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Calculation of various relaxation times and conductivity for a single dielectric relaxation process
Solid State Ionics 42 ( 1990) 2 13-22 1 North-Holland
Calculation of various relaxation times and conductivity for a single dielectric relaxation process Wanqing
Cao and Rosario
Gerhardt
Department of Ceramics, Rutgers University, Piscataway, NJ 08855.0909, Received
5 March
1990; accepted
for publication
USA
I2 June 1990
Various dielectric functions, i.e., c*, tan 6, W, z* and p, have been used to calculate the corresponding time constants for three relaxation models: ( 1) Debye, (2) Cole-Cole and (3) ideal conduction. It has been proved that each dielectric function gives rise to a different relaxation time which follows the general order: T*2 TV> r,,, d > r,2 T,,,. The difference between them depends largely on the dielectric relaxation ratio, t,/c,, as well as the distribution parameter, (Y. Moreover, on a logarithmic scale, the loss tangent relaxation time is equal to the average relaxation time for any pair of inverse complex functions, e.g., t* and W, z* and F. The relationships between the relaxation times illustrate the advantages of using different dielectric functions to analyze experimental data. The relaxation times have also been related to the dielectric conductivity in the Debye and ideal conduction cases.
1. Introduction Analysis of dielectric properties as a function of frequency has generally been called impedance spectroscopy, dielectric relaxation spectroscopy or more appropriately ac spectroscopy. ac spectroscopy reflects the collective response of microscopic polarization processes under an external electrical field. It is a useful and important tool to study defects, microstructure, surface chemistry and electrical conductivity [ 1 ] for materials including dielectrics, ionic conductors, aqueous solutions and adsorbate-adsorbent interfaces [ 2-41. Under an ac field, frequency dispersion or dielectric relaxation is observed due to a number of different polarization mechanisms [ 5 1. The presence of any dielectric relaxation then corresponds to one or more of the possible polarization mechanisms that occur on a microscopic scale. Each relaxation process may be characterized by a relaxation time which describes the decay of its polarization with time in a periodic field. Experimentally, more than one relaxation process can be observed in the frequency range of interest but multiple relaxation times will not be addressed here. Since ac spectroscopy falls within the scope of an 0167-2738/90/S
03.50 0 1990 - Elsevier Science Publishers
interdisciplinary subject, certain dielectric quantities or functions are employed more often than others depending on the particular tield of application. For example, electric modulus is generally used for glasses while impedance is chosen for polycrystalline ionic conductors. In fact, complete data analysis requires skillful utilization and manipulation of several complex dielectric functions. But the relaxation times obtained from different dielectric functions have often been treated improperly. As a result, some confusion has arisen and has caused unnecessary disputes in the literature [ 6,7]. Moreover, the impact of the relaxation time on data presentation and interpretation has been overlooked as well. The present paper attempts to calculate various relaxation times for three single relaxation models: ( 1) Debye, (2) Cole-Cole and (3) ideal conduction. In addition, the dielectric conductivity will be addressed and related to the relaxation time and other parameters.
2. Definition of dielectric functions relaxation times Many dielectric
B.V. (North-Holland
)
functions
and their
have been used to de-
W. Cao, R. Gerhardt / Various relaxation times and conductivity
214
scribe the frequency-dependent properties of a material. Among them the most important are the complex dielectric constant (e*), complex electric modulus (M*), complex impedance (z*) and complex admittance ( r* ). These functions may be expressed in terms of e* t*Lt'-j~~,
(1)
AP=M’+jM”=l/t*,
(2)
z*=.Z’-jZU=l/(jcOCoe*),
(3)
P=
Y’+jy
=joC,e*
,
(4)
where j =fl, o is the angular frequency, 2nf and C, is the geometrical capacitance which, for a thin parallel plate or disk, is given by the vacuum permittivity Q,, area A and thickness t
C,,=q,A/t.
(5)
It should be noted that the real or imaginary part of any function as defined above is always positive for a dielectric or conductive material. Sometimes the complex resistivity (p*=z*CO/eO) and conductivity (a*= r*t,/C,,) may be used in place of z* and P. In addition, the dissipation factor or loss tangent, tan S, is often used to characterize the dielectric loss of a material, which is given by (6) There are two types of plots which may be generated: ( 1) complex plane plots, e.g., M” versus M and eMversus E’, and (2) spectrum plane plots such as t’ versus log w and t” versus log w. The complex plane plots in general reveal a great deal of information from features like the curve shape and intercept (s) on the real axis. Such plots are generally preferred in data analysis, though at the expense of having frequency as an implicit parameter. On the other hand, the spectrum plane plots show the frequency dispersion directly including the location and width of a relaxation peak, if any. As observed experimentally, the real part of any dielectric function varies with frequency monotonically whereas its imaginary counterpart may display a maximum as a function of frequency. But the loss tangent behaves like an imaginary part. Whenever a maximum or peak is spotted, it is desirable to locate its frequency position as well as its width for a par-
titular dielectric function. The peak location may be represented by the relaxation time which is defined as the inverse angular frequency at the maximum of the imaginary part. For instance, T= denotes the impedance relaxation time at which Z” reaches its maximum and or,= 1. Likewise, rE, r*,,, 6, rm and ry may be defined. Thus there are live relaxation times in total, all of which describe the same relaxation process from a different point of view. In order to estimate a particular relaxation time and other important parameters accurately, a least-squares fitting procedure is recommended [8-l 01. Yet the real challenge lies in finding an appropriate analytical expression which can fit the experimental data over a reasonably wide frequency range.
3. Calculation of relaxation times In this Section the relaxation times will be calculated from the functions given above for three simple cases: ( 1) Debye, (2 ) Cole-Cole and (3 ) ideal conduction. Let us first define x= or for E*, z* and tan 6 but x= 1/ (~7) for M* and P where 7 is a time constant and will be specified later. Then all the dielectric functions can be written with x as a parametric variable. As an example, E’(Go)=E’(o+co) but M’ (co ) = M’ (u-+0). The independent variable x is not only dimensionless but also helpful to make dicomparisons between various dielectric rect functions. 3.1. Debye relaxation Debye [ 111 deduced an expression for the ideal frequency response of localized oscillation or motion in a condensed medium, which may be written as E*=f,+
%l!E l+jx
’
1+x*
’
01
E)=E,+
c”=
t,
-es-&a
(7b)
1 +x2x ’ where x=or,
r is the Debye relaxation
time, t, the
W. Cao. R. Gerhardt / Various relaxation times and conductivity
optical dielectric constant (x+co), and E, the static dielectric constant (x+0). (E,- 6,) is the dielectric relaxation strength, AE, and e,/e, is the relaxation ratio, r. It is obvious that the dielectric loss t” becomes a maximum At/2 at x= 1 or
(8)
T,=T.
0.6
L
Since 7, and T are equal for the Debye relaxation, they will be used without distinction unless indicated otherwise. The Debye relaxation may also be expressed in terms of other complex functions
0.4
0.2
0.0-I
I
I
I
1
0.4
0.6
0.8
1.0
Fig. 1. Effect of the relaxation ratio, r, on the impedance R) or admittance (I*= P/G) in the complex plane.
(I* = z*/
’
I
’
0.2
0.0
tan6=
215
I’
9x,
MLM’(co)+-=
1 -jrx
Z*=R.
(10)
’ R
l+jx/reJ p=G
1 -jx
+j
(r--1)x/r’ G ~ (r-1)x’
where M’(co)=l/t,, M’(O)-M’(m), R=rAt/(e,2CO)
M’(O)=l/E,,
(11)
t =I z
(12)
r,=r
AM=
(13)
and G= A&,/r.
As it is widely known, the dielectric constant or modulus function displays a perfect semicircle in the complex plane, though the frequency varies in the opposite direction for the two functions. On the other hand, the complex impedance or admittance diagram becomes a distorted semicircle, particularly at small x due to the second term in eqs. ( 11) or ( 12). Fig. 1 shows that the larger the relaxation ratio, r, the smaller the deviation from the semicircular arc. As a matter of fact, the semicircle will not look perfect until the relaxation ratio becomes fairly large. The relaxation times may be calculated from eqs. (9)-( 12) as follows rt,, s --r/Jr,
(15)
T,=T/r,
(16)
2
“2
“2
2r
(
(17)
’
>
r-3+J(r-l)(r-9) >
(18)
.
In addition to the relaxation peak, the impedance and admittance functions have minimum valleys with respect to frequency given by min
‘5 r-3+
zz
(14)
(r-l)(r-9)
r-3r (
=;
pin = Y
(r-l)(r-9)
“2
2
(
>
r-3-J(r-l)(r-9)
T
(
’
“’ 2r
>
.
(19)
(20)
Evidently there is no maximum or minimum for Z” and I”’ when r< 9. When the relaxation ratio is very large, we have the following simplified expressions r,zr/r,
(17a)
z,=z,
(18a)
T,minz z/Jr,
(19a)
ZYmin
x
r/Jr
.
(20a)
Therefore the minima for impedance and admittance occur roughly at the same frequency while the maxima are shifted by a factor of r. Comparison of the expressions for z* and P suggests that x/r in eq. ( 11) should be equivalent to x in eq. ( 12 ). Noting that x= r/r= at the maximum Z”
W. Cao, R. Gerhardt/ Variousrelaxationtimes and conductivity
216
and
x=7,,/7
at the maximum
I”‘, we have
7z7y=72/r.
(21)
All the relaxation times are therefore other in the form 7~7m=7=7~=7:,,d=72/r,
related to each
(22)
which seems to reflect the inherent relationship between the complex functions, that is, e*AP = z* P = 1. It should be noted that the above result is different from that of Cordaro and Tomozawa [ 121, who dealt with the dielectric loss contributed by a large dc conductivity and a weak Debye relaxation ( rz 1). Several conclusions may be drawn from the comparison between different relaxation times as depicted in fig. 2. First, any relaxation time is not only directly proportional to the Debye relaxation time but also highly dependent on the specific function chosen to be analyzed. Second, all relaxation times follow the order Tc 2 ‘sy > z,,,
6 > tz 2 7,
)
(23)
where the equal sign becomes meaningful only when r 3 1. This means that the impedance and modulus plots place emphasis on the high frequency data whereas the dielectric constant and admittance plots do so on the low frequency data. In addition, the impedance peak is always shifted to longer times or lower frequencies than the modulus peak. The dif-
ference between 7, and 7, becomes more significant with smaller relaxation ratio, e.g., when the electrode double layer or grain-boundary capacitance is of comparable value to the bulk capacitance as demonstrated by Hodge et al. [ 13 1. relaxation
3.2. Cole-Cole
To account for the experimental observation of a depressed semicircle (DS), Cole and Cole [ 141 suggested a distribution of relaxation times for the Debye process, given by Ae 1+ (jx)lPa
t*=t,+
O
1.
(24)
where (Yis the distribution parameter and when (Y= 0, the above equation becomes identical to eq. (7). The value of (Yhas been observed to fall between 0.2 and 0.5 for many materials [ 15,161. Separating eq. (24) into the real and imaginary parts gives Ae( 1 +xIea sin 7c/2a) 1 +2xlea sin 7t/2a+x2(‘-“’
e’=t,+
E”
Pa)
’
AU-” cos n/2a 1+2x’-” sin n/2a+x2(‘-a’
(24b
)
As expected, the complex dielectric constant diagram displays a DS with a depression angle of no/ 2. Consequently the dielectric loss peak becomes lower and broader than the Debye peak, with its full width at half maximum (FWHM) given by
Interestingly, the peak broadening does not affect the area under the curve E” versus log w (integrated to be Aen/2). Once the analytical expression for E* is known, the other dielectric functions may be derived as well tan 6=
w=M’(m)+ Fig. 2. The relaxation times for the Debye model depend on the relaxation ratio as well as the dielectric functions used.
1)~‘~~ cos a/2a 1 )x’-~ sin n/2a+x2(‘-a’
(rr+
(r+
1 +r(
AA4 -jx)lea
’
’
(26)
(27)
217
W. Cao, R. Gerhardt / Various relaxation times and conductivity
zl=Rr-@/[ 1 +(j $)‘Fa](j 5) Rr-@ -j (r-1)x/@
(28)
’ G
G p=
(r-1)x’
[l+(-jx)'-a](_jx)a+j
(29)
where /3= 1/ ( 1-a). Eq. (27) indicates that the complex modulus diagram should also display a DS except that the frequency now increases clockwise. It is not immediately clear what the shape of z* and P will be in the complex plane. For this reason, a mathematical transformation is performed so that, for example, z* may be rewritten as
Z*=R,
&
.
1 +jx,
-J (r-1)(x,-tann/2a)
’
o.o!
R,=
R xa cos n/2a
.
’
.
I
0.6
.
I
0.8
’
I
I
1.0
1.2
(2+a)r+2-a]u3
+r[ (2a-cos
1-a
x r cos n/2a
I
0.4
(30) u4+sinta[
7r/2a +
.
I’ Fig. 3. Effect of the relaxation ratio on the Cole-Cole relation with (~~0.1 in the complex impedance (P=P’(‘-~)ZP/R) or admittance (I*= P/G) plane.
where x, =tan
I
0.2
0.0
m)r+4-2a-cos
7ca]u2
(31) +r2 sinTcu[olr+4-oc]u+r’=O , (32)
Assuming the same relaxation ratio, the 2” versus 2 plot now appears more deviated from the semicircle than that of the Debye model as R, is no longer a constant and increases with reducing x or x,. As shown in fig. 3, the deviation occurs mainly towards smaller x values and becomes less significant with increasing r. This is also true for the admittance since rC can be transformed in the same manner as z*. Eqs. (26)-(29) may be used to calculate the corresponding relaxation times r,,,, d = r/rS/2 ,
(33)
rm = rlrB .
(34)
The remaining two relaxation times, though, have to be obtained by differentiation. For instance, rz may be determined from dZ” /dx= 0 and x= r/r,, which leads to
(35)
where U= ( r/rz)‘-Ol. Numerical calculations may be necessary in order to solve for one of the solutions which gives the maximum Z”. Similar to the Debye case, r, and rY are related to each other since x/P in eq. (28) should be equivalent to x in eq. (29 ), then r,.s,=~~/r~.
(36)
A similar relationship pected, leading to 7~7,=7z7y=7,,6-
2
-k,
between
r, and rm is also ex-
(37)
where k=r2/rfl and depends on the distribution parameter (Y since p= I/( 1 -a). Fig. 4 illustrates that the behavior of the relaxation times for (Y= 0.1 is similar to that of the Debye case. However, the relaxation ratio above which 7= or 7,, exists is shifted to larger values compared to fig. 2. Fig. 5 demonstrates that for r= 100, 75, and 7, do not exist at (Y> 0.16. Moreover, they cannot exist at (II> 0.19 for any value of r because all solutions to eq. ( 3 5 ) are imaginary. Therefore, no impedance or admittance maximum will be observed if the dielectric
218
W. Cao. R. Gerhardt / Various relaxation times and conductivity
where x= 1/ (wr) and 7= rc which is now the conductivity relaxation time [ 131 instead of the Debye relaxation time. The other dielectric functions may be easily deduced Z*=R/( E*=e,(
1
100
10 r
Fig. 4. Dependence of the different relaxation times on the relaxation ratio for the Cole-Cole model with (Y= 0.1.
0.1
0.2
0.3
0.4
a Fig. 5. The relaxation times plotted as a function of the distribution parameter for the Cole-Cole relation with r= 100.
dispersion is due alone. It can also ence between the laxation times parameter.
to localized relaxation or diffusion be seen from fig. 5 that the differlongest (r,) and shortest (TV) reincreases with the distribution
3.3. Non-localized conduction A non-localized diffusion process [ 171 may be represented by a parallel rc circuit for the simplest case, which leads to r*=G(
l+j/x)
,
(38)
1 -j/x)
,
(39)
,
(40)
tand=l/x,
(41)
W=M’(O)/(l-jx),
(42)
where all the quantities are defined as before. It is expected from the above equations that a perfect semicircle will now appear in the complex impedance plane rather than in the dielectric constant plane. The complex modulus diagram is still circular as before. Simple calculations from eqs. (38)-( 42) confirm that z=z,=r=
0.0
1 +jx)
(43)
and r,, 7tan6 and 7y do not exist since their dielectric functions have no maxima at any x value. If a nonlocalized process is regarded as a localized one with an infinitely long time constant, then it may be related to the Debye relaxation with some restrictions. This is accomplished by letting ~,-+co and 7,+c0 while keeping the ratio c,/T~ constant in eq. ( 7 ). This means that the ideal conduction can be treated as an extremely strong Debye relaxation with r,-+co and r-co since E, cannot be less than unity. Consequently, 7ta,,, 6 and t,, are also infinitely large. Eq. (22) may still be regarded as being valid but it no longer carries any significance. The conduction model may include a distribution parameter to account for the common observation of DS in the complex impedance plane. Analogous to the Cole-Cole model, rigid expressions for different dielectric functions may be derived easily. But rm will not be equal to and is slightly smaller than 7, depending on the magnitude of the distribution parameter.
219
W. Cao, R. Gerhardt / Various relaxation times and conductivity
4. Relationship conductivity
between the relaxation times and
Besides the relaxation time, a relaxation process may also be described by the dielectric relaxation conductivity. The dielectric conductivity is usually extrapolated from the circular intercept on the real axis in the complex impedance diagram. It will not be equal to the dc conductivity unless it is due to a non-localized conduction process.
(2)
4.1. Debye relaxation By definition, the dielectric conductivity for the Debye relaxation may be obtained from the complex impedance using eq. ( 13) u=E~/(C~R)=E:E~/(AE~~).
(44)
It can also be acquired from the conductance of view through eq. ( 14)
point
o=Geo/Co
= AEE~/~~.
(45)
In the case of a very strong relaxation, the above two expressions become about equal (less than 10% difference when r> 20), so that
which has already been used empirically by a number of authors [ 6,7,18,19]. For a weak relaxation, the conductivity given by eq. (45 ) is always smaller than that in eq. (44). The discrepancy between them stems from the fact that the Debye relation fails to describe the frequency response of the impedance or admittance function at very low frequencies. For example, as o+O, 2” will approach infinity according to eq. ( 11) but it should be equal to zero as predicted in theory by Syed et al. [ 201. In other words, if the Debye relation strictly applies, then there is no such relaxation conductivity at all. This argument may be better supported by the fact that there are two possible equivalent circuits for the Debye model as schematically drawn in fig. 6. Given the geometrical capacitance C,, a set of relaxation parameters (es, t, and 7) uniquely defines three frequency-independent elements for either of the two circuits, and vice versa. In essence, eqs. (44) and (45) attempt to evaluate the conductivity from the viewpoint of the two circuit models, respectively. It is, therefore,
(b)
Fig. 6. Two equivalent circuits for the Debye model.
justified in theory to use either of the two equations to relate the conductivity to the other relaxation parameters. However, the conductivity will be underestimated from the Z” versus Z’ plot or overestimated from the Y” versus Y’ plot because of the noncircular distortion at small x values (see fig. 1). On the other hand, the value extrapolated from the vertical tail in the complex plane will be in agreement with eq. (44) or (45) depending on the particular function used. 4.2. Cole-Cole
relaxation
Similar to the Debye model, the Cole-Cole relaxation can also be represented by two equivalent circuits. But the pure resistor in fig. 6a or b must be multiplied by a factor of (j,7)-a to account for the distribution in relaxation times [ 14,211. As to the conductivity, it is not possible to estimate its magnitude since both real and imaginary components of z* (eq. (28)) or P (eq. (29)) gotoinfinityasx-+O and CY > 0. If the distribution parameter is small and the relaxation ratio is large, then the approximation given by eq. (44) or (45) may still be employed to relate the conductivity to the relaxation time. As a matter of fact, eq. (45) has been proven to be very satisfactory in predicting the conductivity for many oxide glasses [ 15,161. In addition, both the conductivity and relaxation time are expected to have the same activation energy for a thermally activated process since they depend on the temperature much more
W. Cao. R. Gerhardt / Various relaxation times and conductivity
220
strongly than the other parameters, CY[ 15,16,22-241.
i.e., E, and E, and
tions (e*M* = z*T* = 1) is equal to the loss tangent relaxation time and not related to any other parameters. In addition, they follow the order
4.3. Non-localized conduction Even though the Debye or Cole-Cole model may describe a localized relaxation or diffusion process to a certain extent, it is the non-localized process which generally dominates at low frequencies [ 171. In the absence of interfacial effects, the non-localized conductivity is known as the dc conductivity. As reasoned before, the ideal conduction may be regarded as a very strong Debye relaxation. Consequently, the conductivity can be strictly written as u=E,tO/r,=E,tO/rz,
(47)
which has the same form as eq. (46) but now represents a non-localized or long-range diffusion process. In reality, the ideal long-range conduction is rarely observed in any dielectric or conductor, that is, high dielectric loss is usually accompanied by rising e’ at low frequencies. Noting that a localized relaxation process should have much smaller Ae and TVthan a non-localized one, it is advised that the experimental data be fitted initially in terms of the Debye or ColeCole model. Once the relaxation parameters have been obtained, the next step is to assign a microscopic mechanism, if possible.
Accordingly, of the live possible peaks which describe the same relaxation process, the dielectric loss (E” ) peak appears at the lowest frequency while the modulus (M” ) peak is located at the highest frequency. The difference between them is highly dependent on the relaxation ratio (r= E,/E,) as well as the distribution parameter (a). The above relations are important for analyzing frequency dispersion data properly and may be used to calculate one relaxation time for another if needed. Because of the difference in peak locations, a relaxation peak may be seen in one dielectric function but only partially seen or not seen at all in another within the limited frequency range available experimentally. Furthermore, any peak shift in one type of plot, which may be caused by temperature, humidity and other factors, does not necessarily guarantee the same change in another plot unless both r and cr remain unchanged. For this reason, several or all complex plane plots should be examined before drawing any conclusion about the appearance, disappearance or shift of a relaxation peak. The dielectric conductivity is closely related to the relaxation time for the ideal conduction or strong Debye-type relaxation, and may be given by ls=t,EO/r,
5. Concluding remarks It has been verified that the peak location or relaxation time for the same relaxation process depends on the specific dielectric function chosen for data analysis. Relaxations between various relaxation times have been established for three basic models: ( 1) Debye, (2) Cole-Cole and (3) ideal conduction. In all three cases, the relaxation times, if they exist, are related by
=e,eg/t, .
Here 7m is preferred to 7, since the latter may be affected by the low-frequency distortion of the complex impedance diagram when the relaxation ratio is small.
Acknowledgement This work was supported by the Center of Ceramic Research, a New Jersey Commission on Science and Technology Center.
or (log
7, +1og
T,)/2=
(log r,+log
r,)/2=log
&“a.
This means that, on a logarithmic scale, the average relaxation time for any pair of inverse complex func-
References [ 1 ] L.L. Hench,
in: Proc. 14th Symp. Process Eng. I-2-2 ( 1968 ).
Sot.
Aerosp.
Mater.
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W. Cao. R. Gerhardt / Various relaxation times and conductivity
[2] I.D. Raistrick, Ann. Rev. Mater. Sci. 16 (1986) 343. [3] J.R. Macdonald, J. Electroanal. Chem. 223 ( 1987) 25. [ 41 G. Jones, Dielectric and related molecular processes, Vol. 3 (The Chem. Sot., London, 1977) pp. 176-185. [ 51 A. Von Hipple, ed., Dielectric materials and applications (The Technology Press of M.I.T. and Wiley, New York, 1954) p. 19. [6] J.O. Isard, J. Non-Cryst. Solids 4 (1970) 357. [ 71 M. Tomozawa, J. Non-Cryst. Solids 33 ( 1979) 117. [ 8 ] P.R. Mason, J.B. Hasted and L. Moore, Adv. Mol. Relax. Processes 6 ( 1974) 2 17. [ 91 B.A. Boukamp, Solid State Ionics 20 ( 1986) 3 1. [lo] J.R. Macdonald and L.D. Potter, Jr., Solid State Ionics 23 (1987) 61.
[ 111 C.P. Smith, Dielectric
behavior
and structure
(McGraw-
Hill, New York, 1955) pp. 52-74.
[ 121 J.F. Cordaro (1981)
713.
and M. Tomozawa,
J. Am. Ceram.
Sot. 64
1131 I.M. Hodge,
M.D. Ingram
Chem. 74 (1976)
and A.R. West, J. Electroanal.
125.
[14]K.S.ColeandR.H.Cole,J.Chem.Phys.9(1941)
[ 151 H. Nakajima, Phenomena [ 161 H. Namikawa,
Conf. on Electric Insulation (Nat. Acad. Sci., Willemsburg, J. Non-Cryst.
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Solids 18 (1975)
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[ 171 F.A. Grant, J. Appl. Phys. 29 (1958) 76. [ 181 M. Tomozawa, J. Cordaro and M. Singh, J. Mater. Sci. 14 (1979) 1945. [ 191 J.C. Dyre, J. Non-Cryst. Solids 88 (1986) 271. [20] R. Syed, D.L. Gavin, C.T. Moynihan and A.V. Lcsikar, J. Am. Ceram. Sot. 64 ( 198 I ) C 118. [ 2 11 R.L. Hurt and J.R. Macdonald, Solid State Ionics 20 ( 1986) 111. [22] H. Namikawa, J. Non-Cryst. Solids 14 (1974) 88. [23] B.V.R. Chowdari and K. Radhakrishnan, J. Non-Cryst. Solids 108 (1989) 323. [24] V. Provenzano, L.P. Boesch, V. Volterra, C.T. Moynihan and P.B. Macedo, J. Am. Ceram. Sot. 55 (1972) 492.
Calculation of various relaxation times and conductivity for a single dielectric relaxation process Wanqing
Cao and Rosario
Gerhardt
Department of Ceramics, Rutgers University, Piscataway, NJ 08855.0909, Received
5 March
1990; accepted
for publication
USA
I2 June 1990
Various dielectric functions, i.e., c*, tan 6, W, z* and p, have been used to calculate the corresponding time constants for three relaxation models: ( 1) Debye, (2) Cole-Cole and (3) ideal conduction. It has been proved that each dielectric function gives rise to a different relaxation time which follows the general order: T*2 TV> r,,, d > r,2 T,,,. The difference between them depends largely on the dielectric relaxation ratio, t,/c,, as well as the distribution parameter, (Y. Moreover, on a logarithmic scale, the loss tangent relaxation time is equal to the average relaxation time for any pair of inverse complex functions, e.g., t* and W, z* and F. The relationships between the relaxation times illustrate the advantages of using different dielectric functions to analyze experimental data. The relaxation times have also been related to the dielectric conductivity in the Debye and ideal conduction cases.
1. Introduction Analysis of dielectric properties as a function of frequency has generally been called impedance spectroscopy, dielectric relaxation spectroscopy or more appropriately ac spectroscopy. ac spectroscopy reflects the collective response of microscopic polarization processes under an external electrical field. It is a useful and important tool to study defects, microstructure, surface chemistry and electrical conductivity [ 1 ] for materials including dielectrics, ionic conductors, aqueous solutions and adsorbate-adsorbent interfaces [ 2-41. Under an ac field, frequency dispersion or dielectric relaxation is observed due to a number of different polarization mechanisms [ 5 1. The presence of any dielectric relaxation then corresponds to one or more of the possible polarization mechanisms that occur on a microscopic scale. Each relaxation process may be characterized by a relaxation time which describes the decay of its polarization with time in a periodic field. Experimentally, more than one relaxation process can be observed in the frequency range of interest but multiple relaxation times will not be addressed here. Since ac spectroscopy falls within the scope of an 0167-2738/90/S
03.50 0 1990 - Elsevier Science Publishers
interdisciplinary subject, certain dielectric quantities or functions are employed more often than others depending on the particular tield of application. For example, electric modulus is generally used for glasses while impedance is chosen for polycrystalline ionic conductors. In fact, complete data analysis requires skillful utilization and manipulation of several complex dielectric functions. But the relaxation times obtained from different dielectric functions have often been treated improperly. As a result, some confusion has arisen and has caused unnecessary disputes in the literature [ 6,7]. Moreover, the impact of the relaxation time on data presentation and interpretation has been overlooked as well. The present paper attempts to calculate various relaxation times for three single relaxation models: ( 1) Debye, (2) Cole-Cole and (3) ideal conduction. In addition, the dielectric conductivity will be addressed and related to the relaxation time and other parameters.
2. Definition of dielectric functions relaxation times Many dielectric
B.V. (North-Holland
)
functions
and their
have been used to de-
W. Cao, R. Gerhardt / Various relaxation times and conductivity
214
scribe the frequency-dependent properties of a material. Among them the most important are the complex dielectric constant (e*), complex electric modulus (M*), complex impedance (z*) and complex admittance ( r* ). These functions may be expressed in terms of e* t*Lt'-j~~,
(1)
AP=M’+jM”=l/t*,
(2)
z*=.Z’-jZU=l/(jcOCoe*),
(3)
P=
Y’+jy
=joC,e*
,
(4)
where j =fl, o is the angular frequency, 2nf and C, is the geometrical capacitance which, for a thin parallel plate or disk, is given by the vacuum permittivity Q,, area A and thickness t
C,,=q,A/t.
(5)
It should be noted that the real or imaginary part of any function as defined above is always positive for a dielectric or conductive material. Sometimes the complex resistivity (p*=z*CO/eO) and conductivity (a*= r*t,/C,,) may be used in place of z* and P. In addition, the dissipation factor or loss tangent, tan S, is often used to characterize the dielectric loss of a material, which is given by (6) There are two types of plots which may be generated: ( 1) complex plane plots, e.g., M” versus M and eMversus E’, and (2) spectrum plane plots such as t’ versus log w and t” versus log w. The complex plane plots in general reveal a great deal of information from features like the curve shape and intercept (s) on the real axis. Such plots are generally preferred in data analysis, though at the expense of having frequency as an implicit parameter. On the other hand, the spectrum plane plots show the frequency dispersion directly including the location and width of a relaxation peak, if any. As observed experimentally, the real part of any dielectric function varies with frequency monotonically whereas its imaginary counterpart may display a maximum as a function of frequency. But the loss tangent behaves like an imaginary part. Whenever a maximum or peak is spotted, it is desirable to locate its frequency position as well as its width for a par-
titular dielectric function. The peak location may be represented by the relaxation time which is defined as the inverse angular frequency at the maximum of the imaginary part. For instance, T= denotes the impedance relaxation time at which Z” reaches its maximum and or,= 1. Likewise, rE, r*,,, 6, rm and ry may be defined. Thus there are live relaxation times in total, all of which describe the same relaxation process from a different point of view. In order to estimate a particular relaxation time and other important parameters accurately, a least-squares fitting procedure is recommended [8-l 01. Yet the real challenge lies in finding an appropriate analytical expression which can fit the experimental data over a reasonably wide frequency range.
3. Calculation of relaxation times In this Section the relaxation times will be calculated from the functions given above for three simple cases: ( 1) Debye, (2 ) Cole-Cole and (3 ) ideal conduction. Let us first define x= or for E*, z* and tan 6 but x= 1/ (~7) for M* and P where 7 is a time constant and will be specified later. Then all the dielectric functions can be written with x as a parametric variable. As an example, E’(Go)=E’(o+co) but M’ (co ) = M’ (u-+0). The independent variable x is not only dimensionless but also helpful to make dicomparisons between various dielectric rect functions. 3.1. Debye relaxation Debye [ 111 deduced an expression for the ideal frequency response of localized oscillation or motion in a condensed medium, which may be written as E*=f,+
%l!E l+jx
’
1+x*
’
01
E)=E,+
c”=
t,
-es-&a
(7b)
1 +x2x ’ where x=or,
r is the Debye relaxation
time, t, the
W. Cao. R. Gerhardt / Various relaxation times and conductivity
optical dielectric constant (x+co), and E, the static dielectric constant (x+0). (E,- 6,) is the dielectric relaxation strength, AE, and e,/e, is the relaxation ratio, r. It is obvious that the dielectric loss t” becomes a maximum At/2 at x= 1 or
(8)
T,=T.
0.6
L
Since 7, and T are equal for the Debye relaxation, they will be used without distinction unless indicated otherwise. The Debye relaxation may also be expressed in terms of other complex functions
0.4
0.2
0.0-I
I
I
I
1
0.4
0.6
0.8
1.0
Fig. 1. Effect of the relaxation ratio, r, on the impedance R) or admittance (I*= P/G) in the complex plane.
(I* = z*/
’
I
’
0.2
0.0
tan6=
215
I’
9x,
MLM’(co)+-=
1 -jrx
Z*=R.
(10)
’ R
l+jx/reJ p=G
1 -jx
+j
(r--1)x/r’ G ~ (r-1)x’
where M’(co)=l/t,, M’(O)-M’(m), R=rAt/(e,2CO)
M’(O)=l/E,,
(11)
t =I z
(12)
r,=r
AM=
(13)
and G= A&,/r.
As it is widely known, the dielectric constant or modulus function displays a perfect semicircle in the complex plane, though the frequency varies in the opposite direction for the two functions. On the other hand, the complex impedance or admittance diagram becomes a distorted semicircle, particularly at small x due to the second term in eqs. ( 11) or ( 12). Fig. 1 shows that the larger the relaxation ratio, r, the smaller the deviation from the semicircular arc. As a matter of fact, the semicircle will not look perfect until the relaxation ratio becomes fairly large. The relaxation times may be calculated from eqs. (9)-( 12) as follows rt,, s --r/Jr,
(15)
T,=T/r,
(16)
2
“2
“2
2r
(
(17)
’
>
r-3+J(r-l)(r-9) >
(18)
.
In addition to the relaxation peak, the impedance and admittance functions have minimum valleys with respect to frequency given by min
‘5 r-3+
zz
(14)
(r-l)(r-9)
r-3r (
=;
pin = Y
(r-l)(r-9)
“2
2
(
>
r-3-J(r-l)(r-9)
T
(
’
“’ 2r
>
.
(19)
(20)
Evidently there is no maximum or minimum for Z” and I”’ when r< 9. When the relaxation ratio is very large, we have the following simplified expressions r,zr/r,
(17a)
z,=z,
(18a)
T,minz z/Jr,
(19a)
ZYmin
x
r/Jr
.
(20a)
Therefore the minima for impedance and admittance occur roughly at the same frequency while the maxima are shifted by a factor of r. Comparison of the expressions for z* and P suggests that x/r in eq. ( 11) should be equivalent to x in eq. ( 12 ). Noting that x= r/r= at the maximum Z”
W. Cao, R. Gerhardt/ Variousrelaxationtimes and conductivity
216
and
x=7,,/7
at the maximum
I”‘, we have
7z7y=72/r.
(21)
All the relaxation times are therefore other in the form 7~7m=7=7~=7:,,d=72/r,
related to each
(22)
which seems to reflect the inherent relationship between the complex functions, that is, e*AP = z* P = 1. It should be noted that the above result is different from that of Cordaro and Tomozawa [ 121, who dealt with the dielectric loss contributed by a large dc conductivity and a weak Debye relaxation ( rz 1). Several conclusions may be drawn from the comparison between different relaxation times as depicted in fig. 2. First, any relaxation time is not only directly proportional to the Debye relaxation time but also highly dependent on the specific function chosen to be analyzed. Second, all relaxation times follow the order Tc 2 ‘sy > z,,,
6 > tz 2 7,
)
(23)
where the equal sign becomes meaningful only when r 3 1. This means that the impedance and modulus plots place emphasis on the high frequency data whereas the dielectric constant and admittance plots do so on the low frequency data. In addition, the impedance peak is always shifted to longer times or lower frequencies than the modulus peak. The dif-
ference between 7, and 7, becomes more significant with smaller relaxation ratio, e.g., when the electrode double layer or grain-boundary capacitance is of comparable value to the bulk capacitance as demonstrated by Hodge et al. [ 13 1. relaxation
3.2. Cole-Cole
To account for the experimental observation of a depressed semicircle (DS), Cole and Cole [ 141 suggested a distribution of relaxation times for the Debye process, given by Ae 1+ (jx)lPa
t*=t,+
O
1.
(24)
where (Yis the distribution parameter and when (Y= 0, the above equation becomes identical to eq. (7). The value of (Yhas been observed to fall between 0.2 and 0.5 for many materials [ 15,161. Separating eq. (24) into the real and imaginary parts gives Ae( 1 +xIea sin 7c/2a) 1 +2xlea sin 7t/2a+x2(‘-“’
e’=t,+
E”
Pa)
’
AU-” cos n/2a 1+2x’-” sin n/2a+x2(‘-a’
(24b
)
As expected, the complex dielectric constant diagram displays a DS with a depression angle of no/ 2. Consequently the dielectric loss peak becomes lower and broader than the Debye peak, with its full width at half maximum (FWHM) given by
Interestingly, the peak broadening does not affect the area under the curve E” versus log w (integrated to be Aen/2). Once the analytical expression for E* is known, the other dielectric functions may be derived as well tan 6=
w=M’(m)+ Fig. 2. The relaxation times for the Debye model depend on the relaxation ratio as well as the dielectric functions used.
1)~‘~~ cos a/2a 1 )x’-~ sin n/2a+x2(‘-a’
(rr+
(r+
1 +r(
AA4 -jx)lea
’
’
(26)
(27)
217
W. Cao, R. Gerhardt / Various relaxation times and conductivity
zl=Rr-@/[ 1 +(j $)‘Fa](j 5) Rr-@ -j (r-1)x/@
(28)
’ G
G p=
(r-1)x’
[l+(-jx)'-a](_jx)a+j
(29)
where /3= 1/ ( 1-a). Eq. (27) indicates that the complex modulus diagram should also display a DS except that the frequency now increases clockwise. It is not immediately clear what the shape of z* and P will be in the complex plane. For this reason, a mathematical transformation is performed so that, for example, z* may be rewritten as
Z*=R,
&
.
1 +jx,
-J (r-1)(x,-tann/2a)
’
o.o!
R,=
R xa cos n/2a
.
’
.
I
0.6
.
I
0.8
’
I
I
1.0
1.2
(2+a)r+2-a]u3
+r[ (2a-cos
1-a
x r cos n/2a
I
0.4
(30) u4+sinta[
7r/2a +
.
I’ Fig. 3. Effect of the relaxation ratio on the Cole-Cole relation with (~~0.1 in the complex impedance (P=P’(‘-~)ZP/R) or admittance (I*= P/G) plane.
where x, =tan
I
0.2
0.0
m)r+4-2a-cos
7ca]u2
(31) +r2 sinTcu[olr+4-oc]u+r’=O , (32)
Assuming the same relaxation ratio, the 2” versus 2 plot now appears more deviated from the semicircle than that of the Debye model as R, is no longer a constant and increases with reducing x or x,. As shown in fig. 3, the deviation occurs mainly towards smaller x values and becomes less significant with increasing r. This is also true for the admittance since rC can be transformed in the same manner as z*. Eqs. (26)-(29) may be used to calculate the corresponding relaxation times r,,,, d = r/rS/2 ,
(33)
rm = rlrB .
(34)
The remaining two relaxation times, though, have to be obtained by differentiation. For instance, rz may be determined from dZ” /dx= 0 and x= r/r,, which leads to
(35)
where U= ( r/rz)‘-Ol. Numerical calculations may be necessary in order to solve for one of the solutions which gives the maximum Z”. Similar to the Debye case, r, and rY are related to each other since x/P in eq. (28) should be equivalent to x in eq. (29 ), then r,.s,=~~/r~.
(36)
A similar relationship pected, leading to 7~7,=7z7y=7,,6-
2
-k,
between
r, and rm is also ex-
(37)
where k=r2/rfl and depends on the distribution parameter (Y since p= I/( 1 -a). Fig. 4 illustrates that the behavior of the relaxation times for (Y= 0.1 is similar to that of the Debye case. However, the relaxation ratio above which 7= or 7,, exists is shifted to larger values compared to fig. 2. Fig. 5 demonstrates that for r= 100, 75, and 7, do not exist at (Y> 0.16. Moreover, they cannot exist at (II> 0.19 for any value of r because all solutions to eq. ( 3 5 ) are imaginary. Therefore, no impedance or admittance maximum will be observed if the dielectric
218
W. Cao. R. Gerhardt / Various relaxation times and conductivity
where x= 1/ (wr) and 7= rc which is now the conductivity relaxation time [ 131 instead of the Debye relaxation time. The other dielectric functions may be easily deduced Z*=R/( E*=e,(
1
100
10 r
Fig. 4. Dependence of the different relaxation times on the relaxation ratio for the Cole-Cole model with (Y= 0.1.
0.1
0.2
0.3
0.4
a Fig. 5. The relaxation times plotted as a function of the distribution parameter for the Cole-Cole relation with r= 100.
dispersion is due alone. It can also ence between the laxation times parameter.
to localized relaxation or diffusion be seen from fig. 5 that the differlongest (r,) and shortest (TV) reincreases with the distribution
3.3. Non-localized conduction A non-localized diffusion process [ 171 may be represented by a parallel rc circuit for the simplest case, which leads to r*=G(
l+j/x)
,
(38)
1 -j/x)
,
(39)
,
(40)
tand=l/x,
(41)
W=M’(O)/(l-jx),
(42)
where all the quantities are defined as before. It is expected from the above equations that a perfect semicircle will now appear in the complex impedance plane rather than in the dielectric constant plane. The complex modulus diagram is still circular as before. Simple calculations from eqs. (38)-( 42) confirm that z=z,=r=
0.0
1 +jx)
(43)
and r,, 7tan6 and 7y do not exist since their dielectric functions have no maxima at any x value. If a nonlocalized process is regarded as a localized one with an infinitely long time constant, then it may be related to the Debye relaxation with some restrictions. This is accomplished by letting ~,-+co and 7,+c0 while keeping the ratio c,/T~ constant in eq. ( 7 ). This means that the ideal conduction can be treated as an extremely strong Debye relaxation with r,-+co and r-co since E, cannot be less than unity. Consequently, 7ta,,, 6 and t,, are also infinitely large. Eq. (22) may still be regarded as being valid but it no longer carries any significance. The conduction model may include a distribution parameter to account for the common observation of DS in the complex impedance plane. Analogous to the Cole-Cole model, rigid expressions for different dielectric functions may be derived easily. But rm will not be equal to and is slightly smaller than 7, depending on the magnitude of the distribution parameter.
219
W. Cao, R. Gerhardt / Various relaxation times and conductivity
4. Relationship conductivity
between the relaxation times and
Besides the relaxation time, a relaxation process may also be described by the dielectric relaxation conductivity. The dielectric conductivity is usually extrapolated from the circular intercept on the real axis in the complex impedance diagram. It will not be equal to the dc conductivity unless it is due to a non-localized conduction process.
(2)
4.1. Debye relaxation By definition, the dielectric conductivity for the Debye relaxation may be obtained from the complex impedance using eq. ( 13) u=E~/(C~R)=E:E~/(AE~~).
(44)
It can also be acquired from the conductance of view through eq. ( 14)
point
o=Geo/Co
= AEE~/~~.
(45)
In the case of a very strong relaxation, the above two expressions become about equal (less than 10% difference when r> 20), so that
which has already been used empirically by a number of authors [ 6,7,18,19]. For a weak relaxation, the conductivity given by eq. (45 ) is always smaller than that in eq. (44). The discrepancy between them stems from the fact that the Debye relation fails to describe the frequency response of the impedance or admittance function at very low frequencies. For example, as o+O, 2” will approach infinity according to eq. ( 11) but it should be equal to zero as predicted in theory by Syed et al. [ 201. In other words, if the Debye relation strictly applies, then there is no such relaxation conductivity at all. This argument may be better supported by the fact that there are two possible equivalent circuits for the Debye model as schematically drawn in fig. 6. Given the geometrical capacitance C,, a set of relaxation parameters (es, t, and 7) uniquely defines three frequency-independent elements for either of the two circuits, and vice versa. In essence, eqs. (44) and (45) attempt to evaluate the conductivity from the viewpoint of the two circuit models, respectively. It is, therefore,
(b)
Fig. 6. Two equivalent circuits for the Debye model.
justified in theory to use either of the two equations to relate the conductivity to the other relaxation parameters. However, the conductivity will be underestimated from the Z” versus Z’ plot or overestimated from the Y” versus Y’ plot because of the noncircular distortion at small x values (see fig. 1). On the other hand, the value extrapolated from the vertical tail in the complex plane will be in agreement with eq. (44) or (45) depending on the particular function used. 4.2. Cole-Cole
relaxation
Similar to the Debye model, the Cole-Cole relaxation can also be represented by two equivalent circuits. But the pure resistor in fig. 6a or b must be multiplied by a factor of (j,7)-a to account for the distribution in relaxation times [ 14,211. As to the conductivity, it is not possible to estimate its magnitude since both real and imaginary components of z* (eq. (28)) or P (eq. (29)) gotoinfinityasx-+O and CY > 0. If the distribution parameter is small and the relaxation ratio is large, then the approximation given by eq. (44) or (45) may still be employed to relate the conductivity to the relaxation time. As a matter of fact, eq. (45) has been proven to be very satisfactory in predicting the conductivity for many oxide glasses [ 15,161. In addition, both the conductivity and relaxation time are expected to have the same activation energy for a thermally activated process since they depend on the temperature much more
W. Cao. R. Gerhardt / Various relaxation times and conductivity
220
strongly than the other parameters, CY[ 15,16,22-241.
i.e., E, and E, and
tions (e*M* = z*T* = 1) is equal to the loss tangent relaxation time and not related to any other parameters. In addition, they follow the order
4.3. Non-localized conduction Even though the Debye or Cole-Cole model may describe a localized relaxation or diffusion process to a certain extent, it is the non-localized process which generally dominates at low frequencies [ 171. In the absence of interfacial effects, the non-localized conductivity is known as the dc conductivity. As reasoned before, the ideal conduction may be regarded as a very strong Debye relaxation. Consequently, the conductivity can be strictly written as u=E,tO/r,=E,tO/rz,
(47)
which has the same form as eq. (46) but now represents a non-localized or long-range diffusion process. In reality, the ideal long-range conduction is rarely observed in any dielectric or conductor, that is, high dielectric loss is usually accompanied by rising e’ at low frequencies. Noting that a localized relaxation process should have much smaller Ae and TVthan a non-localized one, it is advised that the experimental data be fitted initially in terms of the Debye or ColeCole model. Once the relaxation parameters have been obtained, the next step is to assign a microscopic mechanism, if possible.
Accordingly, of the live possible peaks which describe the same relaxation process, the dielectric loss (E” ) peak appears at the lowest frequency while the modulus (M” ) peak is located at the highest frequency. The difference between them is highly dependent on the relaxation ratio (r= E,/E,) as well as the distribution parameter (a). The above relations are important for analyzing frequency dispersion data properly and may be used to calculate one relaxation time for another if needed. Because of the difference in peak locations, a relaxation peak may be seen in one dielectric function but only partially seen or not seen at all in another within the limited frequency range available experimentally. Furthermore, any peak shift in one type of plot, which may be caused by temperature, humidity and other factors, does not necessarily guarantee the same change in another plot unless both r and cr remain unchanged. For this reason, several or all complex plane plots should be examined before drawing any conclusion about the appearance, disappearance or shift of a relaxation peak. The dielectric conductivity is closely related to the relaxation time for the ideal conduction or strong Debye-type relaxation, and may be given by ls=t,EO/r,
5. Concluding remarks It has been verified that the peak location or relaxation time for the same relaxation process depends on the specific dielectric function chosen for data analysis. Relaxations between various relaxation times have been established for three basic models: ( 1) Debye, (2) Cole-Cole and (3) ideal conduction. In all three cases, the relaxation times, if they exist, are related by
=e,eg/t, .
Here 7m is preferred to 7, since the latter may be affected by the low-frequency distortion of the complex impedance diagram when the relaxation ratio is small.
Acknowledgement This work was supported by the Center of Ceramic Research, a New Jersey Commission on Science and Technology Center.
or (log
7, +1og
T,)/2=
(log r,+log
r,)/2=log
&“a.
This means that, on a logarithmic scale, the average relaxation time for any pair of inverse complex func-
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