Preface

This book addresses recent advances in electrical resistivity and conductivity modelling, measurement, estimation and sensing methods and implications. Electrical conductivity and electrical resistivity are very important properties for various materials. Water's purity, materials' sorting, chemical composi‐ tion of substances, efficient metals' heat treatment examination, crystalline structures' stress state and heat damage prediction could be measured and estimated from the information of electrical conductivi‐ ty. Electrical resistivity is the inverse of conductivity and acts as resistance of a material to the electrical current's stream through it, bringing about a change of electrical energy into different types of energy. The measure of resistance relies upon the kind of material. Materials with low resistivity are great trans‐ mitters of power, whereas materials with high resistivity are great separators. The number of atomic lattice structure's imperfection such as dislocations, vacancies, interstitial defects and impurity atoms causes temperature's increase and resistivity's increase as well. Motivated by the importance of electri‐ cal resistivity and conductivity, some important experts in this field grasp most recent researches in this book. The chapters are selected for this book to reflect current variable techniques, new concepts and methods related to the book's topic from different perspectives. This book introduces innovative case studies for "Electrical Resistivity Sensing Methods and Implications", "Resistivity Model of Frozen Soil and High-Density Resistivity Method for Exploration of Discontinuous Permafrost", Measurement of the Electrical Resistivity for Unconventional Structures", Estimation of Hydrological Parameters from Geoelectric Measurements" and Assessment of Cryoprotectant Concentration by Electrical Conductivity Measurement and Its Applications in Cryopreservation". These recent advances are well prepared and presented in the form of six chapters as the following:


**Dr. Adel El-Shahat,** Senior IEEE Member Assistant Professor, Department of Electrical Engineering Founder and Director of Innovative Power Electronics and Nano-Grids Research Lab (IPENG) Georgia Southern University, Statesboro, Georgia, USA

## **Chapter 1**

## **Introductory Chapter: Recent Advances**

## Adel El-Shahat

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69046

This book proposes the most important researches in electrical resistivity and conductivity modeling, measurement, estimation and sensing methods, and implications. Electrical conductivity and electrical resistivity are very important properties for different materials. The goal of the book achieved via presenting new and modern case studies for sensing methods of electrical resistivity, resistivity modeling of frozen soil, measurement of the electrical resistivity for unconventional structures, estimation of hydrological parameters from geoelectric measurements, and assessment of cryoprotectant concentration by electrical conductivity measurement. It presents different methods to measure resistivity for both liquid and solid materials by explaining two, three, and four pole as well as toroidal resistivity cells. The special case of sheet material resistivity and resistance is explained in more detail, and equation for that special problem is simplified. It further provides information on common experimental errors, and a short guideline to improve the reliability and accuracy of the measurements. The way to experimentally determine the cell constant of a cell is described and the necessity for calibration is clearly explained too. Also, it provides information to overcome the standard problem of polarization, when the resistivity of solutions with high ionic content is investigated. After that, it explores the conduction characteristics of permafrost. A theoretical model and an experimental study to analyze the factors affecting the resistivity of permafrost are established and implemented. The study region was the permafrost degeneration area in the Northeast China. A permafrost profile map was drawn based on data from engineering drilling and an analysis of factors that influence permafrost resistivity. The reliability of the permafrost profile map was verified by an analysis of temperature data taken at measured points at different depths of the soil profile. Then, it introduces a device for measurement of the concrete structures' electrical volume resistivity. A quench protection active system (QPS) working in tandem with a superconducting coil structure (SCS), in order to prevent the damaging effects when the coil structures passing from the superconducting state in order to switch to normal-conduction state (quench), is presented as well. Moreover, it establishes experimental relationship between hydraulic transmissivity and hydraulic conductivity with Dar-Zarrouk parameter in porous media, transverse resistance (TR), in addition to a characterization of the water quality through the electrical resistivity. The determination

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

of hydraulic transmissivity and hydraulic conductivity is important for the development and management of groundwater exploitation of the study area. Finally, it illustrates an important application of the electrical conductivity measurement in cryopreservation. Cryopreservation is the way to cool the biological materials down to dormant state at low temperatures for longterm storage. This is done in order to reduce the cryo-injury to the cells during cryopreservation, cryoprotective agents (CPAs) should be added before freezing and removed after thawing prior to cell infusion due to the cytotoxicity of CPAs. The electrical conductivity measurement is used to assess the CPA concentration in cryopreservation. Measurement of electrical conductivity is validated as a safer and easier way to online and real-time monitoring of CPA concentration in cell suspensions, as well. Electrical resistivity and conductivity are proposed recently adopted different techniques and trends like: determining the magnitude of soil [1], in investigating frozen soil at Canada [2], in exploring frozen and petroleum polluted soils [3], in examining unfrozen water with ice substance with testing and algebraic equation implementation [4], in checking silty mud at various temperatures [5], in utilizing and examining the spatial appropriation of the island-shaped permafrost layer [6], in investigating the index of chemical weathering [7], in making initial experiments to make a relation in other different natural soil parameters using 79 samples of soil extracted from 10 boreholes [8], in illustrating and correlating clayeysoils' properties using 54 soil samples [9], and different techniques are adopted to estimate the content of water in soil with the aid of efficient electrical resistivity survey [10].

Finally, the rest of the book's chapters with their brief descriptions are shown in the following:

The second chapter, "Electrical Resistivity Sensing Methods and Implications," introduces basic operating principles of different methods to measure resistivity for both liquid and solid materials. It illustrates two, three, and four pole as well as toroidal resistivity cells. The van der Pauw technique is used as a step by step procedure to estimate the resistivity of a material with no arbitrary shape. The special case of sheet material resistivity and resistance is explained in more detail and equation for that special problem is simplified. It further provides information on common experimental errors and a short guideline to improve the reliability and accuracy of the measurements. The implications and challenges faced during resistivity measurements are explored and explained with ways to compensate for errors due to temperature and capacitance changes. In addition, the way to experimentally determine the cell constant of a cell is described and the necessity for calibration is clearly explained. It further provides information to overcome the standard problem of polarization when the resistivity of solutions with high ionic content is investigated.

The third chapter, "Resistivity Model of Frozen Soil and High-density Resistivity Method for Exploration," explores the conduction characteristics of permafrost. A theoretical model and an experimental study to analyze the factors affecting the resistivity of permafrost are established and implemented. The experimental study results are used to validate the rationality of the model of permafrost resistivity. To analyze differences in conductivity between underground media, a high-density resistivity (HDR) method is used, which infers the storage of underground geologic bodies with different resistivity based on the distribution of a conduction current under the electric field action. The study region was the permafrost degeneration area in the Northeast China. A permafrost profile map was drawn based on data from engineering drilling and an analysis of factors that influence permafrost resistivity. The reliability of the permafrost profile map was verified by an analysis of temperature data taken at measured points at different depths of the soil profile.

of hydraulic transmissivity and hydraulic conductivity is important for the development and management of groundwater exploitation of the study area. Finally, it illustrates an important application of the electrical conductivity measurement in cryopreservation. Cryopreservation is the way to cool the biological materials down to dormant state at low temperatures for longterm storage. This is done in order to reduce the cryo-injury to the cells during cryopreservation, cryoprotective agents (CPAs) should be added before freezing and removed after thawing prior to cell infusion due to the cytotoxicity of CPAs. The electrical conductivity measurement is used to assess the CPA concentration in cryopreservation. Measurement of electrical conductivity is validated as a safer and easier way to online and real-time monitoring of CPA concentration in cell suspensions, as well. Electrical resistivity and conductivity are proposed recently adopted different techniques and trends like: determining the magnitude of soil [1], in investigating frozen soil at Canada [2], in exploring frozen and petroleum polluted soils [3], in examining unfrozen water with ice substance with testing and algebraic equation implementation [4], in checking silty mud at various temperatures [5], in utilizing and examining the spatial appropriation of the island-shaped permafrost layer [6], in investigating the index of chemical weathering [7], in making initial experiments to make a relation in other different natural soil parameters using 79 samples of soil extracted from 10 boreholes [8], in illustrating and correlating clayeysoils' properties using 54 soil samples [9], and different techniques are adopted to estimate the

2 Electrical Resistivity and Conductivity

content of water in soil with the aid of efficient electrical resistivity survey [10].

resistivity of solutions with high ionic content is investigated.

Finally, the rest of the book's chapters with their brief descriptions are shown in the following: The second chapter, "Electrical Resistivity Sensing Methods and Implications," introduces basic operating principles of different methods to measure resistivity for both liquid and solid materials. It illustrates two, three, and four pole as well as toroidal resistivity cells. The van der Pauw technique is used as a step by step procedure to estimate the resistivity of a material with no arbitrary shape. The special case of sheet material resistivity and resistance is explained in more detail and equation for that special problem is simplified. It further provides information on common experimental errors and a short guideline to improve the reliability and accuracy of the measurements. The implications and challenges faced during resistivity measurements are explored and explained with ways to compensate for errors due to temperature and capacitance changes. In addition, the way to experimentally determine the cell constant of a cell is described and the necessity for calibration is clearly explained. It further provides information to overcome the standard problem of polarization when the

The third chapter, "Resistivity Model of Frozen Soil and High-density Resistivity Method for Exploration," explores the conduction characteristics of permafrost. A theoretical model and an experimental study to analyze the factors affecting the resistivity of permafrost are established and implemented. The experimental study results are used to validate the rationality of the model of permafrost resistivity. To analyze differences in conductivity between underground media, a high-density resistivity (HDR) method is used, which infers the storage of underground geologic bodies with different resistivity based on the distribution of a conduction current under the electric field action. The study region was the permafrost degeneration area in the Northeast China. A permafrost profile map was drawn based on data from engineering drilling and an analysis of factors that influence permafrost resistivity. The The fourth chapter, "Measurement of the Electrical Resistivity for Unconventional Structures," presents an apparatus for measurement of the concrete structures' electrical volume resistivity to operate at 500 Hz, within range of 5–100 Ω m for probe/concrete sample's interface. Also, a quench protection active system (QPS) working in tandem with a superconducting coil structures (SCS), in order to prevent the damaging effects when the coil structures passing from the superconducting state in order to switch to normal conduction state (quench), is presented. This chapter proposes experimentation of yttrium barium copper oxide (YBCO) tape's SCS with high temperature superconductor (HTS) type at 92 K temperature value as well. Finally, it shows measurement of the electrical resistance of the sensing element (SE) as a part of the resistive type gas sensor.

The fifth chapter, "Estimation of Hydrological Parameters from Geoelectric Measurements," proposes to establish an empirical relationship between hydraulic transmissivity (T) and hydraulic conductivity (K) with Dar-Zarrouk parameter in porous media, transverse resistance (TR), in addition to a characterization of the water quality through the electrical resistivity. This parameter is estimated from surface resistivity measurements, which are more economical in relation to the pumping tests, thus T was characterized in the study area. The reasons behind that are: in the coastal aquifer of the lower part of the right bank of the river Sinaloa, there is a need for fresh water for agricultural development because around 15% of the water used in agricultural irrigation is from underground sources. This situation is exacerbated during periods of drought, which promotes drilling with the risk of finding brackish water in them, besides this, there is the risk of not meeting water demand due to low hydraulic transmissivity (T) of the aquifer, putting at risk the drilling costs implied. In this sense, the determination of T and K (hydraulic conductivity) is important for the development and management of groundwater exploitation of the study area. Generally, by means of pumping tests in wells T is obtained, with high costs, so there are few values of T. K is generally obtained by wells and laboratory test.

The sixth chapter, "Assessment of Cryoprotectant Concentration by Electrical Conductivity Measurement and its Applications in Cryopreservation," presents an important application of the electrical conductivity measurement in cryopreservation. Cryopreservation is the way to cool the biological materials down to dormant state at low temperatures (such as −80 or −196°C, the temperature of liquid nitrogen) for long-term storage and later thaw them back to the normal physiological temperatures before usage with recovered viability and functionalities of the cells and tissues. In order to reduce the cryo-injury to the cells during cryopreservation, cryoprotective agents (CPAs) should be added before freezing and removed after thawing prior to cell infusion due to the cytotoxicity of CPAs. In this chapter, the electrical conductivity measurement was applied to assess the CPA concentration in cryopreservation. The standard correlations between the CPA concentration and the electrical conductivity of the solutions (including CPA-NaCl-water ternary solutions and CPA-albumin-NaCl-water quaternary solutions) were experimentally obtained for a few mostly used CPAs, including dimethyl sulfoxide (DMSO or Me2SO), ethylene glycol (EG), and glycerol. Then, a novel "dilution-filtration" system with hollow fiber dialyzer was designed and applied to remove the CPA from the solutions effectively. Measurement of electrical conductivity was validated as a safer and easier way to online and real-time monitoring of CPA concentration in cell suspensions.

## **Author details**

## Adel El-Shahat

Address all correspondence to: aahmed@georgiasouthern.edu

Department of Electrical Engineering, Georgia Southern University, Georgia, USA

## **References**


## **Electrical Resistivity Sensing Methods and Implications**

Marios Sophocleous

**Author details**

4 Electrical Resistivity and Conductivity

Address all correspondence to: aahmed@georgiasouthern.edu

Journal of Geotechnical Engineering. 1996;**122** (5):397–406

thesis]. Montreal, Quebec, Canada: McGill University; 2010.

Thesis). Lanzhou, China: Lanzhou University; 2012.

Environmental Earth Sciences. 2009;**59**:1319–1326

network. Engineering Geology. 2008;**100**:142–145

of Physical Science. 2010;**5**:47–56

Department of Electrical Engineering, Georgia Southern University, Georgia, USA

[1] Abu-Hassanein ZS, Benson CH, Blotz LR. Electrical resistivity of compacted clays.

[2] Angelopoulos M, Pollard WH, Couture N. Integrated geophysical approach for the detection and assessment of ground ice at Parsons Lake, Northwest Territories [MSc

[3] Delaney AJ, Peapples PR, Arcone SA. Electrical resistivity of frozen and petroleum-contaminated fine-grained soil. Cold Regions Science and Technology. 2001;**32**(2):107–119

[4] Fortier R, LeBlanc AM, Allard M, Buteau S, Calmels F. Internal structure and conditions of permafrost mounds at Umiujaq in Nunavik, Canada, inferred from field investigation and electrical resistivity tomography. Canadian Journal of Earth Science.

[5] Hu ZG, Shan W (2011). Application of geological drilling combined with high-density resistance in island structure permafrost survey. The International conference on electronics, communications, and control, ICECC2011-proceedings: 1898–1904, IEEE

[6] Li L. Study on the characteristics of the electrical resistivity of saline soils (Master's

[7] Son Y, Oh M, Lee S. Estimation of soil weathering degree using electrical resistivity.

[8] Siddiqui FI, Osman SBABS. Simple and multiple regression models for relationship between electrical resistivity and various soil properties. Environmental Earth Sciences.

[9] Das SK, Basudhar PK. Prediction of residual friction angle of clays using artificial neural

[10] Ozcep F, Yildirim E, Tezel O, Asci M, Karabulut S. Correlation between electrical resistivity and soil-water content based artificial intelligent techniques. International Journal

Adel El-Shahat

**References**

2008;**45**(3):367–387

Publisher.

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67748

#### Abstract

This chapter discusses and explains the basic operating principles of various measuring methods of resistivity for materials in both liquid and solid phase. It provides explanations for two-, three-, and four-pole as well as toroidal resistivity cells. The van der Pauw technique is explored as a step-by-step procedure to estimate the resistivity of a material with no arbitrary shape. The special case of sheet material resistivity and resistance is explained in more detail, and equation for that special problem is simplified. It further provides information on common experimental errors and a short guideline to improve the reliability and accuracy of the measurements. The implications and challenges faced during resistivity measurements are explored and explained with ways to compensate for errors due to temperature and capacitance changes. In addition, the way to experimentally determine the cell constant of a cell is described and the necessity for calibration is clearly explained. It further provides information to overcome the standard problem of polarisation when the resistivity of solutions with high ionic content is investigated.

Keywords: resistivity sensors, resistivity measurement techniques, impedance, fringing, cell constant

## 1. Introduction

Electrical resistivity is defined as the ability of the material to resist the flow of electricity. Resistivity is calculated using Ohm's law when dealing with the material is homogeneous and isotropic. To provide a more accurate version of resistivity that can be applied for every material, the more general form of Ohm's law is used [1]:

$$E = \rho \mathbf{J} \tag{1}$$

In this equation, E is a vector that represents the electric field generated in the material (V/m), J is also a vector that represents the current density within the material (A/m�<sup>2</sup> ), and ρ is a tensor which is basically the proportionality coefficient (Ωm).

Eq. (1) is Ohm's law in a more general context where E and J are vectors, and ρ is a tensor. This indicates that the current does not necessarily flow in the direction of the applied electric field. If it is assumed that the sample is homogeneous, meaning that it has the same properties everywhere, and that the material is isotropic, meaning that the material has the same properties in all directions, then ρ becomes a scalar. This is not always a valid assumption though.

In this chapter, isotropic and homogeneous materials are assumed, so ρ is considered to be a scalar. Considering the bar-shaped sample in Figure 1:

The electric field (E) generated in the material is calculated by dividing the potential difference (V) between the two sides, by the distance (l) over which the voltage is applied [1]:

$$E = \frac{V}{l} \tag{2}$$

The current density J is defined as the current I(A) flowing through the material, divided by the cross-sectional area A(m<sup>2</sup> ) through which the current flows [1]:

$$J = \frac{I}{A} \tag{3}$$

Area (A) in Figure 1 is equal to the width w (m) times the height h (m). Combining the equations above, we get [1]:

$$V = \frac{I\rho l}{A} \tag{4}$$

If resistance is defined as:

$$R = \rho \frac{l}{A} \tag{5}$$

Figure 1. General two-point resistivity measuring technique [1].

Then, when Eqs. (4) and (5) are combined [1]:

Eq. (1) is Ohm's law in a more general context where E and J are vectors, and ρ is a tensor. This indicates that the current does not necessarily flow in the direction of the applied electric field. If it is assumed that the sample is homogeneous, meaning that it has the same properties everywhere, and that the material is isotropic, meaning that the material has the same properties in all directions, then ρ becomes a scalar. This is not always a valid assumption though.

In this chapter, isotropic and homogeneous materials are assumed, so ρ is considered to be a

The electric field (E) generated in the material is calculated by dividing the potential difference

<sup>E</sup> <sup>¼</sup> <sup>V</sup>

The current density J is defined as the current I(A) flowing through the material, divided by

Area (A) in Figure 1 is equal to the width w (m) times the height h (m). Combining the

<sup>V</sup> <sup>¼</sup> <sup>I</sup>ρ<sup>l</sup>

R ¼ ρ l

) through which the current flows [1]: <sup>J</sup> <sup>¼</sup> <sup>I</sup>

<sup>l</sup> (2)

<sup>A</sup> (3)

<sup>A</sup> (4)

<sup>A</sup> (5)

(V) between the two sides, by the distance (l) over which the voltage is applied [1]:

scalar. Considering the bar-shaped sample in Figure 1:

the cross-sectional area A(m<sup>2</sup>

6 Electrical Resistivity and Conductivity

equations above, we get [1]:

If resistance is defined as:

Figure 1. General two-point resistivity measuring technique [1].

$$V = IR \tag{6}$$

I is the current (A) flowing through the specific sample, V is the voltage (V) applied across this specific sample, and R is the resistance (Ω) of this specific sample.

Any changes in size and shape of the sample can cause changes in its total resistance, while those changes will not affect the resistivity of the sample since that is a property of the material alone. Conductivity (σ) is in principle the same property of the material, but it is calculated as the inverse of resistivity, and it is measured in Siemens per metre (S/m).

## 2. Resistivity measurement techniques

There are two main techniques to measure the resistivity of a material, either in liquid or in solid phase. The two techniques are the inductive and the contact-based methods.

#### 2.1. Inductive or toroidal resistivity

The toroidal resistivity cell is based on the principle of inducing a current from one coil to another. The level of the induced current will be proportional to the resistivity of the medium inserted within the coils (Figure 2).

The main advantage of toroidal conductivity cell is that the coils do not come in contact with the solution. They are usually surrounded by a polymeric material. This allows the use of the cell in media where direct contact will damage the cell. While this is an advantage, toroidal cells lack sensitivity due to the absence of direct contact. Furthermore, toroidal cells are typically larger and the solution current induced by the toroid occupies a volume around the sensor. Hereafter, toroidal cells need more surrounding space and therefore are mounted in larger pipes [2].

Figure 2. Toroidal resistivity cell [2].

## 2.2. Contacting resistivity

Contacting resistivity cells use two metals or graphite electrodes in contact with the sample, whether that is in liquid or solid phase. An AC current is applied to the electrodes by the electronic instrumentation, and the resulting AC voltage is recorded. This technique can measure down to pure water resistivity. The main downside of this cell type is that the cell is susceptible to coating and corrosion, which severely decreases the performance of the cell. In cases where the sample is a solution of high ionic content, polarisation effects will arise and result in non-linearity of measurements [2]. Further explanation on polarisation is provided later in this chapter.

## 2.2.1. Two-pole cells

In the standard two electrodes cell, an alternating current is applied between the two poles using a current source, while the resulting voltage is recorded (Figure 3).

Knowing the voltage and current across the two electrodes at low frequencies where the capacitance between the electrodes has no effect on the measurement, the resistance between the two electrodes can be calculated. Although the calculated resistance includes the resistance of the electrodes as well, in cases where the sample is a solution, its resistance is much higher than the resistance of the electrodes, and therefore, it can be neglected. Furthermore, in the attempt to measure the sample only, the impedance caused by polarisation of the electrodes and the field effects, interfere with the measurement, and both impedances are measured.

## 2.2.2. Three-pole cells

The three-pole cell is not as popular now as it has been replaced by the four-pole one. The purpose for adding the third pole was to direct and constrain the electric field lines. That minimises the effect of having special field fluctuations and eliminates the influences of external factors, such as the size of the beaker and the distance between the beaker walls and the poles, on the resistivity measurements. It provides more reproducible measurements when determining the cell constant and therefore more reproducible results.

Figure 3. General electronic configuration of a two-pole resistivity cell [2].

## 2.2.3. Four-pole cells

2.2. Contacting resistivity

8 Electrical Resistivity and Conductivity

later in this chapter.

2.2.1. Two-pole cells

measured.

2.2.2. Three-pole cells

Contacting resistivity cells use two metals or graphite electrodes in contact with the sample, whether that is in liquid or solid phase. An AC current is applied to the electrodes by the electronic instrumentation, and the resulting AC voltage is recorded. This technique can measure down to pure water resistivity. The main downside of this cell type is that the cell is susceptible to coating and corrosion, which severely decreases the performance of the cell. In cases where the sample is a solution of high ionic content, polarisation effects will arise and result in non-linearity of measurements [2]. Further explanation on polarisation is provided

In the standard two electrodes cell, an alternating current is applied between the two poles

Knowing the voltage and current across the two electrodes at low frequencies where the capacitance between the electrodes has no effect on the measurement, the resistance between the two electrodes can be calculated. Although the calculated resistance includes the resistance of the electrodes as well, in cases where the sample is a solution, its resistance is much higher than the resistance of the electrodes, and therefore, it can be neglected. Furthermore, in the attempt to measure the sample only, the impedance caused by polarisation of the electrodes and the field effects, interfere with the measurement, and both impedances are

The three-pole cell is not as popular now as it has been replaced by the four-pole one. The purpose for adding the third pole was to direct and constrain the electric field lines. That minimises the effect of having special field fluctuations and eliminates the influences of external factors, such as the size of the beaker and the distance between the beaker walls and the poles, on the resistivity measurements. It provides more reproducible measurements

when determining the cell constant and therefore more reproducible results.

Figure 3. General electronic configuration of a two-pole resistivity cell [2].

using a current source, while the resulting voltage is recorded (Figure 3).

In a four-pole cell (Figure 4), the current is applied to the outer electrodes in such a way that a constant potential difference is maintained between the inner electrodes. As this voltage measurement takes place with a negligible current, these two electrodes are not polarised, and therefore, their resistance is effectively zero. There are cases where the current applied to the outer electrodes is kept constant, and the voltage is measured between the two inner poles. In that case, the resistivity is directly proportional to the voltage measured. The four-pole method is usually used within an insulating tube. This technique minimises the beaker field effect because the electric field is constrained within the tube walls and because the volume of the material is well defined. Simultaneously, this eliminated the problem of the electric field being affected by the beaker walls. Therefore, the position of the cell in the beaker becomes irrelevant.

Ideally, electrodes placed at specific distances with a known effective surface area. The distances between the electrodes can define the cell constant based on the electric fields built up as shown in Figure 5:

The cell constant can be calculated using Poisson's equation:

$$
\Delta V = \frac{I\rho}{2\pi} \left[ \left( \frac{1}{r\_1} - \frac{1}{r\_2} \right) - \left( \frac{1}{r\_3} - \frac{1}{r\_4} \right) \right] \tag{7}
$$

Some specific approaches are shown in Figure 6 where d is the distance between the electrodes (m) and n is an integer to show that it is a multiple of d.

Figure 4. Four-pole resistivity cell [2].

Figure 5. General electrode positioning for the four-pole resistivity cell [3].

Figure 6. Typical electrode configurations for four-pole resistivity [3].

#### 2.2.3.1. Sheet resistance measurements

There are cases where the resistance of sheets or films of various materials is of interest. In those cases, the sheet resistance is usually used to compare between different thin films of materials. The easiest way to measure sheet resistance is to make the material into a square film having equal length and width. Therefore, just like the bar sample in Figure 1, the resistivity can be calculated by [1]:

$$
\rho \equiv \frac{Vwh}{ll} \tag{8}
$$

where ρ is the sample resistivity (Wm), V is the voltage measured by the voltmeter (V), w is the width of the sample (m), h is the thickness of the sample (m), I is the current the ammeter measures flowing through the sample (A), and l is the length of the film (m).

When the width is equal to the length, then Eq. (8) becomes [1]:

$$
\rho\_{sq} \equiv \frac{Vh}{I} \tag{9}
$$

The "sheet resistivity" is the resistivity of a square film of material and is represented by the symbol ρsq. The "sheet resistance" Rs is generally defined by [1]:

$$R\_s \equiv R\_{sq} = \frac{V}{I} \tag{10}$$

where V is the voltage measured by the voltmeter (V) and I is the current the ammeter measures flowing through the sample (A).

General units of sheet resistance are ohms (Ω), but in order to distinguish between resistance and sheet resistance, people most commonly use (Ω per square) or (Ω/square). In reality, sheet resistance is exactly the same as the resistance of a square film of a material. What makes sheet resistance interesting is that it is independent of the size of the square and the thickness of the sheet is not required to measure sheet resistance.

It is also a common technique to measure the resistance of films of arbitrary size and shape. This is usually done by pressing four collinear and equally spaced contacts into the film. The width and length of those contacts must be much greater than the distance between the contacts. In this case, sheet resistance can be calculated using [3]:

$$R\_s = 4.532 \frac{V}{I} \tag{11}$$

where V is the voltage measured across the two inner contacts (V), and I is the current applied through the two outer contacts (A).

It is understood that it will be very difficult to always fulfil these requirements for the contact size and distance between them. Under those circumstances, geometric correction factors are used to compensate in order to accurately measure the sheet resistance. These correction factors are available for the most commonly faced sample geometries [3].

#### 2.2.3.2. Van der Pauw technique

2.2.3.1. Sheet resistance measurements

can be calculated by [1]:

10 Electrical Resistivity and Conductivity

There are cases where the resistance of sheets or films of various materials is of interest. In those cases, the sheet resistance is usually used to compare between different thin films of materials. The easiest way to measure sheet resistance is to make the material into a square film having equal length and width. Therefore, just like the bar sample in Figure 1, the resistivity

<sup>ρ</sup> Vwh

where ρ is the sample resistivity (Wm), V is the voltage measured by the voltmeter (V), w is the width of the sample (m), h is the thickness of the sample (m), I is the current the ammeter

<sup>ρ</sup>sq Vh

measures flowing through the sample (A), and l is the length of the film (m).

When the width is equal to the length, then Eq. (8) becomes [1]:

Figure 5. General electrode positioning for the four-pole resistivity cell [3].

Figure 6. Typical electrode configurations for four-pole resistivity [3].

Il (8)

<sup>I</sup> (9)

Ideally, samples can have or can be made into convenient shapes to allow the use of the fourpole cell to measure their resistivity. There are also cases that the samples are of arbitrary shape and the sample might be damaged in the attempt to make it into the desired shape. Therefore, another technique called van der Pauw technique [1] is used. There are five conditions to be fulfilled in order to correctly use that technique:


When the samples are very small, the dimensional constrains for the van der Pauw method are not feasible, and therefore, some compensation is required.

The general step-by-step procedure for doing a van der Pauw measurement is as follows:


$$\rho\_{\chi} = \frac{\pi d (R\_H + R\_L) f(R\_H / R\_L)}{\ln 4} \tag{12}$$

where ρ<sup>x</sup> is the resistivity (Wm), d is the thickness of the sample (m), resistances RH and RL are measured in W, and ln 4 is approximately 1.3863.

It is not necessary to measure the width or length of the sample.

5. Alter the contact points to measure R43,12 and R14,23. And then repeat steps 3 and 4 to calculate ρ<sup>y</sup> using these new values for RH and RL. If the two resistivities ρ<sup>x</sup> and ρ<sup>y</sup> are not within 10% of each other, then either the contacts are bad or the sample is non-uniform. Try making using new contacts. If the two resistivities are within 10% of each other, the best estimate of the material resistivity ρ is the average:

$$
\rho = \frac{(\rho\_x + \rho\_y)}{2} \tag{13}
$$

The function f(RH/RL) is defined by the transcendental equation:

$$f(\mathbf{R}\_H/\mathbf{R}\_L) = \frac{-\ln 4 (\mathbf{R}\_H/\mathbf{R}\_L)}{[1 + (\mathbf{R}\_H/\mathbf{R}\_L)\ln\{1 - 4^{-\left[(1 + R\_H/R\_L)f\right]^{-1}}\}]} \tag{14}$$

#### 2.2.4. Platinised cells

Platinised cells are most commonly used for measuring the resistivity of solutions. In solutions, the polarisation effect is of high importance due to the accumulation of ions near the surface of the electrodes. One way to minimise the polarisation effect is to decrease the current density. Current density can be decreased by increasing the electrochemical surface area of the electrodes. The most convenient and common way to do that is to cover the electrodes with platinum black. Platinised cells are very powerful because their cell constant is linear over 2–3 decades of resistivity, while without platinum black it is only linear for approximately one decade. If platinum black is damaged or scratched, it will alter the cell constant and the properties of the cell. A minor shortcoming of platinised cells is that the cell constant tends to drift faster when compared with non-platinised cells. It is advisable to use platinum black for measurements in non-viscous samples, without suspensions and frequent calibrations.

## 2.2.5. Flow through cells

When the samples are very small, the dimensional constrains for the van der Pauw method are

1. Define resistance Rab,cd = Vcd/Iab, where Vcd = Vc – Vd and is the voltage between points c

2. Measure the resistances of four points on the sample (R21,34 and R32,41). Define RH as the

<sup>ρ</sup><sup>x</sup> <sup>¼</sup> <sup>π</sup>dðRH <sup>þ</sup> RLÞfðRH=RL<sup>Þ</sup>

5. Alter the contact points to measure R43,12 and R14,23. And then repeat steps 3 and 4 to calculate ρ<sup>y</sup> using these new values for RH and RL. If the two resistivities ρ<sup>x</sup> and ρ<sup>y</sup> are not within 10% of each other, then either the contacts are bad or the sample is non-uniform. Try making using new contacts. If the two resistivities are within 10% of each other, the

<sup>ρ</sup> <sup>¼</sup> <sup>ð</sup>ρ<sup>x</sup> <sup>þ</sup> <sup>ρ</sup>y<sup>Þ</sup>

Platinised cells are most commonly used for measuring the resistivity of solutions. In solutions, the polarisation effect is of high importance due to the accumulation of ions near the surface of the electrodes. One way to minimise the polarisation effect is to decrease the current density. Current density can be decreased by increasing the electrochemical surface area of the electrodes. The most convenient and common way to do that is to cover the electrodes with platinum black. Platinised cells are very powerful because their cell constant

<sup>½</sup><sup>1</sup> þ ðRH=RLÞlnf<sup>1</sup> � <sup>4</sup>�½ð1þRH=RLÞ<sup>f</sup> �

where ρ<sup>x</sup> is the resistivity (Wm), d is the thickness of the sample (m), resistances RH and RL

ln4 (12)

<sup>2</sup> (13)

g� (14)

�1

higher of these two resistances and RL as the lower of these two resistances.

The general step-by-step procedure for doing a van der Pauw measurement is as follows:

and d, while Iab is the current flowing from contact a to contact b.

not feasible, and therefore, some compensation is required.

3. Find ratio RH/RL and solve the function f(RH/RL).

are measured in W, and ln 4 is approximately 1.3863.

best estimate of the material resistivity ρ is the average:

It is not necessary to measure the width or length of the sample.

The function f(RH/RL) is defined by the transcendental equation:

<sup>f</sup>ðRH=RLÞ ¼ �ln4ðRH=RL<sup>Þ</sup>

4. Calculate the resistivity ρ<sup>x</sup> using:

12 Electrical Resistivity and Conductivity

2.2.4. Platinised cells

There are cases when the real-time resistivity of a small volume of flowing liquid is of interest. Flow through type resistivity cells are designed for those cases. These cells are customised for this kind of measurements but show several disadvantages. The most common problem with flow through cells is that they need a closed liquid system protected from air. In particular, for pure water resistivity measurements, it is very important to use a flow cell since contact with air will dissolve carbon dioxide and for carbonate ions changing the resistivity of the sample [4].

## 2.3. Cell-type comparisons and ranges

A short comparison between the classical two-pole resistivity cell and the more advanced fourpole resistivity cell is shown in Table 1:

Different conductivity cells have different properties, and the cell type must be chosen depending on the application. The measurement range over which the cell stays linear gets broader as the number of poles increases. Platinised poles also contribute to increasing the measurement span in which the cell is linear (Figure 7).


Table 1. Comparison between two-pole and four-pole resistivity cells [2].

Figure 7. Operational ranges of several resistivity cell types [2].

## 3. Measurement implications

## 3.1. Calibration and cell constant calculation

Calibration of resistivity cells is important because it calculates the correct value of the cell constant in your working conditions. The cell constant is a factor that is used to convert the measured resistance to resistivity.

The cell constant is calculated by dividing the distance between the two poles by the crosssectional area of the poles. Therefore, the cell constant is ideally, determined by the geometry of the cell. In reality, due to the fact that the cross-sectional area of the poles is not the actual electrochemical area (in case of liquids), the cell constant can only be measured experimentally using samples of known resistivities. In cases where the sample is a solution, cell constant can change due to changes on the electrodes. Those changes are caused due to contamination or due to physical-chemical alteration in case of platinised cells.

If high precision measurements are required, the cell constant needs to be calibrated often in samples of known resistivity at the same temperature as the actual measurements. Furthermore, when using the two-pole cells, the determination of the cell constant must be done at close resistivities to the resistivity of the sample since the cell constant is also resistivity dependent.

When using a two-pole cell, the choice of the cell constant value varies with the linear measurement range of the cell selected. Typically, a cell with K = 0.1 cm<sup>1</sup> is chosen for pure water measurements, while, for environmental water and industrial solutions, a cell with K of 0.4–1 cm<sup>1</sup> is used. Cells with up to K = 10 cm<sup>1</sup> are best for very low resistivity samples. In the case of a four-pole cell, the cell constant value is generally included in the range from 0.5 to 1.5 cm<sup>1</sup> [5].

## 3.2. Polarisation

When attempting to measure the resistivities of solutions, more complications arise. Applying a potential difference or an electrical current to the electrodes of the cell, depending on the polarity, ions will be attracted or repelled from the electrodes. That ionic movement causes and accumulation of ions and therefore charge at the electrodes' surface which can further cause the initiation of chemical reactions (Figure 8). Subsequently, due to the accumulation of charge on the electrodes' surface, the actual resistance of the electrodes changes which is called polarisation. Polarisation can cause error on the measurements as it is a parasitic component to the solution resistance.

## 3.2.1. Preventing polarisation

3. Measurement implications

14 Electrical Resistivity and Conductivity

measured resistance to resistivity.

3.1. Calibration and cell constant calculation

Figure 7. Operational ranges of several resistivity cell types [2].

due to physical-chemical alteration in case of platinised cells.

Calibration of resistivity cells is important because it calculates the correct value of the cell constant in your working conditions. The cell constant is a factor that is used to convert the

The cell constant is calculated by dividing the distance between the two poles by the crosssectional area of the poles. Therefore, the cell constant is ideally, determined by the geometry of the cell. In reality, due to the fact that the cross-sectional area of the poles is not the actual electrochemical area (in case of liquids), the cell constant can only be measured experimentally using samples of known resistivities. In cases where the sample is a solution, cell constant can change due to changes on the electrodes. Those changes are caused due to contamination or

If high precision measurements are required, the cell constant needs to be calibrated often in samples of known resistivity at the same temperature as the actual measurements. Furthermore, when using the two-pole cells, the determination of the cell constant must be done at close resistivities to the resistivity of the sample since the cell constant is also resistivity dependent.

When using a two-pole cell, the choice of the cell constant value varies with the linear measurement range of the cell selected. Typically, a cell with K = 0.1 cm<sup>1</sup> is chosen for pure water measurements, while, for environmental water and industrial solutions, a cell with K of 0.4–1 cm<sup>1</sup> is used. Cells with up to K = 10 cm<sup>1</sup> are best for very low resistivity samples. There are several precautions that can be used to minimise or eliminate polarisation:


Figure 8. Polarisation effect due to ions contaminating the electrodes [1].

If that electrochemical surface area is increased, while the current is constant, it means that the current density on the electrode will decrease and subsequently the polarisation effect will also decrease. A common technique is to use platinum black to cover the electrodes' surfaces because it has a very high electrochemical surface area.


#### 3.3. Geometry and frequency

Different geometries can affect error levels. The most common errors in resistivity measurements are those produced by field effects. A theoretical assumption has been made when designing resistivity cells that the electric field lines are straight lines from one pole to the other and that they are not affected by surrounding objects. In reality, although the majority of the field lines do form in straight lines, some of them form curves (Figure 9). These field lines can affect the measurement especially when another object or another field interferes with them.

Three and four-pole conductivity cells are designed to minimise this effect. There is still some field effect present for the four electrodes cell due to the fact that when field lines do not flow directly to the other electrode, the distance travelled by the current is different from the distance between the two electrodes. That can have a major effect on the cell constant.

In most conductivity metres, the frequency is automatically increased with decreasing resistance of the sample, to avoid polarisation errors at low resistivities [5].

Figure 9. Field lines between the two electrodes [2].

## 3.4. Cable resistance and capacitance

If that electrochemical surface area is increased, while the current is constant, it means that the current density on the electrode will decrease and subsequently the polarisation effect will also decrease. A common technique is to use platinum black to cover the electrodes'

• Since the resistance of the electrodes has no effect on the measurements when using a four-pole cell, it means that polarisation will have no influence on the measurements of

• When using a two-pole cell, deposits on the electrode's surface can have a similar effect to polarisation since the electrodes' resistance changes, while in the case of the four-pole

Different geometries can affect error levels. The most common errors in resistivity measurements are those produced by field effects. A theoretical assumption has been made when designing resistivity cells that the electric field lines are straight lines from one pole to the other and that they are not affected by surrounding objects. In reality, although the majority of the field lines do form in straight lines, some of them form curves (Figure 9). These field lines can affect the

Three and four-pole conductivity cells are designed to minimise this effect. There is still some field effect present for the four electrodes cell due to the fact that when field lines do not flow directly to the other electrode, the distance travelled by the current is different from the

In most conductivity metres, the frequency is automatically increased with decreasing resis-

measurement especially when another object or another field interferes with them.

distance between the two electrodes. That can have a major effect on the cell constant.

tance of the sample, to avoid polarisation errors at low resistivities [5].

surfaces because it has a very high electrochemical surface area.

the four-pole cell.

16 Electrical Resistivity and Conductivity

3.3. Geometry and frequency

cells, contamination has no effect [6].

Figure 9. Field lines between the two electrodes [2].

Cables are made of conductors, meaning that the material has very low resistivities. The total resistance of a cable is proportional to its length. The resistance of the cable can induce an error on the readings of the cell, and therefore, it should be compensated for accurate measurements. The cable resistance becomes significant when the resistance of the sample is lower than (approximately 50 Ω) and when using the two- or three-pole techniques.

For four-pole cells, the cable resistance has no influence. A shielded cable of a given length has a given capacity. When the measured resistance is high, the cable capacitance is not negligible and must be taken into account.

Compensate the cable capacitance when:


## 3.5. Temperature effect

Resistivity measurements are temperature dependent; if the temperature increases, resistivity decreases. The concept of reference temperature was introduced to allow the comparison of resistivity results obtained at different temperature. The reference temperature is usually 20 or 25C. Generally, resistivity metres measure the resistance of the cell and calculate the resistivity by knowing their cell constant. They also measure the temperature of the resistivity measurement, and they use a function to translate the measured resistivity to reference resistivity. Reference resistivity is the resistivity of the sample at a reference temperature. Therefore, resistivity measurements are often associated with temperature sensors for temperature correction to improve resistivity calculations.

There are three common temperature correction methods:


When the measurement requires very high precision and accuracy, the measurements are taken in a temperature controlled environment to ensure temperature stability and for a more accurate determination of the cell constant at that temperature [3].

#### 3.5.1. Linear temperature correction

When measuring resistivity of solutions with medium to low resistivity, the linear temperature correction is used. Linear temperature correction is used, for example, for saline solutions, acids, and leaching solutions. The conductivity of the solution can be calculated by [7]:

$$\kappa\_{T\_{\rm ref}} = \frac{100}{100 + \Theta(T - T\_{\rm ref})} \kappa\_T \tag{15}$$

where κTref is the conductivity at reference temperature, κ<sup>T</sup> is the conductivity at the temperature of the measurement, Tref is the reference temperature, T is the sample temperature, and θ is the temperature coefficient (%/�C). If resistivity is required, then the inverse of the calculated conductivity is the resistivity.

The linear correction method is useful and correct only when the reference temperature and the temperature of the measurement are close. The risk of error for this method is directly proportional to the difference between the reference temperature and the temperature of the measurement.

In order to calculate the temperature coefficient, the resistivity of a sample at temperature T<sup>1</sup> close to Tref and another temperature T<sup>2</sup> is measured. Then, the temperature coefficient is calculated by using the following equation [7]:

$$\theta = \frac{(\kappa\_{T2} - \kappa\_{T1})100}{(T\_2 - T\_1)\kappa\_{T1}} \tag{16}$$

T<sup>2</sup> must be a typical sample temperature, usually room temperature, and should be approximately 10�C different from T1. Indicative ranges for the temperature coefficients of commonly used electrolytes are provided below [2]:

Acids: 1.0–1.6%/�C Bases: 1.8–2.2%/�C Salts: 2.2–3.0%/�C Drinking water: 2.0%/�C Ultrapure water: 5.2%/�C

#### 3.5.2. Non-linear temperature correction

In frequent cases of natural waters, for example, ground water, surface water, drinking water, and waste water, the classical linear temperature correction function is not suitable. The reason is that the temperature dependency of the conductivity for these solutions in nonlinear and can only be defined by a 4th degree polynomial. The basic idea for this correction method is to correct the measured conductivity from the measurement temperature to 25�C to give K<sup>25</sup> [7].

$$K\_{25} = f\_{25}(T)K\_T \tag{17}$$

f25(T) is the temperature correction factor used for the conversion of conductivity values of natural water from T to 25�C. This function is a 4th degree polynomial equation, is provided by ISO/DIN7888 standard, and is valid for measurements between 0 and 35.9C [7].

## 4. Improving measurement reliability

In order to improve the reliability and accuracy of the measurements, the source of errors must be identified. Several common experimental errors are listed below, and a guide to overcome some of the common challenges of resistivity measurements is provided in detail.

## 4.1. Common experimental errors

<sup>κ</sup>Tref <sup>¼</sup> <sup>100</sup>

conductivity is the resistivity.

18 Electrical Resistivity and Conductivity

calculated by using the following equation [7]:

used electrolytes are provided below [2]:

3.5.2. Non-linear temperature correction

measurement.

Acids: 1.0–1.6%/�C Bases: 1.8–2.2%/�C Salts: 2.2–3.0%/�C

Drinking water: 2.0%/�C Ultrapure water: 5.2%/�C

to give K<sup>25</sup> [7].

100 þ θðT � TrefÞ

where κTref is the conductivity at reference temperature, κ<sup>T</sup> is the conductivity at the temperature of the measurement, Tref is the reference temperature, T is the sample temperature, and θ is the temperature coefficient (%/�C). If resistivity is required, then the inverse of the calculated

The linear correction method is useful and correct only when the reference temperature and the temperature of the measurement are close. The risk of error for this method is directly proportional to the difference between the reference temperature and the temperature of the

In order to calculate the temperature coefficient, the resistivity of a sample at temperature T<sup>1</sup> close to Tref and another temperature T<sup>2</sup> is measured. Then, the temperature coefficient is

> <sup>θ</sup> <sup>¼</sup> <sup>ð</sup>κT<sup>2</sup> � <sup>κ</sup>T1Þ<sup>100</sup> ðT<sup>2</sup> � T1ÞκT<sup>1</sup>

T<sup>2</sup> must be a typical sample temperature, usually room temperature, and should be approximately 10�C different from T1. Indicative ranges for the temperature coefficients of commonly

In frequent cases of natural waters, for example, ground water, surface water, drinking water, and waste water, the classical linear temperature correction function is not suitable. The reason is that the temperature dependency of the conductivity for these solutions in nonlinear and can only be defined by a 4th degree polynomial. The basic idea for this correction method is to correct the measured conductivity from the measurement temperature to 25�C

f25(T) is the temperature correction factor used for the conversion of conductivity values of natural water from T to 25�C. This function is a 4th degree polynomial equation,

K<sup>25</sup> ¼ f <sup>25</sup>ðTÞKT (17)

κ<sup>T</sup> (15)

(16)

There are many experimental dangers to avoid when making resistivity measurements. The most common sources of error arise from doing a two-point measurement on a material that has any of the contact problems discussed earlier. Therefore, it is logical to do four-point measurements whenever possible. This section describes experimental practises to avoid errors in measuring resistivity [1, 7]:


#### 4.2. Guidelines for improved resistivity measurements

Other than the common experimental errors and some ways to prevent them, further measurement improvements, can be achieved when following these simple rules [2, 6, 7]:

## 4.2.1. Frequent cell constant calibration

The cell constant is the most important component for accurate resistivity measurements. Although when measuring resistivities of solids, the cell constant is fairly stable, when measuring resistivities of liquids, it can be vital. It is ideal to determine the cell constant of the cell right before any measurement, but a frequent cell constant calibration is advisable. In particular, in the case of platinised cells, calibration should be performed even more frequently due to the elevated risk of contamination and physical or chemical alteration of the platinum layer.

## 4.2.2. Controlling the temperature and maintaining homogeneity

For high accuracy and low resistivity measurements, it is required to have a stable temperature of the sample and the cell itself. If the measurements will be thermostated, then the calibration should be made at the same temperature as the measurements. Furthermore, homogeneity of the sample is critical. All of the equations used for the calculation of resistivity assume homogeneity. In particular, for solution, it is highly advised to stir the solution continuously during both the calibration and the measurement. For resistivity measurements, the resistivity reading can be expressed either at the measuring temperature or at a reference temperature using the pre-mentioned temperature correction factors.

## 4.2.3. Cell positioning

6. An obvious but not trivial point is to ensure the circuit integrity. The most usual cause of meaningless results is when the circuit's integrity is violated. For correct measurements, in any kind of cell, the electrodes must be completely independent of each other and the only thing connecting them must be the sample under investigation. Even in the four-pole cells, all the electrodes should be checked for short circuits before using them for measurements. Any remarkably low resistance value measured between the electrodes should

7. The applied voltage or current can cause self-heating of the material or even the cell itself, which can change the resistivity measurements. To avoid this problem, use as low current as possible while the voltage is still readable on the metre or use a temperature sensor on

8. Depending on the material of the sample, Ohm's law is not always obeyed. There are non-Ohmic materials that change their resistance depending on the applied voltage or current across them. Therefore, before making accurate measurements, the linearity between voltage and current across the sample should be investigated. It is advised to apply voltages or current both above and below the measuring values and to ensure that resistivity measurements will

9. It is good practise to always test the equipment before performing any measurements. Sometimes, voltmeter leads age and their contacts with the voltmeter are damaged. In those cases, the voltmeter gives random values since it operates as an open circuit. The best way to check for open circuits on the voltmeter is to drop the current input to zero and check if the voltage also drops to zero. If it does not fall to zero and gives a random

10. A further check of the equipment is to reverse the leads on the voltmeter and measure the resistance again. The two readings are within 10% of each other; then, the readings are considered as valid. It has to be noted and understood that in this case the current is flowing between the two inner electrodes (in the case of four-pole cell) and that the

11. The resistivity of some materials can light dependent. This is particularly a problem with semiconductors. If there is a chance of this, try blocking all light from the sample during

Other than the common experimental errors and some ways to prevent them, further mea-

The cell constant is the most important component for accurate resistivity measurements. Although when measuring resistivities of solids, the cell constant is fairly stable, when

surement improvements, can be achieved when following these simple rules [2, 6, 7]:

be made within the linear region of the voltage current graph.

voltage is measured between the two outer electrodes.

4.2. Guidelines for improved resistivity measurements

number, then that indicates open circuit.

indicate short circuit.

20 Electrical Resistivity and Conductivity

the resistivity cell itself.

measurement.

4.2.1. Frequent cell constant calibration

Some cell types can be greatly affected by the surrounding materials. In solutions, for example, the distance between the cell and the wall can be a major source of error since the electric field is bounded and altered by the beaker walls. Two-pole cells should always be placed at the centre of the beaker, and all electrodes, no matter what the cell type is, must be completely immersed in the sample.

#### 4.2.4. High resistivity measurements


#### 4.2.5. Low resistivity measurements


## Author details

Marios Sophocleous

Address all correspondence to: sophocleous.marios@ucy.ac.cy

Holistic Electronics Research Lab, Department of Electrical & Computer Engineering, University of Cyprus, Nicosia, Cyprus

## References

