The coaxial probe technique is used to measure the broadband dielectric properties of materials. The complex permittivity of the samples can be extracted using a capacitive model which is shown in Figure 1. The complex permittivity can be calculated from measured reflected coefficient S₁₁ from the reflected signal.

s 11 * = s 11 e j θ = 1 − Y s 1 + Y s e j θ = 1 + j ω z 0 ( ε r c 0 + c f ) 1 − j ω z 0 ( ε r c 0 + c f ) (1)

where Z₀ is the characteristic impedance of the coaxial line (here 50 Ω), C₀ is the capacitance of the probe terminated in free space, C_f is the fringing capacitance, ω is the angular frequency, θ is the phase associated with the electrical length of the coaxial probe and ε is the complex permittivity of the sample. The complex relative permittivity can be derived using.

ε r = 1 − s 11 * j ω z 0 c 0 ( 1 + s 11 * ) c f c 0 (2)

The two unknown capacitances can be calculated by measuring material with known dielectric permittivity such as water [11] [12] [13] [14].

Cavity perturbations is a means of measuring material properties by analysis of the changes in resonant frequency and Q factor when a dielectric material sample is inserted into the cavity. For the TM₀₁₀ mode, the perturbation calculations of the real and imaginary parts of the complex permittivity can be simply derived using two approximate formulas below:

ε r ' ' ≈ 2 J 1 2 ( p 01 ) f 0 − f s f 0 R c 2 R s 2 (3)

ε r ' ' ≈ J 1 2 ( p 01 ) ( 1 Q s − 1 Q 0 ) R c 2 R s 2 (4)

where f is the resonant frequency, Q is the quality factor and the subscripts s and 0 refer to the sample loaded and empty cavities, respectively; R_c is the inner radius of the cavity, R_s is the radius of the cylindrical sample tube (assuming that it occupied the full inner length of the cavity, on-axis), J₁ is the first order Bessel function of the first kind and p₀₁ = 2.405, so that J₁(p₀₁) = 0.52 [15] [16] [17].

Change in the ambient temperature of the cavity will cause change in the resonant frequency, due to change in thermal expansion of the cavity which cause change in the cavity dimensions, radius (a) and length (l). The thermal expansion coefficient α of the aluminum cavity can be used to describe these changes. Therefore, the assumption of that the volume of the cavity is fixed is not valid in the case of changing the temperature. The perturbation Equations (‎3) and (‎4) are valid in the case of the sample size is small compared to the cavity size. Therefore, any change in the sample size due to temperature variation must be included in the calculations.

In this work, the nodal modes were used for temperature correction measurements. Nodal cavity modes are those which have no electric fields in the center of frequency and are therefore unaffected by the presence of the sample. Figure 2 shows the electric and magnetic distributions of some nodal modes for the cavity with 40 mm radius and 46 mm height.

The simplest way to obtain the temperature correction is to subtract the fractional frequency shift of nodal modes such as TM₃₁₀ (of frequency 6.61 GHz) from the fractional shift of the measurement mode TM₀₁₀ [18] [19] [20].Through cavity perturbation, the change in permittivity with respect to a reference temperature can then be expressed as following:

Δ f s ( T ) f s ( 0 ) = [ Δ f 010 , s ( T ) f 010 , s ( 0 ) − Δ f m 10 , s ( T ) f m 10 , s ( 0 ) ] − [ Δ f 010 , u ( T ) f 010 , u ( 0 ) − Δ f m 10 , u ( T ) f m 10 , u ( 0 ) ] (5)

ε 1 ( T ) ≈ − 2 [ f s ( T ) − f 0 ( 0 ) f 0 ( 0 ) ] v c v s G n m p + 1 (6)

ε 2 ( T ) = [ 1 Q s ( T ) − 1 Q 0 ( 0 ) ] v c v s G n m p (7)

The samples measured in these experiments and their description are listed in Table 1. The samples were formed from different particle sizes. They were crushed using a mortar and pestle, and then sieved by 63 µm Lab sieve to be approximately 60 µm size, as illustrated schematically in Figure 3.

The samples were filled in quartz tubes, with inner and outer diameters of 2 mm and 2.4 mm, respectively. The sample tube was centered in the cavity using cylindrical placement ring, as shown in Figure 4. The measurements were repeated at least 10 times for each sample; so that mean values (and standard deviation of the mean) can be estimated reliably for each sample.

The measurements was carried out using coaxial flat probe with a square flange which was connected to an Agilent PNA-L vector network analyzer, after careful calibration was done using an Agilent 85052D calibration Kit; the set-up is shown in Figure 5.

The samples were prepared as mixtures of silicone rubber purchased from MB Fiberglass, Northern Ireland, with different concentrations of powdered activated carbon (AC1) to avoid the uncertainty resulting from the contacts between the powder particles, . Fifteen samples with three different contents of activated carbon were prepared at ratios of mass at 2:1, 4:1 and 8:1. The preparation of samples started by mixing the two parts of silicone and hardener, then the powder was added to the mixture, which was left from 6 - 8 hours at 25˚C to cure.

Pure silicone rubber samples were also measured to investigate the errors resulting from the air bubbles, which were observe to appear as part of the hardening process.

The temperature dependent properties of activated carbon were measured using

the microwave cavity technique at a frequency of 2.5 GH in the temperature range 30˚C to 150˚C.

The cavity loaded sample was placed inside a Memmert UF30 laboratory oven, as shown in Figure 6. The change in the resonant frequency and bandwidth were recorded using a custom LabVIEW program. Temperature was measured using resistance temperature detector PT100 RTDs provided by Omega, interfaced to a National Instruments C-DAQ. At each temperature, the

magnitude of the transmission parameter S₂₁ was measured with the PNA-L network analyzer N5232A to obtain the resonant frequency and quality factor Q of the resonator

The real and imaginary parts of the complex permittivity are shown in Figure 7 and Figure 8, which illustrate the average values of the measurements of complex permittivity for all samples. Figure 7 and Figure 8 show that the dielectric constant and loss factor of the powdered samples for all activated carbon samples are significantly different. The small errors are associated with the small differences in sample volumes each time a powder sample is loaded into the cavity (since they are powders, with random packing nature, and all tubes have a volume size distribution giving an error of about 2%).

The dielectric constant ε' and loss factor ε" measurements of activated carbon samples were performed on samples which were mixed with silicone rubber in three different ratios of mass at 2:1, 4:1 and 8:1. Five samples of each ratio were prepared at different times with separate mixtures, to allow for standard errors to be assessed, and each sample was measured at least 10 times. Figure 9 and Figure 10 show the dielectric constant and loss factor of silicone rubber/activated carbon blends as function of frequency. In the lower frequency region the dielectric constant fluctuated up to 100 MHz. However, in the higher frequency region the dielectric constant values remained approximately constant up to about 5 GHz. The dielectric constant and loss factor change according to the content of activated carbon in the mixture. It has been found that the

increasing filler content of activated carbon leads to increases in both the relative permittivity and dielectric loss factor. This is owing to effective values of complex permittivity being extracted, which is a function of volume fraction. We have not attempted to extract the intrinsic permittivity of the samples using effective medium theory but we would expect these to be approximately the same, irrespective of the concentration. We explain the great variation in loss factor below around 100 MHz as being due to the variations in the contacts between the particles. This gives way to a loss peak at higher frequencies, centered at around 10 GHz and indicative of the intra-particle electrical conductivity of the material.

Figures 11-14 show the change in the dielectric properties of two samples of activated carbon, labelled as AC1 and AC4, at 2.5 GHz and as a function of temperature. The dielectric constant of both samples decreases linearly with temperature in the range between room temperature and about 120˚C, after which there are increases which might due to the release of the water vapor or other agents which adsorbed on the surface of activated carbons, although this is not consistent with the exact temperature of the onset these changes; furthermore, a

loss in water should cause a further decrease of permittivity, owing to its highly polar nature, and this is not what is observed in the experimental data. The origin of the change is currently unknown and will be investigated by other techniques (e.g. downstream mass spectrometry).