Chronicling the early history of thermoelectricity from its discovery to modern times, this text features a considerable amount of experimental data and discusses these findings at length wherever they bear a particular relevance to theory. The author, a well-known authority in this field, draws heavily from his own work on thermoelectrical phenomena as they are observed in the study of metals. Numerous illustrative figures appear throughout the text.
Chronicling the early history of thermoelectricity from its discovery to modern times, this text features a considerable amount of experimental data and discusses these findings at length wherever they bear a particular relevance to theory. The author, a well-known authority in this field, draws heavily from his own work on thermoelectrical phenomena as they are observed in the study of metals. Numerous illustrative figures appear throughout the text.
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Chronicling the early history of thermoelectricity from its discovery to modern times, this text features a considerable amount of experimental data and discusses these findings at length wherever they bear a particular relevance to theory. The author, a well-known authority in this field, draws heavily from his own work on thermoelectrical phenomena as they are observed in the study of metals. Numerous illustrative figures appear throughout the text.
Product Details
| ISBN-13: | 9780486174389 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 02/21/2013 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 144 |
| File size: | 4 MB |
Read an Excerpt
Thermoelectricity
An Introduction to the Principles
By D. K. C. MacDonald
Dover Publications, Inc.
Copyright © 2006 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-17438-9
CHAPTER 1
General survey
1.1 INTRODUCTION
There are two very basic methods for investigating the properties of electrical conductors. We may apply an electric field, a temperature gradient, or both, and observe what occurs. If we apply only the electric field, then an electric current will result, and the ratio of current density to electric field when no temperature gradient is present defines the electrical conductivity (σ) of the material. On the other hand, we may apply a temperature gradient and measure the flow of heat which results when no electric current is allowed to flow (usually by insulating the circuit somewhere); under these conditions the ratio of heat flow per unit area to the temperature gradient defines the thermal conductivity (κ) of the material. These two concepts are very familiar ones, and there seems no difficulty in recognizing that both σ and κ are, generally speaking, bulk properties of any individual substance. Correspondingly, there is no essential difficulty in making experimental arrangements to measure σ and κ so that we may determine, for example, the electrical and thermal conductivity of copper, silver, or sodium.
On the experimental side electrical conductivity has been studied very intensively for about a hundred years, and more recently, experiments at very low temperatures (down to the order of 1° K—liquid helium boils normally at 4.2° K) have proved extremely fruitful in increasing our understanding of many aspects of electrical conduction and the properties of conductors in general. The interest in thermal conductivity was at first less intense (partly because it is rather more difficult to measure with accuracy), but within the last twenty years or so interest in thermal conductivity has increased very much. Furthermore, for insulating materials thermal conductivity remains a useful and informative fundamental quantity to measure, whereas, of course, in any ideal insulator the electrical conductivity becomes zero.
There is, however, a third phenomenon which, at least until rather recently, has received less detailed attention. This is thermoelectricity. When we apply a temperature gradient to a conductor without an electric field, this of itself tends to produce not only an energy flow (i.e., essentially the heat flow) but also an actual electric current, which is a thermoelectric current. But in order to observe an electric current one must have a closed circuit of some kind. If we make a symmetrical circuit from a uniform piece of the same metal, then it is obvious by symmetry (see Fig. 1a) that no net electric current can flow. Looking at Fig. 1a we may say that the two "arms" of the circuit between the temperatures T1 and T2 (which determine the temperature gradient) are the same and therefore "cancel out" each other. In order to observe thermoelectricity, it is in fact necessary to have a circuit composed of two different materials (or perhaps the same material in two different states, for example, one under strain, the other not), and we can then measure the net difference between their thermoelectric properties.
It is the fact that a circuit involving two different materials must be used to observe a thermoelectric current that has led in the past to a great deal of confusion and misunderstanding about thermoelectric effects. Therefore, let us now state categorically that the thermoelectric properties of a conductor are in general just as much bulk properties as are the electrical and thermal conductivities mentioned above. It may then be asked how it is possible to separate out the individual (or "absolute") thermoelectric properties of any given conductor, if it is always necessary to deal with at least two conducting materials in order to observe a thermoelectric current.
It turns out, as we shall see later, that in principle it is always possible to derive absolute thermoelectric properties of a single conductor by starting from measurements of the so-called Thomson heat of one of them in any given circuit. There is, however, a further very useful fact; it appears that a superconductor below its transition temperature (that is, where it becomes superconducting) shows no thermoelectric effects at all. (This conclusion can be argued theoretically from a fairly general model of a superconductor, but it is perhaps better at present to regard it as a well-established experimental result.) Consequently, if we make up a circuit in which one arm is the metal in which we are interested and the other is a superconductor below its transition temperature, we can measure readily and directly the absolute thermoelectric properties of the conductor in which we are interested. This direct method is naturally limited to temperature regions where a superconductor may be found. At present the superconductor with the highest known transition temperature is the compound Nb3Sn which is superconducting below about 18° K. For very low temperature work Lead (having a transition temperature of about 7° K) is often used as the superconducting metal, since Lead wires can readily be made and easily annealed. This is doubly convenient because Lead has also been chosen for a number of reasons as the metal for which an absolute scale of thermoelectricity has been derived up to quite high temperatures (about 400° K; Lead melts at 327° C) from the Thomson heat. This was first done in 1931 by Borelius and his co-workers (cf. Borelius, 1935); more recently W. B. Pearson and colleagues (Pearson and Templeton, 1955; Christian et al., 1958) have improved the accuracy of this absolute thermoelectric scale of Lead below about 20° K. This means that we are in a position to measure readily the absolute thermoelectric properties of any conductor by simply making a circuit with Lead as the other arm; we subtract from our measurements the absolute thermoelectric data appropriate for Lead, and are left with the required data for the substance in which we are interested.
1.2 SEEBECK AND PELTIER
Historically, the Seebeck and Peltier effects were both discovered early in the last century. If one makes a circuit of two metals as in Fig. 1a with a temperature difference applied as shown then in general a thermoelectric current will flow in the circuit. The actual magnitude of the current which results would depend on the temperature difference to be sure, but also on a number of other factors, in particular, on the actual resistance (and hence the specific dimensions) of the conductors concerned. Alternatively, and preferably, we may consider an open circuit as indicated in Fig. 1b and in that case a potential difference will appear between the terminals which will depend on the temperatures at the ends of the couple but not on the shape or dimensions of the conductors themselves. That is to say, the thermocouple is a source of electromotive force, say E12, which is found to be generally a function only of the temperatures at the ends of the couple and of the two particular conducting materials involved. It is not, however, dependent in principle on the actual size or dimensions of the particular conductors making up the circuit. The thermoelectric potential difference is often known as the Seebeck potential in honor of the man to whom the discovery is attributed.
If, as in Fig. 1b, a small temperature difference, ΔT, is applied to the couple, then the derivative, ≈ ΔV12/ΔT, defines the thermoelectric power of the thermocouple concerned. If the observed potential difference, ΔV12, has the polarity shown in
[ILLUSTRATION OMITTED]
Fig. 1b, then we say that the absolute thermoelectric power (S1) of conductor 1 is positive with respect to (i.e., exceeds that of) conductor 2, i.e., S1-S2 [equivalent to] (dV12/dT) > 0. This, of course, refers to the temperature range concerned since the sign of the thermoelectric power of many materials depends on temperature.
It was also discovered by Peltier (a French watchmaker turned physicist) that if an electric current passes from one substance to another as in Fig. 2, then heat may either be given out or absorbed in the junction region, depending on the direction of the current flow. This Peltier heating (or cooling) must be distinguished carefully from the more familiar irreversible (Joule) heating which occurs in all conductors (except superconductors) when an electric current flows in them. The Joule heating is directly related to the electrical resistivity of the substance, and is an entirely irreversible effect which depends only on the square of the current density. This means that the Joule heat is always evolved no matter what the direction of current flow in the substance, and it is therefore always considered as a positive quantity. On the other hand, the Peltier heat depends linearly in magnitude on the current flow, and may be evolved (the junction appears to be heating up), or absorbed (apparent cooling), depending on the relative direction of the current flow and the temperature gradient. The fact that the Peltier heat depends linearly in both magnitude and sign (i.e., either heating or cooling) on the current flow means that in this sense at least it is a reversible phenomenon in contrast to the Joule heating. We define the Peltier heat II12 at a junction as the (reversible) heat developed per unit time per unit electric current flowing in the direction 1 -> 2. We should also note that for a given pair of conductors (1) and (2) and for a given electric current, the Peltier heat II12 depends on the common temperature at the junction of the two conductors.
The Peltier heat is evolved or absorbed in the junction region between two conductors, and this again leads quite frequently to some confusion about the nature of thermoelectric effects. The confusion arises because it tends to be assumed that the Peltier heat is thus intimately connected in some way with the detailed features of the contact itself, and may even sometimes be referred to in error as a "contact phenomenon." Let us therefore now categorically state that, generally speaking, the magnitude and sign of the Peltier heat do not depend in any way on the actual nature of the contact. It is purely a function of the two different bulk materials which have been brought together to form the junction, and each individual bulk substance makes its own characteristic contribution to the Peltier heat. This individual contribution would be directly observed if a junction were made separately between that conductor and a superconductor.
1.3 THE KELVIN (or THOMSON) RELATIONS
If now we look at the circuit in Fig. 1a and visualize that a thermoelectric current is allowed to flow as a consequence of the temperature difference, it is clear that Peltier heating will occur at one junction and Peltier cooling at the other. William Thomson in 1854 decided that there should be a thermodynamical connection between the Peltier heat (II12) and the thermoelectric force of a couple (i.e., the Seebeck potential V12). He analyzed the situation and essentially derived the conclusion that:
dV12/dT = dII12/dT = II12/T (Erroneous)
This leads to the (erroneous) general result that II12 [varies] T. Experimental evidence was not in agreement with this result, and Thomson therefore drew the remarkable conclusion that there must be a further thermoelectric effect thus far not taken into account. This led him to postulate the existence of what is now called the Thomson heat in a conductor, which may be expressed as follows: If an electric current (of current density Jx) is passing through an individual conductor, and a temperature gradient dT/dx is also present (see Fig. 3), then the net heat produced in the conductor per unit volume per second ([??]) is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)
The first term, dependent on the electrical conductivity (σ), dependent on the square of the current, and independent of the temperature gradient, is the irreversible Joule heat. The second term, linearly dependent on current density and temperature gradient, is a thermoelectric heat, whose overall sign (i.e., (+) if the heat is emitted and (-) if absorbed) depends on the directions of the current and the temperature gradient relative to one another. The coefficient μ in the second term is then defined as the Thomson heat for the material concerned, and depends on the temperature of the conductor.
Equation 1 may well be regarded as a fundamental equation for thermoelectricity. Effectively the assumption that the Thomson heat and the irreversible Joule heat may be added to one another independently is equivalent to assuming that the irreversible entropy production corresponding to the Joule heat may be added independently to the reversible entropy associated with the Thomson effect. Thomson himself was well aware that the thermodynamic arguments he employed were not entirely rigorous, and it is often customary today to employ more sophisticated methods under the general heading of so-called irreversible thermodynamics (cf., e.g., de Groot, 1951). Be that as it may, Thomson himself derived in effect the following relations between the Thomson heat (μ), the absolute thermoelectric power (S), and the Peltier heat (II) of a conductor:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
(The net Peltier heat at a junction between two conductors, and the net thermoelectric power in a circuit as observed experimentally are given by II12 = Π1 - Π2, and S12 = S1 - S2.) Integrating the first Kelvin relation we can always in principle determine the absolute thermoelectric power, S, of a conductor from calorimetric measurements of the Thomson heat, μ, i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2a)
It can be argued from an application of the third law of thermodynamics that the thermoelectric properties must vanish as T -> 0, and hence we simply write:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2b)
In this way an absolute thermoelectric scale can be constructed up to high temperatures (cf. Borelius, loc. cit.), but it remains true that direct measurement of S against a superconductor is extremely convenient whenever this is possible.
Thomson undertook the task thereafter of examining the predictions of Eq. 2 experimentally, and since that time there have also been other experimental studies of the Kelvin relations. Although today a certain amount of theoretical discussion still goes on about an entirely satisfactory basis for deriving the Kelvin equations, there is virtually no doubt about their validity or accuracy. We shall henceforth accept them as fundamental in this monograph. A vital feature of Thomson's equations is that they relate the essentially calorimetric quantity (μ) uniquely to the thermoelectric power (S), which in general can be measured with relative ease and considerable accuracy. More generally, the Kelvin relations show us how to derive complete knowledge of all the thermoelectric properties of a conductor from a knowledge of S (i.e., essentially from the Seebeck potential which can be measured rather readily with good accuracy). Moreover the Thomson relations emphasize that we can often usefully invoke the laws of thermodynamics to aid us in studying conduction problems via thermoelectricity, in contrast to the essentially irreversible behavior of electrical and thermal resistivity themselves where thermodynamics is of much less help.
1.4 THOMSON HEAT AND THERMOELECTRIC POWER
Equation 1 may be written alternatively in terms of the charge, q, transported in time, t, through a section of conductor of resistance R under a small temperature difference ΔT as:
Q = q2R/t - μq ΔT (3)
where Q is the heat evolved in this section of conductor in time t. This equation clearly points out that if a given electric charge, q, is moved through a conductor, then the amount of heat, Q, evolved or absorbed depends on the rate at which the electric charge is transported. If the rate is made slow enough (t sufficiently long), then the first (Joule) term will become insignificant in comparison with the Thomson heat term, and it is clear that the amount of heat or entropy involved is then directly related to the electric charge transported and is independent of the rate at which this is done (as long as the rate is slow enough). Thus we can (perhaps best) refer to the Thomson heat as the heat absorbed per unit (positive) charge and unit temperature difference, when this charge is transported sufficiently slowly in the direction of increasing temperature. Looked at in this way it seems very natural to regard the process as reversible, the requirements of carrying out the charge transfer "sufficiently slowly" implying some sort of quasi-equilibrium conditions. The conditions really amount to saying that we first adjust things externally so that no electric current flows under the temperature gradient (i.e., the Seebeck potential in the circuit prevents any charge transfer), and we then allow a slow or "virtual" transport of electric charge to take place without appreciably disturbing the dynamic electrical equilibrium.
(Continues...)
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Table of Contents
Contents
DOVER BOOKS ON PHYSICS,Title Page,
Copyright Page,
Preface,
1 - General survey,
2 - The Boltzmann equation and electron scattering,
3 - Phonon-drag and phonon scattering,
4 - A final miscellany,
References,
Index,