Resistance and specific resistance relationships

[Dario56]

I'm using a four-probe system (nextron) to measure the voltage between the two points of my sample. Sample has a cylindrical shape and it's placed on its base inside the instrument. The probes are placed on its upper surface (the base of the cylinder).

From the voltage measured, I'd like to calculate specific resistance. Resistance is calculated from the Ohm's law (current is kept constant). To calculate specific resistance, sample geometry is required. From what I learned, the relationship between the two is given by: $$R = \rho \frac {l}{A}$$

This relationship holds for the current flowing in the axial direction of the cylinder (where the cross section of the conductor is circular and doesn't change in the direction of the current flow). However, in my case, I'm not able to place the sample like that in the instrument and the question becomes: How can I connect the resistance and specific resistance for such a geometry and measurement?

Are there more general relationships between the two than one given on the equation?

 

[Baluncore]

However, in my case, I'm not able to place the sample like that in the instrument and the question becomes: How can I connect the resistance and specific resistance for such a geometry and measurement?

Why will it not fit?
Is it too big or small?

The probes are placed on its upper surface (the base of the cylinder).

How can the upper surface be the base of the cylinder?

 

[Dario56]

Why will it not fit?
Is it too big or small?

Too big.

How can the upper surface be the base of the cylinder?

Well, upper or bottom, they have the same area. The point is, all probes are on the same surface.

 

[Baluncore]

The equation you give is for two probes at each end of a column. That makes a total of four contacts.
Yet you write:

The point is, all probes are on the same surface.

This does not make sense. Is there something you are not telling us?
What gives?

I'm using a four-probe system (nextron) to measure the voltage between the two points of my sample.

Please provide a web link that shows the sample probe contact geometry of the instrument you use.

 

[Dario56]

The equation you give is for two probes at each end of a column. That makes a total of four contacts.
Yet you write:

This does not make sense. Is there something you are not telling us?
What gives?

Yes, my question is what equation does hold for my experimental setup? I'm aware that the equation I wrote isn't applicable.

Please provide a web link that shows the sample probe contact geometry of the instrument you use.

 

[nasu]

There are formulas and associated tables for various geometries used to measure specific resistance by the four probe method. You have to look up papers where these were published. There is no general formula.
There is also a method called Van der Pauw method. It is for thin sheets, I believe. It uses 4 probes as well. You may try to see if your sample satisfy the conditions of the method. The method is described in a paper as well.

 

[Baluncore]

Most resistivity measurement techniques assume a well-defined equipotential boundary, or a slab of infinite depth and extent. The resistivity and thickness of a slab can be estimated from readings made with four equally spaced electrodes, in a line, on the surface, by varying the electrode spacing.

What geometrical pattern of electrodes, and what boundary conditions, do you have?

 

[Dario56]

There are formulas and associated tables for various geometries used to measure specific resistance by the four probe method. You have to look up papers where these were published. There is no general formula.
There is also a method called Van der Pauw method. It is for thin sheets, I believe. It uses 4 probes as well. You may try to see if your sample satisfy the conditions of the method. The method is described in a paper as well.

I think in my case I'd need to use a cuboid sample geometry as given by the figure.

Again, schematic shows a separation between the electrodes which draw current and the others that measure voltage. Like in the case of the four point probe system. However, schematic also shows that there are multiple probes placed on the various points along the sample length and the voltage is simultaneously measured between the starting point and all other points. I'd need to use a multiple probe system to do this measurement and I'm not sure is there something like four point probe system, but with multiple probes and where to buy it.

For this sample geometry, is resistance to specific resistance relationship given by the equation given in the main post? $$R = \rho \frac {l}{A}$$

 

[Baluncore]

For this sample geometry, is resistance to specific resistance relationship given by the equation given in the main post?

That equation is only applicable for objects that have a parallel current flow.

You have a cylinder but have not identified the axial length, diameter, or ratio.
Please provide a sketch of your sample, with dimensions.

To get a parallel current in a short cylinder, the flat ends of the cylinder would need to be plated with a more conductive material, to connect the current electrodes to the area. Two points on the cylindrical surface could then be probed to measure the voltage.

What material is your sample made from?
Over what temperature range will the resistivity measurements be made?

 

[nasu]

I think in my case I'd need to use a cuboid sample geometry as given by the figure. View attachment 338270

Again, schematic shows a separation between the electrodes which draw current and the others that measure voltage. Like in the case of the four point probe system. However, schematic also shows that there are multiple probes placed on the various points along the sample length and the voltage is simultaneously measured between the starting point and all other points. I'd need to use a multiple probe system to do this measurement and I'm not sure is there something like four point probe system, but with multiple probes and where to buy it.

For this sample geometry, is resistance to specific resistance relationship given by the equation given in the main post? $$R = \rho \frac {l}{A}$$

I was thinking about the case when you have four probes in line, applied on the surface of the rectangular sample. Similar to this paper:

https://www.physicsforums.com/threa...esistance-relationships.1058518/#post-6984160

Look at the section about thick samples for correction factors.

 

[Dario56]

That equation is only applicable for objects that have a parallel current flow.

I'm referring to the cuboid sample shown in the figure.

You have a cylinder but have not identified the axial length, diameter, or ratio.
Please provide a sketch of your sample, with dimensions.

I do have a cylindrical sample, however I realised that my sample is too small to put all these numerous probes on it. I need to make a sample with geometry similar to the figure above. Cuboid or cylindrical sample with the axial dimension much bigger than other dimensions. In essence, I want to do the electrical conductivity relaxation measurement. I'm measuring how specific resistance of the sample is changing with the position in the sample and time as the diffusion of charge carriers (oxide-ion and oxygen vacancies) is imposed by the shift in chemical equilibrium position (oxygen partial pressure step change). Chemical equilibrium concerned includes oxide-ions, oxygen vacancies and oxygen gas. For example, if oxygen partial pressure is increased, equilibrium is shifted towards oxide-ions creation (their concentration is increased) as oxygen molecules receive electrons (reduction) and are then positioned inside the oxygen vacancies (Le Chatelier's principle).

Diffusion changes local concentration of the charge carriers in time causing changes in local ionic conductivity (specific resistance). By measuring voltage between two points in time, I can calculate how does average ionic conductivity (and spatially averaged charge carrier concentration) in the concerned volume change over time.

Van der Pauw method (advised here) can't really measure the specific resistance variation inside the sample as required in this method, if I'm not mistaken. It can only measure spatially averaged specific resistance of the whole sample. That's why I need to change the sample geometry as I'm unaware of the specific resistance measurement method which can measure such a variatiation.

What material is your sample made from?
Over what temperature range will the resistivity measurements be made?

I can't say exactly as the material is NDA protected. I can say only that it's a doped metal oxide. Doping is done to create oxide-ion vacancies in the crystal structure which increases its ionic conductivity.

Temperature range is 500 - 800 C.

 

[Dario56]

That equation is only applicable for objects that have a parallel current flow.

After discussion with my supervisor, I figured that van der Pauw method (or something similar to it) is actually applicable because I'm going to measure spatially averaged conductivity of the whole sample and its response in time. What I didn't know earlier is that this response can also be fitted to the solution of the diffusion equation to yield parameters in question. This is done by integrating out the position variables to get the solution in terms of average conductivity (or average charge carrier concentration).

You have a cylinder but have not identified the axial length, diameter, or ratio.
Please provide a sketch of your sample, with dimensions.

As you noted, sample geometry is simple, a small cylinder. Diameter is 1.1 cm, height is 0.2 cm and therefore ratio 5.5.

 

[Chestermiller]

The basic relationship is $$\rho \vec{q}=-\nabla V$$ where $\vec{q}$ is the current density and V is the voltage.

 

[Dario56]

The basic relationship is $$\rho \vec{q}=-\nabla V$$ where $\vec{q}$ is the current density and V is the voltage.

Microscopic Ohm's law.

 

[Baluncore]

As you noted, sample geometry is simple, a small cylinder. Diameter is 1.1 cm, height is 0.2 cm and therefore ratio 5.5.

That is shaped like a disc, pellet, or coin. I assume that one face only can be accessed.

It should be possible to make 4 point connections, in a line, on a diameter across one face. Those points will be separated by about 3.6 mm. The outer two points will be used to inject the current. The voltage will be measured between the two inner points.

It will be possible to correct the resistivity for the finite thickness of the disc, and for the missing area outside the disc. If the current electrodes are near the edge of the disc, less drive current will be needed, while the outer electric field lines, will follow the circumference of the disc.

 

[Dario56]

That is shaped like a disc, pellet, or coin. I assume that one face only can be accessed.

Yes, nextron is designed to do the measurement by putting all the probes at the same face.

It should be possible to make 4 point connections, in a line, on a diameter across one face. Those points will be separated by about 3.6 mm. The outer two points will be used to inject the current. The voltage will be measured between the two inner points.

Doesn't the van der Pauw require that all four probes are put on the edge of the sample (circumference in my case), not on the line? Line measurement can be done, but it gives the resistivity only in the sensing direction rather than average value (for materials with anisotropic resistivity). How did you get the required spacing?

 

[Baluncore]

This is NOT van der Pauw using parallel currents.
This pattern is from geophysics Earth resistivity measurements, a method used to measure the resistivity of layers from points on the surface.
https://en.wikipedia.org/wiki/Vertical_electrical_sounding

I got the spacing of four electrodes from the diameter of 11 mm, divided by 3, which gives 3.66 mm.
That spacing is greater than the sample thickness, so the full depth is being sampled.
Since the outer electrodes are near the edge, where field lines run parallel with the edge, the field is not significantly distorted by the boundary of the sample, and the majority of the sample is being measured. Only two small volumes, on the opposite side of the disc, near the current electrodes, are not part of the measurement.

Anisotropic resistivity will be a problem, no matter what electrode geometry is used, unless you can rotate the line of four electrodes, to measure a different diameter.