown), under magnetization the peaks shift rightward, while at q40:5 the cusps of w00 ðoÞ move leftward, i.e., to lower frequencies. Another feature of the absorption spectra, also due to the matrix viscoelasticity, is certain narrowing for non-zero q. Quadratic (orientational) susceptibility: The importance of this parameter for probing of magnetic suspensions and colloids follows from the fact that under applied field these systems become birefringent. The resulting optical aniso-
tropy is uniaxial along the direction of the field and at weak excitation is quadratic in its amplitude. A simplest (although not a unique) explanation assumes that the magnetic particles are prolate spheroids. Interpreting experimental data with this concept, one concludes that the geometry axes of the particles on the average coincide with their easy magnetization ones. This simplifies the solution considerably because to find the orientational (quadratic) susceptibility of a dilute suspension it suffices to use expansion (7) retaining there both effective fields. The details of calculation for an arbitrary bias field H 0 will be given elsewhere, they are rather cumbersome. Here we give the final form of the result only for the field-free case which is relatively simple. Namely, the optical anisotropy Dn induced by a harmonic field with the dimensionless amplitude x1 ¼ mH 1 =T is rendered by the expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 ½1 þ 14ðqOÞ2 Þ½1 þ ðqOÞ2 Dn / cosð2ot dÞ 30 ð1 þ O2 Þð1 þ 49 O2 Þ þ
x21 1 þ 12 qð3q 1ÞO2 , 30 1 þ O2
ð11Þ
where tg d ¼
5O þ 12 qO½ð6 5qÞO2 9 3 2O2 þ 12 qO2 ½qð2O2 3Þ þ 15
; O otD .
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1 2 1
0.5
q=0
3
0.6
4
0.4
1
2
0.4 Im ⊥/ 0
Re ⊥/ 0
0.8
5
0.2
q=0
3
0.3
4
0.2
5
0.1 0
0 0
1
1
2
D
3
4
5
0
2
D
3
4
5
0.1
1
2
1
2
1
0.8
3
3
q = 0.8
0.6 0.4 0.2
q = 0.8 0
0 0
1
2
D
3
4
5
0
1
2
D
3
4
5
Fig. 2. Components of the transverse dynamic magnetic susceptibility as functions of frequency at the magnetizing field x ¼ 0:1 (1), 0.5 (2), 1.5 (3), 3 (3), and 5 (4).
2
2
∆ n0 / 1 , arb.units
1
∆ n2 / 1 , arb.units
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 2
4
6
8
10
Ω
0
2
4
6
8
10
Ω
Fig. 3. Frequency dependencies of the components of dynamic birefringence, the values of the viscoelasticity parameter q grow from bottom to top being 0.15, 0.3, 0.6, 0.8, and 1 (dashed).
Eq. (11) shows that this response consists of two harmonics: the zero and double ones. Notably, in the inertialess limit, which we use, only the case of zero bias field admits the high-frequency limit Ob1. For that, one finds for the amplitudes of harmonics of optical anisotropy Dn2o / 3q2 and Dn0 / 12 qð3q 1Þ. This indicates that the constant part changes its sign with q; note also that the
asymptotic values are proportional to q2 . As the latter is linear in tM , they turn to zero together. Thus, viscoelasticity gives birth to a long-living intermediate asymptotics completely unknown for the case of a viscous fluid. The range of validity of this behavior is bounded from above by pffiffiffiffiffiffiffiffiffiffi the condition oo1= tI tM . As tI is normally very small, see remarks given after Eq. (2), the said frequency interval
ARTICLE IN PRESS Y.L. Raikher, V.V. Rusakov / Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233
at moderate tM is rather wide. The graphs of Fig. 3 allow one to get a more clear impression on the role played by viscoelasticity in the field-induced dynamic birefringence of a suspension. Partial financial support from RFBR (Projects 04–02–96034 and 05–02–16949) and CRDF (Award PE–009) is gratefully acknowledged.
e233
References [1] Yu.L. Raikher, V.V. Rusakov, Phys. Rev. E 54 (1996) 3846. [2] Yu.L. Raikher, V.V. Rusakov, J. Magn. Magn. Mater. 258 (2003) 459. [3] Yu.L. Klimontovich, Statistical Theory of Open Systems, vol. 1, Yanus, Moscow, 1995 (in Russian). [4] Yu.L. Raikher, M.I. Shliomis, Adv. Chem. Phys. 87 (1994) 595. [5] Yu.L. Raikher, V.V. Rusakov, Phys. Rev. E, December 2005, to appear.
Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233 www.elsevier.com/locate/jmmm
Magnetic rotary microrheology in a Maxwell fluid Yu.L. Raikher, V.V. Rusakov Institute of Continuous Media Mechanics, Ural Branch of RAS, Perm 614013, Russia Available online 16 November 2005
Abstract A ferroparticle driven by an AC field in a Maxwell fluid is studied with regard to magnetic microrheology. Orientational kinetics is simplified by neglecting the particle rotary inertia. While in a Newtonian fluid the inertialess approximation for nanoparticles is valid practically unboundedly, in a viscoelastic matrix there exists an important restriction on the stress relaxation time. Assuming weak nonequilibrium, the magneto-orientational relaxation times and low-frequency magnetic spectra are found in the presence of a constant (bias) field. Another way of particle-mediated probing of the matrix involves the field-induced birefringence of a suspension; under zero bias field the corresponding (quadratic) susceptibility is found. r 2005 Elsevier B.V. All rights reserved. PACS: 83.10.Pp; 75.50.Mm; 83.60.Bc; 82.70.Gg; 81.05.Qk Keywords: Magnetic nanoparticles; Magnetic fluids; Microrheology; Maxwell fluid
Recent years are marked by growing interest toward microrheology as a tool to explore the structure of complex fluids at the mesoscopic scale. For that, solid particles of the reference sizes 10–100 nm are embedded in a test fluid sample and the info on their dynamic behavior is collected and analyzed. If external forces are absent, the sole cause of the tracer displacements is the Brownian diffusion. Application of an external force modifies the tracer motion and puts it under control. A convenient way to do so is to set out nanoparticles in a regular motion by an AC field HðtÞ, the output signal being the dynamic magnetization MðtÞ of the tested sample. Under apparent reformulation, interpreting of the magnetic microrheology data basically coincides with the task of theoretical description of dynamic magnetization of ferrofluids at high dilution [1,2]. For ferrofluids based on complex media the theory is yet at the start. The pertinent models split into two main classes: (1) coarse systems with the grains of a micron size, where the particle inertia is essential; (2) nanoscopic suspensions, where the inertialess limit well known in the theory of linearly viscous magnetic fluids might apply. In our model, we consider the microrheology test system as a Corresponding author. Tel.: +7 342 237 8323; fax: +7 342 237 8487.
E-mail address: [email protected] (Y.L. Raikher). 0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.10.086
dilute ferrocolloid with a viscoelastic (Maxwell) matrix possessing a single stress relaxation time tM . The colloid particles are identical ‘‘single-domain’’ and magnetically hard. Their magnetic moments have constant values and are rigidly fixed in the particle bodies. Accordingly, the state of each particle is completely determined by the orientation of its unit vector e ¼ l=m. The set of equations describing under this condition the rotary dynamics of a particle has the form [1] d o ¼ JU þ QðtÞ; dt q J ¼ e , qe I
d e ¼ ðx eÞ, dt ð1Þ
d Q þ Q ¼ zx þ yðtÞ, dt hyi ðtÞyj ðt0 Þi ¼ 2zTdij dðt t0 Þ, tM
where J is the rotation operator, yðtÞ a normalized white noise, I the particle moment of inertia, U the interaction energy between the magnetic moment and the external field, Q the friction torque against the carrier fluid, and T the temperature in energy units. Adopting the Stokes flow regime, we set the particle rotary friction coefficient z ¼ 6ZV , where V is the particle volume.
ARTICLE IN PRESS e230
Y.L. Raikher, V.V. Rusakov / Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233
For a Newtonian fluid (tM ¼ 0) Eq. (1) reduces to d x þ zx ¼ JU þ yðtÞ. (2) dt From Eq. (2) one sees that tI ¼ I=z is the ‘‘inertial’’ time which determines the decay of the particle angular velocity. For a sphere of radius a10 nm in water one has tI 10211 s, so that the frequency range of the inertialess approximation in the Newtonian case is rather wide: up to 1010 Hz. To derive the inertialess approximation for a magnetic particle in a Maxwell fluid, we first eliminate from Eq. (1) the viscous torque Q and then set to zero the particle moment of inertia. This results in the stochastic equation I
zij oj ¼ J i U~ þ yi ; zij ¼ zdij þ tM J j J i U, d ~ H . U ¼ U H þ tM dt
(3)
Tilde in the last formula means that the energy function is defined with respect not to a real but to a renormalized (shown as the argument) magnetic field. Note that tensor zij incorporates the usual U. The next step is to associate Eq. (3), following known rules [3], with the kinetic Fokker–Planck equation (FPE) which determines the distribution function W ðe; tÞ of the particle magnetic moment. However, this turns out to be impossible because the so constructed FPE does not admit the equilibrium solution in the Boltzmann form. The conflict is resolved by modification of Langevin equation (3) upon adding an extra drift term and an anisotropic noise. The new equation writes oi ¼ bij J j U~ þ Ai þ zi ðtÞ
(4)
with the notations bij ¼ ðzij Þ1 ¼ ½zdij þ tM J i J j U1 , hzi ðtÞzj ðt0 Þi ¼ 2Dij ðeÞdðt t0 Þ. Constructing FPE associated with Eq. (4) and analyzing it in equilibrium, one determines the drift vector and the angle-dependent diffusivity tensor: Ai ¼ J j Dij and Dij ¼ Tbij . In result, a correct FPE is obtained that in a compact form may be written as q W J i bij WJ j ½U~ þ T ln W ¼ 0. (5) qt Eqs. (4) and (5) show that, in a Maxwell matrix, the particle angular mobility depends on the type of the orientational potential and, thus, is anisotropic. The effective friction coefficient increases at J i J j U40 and diminishes at J i J j Uo0. Therefore, a Brownian particle that ‘‘actively probes’’ the viscoelastic microrheology, lingers near the potential minima and hastens to leave the potential maxima. We solve FPE (5) by the effective-field method [4], taking the solution in the form W ðe; tÞ ¼ W 0 ðe; H 0 Þf ðe; tÞ;
(6)
where W 0 is the equilibrium solution with allowance for a constant (bias) magnetic field H 0 . The coordinate dependence of the function f is preset: f ðe; tÞ ¼ 1 þ ai ðtÞðei hei i0 Þ þ bik ðtÞðei ej hei ej i0 Þ,
(7)
thus reducing the problem to that of finding the timedependent vector a and symmetrical tensor bij ; the subscript 0 here means equilibrium averaging. In the effective-field method, a is assumed to be linear in the perturbing field H 1 , and bij quadratic in that. The afore-presented considerations have in fact a more general relevance, e.g., they could be applied to a nonconducting suspension of electrical dipoles. But here we keep to the magnetic context and specify the orientationdependent part of the particle energy for the case of a magnetically hard particle in the form: UðeÞ ¼ mH 0 e½h þ eðtÞh1 ;
eðtÞ ¼ H 1 ðtÞ=H 0
(8)
with the probing field H 1 . The equilibrium magnetic moment is given then by the Langevin function as hehi0 ¼ hcos Wi0 ¼ LðxÞ; LðxÞ ¼ coth x 1=x; x ¼ mH 0 =T. (9) As a last formal step, FPE (5) is transformed into a set of chain-linked equations for the statistical moments of W ðtÞ. In this set, formally infinite, one or two first equations are retained, closed by means of expansion (7), and then solved for a and bij . Magnetic relaxation times: To find them, one sets H 1 and b to zero. Then the problem reduces to evaluation of the relaxation times for the longitudinal and transverse projections of vector a (we mark them with respect to H 0 ). After necessary calculations one arrives at finite analytical expressions [5], which render tk and t? in units of the Debye time tD ¼ z=2T characterizing the rate of rotary diffusion of the particle in a viscous fluid. The bias-field dependence is described with the aid of parameter x, see Eq. (9): the parameter that measures the viscoelasticity is q ¼ tM =tD . It relates the Maxwell time, which is the characteristics of the matrix, to the Debye time. A novel feature of the modified relaxation times is the ability of the derivatives dt=dx to change sign in the range of small x if q is sufficiently large. A detailed analysis [5] yields that such a behavior is justifiable only for t? . The point is that the basic assumptions of the inertialess approximation, which we use all throughout, hold only for qp1. Linear dynamic susceptibilities: Here tensor b in expansion (7) is once again set to zero while the probing field H 1 is assumed to have both longitudinal and transverse components. Calculation yields 1 iot M wk ¼ 3w0 cos2 W 0 hcos Wi20 , 1 iotk 1 iotM 3 w? ¼ w0 1 cos2 W 0 , ð10Þ 2 1 iot?
ARTICLE IN PRESS Y.L. Raikher, V.V. Rusakov / Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233
e231
1 1
1
0.8
2
q=0
0.6 3 0.4
4
0.2
q=0
2
0.4 Im ||/ 0
Re ||/ 0
0.5
3
0.3 0.2
4
0.1
5
0
5
0 0
1
1
2
D
3
4
5
1
0
0.1
2
D
3
1
2
0.8
1
4
5
q = 0.8 2
3 0.6 0.4
3
q = 0.8
0.2 0
0 0
1
2
D
3
4
0
5
1
2
D
3
4
5
Fig. 1. Components of the longitudinal dynamic magnetic susceptibility as functions of frequency at the magnetizing field x ¼ 0:1 (1), 0.5 (2), 1.5 (3), 3 (3), and 5 (4).
where w0 is the static initial susceptibility of the suspension. In Eq. (10), the stress relaxation effect in the matrix is reflected by the presence of the time tM and the relaxation times tk;? . Basic differences with the conventional case are deduced from Figs. 1 and 2 where, as an example, the frequency dependencies of the real and imaginary parts of the susceptibilities are plotted. The upper panes correspond to a Newtonian fluid while the lower panes render the calculation by formulas (10) at the value of q close to the largest allowed in the inertialess limit. One sees that both figures have much in common: with the growth of q, crossings of the lines in the real part of the spectrum disappear, the lines themselves become more flat and well separated. Flattening of the imaginary parts with growth of x can be expected: enhancement of the bias field ‘‘freezes’’ orientational motion of the particles, this tendency holds for any q. Notably, for the curves w00 ðoÞ the direction of the peak shift with x growth depends on q. At small q (not shown), under magnetization the peaks shift rightward, while at q40:5 the cusps of w00 ðoÞ move leftward, i.e., to lower frequencies. Another feature of the absorption spectra, also due to the matrix viscoelasticity, is certain narrowing for non-zero q. Quadratic (orientational) susceptibility: The importance of this parameter for probing of magnetic suspensions and colloids follows from the fact that under applied field these systems become birefringent. The resulting optical aniso-
tropy is uniaxial along the direction of the field and at weak excitation is quadratic in its amplitude. A simplest (although not a unique) explanation assumes that the magnetic particles are prolate spheroids. Interpreting experimental data with this concept, one concludes that the geometry axes of the particles on the average coincide with their easy magnetization ones. This simplifies the solution considerably because to find the orientational (quadratic) susceptibility of a dilute suspension it suffices to use expansion (7) retaining there both effective fields. The details of calculation for an arbitrary bias field H 0 will be given elsewhere, they are rather cumbersome. Here we give the final form of the result only for the field-free case which is relatively simple. Namely, the optical anisotropy Dn induced by a harmonic field with the dimensionless amplitude x1 ¼ mH 1 =T is rendered by the expression sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 ½1 þ 14ðqOÞ2 Þ½1 þ ðqOÞ2 Dn / cosð2ot dÞ 30 ð1 þ O2 Þð1 þ 49 O2 Þ þ
x21 1 þ 12 qð3q 1ÞO2 , 30 1 þ O2
ð11Þ
where tg d ¼
5O þ 12 qO½ð6 5qÞO2 9 3 2O2 þ 12 qO2 ½qð2O2 3Þ þ 15
; O otD .
ARTICLE IN PRESS e232
Y.L. Raikher, V.V. Rusakov / Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233
1 2 1
0.5
q=0
3
0.6
4
0.4
1
2
0.4 Im ⊥/ 0
Re ⊥/ 0
0.8
5
0.2
q=0
3
0.3
4
0.2
5
0.1 0
0 0
1
1
2
D
3
4
5
0
2
D
3
4
5
0.1
1
2
1
2
1
0.8
3
3
q = 0.8
0.6 0.4 0.2
q = 0.8 0
0 0
1
2
D
3
4
5
0
1
2
D
3
4
5
Fig. 2. Components of the transverse dynamic magnetic susceptibility as functions of frequency at the magnetizing field x ¼ 0:1 (1), 0.5 (2), 1.5 (3), 3 (3), and 5 (4).
2
2
∆ n0 / 1 , arb.units
1
∆ n2 / 1 , arb.units
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 2
4
6
8
10
Ω
0
2
4
6
8
10
Ω
Fig. 3. Frequency dependencies of the components of dynamic birefringence, the values of the viscoelasticity parameter q grow from bottom to top being 0.15, 0.3, 0.6, 0.8, and 1 (dashed).
Eq. (11) shows that this response consists of two harmonics: the zero and double ones. Notably, in the inertialess limit, which we use, only the case of zero bias field admits the high-frequency limit Ob1. For that, one finds for the amplitudes of harmonics of optical anisotropy Dn2o / 3q2 and Dn0 / 12 qð3q 1Þ. This indicates that the constant part changes its sign with q; note also that the
asymptotic values are proportional to q2 . As the latter is linear in tM , they turn to zero together. Thus, viscoelasticity gives birth to a long-living intermediate asymptotics completely unknown for the case of a viscous fluid. The range of validity of this behavior is bounded from above by pffiffiffiffiffiffiffiffiffiffi the condition oo1= tI tM . As tI is normally very small, see remarks given after Eq. (2), the said frequency interval
ARTICLE IN PRESS Y.L. Raikher, V.V. Rusakov / Journal of Magnetism and Magnetic Materials 300 (2006) e229–e233
at moderate tM is rather wide. The graphs of Fig. 3 allow one to get a more clear impression on the role played by viscoelasticity in the field-induced dynamic birefringence of a suspension. Partial financial support from RFBR (Projects 04–02–96034 and 05–02–16949) and CRDF (Award PE–009) is gratefully acknowledged.
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References [1] Yu.L. Raikher, V.V. Rusakov, Phys. Rev. E 54 (1996) 3846. [2] Yu.L. Raikher, V.V. Rusakov, J. Magn. Magn. Mater. 258 (2003) 459. [3] Yu.L. Klimontovich, Statistical Theory of Open Systems, vol. 1, Yanus, Moscow, 1995 (in Russian). [4] Yu.L. Raikher, M.I. Shliomis, Adv. Chem. Phys. 87 (1994) 595. [5] Yu.L. Raikher, V.V. Rusakov, Phys. Rev. E, December 2005, to appear.