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Specific heat, polarization and heat conduction in microwave heating systems: A nonequilibrium thermodynamic point of view
Acta Materialia 54 (2006) 1843–1849 www.actamat-journals.com
Specific heat, polarization and heat conduction in microwave heating systems: A nonequilibrium thermodynamic point of view Paolo Bergese
*
INSTM and Chemistry for Technologies Laboratory, University of Brescia, Via Branze 38, 25123 Brescia, Italy Received 30 September 2005; received in revised form 15 November 2005; accepted 21 November 2005 Available online 10 February 2006 Dedicated to my newborn son, ‘‘il gufetto’’ Leonardo.
Abstract A microwave (MW) field can induce in a dielectric material an oscillatory polarization. By this mechanism part of the energy carried by the waves is converted into chaotic agitation, and the material heats up. MW heating is a nonequilibrium phenomenon, while conventional heating can generally be considered as quasi-static. Excess (or nonthermal) effects of MWs with respect to conventional heating lie in this difference. Macroscopically, MW heating can be described in the framework of linear nonequilibrium thermodynamics (NET). This approach indicates that in a dielectric material under MW heating the specific heat has a dynamic component linked to the variation of polarization with temperature, and that polarization and heat conduction are intertwined. In particular, linear NET provides a new phenomenological equation for heat conduction that is composed of the classic Fourier’s law and an additional term due to polarization relaxation. This term quantitatively describes the excess effect of MWs on thermal conduction. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dielectrics; Thermodynamics; Microwaves
1. Introduction Microwave (MW) heating is today a mature technique which finds wide application in chemistry [1] as well as in materials [2–6] and food [7] processing. Heat is generated at each point of the material by the dissipation of MW energy as a consequence of the interaction of the MW field with the molecular and electronic structure of the material. Therefore, MWs generate in the material volumetric, fast and selective heating, which generally results in process energy and time savings [1–3,6]. Furthermore, MW/ material interaction generates other unique excess effects (also referred as nonthermal or athermal effects), including enhanced chemical reaction rates [1] lower sintering temperatures [6,8], lower diffusion activation energies [6,9] selflimiting reactions [1] and plastic deformations (in ionic *
Tel.: +390303715574; fax: +390303702448. E-mail address: [email protected].
crystals) [10]. Despite the fact that they often represent the overriding advantages of MW processing over conventional heating, excess effects are still not well understood [2,6], and hence are generally omitted in heating models [11]. While much experimental work has shown the overall differences between MW and conventional heating, only few studies have succeeded in isolating a particular excess effect [6]. Also for this reason excess effects remain an open and debated subject in all the fundamental fields touched by MW processing: chemistry [12,13], and materials [6,14] and biomaterials [15] science. MW enhancement of mass transport in ionic crystalline solids is an exception. This phenomenon has been experimentally investigated to some extent, and theoretically justified at the microscopic scale as the consequence of a MW-driven ponderomotive force applied to mobile charged vacancies (see Ref. [6] and references therein). In this paper MW heating of dielectric materials is described macroscopically using linear nonequilibrium
1359-6454/$30.00 Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.11.042
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P. Bergese / Acta Materialia 54 (2006) 1843–1849
thermodynamics (NET). In particular, after introducing the dynamic specific heat of dielectrics under MW heating, phenomenological expressions for MW excess effects on heat conduction and polarization relaxation are presented and discussed. These preliminary theoretical results suggest that the NET framework is a valid candidate for describing MW heating. 2. Linear nonequilibrium thermodynamics and microwave heating of dielectric systems The dominant role in MW heating of dielectrics is played by molecular dipoles, which show relaxation times of the same order of magnitude as MW frequencies. The interaction mechanisms between MW electric field and molecules (electrothermal coupling) consist of rotation and (re)orientation of the molecules with permanent or induced dipoles [3]. These molecules rotate against the resistive forces of the surroundings, a part of the electromagnetic energy is converted into chaotic agitation, and the material heats up. The Debye relaxation equation [16] (1929) is the accepted phenomenological theory to model the dipole orientation mechanism, and, in turn, to describe MW interaction with matter and heating [3]. The molecular dipole is assumed to respond to the torque moment produced by the external electric field as it would do a sphere rotating in a fluid of given viscosity. The model is based on the linear response approximation and holds for systems consisting of molecules that have permanent dipoles and are not deformable (purely polar systems). It predicts a time-independent relaxation time, s, which is determined by the interactions of the dipole with its surroundings. The Debye relaxation equation can also be derived by describing the irreversible interaction between MWs and dielectric materials in the theoretical framework of linear NET [11,17]. As for the Debye phenomenological theory, the linear NET approach is based on the approximation that MW heating occurs via linear response phenomena, or, using the NET formalism, that the irreversible flows occurring in the system are linear functions of the thermodynamic forces. Linear NET relies on the validity of the Onsager law and of the Curie–Prigogine principle, and on the assumption of local equilibrium. They allow one to express in phenomenological equations the correlation between entropy production and thermodynamic flows and forces. In the local equilibrium framework the system is divided in arbitrary mass elements large enough to define the thermodynamic potentials but small enough to neglect the gradients. It is then postulated that even if the system is not in equilibrium it is possible to isolate a mass element that is in equilibrium. This implies that the specific entropy of such mass element is the same function of the extensive thermodynamic variables as it is when the whole system is in equilibrium; that is, the mass element can be described by the (equilibrium) Gibbs equation. In conditions of local equilibrium the thermodynamic variables are then locally
defined, and can be considered continuous functions of space coordinates and time. The basic assumptions for applying linear NET can be fully justified only a posteriori by the validity of the drawn conclusions. However, they were found to be valid for a wide range of macroscopic systems [18] and in the case of MW/matter interaction they lead to the well-accepted Debye equation. Therefore we assume that linear NET gives a reliable description of MW heating systems. Further details of the foundations and validity limits of linear NET or of NET criticisms and alternative formulations are out of the scope of this paper. Extensive information can be drawn from the several (classic [17,19,20], critical [21] and recent [22,23]) monographs on NET and from some very recent review papers [18,24]. 3. Local electric field and dielectric functions In this section some concepts and constitutive relations that will be needed in the following are fixed. In general, the macroscopic electric field inside a polarizable material, Emac, is not the same as the applied external field because the electric dipoles in the material produce a depolarization field which opposes the external field. Moreover, Emac is different from the field at a particular point, Emic (microscopic electric field), and both of them depend on the sample geometry. Analogous considerations hold for magnetic fields. The electric charge density at a point r inside the system, lmic(r), is a very rapidly varying function of r on the length scale of the atomic spacing. Consequently Emic(r), which is related to lmic(r) by one of the Maxwell equations, is also a fast-varying quantity. Emac(r) can be defined as the average of Emic(r) taken over a mass element of volume V. V has a width, s, much larger than the atomic spacing, but appreciably smaller than the material volume, so that the spatial fluctuations of the field at the atomic length scale are averaged out by Emac(r), but spatial variations of the field on a length scale greater than s are retained by Emac(r). In other words, Emac(r) is a local quantity in the meaning of linear NET elucidated in the previous section (note that this concept of local electric field has nothing to do with the one generally encountered in the literature on electromagnetics, where the local, or effective, field is defined as the field at a particular atomic or molecular site [16]). Hereafter, when dealing with internal electric field and related dielectric quantities, we will always intend Emac(r), omitting the dependence from r to lighten notation. The electric polarization vector, P, defined over the mass element of the material in which Emac has established, is related to Emac by [16] P ¼ e0 vE mac
ð1Þ
where e0 is the electric permittivity (also referred to as the dielectric constant) of free space, and v is the linear electric susceptibility of the material. v is generally a tensor in other
P. Bergese / Acta Materialia 54 (2006) 1843–1849
than cubic or isotropic materials. However, a general tensor treatment is out of the scope of this work, which then will be restricted to the isotropic case. P can also be expressed as P ¼ e0 ðe 1ÞE mac
ð2Þ
where e is the linear electric permittivity of the material. Finally, P and Emac are connected by the electric displacement vector, D, which is defined as D ¼ e0 E mac þ P
ð3Þ
The interaction between the material and an alternating electric field of frequency x is a dynamic (nonequilibrium) process, and e and v are frequency dependent. In the frequency domain e and v may have a complex representation [16,25]: 0
00
eðxÞ ¼ e ðxÞ þ ie ðxÞ vðxÞ ¼ v0 ðxÞ þ iv00 ðxÞ ¼ ½e0 ðxÞ 1 þ ie00 ðxÞ
ð4Þ ð5Þ
The real (storage) component, e 0 , characterizes the extent to which charges and dipoles respond to the electric field, and the imaginary component (loss factor), e00 , is related to the ability of the material to dissipate the energy exchanged with the electric field. e 0 and e00 obey the Kramers–Kronig relations. It should be noted that in the frequency domain the relation between polarization and applied field formally retains the expression given by Eq. (1) [16,25], but it describes a nonequilibrium phenomenon. e and v also depend on the material temperature, T, and thus in the case of an alternating Emac of frequency x, by taking into account all the variables on which it depends, P reads (via Eq. (1)) Pðx; T ; rÞ ¼ e0 vðx; T ÞE mac ðx; T ; rÞ
ð6Þ
4. Dynamic specific heat Heating is a time-dependent (nonequilibrium) process which is linked to time-dependent entropy changes. If the rate of change of the external parameters that induce heating is slow enough that the system parameters can instantaneously adjust to them, then the process can be described as a sequence of equilibrium states. In this case heating can be considered reversible and time dependency ruled out. This condition is generally referred to as quasi-equilibrium or quasi-static. When the quasiequilibrium condition is not satisfied the process irreversibility should be accounted for. This may be the case of materials heated by time-modulated heat flows [26] or by MWs. A material that undergoes quasi-equilibrium heating can be described using the static specific heat, related to the equilibrium fluctuations of entropy. Under nonequilibrium heating a dynamic specific heat must be introduced. Besides the static component, the dynamic specific heat has a time-dependent component that accounts for the
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entropy changes out of equilibrium fluctuations, which drive the system through the irreversible process [26,27]. Dynamic specific heat has been the subject of considerable experimental and theoretical studies and is now well accepted [28,29]. Let us now develop and discuss the dynamic specific heat of a multi-component isotropic dielectric material inside a MW field, following the path opened by Adu et al. [11]. Outside the MW field the material is not polarized and the differential of the specific internal energy, u, can be locally expressed by the energy balance equation (first law of thermodynamics) [17]: X du ¼ dq pdv þ lj duj ð7Þ j
where dq is the heat added to a mass element of unit mass, p is the equilibrium pressure, v is the specific volume, and lj and uj are the chemical potential and the weight fraction of the jth component, respectively. Note that if the process is irreversible, dq cannot be replaced by Tds, as is normally done in equilibrium (Gibbsian) thermodynamics. When the material is inside the MW field, the work of electric and magnetic polarization done by the field must be accounted for, and Eq. (7) extends to [17] du ¼ dq pdv þ E mac dðq1 PÞ þ B mac dðq1 MÞ X þ l0j duj
ð8Þ
j
where Emac and Bmac are the instantaneous values of the local macroscopic electric and magnetic fields inside the system which induce the local electric polarization P and the local magnetic polarization M, q is the mass density of the material, and l0j is the chemical potential of the polarized jth component. With few exceptions, the magnetic permeability of dielectrics is close to that of free space [7]. Consequently M 0, and Eq. (8) reads X du ¼ dq pdv þ E mac dðq1 PÞ þ l0j duj ð9Þ j
Eqs. (8) and (9) suggest that polarization due to an external electromagnetic field increases the total internal energy of the material, thus affecting also its other thermodynamic properties. We will now restrict ourselves to the case of the isobaric specific heat, cp, of a closed system (duj = 0). cp is defined as the derivative, at constant pressure, of the specific enthalpy, h, with respect to temperature: dh cp ¼ ð10Þ dT p By substituting Eq. (9) with duj = 0, into the total differential of h, dh ¼ du þ pdv þ vdp
ð11Þ
one obtains dh ¼ dq þ vdp þ E mac dðq1 PÞ
ð12Þ
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P. Bergese / Acta Materialia 54 (2006) 1843–1849
Finally, by merging Eq. (12) and the definition of cp given in Eq. (10) it follows that 1 dq dðq PÞ cp ¼ þ E mac ð13Þ dT p;P dT p The first term on the right-hand side of Eq. (13) represents the isobaric specific heat of the system at constant polarization, i.e., the static component of cp. In particular, outside the polarizing field d(q1P) = 0, and Eq. (13) reduces to the equilibrium definition of specific heat. The polarization P plays the role of the time-dependent internal parameter related to the distance of the system from (global) equilibrium [22]. Hence, the second term on the right-hand side of Eq. (13) identifies the nonequilibrium component of cp. Eq. (13) implies that the specific heat of a material inside and outside an electromagnetic field is different. The relevance of the nonequilibrium component with respect to the equilibrium one indicates if the difference can be neglected or not. Eq. (13) and its interpretation resemble the general definition of the isobaric dynamic specific heat (t is time): cp ðT ; tÞ ¼ cpST ðT Þ þ cpDYN ðT ; tÞ
ð14Þ
where cpST represents the static (equilibrium) component and cpDYN represents the time-dependent (nonequilibrium) component. cpST is determined by the molecular vibrational modes which are fast enough with respect to the experimental (electrothermal coupling) time scale to be considered as quasi-equilibrium dynamics. In contrast, cpDYN is related to slower molecular motions with a time dependence comparable to the experimental time scale, like molecular relaxation, which is the basic mechanism in dipolar molecules rotation, and in turn in MW heating. The concept of dynamic specific heat has been mainly developed in papers dealing with oscillatory calorimetry [26,28], and has been extensively treated with linear NET by Schave [30–33]. cpDYN(T,t) can be handled by the linear response theory [26,32] in the same way as one does for other linear susceptibilities like electric permittivity or compressibility [34]. In particular, if cp(T,t) has a periodic time dependence with angular frequency x, the specific heat in the frequency domain, cp(T,x), is complex and reads [32] cp ðT ; xÞ ¼ ½cpST ðT Þ þ c0pDYN ðT ; xÞ þ ic00pDYN ðT ; xÞ ¼ c0p þ ic00p
ð15Þ
where the real component pertains to the ‘‘storage specific heat’’ and the imaginary one the ‘‘loss specific heat’’, with a physical meaning analogous to that of the complex electric permittivity. A detailed theoretical discussion on complex specific heat can be found in the paper by Nielsen and Dyre [35]. Materials that have experimentally shown dynamic specific heat include glasses at low temperature [36–38] supercooled liquids [26,39] and polymers [27,31,40], supercooled metallic liquids [41] and molecular [42] and polymeric [43]
crystals. All these experiments are linked to oscillatory calorimetry. However, according to Eqs. (13) and (14) one might expect that also some dielectrics under MW heating show a nonnegligible cpDYN. In particular, Eq. (13) indicates that cpDYN may become important when P, and in turn the loss factor e00 , changes with temperature, and this is the case for many MW heating systems [3]. 5. Heat conduction and polarization relaxation As shown by Birge [26], the classic heat equations are not valid for systems with a time-dependent specific heat. In this section we derive and discuss the equilibrium and nonequilibrium heat equations for a closed, isotropic and homogeneous, single-component dielectric system under MW heating at isobaric conditions. These might seem quite severe restrictions, but they are necessary for a first modelling. They include a large number of materials such as pure liquids and pure solids (with negligible evaporation and sublimation, respectively). For the systems considered the local heat balance equation reads [3,44] q
dq ¼ W r jq dt
ð16Þ
where jq is the heat flow vector and W is the average power per unit volume converted into heat by MW/material electrothermal coupling. Assuming that the MW electric field inside the material is sinusoidal with frequency x and neglecting losses by thermal radiation, W reads [3,44] 1 W ¼ xe0 e00 jE mac0 j2 2
ð17Þ
The systems considered can generally be assumed macroscopically motionless, and thus jq is given by the Fourier’s law: j q ¼ jrT
ð18Þ
where j is the thermal conductivity. In the (conventional) quasi-static approximation, considering only isobaric expansive work, dh coincides with dq, and consequently dq can be replaced by cp dT. Here cp is the equilibrium specific heat, or, according to the formalism introduced for the dynamic specific heat, cp = cpST. By these considerations and Eqs. (16)–(18) the quasi-static heat conduction equation reads qcp
dT 1 ¼ xe0 e00 jE mac0 j2 þ jr2 T dt 2
ð19Þ
This is the equation usually encountered in MW heating literature (e.g., see Refs. [6,44]), which, apart from the term accounting for volumetric heat generation (W), is the same for systems heated by a conventional external source in a quasi-static regime [26]. It will be shown now that under nonequilibrium conditions jq depends on both thermal conduction and polarization. Consequently, Eqs. (18) and (19) are no longer valid
P. Bergese / Acta Materialia 54 (2006) 1843–1849
and the dielectric functions should be revisited. Before calculating jq we shall recall some basic equations of linear NET. The variation of the entropy of the system, dS, is postulated to be composed of the sum of the entropy exchanged with the surroundings, deS, and the entropy produced inside the system, diS: dS ¼ de S þ di S
ð20Þ
In this formulation the second law of thermodynamics reads ð21Þ
di S P 0
where diS is zero for reversible processes and positive for irreversible ones. The exchanged entropy deS may be positive, zero or negative, depending on the interaction of the system with the surroundings. diS is generally expressed by the local entropy production (also referred as entropy source strength), r, which is defined as follows: Z di S ¼ r dV ; r P 0 ð22Þ dt V where V is the system volume. The main objective of linear NET is to obtain phenomenological equations by relating r to the various irreversible phenomena occurring in the system. The system considered in this work has a nonuniform temperature distribution and is inside an electric field which produces a polarization P(t). P(t) reaches its equilibrium value, P(0) = e0v(0)Emac, in a period of time related to the system dipolar relaxation time. The irreversible phenomena are then heat conduction and polarization relaxation, and, according to DeGroot and Mazur [17], the entropy source strength is given by r¼
1 1 dP ðP e0 vð0ÞE mac Þ j rT 2 q T e dt vð0Þ T 0
ð23Þ
where v(0) is the equilibrium value of the linear electric susceptibility. The entropy source strength vanishes at equilibrium, when uniform temperature is attained ($T = 0) and P assumes its equilibrium value. Using the flows and thermodynamic forces defined by Eq. (23), the phenomenological equations can be written Lqq LqP ðP e0 vð0ÞE mac Þ rT T e0 vð0Þ T2 dP LPq LPP ¼ 2 rT ðP e0 vð0ÞE mac Þ dt T e0 vð0Þ T
jq ¼
ð24Þ ð25Þ
The flows and forces concerned with the entropy source strength are of the same tensorial rank, and thus can couple (Curie–Prigogine principle) and the cross-coefficients LqP an LPq are not zero. LqP and LPq are connected by the Onsager reciprocal relation LqP ¼ LPq
ð26Þ
and the fact that r is positive definite implies that Lqq > 0;
LPP > 0 and
Lqq LPP > L2qP
ð27Þ
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Lqq, LqP and LPP are generally tensors, but in the isotropic case reduce to scalars. Now, Lqq and LPP can be replaced in Eqs. (24) and (25) by the more familiar thermal conductivity, j, and relaxation time, s: Lqq T2 T e0 vð0Þ s¼ LPP
j¼
ð28Þ ð29Þ
The final expressions for the heat current density and the polarization relaxation are then LqP ðP e0 vð0ÞE mac Þ; T e0 vð0Þ dP LqP ðP e0 vð0ÞE mac Þ ¼ 2 rT dt s T
j q ¼ jrT
ð30Þ ð31Þ
Eq. (30) indicates that the heat flow is determined by both thermal conduction and polarization relaxation, and quantitatively describes a particular excess effect due to MW/ matter interaction. Actually, the second term contributing to the heat flow arises from the polarization induced by the MW field, and vanishes if the case of conventional heating (see Appendix A). On the other hand, Eq. (31) suggests that dipolar relaxation is influenced by the thermal gradient. The main implication of this is that the relation between the instantaneous polarization, P, and the electric field, Emac, is given by the solution of Eq. (31). Consequently, the classic relations between P and Emac are no longer valid. As shown in Appendix A, if $T = 0 Eq. (31) reduces to the Debye relaxation equation. Eqs. (16), (17) and (30) can be cast in order to obtain the heat conduction equation. In the system, together with the expansive work, some work related to electric polarization also appears (Eq. (9)). Consequently, even if the transformation is isobaric, dh and dq do not coincide. However, according to Eqs. (13) and (14), at constant pressure dq can still be replaced by cpSTdT. One might notice that this substitution is just formally analogous to the one that brought us from Eq. (16) to Eq. (19), but it has a different physical meaning. With the help of this consideration and by using Eq. (17), we can rewrite Eq. (16) as follows: qcpST
dT 1 2 ¼ xe0 e00 jE mac0 j r j q dt 2
ð32Þ
The explicit time dependence of the temperature, T(t), can be obtained by solving in the appropriate geometry and experimental conditions the system of equations given by Eqs. (30)–(32). It is straightforward to verify that the system reduces to the conventional Eq. (19) if the contribution of the polarization relaxation to the heat current density disappears. The effectiveness of such a contribution, and in turn the evaluation of the dominant thermodynamic force within the system, can be quantified by the so-called level of coupling [45]. However, this aspect, as more sophisticated NET approaches, is out of the scope of this paper, which is a first tentative striking out in a new direction of theoretical
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P. Bergese / Acta Materialia 54 (2006) 1843–1849
analysis of MW heating systems. Works in the directions mentioned above are necessary and underway. 6. Conclusions The interaction between MWs and matter and MW heating were macroscopically investigated using linear NET. Attention was focused on specific heat, polarization relaxation and heat conduction of closed, isotropic and homogeneous, single-component dielectric systems at isobaric conditions. Results showed that the nonequilibrium polarization induced in the material by the MW field produces modifications of the thermodynamic potentials, and in turn of all the thermodynamic parameters. Such modifications can be interpreted as MW excess (or athermal) effects with respect to quasi-static conventional heating. The specific heat turned out to be a dynamic property (in agreement with the dynamic specific heat introduced in oscillatory calorimetry) with a time-dependent component linked to polarization. Analogously, heat conduction is influenced by polarization relaxation and vice versa. Linear NET allowed the quantification of these effects in the phenomenological equations for heat flow and polarization relaxation. The theory has been developed for materials with particular thermodynamic and symmetry properties in order to focus on the NET description rather than on mathematics, and to take advantage of potentially ‘‘easy’’ experiments. However, the extension of the linear NET theory to nonhomogeneous, multi-component, open systems in which chemical reactions, viscous flows, thermal diffusion and relaxation of the magnetic polarization occur seems possible [17]. Acknowledgments The author thanks for valuable discussions and suggestions Jose` Maria Castillo, Italo Colombo, Laura E. Depero and Francesco Gonella. The author is grateful to Eurand S.p.A for financial support by the post-doctoral grant ‘‘Termodinamica e ottimizzazione di processi di diffusione indotti da microonde (MIND) per l’attivazione di farmaci scarsamente biodisponibili’’. Appendix A This appendix is devoted to show that the conventional phenomenological equations for thermal conduction (Fourier’s law) and polarization relaxation (Debye relaxation equation) are particular cases of the general treatment presented in the paper. If no polarization relaxation exists, then dP/ddt = 0 and thus Eq. (23) reduces to 1 j q rT ðA:1Þ T2 The corresponding phenomenological equation is the Fourier’s law:
r¼
j q ¼ jrT
ðA:2Þ
Of course, no phenomenological equation for polarization relaxation can be obtained. In the absence of a thermal gradient, Eq. (23) becomes r¼
1 dP ðP e0 vð0ÞE mac Þ T e0 vð0Þ dt
ðA:3Þ
and the corresponding phenomenological equation reads dP ðP e0 vð0ÞE mac Þ ¼ dt s
ðA:4Þ
which is the Debye relaxation equation [16]. In this case no phenomenological equation for the heat current density exists. References [1] Kingston HM, Haswell SJ, editors. Microwave-enhanced chemistry. Washington (DC): American Chemical Society; 1997. [2] Committee on microwave processing of materials: an emerging industrial technology. Microwave processing of materials, publication NMAB-473. Washington (DC): National Academy Press; 1994. [3] Metaxas AC, Meredith RJ. Industrial microwave heating. London: Peter Peregrinus; 1993. [4] Metaxas AC. Foundations of electroheat. New York (NY): Wiley; 1996. [5] Bergese P, Colombo I, Gervasoni D, Depero LE. Mater Sci Eng C 2003;23:791. [6] Booske JH, Cooper RF, Freeman SA, Rybakov K, Semenov V. Phys Plasmas 1998;5:1664. [7] Ponne CT, Bartels PV. Radiat Phys Chem 1995;45:591. [8] Lee KY, Ph.D. thesis, Michigan State University, 1998. [9] Gibson C, Metthews I, Samuel A. J Microwave Power E E 1988;23:17. [10] Rybakov KI, Semenov V. Phys Rev B 1994;39:64. [11] Adu B, Otten L, Groenevelt P. J Microwave Power E E 1995;30:90. [12] Stuerga DAC, Gaillard P. J Microwave Power E E 1996;31:87. [13] Stuerga DAC, Gaillard P. J Microwave Power E E 1996;31:101. [14] Jeon JY, Kim YH. Eur Polym J 2000;36:895. [15] Adair RK. Bioelectromagnetics 2003;24:39. [16] Elliot S. Physics and chemistry of solids. Chichester: Wiley; 2000. p. 537. [17] De Groot SR, Mazur P. Non-equilibrium thermodynamics. New York (NY): Dover; 1984. [18] Demirel Y, Sandler SI. J Phys Chem B 2004;108:31. [19] Prigogine I. Termodinamica dei processi irreversibili. Rome: Leonardo Edizioni Scientifiche; 1971 (Translation of the original Introduction to thermodynamics of irreversible processes, 3rd ed. New York (NY): Wiley.). [20] Katchalsky A, Curran PF. Nonequilibrium thermodynamics in biophysics. Cambridge: Harvard University Press; 1965. [21] Lavenda BH. Thermodynamics of irreversible processes. New York (NY): Dover; 1993. [22] Kondepudi D, Prigogine I. Modern thermodynamics. From heat engines to dissipative structures. New York (NY): Wiley; 1999. [23] Demirel Y. Nonequilibrium thermodynamics: transport and rate processes in physical and biological processes. Amsterdam: Elsevier; 2002. [24] Sanfeld A, Velarde MG. J Non-Equilib Thermodyn 2004;29:1. [25] Hasted JB. Aqueous dielectrics. London: Chapman and Hall; 1973. p. 1. [26] Birge NO. Phys Rev B 1986;34:1631. [27] Beiner M, Korus J, Lockwenz H, Schroter K, Donth E. Macromolecules 1996;29:5183.
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Jeong YH. Thermochim Acta 2001;377:1. Simon SL, McKenna GB. J Chem Phys 1997;107:8678. Schawe JEK. Thermochim Acta 1995;260:1. Schawe JEK. Thermochim Acta 1995;261:183. Schawe JEK. Thermochim Acta 1996;287:213. Schawe JEK. Thermochim Acta 1997;304/305:111. Young RJ. Introduction to polymers. London: Chapman and Hall; 1981. p. 229. [35] Nielsen JK, Dyre JC. Phys Rev B 1996;54:1754. [36] Black JL. Phys Rev B 1978;17:2740. [37] Loponen MT, Dynes RC, Narayanamurty V, Garno JP. Phys Rev Lett 1980;45:457.
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Specific heat, polarization and heat conduction in microwave heating systems: A nonequilibrium thermodynamic point of view Paolo Bergese
*
INSTM and Chemistry for Technologies Laboratory, University of Brescia, Via Branze 38, 25123 Brescia, Italy Received 30 September 2005; received in revised form 15 November 2005; accepted 21 November 2005 Available online 10 February 2006 Dedicated to my newborn son, ‘‘il gufetto’’ Leonardo.
Abstract A microwave (MW) field can induce in a dielectric material an oscillatory polarization. By this mechanism part of the energy carried by the waves is converted into chaotic agitation, and the material heats up. MW heating is a nonequilibrium phenomenon, while conventional heating can generally be considered as quasi-static. Excess (or nonthermal) effects of MWs with respect to conventional heating lie in this difference. Macroscopically, MW heating can be described in the framework of linear nonequilibrium thermodynamics (NET). This approach indicates that in a dielectric material under MW heating the specific heat has a dynamic component linked to the variation of polarization with temperature, and that polarization and heat conduction are intertwined. In particular, linear NET provides a new phenomenological equation for heat conduction that is composed of the classic Fourier’s law and an additional term due to polarization relaxation. This term quantitatively describes the excess effect of MWs on thermal conduction. Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dielectrics; Thermodynamics; Microwaves
1. Introduction Microwave (MW) heating is today a mature technique which finds wide application in chemistry [1] as well as in materials [2–6] and food [7] processing. Heat is generated at each point of the material by the dissipation of MW energy as a consequence of the interaction of the MW field with the molecular and electronic structure of the material. Therefore, MWs generate in the material volumetric, fast and selective heating, which generally results in process energy and time savings [1–3,6]. Furthermore, MW/ material interaction generates other unique excess effects (also referred as nonthermal or athermal effects), including enhanced chemical reaction rates [1] lower sintering temperatures [6,8], lower diffusion activation energies [6,9] selflimiting reactions [1] and plastic deformations (in ionic *
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crystals) [10]. Despite the fact that they often represent the overriding advantages of MW processing over conventional heating, excess effects are still not well understood [2,6], and hence are generally omitted in heating models [11]. While much experimental work has shown the overall differences between MW and conventional heating, only few studies have succeeded in isolating a particular excess effect [6]. Also for this reason excess effects remain an open and debated subject in all the fundamental fields touched by MW processing: chemistry [12,13], and materials [6,14] and biomaterials [15] science. MW enhancement of mass transport in ionic crystalline solids is an exception. This phenomenon has been experimentally investigated to some extent, and theoretically justified at the microscopic scale as the consequence of a MW-driven ponderomotive force applied to mobile charged vacancies (see Ref. [6] and references therein). In this paper MW heating of dielectric materials is described macroscopically using linear nonequilibrium
1359-6454/$30.00 Ó 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.11.042
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thermodynamics (NET). In particular, after introducing the dynamic specific heat of dielectrics under MW heating, phenomenological expressions for MW excess effects on heat conduction and polarization relaxation are presented and discussed. These preliminary theoretical results suggest that the NET framework is a valid candidate for describing MW heating. 2. Linear nonequilibrium thermodynamics and microwave heating of dielectric systems The dominant role in MW heating of dielectrics is played by molecular dipoles, which show relaxation times of the same order of magnitude as MW frequencies. The interaction mechanisms between MW electric field and molecules (electrothermal coupling) consist of rotation and (re)orientation of the molecules with permanent or induced dipoles [3]. These molecules rotate against the resistive forces of the surroundings, a part of the electromagnetic energy is converted into chaotic agitation, and the material heats up. The Debye relaxation equation [16] (1929) is the accepted phenomenological theory to model the dipole orientation mechanism, and, in turn, to describe MW interaction with matter and heating [3]. The molecular dipole is assumed to respond to the torque moment produced by the external electric field as it would do a sphere rotating in a fluid of given viscosity. The model is based on the linear response approximation and holds for systems consisting of molecules that have permanent dipoles and are not deformable (purely polar systems). It predicts a time-independent relaxation time, s, which is determined by the interactions of the dipole with its surroundings. The Debye relaxation equation can also be derived by describing the irreversible interaction between MWs and dielectric materials in the theoretical framework of linear NET [11,17]. As for the Debye phenomenological theory, the linear NET approach is based on the approximation that MW heating occurs via linear response phenomena, or, using the NET formalism, that the irreversible flows occurring in the system are linear functions of the thermodynamic forces. Linear NET relies on the validity of the Onsager law and of the Curie–Prigogine principle, and on the assumption of local equilibrium. They allow one to express in phenomenological equations the correlation between entropy production and thermodynamic flows and forces. In the local equilibrium framework the system is divided in arbitrary mass elements large enough to define the thermodynamic potentials but small enough to neglect the gradients. It is then postulated that even if the system is not in equilibrium it is possible to isolate a mass element that is in equilibrium. This implies that the specific entropy of such mass element is the same function of the extensive thermodynamic variables as it is when the whole system is in equilibrium; that is, the mass element can be described by the (equilibrium) Gibbs equation. In conditions of local equilibrium the thermodynamic variables are then locally
defined, and can be considered continuous functions of space coordinates and time. The basic assumptions for applying linear NET can be fully justified only a posteriori by the validity of the drawn conclusions. However, they were found to be valid for a wide range of macroscopic systems [18] and in the case of MW/matter interaction they lead to the well-accepted Debye equation. Therefore we assume that linear NET gives a reliable description of MW heating systems. Further details of the foundations and validity limits of linear NET or of NET criticisms and alternative formulations are out of the scope of this paper. Extensive information can be drawn from the several (classic [17,19,20], critical [21] and recent [22,23]) monographs on NET and from some very recent review papers [18,24]. 3. Local electric field and dielectric functions In this section some concepts and constitutive relations that will be needed in the following are fixed. In general, the macroscopic electric field inside a polarizable material, Emac, is not the same as the applied external field because the electric dipoles in the material produce a depolarization field which opposes the external field. Moreover, Emac is different from the field at a particular point, Emic (microscopic electric field), and both of them depend on the sample geometry. Analogous considerations hold for magnetic fields. The electric charge density at a point r inside the system, lmic(r), is a very rapidly varying function of r on the length scale of the atomic spacing. Consequently Emic(r), which is related to lmic(r) by one of the Maxwell equations, is also a fast-varying quantity. Emac(r) can be defined as the average of Emic(r) taken over a mass element of volume V. V has a width, s, much larger than the atomic spacing, but appreciably smaller than the material volume, so that the spatial fluctuations of the field at the atomic length scale are averaged out by Emac(r), but spatial variations of the field on a length scale greater than s are retained by Emac(r). In other words, Emac(r) is a local quantity in the meaning of linear NET elucidated in the previous section (note that this concept of local electric field has nothing to do with the one generally encountered in the literature on electromagnetics, where the local, or effective, field is defined as the field at a particular atomic or molecular site [16]). Hereafter, when dealing with internal electric field and related dielectric quantities, we will always intend Emac(r), omitting the dependence from r to lighten notation. The electric polarization vector, P, defined over the mass element of the material in which Emac has established, is related to Emac by [16] P ¼ e0 vE mac
ð1Þ
where e0 is the electric permittivity (also referred to as the dielectric constant) of free space, and v is the linear electric susceptibility of the material. v is generally a tensor in other
P. Bergese / Acta Materialia 54 (2006) 1843–1849
than cubic or isotropic materials. However, a general tensor treatment is out of the scope of this work, which then will be restricted to the isotropic case. P can also be expressed as P ¼ e0 ðe 1ÞE mac
ð2Þ
where e is the linear electric permittivity of the material. Finally, P and Emac are connected by the electric displacement vector, D, which is defined as D ¼ e0 E mac þ P
ð3Þ
The interaction between the material and an alternating electric field of frequency x is a dynamic (nonequilibrium) process, and e and v are frequency dependent. In the frequency domain e and v may have a complex representation [16,25]: 0
00
eðxÞ ¼ e ðxÞ þ ie ðxÞ vðxÞ ¼ v0 ðxÞ þ iv00 ðxÞ ¼ ½e0 ðxÞ 1 þ ie00 ðxÞ
ð4Þ ð5Þ
The real (storage) component, e 0 , characterizes the extent to which charges and dipoles respond to the electric field, and the imaginary component (loss factor), e00 , is related to the ability of the material to dissipate the energy exchanged with the electric field. e 0 and e00 obey the Kramers–Kronig relations. It should be noted that in the frequency domain the relation between polarization and applied field formally retains the expression given by Eq. (1) [16,25], but it describes a nonequilibrium phenomenon. e and v also depend on the material temperature, T, and thus in the case of an alternating Emac of frequency x, by taking into account all the variables on which it depends, P reads (via Eq. (1)) Pðx; T ; rÞ ¼ e0 vðx; T ÞE mac ðx; T ; rÞ
ð6Þ
4. Dynamic specific heat Heating is a time-dependent (nonequilibrium) process which is linked to time-dependent entropy changes. If the rate of change of the external parameters that induce heating is slow enough that the system parameters can instantaneously adjust to them, then the process can be described as a sequence of equilibrium states. In this case heating can be considered reversible and time dependency ruled out. This condition is generally referred to as quasi-equilibrium or quasi-static. When the quasiequilibrium condition is not satisfied the process irreversibility should be accounted for. This may be the case of materials heated by time-modulated heat flows [26] or by MWs. A material that undergoes quasi-equilibrium heating can be described using the static specific heat, related to the equilibrium fluctuations of entropy. Under nonequilibrium heating a dynamic specific heat must be introduced. Besides the static component, the dynamic specific heat has a time-dependent component that accounts for the
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entropy changes out of equilibrium fluctuations, which drive the system through the irreversible process [26,27]. Dynamic specific heat has been the subject of considerable experimental and theoretical studies and is now well accepted [28,29]. Let us now develop and discuss the dynamic specific heat of a multi-component isotropic dielectric material inside a MW field, following the path opened by Adu et al. [11]. Outside the MW field the material is not polarized and the differential of the specific internal energy, u, can be locally expressed by the energy balance equation (first law of thermodynamics) [17]: X du ¼ dq pdv þ lj duj ð7Þ j
where dq is the heat added to a mass element of unit mass, p is the equilibrium pressure, v is the specific volume, and lj and uj are the chemical potential and the weight fraction of the jth component, respectively. Note that if the process is irreversible, dq cannot be replaced by Tds, as is normally done in equilibrium (Gibbsian) thermodynamics. When the material is inside the MW field, the work of electric and magnetic polarization done by the field must be accounted for, and Eq. (7) extends to [17] du ¼ dq pdv þ E mac dðq1 PÞ þ B mac dðq1 MÞ X þ l0j duj
ð8Þ
j
where Emac and Bmac are the instantaneous values of the local macroscopic electric and magnetic fields inside the system which induce the local electric polarization P and the local magnetic polarization M, q is the mass density of the material, and l0j is the chemical potential of the polarized jth component. With few exceptions, the magnetic permeability of dielectrics is close to that of free space [7]. Consequently M 0, and Eq. (8) reads X du ¼ dq pdv þ E mac dðq1 PÞ þ l0j duj ð9Þ j
Eqs. (8) and (9) suggest that polarization due to an external electromagnetic field increases the total internal energy of the material, thus affecting also its other thermodynamic properties. We will now restrict ourselves to the case of the isobaric specific heat, cp, of a closed system (duj = 0). cp is defined as the derivative, at constant pressure, of the specific enthalpy, h, with respect to temperature: dh cp ¼ ð10Þ dT p By substituting Eq. (9) with duj = 0, into the total differential of h, dh ¼ du þ pdv þ vdp
ð11Þ
one obtains dh ¼ dq þ vdp þ E mac dðq1 PÞ
ð12Þ
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Finally, by merging Eq. (12) and the definition of cp given in Eq. (10) it follows that 1 dq dðq PÞ cp ¼ þ E mac ð13Þ dT p;P dT p The first term on the right-hand side of Eq. (13) represents the isobaric specific heat of the system at constant polarization, i.e., the static component of cp. In particular, outside the polarizing field d(q1P) = 0, and Eq. (13) reduces to the equilibrium definition of specific heat. The polarization P plays the role of the time-dependent internal parameter related to the distance of the system from (global) equilibrium [22]. Hence, the second term on the right-hand side of Eq. (13) identifies the nonequilibrium component of cp. Eq. (13) implies that the specific heat of a material inside and outside an electromagnetic field is different. The relevance of the nonequilibrium component with respect to the equilibrium one indicates if the difference can be neglected or not. Eq. (13) and its interpretation resemble the general definition of the isobaric dynamic specific heat (t is time): cp ðT ; tÞ ¼ cpST ðT Þ þ cpDYN ðT ; tÞ
ð14Þ
where cpST represents the static (equilibrium) component and cpDYN represents the time-dependent (nonequilibrium) component. cpST is determined by the molecular vibrational modes which are fast enough with respect to the experimental (electrothermal coupling) time scale to be considered as quasi-equilibrium dynamics. In contrast, cpDYN is related to slower molecular motions with a time dependence comparable to the experimental time scale, like molecular relaxation, which is the basic mechanism in dipolar molecules rotation, and in turn in MW heating. The concept of dynamic specific heat has been mainly developed in papers dealing with oscillatory calorimetry [26,28], and has been extensively treated with linear NET by Schave [30–33]. cpDYN(T,t) can be handled by the linear response theory [26,32] in the same way as one does for other linear susceptibilities like electric permittivity or compressibility [34]. In particular, if cp(T,t) has a periodic time dependence with angular frequency x, the specific heat in the frequency domain, cp(T,x), is complex and reads [32] cp ðT ; xÞ ¼ ½cpST ðT Þ þ c0pDYN ðT ; xÞ þ ic00pDYN ðT ; xÞ ¼ c0p þ ic00p
ð15Þ
where the real component pertains to the ‘‘storage specific heat’’ and the imaginary one the ‘‘loss specific heat’’, with a physical meaning analogous to that of the complex electric permittivity. A detailed theoretical discussion on complex specific heat can be found in the paper by Nielsen and Dyre [35]. Materials that have experimentally shown dynamic specific heat include glasses at low temperature [36–38] supercooled liquids [26,39] and polymers [27,31,40], supercooled metallic liquids [41] and molecular [42] and polymeric [43]
crystals. All these experiments are linked to oscillatory calorimetry. However, according to Eqs. (13) and (14) one might expect that also some dielectrics under MW heating show a nonnegligible cpDYN. In particular, Eq. (13) indicates that cpDYN may become important when P, and in turn the loss factor e00 , changes with temperature, and this is the case for many MW heating systems [3]. 5. Heat conduction and polarization relaxation As shown by Birge [26], the classic heat equations are not valid for systems with a time-dependent specific heat. In this section we derive and discuss the equilibrium and nonequilibrium heat equations for a closed, isotropic and homogeneous, single-component dielectric system under MW heating at isobaric conditions. These might seem quite severe restrictions, but they are necessary for a first modelling. They include a large number of materials such as pure liquids and pure solids (with negligible evaporation and sublimation, respectively). For the systems considered the local heat balance equation reads [3,44] q
dq ¼ W r jq dt
ð16Þ
where jq is the heat flow vector and W is the average power per unit volume converted into heat by MW/material electrothermal coupling. Assuming that the MW electric field inside the material is sinusoidal with frequency x and neglecting losses by thermal radiation, W reads [3,44] 1 W ¼ xe0 e00 jE mac0 j2 2
ð17Þ
The systems considered can generally be assumed macroscopically motionless, and thus jq is given by the Fourier’s law: j q ¼ jrT
ð18Þ
where j is the thermal conductivity. In the (conventional) quasi-static approximation, considering only isobaric expansive work, dh coincides with dq, and consequently dq can be replaced by cp dT. Here cp is the equilibrium specific heat, or, according to the formalism introduced for the dynamic specific heat, cp = cpST. By these considerations and Eqs. (16)–(18) the quasi-static heat conduction equation reads qcp
dT 1 ¼ xe0 e00 jE mac0 j2 þ jr2 T dt 2
ð19Þ
This is the equation usually encountered in MW heating literature (e.g., see Refs. [6,44]), which, apart from the term accounting for volumetric heat generation (W), is the same for systems heated by a conventional external source in a quasi-static regime [26]. It will be shown now that under nonequilibrium conditions jq depends on both thermal conduction and polarization. Consequently, Eqs. (18) and (19) are no longer valid
P. Bergese / Acta Materialia 54 (2006) 1843–1849
and the dielectric functions should be revisited. Before calculating jq we shall recall some basic equations of linear NET. The variation of the entropy of the system, dS, is postulated to be composed of the sum of the entropy exchanged with the surroundings, deS, and the entropy produced inside the system, diS: dS ¼ de S þ di S
ð20Þ
In this formulation the second law of thermodynamics reads ð21Þ
di S P 0
where diS is zero for reversible processes and positive for irreversible ones. The exchanged entropy deS may be positive, zero or negative, depending on the interaction of the system with the surroundings. diS is generally expressed by the local entropy production (also referred as entropy source strength), r, which is defined as follows: Z di S ¼ r dV ; r P 0 ð22Þ dt V where V is the system volume. The main objective of linear NET is to obtain phenomenological equations by relating r to the various irreversible phenomena occurring in the system. The system considered in this work has a nonuniform temperature distribution and is inside an electric field which produces a polarization P(t). P(t) reaches its equilibrium value, P(0) = e0v(0)Emac, in a period of time related to the system dipolar relaxation time. The irreversible phenomena are then heat conduction and polarization relaxation, and, according to DeGroot and Mazur [17], the entropy source strength is given by r¼
1 1 dP ðP e0 vð0ÞE mac Þ j rT 2 q T e dt vð0Þ T 0
ð23Þ
where v(0) is the equilibrium value of the linear electric susceptibility. The entropy source strength vanishes at equilibrium, when uniform temperature is attained ($T = 0) and P assumes its equilibrium value. Using the flows and thermodynamic forces defined by Eq. (23), the phenomenological equations can be written Lqq LqP ðP e0 vð0ÞE mac Þ rT T e0 vð0Þ T2 dP LPq LPP ¼ 2 rT ðP e0 vð0ÞE mac Þ dt T e0 vð0Þ T
jq ¼
ð24Þ ð25Þ
The flows and forces concerned with the entropy source strength are of the same tensorial rank, and thus can couple (Curie–Prigogine principle) and the cross-coefficients LqP an LPq are not zero. LqP and LPq are connected by the Onsager reciprocal relation LqP ¼ LPq
ð26Þ
and the fact that r is positive definite implies that Lqq > 0;
LPP > 0 and
Lqq LPP > L2qP
ð27Þ
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Lqq, LqP and LPP are generally tensors, but in the isotropic case reduce to scalars. Now, Lqq and LPP can be replaced in Eqs. (24) and (25) by the more familiar thermal conductivity, j, and relaxation time, s: Lqq T2 T e0 vð0Þ s¼ LPP
j¼
ð28Þ ð29Þ
The final expressions for the heat current density and the polarization relaxation are then LqP ðP e0 vð0ÞE mac Þ; T e0 vð0Þ dP LqP ðP e0 vð0ÞE mac Þ ¼ 2 rT dt s T
j q ¼ jrT
ð30Þ ð31Þ
Eq. (30) indicates that the heat flow is determined by both thermal conduction and polarization relaxation, and quantitatively describes a particular excess effect due to MW/ matter interaction. Actually, the second term contributing to the heat flow arises from the polarization induced by the MW field, and vanishes if the case of conventional heating (see Appendix A). On the other hand, Eq. (31) suggests that dipolar relaxation is influenced by the thermal gradient. The main implication of this is that the relation between the instantaneous polarization, P, and the electric field, Emac, is given by the solution of Eq. (31). Consequently, the classic relations between P and Emac are no longer valid. As shown in Appendix A, if $T = 0 Eq. (31) reduces to the Debye relaxation equation. Eqs. (16), (17) and (30) can be cast in order to obtain the heat conduction equation. In the system, together with the expansive work, some work related to electric polarization also appears (Eq. (9)). Consequently, even if the transformation is isobaric, dh and dq do not coincide. However, according to Eqs. (13) and (14), at constant pressure dq can still be replaced by cpSTdT. One might notice that this substitution is just formally analogous to the one that brought us from Eq. (16) to Eq. (19), but it has a different physical meaning. With the help of this consideration and by using Eq. (17), we can rewrite Eq. (16) as follows: qcpST
dT 1 2 ¼ xe0 e00 jE mac0 j r j q dt 2
ð32Þ
The explicit time dependence of the temperature, T(t), can be obtained by solving in the appropriate geometry and experimental conditions the system of equations given by Eqs. (30)–(32). It is straightforward to verify that the system reduces to the conventional Eq. (19) if the contribution of the polarization relaxation to the heat current density disappears. The effectiveness of such a contribution, and in turn the evaluation of the dominant thermodynamic force within the system, can be quantified by the so-called level of coupling [45]. However, this aspect, as more sophisticated NET approaches, is out of the scope of this paper, which is a first tentative striking out in a new direction of theoretical
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P. Bergese / Acta Materialia 54 (2006) 1843–1849
analysis of MW heating systems. Works in the directions mentioned above are necessary and underway. 6. Conclusions The interaction between MWs and matter and MW heating were macroscopically investigated using linear NET. Attention was focused on specific heat, polarization relaxation and heat conduction of closed, isotropic and homogeneous, single-component dielectric systems at isobaric conditions. Results showed that the nonequilibrium polarization induced in the material by the MW field produces modifications of the thermodynamic potentials, and in turn of all the thermodynamic parameters. Such modifications can be interpreted as MW excess (or athermal) effects with respect to quasi-static conventional heating. The specific heat turned out to be a dynamic property (in agreement with the dynamic specific heat introduced in oscillatory calorimetry) with a time-dependent component linked to polarization. Analogously, heat conduction is influenced by polarization relaxation and vice versa. Linear NET allowed the quantification of these effects in the phenomenological equations for heat flow and polarization relaxation. The theory has been developed for materials with particular thermodynamic and symmetry properties in order to focus on the NET description rather than on mathematics, and to take advantage of potentially ‘‘easy’’ experiments. However, the extension of the linear NET theory to nonhomogeneous, multi-component, open systems in which chemical reactions, viscous flows, thermal diffusion and relaxation of the magnetic polarization occur seems possible [17]. Acknowledgments The author thanks for valuable discussions and suggestions Jose` Maria Castillo, Italo Colombo, Laura E. Depero and Francesco Gonella. The author is grateful to Eurand S.p.A for financial support by the post-doctoral grant ‘‘Termodinamica e ottimizzazione di processi di diffusione indotti da microonde (MIND) per l’attivazione di farmaci scarsamente biodisponibili’’. Appendix A This appendix is devoted to show that the conventional phenomenological equations for thermal conduction (Fourier’s law) and polarization relaxation (Debye relaxation equation) are particular cases of the general treatment presented in the paper. If no polarization relaxation exists, then dP/ddt = 0 and thus Eq. (23) reduces to 1 j q rT ðA:1Þ T2 The corresponding phenomenological equation is the Fourier’s law:
r¼
j q ¼ jrT
ðA:2Þ
Of course, no phenomenological equation for polarization relaxation can be obtained. In the absence of a thermal gradient, Eq. (23) becomes r¼
1 dP ðP e0 vð0ÞE mac Þ T e0 vð0Þ dt
ðA:3Þ
and the corresponding phenomenological equation reads dP ðP e0 vð0ÞE mac Þ ¼ dt s
ðA:4Þ
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