# Measurement of Small Electrochemical Signals

## Overview

Gamry potentiostats are very sensitive scientific instruments. For example, the Reference 600 can theoretically resolve current changes as small as 1.8 femtoamp (1.8 x 10^{-15} amps) without internal gain.

The small currents measured by a potentiostat places 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 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. It will include hints on cell and system design. The emphasis will be on EIS (Electrochemical Impedance Spectroscopy), a highly demanding application.

## Measurement System Model and Physical Limitations

To get a feel for the physical limits implied by very sensitive current measurements, consider the equivalent circuit shown in **Figure 1**. We are attempting to measure the cell impedance given by Z_{cell}.

In** Figure 1: **

E_{s} Is an ideal signal source

Z_{cell} Is the unknown cell impedance

I_{cell} Is the “real” cell current

R_{m} Is the current measurement circuit's current measurement resistance

R_{shunt} Is an unwanted resistance across the cell

C_{shunt} Is an unwanted capacitance across the cell

C_{in} Is the current measurement circuit's stray input capacitance

R_{in} Is the current measurement circuit's stray input resistance

I_{in} Is the measurement circuit's input current

In the ideal current measurement circuit R_{in} is infinite while C_{in} and Iin are zero. All of the cell current, I_{cell}, flows through R_{m}.

With an ideal cell and voltage source, R_{shunt} is infinite and C_{shunt} is zero. All the current flowing into the current measurement circuit is due to Z_{cell}.

The voltage developed across R_{m} is measured by the meter as V_{m}. Given the idealities discussed above, one can use Kirchhoff’s and Ohms law to calculate Z_{cell} using **Equation 1**.

Z_{cell} = E_{s} * R_{m} / V_{m}** Eq. 1**

Unfortunately, technology limits high impedance measurements because:

- Current measurement circuits always have non-zero input capacitance, i.e. C
_{in}> 0. - Infinite R
_{in}cannot be achieved with real circuits and materials. - Amplifiers used in the meter have input currents, i.e. I
_{in}> 0. - The cell and the potentiostat create both a non-zero C
_{shunt}and a finite R_{shunt}.

Additionally, basic physics limits high impedance measurements via Johnson noise, which is the inherent noise in a resistance.

**Johnson Noise in Z _{cell}**

Johnson Noise across a resistor represents a fundamental physical limitation. Resistors, regardless of composition, demonstrate a minimum noise for both current and voltage, per equations 2 and 3.

E = (4 k T R δF)^{1/2} ** Eq. 2**

I = (4 k T δF / R)^{1/2} ** Eq. 3**

Where k is Boltzman's constant 1.38x 10^{-23} J/K, T is temperature in K,δF is noise bandwidth in Hz and R is resistance in ohms.

For purposes of approximation, the Noise bandwidth, δF, is equal to the measurement frequency. Assume a 10^{11} ohm resistor as Z_{cell}. At 300 K and a measurement frequency of 1 Hz this gives a voltage noise of 41 μV rms. The peak-to-peak noise is about 5 times the rms noise. Under these conditions, you can make a voltage measurement of ±10 mV across Z_{cell} with an error of about ±0.4%. 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 about 1.3 μV rms.

Current noise on the same resistor under the same conditions is 0.41 ƒA. To place this number in perspective, a ± 10 mV signal across this same resistor will generate a current of ± 100 ƒA, or again an error of up to ± 0.4%. Again, reducing the bandwidth helps. At a noise bandwidth of 1 mHz, the current noise falls to 0.013 ƒA.

With E_{s} at 10 mV, an EIS system that measures 10^{11} ohms at 1 Hz is about 2 ½ decades away from the Johnson noise limits. At 10 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.

**Finite Input Capacitance**

C_{in} in Figure 1 represents unavoidable capacitances that always arise in real circuits. C_{in} shunts R_{m}, draining off higher frequency signals and limiting the bandwidth that can be achieved for a given value of R_{m}. This calculation shows at which frequencies the effect becomes significant. The frequency limit of a current measurement (defined by the frequency where the phase error hits 45°) can be calculated from Equation 4.

ƒ_{RC} = 1/ ( 2 π R_{m}C_{in} ) ** Eq. 4**

Decreasing R_{m} increases this frequency. However, large R_{m} values are desirable to minimize the effects of voltage drift and voltage noise in the I/E converter’s amplifiers.

A reasonable value for C_{in} in a practical, computer-controlled, low current measurement circuit is 2 pF. For a 6 nA full scale current range, a practical estimate for R_{m} is 10^{7} ohms, giving a cutoff frequency of ca. 8000 Hz, according to Eq. 4.

In general, one should stay two decades below ƒ_{RC} to keep phase shift below one degree. The uncorrected upper frequency limit on a 6 nA range is therefore around 80 Hz.

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**

Both R_{in} and I_{in} in Figure 1 affect the accuracy of current measurements. The magnitude error due to R_{in} is calculated using Equation 5.

Error = 1- R_{in}/(R_{m}+R_{in}) Eq. 5

For an R_{m} of 10^{7} ohms, an error < 1% demands that R_{in} must be greater than 10^{9} ohms. PC board leakage, relay leakage, and measurement device characteristics lower R_{in} below the desired value of infinity.

A similar problem is the finite input leakage current I_{in} 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 10^{12} ohm resistance between +15 volts and the input, the leakage current is 15 pA. Fortunately, most sources of leakage current are DC and can be tuned out in impedance measurements. As a rule of thumb, the DC leakage should not exceed the measured AC signal by more than a factor of 10.

**Voltage Noise and DC Measurements**

Often the current signal measured by a potentiostat shows noise that is not the fault of the current measurement circuits. This is especially true when you are making DC measurements. The cause of the current noise is noise in the voltage applied to the cell.

Assume that you have a working electrode with a capacitance of 40 μF. This could represent a 1 cm^{2} polished bare metal immersed in an electrolyte solution. You can roughly estimate the capacitance of the electrical double layer formed by a metal/electrolyte interface as 20 μF/cm^{2}. The area is the microscopic area of the surface, which is larger than the geometric area, because even a polished surface is rough. The impedance of this 40 μF electrode, assuming ideal capacitive behavior, is given by:

Z = 1/jωC

At sixty Hertz, the impedance magnitude is about 66 Ω.

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 realistic noise voltage, Vn, of 10 μV (this is lower than the noise level of most commercial potentiostats). 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:

I = Vn/Z ≈ 10 x 10^{-6}/ 66 ≈ 150 nA

This rather large noise current will prevent accurate DC current measurement in the low nA or pA ranges.

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.

**Shunt Resistance and Capacitance**

Non-ideal shunt resistance and capacitance arise in both the cell and the potentiostat. Both can cause significant measurement errors.

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 1 to 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 100 pF capacitance, 5 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 10_{12} ohms•m. 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.

**Hints for System and Cell Design**

The following hints may prove helpful.

**Faraday Shield**

A Faraday shield surrounding your cell is mandatory for very low-level measurements. It reduces both current noise picked up directly on the working electrode and voltage noise picked up by the reference electrode.

A Faraday shield is a conductive enclosure that surrounds the cell. The shield can be constructed from sheet metal, fine mesh wire screen, or even conductive paint on plastic. It must be continuous and completely surround the cell. Don't forget the areas above and below the cell. All parts of the shield must be electrically connected. You will need an opening in the shield large enough to allow a cell cable to enter the shield.

Gamry offers a Faraday shield called the VistaShield that features a transparent window so you can see your electrochemical test running. It can be used with any potentiostat.

The Faraday shield must be electrically connected to the potentiostats floating ground terminal. An additional connection of both the shield and the floating ground to an earth ground may also prove helpful.

Only connect floating ground to earth ground if all conductive cell components are well isolated from earth ground. A glass cell is usually well isolated. An autoclave is generally not well isolated. |

Avoid External Noise Sources

Try to keep your system away from electrical noise sources. Some of the worst are:

- Fluorescent lights
- Motors
- Radio transmitters
- Computers and computer monitors

Try to avoid AC powered or computerized apparatus within your Faraday shield.

**Cell Cable Length and Construction**

Cell cables longer than 1 meter will result in degraded instrument performance. Increased noise and decreased stability both can occur. However, with most cells, the instrument will work acceptably with an extended cell cable, so our advice is to go ahead and try it. As a rule, you should not attempt to use current interrupt IR compensation with cell cables longer than 5 meters.

We do not recommend that you use a Gamry potentiostat with any cable not supplied by Gamry Instruments. Our cables include a number of individually shielded wires contained within an overall shield. We pay careful attention to issues such as shield isolation, isolation resistance, and capacitance.

If you do need a special cable, contact us with your requirements.

**Lead Placement**

Many experiments involve cells with small capacitances, the value of which may be important.

In these cases, the capacitance between the cell leads can result in an error. The alligator clips can have 10 pF or more of mutual capacitance if they are run alongside each other.

If you wish to avoid excessive capacitance:

- Place the leads as far apart as possible. Pay special attention to physical separation between the working electrode/ working sense leads and the counter/ counter sense/ reference electrode leads.
- Have the leads approach the cell from different directions.
- Remove the alligator clips from the leads. In extreme cases you can replace the banana plugs and pin jack with smaller connectors. If you do so, be careful not to compromise the isolation between the center conductor and the shield.

The cell leads must not be moved during an experiment measuring small currents. Both microphonic and triboelectric effects can create spurious results when the cell cables are moved.

**Cell Construction**

If you need to measure small currents or high impedances, make sure that your cell construction does not limit your response.

A cell where the resistance between the electrodes is only 10^{10} ohms cannot be used to measure 10^{13} ohm impedances. In general, glass and Teflon are the preferred cell construction materials. Even glass may be a problem when it is wet.

You also must worry about C_{shunt}. Make the "inactive" portion of your electrodes as small as possible. Avoid placing electrodes close together or parallel with each other if you are measuring high impedances.

*Reference Electrode*

Keep your reference electrode impedance as low as possible. High impedance reference electrodes can cause potentiostat instability and excessive voltage noise pickup.

Try to avoid:

- Narrow bore or Vycor tipped Luggin capillaries.
- Poorly conductive solutions - especially in Luggin capillaries.
- Asbestos thread and double junction reference electrodes.

Reference electrodes often develop high impedances as they see use. Anything that can clog the isolation frit can raise the electrode impedance. Avoid using saturated KCl based reference electrodes in perchlorate ion solutions as KClO_{4} has very poor solubility in a number of solvents.

*Instrument Settings*

There are several things to remember in setting up a very sensitive experiment.

- In EIS, use the largest practical excitation. Don't use a 10 mV excitation on a coated specimen that can handle 100 mV without damage.
- Avoid potentials where large DC currents flow. You cannot measure 1 pA of AC current on top of 1 mA of DC current.

*EIS Speed*

In EIS, do not expect a potentiostat to measure 10^{10} ohm impedances at 1 kHz. Many of the factors listed above limit the performance.

As a rule of thumb, the product of Impedance, Z, times frequency, f, should be less than 10^{9} ΩHz for good EIS measurements with an Interface 1000. Z • f < 10^{9} ΩHz *Ancillary Apparatus*

Do not use potentiostats with an ancillary apparatus connected directly to any of the cell leads. Ammeters and voltmeters, regardless of their specifications, almost always create problems when connected to the cell leads.

*Floating Operation*

All Gamry potentiostats are capable of operation with cells where one of the electrodes or a cell surface is at earth ground. Examples of earth grounded cells include: autoclaves, stress apparatus, pipelines, storage tanks and battleships. The internal ground on a Gamry potentiostat is allowed to float with respect to earth ground when it works with these cells, hence the name floating operation.

Instrument performance can be degraded when a potentiostat is operated in a floating mode. The instrument specifications only apply on isolated cells with the earth ground referenced (not floating).

Special precautions must be taken with the cell connections when the potentiostat must float. Make sure that all the cell connections are isolated from earth ground.

Finally, ancillary apparatus connected to the potentiostat must be isolated. External voltmeters, ammeters, FRA's etc. must be isolated.

Cables used with the User I/O Connector must be carefully constructed. We do not recommend connection to the User I/O connector using a 15 conductor shielded cable. The shield will generally be connected to the metal shell of the User I/O connector on the potentiostat end of the cable. The shield will often be connected to an earth ground on the other end of the cable. Connecting the grounds in this way destroys the potentiostats ability to float and to make measurements on earth grounded cells. In extreme cases, this connection can cause damage to the potentiostat.