El anuncio televisivo sobre el Baguazo: la imagen y la palabra

5.2.1 Frequency dependence of the complex permittivity

In setting an expectation for the nature of the permittivity spectra one must consider the features of the material system under investigation in the light of the reviews of Chapter 2. Thus, for a composite formed from conductive particies in a insulating hydrocarbon matrix, it is expected that the only polarisation process with a reiaxation frequency in the range of interest (i.e. 1 Hz to 100 GHz) is the Maxwell-Wagner-Siilars (MWS) polarisation. The matrix material has no permanent dipole moment so no orientation polarisation should be observed. In addition, effects due to bulk free-electron conduction may be present. It is also possible that some hopping conduction may occur when the conductive particles are close to contacting. Some ionic conduction due to impurities (including moisture) in the matrix may also be observed. The permittivity spectra for the microsphereiparaffin wax formulations are presented in Figure 5-3 and Figure 5-4 (the legends list the nominal master-batch filler fractions, refer to Table 4-4). Data for frequencies below 1 MHz have been corrected using equation (5-1) - this is the case for all subsequent data unless otherwise stated. The data do not contain any evidence for the presence of a MWS relaxation. To determine if this is consistent with the theory it is sufficient to determine the relaxation frequency from equation (2-88). For the range of fiiler fractions investigated here and for a filier conductivity of 58800 S/m it foilows that equation (2-88) yieids relaxation frequencies of the order of 10^^ Hz. Thus, the absence of a MWS reiaxation in the measured data is to be expected.

Further inspection of this equation demonstrates that the relaxation frequency is proportional to the filler conductivity and inversely proportional to filler volume fraction as shown in Figure 3-2 and Figure 3-1, respectively. Lowering the fiiler conductivity or increasing the filier fraction lowers the relaxation frequency. For instance, for filler loadings up to the close-packing limit, a fiiier conductivity of 1 S/m would reduce the relaxation frequency to lie in the range 1-10 GHz. Whilst, for a filler conductivity of 58800 S/m the filler fraction wouid have to be greater than 0.9999 to achieve a simiiar reduction in the reiaxation frequency. This value is considered unrealistic since such a composite is entirely metal and its permittivity would be expected to be dominated by free-electron conduction. In practice Figure 5-4 demonstrates that this is the case even for fiiier fractions as iow as 19%. This is consistent with the discussion of the iimitations of the early or traditional effective medium theories presented in Section 2.2.5 and indicates that such theories are only valid for filler fractions up to 0.1. Indeed, as shown by Figure 3-58, they predict that a composite only attains a conductivity of the order of that of the filler phase at a filler fraction very close or equal to unity (i.e. not even at the ciose packing iimit). Nevertheless, this semi-quantitative discussion is not greatiy altered when models including percolation effects are considered. Figure 3-57 shows that the MWS relaxation frequencies are generaily higher for a given filler conductivity in these models. In fact, the discussion is only tempered by the core-shell model of Sihvola and Lindell, which indicates the relaxation frequency wiii be reduced

Filler volume + 5% □ 11% Y 17% 10^ 0 7% X 13% A 9%X 15% “ 1 >10’ r -5 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x ^ m iw gxxmwm 10^ 10 10^ 10' E 10'^ “-10-4 10' - m ? ' ‘ " Ü X ^ 1 0 . :^x 10'^ 10^ 10^ 10® 10^ Frequency [Hz] 109 1 0'

Figure 5-3: Permittivity spectra for silver coated microsphereiparaffin wax formulations below the percolation threshold

Filler volume + 17% D 23% 0 19% X 25% T A 21% X 30% io'- xx: ^ \ x x ^ x x x x x x x x x x x 5 , , ^ ^ ^ G □ □ □ O D D O D Û D O Ü G OoÔ * ’* ^ * ! , °0 OOOGOOOoOoo loK: Frequency [Hz]

Figure 5-4: Permittivity spectra for silver coated microsphereiparaffin wax formulations above the percolation threshold

as the thickness of a conducting shell is reduced. This is directly relevant to the system studied here, however the ratio of silver thickness to particle radius is only expected to reduce the relaxation frequency by two decades, with it remaining above the frequency range of interest to this study.

As stated in the previous paragraph, free-electron bulk conduction is observed in samples with filler volume fractions greater than or equal 0.19. The temperature dependence must be considered to enable further in sight to the conduction mechanism (see Section 5.2.3). It is noted that there is some low frequency dispersion in the imaginary permittivity for low filler fractions; it is considered that this is likely to be due to electrode polarisation or polarisation of adsorbed impurities. There is no evidence to support ionic conduction due to unbound impurities.

5.2.2 Filler fraction dependence of the complex permittivity

5.2.2.1 Effective medium theory

In this section, the filler fraction dependence of the measured permittivity results at an arbitrary but representative frequency of 113 MHz is compared to the range of effective medium theories presented in Chapter 2.3.2. The comparison is also based on the assumption that the complex permittivity of the filler used in the EMT calculations is purely imaginary and derived from a conductivity of 10 S/cm according to equation (2-39). This value is lower than the manufacturer’s value for the powder conductivity, but is considered more representative of the expected maximum conductivity attainable by this material system. This reduced level of conductivity may be due to an increase in the resistance of the inter-particle contacts once the filler is blended with the wax. The final issue to be resolved before accurate conclusions can be drawn from this comparison is that of the true volume fraction of the filler particles. Table 4-14 demonstrates that the test samples appear to possess a significantly higher filler fraction (up to 50% higher) than expected from the master batch formulation.

The possible explanations for this systematic error are: inhomogeneous master batches; loss of wax during pressing; compression of the wax; fracture of the microspheres during pressing. The first source is unlikely since such an occurrence would be expected (by the laws of probability) to produce a random distribution of filler fractions centred about the master batch value. The second source is also unlikely since it would require one third of the volume of material placed in the die to be lost during pressing - such severe loss of material from the sides of the die was not observed. The third source has been eliminated by confirming that the density of wax disks pressed under the same experimental conditions as the microsphere filled test samples is in agreement (to within 1 %) with the manufacturer’s value quoted in Chapter 4.2.1. The fourth source is therefore the most likely and requires confirmation by microscopy.

Figure 5-5: Backscatter SEM image of 5 Figure 5-6: Backscatter SEM image of 15

vol% microsphere:paraffin wax (XC00718) vol% microsphere:paraffin wax (XC00723)

showing a section through the thickness of showing a section through the thickness of

a die pressed sample a die pressed sample

Figure 5-7: Backscatter SEM image of 30 vol% microsphere:paraffin wax (XC00729) showing a section through the thickness of

a die pressed sample

Figure 5-8: SEM image of a broken microsphere within a die pressed microsphere:paraffin wax sample

Figures 5-5 to 5-7 present scanning electron micrographs of three filler fractions (5, 15 and 30 %). These figures demonstrate that microspheres have indeed been broken during the pressing process. An approximate calculation, based on the assumption of monodisperse microspheres, reveals that 20 % (by volume) of the microspheres would need to be broken to result in a 50% increase in the apparent filler volume fraction. Since it is apparent from the figures that it is only the larger spheres that have a been fractured, the percentage by number of microspheres broken would be less than 20 %. This is not qualitatively inconsistent with the images presented in the figures. Quantitative analysis of this issue is further impeded by the fact that the spheres also appear to contain smaller spheres - as demonstrated by Figure 5-8. Thus, the following discussion will be based on the master batch filler fractions, although an alternative non-destructive method of determining filler fraction is desirable.

- a — B ru g g e m a n (s y m m e tric )

- A— L ic h te n e c k e r R e fra c tiv e Index

X B ( p é rim e n t

- 0 — B ru g g e m a n (u n s y m m e tric ) - e — L a n d a u , L its h itz, L o o y e n g a

T o b ia s & M e red itti

< — B o ttc tie r M a x w e ll-G a rn e tt - G r e f f e £ 1.0Ef01 1 .0 E -0 1 0 0.2 0 .4 0.6 0.8 1 10 5 0 0 0 .0 5 0.1 0 .1 5 0.2 0 .2 5

Aller volume fraction Aller volume fraction

Figure 5-9: Filler fraction dependence of the real permittivity for microsphereiparaffin wax formulations based on master batch quantities compared to Effective Medium

Theories (EMTs)

- o — B ru g g e m a n (s y m m e tric )

- A— L ic fite n e c k e r R e fra c tiv e Index

X E xp e rim en t

- 0 B ru g g e m a n (u n s y m m e tric ) - e — L a n d a u , L ifs fiitz, L o o y e n g a

T o b ia s & M e red itti

■ B o ttc tie r • M a x w e ll-G a rn e tt G r e f fe I.O E fO e 1.0E+O 5 I.O E K M 1 .0 E f0 3 1 .0 E f0 2 I.O E fO I 1 .0 E + 0 0 1 .0 5 -0 1 1 .0 5 -0 2 1 .0 5 -0 3 1 .0 5 + 0 0 1 .0 5 -0 1 o ) 1 .0 5 0 2 1 . 0 5 0 3 0 .4 0 .6

Aller volume fraction

0 .0 5 0.1 0 .1 5 0 .2

Aller volume fraction

0 .2 5

Figure 5-10:Flller fraction dependence of the Imaginary permittivity for microsphereiparaffin wax formulations based on master batch quantities compared to

Effective Medium Theories (EMTs)

T h e comparison of the m easured real and Im aginary permittivity to the E M T s is presented in Figure 5 -9 and Figure 5 -1 0 , respectively. T h e figures show that th ese m odels are in fact poor approxim ations for the effective permittivity of this particular conductor-insulator com posites for filler volum e fractions beyond 0.1. T h e lack of ag reem en t is to be expected since these models

generally consider the filler particles to be non-interacting (true only for dilute mixtures) and the distribution of the filler particles is assumed to be truly random or at least a randomly filled regular lattice. It is further noted that the models of Tobias and Meredith and of Greffe, which are second order extensions to the Maxwell-Garnett and Bottcher models, respectively, also fail to significantly improve the level of agreement. The interest in EMTs is, however, perpetuated by their simplicity, since one only needs to know the permittivity and respective volumes of the component materials. Whilst improved models have been developed, one generally needs to know more detail about the microstructure of the mixture with the disadvantage of increasing the amount of analysis required before predictions can be made. This is the case when producing ever more accurate bounds on the effective properties of the mixture (see Chapter 2.3.3). The question of, at what point an empirical study becomes the most efficient design option remains important and unanswered.

A more striking observation from this comparison is the rate at which the experimental results diverge from the EMTs. In fact, it is the further assumption of these models, namely that of uniform transport in the material, that needs to be questioned. The Bruggeman symmetric (or Bottcher) and Greffe models provide the best qualitative fit to the experimental data in that they replicate the sharp transition observed in the measured imaginary permittivity. However, where these models fail is in determining the filler fraction at which this transition occurs. They also under estimate the peak value of the real permittivity. This feature, of a sharp transition in properties at a specific filler fraction, was also evident in the frequency dependence of the imaginary permittivity (Figure 5-3 and Figure 5-4). Above this critical filler fraction the imaginary permittivity exhibits a frequency dependence which is characteristic of the material possessing a bulk dc conduction mechanism. With this in mind, the comparison is now extended to statistical percolation models (as described in Chapter 2.5.2).

5.2.2.2 Percolation theory

An alternative formalism potentially yielding simple analytical expressions for the filler fraction dependence of the mixture permittivity is percolation theory (see Chapter 2.5). This is particularly relevant to conductor-insulator composites which often exhibit a sharp transition in their properties at a given filler fraction - known as the percolation threshold. Such a sharp transition is evidenced in Figure 5-4 with the imaginary permittivity for the sample with highest filler fraction showing a frequency dependence characteristic of a bulk material exhibiting a dc conduction mechanism. Figure 5-11 compares the experimental results (in terms of pseudo-dc conductivity at 1 Hz) to three statistical percolation models.

It is noted that the models of Kirkpatrick and Zallen [85] and McLachlan [100] provide a much better fit than that of Beuche [102]. It is clear that all of the models provide a similar qualitative representation of the transition but with Beuche’s model not immediately providing an accurate estimate of the threshold. The reason for this lies in the fact that, of the three models presented, Beuche’s is the only model to attempt to predict the percolation threshold; the other models use the threshold as a fitting parameter. Indeed, it is this methodology that has

provided the excellent fit in the figure. In B euche’s model the threshold is determ ined by assum ing a regular lattice type with a corresponding num ber of nearest neighbours. O ther m odels are available to estim ate the threshold (as described in C h a p te r 2 .5 ) but the trade-off b etw een microstructural knowledge of the mixture system and accuracy of the m odel is again apparent. (0 .i > 3 ■o c o Ü 1.E+03 1.E+00 1.E-03 1.E-06 1.E-09 1.E-12 1.E-15 1 0.01 0.1 Kirkpatrick & Zallen ““— Bueche (hep) “♦“ McLachlan X Experiment

Filler volume fraction

Figure 5-11 : Filler fraction dependence of the conductivity (at 1 Hz) for microsphereiparaffin wax formulations based on master batch densities compared to

statistical percolation theories

0.5 - - 0.8 1 ■o -0.5 - - 0.6 « t = 1.97 S = 0 .7 0 — J - 0 . 2 -2.5 -2.5 -1.5 -0.5 LOG|v-vc|

Figure 5-12:Determination of the critical percolation exponents, s and t, for the conductivity and real permittivity from the experimental data near the percolation

— 7 vol% 15vo l% o 16 vol%

• • ❖ ■ 1 7 vol% * 19 vol% • • • - - 25 vol%

1.E+03

1.E+01

o 1.E-01

1.E-03

1.E-05

1.E+07 1.E+09 1.E+11

Frequency (Hz)

tan 6