Modelling of interaction effects in fine particle systems

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Journal of Magnetism and Magnetic Materials 157/158 (1996) 250-255

Invited paper

ELSEVIER

journal of ".T-" magnetism and magnetic materials

Modelling of interaction effects in fine particle systems R.W. Chantrell a,*, G.N. Coverdale a, M. E1 Hilt b, K. O'Grady c Physics Department, Keele University, Keele, Staffs ST5 5BG, UK b Jordan University for Women, Amman, Jordan c SEECS, UCNW, Dean Street, Bangor, Gwynedd, UK

Abstract A model of magnetisation processes in interacting fine particle systems has been developed. This is based on a Monte Carlo approach and is capable of simulations of temperature and time dependent magnetic properties. A separate model has been developed to predict the microstructure of fine particle systems such as those obtained by the solidification of magnetic fluids. These simulations generate realistic microstructures, which are used as the central cell in the Monte Carlo calculations of magnetic properties. Initial calculations of hysteresis loops and remanence curves have been carried out which demonstrate the importance of the physical microstructure on the interaction problem. Keywords: Fine particles; Interactions; Monte Carlo calculations

1. Introduction There is considerable interest in the study of fine particle systems, both from the fundamental viewpoint and also because of their potential for applications. One especially interesting aspect of the behaviour of magnetic fine particle systems is the effect of magnetostatic interparticle interaction on their behaviour. A number of authors [1-4] have theoretically studied interaction effects in fine particle systems and recording media, using approaches ranging from Monte Carlo methods to the direct solution of the Landau-Lifshitz equation of motion. However, all treatments have the disadvantage of assuming very idealised physical microstructures, normally either complete disorder or a regular lattice. The exception is Vos et al. [3] who created microstructures to empirically represent the extremes of alignment in the case of standard particulate media and 'stacking' in systems of barium ferrite platelets. It might be expected that the physical microstructure of a fine particle system would have a strong bearing on the interaction problem. Consequently we have developed a model of an interacting fine particle system in parallel with a computer simulation of the microstructure of such a system prepared by solidification of a fluid dispersion. Experimentally this corresponds to the solidification of a magnetic fluid, or to an approximate model of a particulate recording medium. Here we outline the overall model and give some initial results of hysteresis and remanence curves

* Corresponding author. Email: [email protected].

which demonstrate the importance of using a realistic physical microstructure in computer simulations. These represent the first such calculations and a preliminary discussion of their relation to current experimental studies is also given. 2. Theoretical model

The present work is specifically aimed at understanding the behaviour of 'model' fine particle systems such as those produced by the solidification of a magnetic fluid. For such materials the microstructure is variable by for example varying the packing density and also by the application of a magnetic field during the solidification process. The model essentially consists of 2 parts, the first used to calculate the magnetic properties of a solid fine particle system and the second to simulate the microstmcture of the system via studies of the precursor fluid dispersions. Both models are described in the following sections. 2.1. Magnetic properties o f the solid

This is essentially a model described previously [5] where it was used to study the effects of interactions on the giant magnetoresistance of a granular magnetic solid. A cubic simulation cell is assumed, with periodic boundary conditions extending the cell into 3-D. The cell contains N particles having a lognormal volume distribution which are distributed spatially in a configuration determined by the method given in the following section. The particles are assumed to have uniaxial anisotropy with easy axes dis-

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R. W. Chantrellet al. / Journal of Magnetism and MagneticMaterials 157/158 (1996) 250-255 tributed randomly. The energy of the ith particle within the cell is then E - KV/sin2c~ - / x i "/-/tot,

sizes greater than the SPM limit are treated using modified Stoner-Wohlfarth ( S - W ) [7] theory. Essentially we calculate the energy harrier to reversal using the numerical approximation of Pfeiffer [8],

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where K is the anisotropy constant and/x~ is the magnetic moment of particle i. In order to take into account the magnetostatic interactions between particles the field //tot includes an interaction field calculated as the vector sum of the dipolar fields from the remaining particles, i.e. /-/t°t=Ha+

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where htot = Htot/H K with H K the anisotropy field and, g(~O) = [cosZ/Sqr + sin2/3~0] 2/s. From consideration of the relaxation time it can be shown that a transition between states in the modified S - W model occurs when A E = 25kT. This criterion allows one to define the minimum in which the magnetic moment lies. Because//to t can have any arbitrary direction in space it is most convenient to determine the moment orientation in a coordinate system using Htot as the polar axis, followed by a coordinate transformation into the laboratory frame. Having placed the moment in the relevant energy minimum, small angle M - C moves are carried out in order to simulate thermal activation within the minimum. This process is continued until equilibrium is reached at a given field.

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where the second term represents the interaction field, and is calculated for all neighbours within a cut off radius, rij and d~j are the vector and scalar separations of particles i and j. The calculation proceeds firstly by calculation of the particle energy barriers. In the case of particles which exhibit superparamagnetic (SPM) behaviour, i.e. where the intrinsic energy barrier KV < 25kT, the magnetisation is determined by a standard Metropolis [6] algorithm based on energy changes determined from Eq. (1). Particles with

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R. W. Chantrell et al. /Journal of Magnetism and Magnetic Materials 157/158 (1996) 250-255

This forms the basis of all the calculations, which proceed by subjecting the sample to the relevant magnetic field sequence, for example a continuous field cycle for the hysteresis loops. Isothermal Remanent Magnetisation (IRM) curves were produced by applying a field H to an initially demagnetised sample after which the field was removed and the remanence measured. IRM curves were studied because it is known that remanence curves are characteristic of interaction effects. 2.2. Microstructural simulations

We are interested in materials produced from a dispersed colloidal precursor, since experimentally these are likely to give the best defined and characterised systems. We have developed a 3-D simulation of small ( ~ 10 nm) particles based on our model of a strongly interacting particulate dispersion [9,10]. This uses a Monte Carlo technique including force-bias to improve the rate of convergence. The energy of a given particle is the sum of all the dipolar fields within a cut off radius and also a hard-sphere repulsion term preventing overlap of neigh-

bouring particles. Each Monte Carlo move involves the simultaneous translation of a given particle and rotation of its magnetic moment, giving rise to an energy change A E, following which the trial move is allowed with a probability min(1, e x p ( - A E / k T ) ) . After many such moves the system evolves into thermal equilibrium, physically representing the configuration of a magnetic fluid containing interacting particles. We now assume that the magnetic fluid is carefully solidified (possibly by polymerisation of the fluid matrix) thus preserving the physical microstructure. In terms of the current approach this means that the spatial coordinates of the particles produced by the dispersion simulation essentially define the positions of the particles in the model outlined in the previous section. 3. Results We have carried out simulations of Cobalt fine particle systems with median diameter of 6 and 10 nm and standard deviation of the (lognormal) distribution of 0.15, over a concentration range of 1 to 35 vol%. The first step is to produce the equilibrium configurations in the fluid state

Fig. 2. Equilibrium configurations of a system with median diameter 6 nm close to magnetic saturation.

R. W. Chantrell et al. / Journal of Magnetism and Magnetic Materials 157/158 (1996) 250-255

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magnetisation curves, representing the magnetisation after the application of a given field to an initially demagnetised sample. It should be noted that the demagnetised states were produced by a simulated annealing approach which has been shown in studies of longitudinal thin films [11] to produce a highly magnetically correlated, low energy demagnetised state. At r o o m temperature the magnetic behaviour of the 6 nm system was essentially SPM, so more detailed calculations were made at T = 77 K. Initial magnetising curves for the 6 nm system are shown in Fig. 3. It can be seen that there is a significant reduction in the initial slope of the curve, which can be ascribed to the demagnetising effects arising from flux closure configura-

which we then associate with the microstructure of the solidified fine particle system. For the purposes of the current study the solidification is assumed to take place at room temperature. As might be expected the dipolar interactions between the particles result in significant correlations as shown in Fig. 1. The correlation effects can be more clearly seen in Fig. 2 which shows the same system close to magnetic saturation. The chains produced by the dipolar interactions are clearly evident. The physical microstructures produced by the dispersion simulations were then introduced to the computational cell for the determination of the bulk magnetic properties of the fine particle system. First we studied the initial

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R. W. Chantrell et aL /Journal of Magnetism and Magnetic Materials 157/158 (1996) 250-255

tions in the M - C simulated microstructures (which are not expected to be as pronounced in systems produced by purely random placement of particles). The effects of the physical microstructure can easily be demonstrated by comparing results for a realistic micros~ucture with those obtained assuming a random packing (but with no particle overlap). Initial magnetisation curves and IRM curves for a system with 6 nm median diameter and 1% packing density are shown in Fig. 4 for the realistic and random microstructures. All curves tend to have a lower initial slope than the non-interacting system, however the realistic microstructure gives a significantly reduced slope in relation to the random state. This can be ascribed to the effects of aggregation which are significant even at such low densities as 1%. Essentially the colloidal simulations show a phase separation, with small clusters of particles being formed under the influence of magnetostatic interactions. These naturally tend to promote flux closure leading to a reduction in both the initial magnetisation and remanence. The structures are not present in the random system in which the interaction effects are consequently much weaker, as might be expected given an increase in the average particle separation. The inference of these results is that in systems with significant inter-particle interactions is it probably not possible to remove interaction effects by dilution - an effect which has long been intutitively understood but which this study is the first to quantify. The dilution process thus results in a complex modification of the physical microstructure which appears to be important in interaction studies. Finally, we have calculated IRM curves in order to study the effects of interactions on the irreversible magnetic behaviour. Fig. 5 shows IRM curves for a l0 nm median diameter system. At 1% concentration the remanence is similar in form to that expected for a non-interacting system having these physical parameters, although

the remanence is significantly lower (the saturation remanence for this system, in reduced units should have the maximum possible value of 0.5). The effect of increasing the packing density is to lower the remanence even further and somewhat change the character of the curve, which becomes much more broad. The IRM curve is often differentiated to reveal a switching Field Distribution (SFD) which is related to the intrinsic energy barrier distribution. The results here indicate that, as might be expected, the interaction effects tend to broaden the distribution of energy barriers. Interestingly, the results at 5% packing fraction indicate a relatively steep initial slope of the IRM curve. It is currently not clear whether this is an effect of the microstructure or a relatively poor initial demagnetised state, both of which could explain the data. This effect is currently under investigation.

4. Discussion W e have given the first calculations of the magnetic properties of a fine particle system produced by solidification of a colloidal dispersion in which the microstructure of the system has itself been simulated by a Monte Carlo method. It has been shown that dilution of the system does not necessarily reduce the interactions, rather it can lead to a phase separation in which the behaviour of small clusters dominates the magnetic behaviour. Although dilution experiments remain valuable, the development of theoretical models taking account of the microstructure would appear to be important for comparison with experimental data and to quantify the effects of phase separation. Our resuks comparing systems with realistic and random microsmactures suggest an interesting study of disorder effects in dipolar systems, if such a random system could be prepared, possibly by rapid cooling. W e expect significant differences in the static and dynamic behaviour of such

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R.W. Chantrell et aL /Journal of Magnetism and Magnetic Materials 157/158 (1996) 250-255 systems, which are currently under investigation. It has been shown that interactions lead to a significant reduction in the saturation remanence, which is enhanced by the physical microstructure produced by the current model, which naturally enhances flux closure configurations. The shape of the IRM curve is also changed, leading to an apparent increase in the width of the energy barrier distribution. It is interesting, finally, to consider our simulations in relation to previously published work [12] giving the results of F M R simulations. Here it was shown that at fields close to magnetic saturation the nearest neighbours apparently tended to support the magnetised state, which is contrary to the experience of the current work, but in agreement with a previous study using mean-field theory [13] and recent experimental data [14]. The resolution of this apparent discrepancy probably lies in the fact that close to saturation flux closure is not possible and one can conceive of interactions between nearest neighbours as having a magnetising effect in this case. However, we note that the simulations in [12] were carried out on a random system which might be expected to exhibit a large effect close to saturation relative to our realistic microstructures. Currently, F M R calculations are underway to check whether a magnetising effect is apparent in our simulations at high fields. Finally, we can conclude that interaction effects in fine particle systems strongly depend on the combination of the magnetic state and the physical microstructure. Experimental studies of model fine particle systems in conjunction with models such as are described in this paper might be

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expected to make useful advances in the understand!ng of interaction effects in fine particle systems. Acknowledgement: This work was supported by the UK EPSRC and carried out within the E U CAMST projecf.

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