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Phenomenological approach on wave propagation in dielectric media with two relaxation times
ARTICLE IN PRESS Physica B 404 (2009) 320–324
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Physica B journal homepage: www.elsevier.com/locate/physb
Phenomenological approach on wave propagation in dielectric media with two relaxation times V. Ciancio a,, F. Farsaci b, P. Rogolino a a b
Department of Mathematics, University of Messina, Italy IPCF-C.N.R., Messina, Italy
a r t i c l e in f o
a b s t r a c t
Article history: Received 21 October 2008 Received in revised form 4 November 2008 Accepted 4 November 2008
In this paper dielectric phenomena with two relaxation times are discussed. By assuming a sinusoidal form for induction vector D a sinusoidal electric field is generated and it depends on unknown phenomenological coefficients whose expressions together to their numerical values as functions of frequency are obtained. Moreover, electromagnetic wave propagation is analysed obtaining wave vector as function of the aforementioned coefficients. The results are applied to a Vinylidene Chloride-Vinyl Chloride (VDC–VC) to test the applicability of the model. & 2008 Elsevier B.V. All rights reserved.
Keywords: Irreversible thermodynamics Relaxation phenomena Wave propagation
1. Introduction In a previous paper [1] the connection between phenomenological coefficients and quantities experimentally measurable, e.g. real and imaginary parts of complex dielectric constant, has been obtained for media with dielectric relaxation phenomena and in the case in which just one relaxation time is considered. In Refs. [2–4] a phenomenological equation was proposed in which two relaxation times occur and this is connected to physical behaviour of materials. In fact the instantaneous increasing or decreasing in the polarization is impossible because any change of the polarization is related to the motion of any kind of microscopic particles which cannot be infinitely fast. The phenomenological equation will be discussed in Section 2 and we will refer to the case in which two relaxation times are taken into account and in such a context we will study electromagnetic wave propagation obtaining wave vector as function of the aforementioned quantities experimentally measurable. Our aim is to study the system under a sinusoidal perturbation represented by the induction vector D, which is an extensive variable (cause), and to analyse the relative electric field as an intensive variable (effect) inside the medium [5]. In particular, if we consider a generic dielectric medium placed between the plain plate of a capacitor where a sinusoidal voltage has been applied, then on the plates a sinusoidal surface charge arises whose density is characterized by the normal component of induction
Corresponding author. Tel.: +39 90 6765061; fax: +39 90 393502.
E-mail address: [email protected] (V. Ciancio). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.11.070
vector D ¼ D n (n is the unit normal to the plates) generating a sinusoidal electric field inside the capacitor [6]. The linear-response theory establishes that if D (cause) evolves harmonically [5], i.e. D ¼ D0 eiot
(1.1)
where D0 is the displacement amplitude and o the angular frequency, then the normal component (E ¼ E n) of the electric field inside the capacitor is also harmonic and characterized by the same frequency but different phase and amplitude: E ¼ E0 eiðotþfðoÞÞ
(1.2)
where E0 is the field amplitude and f the phase lag. Furthermore we have D ¼ E ¼ ð0 i00 ÞE
(1.3)
where is the complex dielectric constant and
0 ¼ j j cos f;
00 ¼ j j sin f;
00 ¼ tan f 0
(1.4)
the quantities 0 and 00 are the real and imaginary components of the complex dielectric constant [7]. The quantities 0 and 00 are related to the relative dielectric constants 1 , 2 by the following expressions:
0 ¼ 0 1 ;
00 ¼ 0 2
(1.5)
where 0 is the dielectric constant in vacuum. These are usually called the dielectric storage factor and dielectric loss factor, respectively, and tan f is termed the loss tangent. Let us remark that in a relaxation region 0 is decreasing with frequency from a value of R to U . This decrease represents the dispersion of the
ARTICLE IN PRESS V. Ciancio et al. / Physica B 404 (2009) 320–324
dielectric constant [6], the difference ðR U Þ is known as the magnitude of the relaxation and it expresses a measure of the orientation polarization [8]. In relaxation region 00 passes through a maximum at a frequency o00 . By computing the real part of Eq. (1.2) one has E ¼ D0 s1 sinðotÞ þ D0 s2 cosðotÞ
(1.6)
where (1.7) (1.8)
If electric charge density on the plates (extensive quantity) is viewed as the cause determining the electric field inside capacitor (intensive variable), it allows us to study dielectric relaxation phenomena. By defining the reciprocal complex dielectric constant s ¼ E =D ¼ s1 þ is2 the complex dielectric constant is related to it as 1 ¼ ¼ 0 i00 s
(1.9)
; 0 2 þ 002
0 2 þ 002 00
s2 ¼
(2.3)
where 0 is the dielectric constant in vacuum; by substituting it in Eq. (1.3) the equation for dielectric relaxation reduces to 2
ð0Þ wðEDÞ E þ wð1Þ ðEDÞ
2
dE d E dD d D ð0Þ þ wð2Þ þ wð2Þ ¼ wðDEÞ D þ wð1Þ ðEDÞ ðDEÞ ðDEÞ dt dt dt 2 dt 2
ð0Þ ð0Þ ð0Þ wðEDÞ ¼ wðEPÞ þ 0 wðPEÞ ð1Þ ðEDÞ
(1.10)
Taking into account Eqs. (1.7) and (1.8) the following expressions are obtained:
w
ð2Þ ðEDÞ
w
¼1þw
(2.6)
ð2Þ ðPEÞ 0
¼w
(2.7)
ð0Þ ð0Þ wðDEÞ ¼ wðPEÞ ð1Þ ðDEÞ
w
ð2Þ ðDEÞ
w
(2.8)
ð1Þ ðPEÞ
¼w
(2.9)
ð2Þ ðPEÞ
¼w
(2.10) ð2Þ ðEDÞ a0
By dividing w form:
Eq. (2.4) one obtains the following normal
ð1Þ
2
ð0Þ
(1.11) (1.12)
where
Since the phase difference f depends on frequency, it follows that for values of o sufficiently small f approaches to zero obtaining from Eqs. (1.12) and (1.12):
a ¼ D0
00 ffi 0
(1.13)
where E0R is the value of E0 ðoÞ for sufficiently small values of o and R is the relaxed value of 0 . Analogously, for sufficiently large values of o the phase f goes to zero and one has
0 ffi
D0 ¼ U ; E0U
!
ð0Þ wðDEÞ o2 ; ð2Þ wðEDÞ 0
"
(1.14)
where E0U is the value of E0 ðoÞ for large values of o, and U is the un-relaxed value of 0 .
wð1Þ ðDEÞ o . wð2Þ ðEwDÞ
#
bðl1 l2 o2 Þ þ aoðl1 þ l2 Þ cosðotÞ 2 2 ðl1 þ o2 Þðl2 þ o2 Þ
wð1Þ wð0Þ ðEDÞ l þ ðEDÞ ¼0 ð2Þ wðEDÞ wð2Þ ðEDÞ
P ¼ Pð0Þ þ Pð1Þ ð0Þ
(2.1) ð1Þ
where P and P are the reversible (elastic) and irreversible parts of P, respectively. In the linear approximation and by neglecting cross effects as the influence of electric conduction, heat conduction and (mechanical) viscosity on electric relaxation, the following relaxation equation may be derived [9]: 2
wð0Þ EP E þ
dE d P ð1Þ dP ¼ wð0Þ þ wð2Þ PE P þ wPE ðPEÞ dt dt dt 2 ð0Þ ðEPÞ ,
(2.2) ðiÞ ðPEÞ
where E is the electric field and w w ði ¼ 0; 1; 2Þ are algebraic functions of the coefficients occurring in the phenomenological equations (describing the irreversible processes) and in the equations of state.
(2.14)
(2.15) 1
Following Refs. [9–15], a general hypothesis concerning the entropy allows us to decompose the polarization vector P as
(2.13)
where c1 and c2 are two arbitrary integration constants, l1 and l2 are solutions of the homogeneous equation associated to (2.12)
l2 þ 2. Phenomenological coefficients
b ¼ D0
(2.12)
The integration of differential equation (2.12) gives the following general solution that gives the electric field: " # aðl1 l2 o2 Þ boðl1 þ l2 Þ l1 t l2 t EðtÞ ¼ c1 c1 e þ c2 e þ sinðotÞ 2 2 ðl1 þ o2 Þðl2 þ o2 Þ þ
00 ffi 0
(2.11)
By computing the time derivative of D expressed by the real part of Eq. (1.1) having used the normal component of electric field E ¼ E n, the differential equation (2.11) can be written as d E wðEDÞ dE wðEDÞ þ þ E ¼ a sinðotÞ þ b cosðotÞ dt wð2Þ dt 2 wð2Þ ðEDÞ ðEDÞ
D ¼ 0 cosðoÞ E0 ðoÞ D 00 ¼ 0 sin fðoÞ E0 ðoÞ 0
D 0 ffi 0 ¼ R ; E0R
(2.4)
(2.5)
ð1Þ ðPEÞ 0
ð0Þ ð0Þ 2 2 wðEDÞ wð1Þ wð1Þ d E wðDEÞ 1d D ðEDÞ dE ðDEÞ dD þ 2 ¼ ð2Þ D þ ð2Þ þ E þ ð2Þ ð2Þ wðEDÞ wðEDÞ dt dt wðEDÞ wðEDÞ dt 0 dt2
where s1 ¼
By considering the expression for the polarization vector: P ¼ D 0 E
where we are setting
E0 ðoÞ cos fðoÞ D0 E0 ðoÞ sin fðoÞ s2 ¼ D0
s1 ¼
0
321
1
moreover the quantities l1 and l2 represent the two relaxation times. Since Eqs. (1.6) and (2.14) describe mathematically the same phenomenon, by neglecting any transitory effect, the identification of two equations leads to the following:
aðl1 l2 o2 Þ boðl1 þ l2 Þ 2 2 ðl1 þ o2 Þðl2 þ o2 Þ bðl1 l2 o2 Þ þ aoðl1 þ l2 Þ D0 s2 ¼ 2 2 ðl1 þ o2 Þðl2 þ o2 Þ D0 s1 ¼
(2.16) (2.17)
By using Eqs. (1.10) and (2.13), from Eqs. (2.16) and (2.17) we obtain 00 0 wð0Þ ðEPÞ þ oð 0 Þ o½ð0 0 Þ2 þ 002 ð0Þ wð0Þ wðEPÞ ð0 0 Þ þ 00 o ðPEÞ wð2Þ þ ðPEÞ ¼ 2 o o2 ½ð0 0 Þ2 þ 002
wð1Þ ðPEÞ ¼
(2.18) (2.19)
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This is an algebraic system of two equations with four unknown ð0Þ 0 00 functions wðiÞ ðPEÞ ði ¼ 0; 1; 2Þ and wðEPÞ where and are experimentally measured. Since we will express these coefficients as function of 0 and 00 we need other two equations to complete the system. We will select our analysis only for low and high frequencies and it is not considered the range of frequency in which glass transition occurs. The first equation to complete the system is obtained by observing that under a sufficiently low frequency the phase lag between D and E is close to zero value; moreover the variation of such vectors can be neglected and from Eq. (2.4) it follows: ! ð0Þ wð0Þ ðEPÞ þ 0 wðPEÞ D¼ E (2.20) ð0Þ
ð0Þ wð2Þ ðPEÞ ¼ wðEPÞ
R=U
ð0Þ ð0Þ wðEPÞ þ 0 wðPEÞ ¼ wð0Þ ðPEÞ
(2.21)
where it one can select the value R or U in R=U depending on it and referring to low or high frequencies, respectively. From Eq. (2.21) one has ð0Þ wð0Þ ðEPÞ ¼ wðPEÞ ðR=U 0 Þ
(2.22)
This is one of the two equations to complete the system; the second one can be obtained by taking into account that from classical considerations about the roots of second degree equation applied to Eq. (2.15), the following relations hold: 1
t1
þ
1
t1 t2
1
t2
¼
¼
wð1Þ ðEDÞ wð2Þ ðEDÞ
! þ
00 o½ð 0 Þ2 þ 002 0
(2.27) As it is well known [2–16], since the entropy production is a nonnegative quantity coefficients (2.24)–(2.27) have to be positive; ð0Þ ð0Þ then the positivity of the coefficients wðEPÞ , wðPEÞ , wð1Þ ðPEÞ provides the following conditions:
oðt1 þ t2 ÞX
00 0 ðR=U 0 Þ R=U ½ð0 0 Þ2 þ 002
oðt1 þ t2 ÞX
0 002 ðR=U 0 Þ ½0 ð0 0 Þ þ 002 ½ð0 0 Þð0 R=U Þ þ 002 R=U 00 ½ð0 0 Þ2 þ 002 (2.29)
It is easy to show from Eqs. (2.24) to (2.27) that for sufficiently low and high frequencies wð2Þ ðPEÞ approach zero whereas the other ð1Þ coefficients wð0ÞðEPÞ ; wð0Þ ðPEÞ and wðPEÞ approach a positive value. This means that in Eq. (2.2) the term connected with the second derivative vanishes and it reduces to the Debye equation. This is confirmed by experiments carried out on Vinylidene Chloride– Vinyl Chloride (VDC–VC) 80% VDC and 5% Plasticizer (see Fig. 1).
3. Electromagnetic waves In this section the expressions which assume some wave characteristic entity as function of dielectric complex constant are investigated. Let us consider the following Maxwell equations in the Gaussian system of unity, in the case in which the electric currents and electric charges are neglected rot H
1 qD ¼0 c dt
rot E þ
1 qB ¼0 c qt
w
div B ¼ 0
ð0Þ wðEDÞ wð2Þ ðEDÞ
(2.23)
B ¼ mH
(3.2)
o½ð0 0 Þ2 þ 002 þ ð0 0 Þ0
1
ðR=U 0 Þ t1 t2 ¼ 1 1 o½ð0 0 Þ2 þ 002 R=U þ 00 t11t2 0 ðR=U 0 Þ t1 t2 (2.24)
ð0Þ ðPEÞ
w
1 ¼ wð0Þ R=U 0 EP
wð1Þ ðPEÞ ¼
(3.1)
div D ¼ 0
where t1 and t2 are the two relaxation times; moreover, it is useful to observe that these relations can be obtained experimentally. Obviously these last equations are not independent. Relations (2.18), (2.19), (2.22) and ð2:23Þ1 or ð2:23Þ2 represent two complete systems of equations: one for low frequency where in Eq. (2.21) R is selected and the other one for high frequency in which the value U is selected. Therefore the solution of system is resumed as follows:
ð0Þ ðEPÞ
(2.28)
while the coefficient wð2Þ ðPEÞ is positive if the last inequality (2.28) and the following together hold:
wðPEÞ
On the other hand, even for sufficiently high frequency, the phase lag between D and E is close to zero value since the density charge changes so rapidly that no relaxation has time to occur; this implies that the medium is not sensible to time variations of D and consequently variations of E are very small; in such a context it is possible to neglect the time derivatives in Eq. (2.4) obtaining a relation as Eq. (2.20). More precisely, it is assumed that there exist two ranges of o, one for low frequency and the other one for high frequency, such that the relative values of 0 will assume a value close to R and U , respectively. It is reasonable to identify approximatively these quantities with the value into the bracket of Eq. (2.20) setting
ðR=U
1 ð 0 0 Þ 2 2 0 0 Þ o o ½ð 0 Þ2 þ 002
(2.25)
ð 0 Þ wð0Þ þ o½ð0 0 Þ2 þ 002 ðEPÞ ð0 0 Þ2 þ 002 00
0
(2.26) Fig. 1. Phenomenological coefficients for VDC–VC.
ARTICLE IN PRESS V. Ciancio et al. / Physica B 404 (2009) 320–324
323
where H is the magnetic displacement field and B the magnetic field strength and m the magnetic permeability; moreover magnetic relaxation phenomena are left out of consideration and it is considered that the medium is at rest; then the material derivative with respect to time may be replaced by the local time derivative [9]. It is well known that Eqs. (2.4) and (3.1) have wave solution. In particular, we consider plane waves which propagate in the direction of the x-axis. The hypotheses are that the generic vector W ðH; E; D; BÞ has the form W ¼ W0 eiðkxotÞ
(3.3)
where k is the complex wave number, o is the real angular frequency, i2 ¼ 1 and W0 is a constant which may be complex. It is important to assume that these values of o have to be the same of those relative to sinusoidal voltage applied between the plates of capacitor. Since in a medium with dielectric relaxation attenuation wave occurs, it is useful to introduce a complex wave number k ¼ k1 þ ik2
(3.4)
where k1 and k2 are real and connected to phase velocity vs ¼ o=k1 and to attenuation k2 o0, respectively. Following Ref. [9], by substituting the expression (3.3) into ð3:1Þ1;2 Eq. (2.4) and by taking into account that the components of E and D are different from zero one has ( ) 2 c2 k ð1Þ ð1Þ 2 ð2Þ 0 2 ð2Þ E0 ¼ 0 ðwð0Þ ðEDÞ iowðEDÞ o wðEDÞ Þ ðoðDEÞ iowðDEÞ o wðDEÞ Þ 2
mo
Fig. 2. Wave number and phase velocity for VDC–VC.
a value R and to zero, respectively, then pffiffiffiffiffiffiffiffi mR k1 ffi o c
(3.12)
k2 ffi 0
(3.13)
while for values of o sufficiently large, and go to a value U and zero, respectively, and one has pffiffiffiffiffiffiffiffiffi mU (3.14) k1 ffi o c 0
00
(3.5) Therefore from Eq. (3.5) one has ( ð0Þ ) ð2Þ ð1Þ mo2 wðEDÞ o2 wðEDÞ iowðEDÞ 2 k ¼ 2 ð0Þ ð1Þ c wðDEÞ o2 wð2Þ ðDEÞ iowðDEÞ
k2 ffi 0
(3.6)
Taking into account the decomposition of k as complex wave number, it follows: ( ) ð2Þ ð2Þ ð1Þ ð1Þ ð0Þ ð2Þ ð2Þ ð0Þ mo2 o4 wðEDÞ wðDEÞ þ o2 wðEDÞ wðDEÞ wðEDÞ wðDEÞ wðEDÞ wðDEÞ 2 2 k1 k2 ¼ 2 2 ð2Þ 2 2 ð1Þ 2 c ðwð0Þ ðDEÞ o wðDEÞ Þ þ o ðwðDEÞ Þ (
)
ð1Þ ð2Þ ð2Þ ð1Þ ð0Þ ð1Þ ð1Þ ð0Þ mo3 o2 wðEDÞ wðDEÞ o2 wðEDÞ wðDEÞ þ wðEDÞ wðDEÞ wðEDÞ wðDEÞ . k1 k2 ¼ 2 ð2Þ 2 2 ð1Þ 2 2c2 ðwð0Þ ðDEÞ o wðDEÞ Þ þ o ðwðDEÞ Þ
(3.7) By substituting relations (2.25), after some calculations expressions (3.7) can be written as follows: 2
2
k1 k2 ¼ k1 k2 ¼
mo2 c2
mo2 2c2
00
0
(3.8)
Finally, the phase velocity of the waves is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 2c 0 þ 0 2 þ 002 ¼ pffiffiffiffiffiffi v¼ k1 2m00
(3.15)
(3.16)
while the term ek2 x , with k2 o0, is responsible of the attenuation of the amplitude. More precisely, the behaviours of coefficients k1 , k2 and phase velocity as functions of angular frequency can be analysed observing Fig. 2.
4. Experimental applications In this paper the phenomenological coefficients (2.24)–(2.27) for a polymeric material as VDC–VC have been calculated. The dielectric measurements were performed scanning the range of frequency 102 2106 H2 by Rheometric Scientific Analyser (DETA). The VDC–VC material sample in the shape of suitable disk was previously metallized with gold to ensure a good contact with stainless-steel blocking electrodes of the DETA; the quantity 0 and 00 were determined at temperature of 72 1C.
(3.9)
By solving Eqs. (3.8) and (3.9) with respect to k1 and k2 , the following relations, as functions of real and imaginary parts of complex dielectric constant, are obtained: pffiffiffiffiffiffi 2m o00 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 ¼ (3.10) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c 0 þ 0 2 þ 002 pffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 mo (3.11) 0 þ 0 2 þ 002 k2 ¼ pffiffiffi c 2 Moreover, it is possible to remark that for angular frequency o sufficiently small, as it was stated in a previous section, the real and imaginary parts of complex dielectric constants 0 and 00 go to
5. Conclusions From a theoretical point of view the results obtained in this paper give a concrete contribution to complete the fundamental equations which hold in relaxation phenomena making possible the integration. Moreover, these contribute to the experimental determination of phenomenological and state coefficients which appear in Kluitenberg–Ciancio theory for dielectric media with relaxation phenomena. On the other hand, from the aforementioned expression of phenomenological coefficients it has been possible to obtain a very simple form for the wave vector which reduces to classical expression for sufficiently low and high frequencies. It is important
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to remark that in our calculation this occurs for sufficiently low and high frequencies and not for o ¼ 0 and 1 which have no physical sense. The experimental verification of the conclusions of Section 2 on a polymeric material as VDC–VC, after locating the zone in which the medium has an approximative linear behaviour, is in agreement with some theoretical predictions. In particular, the approach to zero of coefficient wð2Þ ðPEÞ in these regions has been observed. Moreover, the positivity of phenomenological coefficients discussed above shows a region of low frequency with negative values. This fact can be justified if in this range some relaxation phenomenon occurs which has been neglected. This result can be utilized as a test for selecting the region in which the principle of entropy production is true. References [1] V. Ciancio, F. Farsaci, G. Di Marco, Physica B 387 (2007) 130. [2] G.A. Kluitenberg, Physica A 109 (1–2) (1981) 91.
[3] V. Ciancio, J. Non-Equilib. Thermodyn. 14 (1989) 239. [4] V. Ciancio, Afrika Math. Ser. 3 4 (1994). [5] D.C. Champeney, Fourier Transforms and their Physical Applications, Academic Press, London, New York, 1973. [6] N.G. McCrum, B.E. Read, G. Williams, An elastic and Dielectric Effects in Polymeric Solids, Wiley, London, New York, Sydney, 1967. [7] P. Hedvig, Dielectric Spectroscopy of Polymers, Adam Hilger, LTD Bristol, 1977. [8] W. Graham, in: G. Allen, J.C. Bevington (Eds.), Comprehensive Polymer Science, vol. 2, Pergamon Press, Oxford, 1989 (Chapter 16). [9] G.A. Kluitenberg, Physica 68 (1) (1973) 75. [10] G.A. Kluitenberg, Physica A 87 (2) (1977) 302. [11] G.A. Kluitenberg, V. Ciancio, Acta Phys. Hungar. 66 (1989) 251. [12] V. Ciancio, Afrika Matematika Serie 3 4 (1994) 89. [13] V. Ciancio, G.A. Kluitenberg, J. Non-Equilib. Thermodyn. 15 (1990) 157. [14] V. Ciancio, C. Papenfuss, P. Rogolino, Tech. Mech. 22 (2) (2002) 132. [15] S.R. De Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam, 1962. [16] G.D.C. Kuiken, Thermodynamics of Irreversible Processes, Application to Diffusion and Rheology, Wiley, New York, 1994.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Phenomenological approach on wave propagation in dielectric media with two relaxation times V. Ciancio a,, F. Farsaci b, P. Rogolino a a b
Department of Mathematics, University of Messina, Italy IPCF-C.N.R., Messina, Italy
a r t i c l e in f o
a b s t r a c t
Article history: Received 21 October 2008 Received in revised form 4 November 2008 Accepted 4 November 2008
In this paper dielectric phenomena with two relaxation times are discussed. By assuming a sinusoidal form for induction vector D a sinusoidal electric field is generated and it depends on unknown phenomenological coefficients whose expressions together to their numerical values as functions of frequency are obtained. Moreover, electromagnetic wave propagation is analysed obtaining wave vector as function of the aforementioned coefficients. The results are applied to a Vinylidene Chloride-Vinyl Chloride (VDC–VC) to test the applicability of the model. & 2008 Elsevier B.V. All rights reserved.
Keywords: Irreversible thermodynamics Relaxation phenomena Wave propagation
1. Introduction In a previous paper [1] the connection between phenomenological coefficients and quantities experimentally measurable, e.g. real and imaginary parts of complex dielectric constant, has been obtained for media with dielectric relaxation phenomena and in the case in which just one relaxation time is considered. In Refs. [2–4] a phenomenological equation was proposed in which two relaxation times occur and this is connected to physical behaviour of materials. In fact the instantaneous increasing or decreasing in the polarization is impossible because any change of the polarization is related to the motion of any kind of microscopic particles which cannot be infinitely fast. The phenomenological equation will be discussed in Section 2 and we will refer to the case in which two relaxation times are taken into account and in such a context we will study electromagnetic wave propagation obtaining wave vector as function of the aforementioned quantities experimentally measurable. Our aim is to study the system under a sinusoidal perturbation represented by the induction vector D, which is an extensive variable (cause), and to analyse the relative electric field as an intensive variable (effect) inside the medium [5]. In particular, if we consider a generic dielectric medium placed between the plain plate of a capacitor where a sinusoidal voltage has been applied, then on the plates a sinusoidal surface charge arises whose density is characterized by the normal component of induction
Corresponding author. Tel.: +39 90 6765061; fax: +39 90 393502.
E-mail address: [email protected] (V. Ciancio). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.11.070
vector D ¼ D n (n is the unit normal to the plates) generating a sinusoidal electric field inside the capacitor [6]. The linear-response theory establishes that if D (cause) evolves harmonically [5], i.e. D ¼ D0 eiot
(1.1)
where D0 is the displacement amplitude and o the angular frequency, then the normal component (E ¼ E n) of the electric field inside the capacitor is also harmonic and characterized by the same frequency but different phase and amplitude: E ¼ E0 eiðotþfðoÞÞ
(1.2)
where E0 is the field amplitude and f the phase lag. Furthermore we have D ¼ E ¼ ð0 i00 ÞE
(1.3)
where is the complex dielectric constant and
0 ¼ j j cos f;
00 ¼ j j sin f;
00 ¼ tan f 0
(1.4)
the quantities 0 and 00 are the real and imaginary components of the complex dielectric constant [7]. The quantities 0 and 00 are related to the relative dielectric constants 1 , 2 by the following expressions:
0 ¼ 0 1 ;
00 ¼ 0 2
(1.5)
where 0 is the dielectric constant in vacuum. These are usually called the dielectric storage factor and dielectric loss factor, respectively, and tan f is termed the loss tangent. Let us remark that in a relaxation region 0 is decreasing with frequency from a value of R to U . This decrease represents the dispersion of the
ARTICLE IN PRESS V. Ciancio et al. / Physica B 404 (2009) 320–324
dielectric constant [6], the difference ðR U Þ is known as the magnitude of the relaxation and it expresses a measure of the orientation polarization [8]. In relaxation region 00 passes through a maximum at a frequency o00 . By computing the real part of Eq. (1.2) one has E ¼ D0 s1 sinðotÞ þ D0 s2 cosðotÞ
(1.6)
where (1.7) (1.8)
If electric charge density on the plates (extensive quantity) is viewed as the cause determining the electric field inside capacitor (intensive variable), it allows us to study dielectric relaxation phenomena. By defining the reciprocal complex dielectric constant s ¼ E =D ¼ s1 þ is2 the complex dielectric constant is related to it as 1 ¼ ¼ 0 i00 s
(1.9)
; 0 2 þ 002
0 2 þ 002 00
s2 ¼
(2.3)
where 0 is the dielectric constant in vacuum; by substituting it in Eq. (1.3) the equation for dielectric relaxation reduces to 2
ð0Þ wðEDÞ E þ wð1Þ ðEDÞ
2
dE d E dD d D ð0Þ þ wð2Þ þ wð2Þ ¼ wðDEÞ D þ wð1Þ ðEDÞ ðDEÞ ðDEÞ dt dt dt 2 dt 2
ð0Þ ð0Þ ð0Þ wðEDÞ ¼ wðEPÞ þ 0 wðPEÞ ð1Þ ðEDÞ
(1.10)
Taking into account Eqs. (1.7) and (1.8) the following expressions are obtained:
w
ð2Þ ðEDÞ
w
¼1þw
(2.6)
ð2Þ ðPEÞ 0
¼w
(2.7)
ð0Þ ð0Þ wðDEÞ ¼ wðPEÞ ð1Þ ðDEÞ
w
ð2Þ ðDEÞ
w
(2.8)
ð1Þ ðPEÞ
¼w
(2.9)
ð2Þ ðPEÞ
¼w
(2.10) ð2Þ ðEDÞ a0
By dividing w form:
Eq. (2.4) one obtains the following normal
ð1Þ
2
ð0Þ
(1.11) (1.12)
where
Since the phase difference f depends on frequency, it follows that for values of o sufficiently small f approaches to zero obtaining from Eqs. (1.12) and (1.12):
a ¼ D0
00 ffi 0
(1.13)
where E0R is the value of E0 ðoÞ for sufficiently small values of o and R is the relaxed value of 0 . Analogously, for sufficiently large values of o the phase f goes to zero and one has
0 ffi
D0 ¼ U ; E0U
!
ð0Þ wðDEÞ o2 ; ð2Þ wðEDÞ 0
"
(1.14)
where E0U is the value of E0 ðoÞ for large values of o, and U is the un-relaxed value of 0 .
wð1Þ ðDEÞ o . wð2Þ ðEwDÞ
#
bðl1 l2 o2 Þ þ aoðl1 þ l2 Þ cosðotÞ 2 2 ðl1 þ o2 Þðl2 þ o2 Þ
wð1Þ wð0Þ ðEDÞ l þ ðEDÞ ¼0 ð2Þ wðEDÞ wð2Þ ðEDÞ
P ¼ Pð0Þ þ Pð1Þ ð0Þ
(2.1) ð1Þ
where P and P are the reversible (elastic) and irreversible parts of P, respectively. In the linear approximation and by neglecting cross effects as the influence of electric conduction, heat conduction and (mechanical) viscosity on electric relaxation, the following relaxation equation may be derived [9]: 2
wð0Þ EP E þ
dE d P ð1Þ dP ¼ wð0Þ þ wð2Þ PE P þ wPE ðPEÞ dt dt dt 2 ð0Þ ðEPÞ ,
(2.2) ðiÞ ðPEÞ
where E is the electric field and w w ði ¼ 0; 1; 2Þ are algebraic functions of the coefficients occurring in the phenomenological equations (describing the irreversible processes) and in the equations of state.
(2.14)
(2.15) 1
Following Refs. [9–15], a general hypothesis concerning the entropy allows us to decompose the polarization vector P as
(2.13)
where c1 and c2 are two arbitrary integration constants, l1 and l2 are solutions of the homogeneous equation associated to (2.12)
l2 þ 2. Phenomenological coefficients
b ¼ D0
(2.12)
The integration of differential equation (2.12) gives the following general solution that gives the electric field: " # aðl1 l2 o2 Þ boðl1 þ l2 Þ l1 t l2 t EðtÞ ¼ c1 c1 e þ c2 e þ sinðotÞ 2 2 ðl1 þ o2 Þðl2 þ o2 Þ þ
00 ffi 0
(2.11)
By computing the time derivative of D expressed by the real part of Eq. (1.1) having used the normal component of electric field E ¼ E n, the differential equation (2.11) can be written as d E wðEDÞ dE wðEDÞ þ þ E ¼ a sinðotÞ þ b cosðotÞ dt wð2Þ dt 2 wð2Þ ðEDÞ ðEDÞ
D ¼ 0 cosðoÞ E0 ðoÞ D 00 ¼ 0 sin fðoÞ E0 ðoÞ 0
D 0 ffi 0 ¼ R ; E0R
(2.4)
(2.5)
ð1Þ ðPEÞ 0
ð0Þ ð0Þ 2 2 wðEDÞ wð1Þ wð1Þ d E wðDEÞ 1d D ðEDÞ dE ðDEÞ dD þ 2 ¼ ð2Þ D þ ð2Þ þ E þ ð2Þ ð2Þ wðEDÞ wðEDÞ dt dt wðEDÞ wðEDÞ dt 0 dt2
where s1 ¼
By considering the expression for the polarization vector: P ¼ D 0 E
where we are setting
E0 ðoÞ cos fðoÞ D0 E0 ðoÞ sin fðoÞ s2 ¼ D0
s1 ¼
0
321
1
moreover the quantities l1 and l2 represent the two relaxation times. Since Eqs. (1.6) and (2.14) describe mathematically the same phenomenon, by neglecting any transitory effect, the identification of two equations leads to the following:
aðl1 l2 o2 Þ boðl1 þ l2 Þ 2 2 ðl1 þ o2 Þðl2 þ o2 Þ bðl1 l2 o2 Þ þ aoðl1 þ l2 Þ D0 s2 ¼ 2 2 ðl1 þ o2 Þðl2 þ o2 Þ D0 s1 ¼
(2.16) (2.17)
By using Eqs. (1.10) and (2.13), from Eqs. (2.16) and (2.17) we obtain 00 0 wð0Þ ðEPÞ þ oð 0 Þ o½ð0 0 Þ2 þ 002 ð0Þ wð0Þ wðEPÞ ð0 0 Þ þ 00 o ðPEÞ wð2Þ þ ðPEÞ ¼ 2 o o2 ½ð0 0 Þ2 þ 002
wð1Þ ðPEÞ ¼
(2.18) (2.19)
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This is an algebraic system of two equations with four unknown ð0Þ 0 00 functions wðiÞ ðPEÞ ði ¼ 0; 1; 2Þ and wðEPÞ where and are experimentally measured. Since we will express these coefficients as function of 0 and 00 we need other two equations to complete the system. We will select our analysis only for low and high frequencies and it is not considered the range of frequency in which glass transition occurs. The first equation to complete the system is obtained by observing that under a sufficiently low frequency the phase lag between D and E is close to zero value; moreover the variation of such vectors can be neglected and from Eq. (2.4) it follows: ! ð0Þ wð0Þ ðEPÞ þ 0 wðPEÞ D¼ E (2.20) ð0Þ
ð0Þ wð2Þ ðPEÞ ¼ wðEPÞ
R=U
ð0Þ ð0Þ wðEPÞ þ 0 wðPEÞ ¼ wð0Þ ðPEÞ
(2.21)
where it one can select the value R or U in R=U depending on it and referring to low or high frequencies, respectively. From Eq. (2.21) one has ð0Þ wð0Þ ðEPÞ ¼ wðPEÞ ðR=U 0 Þ
(2.22)
This is one of the two equations to complete the system; the second one can be obtained by taking into account that from classical considerations about the roots of second degree equation applied to Eq. (2.15), the following relations hold: 1
t1
þ
1
t1 t2
1
t2
¼
¼
wð1Þ ðEDÞ wð2Þ ðEDÞ
! þ
00 o½ð 0 Þ2 þ 002 0
(2.27) As it is well known [2–16], since the entropy production is a nonnegative quantity coefficients (2.24)–(2.27) have to be positive; ð0Þ ð0Þ then the positivity of the coefficients wðEPÞ , wðPEÞ , wð1Þ ðPEÞ provides the following conditions:
oðt1 þ t2 ÞX
00 0 ðR=U 0 Þ R=U ½ð0 0 Þ2 þ 002
oðt1 þ t2 ÞX
0 002 ðR=U 0 Þ ½0 ð0 0 Þ þ 002 ½ð0 0 Þð0 R=U Þ þ 002 R=U 00 ½ð0 0 Þ2 þ 002 (2.29)
It is easy to show from Eqs. (2.24) to (2.27) that for sufficiently low and high frequencies wð2Þ ðPEÞ approach zero whereas the other ð1Þ coefficients wð0ÞðEPÞ ; wð0Þ ðPEÞ and wðPEÞ approach a positive value. This means that in Eq. (2.2) the term connected with the second derivative vanishes and it reduces to the Debye equation. This is confirmed by experiments carried out on Vinylidene Chloride– Vinyl Chloride (VDC–VC) 80% VDC and 5% Plasticizer (see Fig. 1).
3. Electromagnetic waves In this section the expressions which assume some wave characteristic entity as function of dielectric complex constant are investigated. Let us consider the following Maxwell equations in the Gaussian system of unity, in the case in which the electric currents and electric charges are neglected rot H
1 qD ¼0 c dt
rot E þ
1 qB ¼0 c qt
w
div B ¼ 0
ð0Þ wðEDÞ wð2Þ ðEDÞ
(2.23)
B ¼ mH
(3.2)
o½ð0 0 Þ2 þ 002 þ ð0 0 Þ0
1
ðR=U 0 Þ t1 t2 ¼ 1 1 o½ð0 0 Þ2 þ 002 R=U þ 00 t11t2 0 ðR=U 0 Þ t1 t2 (2.24)
ð0Þ ðPEÞ
w
1 ¼ wð0Þ R=U 0 EP
wð1Þ ðPEÞ ¼
(3.1)
div D ¼ 0
where t1 and t2 are the two relaxation times; moreover, it is useful to observe that these relations can be obtained experimentally. Obviously these last equations are not independent. Relations (2.18), (2.19), (2.22) and ð2:23Þ1 or ð2:23Þ2 represent two complete systems of equations: one for low frequency where in Eq. (2.21) R is selected and the other one for high frequency in which the value U is selected. Therefore the solution of system is resumed as follows:
ð0Þ ðEPÞ
(2.28)
while the coefficient wð2Þ ðPEÞ is positive if the last inequality (2.28) and the following together hold:
wðPEÞ
On the other hand, even for sufficiently high frequency, the phase lag between D and E is close to zero value since the density charge changes so rapidly that no relaxation has time to occur; this implies that the medium is not sensible to time variations of D and consequently variations of E are very small; in such a context it is possible to neglect the time derivatives in Eq. (2.4) obtaining a relation as Eq. (2.20). More precisely, it is assumed that there exist two ranges of o, one for low frequency and the other one for high frequency, such that the relative values of 0 will assume a value close to R and U , respectively. It is reasonable to identify approximatively these quantities with the value into the bracket of Eq. (2.20) setting
ðR=U
1 ð 0 0 Þ 2 2 0 0 Þ o o ½ð 0 Þ2 þ 002
(2.25)
ð 0 Þ wð0Þ þ o½ð0 0 Þ2 þ 002 ðEPÞ ð0 0 Þ2 þ 002 00
0
(2.26) Fig. 1. Phenomenological coefficients for VDC–VC.
ARTICLE IN PRESS V. Ciancio et al. / Physica B 404 (2009) 320–324
323
where H is the magnetic displacement field and B the magnetic field strength and m the magnetic permeability; moreover magnetic relaxation phenomena are left out of consideration and it is considered that the medium is at rest; then the material derivative with respect to time may be replaced by the local time derivative [9]. It is well known that Eqs. (2.4) and (3.1) have wave solution. In particular, we consider plane waves which propagate in the direction of the x-axis. The hypotheses are that the generic vector W ðH; E; D; BÞ has the form W ¼ W0 eiðkxotÞ
(3.3)
where k is the complex wave number, o is the real angular frequency, i2 ¼ 1 and W0 is a constant which may be complex. It is important to assume that these values of o have to be the same of those relative to sinusoidal voltage applied between the plates of capacitor. Since in a medium with dielectric relaxation attenuation wave occurs, it is useful to introduce a complex wave number k ¼ k1 þ ik2
(3.4)
where k1 and k2 are real and connected to phase velocity vs ¼ o=k1 and to attenuation k2 o0, respectively. Following Ref. [9], by substituting the expression (3.3) into ð3:1Þ1;2 Eq. (2.4) and by taking into account that the components of E and D are different from zero one has ( ) 2 c2 k ð1Þ ð1Þ 2 ð2Þ 0 2 ð2Þ E0 ¼ 0 ðwð0Þ ðEDÞ iowðEDÞ o wðEDÞ Þ ðoðDEÞ iowðDEÞ o wðDEÞ Þ 2
mo
Fig. 2. Wave number and phase velocity for VDC–VC.
a value R and to zero, respectively, then pffiffiffiffiffiffiffiffi mR k1 ffi o c
(3.12)
k2 ffi 0
(3.13)
while for values of o sufficiently large, and go to a value U and zero, respectively, and one has pffiffiffiffiffiffiffiffiffi mU (3.14) k1 ffi o c 0
00
(3.5) Therefore from Eq. (3.5) one has ( ð0Þ ) ð2Þ ð1Þ mo2 wðEDÞ o2 wðEDÞ iowðEDÞ 2 k ¼ 2 ð0Þ ð1Þ c wðDEÞ o2 wð2Þ ðDEÞ iowðDEÞ
k2 ffi 0
(3.6)
Taking into account the decomposition of k as complex wave number, it follows: ( ) ð2Þ ð2Þ ð1Þ ð1Þ ð0Þ ð2Þ ð2Þ ð0Þ mo2 o4 wðEDÞ wðDEÞ þ o2 wðEDÞ wðDEÞ wðEDÞ wðDEÞ wðEDÞ wðDEÞ 2 2 k1 k2 ¼ 2 2 ð2Þ 2 2 ð1Þ 2 c ðwð0Þ ðDEÞ o wðDEÞ Þ þ o ðwðDEÞ Þ (
)
ð1Þ ð2Þ ð2Þ ð1Þ ð0Þ ð1Þ ð1Þ ð0Þ mo3 o2 wðEDÞ wðDEÞ o2 wðEDÞ wðDEÞ þ wðEDÞ wðDEÞ wðEDÞ wðDEÞ . k1 k2 ¼ 2 ð2Þ 2 2 ð1Þ 2 2c2 ðwð0Þ ðDEÞ o wðDEÞ Þ þ o ðwðDEÞ Þ
(3.7) By substituting relations (2.25), after some calculations expressions (3.7) can be written as follows: 2
2
k1 k2 ¼ k1 k2 ¼
mo2 c2
mo2 2c2
00
0
(3.8)
Finally, the phase velocity of the waves is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 2c 0 þ 0 2 þ 002 ¼ pffiffiffiffiffiffi v¼ k1 2m00
(3.15)
(3.16)
while the term ek2 x , with k2 o0, is responsible of the attenuation of the amplitude. More precisely, the behaviours of coefficients k1 , k2 and phase velocity as functions of angular frequency can be analysed observing Fig. 2.
4. Experimental applications In this paper the phenomenological coefficients (2.24)–(2.27) for a polymeric material as VDC–VC have been calculated. The dielectric measurements were performed scanning the range of frequency 102 2106 H2 by Rheometric Scientific Analyser (DETA). The VDC–VC material sample in the shape of suitable disk was previously metallized with gold to ensure a good contact with stainless-steel blocking electrodes of the DETA; the quantity 0 and 00 were determined at temperature of 72 1C.
(3.9)
By solving Eqs. (3.8) and (3.9) with respect to k1 and k2 , the following relations, as functions of real and imaginary parts of complex dielectric constant, are obtained: pffiffiffiffiffiffi 2m o00 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 ¼ (3.10) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c 0 þ 0 2 þ 002 pffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 mo (3.11) 0 þ 0 2 þ 002 k2 ¼ pffiffiffi c 2 Moreover, it is possible to remark that for angular frequency o sufficiently small, as it was stated in a previous section, the real and imaginary parts of complex dielectric constants 0 and 00 go to
5. Conclusions From a theoretical point of view the results obtained in this paper give a concrete contribution to complete the fundamental equations which hold in relaxation phenomena making possible the integration. Moreover, these contribute to the experimental determination of phenomenological and state coefficients which appear in Kluitenberg–Ciancio theory for dielectric media with relaxation phenomena. On the other hand, from the aforementioned expression of phenomenological coefficients it has been possible to obtain a very simple form for the wave vector which reduces to classical expression for sufficiently low and high frequencies. It is important
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to remark that in our calculation this occurs for sufficiently low and high frequencies and not for o ¼ 0 and 1 which have no physical sense. The experimental verification of the conclusions of Section 2 on a polymeric material as VDC–VC, after locating the zone in which the medium has an approximative linear behaviour, is in agreement with some theoretical predictions. In particular, the approach to zero of coefficient wð2Þ ðPEÞ in these regions has been observed. Moreover, the positivity of phenomenological coefficients discussed above shows a region of low frequency with negative values. This fact can be justified if in this range some relaxation phenomenon occurs which has been neglected. This result can be utilized as a test for selecting the region in which the principle of entropy production is true. References [1] V. Ciancio, F. Farsaci, G. Di Marco, Physica B 387 (2007) 130. [2] G.A. Kluitenberg, Physica A 109 (1–2) (1981) 91.
[3] V. Ciancio, J. Non-Equilib. Thermodyn. 14 (1989) 239. [4] V. Ciancio, Afrika Math. Ser. 3 4 (1994). [5] D.C. Champeney, Fourier Transforms and their Physical Applications, Academic Press, London, New York, 1973. [6] N.G. McCrum, B.E. Read, G. Williams, An elastic and Dielectric Effects in Polymeric Solids, Wiley, London, New York, Sydney, 1967. [7] P. Hedvig, Dielectric Spectroscopy of Polymers, Adam Hilger, LTD Bristol, 1977. [8] W. Graham, in: G. Allen, J.C. Bevington (Eds.), Comprehensive Polymer Science, vol. 2, Pergamon Press, Oxford, 1989 (Chapter 16). [9] G.A. Kluitenberg, Physica 68 (1) (1973) 75. [10] G.A. Kluitenberg, Physica A 87 (2) (1977) 302. [11] G.A. Kluitenberg, V. Ciancio, Acta Phys. Hungar. 66 (1989) 251. [12] V. Ciancio, Afrika Matematika Serie 3 4 (1994) 89. [13] V. Ciancio, G.A. Kluitenberg, J. Non-Equilib. Thermodyn. 15 (1990) 157. [14] V. Ciancio, C. Papenfuss, P. Rogolino, Tech. Mech. 22 (2) (2002) 132. [15] S.R. De Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam, 1962. [16] G.D.C. Kuiken, Thermodynamics of Irreversible Processes, Application to Diffusion and Rheology, Wiley, New York, 1994.