Complex susceptibility measurements of a suspension of magnetic beads

Abstract

Measurements of the frequency and field dependence of the complex magnetic susceptibility, ${\chi }_{\mathrm{s}}(\omega ,H)={\chi }_{\mathrm{s}}^{\prime }(\omega ,H)-\mathrm{i}{\chi }_s^{″}(\omega ,H)$, of a suspension of magnetic beads in water over the frequency range 200 Hz to 1 MHz are presented. The magnetic polarizing field, H, is applied to the sample, first in a forward direction and then in a reverse direction and from a plot of the static susceptibility, ${\chi }_{0\mathrm{S}}$, against polarizing field H, the existence of a hysteresis effect is demonstrated.

Introduction

The use of magnetic colloids with particles (beads) coated with antibodies that specifically bind to specific proteins if of current interest in the area of medical research [1].

The magnetic beads investigated here are spherical and have a diameter of 200 nm [2] and contain 50% volume fraction of maghemtite particles of 10 nm radius and 50% volume fraction of olaic acid. The maghemite particles are extremely confined inside the spheres (almost close-packed) and are considered not to experience Brownian motion thereby resulting in Néel relaxation [3] being the dominant relaxation mechanism of the beads, as reported in Ref. [4]. Here we use measurements of the frequency-dependent complex susceptibility, $\chi (\omega )={\chi }^{\prime }(\omega )-\mathrm{i}{\chi }^{″}(\omega )$, which according to Debye's theory [5], $\chi (\omega )$, has a frequency dependence as given by the equation,

$$\chi (\omega )={\chi }_0(1/(1+{\omega }^2{\tau }^2)-\mathrm{i}\omega \tau /(1+{\omega }^2{\tau }^2))\text{,}$$

where the static susceptibility,

$${\chi }_0=n{m}^2/3\mathit{kT}{\mu }_o$$

and

$$\tau =1/{\omega }_{\max }=1/2\pi f_{\max }\text{.}$$

$f_{\max }$ is the frequency at which ${\chi }^{″}(\omega )$ is a maximum, n is the particle number density and k is Boltzmann's constant.

The relationship between ${\chi }^{\prime }(\omega )$ and ${\chi }^{″}(\omega )$ and their dependence on frequency, $\omega \text{/}2\pi$, can be displayed by means of the magnetic analogue of the Cole–Cole plot [6]. In the Cole–Cole case the circular arc cuts the ${\chi }^{\prime }(\omega )$ axis at an angle of $\alpha \pi \text{/}2$; $\alpha$ is referred to as the Cole–Cole parameter and is a measure of the distribution of relaxation times.