Viscosity of suspension of HTSC particles

Journal of Magnetism and Magnetic Materials 149 (1995) 50-52

~ l ~ Journalof magnetism and magnetic materials

ELSEVIER

Viscosity of suspension of HTSC particles Itaru Kobori *, Hiroshi Yamaguchi Faculty of Engineering, Doshisha University, Tuzuki-gun, Tanabe-cho, Kyoto 610-03, Japan Abstract

The stress tensor of a suspension of high-T~ ceramic particles is investigated theoretically. The effect of Brownian motion is considered by using the Fokker-Planck equation. The increment in the effective viscosity is shown for Couette flow between two parallel plates which are perpendicular to the applied external magnetic field•

I. Introduction

Recently, suspensions of high-T¢ superconducting particles have been studied by several authors [1]. The suspended particles were treated as complete diamagnetic particles without taking into account the characteristics of so-called high-temperature superconductors (HTSCs). Kalikmanov [2] investigated the behavior of dilute suspensions of single-crystal high-T¢ superconducting particles in an alternating magnetic field. Although he took into account the uniaxial characteristics of HTSC particles, the problem was confined to the dynamics of a single particle in a fluid. Recently, we have studied the fluid mechanical properties of a dilute HTSC particle suspension in a nonconducting liquid theoretically [3]. The constitutive equation is obtained by using the formalism analogous to the theory of ferrofluids of Shliomis [4]. The formalism has a character of mean field theory and it is not easy to take into account the individual motions of suspended particles and the hydrodynamic interactions among suspended particles. In this paper, we formulate the same problem in the theological point of view of Kirkwood [5]. A single crystal of HTSC is strongly anisotropic, and an external field exerts a torque on the crystal below the critical temperature Tc of the superconducting transition. Since HTSCs are type-II superconductors, they show different behaviors depending on the strength of the external magnetic field H. We restrict the discussion to the domain H¢1 < H < H¢2 as we have the analytic expression of the reversible magnetic moment of a particle which agrees with available experimental results. In these formulae the anisotropy is introduced by the anisotropic effective mass tensor of the electron in the crystal [6].

* Corresponding author. Fax: +81-7746-5-6831.

The increment in the stress tensor in the external magnetic field is obtained as nfi k, where n is the number density of particles and __

*

• ,.~/(H2m33)1/2)12. In this equation, the vector e is the unit vector of the crystal axis (orthogonal to the CuO plane) of a particle, H is the external field, and O is the local angular velocity of the fluid, m33 = m 1 + (m 3 - ml)(e. B ) Z / B 2 is a component of the mass tensor of an electron, where ml, m2, and m 3 are the diagonal elements of the mass tensor in the crystal frame and B is the magnetic flux density, r and V are the Brownian relaxation time of rotation and the volume of a particle, respectively• ( )~ means the average on the stationary distribution function of rotating states which is determined by the Fokker-Planck equation• 2. Torque on a fluid element

All known HTSCs are strongly anisotropic. In the orthorhombic Y1Ba2Cu307, the c-axis of the primitive cell is about three times larger than either the a or b dimension in the basal CuO plane. In the CuO plane the anisotropy is relatively weak. Therefore the major anisotropy effects can be described by a uniaxial model. The coherence length ~ of these materials is considerably shorter than the penetration depth A. This means that the vortex core size (of order sc) is small with respect to the characteristic size of the current and field distributions. For the intermediate field He1 < H < H¢2 , where the Ginzburg-Landau equations cannot be solved in a closed form due to their nonlinearity, the London model has provided the only detailed phenomenological description of superconductors for which the Ginzburg-Landau parameter K = A / sc is much larger than unity (the second kind or type-II superconductors). In the London model the

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2

fik -- eikjT( V H / 8 7 r ) (m 3 - m, ) ( ( e × H ) j ( H X e)

L Kobori, H. Yamaguchi/Journal of Magnetism and Magnetic Materials 149 (1995) 50-52 local magnetic flux density of type-II superconductors is represented by a superposition of the fields of isolated vortices. The suspended particles are assumed to be spherical single crystals. In the crystal frame of reference, the effective mass tensor of the electron is m i k = mitSik. In order to deal with the field characteristics, it is convenient to use the vortex frame of reference in which the vortex axis is parallel to the z-axis. In order to represent the orientation of the crystal, we introduce the fundamental orthogonal vectors eck (k = 1, 2, 3) in the crystal frame, where e,.3 is parallel to the c-axis of crystal. When a particle rotates, the torque experienced by the particle is, in the vortex frame [3],

N = (VHoH */4"rr)(m 3 - m 1 ) ( e . B ) ( B -,

/(B-m33 )

1/'2

×

.

(2.1)

(2.2)

where Y is the operator of rotation, ie X 0/0e, and D is the rotational diffusion constant which is equal to 1/2"r B in Ref. [4]. The Brownian rotational diffusion time is r e = 3Vrl/kT, where V, r/, k and T are the volume of a particle, the viscosity, the Boltzmann constant and the absolute temperature, respectively. In the following, we use 1/2"r B instead of D. The equilibrium distribution function is the solution to Eq. (2.2) in the absence of vorticity, 0 = A exp[ - ( V H o H * / 4 ~ r k T ) × {m, + (m 3 -- m l ) ( e .B)2}1/2] ,

The equation of ( e ) is obtained by averaging e with the nonstationary distribution which satisfies Eq. (2.2). Multiplying Eq. (2.2) by e and integrating over the phase space (using the anti-Hermitian character of the operator i 2 , integration by parts and flux-free condition on the boundary of phase space), we obtain the equation for e after some straightforward algebra: 27"B O(e)/3t = - - 2 ( e ) -- ( N X e ) / k T + 2"rB(~ X e).

(2.4) The average value in the steady state is given by the functional equation - 2 ( e ) - ( N X e ) / k T + 2 % ( / ~ x e) = 0.

(2.5)

e)

The torque expressed by Eq. (2.1) is invariant under the inversion of e. This property differs from that of the anisotropic ferromagnetic particle. The torque experienced by a fluid element is the sum of the torques experienced by each particle in that fluid element. This is obtained as the sum of the statistically averaged values of single-particle quantities, and is equivalent to the average over the statistical ensemble of fluid elements. The statistical distribution function is obtained as the solution of the Fokker-Planck equation. The fluid element rotates with local angular velocity O = to/2 (to = vorticity). The torque exerted on a single particle (by the external field) is N. Then the distribution function tk(e) of the orientation of suspended particles satisfies [7]

O~b/~t=Di~[i_~-N/kT]~b-i.c2(12qz),

51

(2.3)

3. Stress tensor

The exertion of the torque on the particles by the external field generates an antisymmetric contribution to the bulk stress. The torque exerted by the external field per unit volume is

N i = eijko'jk,

(3.1)

which gives O'ik

=

--

neik l ( N l / 2 ) .

(3.2)

From Eqs. (2.l) and (3.2) one can write o'i, = - neik,( ( VHoH * / 4 a r ) ( m 3 - mj )(e" B ) ( B X e)l

/(B2m33)'/2),

(3.3)

where n is the number density of particles and eij t is the Levi-Civita tensor. ( n N / 2 ) is determined using the equation of ( ( e - H ) ( H × e)). In the same way as in the case

of (e), 2% a((e-H)(Hx

e))/Ot

= -6<(e-H)(Hxe)>

- 2 < N ( e . H ) 2 ) / k T + 2"rB<(HXe)2gl).

(3.4)

In the following, ( )a denotes the average over the steady-state ensemble with angular velocity O, and ( )o denotes the average over the equilibrium ensemble (2.3). Then, from (2.13), we have

( ( e . H ) ( n x e)),~ = - ( N( e- H ) 2 >xJ3 kT + rB/3{ ( H X e )2 a )~.

where A is the normalization constant. In the following, we consider the flow for which the vorticity is perpendicular to the external field. The average of the square of e ~_, which is the component of e perpendicular both to the external field and the vorticity, is necessary in the calculation of the viscosity. In the following, we need the average over the fluid element in steady flow with finite vonicity. This average is calculated by making use of Eq. (2.2).

(3.5) Although it is difficult to obtain the explicit expression for the steady-state distribution function, it is not difficult to calculate the average to first order in ,O. ( N ) o is considered to be a quantity of order 1 in /2, because there is no macroscopic torque in the equilibrium state. We replace ( N ( e . H ) 2 ) o by ( N ) a ( ( e . H ) 2 ) o . This is assumed to

L Kobori, H. Yamaguchi /Journal of Magnetism and Magnetic Materials 149 (1995) 50-52

52

be reasonable within the approximation to first order in g2. We then have ((e.H)(H×

e))s~ = r / 3 ( ( H ×

e)(H×

e). a)0, (3.6)

where

1 ((e.H)2)oVH 7"= ~'B/ 1 + -~ 4~kT

* (m3-ma) ] ~ "

(3.7)

where ~o is the volume ratio of suspended particles to the entire volume of the fluid, and is equal to nV. Eq. (4.1) shows that the effective viscosity increases linearly with the external field. For typical values of variables, A = 10 -5 cm, ~ = 10 7 cm, C ~ o = h c / 2 e = 2.07 × 10 -7 G cm 2, H* = (1/8"rr) × 103 G, V = (4'rr/3) X 10 -19 cm 3, T = 100 K, and m 3 = 10 m 1, we get the effective viscosity as

aqef~/rt

The above formulae give the explicit expression of the stress tensor

= [1+q~{5/2+1.8×10

3×H×ln(~Hc2/H)}

]. (4.2)

o.ik = nfi k = neikj( N j / 2 ) o , where

fik = eikff( V2H * / 8 1 r ) ( m 3 - m l ) < ( e

×

H ) j ( H × e)

• ~Q/(n2m33)l/2)[].

(3.8)

Eq. (3.8) shows that the stress increment which comes from the rotational motion of the particles is proportional to the external field. The explicit expression for the effective viscosity for Couette flow is given in Section 4.

4. Application and discussion In an external magnetic field the stable orientation of the c-axis of a suspended HTSC particle is perpendicular to the field direction. In the equilibrium state, the c-axes of particles in the fluid are ordered parallel to the plane perpendicular to the field. When there are flows in the fluid, the orientation of particles is disturbed, and the particles return to the ordered states in the relaxation time ~- (characteristic time) shown by (3.7). This is the mechanism of the dissipation of energy and causes the increase in viscosity under an external field. When the external field and vorticity of the fluid are parallel the above-mentioned effect disappears, at least in the first approximation. The Couette flow between two parallel plates which are perpendicular to the external field is an example in which vorticity is expected to be perpendicular to the field. The flow of an incompressible fluid contained between two parallel plates is considered. One of the planes at z = 0 is immobile and the other at z = a moves along the x-axis with velocity 2J2a. We assume that J2~- is small. The steady state, v=(2g2z, 0,0),

g ~ = ( 0 , g2,0),

U=(0,0,

n),

satisfies the hydrodynamic equations and Maxwell equations in the case of a macroscopically nonconducting fluid. The viscous force acting on a plane surface at z = 0, per unit area, is 2,O"011 + ~o{5/2 + ( V / 6 4 a x k T ) H o H *(m 3 - ml) /1/-~-1 }].

(4.1)

where H is the external field and ~" may be considered to be an adjustable parameter. The above calculation is based on the model which assumes that the dimensions of particles are large in comparison with the radius of the vortex core. From the data of the coherence length of the HTSC, the diameter of the vortex core of the material is assumed to be several angstroms. On the other hand, the dimensions of the suspended particles in the ordinary fluid are 10-50 times larger than the dimension of the vortex core. We have shown [3] that the particle in which a single vortex is present, reveals rather similar angular dependences of the magnetization and torque characteristics. The angular dependence of the critical field is taken into account by replacing the critical field H c with Hc/m33. The angular dependence of the critical field suggests the interesting phenomenon of dependence on the velocity field of the fluid. Spherical Couette flow will show interesting phenomena that differ from the case of a ferrofluid. As for the shape of particles, the results for the spherical particles provide insight into the particles of other shapes. In an actual powder of HTSC, there will be particles of various shapes. It is thought that the shape of particles will not necessitate any qualitative modification of the theory. However, the Landau-Ginzburg approach is necessary for a detailed discussion of the shape dependence of the magnetization.

References [1] V.I. Kalikmanov and I.G. Dyadkin, J. Phys.: Condens. Matter 1 (1989) 993; V.I. Kalikmanov and D.I. Sementsov, J. Magn. Magn. Mater. 85 (1990) 71. [2] V.I. Kalikmanov, J. Magn. Magn. Mater. 122 (1993) 154. [3] I. Kobori and H. Yamaguchi, J. Phys. Soc. Jpn. 63 (1994) 2691. [4] M.J. Shliomis, Sov. Phys. Usp. 17 (1974) 153. [5] J.G. Kirkwood, Selected Topics in Statistical Mechanics, in: Document on Modern Physics (Gordon and Breach, New York, 1967). [6] V.G. Kogan, Phys. Rev. B38 (1988) 7049. [7] M. Doi and S.F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, Oxford, 1986) p. 293.