Thermodynamics 2012 Part 11 pot

Our analysis recognizes that the Helmholtz free energy, ˆA, is an explicit function of the deformed crystallographic or lattice vectors defining the deformed crystalline structure.Edelen

Fig 6 Reduce Efficiency.

factor gives an insight of the sensibility of the thermoelectric system to varying workingconditions It shows that for large ZT this sensibility becomes very high and strong reduction

of the efficiency can rapidly occur For an optimal efficiency, obtained for u=s it shows that

Φ precises the intrinsic irreversibilities contribution for a given material

7 Conclusion

The thermoelectric process has been described using a classical fluid approach The ”Fermigas”’ of electrons takes place of the traditional ”steam” in the thermodynamical cycle, givingstrong similarities in the description of the underlying mechanismes, which are based onthe linear Onsager theory of ”‘out of equilibrium thermodynamics” It is shown that the socalled ”‘figure of merit ZT”’ of the thermoelectric material can be directly derived from thisapproach Finally, the importance of the working conditions is demonstrated, leading to theconcept of thermoelectric potentialΦ which is an extension of the concept of free energy forthe gas, under out of equilibrium conditions

8 References

[Callen 1952] Herbert B Callen and Richard F Greene, Phys Rev 86 (5) 702-710 (1952).

[Callen 1954] Callen, H B., Irreversible thermodynamics of thermoelectricity, Rev Mod.Phys., 26, 237, 1954

[Callen 1948] Herbert B Callen, Phys Rev 73 (11) 1349-1358 (1948)

[Domenicali 1954] C.A Domenicali, Rev Mod Phys 26, 237-275, (1954).

[Goupil 2009] C Goupil, J Appl Phys 106, 104907 (2009) 5

[Greene 1952] Richard F Greene and Herbert B Callen, Phys Rev 88 (6) 1387-1391 (1952).

[Ioffe 1960] A.F Ioffe, Physics of Semiconductors (Infosearch, London, 1960)

[Landau 1984] L D Landau and E M Lifshitz, Electrodynamics of Continuous Media, 2ndEdition, Butterworth Heinemann (Oxford, 1984)

[Onsager 1931a] L Onsager, Phys Rev 37, 405 - 426 (1931)

[Onsager 1931b] L Onsager, Phys Rev 38, 2265 - 2279 (1931)

[Peltier 1834] Peltier, J C A., Nouvelles experiences sur la caloricit´e des courants electrique,Ann Chem Phys., 56, 371, 1834

[Pottier 2007] No¨elle Pottier, Physique statisitique hors ´equilibre, processus irr´eversibleslin´eaires Savoirs Actuels EDP Sciences/CNRS Editions, (2007).

[Rocard 1967] Y Rocard ; Thermodynamique, Masson (2e ´edition-1967)

[Seebeck 1821] Seebeck, T J., Ueber den magnetismus der galvenische kette, Abh K Akad.Wiss Berlin, 289, 1821

[Seebeck 1826] Seebeck, T J., Ann Phys (Leipzig), 6, 1, 1826 4 Seebeck, T J.,Methode, Platinatiegel auf ihr chemische reinheit durck thermomagnetismus zuprufen,Schweigger’s J Phys., 46, 101, 1826

[Seebeck 1823] Seebeck, T J., Magnetische polarisation der metalle und erze durcktemperatur-differenz, Abh K Akad Wiss Berlin, 265, 1823

[Snyder 2003] G Jeffrey Snyder and Tristan S Ursell,Phys Rev Lett 91 148301 (2003).[Thompson 1848] Thomson, W., On an absolute thermometric scale, Philos Mag., 33, 313,1848

[Thompson 1849] Thomson, W., An account of Carnot’s theory of the motive power of heat,Proc R Soc Edinburgh, 16, 541, 1849

[Thompson 1852] Thomson, W., On a mechanical theory of thermo-electric currents, Philos.Mag [5], 3, 529, 1852

[Thompson 1854] Thomson, W., Account of researches in thermo-electricity, Philos Mag [5],8,62, 1854

[Thompson 1856] Thomson, W., On the electrodynamic qualities of metals, Philos Trans R.Soc London, 146, 649, 1856

[Vinning 1997] C B Vining Materials Research Society Symposium Proceedings:Thermoelectric Materials - Mater Res Soc 278 (1997) 3-13

Application of the Continuum-Lattice

Thermodynamics

Eun-Suok Oh

University of Ulsan South Korea

1 Introduction

Through the continuum-lattice thermodynamic approach, the thermodynamic behaviors oftwo- and three-dimensional multicomponent, elastic, crystalline solids are developed Webegin with a discussion of non-equilibrium thermodynamics of an isolated body that is notundergoing a phase transformation

Our analysis recognizes that the Helmholtz free energy, ˆA, is an explicit function of the

deformed crystallographic or lattice vectors defining the deformed crystalline structure.Edelen (1975) gave a similar discussion with the assumption that ˆA was an explicit function

of the deformation gradient rather than a function of the deformed lattice vectors For thisreason, his analysis requires an additional step in which the implications of the isotropygroup are observed (Truesdell & Noll, 1965, p 310); the requirements of the isotropygroup are automatically accounted for in our analysis through the use of the lattice vectors(Slattery & Lagoudas, 2005)

As applications, we obtain the stress-deformation behaviours of graphene, carbonnanotubes(CNTs), boron-nitride nanotubes(BNNTs) which are composed of a regulartwo-dimensional array of hexagonal lattices of atoms (Oh et al., 2005; Oh, 2010), andthe stress-deformation behaviours of face-centred cubic crystals such as diamond, silicon,silicon-carbide, and boron-nitride (Oh & Slattery, 2008) Using an interatomic potential, theTersoff (Tersoff, 1988; 1989) or Tersoff-like potential (Brenner, 1990; Albe & Moller, 1998;Brenner et al., 2002) to describe interaction between atoms, we compute the elastic propertiesfor the crystals These are compared with the available experimental and theoretical values

2 Continuum-lattice thermodynamics

A simple two- or three-dimensional crystal is one in which two or three primitive latticevectors, vectors drawn between immediate neighbor atoms, can express all of the latticepoints as shown in Figs 1(a) and 1(b) for two-dimensional crystals and in Figs 2(a) and 2(b)for three-dimensional crystals Generally, these primitive lattice vectors are not sufficient todescribe more complicated structures such as those depicted in Figs 1(c), 1(d), 2(c), and 2(d)

The primitive lattice vectors e(1)and e(2)in Fig 1 and e(1), e(2)and e(3)in Fig 2—so calledexternal lattice vectors—determine the external structure of the unit cell In order to describethe internal structure of the unit cell, one or more additional lattice vectors—so called internal

lattice vectors—are required, such as e(3)in Figs 1(c) and 1(d) or e(4)in Figs 2(c) and 2(d)

14

Fig 1 Two-dimensional lattices: (a) a square Bravais lattice, (b) a diamond Bravais lattice, (c)

a more general lattice having a square unit cell, and (d) a hexagonal lattice (Oh et al., 2005)

Our particular interest in what follows is complicated two- or three-dimensional crystals thatare not undergoing a phase transformation As seen in Fig 3,

we will assume that the adjoining phases are not crystalline, and that they are not alsoundergoing a phase transformation

For the adjoining non-crystalline phase (gas, liquid, or amorphous solid), we will assume thatthe Helmholtz free energy per unit mass is given by

ˆ

A=Aˆ

T,ρ,ω(1), ,ω (N−1) (1)

Here T is the temperature, ρ is the total mass density, ω (A)=ρ (A)/ρ is the mass fraction

of species A, and ρ (A) is the mass density of species A For the two- or three-dimensional

multicomponent crystal, let us initially assume that the surface Holmhotz free energy permass is

Here E(i) is a primitive lattice vector in the undeformed configuration and its length will

be determined by the equilibrium bond length Its corresponding lattice vector in the

configuration deformed by an in-plane homogeneous deformation, F, is denoted as e(i) Thedeformation gradient is defined as

(a) (b)

Fig 3 An isolated body consisting of a crystalline solid and its adjoining phase: (a)

two-dimensional crystal and (b) three-dimensional crystal

where the gradient operation is performed in the undeformed configuration and z is a

position vector Taking a continuum point of view, we will regard these lattice vectors

as being continuous functions of position on the space Using the Born rule (Ericksen,1992; Zanzotto, 1992; Klein, 199; James & Hane, 2000; Zhang et al., 2002), we can express thedeformed external lattice vectors introduced to describe the external structure of the deformedunit cell as

e(i)≡FE(i), i=1, 2, for 2-D crystals or i=1, 2, 3 for 3-D crystals (4)

It will be shown in equation (40) that the deformed internal lattice vectors e(i) (i=3, , k for 2-D crystals or i=4, , k for 3-D crystals) describing the internal structure of the unit cell are

decided by minimizing the Helmholtz free energy at equilibrium

The principle of frame indifference (Truesdell & Noll, 1965, p 44) requires that the mostgeneral form of such a function is one in which all possible scalar products of the variouslattice vectors appear (Truesdell & Noll, 1965, p 29) We will eliminate scalar products of

the form E(m)···e(n), since we will show later that they would lead to a non-symmetricstresstensor [see the discussion concluding with equation (38)], which we will not allow in this

development Scalar products of the form E(m)···E(n) may be retained in the background

contributing to the behaviour of the surface In view of this reasoning, we will write

Here all other scalar products are excluded from the dependence of ˆA since they are not

independent variables It will be more convenient to represent the scalar products appearing

in equation (5) as

I (mn)≡e(m)···e(n)−E(m)···E(n) m=1, 2, 3 and n=1,· · ·, k, (6)and equation (5) becomes

ˆ

A=Aˆ

T,ρ,ω(1),· · ·,ω (N−1) , I(11),· · ·, I (1k) , I(22),· · ·, I (2k) , I(33),· · ·, I (3k)

Using equation (4) and the definition of the right Cauchy–Green strain tensor (Slattery et al.,

for three-dimensional crystals

In a very similar way, the Holmholtz free energy per unit mass for a two-dimensionalmulticomponent crystal can be expressed as

2.1 Euler, Gibbs, and Gibbs–Duhem equations

From the differential entropy inequality (Slattery, 1999, p 438), we conclude

With these expressions, the differentiation of equation (11) can be expressed as

This is referred to as the modified Gibbs equation

In view of the Euler’s equation (Slattery et al., 2007, p 310)

3 Equilibrium: constraints on isolated systems

We define equilibrium to be achieved by an isolated body, when the entropy inequality becomes

an equality In the following sections, we wish to develop necessary and sufficient criteriafor the achievement of equilibrium in the isolated body shown in Fig 4 The followingassumptions will be made

1 Once the body is isolated, it is totally enclosed by an impermeable, adiabatic boundary, thevelocity of which is zero

2 There is no mass transfer between the crystalline solid and its adjoining phase in theisolated body

3 No chemical reactions occur

Let us begin by examining the constraints imposed upon the isolated body by the massbalance, by the momentum balance, by the energy balance, and by the entropy inequality

3.1 Species mass balance

Since no chemical reactions occur in the isolated body, the mass balance for each speciesrequires (Slattery et al., 2007, p 269)

d

dt



Here dV indicates that a volume integration is to be performed Applying the transport

theorem for a multiphase body (Slattery, 1999, p 433), we conclude

ρω (A)(v−u) ···ξξξ dA

=0

(21)

Here d(m) /dt is the derivative following a material particle within a phase (Slattery, 1999, p.

4), v is the mass average velocity, and u is the time rate of change of position following a

surface point (Slattery, 1999, p 23) The boldface brackets denote the jump quantity enclosedacross the interface between phasesα and β

Fig 4 An isolated body consisting of a three-dimensional crystalline solid and its adjoining

phase R indicates a region occupied by the body, S is a bounding surface of the body, andΣ

denote a phase interface between the crystal and its adjoining phase n andξξξ are unit normal

vectors to the surfaces S andΣ, respectively

where T is the Cauchy stress tensor, and b(A)is the body force per unit mass acting at eachpoint within each phase Here we have employed the divergence theorem (Slattery, 1999, p.682) Again applying the transport theorem (Slattery, 1999, p 433), we see

We can summarize equation (25) as



ρ

ˆ

For the isolated body under consideration here, the entropy inequality says that the time rate

of change of the body’s entropy must be greater than or equal to zero (Slattery et al., 2007, p.295):

4 Implications of equilibrium

As explained above, if equilibrium is to be achieved, the left side of equation (27) must beminimized and approaches to zero within the constraints imposed by conservation of massfor each species, by the momentum balance, and by the energy balance as developed in theprior section

In view of equations (21), (24), and (26), there is no loss in generality in writing equation (28)

From the modified Gibbs equation (16) and the definition of ˆA≡Uˆ −T ˆ S (Slattery et al., 2007,

and e(i)⊗e(j)is the tensor product or dyadic product of two vectors e(i)and e(j).

After rearranging equation (29) by means of equations (21) through (31), we have

In arriving at this result, we have recognized the differential momentum balance (Slattery,

μ (A)=Tλ (A) for each species A=1, 2,· · ·, N, (35)

v(A)=v for each species A=1, 2,· · ·, N, (36)

μ (I,mn)=0 for m=1, 2, 3 and n=4,· · ·, k, (37)

These relations determine e(4), , e(k) at equilibrium Notice that, if we had retained

scalar products of the form E(m)···e(n) in equation (5), the stress tensor would have been

5 Stress-deformation behaviour at equilibrium in the limit of infinitesimal deformations

Since it is common to consider infinitesimal deformations, let us consider how equation (38)reduces in this limit:

to be the displacement vector, and z and zκ are the position vectors of a material particle in

the deformed and undeformed configurations It follows that the displacement gradient is

Let be a very small dimensionless variable characterizing an infinitesimal deformation

process We will seek a solution of the form

w=w(0)+w(1)+2w(2)+ · · · (46)Let us recognize that in the absence of an infinitesimal deformation or as→0

This means that

H=H(1)+2H(2)+ · · ·, where H(i)=grad w(i), (48)and thus the right Cauchy-Green strain tensor becomes

C≡FTF

= (I+H(1)+ · · ·)T(I+H(1)+ · · ·)

=I+H(1)+H(1)T

+O



2

(50)

Since, in the limit of small deformations, the strain

εεε≡12

=εεε(1)+ · · ·,

(51)which allows us to also express equation (50) as

ρ=ρ κ1−trεεε(1)+ · · · (52)

Equations (45), (48), and (52) also permit us to rewrite equation (43) to the first order in as

(for the moment not addressing the order ofμ (I,mn))



E(m)⊗E(n)+E(n)⊗E(m)

 ... Helmholtz free energy and the intermolecularpotential energy at equilibrium.

We will employ two types of potentials, Tersoff (1988; 1989) and modified Tersoff potentials(Brenner, 1990; Albe &... interatomicpotentials for the crystalline solids However, the choice of the interatomic potential is ratherarbitrary and is dependent of atoms consisting of the crystals

5.2.1 Tersoff potential... (21) through (31), we have