Table 1 Equations used in determining the heating power

1

Néelian relaxation time, \(\tau_{N}\) (s)

\(\tau_{N} = \frac{\sqrt \pi }{2}\tau_{0} \frac{\exp \varGamma }{{\varGamma^{1/2} }}\)

\(\tau_{0}\) time constant (s)

\(K\) anisotropy constant (J m⁻³)

\(V_{M}\) magnetic volume (m³)

\(k_{B}\) Boltzmann constant

\(T\) absolute temperature (K)

2

Gamma, \(\varGamma\)

\(\varGamma = \frac{{KV_{M} }}{{k_{B} T}}\)

3

Brownian relaxation time, \(\tau_{B}\) (s)

\(\tau_{B} = \frac{{3\eta V_{H} }}{{k_{B} T}}\)

\(\eta\) dynamic viscosity of the fluid (Pa s)

\(V_{H}\) hydrodynamic volume (m³)

4

Effective relaxation time, \(\tau\) (s)

\(\frac{1}{\tau } = \frac{1}{{\tau_{N} }} + \frac{1}{{\tau_{B} }}\)

5

Volumetric power dissipation, \(P\) (W m⁻³)

\(P = \pi \mu_{0} \chi_{0} H_{0}^{2} f\frac{2\pi f\tau }{{1 + (2\pi f\tau )^{2} }}\)

\(\mu_{0}\) free space permeability

\(H_{0}\) magnetic field strength (A m⁻¹)

\(f\) magnetic field frequency (s⁻¹)

\(\phi\) nanoparticles volume fraction

\(M_{d}\) domain magnetization (A m⁻¹)

6

Equilibrium susceptibility, \(\chi_{0}\)

\(\chi_{0} = \chi_{i} \frac{3}{\xi }(\coth \xi - \frac{1}{\xi })\)

7

Initial susceptibility, \(\chi_{i}\)

\(\chi_{i} = \frac{{\mu_{0} \phi M_{d}^{2} V_{M} }}{{3k_{B} T}}\)

8

Langevin parameter, \(\xi\)

\(\xi = \frac{{\mu_{0} M_{d} H_{0} V_{M} }}{{k_{B} T}}\)

9

Specific loss power (W kg⁻¹)

\(SLP = \frac{P}{\rho \phi }\)

\(\rho\) nanoparticle density (kg m⁻³)

10

Volumetric power dissipation of a polydispersion, \(\bar{P}\) (W m⁻³)

\(\bar{P} = \mathop \smallint \limits_{0}^{\infty } Pg\left( D \right)dD\)

\(D\) particle diameter (m)

\(\ln D_{0}\) median of \(\ln D\)

\(\sigma\) standard deviation of \(\ln D\)

11

Particle size distribution function \(g\left( D \right)\)

\(g\left( D \right) = \frac{1}{{\sqrt {2\pi } \sigma D}}\exp \left[ {\frac{{ - \left( {\ln \left( {D/D_{0} } \right)} \right)^{2} }}{{2\sigma^{2} }}} \right]\)