Dielectric relaxation and distribution function of relaxation times in dilute colloidal suspensions

COLLOIDS AND ELSEVIER

Colloids and Surfaces A: Physicochemical and Engineering Aspects 97 (1995) 141-149

A

SURFACES

Dielectric relaxation and distribution function of relaxation times in dilute colloidal suspensions F..Carrique a,1, A. Quirantes a, A.V. Delgado

"'*

a Departamento de Fiscia Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada. Spain

Received 3 July 1994; accepted 11 December 1994

Abstract

This paper describes an empirical approach to the theoretical distribution function of relaxation times (DFRT) that best characterizes the dielectric behaviour of colloidal suspensions, according to the model proposed by DeLacey and White (E.H.B. DeLacey and L.R. White, J. Chem. Soc., Faraday Trans. 2, 77 (1981) 2007; DW hereafter). It is well known that this model predicts a dielectric response that cannot be given in terms of a single relaxation time for the system, even if the colloidal particles are homogeneous spheres. Hence, the suspension must be characterized by a distribution of relaxation times; the dielectric response of a dilute colloidal suspension is described in this work using a reasonable DFRT, and the results are compared to the predictions of the DW model. It is demonstrated that such a DFRT is also a suitable approximation to the results obtained using more recent theories, incorporating into the explanation of the dielectric properties of the suspension the possibility of surface conductivity in the inner part of the double layer. Keywords: Dielectric relaxation; Distribution function of relaxation times; Dynamic Stern layer model; Standard electrokinetic model

1. Introduction

Both experimental data and theoretical models on the conductivity and dielectric response of a colloidal suspension in the presence of alternating electric fields demonstrate the existence of important relaxation processes in the radiofrequency range of the spectrum [ 1-14]. These processes are related to the polarization of the electric double layers surrounding the particles, as a consequence of the finite response time of the ion clouds to field variations. As the frequency of the external field * Corresponding author. 1Present address: Departamento de Fisica Aplicada I, Facultad de Ciencias, Universidad de Mfilaga, 29071 Mfilaga, Spain. 0927-7757/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved S S D I 0927-7757(94)03065-0

increases, a phase difference occurs between the polarization of the system and the field; such a phase difference is responsible for the strong dielectric dispersion observed [ 1 ]. It is c o m m o n practice in dielectric studies to characterize the dielectric relaxation patterns by plotting the imaginary part of the dielectric constant, e"(m), against its real part, e'(o~). The simplest theoretical way of fitting such data uses the Debye model [15], according to which the plot is a semicircle with its center on the real axis, and is described by a simple parameter, the so-called relaxation time, ~0. However, many systems, particularly colloidal suspensions, do not follow the pattern predicted by this simple model. An empirical generalization of the Debye theory, the Cole-Cole model, seems to agree reasonably

F. Carrique et aL/Colloids Surfaces A: Physicochem. Eng. Aspects 97 (1995) 141 149

142

well with the experimental data [9,16]; according to this model, the plot 6 " - e ' is a semicircle with its center below the real axis. Two parameters, c~ and to are needed in this case:

6"(o9) = e'(og) - ie"(o•) = e~ +

6 s - - 6"~

1+

(ito'C0) 1 - a

(~<1)

(l)

where es and e~, respectively, are the low- and high-frequency limiting values of e'(~o) and ~o is the frequency of the field. The characteristic time to is related to (~0max (the frequency at which e" is a maximum) by ~0maxr0= 1. One of the most general standard electrokinetic models of the dielectric response of dilute colloidal suspensions was published by DeLacey and White [ l ] . Predictions of the shape of e" - e' plots from this model (DW hereafter) have been analyzed and compared to Cole-Cole type relationships [ 16,1 7]; it has been demonstrated that theoretical DW values of 6*(09) do not follow a Cole-Cole relaxation pattern, and hence disagree with experimental 6 " - e ' plots too. As a rule, when the Debye model (using a single relaxation time) is not applicable, it should be possible to find a distribution function of relaxation times (DFRT) to reproduce the characteristics of the plots, which can be considered as a true fingerprint of the dielectric behavior of a system. We shall call g(z) the D F R T of the system; if it is properly chosen, one should be able to reproduce all the aspects of the 6" - e' plot. It can be shown [15] that

e*(co) =

coo + (es - e~) i 1 g(z)+io~z dz

(2)

general treatment based on standard electrokinetic models. In a previous work [17], we showed that the shape of e " - e ' plots is very similar in DW theory and in the so-called dynamic Stern layer model (DSL), proposed by Rosen et al. [8]; the latter can be considered as more rigorous than DW theory in that it takes into account adsorption and ionic transport in the inner part of the double layer (the Stern layer). It has been shown [8,18] that the agreement between theory and experimental data is significantly improved, compared to DW predictions, when reasonable choices are made for the parameters of the model. Similar conclusions were reached by Kijlstra et al. [19], who elaborated on an approximate theory for the dielectric behavior of colloidal dispersions of spherical particles with thin double layers. For these reasons, the type of e " - e ' plots, and the corresponding DFRT, for DSL theory will also be considered.

2. Distribution function of relaxation times (DFRT) The g(r) function can be computed from the frequency spectrum of the complex dielectric constant, e*(co) [15]. In fact, if we define Q(i(o) as Q(io))=

6*(~)--E~

(4)

£s - - 6co

then [15,20] 1 g(z) = - • - l L f "C

X[Q(ie))]

(5)

0

with

i g(z) dr = l

(3)

0

Our aim in this paper is to find the z-dependence of g, in order to reproduce the 6" - 6' relationships predicted by theoretical models. We will consider D W theory in particular, because it is a very

where LP 1 denotes the inverse Laplace transform operator. This type of calculation is very complicated in our case, since we only have a discrete set of values of e'(~o) and e"(e)). Other methods involve the calculation of the inverse Laplace transform of discrete numbers of data, using for instance Provencher's Contin program [21]. However, we will use here a more empirical approach. We seek to obtain the distinctive characteristics of the e " - e ' plots deduced from D W and DSL models.

[( Carrique et aL/Colloids Surfaces A: Physicochem. Eng. Aspects 97 (1995) 141 149

Hence we propose a D F R T of the form g(z) = rF(?/2c0

exp [ - (Zo/Z)"] (z/z0)7/2

(6)

or, in logarithmic scale, G(ln r) = zg(z)

143

of both c~ and 7. The D F R T considered in this work is a version of that proposed by Williams and Watts [ 15,22]; in order to qualitatively match the shape of the theoretical Cole-Cole plot for the dielectric constant of suspensions, the WilliamsWatts equation,

(7)

with

G(ln z) = rg(z)-- \ 4 ~ % ]

f G(ln z) d i n z = 1

(8)

In Eq. (6), ~ and 7 are two positive non-zero parameters, F is Euler's gamma function, and zo is some characteristic time that we will demonstrate to be the Debye's relaxation time for large values

exp[-(~/4%)]

(9)

was modified by substituting the (r/%) dependence in the exponential by (%/r), and introducing the new parameters e and 7 and the corresponding normalization factor. Thus, our D F R T is, to some extent, the mirror image of the Williams-Watts function for fl = 1/2. It is useful to use the D F R T in a different form by introducing a new variable, x = ln(~/%),

0,50 1,6

0,45 1,4 0,40

1,2 0,35

0,30

1,0

q,

q"

0,25

0,8 0,20

0,6 0,15

0,4 0.10

0,05

0,2

0,00 -10

-5

0

5

10

0,0 -5,0

-2,5

0,0

2,5

5,0

7,5

X X Fig. 1. D F R T [Eq. (9)] p l o t t e d as a function of x = ln(z/Zo) for y = 1 a n d different values of ~.

Fig. 2. Same as Fig. 1, for ~ = 1 a n d variable ),.

10,0

F. Carrique et aL/Colloko 5;:,Jaces A. Physicochem. Eng. Aspects 97 (1995) 141 149

144

defined as

H(x)= G[ln(v/Vo)] - F(-~exp

x+exp(-~:>; i ~

-

(10)

distribution function is not symmetrical about its maximum, unlike the Cole-Cole distribution [ 15]. The position of the maximum of the former depends on c~ and ~ as 1 Xmax = -

with

i H(x)dx=l

(11)

oo

(12)

with H(xmax)

Figs. 1 and 2 show, respectively, the D F R T [Eq. (9)] for ~ = 1 and different values of ~, and = 1 with ~ as a parameter. As observed, the

ln(y/2c0

-

- -

r(7/2 )

(]~/2~) e/2~

exp(-7/2c 0

(13)

The proposed general distribution contains other distributions [15], as demonstrated by the limiting values of for large ~ and 7. In

H(x),

1,1

1,0

~=o:25 :=0.3:

0,9

0,8

tO

,

,

•

1,0

0,9

I. . . . . . .

Off

0,8

0,6

0,7

ct=0.5 8

tO

0,5

T y=l.5

7=0.6 i / y=0.8

0,6

i

3

8

0,4

0,5

a=0"6 < I

s 0,4

0,3

~=0.7

J

0,2

0,3

0,2

0,1 ~ = 1 0 , 50, ~,

0,1

0,0 0,1 i ............... ~ ....................... , . . . . . . . . . . . . . . . . . . . . . . . . , 10 -4

10 - 3

10 "2

10

10 0

10 1

10 2

10 3

10 4

0,0 t . o,1

. ........

10-5

.

. i

.

. . . . . . . .

10-4

.

. . . . . . . .

10-3

.

10-2

10-1

10 0

10 1

10 2

COTo F i g . 3. N o r m a l i z e d real part of the dielectric constant as a function o f OgZo, calculated from the D F R T for T = 1 and different values o f ~.

LOTo F i g . 4. S a m e as F i g . 3, f o r ~ = 1 and different values o f 7.

F. Carrique et al./Colloids Surfaces A: Physicochem. Eng. Aspects 97 (1995) 141-149

into eXH(x) 1 +~2"c2e2xdx

%

particular, it can be shown that

e"(m)

H(x) = 5(x)

lira

[ O

145

(14)

¢s--£~

--

(16)

-oo

which is, in fact, the Debye relaxation model, since

&(x) (the Dirac delta function) is the DFRT of this model. If H(x) is known, the dielectric spectrum of the colloidal system can be obtained from Eq. (2):

e'(oo) Go

~

-

¢s--6~

--

J

H(x) l+¢oZzZe

(is)

2xdx

oo

0,33

[ ....... ! ....... 7

t

In Figs. 3-6 we have plotted the dielectric spectra deduced from Eqs. (14) and (15) for different values of ~ and ),. Note the lack of dependence of the shapes of the curves on %, and the asymmetry of the e" - log(oot0) plots, a consequence of the lack of symmetry in the In(t/to) dependence of the DFRT proposed. It is also worth mentioning that, for given t0 and ? (Fig. 3), the frequency with which e"/(G- e~) is a maximum decreases with increasing ~, whereas it increases with ~ at constant ~, as shown in Fig. 4. When e" is plotted against e', curves such as those in Figs. 7 and 8 are obtained. Our previous analysis of the

0,30

0,33 0127

0,30

v=1.5

0,24

0,27

0,21

0,24

',0 ~

0,18

0,21

'3

0.15

0,18

%

3

0,12

0,15

% 0,09

0,12

0,06

0,09

0,03

0,06

¥=0.6 -

7=0.8 -

0,03

0'0010-4

10-3

10-2

10 -1

10 0

10 1

10 2

10 3

10 4 0,00

10"5

~

10-4

10-3

10-2

10 -1

10 0

10 1

J 10 2

~0ro Fig. 5. Normalized imaginary part of the dielectric constant as a function of OgZo, calculated from the D F R T for ~ = 1 and different values of ~.

COTo

Fig. 6. Same as Fig. 5, for a = 1 and different values of 7.

146

F Carrique et al./Colloids Surfaces A: Physicochem. Eng. Aspects 97 (1995) 141-149 0,35 0,30

"S

0,25

I 0,20

3

%

0.15 0,10 0,05 0,00 -0.

-0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

(s'(~0)-s~)/(s,-~) Fig. 7, Normalized imaginary part of the dielectric constant versus its real part, calculated from the DFRT for ~ = 1 and several values of ~. 0,35

!

0,30 0,25 U3 t~,,1 u3

0,20

3

0,15

%

0,10 0,05 0,00 -0

-0,0

0,1

0,2

03

0,4

0,5

06

0,7

0,8

0,9

1,0

,1

(e'(~)-s~)/(s~-~) Fig. 8. Same as Fig. 7, for c~= 1 and different values of 7D W and D S L models demonstrated [ 1 7 ] that such curves can be characterized as follows: (i) the curves are depressed, i.e. max [e"(m)] < (e s -Coo)/2; (ii) their tangents at low frequencies intersect the real axis at an angle of - ~/4 rad; (iii) their tangents at high frequencies form an angle of ~/2 rad with the e' axis. O u r results in Figs. 7 and 8 show the expected asymmetry. When I' = 1 (Fig. 7), the three abovementioned characteristics are fulfilled whatever the value of e; the magnitudes of the maxima increase when ~ is raised, the curves approaching a limiting shape. W h e n ~ = 1 (Fig. 8), the previous conditions are only verified if 7 = 1; for larger 7 values, the low-frequency slope of the curves tends to increase. This result suggests that 1~= 1 should be kept

constant when reproducing the D W or D S L results, using e and % as the only parameters of the distribution. These points will be considered in the next paragraph.

3. Dielectric response for the DW and DSL models. Comparison with the proposed DFRT We shall now check our D F R T with results deduced from both classical and D S L theories using standard conditions often found in experimental situations. Pertinent data are shown in Figs. 9 and 10 ( D W model), and 11 and 12 ( D S L theory). In these figures, Ae" =e"/~, and A e ' = (e' -- er¢)/¢, where ¢ is the volume fraction of solids

F. Carrique et al./Colloids Surfaces A: Physicochem. Eng. Aspects 97 (1995) 141-149 1200

[

20000

...............

147

!

]

-

----o-- DSL

DW

~8ooo

DFRT

1000

DFRT 16000

l

800

~ ........

600

. . . . .

14000 ~ F 12000

*

-

•

i

F

i [

10000 - - -

% .4

'4 400

]

-,.o <1

8000

4

---

i

,

8000 200

:

~--

-

......

i

4000

i

2000

0

_20 n

01

........

102

......................

103

104

105

106

--

20001 01

10 7

/_>

!

1 02

to (rad/s)

103

104

105

10 5

107

to (rad/s)

Fig. 9. Increments of the real and imaginary parts of the dielectric constant plotted as a function of frequency, calculated from the DW model and the DFRT. Electrolyte: barium acetate; Ka= 10; ( = 5 0 m V . DFRT parameters: ~=0.9; ~o=1.42x 10 5s.

Fig. 11. Increments of the real and imaginary parts of the dielectric constant as a function of frequency, calculated from the DSL model and the DFRT. Electrolyte: KCI; ~ca = 20; =-103mV; c = 5 . DFRT parameters: c~=0.63; ro= 1.14x10 5s.

360 i

i

i

300 240 tO <]

180 120 60 0 -200

-100

0

100

200

300

400

500

A~'(t0)

600

700

+

800

DW

900

~

1000

1100

DFRT

Fig. 10. Increment of the imaginary part of the dielectric constant plotted as a function of its real part, calculated from the DW model and the DFRT. Same conditions as in Fig. 9.

in s u s p e n s i o n , a n d ere is the dielectric c o n s t a n t of the electrolyte. In the case of the D W m o d e l (Figs. 9 a n d 10), spherical particles o f radius a are s u p p o s e d to be dispersed in b a r i u m acetate s o l u t i o n w i t h ~a = 10

( ~ - ~ is the t h i c k n e s s o f the d o u b l e layer s u r r o u n d ing the particles), a n d a zeta p o t e n t i a l ~ = 50 m V . Since the m a x i m u m of e" d e p e n d s on ~ (Fig. 5), a suitable v a l u e 6 f ~ m u s t be c h o s e n to fit the D W results; excellent a g r e e m e n t b e t w e e n the D F R T

148

F. Carrique et al./Colloids SurJitces A: Physicochem. Eng. Aspects 97 (1995) 141 149 5000

r

I

''

' i '

''

i'

''

4000 3 =

3000 2000 1000 0 -2000

0

2000

4000

6000

8000

10000

AC'(~)

12000

+

14000

DSL

16000

18000

20000

---,--- DFRT

Fig. 12. Increment of the imaginary part of the dielectric constant versus its real part, calculated from the DSL model and the DFRT. Same conditions as in Fig. 1l. and D W predictions shown in Fig. 10 was obtained for ~ = 0.9. The value of to affects the frequency at which maxima appear; the data in Figs. 9 and 10 were obtained assuming to = 1.42 x 10 -s s. It must be noted that the whole plots move along the horizontal axis when to is changed, but no modification in their shapes occurs unless c~ and/or 7 are modified. As observed in Figs. 9 and 10, our D F R T agrees very closely with the D W predictions for both Ae' and Ae", over the whole frequency range studied. Data corresponding to the DSL model are shown in Figs. 11 and 12; it can be seen that the relaxation pattern predicted by this model is very similar to that of the D W model; the results included in these figures were computed assuming xa = 20 (electrolyte KC1), ~ = - 1 0 3 mV, and c = 5. Hence c is a factor describing the relationship between the surface conductivity linked to K + cations, and their conductivity in the bulk of the double layer [8,18]. Theoretical Ae' data were taken from Ref. [18], whereas the corresponding Ae" spectrum was calculated from Ae' by means of the K r a m e r s - K r 6 n i g relations [ 15]. The values of c~ and % that best fit the data are ct = 0.63 and t o = 1.14 x 10 -5 s. As before, the agreement between DSL and D F R T results is excellent, although somewhat poorer in the high-frequency range owing to the limited data available to the authors.

relaxation times and the actual relaxation processes taking place in the suspensions studied, in spite of the excellent quantitative agreement reached. Since such processes are basically diffusive, it has been suggested [23] that different ions take different times to move back and forth in the double layer under the influence of the applied AC field. We have definitely demonstrated that a symmetrical function, like the Cole-Cole function, cannot reproduce the theoretical dielectric response of colloidal suspensions. Rather, a non-symmetrical DFRT, similar to that proposed here, must be used. It is interesting to mention, however, that if the existence of interactions between the particles (not contemplated by either the D W or DSL model) is assumed in the theory, the predicted dielectric relaxation pattern is closer to that observed experimentally [24]. Although this does not mean that particle interactions are the only source of discrepancy between the D W and DSL models and experimental results, the effect of such interactions seems to be an interesting topic for further research. In fact, as reported by Dunstan [13], significant interactions between dispersed particles do indeed exist in suspensions commonly considered as dilute, for interparticle distances as large as 10•- 1.

4. Conclusion

Acknowledgments

It is not easy to establish a phygical connection between our proposed distribution function of

Financial suport by Fundaci6n Ram6n Areces, Spain, is gratefully acknowledged.

F. Carrique et al./Colloids Surfaces A: Physicochem. Eng. Aspects 97 (1995) 141 149

References Eli E.H.B. Delacey and L.R. White, J. Chem. Soc., Faraday Trans. 2, 77 (1981) 2007. [2] S. Dukhin and V.N. Shilov, Dielectric Phenomena and the Double-Layer in Disperse Systems and Polyelectrolyres, Wiley, New York, 1974. E3] M. Fixman, J. Chem. Phys., 72 (1980) 5177. [4] R.W. O'Brien, Adv. Colloid Interface Sci., 16 (1982) 281. [5] M. Fixman, J. Chem. Phys., 78 (1983) 483. [6] W.C. Chew, J. Chem. Phys., 80 (1984) 4541. [7] E. Vogel and H. Pauly, J. Chem. Phys., 89 (1988) 3823. [8] L.A. Rosen, J.C. Baygents and D.A. Saville, J. Chem. Phys., 98 (1993) 4183. [9] M.M. Springer, Ph.D. Thesis, Wageningen University, The Netherlands, 1979. El0] LA. Rosen and D.A. Saville, J. Colloid Interface Sci., 140 (1990) 82. El 1] L.A. Rosen and D.A. Saville, Langmuir, 7 (1991") 36. [12] L.A. Rosen and D.A. Saville, J. Colloid Interface Sci., 149 (1992) 542. [13] D.E. Dunstan, J. Chem. Soc., Faraday Trans. 3, 89 (1993) 521.

149

[-14] F. Carrique, L. Zurita and A.V. Delgado, J. Colloid Interface Sci., 169 (1995). [15] C.J.F. Battcher and P. Bordewijk, Theory of Electric Polarization, Vol. II, Elsevier, Amsterdam, 1978. [16] F. Carrique, L. Zurita and A.V. Delgado, Colloids Surfaces A: Physicochem. Eng. Aspects, 92 (1994) 9. [ 17] F. Carrique, Ph.D. Dissertation, University of Granada, 1993. [18] D.A. Saville, Colloids Surfaces A: Physicochem. Eng. Aspects, 92 (1994) 29. [19] J. Kijstra, H.P. Van Leewen and J. Lyklema, J. Chem. Soc., Faraday Trans., 88 (1992) 441. [20] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectric Group, London, 1983. [21] S.W. Provencher, Comput. Phys. Commun., 27 (1982) 213. E22] G. Williams and D.C. Watts, Trans. Faraday Soc., 66 (1970) 80. E23] K. Lim and E.I. Franses, J. Colloid Interface Sci., 110 (1986) 201. [24] E. Vogel and H. Pauly, J. Chem. Phys., 89 (1988) 3830.