The impedance spectroscopy versus one-frequency measurements
Measurements of electrical conductivity of materials are tricky when conduction is due to the migration of ions (ionic conductivity). When ionic conductivity dominates, ions migrate due to the applied electric field towards the electrodes. At the electrode, ions must exchange electrons with the electrode. This takes finite time, and hence, some electric charge is built up at the electrode. This charge will retard further migration of ions and gives apparently low electrical conductivity. Consequently, a DC (or one low frequency) measurement of electrical conductivity gives erroneous results when conduction is due to the migration of ions.
In order to avoid this problem, impedance spectroscopy is usually used. In this approach, one applies AC voltage with a range of frequencies and uses a model where a pair of electrodes is modeled as a capacitor. After taking the impedance response of a specimen for a broad range of frequencies, one can calculate the DC conductivity. This method has been used by most of researchers (e.g., Huang et al. 2005; Xu et al. 1998), but in early studies, Yoshino and his co-workers used one low frequency (0.01 or 0.1 Hz) in their studies (Manthilake et al. 2009; Yoshino et al. 2008, 2006).
The use of this inappropriate method leads to two important misconceptions: (i) in most cases, this method gives systematically higher resistivity than true resistivity (see Fig. 1a). The degree to which this happens is larger at higher temperatures where more charge accumulation occurs at electrodes. Also the degree to which this artifact occurs is larger for a sample with higher water content. Consequently, this leads to an apparent concentration dependence of activation enthalpy (Fig. 1b; see section Parameterization of electrical conductivity). (ii) When such results are extrapolated to higher temperatures, there is systematic under-estimation of conductivity. The “discrepancy” between Karato group’s results and Yoshino’s results is largely due to the use of this incorrect method in Yoshino’s group as shown by Karato and Dai (2009). Yoshino later started to use impedance spectroscopy, and obtained results that are not far from our results (e.g., Yoshino et al. 2012) although that paper contains another problem for “dry” samples as will be discussed in the next section.
Hydrogen loss/gain
Measurements of electrical conductivity of minerals containing hydrogen are challenging for several reasons. Firstly, measurements of any defect-related properties (electrical conductivity, high-temperature creep) are complex because these properties are sensitive to many physical and chemical parameters such as temperature, pressure, oxygen fugacity, and hydrogen fugacity. Secondly, however, in almost all laboratory studies, a sample is not in chemical equilibrium during electrical conductivity measurement.
And more importantly, the behavior of hydrogen is delicate because it can be lost from a sample or a sample may gain hydrogen during an experiment. I will show some examples of hydrogen loss and gain during electrical conductivity measurements in sections Hydrogen loss and Hydrogen gain in a nominally "dry" experiment.
Another difficulty that is unique to electrical conductivity measurements of hydrogen-bearing mineral is that it is hard to conduct experiments where a sample is under well-defined thermochemical conditions. In an ideal experiment, a sample must contain the hydrogen-related defect whose concentration corresponds to the thermochemical conditions under which conductivity is measured. In order to achieve those conditions, a sample must exchange key elements such as hydrogen with the surroundings (the open system behavior). It is possible to realize such a sample assembly at high pressures using hydrogen fugacity buffer such as developed by Otsuka and Karato (2011), but the use of such a buffer in electrical conductivity measurements is difficult because the buffer materials have low electrical resistance.
An alternative way to keep hydrogen in a sample is to seal a sample using some metal jacket or a capsule. A commonly used metal for this purpose is Au-Pd (e.g., Kawamoto and Hirose 1994), but the use of a metal jacket in the conductivity measurement is difficult and I do not know any successful examples (Yang (2012) used a BN capsule to keep hydrogen in the sample assembly with some success).
Considering these difficulties, a common practice is to conduct an experiment in which the composition of a sample is nearly fixed (nearly closed system). In these experiments, the concentration of defects such as the total hydrogen content is fixed, and the fixed value is determined at the conditions where a sample was prepared (high T, P annealing with a given chemical condition). When one changes temperature during an experiment, defect mobility changes and the relative population of various defects may also change (e.g., Karato 2013, 2015a) but not the total number of hydrogen.
When one applies these laboratory results to estimate water content in Earth, one also assumes that Earth also behaves like a closed system. I consider that this is a valid approach, because diffusion time scale for a region larger than ~ 10 km exceeds the age of Earth, and therefore, large regions can be considered as a closed system: chemical composition of these large regions can change only through large-scale melt segregation upon partial melting. After partial melting, the chemical composition of the region is essentially fixed.
However, the assumption of the closed system behavior is not valid when one considers the exchange of hydrogen among co-existing minerals. In such a case, the space scale is several millimeter to centimeter, and chemical equilibrium is likely established. Therefore, the influence of hydrogen partitioning among co-existing minerals is important in assessing the water content in various regions (e.g., Dai and Karato 2009a, see also section Conductivity of the lithosphere).
Hydrogen loss
Hydrogen loss during a conductivity measurement is an obvious possibility, and in the first study of electrical conductivity of hydrogen-bearing minerals, Huang et al. (2005) recognized that hydrogen loss occurred in some of their samples (in some runs, a large fraction of hydrogen was lost. In these cases, the results were rejected). Similarly, Yoshino et al. (2006) reported the evidence of hydrogen loss during their measurements of electrical conductivity in single crystal olivine.
When a sample containing hydrogen is placed in the environment with low hydrogen fugacity, hydrogen tends to escape from a sample to the surroundings via diffusion. The hydrogen loss also occurs due to the migration of protons (by the electric field) in a sample to the electrode where protons are accumulated and lost.
The degree of hydrogen loss is high at high temperatures where hydrogen mobility is high. Yoshino and Katsura argue that hydrogen always escapes from the sample above ~ 1000 K, and therefore, results above this temperature are unreliable. Indeed, their conductivity measurements are always below ~ 1000 K (e.g., Yoshino 2010; Yoshino and Katsura 2013). These authors consider that low temperature results are reliable and extrapolate these data to asthenosphere (~ 1600 K) and argue that hydrogen cannot explain high conductivity in the asthenosphere (see also Naif et al. 2013).
Let us consider how hydrogen loss might occur. Hydrogen loss could occur either by diffusion or by the migration of hydrogen caused by the electric field. Diffusion loss time scale can be calculated from the effective diffusion coefficient for hydrogen loss from \( \tau \approx \frac{{\left(d/2\right)}^2}{\pi^2D} \). Using the lab data (e.g., Kohlstedt and Mackwell 1998), I get τ≈ 1 h at 1400 K. A typical time duration for a conductivity measurement at one temperature is ~ 10 min. Therefore, for a typical conductivity measurement, diffusion loss is not important (although diffusion loss near the electrode could affect the conductivity measurement).
Another important mechanism of hydrogen loss is the hydrogen loss at the electrode. When protons reach to an electrode, unlike electrons (or electron holes), protons cannot go into the electrode, and therefore, they will be accumulated near the electrode (this gives capacitance behavior seen in the Z′ − Z″ plot). When the concentration of protons goes beyond the solubility limit, they will diffuse out of the sample.
If high-frequency field is used, then the mean distance of migration of protons is shorter, and hence, a majority of protons will be preserved. In contrast, when a low-frequency field is applied, a large fraction of protons will reach the electrode and will be lost.
The degree to which this occurs can be estimated roughly by the mean distance by which a proton migrates driven by the applied electric field. Therefore, we can use the following non-dimensional parameter to evaluate the degree of hydrogen loss, viz.,
$$ \xi \equiv \frac{l}{d}\approx \frac{\mu }{\omega d}E=\left|\frac{\sigma }{\omega {C}_H qd}\right|E $$
(1)
where l is the mean distance of migration of proton, d is the sample thickness, μ is the mobility of proton, E is the electric field, and ω is the frequency. If ξ > 1, a substantial hydrogen could occur (assuming the fast reaction at the electrode), whereas if ξ ≪ 1, hydrogen loss is small.
Figure 2 shows a plot of ξ as a function of (the lowest) frequency (ω) using the following values, q = 1.9 × 10− 19 C, d = 0.5 mm, and E = 2000 V/m (1 V across 0.5 mm). Since σ ∝ CH approximately, the results do not depend strongly on the hydrogen content. Since conductivity increases with temperature, tendency for hydrogen loss is large at high temperatures. For this voltage (1 V) and the lowest frequency of 0.01 Hz (typical parameters in an experiment), substantial hydrogen loss occurs (ξ > 1) above ~ 1000 K. This is consistent with some experimental observations (e.g., Yoshino et al. 2006). In contrast, if the lowest frequency is 1 kHz, hydrogen loss is much smaller. Also, if one uses a smaller voltage, the degree of hydrogen loss is smaller (for 0.1 V, the critical temperature for hydrogen loss increases by ~ 300 K compared to 1 V).
Although the above analysis suggests that one can minimize the hydrogen loss using high frequencies and/or low voltage, it is important to confirm that a substantial hydrogen loss (or gain) did not occur during a conductivity measurement. To do this, one must measure the hydrogen content both before and after a measurement. It is also very useful to test if a substantial hysteresis is present during a measurement. If hydrogen loss (or gain) occurs during the conductivity measurement, there should be hysteresis, i.e., discrepancy in the results during a temperature cycling. In all our studies, these steps were followed, and when a substantial change in water content occurred, we rejected the data. But Poe et al. (2010) did not check the water loss or gain during their experiments. They only measured the water content after the experiments. As will be discussed later, their results are anomalous (the dependence of conductivity on water content including the water content dependence of activation energy is much stronger than those reported in all other publications) and I suspect that changes in water content or state of water during their experiments might be the cause for these anomalous results.
Hydrogen gain in a nominally “dry” experiment
The possibility of hydrogen loss during an experiment is well recognized, but the evidence of hydrogen gain was also recognized after the first successful study of hydrogen-enhanced electrical conductivity by Huang et al. (2005). A remarkable example is the study by Xu et al. (1998) in which they showed that nominally “dry” wadsleyite and ringwoodite have substantially (a factor of ~ 300) higher conductivity than “dry” olivine (Fig. 3). We measured the water content in the samples studied by Xu et al. (1998) and noted that a large fraction of the difference in conductivity is due to the difference in water content: wadsleyite and ringwoodite in their samples contain a substantially larger amount of water than olivine, and that the reported large difference in conductivity is mostly due to the difference in water content (Huang et al. 2006). The degree to which water (hydrogen) can be acquired by a sample depends strongly on the hydrogen solubility. For this reason, wadsleyite acquires more hydrogen than olivine, and Fe-rich olivine acquires more water than Fe-poor olivine.
Another observation is a comparison of nominally “dry” olivine conductivity. Figure 4 summarizes the published results on electrical conductivity of “dry” olivine with different FeO content (modified from Dai and Karato (2014c)). In this Figure, the results from truly dry data are marked. They include the results from room pressure high-temperature studies (Hirsch et al. 1993; Constable et al. 1992) where there is no chance to get a substantial amount of hydrogen, and the results by Dai and Karato (2014c) where the water content of the samples was determined to be small (less than ~ 10 ppm wt).
Although these truly dry data agree well, much higher and scattered conductivity is observed in the results obtained at high pressures where the water content of the samples was not measured. In these cases, the results obtained at high pressures show much higher conductivity than those for truly dry samples and that the deviation from truly dry data tends to be larger for higher FeO content. It is possible that this large scatter and high conductivity are due to water in their samples. Dissolution of hydrogen into a sample during a high-pressure experimentation is well known, and the tendency is higher for a sample with higher hydrogen solubility. In case of olivine, hydrogen solubility systematically increases with FeO content (Zhao et al. 2004), that is consistent with the trend seen in Fig. 4. However, the water content of samples from these studies was not determined that makes difficult to understand the causes for the discrepancy between these results (“dry?” data in Fig. 4) and the results where water content was shown to be low (“truly dry” data in Fig. 4). It is essential to measure the water content to confirm that the results correspond to “dry” conditions.
In the similar context, the “dry” samples studied by Yoshino et al. (2008) are not truly dry. Their FTIR spectra for water-poor samples are not clearly shown and it is impossible to quantify the water content in these samples (Fig. 5a). If wadsleyite really contains very low hydrogen content, the peak frequency of infrared absorption moves from ~ 3330 cm− 1 to ~ 3205 cm− 1 or ~ 3620 cm− 1 depending on the oxygen fugacity (Nishihara et al. 2008) (see Fig. 5b). Yoshino et al. (2008)’s IR absorption spectra have poor resolution and such a shift cannot be identified. For wadsleyite, Dai and Karato (2009b) showed that the water content as small as 23 ppm wt (0.0023 wt%) can enhance electrical conductivity (by a factor of ~ 10 at T = 1000 K) compared to truly dry sample that contains water less than 3 ppm wt. The low hydrogen content in wadsleyite can be clearly seen in Fig. 5b from the shift of the peak frequency as demonstrated by Nishihara et al. (2008). In contrast, the infrared absorption spectra published by Yoshino et al. (2008) do not show the details to convincingly show low water content. From their figure, I can only conclude that their “dry” sample has water content less than 0.1 wt% (Fig. 5b). Indeed, Yoshino et al. (2008)’s “dry” wadsleyite has ~ 10 times higher conductivity than our truly dry sample.
The situation is worse for ringwoodite. From the published results of IR absorption spectra, one could roughly estimate the water contents of their “dry” samples by 100–1000 ppm wt (compared to the dry sample studied by Dai and Karato (2009b) (~ 2 ppm wt, see their Fig. 1)).
I should also comment on the water content in samples studied by Poe et al. (2010). Firstly, they measured the water content of samples only after each experiment. Consequently, possible water loss or gain during an experiment cannot be assessed. More importantly, the reported values of water content by Poe et al. (2010) are large (363–2215 ppm wt based on Bell et al. (2003) calibration (121–746 ppm wt by Paterson (1982) calibration)), and most values exceed the solubility limit determined by Kohlstedt et al. (1996) and Zhao et al. (2004) (Fig. 6). For instance, according to Kohlstedt et al. (1996), the solubility of water at 8 GPa and 1373 K is ~ 2500 ppm wt (Bell calibration) (850 ppm wt (Paterson calibration)). Poe et al. (2010) conducted their measurements at P = 8 GP and T = 715–973 K. Under these conditions, the water solubility in olivine is 45–300 ppm wt based on Bell calibration (15–100 ppm wt based on Paterson calibration) (Zhao et al. 2004). It appears that the Poe’s samples were nearly water-saturated at the conditions where they added water (8 GPa, 1373 K), but under the conditions where they measured electrical conductivity, the samples were super-saturated with water. In these cases, behavior of hydrogen during a conductivity measurement could be different from those measurements under the conditions where hydrogen content is below the solubility limit. Indeed, similar studies on olivine single crystals by Yang (2012) and Dai and Karato (2014b) did not show strong anisotropy reported by Poe et al. (2010).
These observations show that it is essential to measure the water content of a sample both before and after each high-pressure measurement to make sure that the water content of the sample did not change much during an experiment. To be sure, it is also very useful to measure electrical conductivity through a few cycles. If hydrogen loss occurs during the measurement, one would detect hysteresis.