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Measurement of Small Electrochemical Signals |
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Overview Potentiostats are often called on to work near the limits of measurement science. The Gamry Instruments FAS2 Femtostat can resolve current changes as small as 1 femtoamp (10-15 amps). To place this current in perspective, 1 fA represents the flow of about 6000 electrons per second! Small current measurements place demands on the instrument, the cell, the cables and the experimenter. Many of the techniques used in higher current electrochemistry must be modified when used to measure pA and fA currents. In many cases, the basic physics of the measurement must be considered. This Application Note will discuss the
limiting factors controlling low current measurements. The emphasis will be on EIS
(Electrochemical Impedance Spectroscopy), a major application for low current
potentiostats. This model is valid for analysis purposes even when a real potentiostat’s circuit topology differs significantly. In Figure 1:
In the ideal current measurement circuit Rin is infinite while Cin and Iin are zero. All the cell current, Zcell, flows through Rm. With an ideal cell and voltage source, Rshunt is infinite and Cshunt is zero. All the current flowing into the current measurement circuit is due to Zcell. The voltage developed across Rm is measured by the meter as Vm. Given the idealities discussed above, one can use Kirchoff's and Ohms law to calculate Zcell:
Figure 1 Unfortunately technology limits high impedance measurements because:
Additionally, basic physics limits
high impedance measurements via Johnson noise, which is the inherent noise in a
resistance.
For purposes of approximation, the noise bandwidth, d F, is equal to the measurement frequency. Assume a 1012 ohm resistor as Zcell. At 300oK and a measurement frequency of 1 Hz this gives a voltage noise of 129 mV rms. The peak to peak noise is about 5 times the rms noise. Under these conditions, you cannot make a voltage measurement of ± 10 mV across Zcell without an error of up to ± 3%. Fortunately, an AC measurement can reduce the bandwidth by integrating the measured value at the expense of additional measurement time. With a noise bandwidth of 1 mHz, the voltage noise falls to 4 mV rms. Current noise on the same resistor under the same conditions is 0.129 fA. To place this number in perspective, a ± 10 mV signal across this same resistor will generate a current of ± 10 fA, or again an error of up to ± 3%. Again, reducing the bandwidth helps. At a noise bandwidth of 1 mHz, the current noise falls to 0.004 fA. With Es at 10 mV, an EIS system that measures 1012 ohms at 1mHz is about 3 decades away from the Johnson noise limits. At 0.1 Hz, the system is close enough to the Johnson noise limits to make accurate measurements impossible. Between these limits, readings get progressively less accurate as the frequency increases. In practice, EIS measurements usually
cannot be made at high enough frequencies that Johnson noise is the dominant noise source.
If Johnson noise is a problem, averaging reduces the noise bandwidth, thereby reducing the
noise at a cost of lengthening the experiment.
Decreasing Rm increases this frequency. However, large Rm values are desirable to minimize voltage drift and voltage noise. A reasonable value for Cin in a practical, computer controllable low current measurement circuit is 5 pF. For a 30 pA full scale current range, a practical estimate for Rm is 1010 ohms.
In general, one should stay two decades below fRC to keep phase shift below one degree. The uncorrected upper frequency limit on a 30 pA range is therefore around 30 mHz. One can measure higher frequencies using the higher current ranges (i.e. lower impedance ranges) but this would reduce the total available signal below the resolution limits of the "voltmeter". This then forms one basis of statement that high frequency and high impedance measurements are mutually exclusive. Software correction of the measured response can also be used to improve the useable bandwidth, but not by more than an order of magnitude in frequency. Leakage
Currents and Input Impedance
For an Rm of 1010 ohms, an error < 1% demands that Rin must be > 1012 ohms. PC board leakage, relay leakage, and measurement device characteristics lower Rin below the desired value of infinity. A similar problem is the finite input leakage current (Iin) into the voltage measuring circuit. It can be leakage directly into the input of the voltage meter, or leakage from a voltage source (such as a power supply) through an insulation resistance into the input. If an insulator connected to the input has a 1012 ohm resistance between +15 volts and the input, the leakage current is 15 pA. Fortunately, most sources of leakage current are DC and thus they disappear in impedance measurements. As a rule of thumb, the DC leakage should not exceed the measured signal by more than a factor of 10. The best commercially available input amplifier has input current of around 50 fA. Other circuit components may also contribute leakage currents. You therefore cannot make absolute current measurements of low fA currents. In practice, the input current is approximately constant, so current differences or AC current levels of a few fA can be measured. Voltage
Noise and DC Measurements Assume that you have a working electrode with a capacitance of 1 mF. This could represent a passive layer on a metal specimen. The impedance of this electrode, assuming ideal capacitive behavior, is given by
At 60 Hertz, the impedance value is about 2.5 kW. Apply an ideal DC potential across this ideal capacitor and you get no DC current. Unfortunately, all potentiostats have noise in the applied voltage. This noise comes from the instrument itself and from external sources. In many cases, the predominant noise frequency is the AC power line frequency. Assume a reasonable noise voltage, Vn, of 10 mV. Further, assume that this noise voltage is at the US power line frequency of 60 Hz. It will create a current across the cell capacitance:
This rather large noise current will prevent accurate DC current measurement in the pA range. In an EIS measurement, you apply an AC
excitation voltage that is much bigger than the typical noise voltage, so this is not a
factor. Parallel metal surfaces form a capacitor. The capacitance rises as either metal area increases and as the separation distance between the metals decreases. Wire and electrode placement have a large effect on shunt capacitance. If the clip leads connecting to the working and reference electrodes are close together, they can form a significant shunt capacitor. Values of 10 pF are common. This shunt capacitance cannot be distinguished from "real" capacitance in the cell. If you are measuring a paint film with a 30 pF capacitance, 10 pF of shunt capacitance is a very significant error. Shunt resistance in the cell arises because of imperfect insulators. No material is a perfect insulator (one with infinite resistance). Even Teflon, which is one of the best insulators known, has a bulk resistivity of about 1014 ohms. Worse yet, surface contamination often lowers the effective resistivity of good insulators. Water films can be a real problem, especially on glass. Shunt capacitance and resistance also occur in the potentiostat itself. In most cases, the cell's shunt resistance and capacitance errors are larger than those from the potentiostat.
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