discrete particle size distribution in ferrofluids

Z. Phys. B - Condensed Matter 59, 253-256 (1985) Gon Zeitschrift Matter ~r Physik B

�9 Springer-Verlag 1985

Discrete Particle Size Distribution in Ferrofluids

T. Weser and K. Stierstadt Sektion Physik der Universitiit Miinchen, Federal Republic of Germany

Received February 4, 1985

We present a method to obtain stable discrete particle size distributions from the room temperature magnetization curves of ferrofluids. This method has been successfully applied to a hydrocarbon base magnetite ferrofluid.

1. Introduction

A ferrofluid is a suspension of single domain magnetic particles which are coated with a layer of long chain molecules to inhibit agglomeration. Typical particle sizes are of the order of 10 nm. For particles of this diameter Brownian motion is sufficient to prevent sedimentation in the earth's gravitational field.

Superparamagnetic Mean Field Theory

According to the Langevin theory of superparamag- netism, the magnetization m i of an isolated isotropic subdomain particle of volume v~ in an applied field Ho~ is given by

m i = M D langv \(#~ MDkT v-i Hex ) (1)

with langv (x), = coth (x) - 1/x ("Langevin Func- tion"). In this equation k is the Boltzmann constant, T the absolute temperature, #0 the permeability of free space and M D the domain magnetization. Equation (1) can be extended to consider the magnetic properties of a volume V, in which particles of different size are suspended. If the particles are non- interacting the individual contribution M~ to the magnetization M of this volume due to n~ particles of size v~ is equal to the sum of the individual moments of these particles averaged over the whole volume, d2~rnl; here Oi=nivi/Vis the volumetric concentration of the particles of the size v i. To obtain

the overall magnetization M the contributions M i of all size fractions have to be summed up and one obtains

M = M ~ ~ langv \ kT ] (2)

where ~ = ~ b ~ is the total volumetric concentration

of solids and Ms=el)M D the saturation magnetization of the ferrofluid. This conventional procedure is described for example by Kaiser [-1]. In concentrated ferrofluids the particle interaction has to be considered. The particles are of spherical shape and their interactions may be treated in the framework of the following mean field theory: One replaces the applied field Hex by the local field Hlo r inside a spherical cavity within a homogeneous material of magnetization M and with the shape of the ferrofluid sample. If the latter is approximately an ellipsoid, the local field Hto ~ is given by

H~o~ = Hex -DM, (3)

where D is the combined demagnetization factor of the inner and outer material surfaces. For a proof of this result in the analogous situation of a dielectric material in an electric field see for example Kittel [-2]. Substituting (3) into (2) the following relation is obtained

M=Ms~ ~ langv ( ~~ )" (4)

254 T. Weser and K. Stierstadt: Discrete Particle Size Distribution in Ferrofluids

In the case of a continuous particle size distribution qS(v) one has ~b=SdqS(v ) and

(#o (Hex -DM) vM D ). M = M s ~ ~ langv \ (4')

In (4') He~ and M can be measured, Mo, D and T are supposed to be known; therefore (4') is a linear integral equation of first kind (LIE 1) for ~b(v). It is well known that the solution of a LIE1 is an "ill-posed problem" in the following sense: The solution operator is not continuous, i.e. the solution of (4') for q5 is not stable under small variations of the experimental values of magnetization and field. With other words, because of their finite number and principal imperfection, the experimental data are compatible with a variety of possible solutions and do not permit a unique determination of a distribution function. Therefore the treatment of (4') with numerical standard methods (for example discreti- zation of q5 like in (4); least squared deviations) produces solutions which are bare of any physical sense. To obtain nevertheless some kind of particle size distribution an additional selection principle has to be considered which leads to a somehow preferred solution among the whole variety, and which makes the solution process stable under variations of the experimental data. This can be done in one of the following ways: 1. An a priori given size distribution is fitted to the magnetization curve or only parts of it (initial sus- ceptibility; saturation magnetization). Many authors (for example [31) prefer a log-normal distribution function, but the fit is as well possible with a rect- angular or a triangular or some other function (see for example [4]). 2. The parameters of a discrete particle size distribution are determined by fitting the magnetization curve under an additional extremal condition (maximum entropy: Potton [5]). The hence obtained solutions show some desired qualities; the maximum entropy technique selects the solution with the larg- est uncertainty ("entropy") in the sense of Shannon [6] among all compatible functions. This leads to non-prejudiced, believable size distributions. 3. Tichonov's [7] regularization method is similar to the second strategy, but numerically easier, because the additional extremal condition is treated only in an heuristic way: One minimizes a suitable linear combination of the two terms representing the two extremal principles (data fit and selection rule). Hence one obtains stable non-prejudiced results.

Regularization Method

In this paper we present the application of the regularization method of Tichonov [7] - the third of the above mentioned strategies. Referring to (4) we define the magnetization fraction wi:=MD~)i=ms(gi/~ (i=1, ,I) and obtain the following ansatz for the total magnetization

I

Md~',','d~(Hex --DM) = ~, w, langv (cd3(H~ - D M ) ) (5) i = 1

with c: =#o 7rM~/6kT. In this equation d i denotes the diameter of the particles (supposed to be spherical) of volume v~ (i= 1, , I). In this ansatz with given dl (i= 1, , I) the contributions w~ of each fraction to the magnetization will be determined using N experimental data (H~x, M") (n = 1, , N), N > I. The data fit corresponds to the principle of least sum a of squared deviations

N W l , . . . , w i n ~= ~', {Md . . . . e~ ( H e x - D M " ) - M " } 2, (6)

n = l

but - according to the above mentioned difficulties - the extremum conditions

O=ow j= 3 ,, n . Oa 2 wilangv(cd i ( H ~ x - D M ) ) - M n = l i

3 n ?/ �9 �9 langv (cdj (Hex - D M )) (j = 1 . . . . I) (7)

lead to a system of I linear equations for the un- known w i ( i = 1 , , I ) the system matrix of which is nearly singular. Therefore the results for w~ (i = 1, , t) have large pos- itive and negative values which are unstable under the variation of M" and H~x ( n = l , , N ) and are obviously bare of any physical sense. This shows the need for an additional selection principle which discriminates against these unphysical solutions. As they are characterized by large values of their squared norm

I

f l : = 2 W2 ~ , (8/ i = l

we chose the demand for small values of p as additional, stabilizing principle. Following Tichonov [7] we minimize

cr':=cr+ c~- p, c~>0, (9)

and call p the "regularizing operator" and e the "regularizing parameter". As above we obtain again

T. Weser and K. Stierstadt: Discrete Particle Size Distribution in Ferrofluids 255

I equations for w~, , w~

([ ~, w, langv(cd~(H:,~-DM"))-M"] n= I I. L i= 1

d~ (H"e x - DM"))I + o~ wj = 0 (j = 1, , I). langv (c 3

(10)

The case ~ =0 corresponds to the original ill-posed problem. By construction of the regularizing operator the increase of c~ has a stabilizing effect, because the above mentioned unstable results for w i (i= 1, , I) are suppressed. More precisely there exists an interval of finit length for ~ within which the minimum conditions for a' have unique solutions w i (i= 1, , I) which are stable under variations of e and (He~, M") (n = 1, , N). The quality of these results as approximative solutions for the original problem (e= 0) is determined by the value of a and can be controlled by inserting the wl (1 = 1, , 1) into (6). Using small values of c~ one obtains accurately fitted magnetization curves (small values of o-), but crude and unstable particle size distributions (great values of p). On the other hand large values of ~ lead to smooth and stable distributions, but to larger values of a. Between these extremal cases a reasonable value for c~ has to be found. Helpful in this situation is the fact that the

mean standard deviation I / ~ / N has to be compara- ble to the accuracy of measurement. From this condition one obtains a guess for ~r and hence a suitable value for e. The choice of the regularizing operator is in some way arbitrary - as arbitrary as any other selection mechanism. This must not be regarded as a lack of Tichonov's method, but is due to the nature of the original problem (LIE1). Our choice of p leads to simple equations (the equations (10) are linear in w~) and to stable, unprejudiced and substantially posi- tive results. Possibly appearing negative values of w~ do not exceed the error boundaries. A more elab- orated treatment of freedom from prejudice and sen- sible positivity is given by the maximum entropy method - at the cost of a considerably higher numerical effort. On the other side in comparison with an a priori chosen size distribution function the parameters of which fit both ends of the magnetization curve, the results obtained by the regularization method have two advantages: They contain more information because they are obtained by evaluating and fitting the whole magnetization curve, and they are less prejudicious because the shape of the resulting particle size distribution is not a priori fixed.

2. Results

We prepared a hydrocarbon base magnetite ferrofluid using Reimers' [8] precipitation technique. This ferrofluid has been investigated at room temperature with Bissell's [9] cavity field technique. Experimen- tal details are described in [10]. The experimental data for the magnetization M(Hloo) of our ferrofluid and a theoretical curve obtained by using the regularization method with = 1/16 are shown in Fig. 1. From these sets of data we determined discrete particle size distributions using (10) with various regularizing parameters e ( e = 2 -p, p =0 , 1 . . . . ,10). The c~-dependence of the results was as expected (see Fig. 2a-e ; these data refer to the magnetic nuclei of the particles ignoring their nonmagnetic surface coating). The above mentioned adaptation of the mean stand-

ard deviation 1/~/N to the accuracy of measurement led to e = l / 1 6 as optimal regularizing parameter. The corresponding particle size distribution is shown in Fig. 3 where error boundaries - obtained by variation of the experimental data - are included. We propose the following interpretation: ( I ) The results reflect clearly the fact that very small particles (d=2 nm) are not "seen" by the examination method because they have no magnetic remanence. (2) The poor significance of the contribution of d =4 nm can be understood from the lack of experimental data corresponding to higher fields ( / - / ~ 50 .104 A/m) where the small particles dominate the magnetization curve. (3) The negative value at d = 2 0 n m is unphysical but not significant. (It is a general property of Tichonov's method that an im- perfect regularization is indicated by an increasing error.) (4) The main part of the distribution curve is quite satisfying. (Note also its stability with respect to ~, Fig. 2 b-d).

u~

o

r- u~ o

.N N o r

~33

c5

.x- x~

x- x"

x" x:

,x

•

' . 2 .r ; .; ' 1~0' 1[2 ' 1'.s 116' 1 :8 '2 ' .0 2'.2 Local H -F ie ld [ 10SA /m)

Fig. 1. Magnetization of hydrocarbon base magnetite ferrofluid. Temperature 20~ (x) Experimental values, (...) theoretical curve (regularization method)

256 T. Weser and K. Stierstadt: Discrete Particle Size Distribution in Ferrofluids

C 1 ) ~ = 1

cr = 1 , 9 . 1 0 7 A 2 / m 2

b) o~ = 1/8

o" = 1,6"106A2/m 2

C) ct = 1/16 = 9,9" 105 A2/m 2

d ) ,~ = 1/32 o" = 7,1 �9 105 A2/m 2

e) a = 111024 o" = 1,4.105 A2/m 2

Particte Diameter

di(nm) 24

6 8

10 12 14 16 18 20

2 4 5 8

10 12 14 16 18 20

Magnetization C~ wi

I i I

5000Aim

--1 I

I I

I

4 6 8

10 12 14 16 18

20

2 /,

5 8

10 12 14 16 18 20

21 4

6 %

10 12 14 16

2O

5000Aim

-1 i

i I I

5000Aim

- - - - - q I

1

5000Aim

I I I

- q

Fig. 2. Particle size distributions with various values of

3. Conclusions

The principal conclusions are: (1) The magnetization of colloidal magnetite ferrofluids at room temperature is in good agreement with superparamagnetic mean field theory. (2) A stable determination of a discrete particle size distribution from the magnetization curve is possible by using the regularization technique of Tichonov.

c- O

ca "r-

0 (D t- O

N

c-

O

7

• J- 12 ' 2 4 6 8 10

Part icte d iameter di (nm)

Fig. 3. Particle size distribution (e = 1/16)

We are much indebted to L. Schwab for valuable comments and discussions. This work was supported by the Stiftung Volkswa- genwerk of the German Federal Republic.

References

1. Kaiser, R., Miskolczy, G.: J. Appl. Phys. 41, 1064 (1970) 2. Kittel, C.: Introduction to Solid State Physics. 4th Edn., pp.

449-463. New York: Wiley 1971 3. Chantrell, R.W., Popplewell, J., Charles, S.W.: IEEE Trans.

Magn. 14, 975 (1978) 4. Kneller, E.: In: Ferromagnetism. In: Handbuch der Physik.

Fltigge, S. (ed.), Bd. XVIII, 2. Teil, pp. 479-480. Berlin, Heidel- berg, New York: Springer 1966

5. Potton, J.A., Daniell, G.J., Melville, D.: J. Phys. D17, 1567 (1984)

6. Shannon, C.E.: Bell Syst. Tech. J. 27, 379 (1948) 7. Tichonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Prob-

lems. New York: Wiley 1977 8. Reimers, G.W., Khalafalla, E.: US pat. 3,843,540 (1974) 9. Bissell, P.R., Bates, P.A., Chantrell, R.W.: J. Magn. Magn.

Mater. 39, 27 (1983) 10. Weser, T.: Magnetoviskosidit yon Ferrofluiden. Diplomarbeit

an der Universit~it Mtinchen (1984)

T. Weser K. Stierstadt Sektion Physik UniversitSt Miinchen Schellingstrasse 4 D-8000 Miinchen 40 Federal Republic of Germany

discrete particle size distribution in ferrofluids

Documents