A model of the properties of colloidal dispersions of weakly interacting fine ferromagnetic particles

166

Journal of Magnetism and Magnetic Materials 43 (1984) 166-176 North-Holland, Amsterdam

A MODEL OF THE PROPERTIES OF COLLOIDAL DISPERSIONS OF WEAKLY INTERACTING FINE FERROMAGNETIC PARTICLES S. M E N E A R , A. BRADBURY and R.W. C H A N T R E L L * Department of Physics, U.C.N.W., Bangor, Gwynedd, UK Received 13 January 1984

A Monte Carlo technique has been used to simulate the magnetic properties of a colloidal dispersion of weakly interacting fine ferromagnetic particles. The initial susceptibility is shown to obey a Curie-Weiss like law in its variation with temperature. The ordering temperature in the Curie-Weiss law is found to increase with the diameter of the particles, the increase being associated with an increase in the local order in the system. Data from the Monte Carlo simulation is also used to assess the effects of interactions on the determination of particle size parameters from magnetic measurements. Investigation of the spatial correlation within the system reveals evidence of field induced particle agglomeration.

1. Introduction

Magnetic fluids are colloidal dispersions of small single domain particles in non-magnetic carrier fluids. Colloidal stability is achieved in two ways. Firstly, the particles are coated with long chain surfactant molecules which prevent close approach of the particles thereby reducing the possibility of aggregation via the short-range Van der Waals force. Secondly, the particles are made as small as possible (normally < 10 nm) in order to reduce the effect of the long-range magnetic dipolar interactions between the particles. These fluids thus resemble a paramagnetic gas and their associated properties are described as superparamagnetic, since the magnetic moments of the particles are orders of magnitude larger than atomic magnetic moments. The effect of interactions on the magnetic prop: erties of ferrofluids has generally been neglected. However, re-interpretation of the work of Soffge and Schmidbauer [1] by O'Grady [2], and the results of O'Grady et al. [3] have shown the interesting possibility of a description of interactions in terms of a thermodynamic ordering temperature. * Division of Physics and Astronomy, Preston Polytechnic, Preston, Lanes. PR1 2TQ, UK.

Refs. [1,3] reported low field susceptibility measurements on ferrofluids consisting of ferrite particles in a hydrocarbon base and of cobalt particles in toluene. In each case the variation of initial susceptibility with temperature was found to obey a Curie-Weiss law Xi =

C(Z- To)-'

(1)

It was shown by the Monte Carlo calculations of O'Grady et al. [3] that the ordering temperature TO could be explained in terms of interparticle interactions. The magnitude of the ordering temperature depends only on the interaction strength. Thus, TOis an important quantity in the investigation of the effects of interactions since it may be compared with the equivalent experimental parameter to quantify the effects of interactions in a practical magnetic fluid. Any correlation should be of general interest since interactions not only affect the fluid stability but may also provide an important contribution to the field-induced anisotropy which has been associated with such properties as viscosity [4], transparency [5-7] ultrasonic attenuation [8,9] and birefringence [10-15] of a magnetic fluid. In this paper the results of Monte Carlo calculations of the effect of interactions on a dispersion of magnetic particles in the weak interaction reg-

0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

S. Menear et al. / A model of the properties of colloidal dispersions

ime are presented. In particular the variation of To with particle diameter for a monodisperse system has been investigated, and is shown to be related to the onset of limited local order in the form of dimers and trimers of particles. It is also shown that the transition to longer range order at larger particle diameters leads to the reduction in susceptibility predicted by previous Monte Carlo calculations [16,17].

2. Basis of the method used The magnetic moment # of a particle in a magnetic field can rotate by one of two mechanisms. Firstly, if the magnetic anisotropy energy KV (with K the anisotropy constant and V the particle volume) is large there is a strong coupling between the magnetic moment and the crystallographic axes of the particle. In this case the magnetic moment and the particle rotate together and the anisotropy energy has no bearing on the magnetisation. If KV is small, however, the moment can rotate independently of the particle. It has been shown by Krueger [18] that the anisotropy energy favours alignment of the anisotropy easy axis along the magnetic moment direction but that this alignment has no effect on the magnetisation. Both rotation mechanisms have characteristic relaxation times much shorter than typical experimental measurement times. Thus, the system can be considered as being in thermal equilibrium with the magnetic anisotropy of the particles having no bearing on the magnetisation. Hence the latter is determined simply by the energy of the particle in the applied magnetic field H and is given by the Langevin function,

L(a) = c o t h ( a ) - I/a,

(2)

where a = I.tH/kT (k is the Boltzmann constant and T the absolute temperature). The thermal average of the magnetisation is calculated taking into account the effects of interparticle interactions. The thermodynamic average of a function f(S) is

( f ) = fJ d s f ( S ) exp( - E J k T )

/ f ds exp( -

EJkT)

where E s is the potential energy of state S. The simplest Monte Carlo technique would evaluate these integrals over a random sample of points. However, by utilising the technique of importance sampling, Metropolis et al. [19] devised a method of generating Markov chains for which ( f ) converges. If Pij is the transition probability from state S~ to state Sj, then

P,+ =

[ exp(-AE/kT) 1

for AE > 0, for A E ~<.0,

(4)

where A E is the energy difference between states Si and Sj, i.e. aE = E(Sj)-

(5)

In practice, each particle in an initially random configuration experiences in turn a random displacement, and if the change in energy as a result of this move is non-positive (AE ~< 0) the move is allowed: if the energy increases (AE > 0) the move is allowed with probability exp(-AE/kT). Each configuration produces a value f o r f ( S ) , and for N configurations 1 u ( f ) = -~ ~_, f(Sg).

(6)

i=l

A two-dimensional computer model has been developed which generates typical instantaneous pictures of a small section through a monodisperse system of cobalt particles of diameter D. Each particle is assumed to have a surfactant coating thickness 8 of 2 nm (corresponding to an adsorbed monolayer of oleic acid). For the range of particle sizes studied, the criterion 6/D < 0.2 quoted by Fiedler [20] is satisfied. The energy of each particle has three components: (i) the energy E , due the presence of an externally applied magnetic field H,

EH=p'H.

(7)

(ii) the energy E M of magnetic interaction with neighbouring particles, a summation of terms of the form

p,.l~j

(3)

167

eM = - -

3(p,.r)(#j.r)

,

(8)

S. Menear et at / A model of the properties of colloidal dispersions

168

where Pi and pj are the magnetic moments of particles i and j, joined by the vector %. (iii) the energy E R due to entropic repulsion which occurs if the surfactant coating overlaps that of a neighbouring particle [21],

~rDZN'{2 E R = ~

(h+2)1 n (l+b) b (l+h/2)

h}

b '(9)

where h = ( 2 r i J D - 2 ) and b = 28/D, with N ' surfactant molecules per unit area of particle surface. The total energy of a particle is simply the sum of the contributions EH, E M and E R. The value for the magnetic moment of each particle was taken to be tL = IsV with V the particle volume and I~ the saturation magnetisation of the bulk material, and using I s = 1400 e m u / c m a for cobalt. In practice I s is slightly temperature dependent and may indeed depend upon the particle size. In addition the surface layers of the particle may be non-magnetic due to chemical reaction with the surfactant [2,22] or spin pinning at the metal/surfactant interface [23]. Taking # = IsV is thus a simplifying assumption, but one which is unlikely to have a significant effect on the predictions of the model. The magnetic interaction energy of a particle is calculated by a summation of terms like eq. (8) over nearest neighbours within a distance of typically five particle diameters. This truncation of the summation is necessary for computational efficiency but clearly neglects the longer range interactions. These interactions can be taken into account by approximating the sample outside the near neighbour region to a uniformly magnetised medium. The long range interactions are then represented by a term

E = - # ( N s - N¢)I

(10)

with I the magnetisation of the sample. N~I is the Lorentz cavity field and NcI is the demagnetising field. Ns and Nc are factors dependent on the geometry of the sample and container, respectively. Clearly E = 0 for N s = No, hence the results presented here (which neglect E ) are valid for containers having spherical symmetry. In a weakly interacting system an important

parameter is the average interparticle separation ( d ) , which determines the average strength of the interaction between pairs of particles. Experimentally this parameter is determined via the volumetric packing fraction where ( d ) c~ (c)-1/3. In order to facilitate comparison with experiment effective packing fractions will be quoted for all the systems investigated and calculated using c = ~rD3/6(d )3. The computations have been carried out using a two dimensional cell containing 1000 particles. This ensures that the cell dimensions are very much greater than the mean distance r m over which significant local ordering occurs (typically rm = 2-3 particle diameters in these simulations), thus minimising the effect of the periodic boundary conditions on the results. The susceptibility calculations were made at temperatures >> To, the ordering temperature in the Curie-Weiss law (eq. (1)), so it is unlikely that the dimensionality of the system will have a great bearing on the results obtained. Since a very large number of random numbers are used, it is essential that any random number generator produces values with good statistical properties. A routine from the Numerical Algorithms Group (NAG) library, available to all the computer systems used, generates a sequence of over 10 ~° quasi-random numbers using the multiplicative congruential relation, X,+ 1 = 1313X, mod 259. The default initial value is 123456789 * (212 + 1), although this can be chosen randomly as an alternative. The computer model has been executed on the DEC-10 computer at Bangor, the CDC 7600 at UMRCC, Manchester, and the CRAY computer previously at Daresbury, now at ULCC.

3. Behaviour of non-interacting particles The magnetic properties of non-interacting superparamagnetic particles are described by the Langevin function:

] = L(a) = coth(a) - 1/a,

S. Menear et a L / A model of the properties of colloidal dispersions where i is the reduced magnetisation and a = g H / k T . Since the Monte Carlo calculations were carried out for the two dimensional case, it is necessary to compare the results with a two dimensional equivalent of the Langevin function. It is shown in the appendix that for the two dimensional case,

i = L 2 ( a ) = I , ( a ) / I o ( a ),

30

20

10

(11)

where I t ( a ) and Io(a ) are modified Bessel functions of the first kind, of order 1 and O, respectively. In small fields where a << 1 eq. (11) approximates to

] = a/2 = #H/2kT.

169

200

600

1000

lqO0

1800

2200

2600

TEMPERATURE(KELVIN)

Fig. 1. Curie-Weiss behaviour, predicted for 6.5 nm diameter particles.

(12)

Thus the reduced initial susceptibility X i ( = d i / d H ) in the absence of interactions obeys the Curie law Xi = C / T with a Curie constant C =

t,/2a:. 4. R e s u l t s

The computer simulation outlined previously has been used to calculate the variation of reduced initial susceptibility ~ i with temperature for monodisperse systems having particle diameters in the range 5 to 7.5 nm. This was accomplished by first calculating many values of reduced magnetisation over the initial portion of the magnetisation curve. From the initial slope of the magnetisation curve Xi could be accurately calculated over a range of temperatures. For each particle diameter it was found that the variation of X i with temperature obeyed a Curie-Weiss law at temperatures T >> T0. This is to be expected since the prediction via eq. (1) of an infinite susceptibility at T = TO is clearly unrealistic. It was found that the ordering temperatures were generally larger than values calculated from experimental measurements. This may well result from the fact that the value for the volumetric packing fraction c used in the computations was significantly larger than values used experimentally. This point will be discussed more fully later. Fig. 1 shows a typical plot of 1//Xi vs. T. It can be seen that for sufficiently large values of T the

variation is linear, extrapolating to an ordering temperature To. In order to obtain an accurate value of TO the calculations were extended to large values of T, most of which would be out of range of experimental measurement. In all cases the best straight line fit to the variation of 1 / ~ i with T was parallel to that for the equivalent relation in the non-interacting case, indicating that the sole effect of interactions is to introduce a finite intercept at an ordering temperature To. The value of the Curie constant C is relatively unaffected as shown in table 1, which gives values of C for both interacting and non-interacting cases. The agreement is good for all particle diameters. The variation of ordering temperature with diameter is shown in fig. 2 for cobalt particles in the range 5 to 7.5 nm. This covers the range of particle sizes associated with stable cobalt ferrofluids. Over this range the variation of TOwith diameter is seen to be approximately 'linear. However, a least squares fit to the data does not extrapolate to zero

Table 1 Particle diameter (nm)

C-value interactions

no interactions

5.0 5.5 6.0 6.5 7.5

3.20 2.31 1.79 1.37 0.86

3.01 2.26 1.74 1.37 0.89

170

S. Menear et al. / A model of the properties of colloidal dispersions

A version of the spatial distribution function was used to indicate the degree of order within a system,

140 o

S D F ( X , Y) = K N ( X , Y ) ,

(13)

5. Spatial ordering

where N ( X , Y ) is defined as the number of particles/unit area at the point (X, Y), and K is a normalising factor. X and Y are the Cartesian coordinates with respect to each particle. Fig. 3 shows a three dimensional plot of this function for 5 n m particles in zero applied field, at a temperature of 300 K. This figure was obtained using a polynomial surface fitting routine, to reduce rand o m effects, and a three dimensional plotting routine. The toroid surrounding the central particle portion is indicative of dimer formation. Such dimers can be seen in the associated configuration (fig. 4). It should be noted that the apparent structure in this peak is simply a consequence of the polynomial fitting routine, and is not an inherent feature of the Monte Carlo analysis. Fig. 5 shows a typical equilibrium configuration for 7.5 nm particles. The increased local order can clearly be seen. This increased order results in a second peak in the spatial distribution function, which is shown in fig. 6. Here also, some irregularities have been introduced by the fitting procedure. We have also investigated the configurations when a large field is applied to the system. Fig. 7

As stated previously the Monte Carlo method is used to calculate thermodynamic averages of quantities such as the reduced magnetisation. However, after many Monte Carlo moves per particle the system is in a configuration which represents a typical instantaneous picture of a magnetic fluid in thermal equilibrium. The configurations generated by the Monte Carlo method were used to determine the degree of ordering at each particle diameter so that the relation between TO and the local ordering within the system could be investigated. The degree of spatial ordering within the system was investigated by two methods, (a) using the spatial distribution function (SDF) of Chantrell et al. [16] and (b) by evaluation of the mean nearest neighbour distances.

Fig. 3. Spatial distribution function for 5 nm particles in zero applied field at 300 K.

ioo

60 4,0

5,o

6,0

7,o

DIAMETER (NM)

Fig. 2. Ordering temperature TO as a function of particle diameter.

as might be expected, since TO arises from nearest neighbour interactions which tend to zero as the particle diameter tends to zero. This suggests that over the whole range of diameters the relationship between TO and diameter may be rather more complex than appears at first sight. However, this is left to a later investigation since the purpose of this paper is to describe the behaviour at larger diameters and to relate the variation of TO to the increase in local order as the particle size increases. The latter is discussed in the next section.

S. Menear et al. / A model of the properties of colloidal dispersions

171

Fig. 6. Spatial distribution function for 7.5 n m particles in zero applied field at 300 K.

Fig. 4. A typical equilibrium configuration for 5 n m particles in zero applied field at 300 K.

shows the configuration for particles of 7.5 nm diameter in a field of 0.5 T. The anisotropy of the system can be seen by visual inspection, and is clearly evident in the spatial distribution function, which is shown in fig. 8. This effect could have an

important bearing on measurements made after solidifying a magnetic fluid (by freezing or polymerisation) in a large magnetic field, since if there is appreciable induced anisotropy the interactions will have a very different effect in samples solidified in zero field compared to those solidified in large fields. We have also found some evidence in

......

_

~

~

Ix

~

+

. ~ , t ' - - ',+'' ' G ' ~

++'~+ +~o.o+++..~+~<++o+-o,p~_ + ,,~_+

~+~o++++:o'~N+~+++;-++++<~

P+ +

,+ m

++ +"+ ~'++'+_%++'+m) " - '+~'+~# ++,.,m.e~

E°+~+<+~+,;~ +

o++;,+~

o++m++++ ++

p~ <~+"++++ ++% o

o'm~_ ,m,+..+++ ,+_~%~_< (P"

o

<~~ m p~..-+_o , + +L£~'~mr "

+++o+++.+

Fig. 5.7.5 n m particles in zero field at 300 K.

i

:_ff~~~.~~o

Fig. 7. 7.5 n m particles in an applied magnetic field of 0.5 T 300 K.

S. Menear et a L / A model of the properties of colloidal dispersions

172

Fig. 8. Spatial distribution function for 7,5 n m particles in an applied magnetic field of 0.5 T at 300 K.

our data of field-induced aggregation, in which the degree of ordering is increased significantly by the application of an applied field. This effect has an important bearing on the stability of magnetic fluids and will be discussed in more detail later. The second method used to detect the degree of local ordering was to find values for the mean reduced particle separations over a large number of configurations, M, 1 M (r~,=~)'.

((R,j),/D}k,

(14)

k~l

( R i j ) , represents the mean value for the separation distance of the nth nearest neighbour of a particle, over the whole ensemble, n has been given values from 1 to 5, i.e. the mean reduced distances of the first to fifth nearest neighbours have been found. For a truly random system of non-interacting spheres, with c = 0.05, the values of (r), are listed in table 2 (by comparison, for particles arranged on a simple cubic lattice, (r), = ('~/6c) 1/3= 2.12

for n = 1 to 8). Note that in the absence of interactions the mean reduced particle separation is independent of particle diameter. Thus, any observed variation of ( r ) , with diameter for an interacting system therefore would provide a means of studying the local order within the system. Fig. 9 shows the variation of { r ) , with particle diameter in zero applied magnetic field and at a temperature of 500 K. It can be seen that { r ) l is monotonically decreasing, indicating an increase in local spatial ordering of the particles. {r)5 increases marginally over the region 5 to 7.5 nm implying that although dimers and some trimers are present, chains of five or more particles are not. It is interesting to note that values for { r ) l at smaller diameters are larger than the value of ( r ) l calculated for the non-interacting case. This is interpreted as a small degree of lattice ordering arising from the repulsive interaction associated with the surfactant coating. Of the two methods used here to investigate the local order predicted by the monte Carlo method, the SDF (the pair correlation function) is more likely to be measurable experimentally, for example using scattering techniques. The mean reduced particle separation is, however, useful in interpreting the computer results. The parameter ~ = 1/ ~r ) has been chosen as representative of the degree of order within the system. Although ~ is not a particularly sensitive parameter it can be calculated with great accuracy from configurations generated by the model. To illustrate the relation between TO and local 2,6 . ' ° o ° ° ° ° ° ° ° ° ° ° ° a a ~ ° ° ° m ° a ~ ° ° ° ° ° ° o°o°°°°°°°°a°QOaoQoooooo~ooo

006 2.2

1.8

n

°


1

1.691

2 3 4 5

1.961 2.235 2.536 2.828

o

~0~ 00 °

1.4

o

°°a o

.r°°a°Q°°°~°~°°°°°oooooo 'O~Q*OOOOOOOOQOOOOO0

Table 2

°60 °

0o OQ Q~OO0 Q ~O~ ~O0

O 0

1,0 6

7

8

9

10

11

DIAMETER (NM)

Fig. 9. Variation of (r)n with particle diameter for n = 1, 2, 3, 4, 5, showing a steady increase in local spatial ordering.

173

S. Menear et aL / A model of the properties of colloidal dispersions T = I/I s

0,63

0,8

0,62

8 NM

O O

7 NM

Q

0,61

O

0,6

6N

M

~ ~NOINTERACTION

Q

0.60

~'//

O

/

O

0.59 0.58 5.0

Q O

0,4

O

5,5

6,0

6,5 DA IMETER(NM)

1,0

7,5 0,2

Fig. 10. Increase in ~ with particle diameter.

0

ordering, fig. 10 shows the variation of ~ with diameter for the range of particle sizes covered in this investigation. Clearly there is a correlation between To and the degree of local order within the system. Presumably this correlation will depend on the volumetric packing fraction and the surfactant thickness, and is worthy of further study since it seems possible that measurement of TO might be used to predict the onset of instability in a magnetic fluid.

6. Magnetisation curves

0,2

1,0

1,4 ~( =

1,8

~/kT

Fig. 11. The effect of interparticle interactions on the magnetisation curve. of particle sizes and uses the low and high field magnetic data to calculate the median diameter, Dv, and standard deviation of log(D), o, of the distribution. The method described in ref. [24] has been used to derive similar relations, but using the two dimensional equivalent of the Langevin function. The resulting equations are,

6kT

D v = { -~-~- ~ The full magnetisation curves for particle sizes of 5, 6, 7 and 8 nm, at a temperature of 300 K have been calculated and are shown in fig. 11. F r o m eq. (11) it can be seen that in the absence of interactions the reduced magnetisation depends only on the parameter a = # H / k T . Fig. 11 shows however that the magnetisation curves plotted as a function of a do not superimpose when interactions are taken into account. Thus the presence of dimers and trimers enhances the magnetisation to an extent which increases with increasing particle size. The room temperature magnetisation curve of a ferrofluid is important in that it provides information about the particle size within the fluid. To date, however, all methods of extracting this information neglect the effects of interactions. One such method [24] assumes a lognormal distribution

0,6

~1/3 )

o = ½(ln(4H0~ i ) } 1/2,

,

(15) (16)

Xi is the reduced initial susceptibility and 1 / H o is

the value of the field at which a plot of I against 1 / H (for large H ) extrapolates to I = 0. Eqs. (15) and (16) are applicable in the absence of interactions. It has been shown earlier that the value of Xi is altered by interactions. However, the effect of this on the high field variation of I with 1 / H appears to be negligible to a first approximation. To assess the magnitude of the errors introduczd by interactions into the particle size determination values for Xi and 1 / H o calculated using the Monte Carlo method have been substituted into eqs. (15) and (16). As a typical example, the values obtained for 7.5 nm particles were

174

S. Menear et a L / A model of the properties of colloidal dispersions

D v = 8.0 nm and a = 0.2. The error in the diameter due to the presence of interactions is around 7%. However, this is not significant since the magnetisation is more sensitive to the diameter than to the effect of interactions. Another factor to be taken into account is the large value of ( used in the calculations. It is reasonable to expect that the interaction strength will scale with c so that for smaller packing fractions used in experimental fluids the effect of interactions on the calculation of the median particle diameter should be less pronounced. The deviation in the value of o from zero is far more significant since it is comparable with actual values obtained experimentally, which are typically in the range 0.2 to 0.4 for cobalt particles. There are, however, two factors which suggest that in practice the error in a may not be as large as that predicted here. Firstly, as mentioned previously, most magnetic fluids containing cobalt particles have smaller packing fractions than the value ( = 0.05 used here, and as a consequence their magnetic properties should be less affected by interactions than were the magnetisation curves predicted by the model. Secondly, it is straightforward to show by partial differentiation of eq. (16) that the error in a corresponding to small changes in Xi and 1//n0(A~i and A(1/H0), respectively) is proportional to 1 / o . This arises from the logarithmic dependence of o on Xi and 1 / H o. For a real magnetic fluid with a finite value of o, it is to be expected that Aa will be reduced from the value obtained here as long as m~i and A(1/H0) are not greatly affected by the introduction of a particle size distribution. A complicating factor here is the possibility that the introduction of a few larger particles into the system may cause a disproportionately large increase in the effect of the interactions resulting in an increase of AXi and A(1/H0) with o. A model which includes a particle size distribution is being developed in order to investigate this situation.

occurs when the moments are oriented along the line joining the centres of the particles. However, Scholten and Tjaden [25] have shown that because of thermal disorientation of the moments the actual energy is significantly smaller than Um. Application of a magnetic field aligns the magnetic moments thereby increasing the attractive interaction energy between particles. Thus, it is possible that an applied field will induce the formation of aggregates of particles in a magnetic fluid. Although large scale field-induced agglomeration has been observed by Peterson [26], Krueger [27] and Hayes [6,7], it is not evident in our Monte Carlo simulation, presumably because of the smaller particle sizes involved. We have, however, found evidence of a limited increase in average interaction energy with applied field. The effect is more pronounced for the larger particle sizes, and as an example data for 7.5 nm particles is shown in fig. 12. Here the parameter ~ = 1 / ( r ) l is plotted as a function of the logarithm of applied field. The overall change in ~ is quite small, hence the scatter in the data appears relatively large. There is, nonetheless, an increase in ~ with In(H), indicating that the average interaction energy of a pair of particles increases with field. This may be an important effect as regards the colloidal stability of magnetic fluids. In conjunction with the field-induced spatial anisotropy mentioned previously it may have a bearing also on the observed birefringence and field dependent viscosity of these materials.

O.656

o

o

O,654 o

o

O,652 o

O,650

o

o

o

0.648

7. Field induced aggregation The maximum attractive interaction energy between a pair of particles is Um = 2p2//R 3, and

0,646 4.0

5.0

6,0

7.0

8.0

Fig. 12. Increase in local spatial ordering with applied magnetic field (In(H)) for particles of 7.5 nm diameter at 300 K.

175

S. Menear et al. / A model of the properties of colloidal dispersions

8. Conclusions A model using the Monte Carlo method to investigate the effects of interparticle interactions on the properties of a colloidal dispersion of fine magnetic particles, or magnetic fluid, has been developed. The model has been used to investigate the variation of the initial susceptibility with temperature for such a fluid. This was found to obey the Curie-Weiss law, in agreement with previous experimental work. The values of ordering temperature predicted by the model were rather larger than experimental values. The probable origin of this discrepancy lies in the large value of the volumetric packing fraction (c) used in the computations. It is reasonable to expect that the ordering temperature will scale with E, so that computations carried out at lower experimental values of c may well give better agreement with experiment. An investigation of the variation of TO with c is currently under way. It was found that the value of TOincreased with increasing particle diameter. This increase was associated with an increase in limited local ordering in the form of dimers and trimers of particles. Thus, it is possible that measurement of TO could be used to predict the tendency of a magnetic fluid to agglomeration and consequent long-term instability. Further work is necessary to clarify this point. The effect of interactions on the values of the particle size distribution parameters determined from the magnetisation curve has been investigated. The value of the median diameter is found to be relatively unaffected by the interactions. However, a more important error was found in the value of the standard deviation. Since the value obtained for a is sensitive to small changes in the magnetisation curve, errors caused by interactions are likely to be significant in general. The determination of particle sizes from the magnetisation curves of magnetic fluids is an important and commonly used technique. The initial results presented here suggest that interactions could cause a large uncertainty, at least in the value of the standard deviation. For this reason a further investigation is necessary to further clarify the situation.

Analysis of the spatial configurations generated by the model has also produced evidence to support the theory that the average attractive interaction strength between particles is increased by the application of a magnetic field. This effect is thought to contribute to the field-induced aggregation which occurs in magnetic fluids. Further investigation of this phenomenon is desirable since it could cause long-term stability problems in magnetic fluid applications. The calculations have been carried out in the weak interaction regime, i.e. with small particle sizes at temperatures well above the ordering temperature T0. The interesting possibility of a type of phase transition at To cannot be properly investigated with a two dimensional model, although computations have been carried out which show the existence of long range order below TO. Equivalently, the onset of long range order appears at a critical diameter Dcr as the particle size is increased. For small particle diameters the initial susceptibility is enhanced by interactions, leading to the Curie-Weiss law. As the diameter increases the susceptibility enhancement becomes larger. At the diameter Dcr however, the initial susceptibility begins to decrease rapidly and over a very narrow range becomes smaller than the value for non-interacting particles. This change is associated with a transition from limited local order in the form of dimers and trimers of particles to the long range order in the strong interaction regime observed by Chantrell et al. [16,17].

Appendix Consider a system which contains magnetic moments restricted to two dimensions. The energy of a moment oriented at an angle 0 to the applied magnetic field H is - # H cos 0. The orientations of the moments will be Maxwell-Boltzmann distributed and the reduced magnetisation of the system is simply the weighted average of cos 0 thus

fo

2~COS 0 exp(/~H cos O / k T ) d O

i =

,

fo

2~rexp(btH cos O / k T ) d O

(A.1)

S. Menear et al. / A model of the properties of colloidal dispersions

176

where the d e n o m i n a t o r is the p a r t i t i o n function of the system. The integrals with respect to 0 in eqs. (A.1) a n d (A.2) are of a s t a n d a r d form for repres e n t a t i o n o f Bessel functions of the first kind:

jn(z )

(-i)" eft

( ~ e x p ( i z cos 0 ) c o s " 0 d 0 . a0

(A.2)

H e n c e eq. (A.1) can be written

i= iJl(-i#H/kT)/Jo(-i#H/kT

).

N o w J n ( i z ) = i " I . ( z ) where I , ( z ) is the m o d i f i e d Bessel function of the first kind. Hence 11(-taH/kT)

]= - io(_l~H/kT

1,(a) ) = io( a) ,

(A.3)

where I i ( a ) a n d Io(a ) are the m o d i f i e d Bessel functions of the first k i n d of o r d e r 1 a n d 0, respectively. T h e a s y m p t o t i c solutions for eq. (A.3) in small a n d large fields are i m p o r t a n t . F o r small fields such that a < < 1, I , ( a ) = a " / 2 " n ! W i t h this app r o x i m a t i o n eq. (A.3) b e c o m e s

i = a/2 = ttH/kT

(A.4)

a n d the r e d u c e d initial susceptibility of the system is

f(i = d I / d H

= 1~/2kT

(A.5)

which should be c o m p a r e d with the three d i m e n sional value of Xi = # / 3 k T . F o r a large the v a r i a t i o n of the i n t e g r a n d s in eqs. (A.1) a n d (A.2) are very r a p i d as O increases f r o m zero. Thus we can o b t a i n an a p p r o x i m a t e value for I b y e x p a n d i n g cos O as a p o w e r series a n d retaining the term in 0 2. T h e u p p e r limit of the integral is set to infinity since this has very little b e a r i n g on the value of the integral if a is large. This gives L~(1 - 02/2) exp(a(1 - 02/2))d0

i f o ° ° e x p ( a (1 -- 0 2 / 2 ) ) d 0 M a k i n g the s u b s t i t u t i o n y = Ov/a/2 gives

= 1 - S(1)/aS(O), where S ( n ) is the s t a n d a r d integral

(A.6)

S(n) = fy2, exp(-y2)dy

=

1.3.5... ( 2 . - a) ¢g. 2 ~+1

H e n c e S ( 1 ) = v~-~/4 a n d S ( 0 ) - - ~ - ~ / 2 . tion of these values into eq. (A.6) gives

=- 1 - 1 / 2 a

Substitu-

(A.7)

as the high field a p p r o x i m a t i o n . This should be c o m p a r e d with the expression i - - - 1 - 1 / a obt a i n e d f r o m the t h r e e - d i m e n s i o n a l Langevin function.

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