Non-Equilibrium Thermodynamics of Thermoelectric Effects

The Thermoelectric Effects and Non-Equilibrium Thermodynamics:

The thermoelectric effects deals with the interrelations between heat flow and electric conduction in some electrical conductor systems (e.g., in the thermoelectric devices). From the viewpoint of non-equilibrium thermodynamics, the thermoelectric effects arise from the coupled dependence (as per the Onsager reciprocity relations) of the thermal flow (heat flow) and electrical flow (current) on the generalised thermal force (difference of temperature) and the generalised electrical force (difference of electrical potential), generally between the two junctions of two different conductors joined together (the Seebeck and the Peltier effects), though there is also the Thomson effect dealing with a kind of thermal flow (heat generation or absorption) associated with current flowing through one conductor having a temperature gradient along its direction of current. However, the effects are varyingly formulated (as detailed below) in forms of interrelations among the two generalised forces (in case of Seebeck effect) or among the two generalised flows (in case of Peltier effect) etc., thus differing from the formal forms of linear phenomenological relations strictly between the generalised forces C_k on one hand and the generalised flows J_k on the other.

The Seebeck Effect

The Seebeck effect deals with the development of an emf (i.e., a potential difference), called the thermoelectric emf, created by the presence of a temperature difference between two junctions (see figure below) of two different conductors (made of conducting materials A & B) joined together at two ends. This arrangement is called a thermocouple, which is generally used to measure temperatures within various kinds of instruments, and sometimes even for useful conversion of thermal energy to electrical one.

The Seebeck effect arises because the applied temperature difference causes the charged carriers of electricity (i.e., the mobile electrons or the mobile holes -- a hole meaning a lack of an electron) present in the materials to diffuse from the hot region (i.e., hot junction) towards the cold region, similar to what gas molecules would do. Mobile charged carriers moving towards the cold-junction region leave behind their oppositely charged and immobile nuclei at the hot-junction region, thus giving rise to the thermoelectric emf.

As per the Seebeck effect, the potential difference DV generated between the two junctions maintained at two different temperatures T₁ and T₂ (with T₂ > T₁) is given by the following relation (where S_A & S_B are the Seebeck coefficients or thermoelectric power of the conducting materials A & B respectively):
        DV {i.e., V(T₂)V(T₁)} =  {S_A(T)S_B(T)}dT  =  S_AB(T) dT
where S_AB(T) = S_A(T) – S_B(T), the Seebeck coefficient for the thermocouple is the difference of the individual Seebeck coefficients (i.e., S_A and S_B) for the two materials.
For small temperature-differences, S_A & S_B may be assumed to be constants, so that approximately,
        DV ≈ {S_A – S_B}(T₂ – T₁), or,   DV ≈ S_AB DT 
For junctions made of copper and constantan (it is a Cu-Ni alloy), the Seebeck coefficient difference S_AB has one of the largest values of around 41 mV K^–1, so that for a temperature difference of 100 K, the Seebeck potential difference is around 4.1 mV. Thus in general the Seebeck voltages are rather small quantities, and so large output voltages are sometimes obtained by connecting several thermocouples in series forming a thermopile. Seebeck effect gets utilized in thermocouples which are generally used for measuring temperatures or differences in temperatures.

Peltier Effect

The Peltier effect deals with the thermal consequence (i.e., heating/cooling effect) of an electrical current flowing through a junction of two different conducting materials, as in the case of the two junctions within the aforementioned figure of a thermocouple. When a current I is made to flow through the circuit in the said figure, heat is absorbed at one junction (called thermoelectric cooling), while heat would be evolved (with a rate of equal magnitude) at the other junction. The Peltier effect is measured by the Peltier heat or the Rate of Peltier cooling which is defined as the time rate of heat absorbed (i.e., δQ/dt) at one junction; it is proportional to the current I that flows through it: 

δQ/dt = I. Π_AB = I. (Π_A – Π_B)
 (where the current I is flowing from material A towards material B; note that here Π_AB equals Π_A–Π_B, not  Π_B–Π_A)

Where Π_AB is the Peltier coefficient of the thermocouple as a whole, and Π_A and Π_B are the Peltier coefficients of the conducting materials A & B respectively (Π_A and Π_B are also understood as their electrical potentials at both sides of the junction, so that Π_AB is the potential difference for the junction). Here the direction of heat transfer gets controlled by the direction of the current, reversing the direction would change the direction of heat transfer (as in the above equation, Π_BA would then appear in place of Π_AB -- obviously, Π_BA = Π_B–Π_A equals  – Π_AB). Peltier effect could cool a junction because when charged carriers of electricity move from a region of high electrical potential (say, material A) to a region of low electrical potential (say, material B), they would get accelerated, for which the required energy must be absorbed from the environment (thereby resulting in the cooling effect).

The values of the Peltier coefficients Π for various materials are obtainable from their Seebeck coefficients S by referring to the First Thomson Relation  Π = S.T, where T is the absolute (i.e., Kelvin-scale) temperature. This means that for a copper-constantan thermocouple at room temperature (25 °C) through which a current of 0.1 Ampere (a usual value) is flowing, the Peltier coefficient Π_AB is 12.22 mV (how?) while the rate of Peltier cooling is only 1.222 x 10^–3 J s^–1. Thus, the rate of heat transfer for an individual thermocouple is rather low; to enhance the heating/cooling effect, several thermocouples may be connected in series. Peltier effect is utilized in thermoelectric heat pumps (Peltier coolers), which are solid-state devices transferring heat from one side ( the cooler side) of the device to the other side.

The Interrelation of the Seebeck and the Peltier Effects:

The Seebeck and the Peltier effects for a thermocouple are interrelated not only by their coefficient values (as per the relation Π_AB = T.S_AB, readily obtainable from the first Thomson relations for the two constituent materials -- just do subtract them!), but also in their directions of operation. Such interrelation arises because they are nothing but irreversible cross-phenomena between thermal and electrical driving forces and fluxes: Seebeck effect is about the electrical effect of thermal driving force, whereas Peltier effect is about the thermal effect of electrical driving force (detailed in a later section). Their directions are interrelated as follows: (i) Peltier effect due to the current arising from the Seebeck emf would cool the hotter junction and heat the colder junction (thereby decreasing the temperature difference), (ii) The Seebeck emf arising out of the Peltier current would tend to decrease the Peltier current itself. Had the directions been otherwise, the two effects would not decrease the causes (i.e., the driving forces of the temperature difference or of the Peltier current) causing the irreversible flows, but would rather have increased these causes (thus leading to a catastrophic situation, isn't it!). We, however, know that the consequences of any irreversible flows are to decrease their causes, not to augment them! 

Thomson Effect

The Thomson effect discusses the thermal (i.e., heating/cooling) effect of an electric current passing through a conductor material (yes, here only one conductor material) having a temperature gradient along the direction of the current. If dT/dx is the temperature gradient along the direction of current through the conducting wire, then the heat production (δq/dt) per unit time (per unit volume of the conductor material) due to the Thomson effect is given as:

δq/dt  =  – m J dT/dx

Where m is the Thomson coefficient and J is the current density (i.e., electric current per unit area). Thus we see that if the temperature increases along the direction of the current, the temperature gradient dT/dx is positive and so δq/dt will be negative (i.e., cooling effect will be observed) for a material having a positive value of the Thomson coefficient. This is the case with zinc and copper conductors. But with cobalt, nickel and iron, the Thomson coefficient is negative. The Thomson effect, however, generally has an even smaller magnitude compared to the Peltier effect.

Further Considerations of Non-Equilibrium Thermodynamics: 

The first Thomson relation (Π = S.T) mentioned above is obtainable from rather simple considerations of non-equilibrium thermodynamics only. Clearly, the thermocouple with a temperature difference between the two junctions and an electric current flowing through them is an example of a system in which two irreversible processes are simultaneously occurring: (i) heat flow between the two junctions, and (ii) current flow through the two junctions. Thus it is a coupled flow situation, and the pair of generalized forces and generalized flows here are: (i) the generalized force C_th = D(1/T) and the generalized flow J_th = δQ/dt corresponding to the thermal (heat) flow (reminding us about the earlier example of heat flow between two parts of one system), and (ii) the generalised force C_el = –D(V/T) and the generalized flow J_el = I corresponding to the electrical flow (here V is the electrical potential and I is the electrical current). Note that the two generalized forces above must be properly defined {i.e., as D(1/T) and –D(V/T) respectively, surely not as, say, DT and DV}, so that the criteria d_iS/dt = J_th.C_th + J_el.C_el regarding the rate of internal entropy production in irreversible processes gets duly satisfied (also, the electrical driving force above must have the negative sign which would imply that the electrical current, at least at isothermal conditions, spontaneously flows only from a region with high V to one with low V). Now the two linear phenomenological relations are:  (i) J_th = L_th,th.C_th + L_th,el.C_el  and (ii) J_el = L_el,th.C_th + L_el,el.C_el. So, the Onsager reciprocal relation implies that the two cross-coefficients are equal, i.e., L_th,el = L_el,th. 

Now, let R be the electrical resistance of the entire thermocouple system, so that the current I passing through it is related to the potential difference DV as I = DV/R. As the Peltier effect relates J_th = δQ/dt and the current I, so we have (assuming the temperature T to be constant) for the Peltier effect:  J_th = Π_AB.DV/R = (–T.Π_AB/R).{–D(V/T)} = (–T.Π_AB/R).C_el. Also at constant temperature, C_th = D(1/T) is zero, so that comparing this relation with the first phenomenological relation above we get L_th,el = –T.Π_AB/R. Next, for the Seebeck effect we have DV = S_AB.DT for a very small temperature difference DT. This means DV/R = (S_AB/R).DT = (S_AB/R).{–T²D(1/T)} = (–T²S_AB/R).D(1/T) = (–T²S_AB/R).C_th (as for a small temperature difference, D(1/T) equals –DT/T²). However, DV/R = I, which is J_el. This means that  J_el = (–T²S_AB/R).C_th, in the absence of an independent C_el. Comparing this with the second phenomenological relation, we immediately get  L_el,th = –T²S_AB/R. So the Onsager reciprocal relation above (i.e., L_th,el = L_el,th) gives that –T.Π_AB/R = –T²S_AB/R, or that Π_AB = T.S_AB. 

After obtaining the above first Thomson relation (i.e., Π_AB = T.S_AB) for the thermocouple as a whole, we may extend this relation for the two constituent materials by assuming that Π_A = T.S_A, Π_B = T.S_B, so that (Π_A – Π_B) equals T(S_A – S_B), i.e., Π_AB = T.S_AB. It is interesting to note that the individual Peltier coefficients and the individual Seebeck coefficients can't be experimentally determined (only their inter-material differences can be determined), just as in the case of the individual electrode potentials in a Galvanic cell! Thus, a suitable reference material need be chosen to have these two coefficients exactly zero, and the pairs of coefficients for the other materials experimentally determined with reference to that material.

Sources:

"Peltier Effect", The Internet Encyclopaedia of Science. <http://www.daviddarling.info/encyclopedia/P/Peltier_effect.html>