Specific Power Absorption & Intrinsic Loss Power Values

01 Dec Specific Power Absorption & Intrinsic Loss Power Values

One of the first steps when characterizing any sample of magnetic nanoparticles as a heating agent is to determine its capacity of absorbing power from the applied magnetic field. The parameter that measures this property is the Specific Power Absorption (SPA), also known as Specific Absorption Rate (SAR). In this text, we give a basic explanation on how to interpret this parameter and other related concepts used in calorimetry.

Magnetic heating (MnH) refers to the process of increasing the temperature of a magnetic material by the application of an alternate magnetic field (AMF), and nowadays many areas of nanotechnology as biomedicine, catalysis… There are a broad range of magnetic materials, mainly in magnetic colloids which refer to a magnetic fluid made of ferromagnetic nanoparticles suspended in a non-magnetic carrier liquid. This MnH is produced by the transformation of the energy absorbed by the ferromagnetic nanoparticles into heat, when submitted to an AMF of certain amplitude and frequencies.

One way of characterizing the heating capability of magnetic colloids is by means of the Specific Power Absorption (SPA) or the Intrinsic Loss Power (ILP) values.

Calculation of Specific Power Absorption & Intrinsic Loss Power Values on Experiments of Magnetic Heating of Nanoparticles

Specific Power Absorption (SPA)

The capacity of a magnetic material to absorb energy from an alternating magnetic field is quantified by the SAR or SPA rate (Specific Absorption Rate // Specific Power Absorption). SPA is defined as the amount of energy/power absorbed by the sample per mass unit (W/kg).

Equation for Specific Power Absorption

(Eq. 1)

where P is the absorbed power (in W) and mnp is the mass of nanoparticles in the sample (in kg).

The power absorbed by the magnetic colloid during a MnH experiment can also be defined as the amount of energy converted into heat per unit of time and mass

Equation of Specific Power Absorption depending on heat

(Eq. 2)

where Q is the heat (in Joules) generated by the MNPs within a time Δt (in seconds) by a mass mnp (in kg) of magnetic nanoparticles. This is an intensive property of the material.

For an adiabatic system, a calorimetric approach relates the heat dissipated (Q) by the nanoparticles in a colloid and the observed increase in temperature (ΔT) by:

(Eq. 3)

where mnp and ml are the mass of nanoparticles and the liquid carrier respectively, and cnp and cl are their specific heat capacities (in J/(K·kg)) which are intrinsic characteristics for each material.

Substituting the equation for Q in equation 2, the SPA is expressed as:

(Eq. 4)

Temperature increase with time is calculated by the temperature data obtained during the test as observed in Image 1. The value of increase in temperature with respect to time, which is taken to calculate the SPA, is the maximum value of the gradient of the temperature curve, which, in a test of MnH rate normally corresponds to its initial gradient (see Image 1).

Temperature curve of a magnetic colloid submitted to magnetic heating done with Zar software

Image 1. Zar Analysis software: main window showing a temperature curve of a magnetic colloid submitted to a magnetic nanoHeating test.

 

Thus, finally, the SPA index for a magnetic colloid submitted to a MnH test is calculated by the following expression:

(Eq. 5)

where the (δT/δt) is the maximum heating rate of the colloid submitted to an MnH test [K/s], as could be observed in Image 2.

This equation can be further simplified assuming mnp·cnp << ml·cl and defining de sample concentration as φ=mnp/Vl where Vl is the volume (in L) of colloid having a mass of nanoparticles mnp (in kg).

The SPA can be expressed as:

(Eq. 6)

SPA calculation according to lineal fitting with Zar softwareImage 2. Zar Analysis software: Lineal fitting to calculate SPA and ILP

Dependence  of  temperature  with  time  should  include  heat  exchange  with  the  medium, yielding  an  asymptotic  thermal  equilibrium  for  long  times  due  to  adiabaticity  is  not  complete as shown in Image 1. The first method to analyse a non-linear model was proposed by Box and Lucas et al. in 1959.[1] The heating rate of a  magnetic  colloid placed  under  an  AMF is  described  by the  following function:

SPA calculation box and lucas equation 7(Eq. 7)

A  and  B  are  experimental  parameters  related  with  saturation  temperature (T(t=∞)), and C is related by C=T(t=0)=T0, A+C=T(t=∞)=Teq and B=1/τ. Then,  replacing  the  relevant  parameters T(t) is expressed as:

SPA calculation box and lucas equation 8(Eq. 8)

where:

T0: Initial Temperature of the colloid [°C]
Teq: Equilibrium Temperature of the colloid [°C].
τ
: Characteristic heating time that depends on sample properties [s].  

These approximations are the modified-Box-Lucas model (MBL) which describes the temperature dependence with time, T(t):

SPA calculation box and lucas equation 9(Eq. 9)

The maximum value is obtained when t=0:

SPA calculation box and lucas equation 10

(Eq. 10)

The  product  of  these  parameters,  A  and  B,  is  equivalent  to  the  initial  heat  rate  and  could  be used  to  calculate  the  SPA  comparing  the  simple  linear  fit of  T(t) curves  and the  MBL fitting model as shown in Image 3.

SPA calculation box and lucas equation 11(Eq.11)

SPA calculation according to exponential fitting with Zar software

Image 3. Zar Analysis software: Exponential fitting to calculate SPA and ILP

Intrinsic Loss Power (ILP)

ILP was proposed[2] to compare heating efficiency at different experimental conditions of amplitude and frequency of the magnetic field. It assumes a quadratic dependence with magnetic field (H) and linear dependence with frequency (f). Therefore, by normalizing the power absorption with these dependences as

(Eq. 12)

this parameter (in nH·m2/kg) allows a direct comparison of data from different laboratories or with different devices.

Our approach…

nanoScale Biomagnetics has developed ZaR application to analyze the temperature curves once the test is finished and the heating curve has been completed. This application enables the analysis of temperature curves by both methods, linear and MBL (exponential) as shown in Images 2 and 3. The Calculation menu allows to introduce the parameters required to calculate SPA/ILP values as time range, colloid concentration, solvent, and test conditions (see Image 4) to carry out these analyses.

Zar software window for SPA and ILP calculationImage 4. ZaR analysis software: SPA and ILP calculation menu

Once the calculation is properly done, the SPA and ILP values will appear in the main window as observed in Image 5.

SPA and ILP results in Zar softwareImage 5. Zar analysis software: SPA and ILP results

References

    1. Box, G. E., & Lucas, H. L. (1959). Design of experiments in non-linear situations. Biometrika, 46(1/2), 77-90
    2. Kallumadil, M., Tada, M., Nakagawa, T., Abe, M., Southern, P., & Pankhurst, Q. A. (2009). Suitability of commercial colloids for magnetic hyperthermia. Journal of Magnetism and Magnetic Materials, 321(10), 1509-1513.

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