Effect of Sample Concentration on the Determination of the Anisotropy Constant of Magnetic Nanoparticles

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Techniques such as equilibrium (DC) and dynamic (AC) magnetic measurements have been used to determine the anisotropy constant of a variety of magnetic nanoparticles, obtaining values that are often an order of magnitude higher than the corresponding bulk material. Unfortunately, the effect of particle–particle interactions is often neglected, hence the reported values are an effective collective parameter , rather than an intrinsic property. In this study we used DC and AC magnetic measurements to determine the anisotropy constant of magnetic nanoparticles fixed in a cross-linked polymer matrix. We used nanoparticle concentrations of 0.1%, 1%, 6%, and 33%(w/w) in order to determine the effect of concentration, and hence interactions, on the value determined for the anisotropy constant. The effect of interaction on determining the magnetocrystalline anisotropy constant was accounted for by using the Néel and Vogel-Fulcher relaxation models, the latter with an effective interaction temperature determined from independent measurements. A decrease in the anisotropy constant determined from these measurements was observed with decreasing particle concentration from <formula formulatype="inline"><tex Notation="TeX">${\sim} 6100$</tex></formula> kerg/cm<formula formulatype="inline"><tex Notation="TeX">$^{3}$</tex></formula> to <formula formulatype="inline"><tex Notation="TeX">${\sim} 500$</tex></formula> kerg/cm<formula formulatype="inline"><tex Notation="TeX">$^{3}$</tex></formula> using the Vogel-Fulcher model and from <formula formulatype="inline"><tex Notation="TeX">${\sim} 9100$</tex></formula> kerg/cm<formula formulatype="inline"><tex Notation="TeX">$^{3}$</tex> </formula> to <formula formulatype="inline"><tex Notation="TeX">${\sim} 1100$</tex> </formula> kerg/cm<formula formulatype="inline"><tex Notation="TeX">$^{3}$</tex> </formula> using the Néel model. The values obtained for the most dilute samples using the Vogel-Fulcher model are of comparable magnitude to bulk values for magnetite. More importantly, the value of <formula formulatype="inline"> <tex Notation="TeX">$\tau_{0}$</tex></formula> obtained from AC susceptibility decreased from <formula formulatype="inline"><tex Notation="TeX">$10^{- 32}$</tex> </formula> to <formula formulatype="inline"><tex Notation="TeX">$10^{- 9}$</tex> </formula> s with decreasing particle concentration. As <formula formulatype="inline"> <tex Notation="TeX">$\tau_{0}$</tex></formula> is expected to be of the order of <formula formulatype="inline"><tex Notation="TeX">$10^{- 9}$</tex></formula> s, these results indicate that the value of magnetic anisotropy determined for the dilute sample represents an intrinsic rather than effective property. These measurements and analysis illustrate the importance of particle concentration/interactions in determining the intrinsic magnetic properties of nanoparticles, particularly the anisotropy constant . </para>