- Email: [email protected]
Effects of gradient of polarization on dielectric relaxation
Physica A 268 (1999) 469–481
www.elsevier.com/locate/physa
E ects of gradient of polarization on dielectric relaxation a Instituto
L.F. Del Castilloa; ∗ , R. Daz-Callejab
de Investigaciones en Materiales, UNAM, Apartado Postal 70-360, 04510 MÃexico, D.F., Mexico b Departamento de Termodinà amica Aplicada, Universidad PolitÃecnica de Valencia, Valencia, Spain Received 2 June 1998
Abstract We present a non-equilibrium thermodynamic model to describe di usion e ects in dielectric materials with spatial inhomogeneities in the polarization vector and with local viscoelastic e ects. The model presented here is a generalization of the Debye relaxation equation including inertial e ect, and contributions from the spatial inhomogeneities of the polarization vector, together with a contribution from the antisymmetric stress tensor via the divergence operator. At the same time, a Maxwell viscoelastic-relaxation type equation for the antisymmetric stress tensor is proposed to describe the time evolution of this tensor. On the other hand, a generalized hydrodynamics model for the dielectric memory for short wave length and high frequencies is obtained, by considering the linearized complete set of di erential equations of the model. By working in the Fourier–Laplace space, the dielectric susceptibility is obtained and their main feac 1999 Elsevier Science tures are described and compared with and dielectric relaxation. B.V. All rights reserved. PACS: 05.70Ln; 05.40+j; 44.60+k
1. Introduction Previous models for di usion dielectric relaxation following the continuum formalism have been given by Felderhof [1], Kroh and Felderhof [2], Hubbard and Stiles [3,4] and Van der Zwan and Hynes [5]. The new aspect we deal with is related to the coupling between the tensor and vector quantities which are forbidden using the lineal non-equilibrium thermodynamics [6]. In our case, we use the rules provided by ∗
Corresponding author. Fax: +52-5-6161201. E-mail address: [email protected] (L.F. Del Castillo)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 0 5 7 - 6
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extended irreversible thermodynamics (EIT) [7–9] to introduce the modi cations into the relaxing equations. In a previous paper using EIT, del Castillo and Davalos-Orozco [10] described the velocity e ect on the dielectric relaxation memory. They, used a modi ed Debye relaxation type-equation including spatial inhomogeneities coupling terms. In another paper, Davalos-Orozco and del Castillo [11] included the rotation of the dipole domain to show that di usion of polarization charges is induced when translation of localities takes place. In this paper we study the case of polarization di usion produced by spatial inhomogeneities, when the polarization is produced by an external electric eld. Here, we take into account the di usion of polarization through the system. From the microscopic point of view, the polarization di usion process may be understood as a manifestation of the long range interaction due to dipoles, where the orientation of one dipole is a ected by the orientation of other dipoles. In the present paper, we describe the coupling between the polarization vector and the antisymmetric stress tensor, which account for the relative rotation of a group of particles in a locality. The obtained result can be used to describe the coupling between the antisymmetric stress tensor uctuation in polymeric systems and the polarization dynamics; an e ect which can be recognized in the spectra of dielectric relaxation as -relaxation. We begin in Section 2 considering the relation of the memory kernel with the electric susceptibility in the transformed Fourier representation. In Section 3, we present a generalized hydrodynamics model through the used of two coupled di erential equations. One for the polarization vector, and the other for the antisymmetric stress tensor, the derivation of these equations are given in Appendix A. In Section 4, we describe the complex electric susceptibility tensor with transversal and longitudinal components. The comparison between our results and the spectral response with and dielectric relaxation is presented at the end of Section 4.
2. Non-equilibrium thermodynamic model for non-uniform polarization systems To describe the non equilibrium state of a polar viscoelastic-liquid in the frame of Extended Irreversible Thermodynamics (EIT) [7–9], it is necessary to specify the set of independent variables as slow and fast variables of the thermodynamic system. The slow variables are de ned as the local internal energy (u) and the polarization vector P, which includes the e ect of the polarization when an external electric eld is applied. For the fast variables we consider the polarization ux, namely Jp =
dP dt
(1)
and the e ective-antisymmetric stress tensor, Q a , acting on a group of rotating dipole ∼ particles. a − Ta ; Qa = ∼ ∼
∼
(2)
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a is the local antisymmetric stress tensor and T a is the antisymmetric part of where ∼ ∼
the Maxwell electromagnetic stress tensor. : Ta = E × P ; ∼
≈
(3)
where E is the local electric eld and is the alternating tensor. ≈ The e ective antisymmetric stress tensor plays the role of a net force on a locality, as is indicated in the energy density and in the angular momentum balance equations, namely
du = E · Jp − Q a : ·! ; ≈ ∼ dt
(4)
I
d! = · Qa ; ≈ ∼ dt
(5)
where is the local mass density, ! the angular velocity vector and I the average moment of inertia of a collection of dipole-particles involved in a locality. It should be understood that, when the e ective antisymmetric stress tensor is equal to zero, there is no inertial e ect (see Eq. (5)) and the Debye equation for the polarization evolution is obtained, considering mechanical equilibrium and non interacting e ects between dipoles. Nevertheless, when the interaction e ect is present, the e ective Q a ∼ appears producing an additional e ect on the polarization evolution. This new aspect of the polarization dynamics is the aim of the present work. Up to now, the set of independent variables in the present theory is given by {u; P; Jp ; Q a } : ∼
(6)
Therefore, the time evolution equations for the fast variables are obtained by the procedure prescribed by EIT. Such calculations are outlined in Appendix A and these equations are the following 1 d dP d = (P − 0 E) + D2 ∇ · Q a + D3 1 + 1 ∇ · (∇P) ; 1 + 1 ∼ dt dt D dt (7) d d Q a − D4 1 + 2 (∇P)a = −0 · ! : (8) 1 + 2 ≈ dt ∼ dt The role played by this two coupled equations is that, they describe the interdependent dynamics between the vector P and the antisymmetric stress tensor Q a . The ∼ reciprocal coupling between these two quantities arises when the polarization e ect, induced by the action of an external electric eld, is coupled to spatial inhomogeneities in the polarization, such that the (∇P) is present in the polarization evolution equation. It should be stressed that the viscoelastic e ects of Eqs. (7) and (8) lie in the parameters 2 and 2 , which are the relaxation time for the evolution of the antisymmetric stress relaxation and the retardation time of the same quantity, respectively.
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Another interesting point presented in Eqs. (7) and (8) is that they are a generalization of the Debye equation. In fact, if D2 and D3 are zero, the well known result of Debye is obtained. However, if D2 =0 and D3 is di erent from zero the inhomogeneous case without coupling is obtained. In this case Eq. (7) becomes 1
d2 P dP 0 D3 D3 2 1 + + P= E+ ∇p + ∇ P; dt 2 dt D D 2 2
(9)
where p = −∇ · P: Eq. (9) predicts a non-instantaneous transmission for any perturbation in the polarization in the system, with a velocity of propagation given by C2 =
D3 : 21
(10)
Therefore, we have obtained in Eq. (9) the telegrapher’s equation for signal propagation, which was also obtained from a basic formalism of persistent random walk [12]. The wave propagation produces perturbations on P in the same way that spin waves are produced in a magnetic sample [13]. The most general case of Eq. (9) is obtained when D2 is di erent from zero, and the coupling e ect enter through the term of ∇ · Q a 6= 0. If we consider that viscoelastic e ects are absent in the sample, that is ∼ when 2 = 1 = 2 = 0, Eq. (9) becomes 1
1 d2 P dP 0 D3 − D2 D4 + P= E+ ∇p + 2 dt dt D D 2 +
D3 − D2 D4 2 ∇ P − D2 0 ∇ × !: 2
(11)
Therefore, it keeps the general form of Eq. (9), and the velocity of propagation of the polarization disturbance is the same, but the sources of polarization are modi ed due to the appearance of the term ∇ × !. The contribution of this last term of Eq. (11) becomes important when a mechanical e ect of twist is present in the sample. An e ect of drift is then produced in the dipole system which makes a contribution to the evolution of the polarization vector. The most general case described by the set of di erential equations given in Eqs. (7) and (8) is obtained when viscoelastic e ects are present in the sample. In this case, several new terms appears which have to be added to Eq. (11). They are 0 2 d d3 P d2 P 2 dP + E − + 2 dt 3 dt 2 D dt D dt d d 1 D2 D4 2 + D3 2 + D3 2 1 (∇p + ∇2P) : + 2 dt dt
−2 1
Note should be made that in the viscoelastic contributions to the polarization evolution, the libration friction appears in the term with the coecient 1 2 . That is, the viscoelastic relaxation induces a force to delay the change of the polarization. Other e ects appears in these contributions, but are not considered in this paper. The linear
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response of a system obeying the set of di erential Eqs. (7) and (8) will be studied in the Section 4. 3. The memory functions for dielectric relaxation The response of a material to an external electric eld can be formally treated in terms of the dielectric memory function [14]. In fact, within the framework of linear response theory the electric susceptibility is given in terms of the transformed memory e !), namely function K(k; 1 (k; !) : = e 0 1 − i!K −1 (k; !)
(12)
The transformed dielectric memory in the hydrodynamic limit (k → 0; ! → 0) tends to a single relaxation time, which is expressed by Debye’s theory of Brownian dipoles. This relaxation time is related to the Stokes–Einstein relationship with the rotational-viscosity. 1 e K(!; k) = ; D
k → 0; ! → 0 :
(13)
On the other hand, when hydrodynamic memory is present, the transformed dielectric memory has contribution which depend on the values of k and !. Di erent types of such contributions have been called the inertial e ect, the dielectric friction (Nee– Zwanzig contribution) and polarization di usion. All of these may be considered as separate contributions in the linear approximation, namely 1
e !) K(k;
= D − i! 1 −
1
e1 (!) K
−
k2 1 : − e2 e3 (!; k) i!K K
(14)
The rst term of rhs is the Debye relaxation time, considered as a constant, next is the inertial term [15]. The third term gives the dielectric friction given previously by Nee–Zwanzig [16]. The fourth term represents the polarization di usion contribution and K2 corresponds to the inverse of the di usion coecient. We will show that this term does not make a signi cant contribution to the memory function. The last term represents a dielectric memory produced by the coupling of the antisymmetric stress relaxation and the polarization di usion. The actual contribution of this last term to the dielectric memory is equivalent to a polarization di usion process, coupled with the stress relaxation, with a di usion coecient given by the inverse of K3 , and this is the new contribution to study in the next section. 4. The complex electric susceptibility calculation We have presented in Eqs. (7) and (8) the time and space evolution equations for the quantities P and Q a . In this section use is made of the Fourier–Laplace analysis to ∼
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study the general features of these equations in frequencies–wave number vector space. To this aim, we rst linearize those equations and then calculate the complex electric susceptibility ( ij ) de ned by Pi = ij (!; k)Ej ;
(15)
where Pi and Ej are the Fourier transformed quantities. In the linearization step, we consider that the applied electric eld is not very strong thus we can neglect nonlinear e ects in the thermodynamic variables and the angular velocity. To nd the complex electric susceptibility we follow the same algebraic procedure for tensors as in the calculation of the generalized response coecient given elsewhere [11]. Then, considering the wave vector as k = (0; 0; k) the longitudinal electric susceptibility is L (k; !) = [1 − i! DL (1 − i!1 ) + D3L k 2 DL (1 − i! 1 )]−1 0L
(16)
and for the transversal electric susceptibility T (k; !) = [1 − i! TD (1 − i!1 ) + D3T k 2 DL (1 − i! 1 ) − i!A(k; !)]−1 ; 0T
(17)
where A(k; !) =
k 2 D1 D2 (1 − i! 2 ) TD : i!(1 − i!2 ) + 20 =I
(18)
These results have been obtained from Eqs. (7) and (8) using Eq. (5). Now, we compare with the results of Sullivan–Dutch [20], Fulton [17,18] and Kivelson–Madden [19] approaches. To this aim, we consider the limit of ! → 0 Eqs. (16) and (17) to be L (k; 0) = [1 + D3L k 2 DL ]−1 ; 0L
(19)
T (k; 0) = [1 + D3T k 2 TD ]−1 : 0T
(20)
Accordingly, the electric susceptibility is k dependent by two reasons: First, if we consider that the electric susceptibility should be related not to the local electric eld as in Eq. (15), but the external applied eld, a dependence in k is introduced for uniform and isotropic systems. This dependence can be cast as a function of the factor (0), whereas for the longitudinal and transverse susceptibilities (!) − ∞ (0) L = · ; 0 (0) − ∞ (!)
(21)
(!) − ∞ T = ; 0 (0) − ∞
(22)
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where those susceptibilities are related to the external electric eld. Therefore, there is no any additional contribution of the long-range character and consequently, D3 has to be taken equal to zero. Beside of the property of the electric susceptibility given in Eqs. (19) and (20), the reorientation time for the collective dipole relaxation T (k; !) and L (k; !) is di erent in each mode, as can be noted in the following expressions: L (k; !) = T (k; !) =
((0) − 1) 1 ; K3L (k; !) (0) 1 K3T (k; !)
((0) − 1) :
(23) (24)
Note that the consequence of having D3 equal to zero implies, from Eqs. (9) and (10), that the wave propagation e ect is always negligible. This fact was also pointed out by Fulton [17] and Madden and Kivelson [19]. On the other hand, the second reason to have a k-dependent electric susceptibility is when in the memory function there are contributions arised from spatial non-uniformity induced in the material by the interaction e ect between dipoles. Therefore, according to Eq. (17) there are di usion contributions to the transverse mode and the dielectric memory for this mode is di erent from that for the longitudinal one: K3L (k; !) 6= K3T (k:!) :
(25)
The dielectric memory are not longer equals and implies that the correlation function of the polarization vector contains a contribution of long range character. In the uniform and isotropic limit, the equality of the two memory functions prevails when ! → 0, as it was demonstrated by Sullivan and Dutch [20]. Finally, we obtain a new result for the transverse mode in terms of a coupled-di usion term, which is present in the dielectric memory, identi ed according to Eq. (14) as D (1 − i! 2 ) 1 = ; e K3 (k; !) −i! 2 (1 − i!2 ) + ( 2 = R )
(26)
where 2 −1 2 = D1 D2 k
and
R =
I : 20
These are characteristic relaxation times related to the polarization-di usion coupling process and the characteristic rotational time, respectively. These contributions produce important modi cations to the absorption band of Debye as we will see in the next subsection. 4.1. Comparison with experiment We consider that the transverse mode is that observed experimentally, since according to the usually experimental arrangement the electric eld is directed along the normal to the surface of the sample. Then Eqs. (17) and (18) with D3T = 1 = 0 are the proper
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Fig. 1. The imaginary part of the complex dielectric constant as a function of log (!). The dashed line is the ideal Debye case obtained using Eq. (27) for D = D1 D2 = 0. The continuous line is the non-ideal case for Dk 2 = 30 s−1 ( 2 = 3:3 × 10−2 s); 2 = 4:7 × 10−4 s; 2 = 1:5 × 10−7 s; D = 2:3 × 10−2 s and R = 0:2 s.
form to use to compare with experimental results. Then the complex dielectric constant for our di usion-coupling model is 1 (k; !) − ∞ : = (1−i! 2 )Dk 2 (0) − ∞ 1 − i! D 1 − −i!(1−i! )+(1= ) 2 R
(27)
The coupling term, D = D1 D2 controls the value of the peak of the imaginary part of the electric susceptibilidad as shown in Fig. 1. In this gure, the D-peak represents the case where the coupling term was zero, and Eq. (27) gives the well known complex dielectric constant of Debye. The -relaxation is predicted by Eq. (27) when this coupled-di usion term is di erent from zero. Beside this absorption band, the -peak appears given by the parameter R , which represents the e ect of dipole-drift produced by a twist of the material. Bartenev and Bartenava [21] have veri ed the experimental possibility of the inclusion of the -relaxation in the spectra of mechanical relaxation. However, the inclusion of this -peak into the dielectric relaxation spectra is not simple, since the -band is so broad that it overlaps this peak. Fig. 2 shows a possible result of Eq. (27), in which and peaks tends to overlap each other in the same absorption band. The shoulder at the left side of the curve of the gure is due to the contribution. Furthermore, the parameter 2 in Eq. (27)
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477
Fig. 2. The continuous line is the frequency dependent dielectric loss factor obtained using Eq. (27) with parameters value given by D = 1 × 10−2 s, R = 3:2 × 10−2 s, 2 = 0:80 × 10−2 s, 2 = 0:4 × 10−2 s, Dk 2 = 47:0 s−1 ( 2 = 2:1 × 10−2 s).
produces a small peak at the right side of the curve of the Fig. 2, which represents the local viscoelastic relaxation with axial symmetry (other than normal mode or Rouse relaxation) coupled to the local polarization. From the molecular point of view, this relaxation involves a collective mobility of dipoles attached to segmental chains. It should be noted that the parameter 2 does not make any appreciable modi cation to these absorption bands. On the other hand, to provide greater detail of the -dielectric relaxation within the present approach, we now consider the case in which the rotation-velocity e ect does not contribute to the dipolar relaxation process. Therefore, we introduce from the beginning of the model formulation a zero angular velocity in Eq. (8). The new result for the electric susceptibility is the same as Eq. (27), but with 1= R =0. We also consider that the -peak is encroached on the -relaxation. Fig. 3 shows the prediction of this result, comparing it with the experimental data reported for the poly(propylene glycol) and the poly(propylene oxide). However, it is not possible to give a full description of the experimental data because the predicted bandwidth is smaller than the experimental one (see dashed line in Fig. 3). Nevertheless, an improvement of the t to the data can be obtained if a dispersion relationship is introduced. Accordingly, we assume that k 2 = k02 (!=!0 )n , accounts for the dispersive e ect of the anomalous di usion (an e ective di usion of t −1−n has a dispersive dependence of !n with n ≈ 0:5, see Resibois and Leener [22], p. 632). We show in Fig. 3, the corresponding t by continuous line, with the corresponding parameters given in Table 1. However, the deviation of the experimental data from the continuous line still persists and this might be explained by considering the in uence of the -relaxation on the high frequency
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L.F. Del Castillo, R. DÃaz-Calleja / Physica A 268 (1999) 469–481
Fig. 3. The relative loss modulus as a function of the log (! D ). Dashed line is the t obtained considering Eqs. (27) and parameters given in Table 1. The continuous line is the best t considering the Eq. (27) modi ed by a dispersion relationship and −1 R = 0, (see the text) with parameters given in Table 1. The experimental points correspond to the poly(propylene glycol) (black points) and poly(propylene oxide) (white squares). Data from W.H. Stockmayer, Pure Appl. Chem. 15 (1967) 539 and T. Alper et al., Polymer 17 (1976) 665.
Table 1 Parameters to t experimental data of Fig. 3 Line
Dk 2
Dk0
2 (s)
2 (s)
D (s)
2 (s)
!0
n
Dashed Continuous
0.565 s−1 –
– 11.1 s−1
0.07 0.05
0.44 0.34
2.80 1.58
1.77 –
– 55.43 s−1
– 0.7
wing of the -relaxation. In fact, in the present model we have not considered any secondary relaxation.
5. Final discussion In this paper we present a dielectric relaxation approach based on the result obtained by using EIT. To this end, we consider as new independent variables the polarization
ux and the antisymmetric stress tensor. The result of this theory is expressed in terms of a set of di erential equation, namely Eqs. (7) and (8). These equations describe the time and the spatial behavior of the polarization and the antisymmetric stress tensor.
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It should be emphasized that the coupling between the antisymmetric stress tensor and the polarization vector is given in terms of the gradient of the polarization, which represents the e ect of spatial inhomogeneities on the macroscopic polarization. On the other hand, when the di erential equations are linearized we consider that the system makes only small uctuations from the equilibrium. The regression of the
uctuations are described by the linearized equation and if any external disturbance is present, by the action of an oscillatory electric eld, the system response is described by the complex electric susceptibility. The present model gives us some insight of the dipolar di usion mechanism evolved on the -relaxation. In fact, when the external application of an electric eld takes place, the polarization e ect induces transitions in the molecular structure and produces spatial inhomogeneities in polarization and local stresses. Through the uctuation mechanism, both the polarization gradient and local stresses relax together. The result which expresses this fact is given in Eq. (27) by the di usion coupling term involving the parameter D = D1 D2 . Finally, according to the tting procedure to represent the experimental data with Eq. (27) the parameter Dk 2 is determined, and it is shown that the corresponding di usion time 2 = 1=Dk 2 is of the same order of magnitude than D (see the values of such parameters on gure captions and Table 1). It means that the polarization di usion takes place in the -dielectric relaxation and modulate it. Acknowledgements We thank professor L.A. Davalos-Orozco for helpful preliminary discussions. One of the authors (L.F. del Castillo) acknowledges support from DGAPA-UNAM IN106797. Appendix A According to the usual procedure of EIT, we assume that there exists a suciently regular and continuous function , the extended entropy, de ned over the complete set of independent variables = (u; P; Jp ; Q) : ∼
(A.1)
The aim of this assumption is to generate a di erential form which in a formal sense will generalize the Gibbs relation of local equilibrium thermodynamics. If we restrict ourselves to lowest order in the fast variables the extended entropy takes the form d Qa dJp ∼ du E0 dP d + a2 Q a : ; + · + a1 Jp · = ∼ dt T dt T dt dt dt
(A.2)
where T is the absolute temperature, E0 is the local electric eld observed in the material at any time, which in general is di erent from the Maxwell electromagnetic eld.
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At equilibrium (t → ∞) E0 =E. On the other hand, a1 and a2 are coecients dependent on the local thermodynamic state of the material, for instance ai = ai (T; E0 ) (i = 1; 2). We assume that the response of the material is so linear that coecients do not depend on the electric eld or on any other variable, such that we consider ai as constants. The corresponding extended entropy ux is given by J = 1 Jp · Q a :
(A.3)
∼
Similarly, the extended entropy production is the scalar de ned by the linear combination of the product of two fast variables and the product of two quantities related to the response of the system, such as is the spatial inhomogeneities of P and its ux. = Jp · Xp + Q a : X a ;
(A.4)
˙ ; Xp = 1 Jp + 2 ∇ · (∇P) + 3 ∇ · (∇P)
(A.5)
a ˙ a: = 4 Q a + 5 (∇P)a + 6 (∇P) X ∼
(A.6)
∼
∼
∼
We now consider the extended entropy balance equation d + ∇ · J = (A.7) dt from this equation, by using Eqs. (A.2), (A.4) with (A.5) and (A.6), (A.3) and Eq. (5) of the text, the generalized constitutive equations are obtained. These equations are given in Eqs. (7) and (8). The corresponding identi cation parameters in these equations are
D = T1 0 ;
1 = a1 =1 ;
2 = a2 =4 ;
D4 = −5 =4 ;
D2 = 1 =1 ;
D3 = −2 =1 ;
2 = 6 =5 ;
0 = 1=T4 :
In writing Eq. (7) we have used the equality P = 0 E0 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
U. Felderhof, Mol. Phys. 48 (1983) 1283. H.J. Kroh, B.V. Felderhof, Z. Phys. B 66 (1987) 1. J.B. Hubbard, P.J. Stiles, Chem. Phys. Lett. 114 (1985) 121. P.J. Stiles, J.B. Hubbard, J. Chem. Phys. 84 (1984) 431. G. Van der Zwan, J.T. Hynes, Physica A 121 (1983) 227. S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962. D. Jou, J. Casas-Vazquez, G. Lebon, Rep. Prog. Phys. 51 (1988) 1108. D. Jou, J. Casas-Vazquez, G. Lebon, Extended Irreversible Thermodynamics, Springer, Berlin, 1996. L.S. Garca-Coln, Mol. Phys. 86 (1995) 697. L.F. del Castillo, L.A. Davalos-Orozco, J. Chem. Phys. 93 (1990) 5147. L.A. Davalos-Orozco, L.F. del Castillo, J. Chem. Phys. 96 (1992) 9102. J. Masoliver, G.H. Weiss, Phys. Rev. E. 49 (1994) 3852. J.M. Rubi, J. Casas-Vazquez, J. Non-equilibrium Thermodyn. 5 (1980) 155. J.P. Boom, S. Yip, Molecular Hydrodynamics, McGraw-Hill, New York, 1980. J. McConnell, Rotational Brownian Motion and Dielectric Theory, Academic Press, New York, 1980.
L.F. Del Castillo, R. DÃaz-Calleja / Physica A 268 (1999) 469–481 [16] [17] [18] [19] [20] [21] [22]
T.W. Nee, R. Zwanzig, J. Chem. Phys. 52 (1970) 6353. R.L. Fulton, Mol. Phys. 29 (1975) 405. R.L. Fulton, J. Chem. Phys. 62 (1975) 4355. P. Madden, D. Kivelson, Adv. Chem. Phys. 56 (1984) 467. D.E. Sullivan, J.M. Deutch, J. Chem. Phys. 62 (1975) 2130. G.M. Bartenev, A.G. Barteneva, Polymer Science, Ser. A 37 (1995) 426. P. Resibois, M. De Leener, Classical Kinetic theory of Fluids, Wiley, New York, 1977.
481
www.elsevier.com/locate/physa
E ects of gradient of polarization on dielectric relaxation a Instituto
L.F. Del Castilloa; ∗ , R. Daz-Callejab
de Investigaciones en Materiales, UNAM, Apartado Postal 70-360, 04510 MÃexico, D.F., Mexico b Departamento de Termodinà amica Aplicada, Universidad PolitÃecnica de Valencia, Valencia, Spain Received 2 June 1998
Abstract We present a non-equilibrium thermodynamic model to describe di usion e ects in dielectric materials with spatial inhomogeneities in the polarization vector and with local viscoelastic e ects. The model presented here is a generalization of the Debye relaxation equation including inertial e ect, and contributions from the spatial inhomogeneities of the polarization vector, together with a contribution from the antisymmetric stress tensor via the divergence operator. At the same time, a Maxwell viscoelastic-relaxation type equation for the antisymmetric stress tensor is proposed to describe the time evolution of this tensor. On the other hand, a generalized hydrodynamics model for the dielectric memory for short wave length and high frequencies is obtained, by considering the linearized complete set of di erential equations of the model. By working in the Fourier–Laplace space, the dielectric susceptibility is obtained and their main feac 1999 Elsevier Science tures are described and compared with and dielectric relaxation. B.V. All rights reserved. PACS: 05.70Ln; 05.40+j; 44.60+k
1. Introduction Previous models for di usion dielectric relaxation following the continuum formalism have been given by Felderhof [1], Kroh and Felderhof [2], Hubbard and Stiles [3,4] and Van der Zwan and Hynes [5]. The new aspect we deal with is related to the coupling between the tensor and vector quantities which are forbidden using the lineal non-equilibrium thermodynamics [6]. In our case, we use the rules provided by ∗
Corresponding author. Fax: +52-5-6161201. E-mail address: [email protected] (L.F. Del Castillo)
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 0 5 7 - 6
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extended irreversible thermodynamics (EIT) [7–9] to introduce the modi cations into the relaxing equations. In a previous paper using EIT, del Castillo and Davalos-Orozco [10] described the velocity e ect on the dielectric relaxation memory. They, used a modi ed Debye relaxation type-equation including spatial inhomogeneities coupling terms. In another paper, Davalos-Orozco and del Castillo [11] included the rotation of the dipole domain to show that di usion of polarization charges is induced when translation of localities takes place. In this paper we study the case of polarization di usion produced by spatial inhomogeneities, when the polarization is produced by an external electric eld. Here, we take into account the di usion of polarization through the system. From the microscopic point of view, the polarization di usion process may be understood as a manifestation of the long range interaction due to dipoles, where the orientation of one dipole is a ected by the orientation of other dipoles. In the present paper, we describe the coupling between the polarization vector and the antisymmetric stress tensor, which account for the relative rotation of a group of particles in a locality. The obtained result can be used to describe the coupling between the antisymmetric stress tensor uctuation in polymeric systems and the polarization dynamics; an e ect which can be recognized in the spectra of dielectric relaxation as -relaxation. We begin in Section 2 considering the relation of the memory kernel with the electric susceptibility in the transformed Fourier representation. In Section 3, we present a generalized hydrodynamics model through the used of two coupled di erential equations. One for the polarization vector, and the other for the antisymmetric stress tensor, the derivation of these equations are given in Appendix A. In Section 4, we describe the complex electric susceptibility tensor with transversal and longitudinal components. The comparison between our results and the spectral response with and dielectric relaxation is presented at the end of Section 4.
2. Non-equilibrium thermodynamic model for non-uniform polarization systems To describe the non equilibrium state of a polar viscoelastic-liquid in the frame of Extended Irreversible Thermodynamics (EIT) [7–9], it is necessary to specify the set of independent variables as slow and fast variables of the thermodynamic system. The slow variables are de ned as the local internal energy (u) and the polarization vector P, which includes the e ect of the polarization when an external electric eld is applied. For the fast variables we consider the polarization ux, namely Jp =
dP dt
(1)
and the e ective-antisymmetric stress tensor, Q a , acting on a group of rotating dipole ∼ particles. a − Ta ; Qa = ∼ ∼
∼
(2)
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a is the local antisymmetric stress tensor and T a is the antisymmetric part of where ∼ ∼
the Maxwell electromagnetic stress tensor. : Ta = E × P ; ∼
≈
(3)
where E is the local electric eld and is the alternating tensor. ≈ The e ective antisymmetric stress tensor plays the role of a net force on a locality, as is indicated in the energy density and in the angular momentum balance equations, namely
du = E · Jp − Q a : ·! ; ≈ ∼ dt
(4)
I
d! = · Qa ; ≈ ∼ dt
(5)
where is the local mass density, ! the angular velocity vector and I the average moment of inertia of a collection of dipole-particles involved in a locality. It should be understood that, when the e ective antisymmetric stress tensor is equal to zero, there is no inertial e ect (see Eq. (5)) and the Debye equation for the polarization evolution is obtained, considering mechanical equilibrium and non interacting e ects between dipoles. Nevertheless, when the interaction e ect is present, the e ective Q a ∼ appears producing an additional e ect on the polarization evolution. This new aspect of the polarization dynamics is the aim of the present work. Up to now, the set of independent variables in the present theory is given by {u; P; Jp ; Q a } : ∼
(6)
Therefore, the time evolution equations for the fast variables are obtained by the procedure prescribed by EIT. Such calculations are outlined in Appendix A and these equations are the following 1 d dP d = (P − 0 E) + D2 ∇ · Q a + D3 1 + 1 ∇ · (∇P) ; 1 + 1 ∼ dt dt D dt (7) d d Q a − D4 1 + 2 (∇P)a = −0 · ! : (8) 1 + 2 ≈ dt ∼ dt The role played by this two coupled equations is that, they describe the interdependent dynamics between the vector P and the antisymmetric stress tensor Q a . The ∼ reciprocal coupling between these two quantities arises when the polarization e ect, induced by the action of an external electric eld, is coupled to spatial inhomogeneities in the polarization, such that the (∇P) is present in the polarization evolution equation. It should be stressed that the viscoelastic e ects of Eqs. (7) and (8) lie in the parameters 2 and 2 , which are the relaxation time for the evolution of the antisymmetric stress relaxation and the retardation time of the same quantity, respectively.
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Another interesting point presented in Eqs. (7) and (8) is that they are a generalization of the Debye equation. In fact, if D2 and D3 are zero, the well known result of Debye is obtained. However, if D2 =0 and D3 is di erent from zero the inhomogeneous case without coupling is obtained. In this case Eq. (7) becomes 1
d2 P dP 0 D3 D3 2 1 + + P= E+ ∇p + ∇ P; dt 2 dt D D 2 2
(9)
where p = −∇ · P: Eq. (9) predicts a non-instantaneous transmission for any perturbation in the polarization in the system, with a velocity of propagation given by C2 =
D3 : 21
(10)
Therefore, we have obtained in Eq. (9) the telegrapher’s equation for signal propagation, which was also obtained from a basic formalism of persistent random walk [12]. The wave propagation produces perturbations on P in the same way that spin waves are produced in a magnetic sample [13]. The most general case of Eq. (9) is obtained when D2 is di erent from zero, and the coupling e ect enter through the term of ∇ · Q a 6= 0. If we consider that viscoelastic e ects are absent in the sample, that is ∼ when 2 = 1 = 2 = 0, Eq. (9) becomes 1
1 d2 P dP 0 D3 − D2 D4 + P= E+ ∇p + 2 dt dt D D 2 +
D3 − D2 D4 2 ∇ P − D2 0 ∇ × !: 2
(11)
Therefore, it keeps the general form of Eq. (9), and the velocity of propagation of the polarization disturbance is the same, but the sources of polarization are modi ed due to the appearance of the term ∇ × !. The contribution of this last term of Eq. (11) becomes important when a mechanical e ect of twist is present in the sample. An e ect of drift is then produced in the dipole system which makes a contribution to the evolution of the polarization vector. The most general case described by the set of di erential equations given in Eqs. (7) and (8) is obtained when viscoelastic e ects are present in the sample. In this case, several new terms appears which have to be added to Eq. (11). They are 0 2 d d3 P d2 P 2 dP + E − + 2 dt 3 dt 2 D dt D dt d d 1 D2 D4 2 + D3 2 + D3 2 1 (∇p + ∇2P) : + 2 dt dt
−2 1
Note should be made that in the viscoelastic contributions to the polarization evolution, the libration friction appears in the term with the coecient 1 2 . That is, the viscoelastic relaxation induces a force to delay the change of the polarization. Other e ects appears in these contributions, but are not considered in this paper. The linear
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response of a system obeying the set of di erential Eqs. (7) and (8) will be studied in the Section 4. 3. The memory functions for dielectric relaxation The response of a material to an external electric eld can be formally treated in terms of the dielectric memory function [14]. In fact, within the framework of linear response theory the electric susceptibility is given in terms of the transformed memory e !), namely function K(k; 1 (k; !) : = e 0 1 − i!K −1 (k; !)
(12)
The transformed dielectric memory in the hydrodynamic limit (k → 0; ! → 0) tends to a single relaxation time, which is expressed by Debye’s theory of Brownian dipoles. This relaxation time is related to the Stokes–Einstein relationship with the rotational-viscosity. 1 e K(!; k) = ; D
k → 0; ! → 0 :
(13)
On the other hand, when hydrodynamic memory is present, the transformed dielectric memory has contribution which depend on the values of k and !. Di erent types of such contributions have been called the inertial e ect, the dielectric friction (Nee– Zwanzig contribution) and polarization di usion. All of these may be considered as separate contributions in the linear approximation, namely 1
e !) K(k;
= D − i! 1 −
1
e1 (!) K
−
k2 1 : − e2 e3 (!; k) i!K K
(14)
The rst term of rhs is the Debye relaxation time, considered as a constant, next is the inertial term [15]. The third term gives the dielectric friction given previously by Nee–Zwanzig [16]. The fourth term represents the polarization di usion contribution and K2 corresponds to the inverse of the di usion coecient. We will show that this term does not make a signi cant contribution to the memory function. The last term represents a dielectric memory produced by the coupling of the antisymmetric stress relaxation and the polarization di usion. The actual contribution of this last term to the dielectric memory is equivalent to a polarization di usion process, coupled with the stress relaxation, with a di usion coecient given by the inverse of K3 , and this is the new contribution to study in the next section. 4. The complex electric susceptibility calculation We have presented in Eqs. (7) and (8) the time and space evolution equations for the quantities P and Q a . In this section use is made of the Fourier–Laplace analysis to ∼
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study the general features of these equations in frequencies–wave number vector space. To this aim, we rst linearize those equations and then calculate the complex electric susceptibility ( ij ) de ned by Pi = ij (!; k)Ej ;
(15)
where Pi and Ej are the Fourier transformed quantities. In the linearization step, we consider that the applied electric eld is not very strong thus we can neglect nonlinear e ects in the thermodynamic variables and the angular velocity. To nd the complex electric susceptibility we follow the same algebraic procedure for tensors as in the calculation of the generalized response coecient given elsewhere [11]. Then, considering the wave vector as k = (0; 0; k) the longitudinal electric susceptibility is L (k; !) = [1 − i! DL (1 − i!1 ) + D3L k 2 DL (1 − i! 1 )]−1 0L
(16)
and for the transversal electric susceptibility T (k; !) = [1 − i! TD (1 − i!1 ) + D3T k 2 DL (1 − i! 1 ) − i!A(k; !)]−1 ; 0T
(17)
where A(k; !) =
k 2 D1 D2 (1 − i! 2 ) TD : i!(1 − i!2 ) + 20 =I
(18)
These results have been obtained from Eqs. (7) and (8) using Eq. (5). Now, we compare with the results of Sullivan–Dutch [20], Fulton [17,18] and Kivelson–Madden [19] approaches. To this aim, we consider the limit of ! → 0 Eqs. (16) and (17) to be L (k; 0) = [1 + D3L k 2 DL ]−1 ; 0L
(19)
T (k; 0) = [1 + D3T k 2 TD ]−1 : 0T
(20)
Accordingly, the electric susceptibility is k dependent by two reasons: First, if we consider that the electric susceptibility should be related not to the local electric eld as in Eq. (15), but the external applied eld, a dependence in k is introduced for uniform and isotropic systems. This dependence can be cast as a function of the factor (0), whereas for the longitudinal and transverse susceptibilities (!) − ∞ (0) L = · ; 0 (0) − ∞ (!)
(21)
(!) − ∞ T = ; 0 (0) − ∞
(22)
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where those susceptibilities are related to the external electric eld. Therefore, there is no any additional contribution of the long-range character and consequently, D3 has to be taken equal to zero. Beside of the property of the electric susceptibility given in Eqs. (19) and (20), the reorientation time for the collective dipole relaxation T (k; !) and L (k; !) is di erent in each mode, as can be noted in the following expressions: L (k; !) = T (k; !) =
((0) − 1) 1 ; K3L (k; !) (0) 1 K3T (k; !)
((0) − 1) :
(23) (24)
Note that the consequence of having D3 equal to zero implies, from Eqs. (9) and (10), that the wave propagation e ect is always negligible. This fact was also pointed out by Fulton [17] and Madden and Kivelson [19]. On the other hand, the second reason to have a k-dependent electric susceptibility is when in the memory function there are contributions arised from spatial non-uniformity induced in the material by the interaction e ect between dipoles. Therefore, according to Eq. (17) there are di usion contributions to the transverse mode and the dielectric memory for this mode is di erent from that for the longitudinal one: K3L (k; !) 6= K3T (k:!) :
(25)
The dielectric memory are not longer equals and implies that the correlation function of the polarization vector contains a contribution of long range character. In the uniform and isotropic limit, the equality of the two memory functions prevails when ! → 0, as it was demonstrated by Sullivan and Dutch [20]. Finally, we obtain a new result for the transverse mode in terms of a coupled-di usion term, which is present in the dielectric memory, identi ed according to Eq. (14) as D (1 − i! 2 ) 1 = ; e K3 (k; !) −i! 2 (1 − i!2 ) + ( 2 = R )
(26)
where 2 −1 2 = D1 D2 k
and
R =
I : 20
These are characteristic relaxation times related to the polarization-di usion coupling process and the characteristic rotational time, respectively. These contributions produce important modi cations to the absorption band of Debye as we will see in the next subsection. 4.1. Comparison with experiment We consider that the transverse mode is that observed experimentally, since according to the usually experimental arrangement the electric eld is directed along the normal to the surface of the sample. Then Eqs. (17) and (18) with D3T = 1 = 0 are the proper
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Fig. 1. The imaginary part of the complex dielectric constant as a function of log (!). The dashed line is the ideal Debye case obtained using Eq. (27) for D = D1 D2 = 0. The continuous line is the non-ideal case for Dk 2 = 30 s−1 ( 2 = 3:3 × 10−2 s); 2 = 4:7 × 10−4 s; 2 = 1:5 × 10−7 s; D = 2:3 × 10−2 s and R = 0:2 s.
form to use to compare with experimental results. Then the complex dielectric constant for our di usion-coupling model is 1 (k; !) − ∞ : = (1−i! 2 )Dk 2 (0) − ∞ 1 − i! D 1 − −i!(1−i! )+(1= ) 2 R
(27)
The coupling term, D = D1 D2 controls the value of the peak of the imaginary part of the electric susceptibilidad as shown in Fig. 1. In this gure, the D-peak represents the case where the coupling term was zero, and Eq. (27) gives the well known complex dielectric constant of Debye. The -relaxation is predicted by Eq. (27) when this coupled-di usion term is di erent from zero. Beside this absorption band, the -peak appears given by the parameter R , which represents the e ect of dipole-drift produced by a twist of the material. Bartenev and Bartenava [21] have veri ed the experimental possibility of the inclusion of the -relaxation in the spectra of mechanical relaxation. However, the inclusion of this -peak into the dielectric relaxation spectra is not simple, since the -band is so broad that it overlaps this peak. Fig. 2 shows a possible result of Eq. (27), in which and peaks tends to overlap each other in the same absorption band. The shoulder at the left side of the curve of the gure is due to the contribution. Furthermore, the parameter 2 in Eq. (27)
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Fig. 2. The continuous line is the frequency dependent dielectric loss factor obtained using Eq. (27) with parameters value given by D = 1 × 10−2 s, R = 3:2 × 10−2 s, 2 = 0:80 × 10−2 s, 2 = 0:4 × 10−2 s, Dk 2 = 47:0 s−1 ( 2 = 2:1 × 10−2 s).
produces a small peak at the right side of the curve of the Fig. 2, which represents the local viscoelastic relaxation with axial symmetry (other than normal mode or Rouse relaxation) coupled to the local polarization. From the molecular point of view, this relaxation involves a collective mobility of dipoles attached to segmental chains. It should be noted that the parameter 2 does not make any appreciable modi cation to these absorption bands. On the other hand, to provide greater detail of the -dielectric relaxation within the present approach, we now consider the case in which the rotation-velocity e ect does not contribute to the dipolar relaxation process. Therefore, we introduce from the beginning of the model formulation a zero angular velocity in Eq. (8). The new result for the electric susceptibility is the same as Eq. (27), but with 1= R =0. We also consider that the -peak is encroached on the -relaxation. Fig. 3 shows the prediction of this result, comparing it with the experimental data reported for the poly(propylene glycol) and the poly(propylene oxide). However, it is not possible to give a full description of the experimental data because the predicted bandwidth is smaller than the experimental one (see dashed line in Fig. 3). Nevertheless, an improvement of the t to the data can be obtained if a dispersion relationship is introduced. Accordingly, we assume that k 2 = k02 (!=!0 )n , accounts for the dispersive e ect of the anomalous di usion (an e ective di usion of t −1−n has a dispersive dependence of !n with n ≈ 0:5, see Resibois and Leener [22], p. 632). We show in Fig. 3, the corresponding t by continuous line, with the corresponding parameters given in Table 1. However, the deviation of the experimental data from the continuous line still persists and this might be explained by considering the in uence of the -relaxation on the high frequency
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Fig. 3. The relative loss modulus as a function of the log (! D ). Dashed line is the t obtained considering Eqs. (27) and parameters given in Table 1. The continuous line is the best t considering the Eq. (27) modi ed by a dispersion relationship and −1 R = 0, (see the text) with parameters given in Table 1. The experimental points correspond to the poly(propylene glycol) (black points) and poly(propylene oxide) (white squares). Data from W.H. Stockmayer, Pure Appl. Chem. 15 (1967) 539 and T. Alper et al., Polymer 17 (1976) 665.
Table 1 Parameters to t experimental data of Fig. 3 Line
Dk 2
Dk0
2 (s)
2 (s)
D (s)
2 (s)
!0
n
Dashed Continuous
0.565 s−1 –
– 11.1 s−1
0.07 0.05
0.44 0.34
2.80 1.58
1.77 –
– 55.43 s−1
– 0.7
wing of the -relaxation. In fact, in the present model we have not considered any secondary relaxation.
5. Final discussion In this paper we present a dielectric relaxation approach based on the result obtained by using EIT. To this end, we consider as new independent variables the polarization
ux and the antisymmetric stress tensor. The result of this theory is expressed in terms of a set of di erential equation, namely Eqs. (7) and (8). These equations describe the time and the spatial behavior of the polarization and the antisymmetric stress tensor.
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It should be emphasized that the coupling between the antisymmetric stress tensor and the polarization vector is given in terms of the gradient of the polarization, which represents the e ect of spatial inhomogeneities on the macroscopic polarization. On the other hand, when the di erential equations are linearized we consider that the system makes only small uctuations from the equilibrium. The regression of the
uctuations are described by the linearized equation and if any external disturbance is present, by the action of an oscillatory electric eld, the system response is described by the complex electric susceptibility. The present model gives us some insight of the dipolar di usion mechanism evolved on the -relaxation. In fact, when the external application of an electric eld takes place, the polarization e ect induces transitions in the molecular structure and produces spatial inhomogeneities in polarization and local stresses. Through the uctuation mechanism, both the polarization gradient and local stresses relax together. The result which expresses this fact is given in Eq. (27) by the di usion coupling term involving the parameter D = D1 D2 . Finally, according to the tting procedure to represent the experimental data with Eq. (27) the parameter Dk 2 is determined, and it is shown that the corresponding di usion time 2 = 1=Dk 2 is of the same order of magnitude than D (see the values of such parameters on gure captions and Table 1). It means that the polarization di usion takes place in the -dielectric relaxation and modulate it. Acknowledgements We thank professor L.A. Davalos-Orozco for helpful preliminary discussions. One of the authors (L.F. del Castillo) acknowledges support from DGAPA-UNAM IN106797. Appendix A According to the usual procedure of EIT, we assume that there exists a suciently regular and continuous function , the extended entropy, de ned over the complete set of independent variables = (u; P; Jp ; Q) : ∼
(A.1)
The aim of this assumption is to generate a di erential form which in a formal sense will generalize the Gibbs relation of local equilibrium thermodynamics. If we restrict ourselves to lowest order in the fast variables the extended entropy takes the form d Qa dJp ∼ du E0 dP d + a2 Q a : ; + · + a1 Jp · = ∼ dt T dt T dt dt dt
(A.2)
where T is the absolute temperature, E0 is the local electric eld observed in the material at any time, which in general is di erent from the Maxwell electromagnetic eld.
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At equilibrium (t → ∞) E0 =E. On the other hand, a1 and a2 are coecients dependent on the local thermodynamic state of the material, for instance ai = ai (T; E0 ) (i = 1; 2). We assume that the response of the material is so linear that coecients do not depend on the electric eld or on any other variable, such that we consider ai as constants. The corresponding extended entropy ux is given by J = 1 Jp · Q a :
(A.3)
∼
Similarly, the extended entropy production is the scalar de ned by the linear combination of the product of two fast variables and the product of two quantities related to the response of the system, such as is the spatial inhomogeneities of P and its ux. = Jp · Xp + Q a : X a ;
(A.4)
˙ ; Xp = 1 Jp + 2 ∇ · (∇P) + 3 ∇ · (∇P)
(A.5)
a ˙ a: = 4 Q a + 5 (∇P)a + 6 (∇P) X ∼
(A.6)
∼
∼
∼
We now consider the extended entropy balance equation d + ∇ · J = (A.7) dt from this equation, by using Eqs. (A.2), (A.4) with (A.5) and (A.6), (A.3) and Eq. (5) of the text, the generalized constitutive equations are obtained. These equations are given in Eqs. (7) and (8). The corresponding identi cation parameters in these equations are
D = T1 0 ;
1 = a1 =1 ;
2 = a2 =4 ;
D4 = −5 =4 ;
D2 = 1 =1 ;
D3 = −2 =1 ;
2 = 6 =5 ;
0 = 1=T4 :
In writing Eq. (7) we have used the equality P = 0 E0 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
U. Felderhof, Mol. Phys. 48 (1983) 1283. H.J. Kroh, B.V. Felderhof, Z. Phys. B 66 (1987) 1. J.B. Hubbard, P.J. Stiles, Chem. Phys. Lett. 114 (1985) 121. P.J. Stiles, J.B. Hubbard, J. Chem. Phys. 84 (1984) 431. G. Van der Zwan, J.T. Hynes, Physica A 121 (1983) 227. S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962. D. Jou, J. Casas-Vazquez, G. Lebon, Rep. Prog. Phys. 51 (1988) 1108. D. Jou, J. Casas-Vazquez, G. Lebon, Extended Irreversible Thermodynamics, Springer, Berlin, 1996. L.S. Garca-Coln, Mol. Phys. 86 (1995) 697. L.F. del Castillo, L.A. Davalos-Orozco, J. Chem. Phys. 93 (1990) 5147. L.A. Davalos-Orozco, L.F. del Castillo, J. Chem. Phys. 96 (1992) 9102. J. Masoliver, G.H. Weiss, Phys. Rev. E. 49 (1994) 3852. J.M. Rubi, J. Casas-Vazquez, J. Non-equilibrium Thermodyn. 5 (1980) 155. J.P. Boom, S. Yip, Molecular Hydrodynamics, McGraw-Hill, New York, 1980. J. McConnell, Rotational Brownian Motion and Dielectric Theory, Academic Press, New York, 1980.
L.F. Del Castillo, R. DÃaz-Calleja / Physica A 268 (1999) 469–481 [16] [17] [18] [19] [20] [21] [22]
T.W. Nee, R. Zwanzig, J. Chem. Phys. 52 (1970) 6353. R.L. Fulton, Mol. Phys. 29 (1975) 405. R.L. Fulton, J. Chem. Phys. 62 (1975) 4355. P. Madden, D. Kivelson, Adv. Chem. Phys. 56 (1984) 467. D.E. Sullivan, J.M. Deutch, J. Chem. Phys. 62 (1975) 2130. G.M. Bartenev, A.G. Barteneva, Polymer Science, Ser. A 37 (1995) 426. P. Resibois, M. De Leener, Classical Kinetic theory of Fluids, Wiley, New York, 1977.
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