Polydisperse size data from single electric birefringence transients

63

Powder Technology 72 (1992) 63-69

Polydisperse

size data from single electric birefringence

R. M. J. Watson

and B. R. Jennings*

transients

Electra-Optics Group, J. J. i%omson Physical Laboratory, Reading University, Reading, Berkshire, RG6 2AF (UK) (Received

July 29, 1991)

Abstract When a suspension of anisotropic, sub-micron sized particles is subjected to an electric pulse, the medium becomes birefringent in a transient manner. An analysis is presented of the post field relaxation of the transient response from a dilute, polydisperse suspension of rigid colloidal particles. Two averaged parameters relating to the rotational diffusion coefficient D, namely (D) and (l/D) are defined which are obtained from the initial rate of the decay data and from the area under the decay curve respectively. The averages are weighted according to the geometric shape and electrical properties of the particles. Measurement of these averages leads directly to the evaluation of an equivalent log-normal distribution of particle major dimension. By considering the implications of the theory for both high and low field data, a method of evaluating the electrical characteristics of the particles is indicated. Representative results are given.

Introduction

using the relation

In response to the application of an electric pulse, anisotropic, anisodiametric particles in suspension orientate and align and the medium exhibits birefringence up to an equilibrium value. Termination of the pulse is followed by a diffusion controlled disorientation regime. The rate of this process relates to the medium viscosity, the temperature and the particle size and shape. This paper presents the method of characterising the particle size distribution for a dilute solution of rigid, electrically anisotropic submicrometer colloidal particles from an analysis of the free field relaxation of electrically induced birefringence transient responses, which are generated under given electric field strength conditions. The birefringence, hn(t) at time t after switching off the field for a monodisperse dispersion is a single exponential decay which is independent of the initial orientation state of the particles. At arbitrary field strengths this is given by [l]

D=

An(t) = An(0)e-6D’

(1)

where hn(0) denotes the steady-state birefringence at time t = 0 when the field is switched off and D is the particle rotational diffusion coefficient. A knowledge of D can be related to the major dimension 1 for rigid, regular geometries of the particles *AIso: Research Department, ECC International tell, Cornwall PL25 4DJ (UK)

0032-5910/92/$5.00

Ltd., St. Aus-

A%4 P

[2],

(2)

where A = 3kTh-q with k Boltzmann’s constant, T the absolute temperature, 77the solvent viscosity and F(p) a function of the axial ratio p of the particles. Expressions for F(p) are given in the appendix. Measurement of the rotational diffusion coefficient provides a sensitive indication of particle size through the predominant Z3 dependence. For a polydisperse system, the birefringence decay is multi-exponential, with contributions from each dispersed particle species. The suspension medium is assumed to be inactive. A weighted average value of (D) can thus generally be obtained. It is convenient to analyse the initial rate of the decay as these represent the highest amplitude response data. Some experimenters have fitted a multi-exponential curve to the free-field relaxation data and used Laplace transforms to deconvolute the integral expressing the decay for the polydisperse system to obtain a distribution of relaxation times [4, 5, 61. This distribution can then be transformed in principle to a size distribution. The method is limited by the number of components of the multi-exponential decay that can be isolated. If more than two or three components of the multiexponential are to be fitted to the full decay curve, there is an infinite number of solutions, many of which are unstable [7] owing to the inherent noise in the experimental data. It is therefore necessary to introduce

0 1992 - Elsevier Sequoia. All rights reserved

64

mathematical constraints to the analysis if a physically meaningful solution is to be obtained. By using a parameterised model of the distribution, a stable solution may be found. At low field strengths, the Kerr law is applicable and An(O) is proportional to E2 where E is the electric field strength. For this case, (D) is weighted towards the larger particles in the distribution which are more easily orientated, and (D) attains a minimum value in the limit of low field strengths. At high field strengths a state of saturation of the birefringence is reached and the particles can be regarded as fully orientated. In this case (D) is weighted for all the particles in the distribution and (D) thus attains a maximum value in the limit of high field strengths. Recent research shows that, by measuring the rotational diffusion coefficient from the initial rate of decay of the birefringence for specific low and high field strength conditions, two distinct weighted average rotational diffusion coefficients are obtained which can be related to the two parameters defining a log-normal distribution of particle sizes [S]. This is referred to as the ‘two-field’ method. In principle, all particles present in the sample contribute to the observed birefringence and to the rate of the transient decay. In response to a desire to obtain the distribution data from a single transient response, a procedure is described herein. It is shown that for a single transient, two distinct weighted averages of the rotational diffusion coefficient namely (D) and (l/ 0) may each be evaluated from separate analyses of the free field relaxation of the birefringence. Here (D) is found from the initial rate of decay and (l/D) from the integrated area under the time-dependent decay curve. These are then used as the basis for the generation of a two-parameter function to describe the size distribution for the particles. To test the concept, experimental data are presented for a polydisperse dispersion of polytetrafluoroethylene (PTFE) particles of known anisometry. The particle size distribution and relevant electrical properties are measured and a corroborating comparison is made with electron microscopy data.

Theory

(i) Electric field birefringence The steady-state electric field birefringence of a solution of dilute rigid, axially symmetric particles orientated by a pure induced electrical dipole moment mechanism, such as that excited by an alternating electric field of frequency exceeding the Debye frequency [9] is given by [lo]

where C, is the volume fraction, equal to the product of the number of particles per unit volume N, and the solute particle volume V. Here II is the refractive index of the dispersion and Ag is the intrinsic optical anisotropy. For submicron particles & is regarded as a constant optical factor. The orientation function q-y) depends on the parameter, y where, ‘=

ALXE2 2kT

(4)

with E the rms electric field strength. Here, the anisotropy of the electric polarisability between the particle symmetry axis and transverse axes (indicated by 1 and 2 respectively) is Aa = (a1 - cr2). For positively anisotropic rod-like and prolate particles the orientation function can be approximated in its limits by [ill @$+:Yl5

as E-+0 asE+m

(5a)

For negatively discotic and oblate particles, the related limits are G(y)= -2~115 Q(y)= -l/2

as E-t0 asE+m

(5b)

The analysis that follows is given for the prolate particles. The same final equations for the particle size analysis would be obtained if the same procedure were to be followed for oblate particles. The nature of the dependence of LYand hence Aa on particle size is unsure. For all geometries, it can be expressed in the form Acy= ALYJ

(6)

where A% is a reduced polarisability and s is a positive integer. For rods, values from 1 to 3 have been suggested in the literature [12, 13, 141. From eqn. [5a] at low field strengths, the steady-state birefringence b(O) and the rate of its decay will be weighted according to s throughout the particle size distribution. Many colloid systems can be modelled on a lognormal distribution of sizes. The probability distribution f(l) as a function of the major particle dimension, is given by [15].

where /; f(Z) dl= 1. Here m is the median and u is a breadth parameter of the distribution. The nth moment of such a distribution is denoted,

6.5

I(n)= j

-wo n=s

Pf(Z) dl

o An(O)

0

If the number density of species i is Ni then,

s

(9)

Z”f(l) dZ= x Njri 1

(IO)

(ii) Initial rate of decay

By considering the moments of the distribution expressions have been derived [8] for (D) which can be evaluated from measurement of the initial slope, So obtained from a semi-logarithmic plot of the birefringence decay, ln(L\n(t)lAtz(O)) against time. The average (D) so obtained is defined as, -6(D)

(11)

The average parameter (D) can be expressed implicitly in terms of the geometrical and electrical dependent parameters, u and s respectively where the particle volume and Acr are proportional to I” and I” respectively. Under low and high field conditions (subscripts L and H respectively),

024 and

05

In terms of the log-normal distribution, these equations become exp i (9 - 6(u +s))aZ I

L= AF(p)

exp( i (9-6u)2]

(16a)

I: Nil,(u+” L

Wb) or, in terms of the moments of a continuous distribution, 1

0 5

L

I(u+s+3)

1

=-

A&))

(174

Z(u +s>

and 1 5

0

_ H

1 A%)

@+3) Z(u)

(17b)

One obtains, for the log-normal case

0

m3 L =- A%))

exp i [9 + 6(u +s)]2

(184

and

05 1

= H

m3 exp[jj

-wd

[9+6u]d]

WI

Wa) (iv) Determination of a log-normal particle size distribution from a single transient

and @$

F Nili(u+5+3’

1

1

1

Wb)

(D)H=

(15)

Substituting for Ci and G(x) for low and high electric field strengths leads to

5

9

diffusion coef-

F ci@CYi>

Z(n) = m”e(“W

(D)==

(14)

F Ci @(Yi)/Di

so that the nth moment becomes,

-$ln(z)]t_F&=

E Ci@(Yi)/Di i 6 7 Ci@CYi>

For a discrete distribution, a reciprocal ficient can be defined as,

OD

0

*,A

(13b)

(iii) Area under the decay

The characteristics of the decay curve are reflected in the integrated area under the decay. An alternative set of weighted averages can be deduced by considering this area. If s2 represents the normalised birefringence integrated in the limits from t =0 to t = m, then from equations (1) and (3),

It is of interest to consider the product of the average parameters (D) and (l/D) under each of conditions of high and low field strength. From couplet eqns. (13a), (18a) and (13b) with (18b), dimensionless product

two the the the

WDMD)L=

(19)

WDMDk=

exp(94

is obtained in each case. A means of evaluating a is thus apparent. The conditions of low and high fields can be evaluated from a supplementary experiment. A single transient under either of these conditions is

66

recorded from a one-shot response and (D) and (l/ D) evaluated from So and a. The product yields (+ directly. The above analysis has advantages over former procedures. Firstly, use of a single transient enables advantages of speed to be achieved and avoids problems associated with unnecessary dual or multiple pulsing of the media. Secondly, if one assumes a value for s, and knows the particle shape and hence a value for u, then m can be obtained using either (D),_ or (l/ D),_ from a low field transient. For spheroids, an approximate estimate for p is also needed. Thirdly, if a high field transient can be realised, then u and m can be obtained directly from a single transient without any pre-knowledge of s. This is an important advantage. A fourth feature of the theory is evident from a comparison of the high and low field data. By considering the sets of eqns. (13) and (18) the following high-tolow field ratios give the same value, i.e.

(D)H_ -WD>, (D>L

(lJD>H

=exp(3s2)

(20)

Hence, if each of the parameters S, and R are measured at high and low field strengths, the values of the four average quantities can be isolated and the parameter s also enumerated from the appropriate relations. This is an extremely attractive feature of the theory. The following procedure is therefore proposed. For a dilute suspension of rigid colloids, pulses of high frequency potential are applied across the sample cell. For near-micron sized particles, frequencies of 1 kHz are adequate. Transient birefringence responses are recorded for increasing applied field strengths and the parameters So and fi measured from which (D) and (l/D) are enumerated. For the low field data, assuming the particle shape, u is evaluated from the use of eqn. (19). If s has been assumed or predetermined for the material, m is calculated from eqn. (13a) or (18a) and the log-normal distribution obtained. Experimentally, where possible, low field data is preferred to avoid the use of the high applied voltages. To evaluate s, both the high and low field limiting values of (D) and (l/D) are measured, and the expressions of eqn. (20) used. This is a valuable method to obtain s experimentally and hence to appraise the various theories for the electric polarisability and surface polarisation of colloidal particles in suspension. In principle, the use of the double ratios in eqns. (19) and (20) afford a consistency check on the data obtained.

Representative data Materials and measuring

Electric field birefringence measurements were carried out on a 40 pg ml-’ aqueous dispersion of emulsion polymerised PTFE particles. Electron microscopic data revealed the particles to be highly crystalline prolate ellipsoids, with an average axial ratio of 1.5. Field strengths of up to 350 Vm-’ and of 1 kHz alternating frequency were applied via pulses of up to 150 ms duration. The electric field birefringence of the solution was measured using a He-Ne laser of 633 nm wavelength as a probe beam. After being initially linearly polarised at 45” to the applied electric field direction by a polarising prism, the beam travelled a pathlength of 50 mm between two stainless steel electrodes set 2 mm apart within the sample cell. Emerging from the cell the beam passed through a quarter wave plate then a high quality GlanThompson polarising prism which was offset some 10” from the crossed position. This arrangement provides a linear response between the birefringence and the recorded signal [17]. The beam was then detected using a Centronic Ltd silicon photodiode type (OS15-5T) connected to an amplifier circuit. The free-field relaxation of the birefringence was evaluated from an analysis of the photodiode output voltage together with an alternating electric field pulse sampled with a 1OOO:lvoltage probe. Both signals were captured and digitised using a Datalab D 1200 transient recorder. The digitised information was computer processed. The rotational diifusion coefficient (D) was calculated from a linear-least squares fit of the first few data points of ln(b(t)/h(O)) against time t after field removal. This yielded the initial slope, S,. The reciprocal rotational diffusion coefficient (l/D) was calculated from the total area R under the decay response of An(t)lAn(O) against time. Both (D) and (l/D) were temperature corrected to 20 “C, and were measured over a range of field strengths. Data for (D) and (l/D) measured at various field strengths are shown in Figs. 1 and 2. Each parameter attained a plateau region in the limits of high and low field strength conditions. These curves enabled the low and high field limiting conditions to be identified for this system. Both sets of data corresponded to a lowfield upper limit of 20 kVm_l and an onset of the high field condition at 150 kVm-‘. Figure 3 shows a graph of the field dependence of the product (D)(l/ 0). As this is a sensitive parameter, data were recorded for each transient response signal-averaged ten times. The resultant curve expressed constant values of the product at the high and low field limits, but a minimum at intermediate conditions. It is demonstrated that eqns. (19) and (20) are not applicable under arbitrary field

67 70

From the relevant products, at low field strengths 1.76 whilst at high field strengths (D)L(l/D)L=egd= (D)H(1/D)H=ego2= 1.66. These correspond via eqn. (19) to values of 0.25 and 0.24 respectively for u. It is of interest to consider the ratios of the relevant averaged parameters under high and low field conditions. From the initial slopes,

60

-i "50 ;: V

40

.j_/

n /

-(D>H =e3”“=1.62

10"

10’0

109

108

(4~

-I

1

I

30

From the reciprocal measurements:

Ez(Vm-')2

Fig. 1. Variation of the average parameter for a dilute aqueous PTFE suspension.

50

. 45

. 7 ,,

25

108

-(llD)L (l/D),

,t‘.... .

10'0

109

,c

10"

Ez(Vm-I)2

Fig. 2. Variation of the parameter the aqueous PTFE suspension.

(l/D)

with field strength

for

1.7

1.4 108

I

10'0

Fig. 3. Experimental data for the product (0) (l/D) with field strength for the PTFE suspension. Note the plateau regions at high and low fields.

conditions. From the experimental data, the following limiting parameters were obtained. and s-’

(D)H =54.5 s-l (l/D), = 0.03054 s

Using an experimental value of a=0.24 ( f 0.02), these two ratios correspond to values of the electrically dependent parameter s of 2.8 and 3.1 respectively. It is of interest to note two points. Firstly, the two ratios again show a high consistency in the result obtained. Secondly, for this system under these conditions, a value of s= 3 is indicated. Using these values of CTand s enable m to be found from the various equations (13) and (18). Substituting for the known parameters, i.e. p= 1.5, u = 3, s=3 in these equations, each of the experimental parameters leads to the value of m and CTas indicated in Table 1. It should be noted that in this analysis, if a value of 2 had been assigned to s, then an inconsistent set of u and m values would have been obtained from the four measured diffusion coefficients. Finally, for comparative purposes, an electron microscopic analysis of 625 particles gave a median size of 319 nm and u= 0.26. Figure 4 compares the frequency histogram with the log-normal distribution derived from a single high field electric birefringence transient. The agreement is good.

calculated

from

10"

Ez(Vm-')2

(D),=33.7 s-’ (l/D),=O.O5242

= e3”+ = 1.72

TABLE 1. Log-normal distribution parameters the rotational diffusional coefficients

I

109

diffusion coefficient

(D) with field strength

.

30

rotational

Method

(T

m (nm)

A. Single transient, high field B. Single transient, low field (s=3) C. High and low transients for (D) D. High and low transients for (l/D)

0.24 0.25

325 312

0.23

327

0.25

320

Combination of A and B data simultaneously yield s=3. For C and D, a value of s=3 has had to be assumed, not derived.

68 30

r

1

\

6 : 37-O

\

4 .k ? .'0 --,lO E

-

i

0 50

150

250

350

consistent results for the size distrib.ution parameters based on high and low field single transient analyses only if s =3 for this particular system. This result and its general applicability to relevant non-rod shaped particles is a topic for future studies. Theoretical interpretation of the induced electric dipole moment depends greatly on the assumed nature of the counter-ion atmosphere surrounding the solute particles. By considering repulsive effects between ions and counter-ions surrounding a rod-like polyion, Gibbs and McTague [12] derived a size dependence of the electric dipole moment which was linear for small degrees of polymerisation but having an F dependence for large degrees of polymerization. An P dependence was predicated by Schurr [13] for rod-like polyions when repulsive effects between the counter-ions was included. Fixman [14] also predicted an P dependence at low field strengths by assuming a thin double layer surrounding rod-like particles. The experimental result of s=3 concurs with these theories. The PTFE surface [20] consists of some 2% coverage with carboxylateions introduced during the manufacturing process. These may well result in a thin surface double layer appropriate for the Fixman theory. Also, from electron microscope pictures, many of the PTFE particles indicate nearly spherical morphology. The appropriation of theories for rod-like particles has limited value and must be treated with caution. Apart from the availability of size data from a single birefringence transient, the present analysis has value in those situations where it proves difficult to obtain responses at both high and low fields, as required for the former method. Three cases are noteworthy. Firstly, large colloidal particles which have large electrical polarisabilities and require very low field strengths to satisfy the Kerr law. The low field rise of the birefringence is diffusion controlled, therefore large particle sizes require a relatively long time usually of the order of seconds to attain orientational equilibrium. Concentration effects can be encountered. Therefore experimentally, it is difficult to isolate (D),_ due to the narrow field strength range and long pulse durations that are necessary. Secondly, particles with small Ag and hence a small signal response may also be studied under high fields where the response is enhanced. Thirdly, in more highly conducting media, the use of low fields is necessary. In conclusion, measurement of the diffusion coefficients (D) and (l/D) from a single transient at high field strengths permits an evaluation of the particle size distribution without prior knowledge of the induced electric dipole mechanism. Assumption as to the particle shape is needed. When low field strength measurements of (D) or (l/D) are invoked, a knowledge of the polarisability mechanism is required to obtain the par-

I,::,_

Size

450

(nm)

550

650

750

Fig. 4. Electron microscopy histogram for 625 particles of PTFE. The full curve represents the birefringence determined distribution where m=325 nm and cr=O.24.

Discussion

Three important factors evolve from this study. Firstly, it is apparent that the characteristic parameters of the log-normal distribution are discernible from an analysis of the decay characteristic of a single induced birefringence transient. The measurement of (D) from the initial decay and (l/D) from the time integrated area under the relevant decay curve of the transient provide the data required. Measurements of these two averages under conditions of either low (that is in the Kerr law region) or high (full particle orientation region) amplitude force fields suffice. By using an electric pulse of high frequency alternating current, particle orientation is due to an induced dipole moment alone. Hitherto it has been assumed that the electrical polarisability has an approximately quadratic dependence on particle size. In reality the induced electrical polarisability is related to both the size and shape of the particles in a complicated way. With conducting suspensions the polarisability is generally understood to arise from a surface interfacial mechanism in which charge distributed on a particle surface is associated with an atmosphere of counter-ions. This becomes distorted in the electric field. Mandel [18] derived an e dependence of the longtitudinal polarisability for rod shaped particles of length 1. Electra-optic studies made in this research group [16] of needle shaped sepiolite clay in aqueous dispersions confirmed this. Until recently this has been taken to be generally applicable to the particle size distributions for rigid colloids of variable shape. This corresponds to s=2 in eqn. (20) so the (D&J(D),_= exp(62) was used to obtain representative size distributions [19]. In this work an independent assessment has been made of s for the near spherical lozenge shaped particles of PTFE in water, for which the prolate ellipsoid model with p= 1.5 has been used. The data lead to self

69

Acknowledgements

11 C. T. O'Konski, K. Yoshioka and W. H. Orttung, J. Phys. Chem., 63 (1959) 1558. 12 J. H. Gibbs and J. P. McTague, J. Chem. Phys., 44 (1966) 4295. 13 J. M. Schurr, Biopolymers, 10 (1971) 1371. 14 M. Fixman, Macromolecules, 13 (1980) 711. 15 N. A. J. Hastings and J. B. Peacock, Statistical Distributions, Butterworth and Co., London, 1975. 16 V. J. Morris, G. J. Brownsey and B. R. Jennings, in B. R. Jennings (ed.), Electro-optics and Dielectrics of Macromolecules and Colloids, Plenum, New York, 1979, p. 311. 17 E. Fredericq and C. Houssier, Electdc Dichroism and Electric Birefringence, Clarendon Press, Oxford, 1973. 18 M. Mandel, Mol. Phys., 4 (1961) 489. 19 B. R. Jennings and D. M. Oakley, Appl. Opt., 21 (1981) 1519. 20 R. H. Ottewill and D. Rance, Coat. Chim. Acta, 50 (1977) 65.

R. M. J. Watson thanks the S E R C and Messrs. Kodak Ltd. for a C A S E studentship during the t e n u r e of which this work was undertaken.

Appendix - The shape factor F(p) for various geometries

References

Geometry

F(p)

(a) PPr°late= a/b >spher°idl

2(P404-1) [[(2p2- 1 ) pln[p ~ + (P2- 1)I~1 - 1 ]

(b) p=b/a>lOblate spheroid

2 ( 14)pa - 0 L[(2-°2) ~ t a n 1[(P2 - -- 1)Irz] --1 ]

(c) Thin disc

~r 4

ticle size distribution. Both the size distribution characterisation parameters and the nature of the polarisation mechanism dependence on size can be evaluated simultaneously if all the parameters (D)x-i, (D)L, (1/ D)i-t and (1/D)L are recorded from a pair of transients under the two appropriate conditions. These results indicate that Aot has a cubic dependence on the major dimension for the prolate spheroids of PTFE particles and that the use of (D) and (l/D) to calculate the particle size distribution is justified.

1 2 3 4 5 6 7 8 9 10

H. Benoit, Ann. Phys., 6 (1951) 561. F. Perrin, J. Phys. Rad., Ser vii (1934) 497. S. Broersma, Z Chem. Phys., 32 (1959) 1626. M. Matsumoto, H. Watanabe and K. Yoshioka, Kolloid z~z. Poem., 250 (1972) 298. B. Chu, R. Xu and A. Dinapoli, J. Colloid Interface Sci., 116 (1987) 182. K. Kikuchi, Polym. Commun., 28 (1987) 40. S. W. Provencher, Comput. Phys. Commun., 27 (1982) 213. B. R. Jennings, A. R. Foweraker and V. J. Morris, Adv. MoL Relaxation Processes, 12 (1978) 211. P. Debye, PoiarMolecules, Dover, New York, 1929 (reprinted from Chemical Catalogue Co.). A. Peterlin and H. A. Stuart, Z. Phys., 112 (1939) 1.

(d) Thin rod length, l, (ln~2p) )2 breadth, p=l/b>4 ln(2p)-l.57+7 -0.28

Spheroids have major axes a, minor axes b and axial ratios p. Equations from [2] for cases (a), (b) and (c) and ref. [3] for case (d).