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Photon correlation and sedimentation experiments on interacting PMMA particles in benzene
Advancesin ColloidandInterfaceScience,16(1982)161-174 Elsevier Scientific Publishing Company, Amsterdam -Printed
161
in The Netherlands
PHDTCN CORRELATION AND SEDIllENTATION EXPERIMENTS ON INTERACTING PMMA PARTICLES IN BENZENE
H.Fl. FIJNAUT, J.K.G. DHONT and E-A. NIEUWENHUIS Van't Hoff Laboratory for Physical and Colloid Chemistry, Transitorium
3, Padualaan
3, 3584 CH Utrecht, The Netherlands
CONTENTS I.
AGSTRACT
II.
INTRODUCTION
. . . .. ... .. . . . . .. . ..._________________________________________161
III_
THEORY
IV.
EXPERIMENTS
V.
DISCUSSION
.. . . . .._.___......._____________._________________~______~6l
.____...____.._..__._________.______.___...____.___.__._.._._._16~ . . ..._........._....__.______~___~~___~~~__.~~____________16~ . ...___....__..____________________________________.___.___l69
VI.
CONCLUSIONS
VII.
ACKNOWLEDGEMENTS
__..___._____._._____________._.___.........................173
VIII.
REFERENCES
I.
ABSTRACT
__...__.______._.._.____~___~~___~~~___~____.~_______173
. .. .. . .. . . . .. . .._...~___.__~_.~_~~.~._~.~~~~~~~.~~~~_~~_.__173
Photon correlation
(PCS) experiments on interacting
PMHA (polymethylmethacrylate)
latexes show the existence of static (thermodynamic) and hydrodynamic The interactions are observed through the measurement
interactions.
of an effective diffusion
coefficient as a function of concentration and scattering wave vector.
For the
largest particles the wave vector dependence of the effective diffusion coefficient confirms theoretical predictions that from PCS experiments mutual and self-diffusion coefficients can be observed_
The combination of PCS experiments with sedimentation
experiments enables the separation of hydrodynamic and static interactions.
The
hydrodynamic interactions contain two physical significant terms which can be deduced separately only by the combination of PCS experiments at large wave vectors with the sedimentation experiments._ The combined experimental data show that the interaction is too complicated to be described by simple model potentials of hard sphere repulsion with either an attractive or a repulsive disturbance. II_
INTRODUCTION Nowadays the study of colloidal interactions is subject to a large number of
theoretical and experimental
investigations.
Part of the work is concerned with
colloidal suspensions as, for example, electrically OOOl-8686/82/0000-00001$03.500
charged particles in water
1982Elsevier ScientificPublishingCompany
(ref. 11, microemulsion systems (ref. 2) and sterically stabilized particles in non-polar solvents (ref. 1, 3). These systems are, among other things, intensively studied by means of light scattering techniques and, to a somewhat lesser extent, by means of ultra centrifugal sedimentation. The ligiitscattering methods include time-averaged intensity, photoncorrelationspectroscopy (PCS) and forced Rayleigh scattering. Time-averaged light scattering gives information about particle interactions and possible (static) structures as a result of these interactions. PCS and forced Rayleigh scattering give information about interaction effects on the particle dynamics_ Sedimentation experiments can give the concentration dependence of the friction coefficient of the suspended particles_ In this paper we will deal with PCS and sedimentation experiments on polymethylaethacrylate (PUMA) latexes dispersed in benzene. The investigations are concerned +lithsuspensions of latex volume fractions up to 25 volume percent. Some of the latex samples have been studied earlier by means of time-averaged light scattering (ref- 3, 4). The concentration regime studied is such that, apart from static interactions, hydrodynamic interactions also play an important role in the particle dynamics. This study, therefore, is intended to increase our understanding of the hydrodynamic interactions from an experimental point of view_ Theories about hydrodynamic interactions, resulting in concentration dependent friction and viscosity, appear to be very complicated_ Nowadays only pair interfor PCS actions are reasonably well understood (ref. 5, 6, 7). The predictions (ref. 7) and friction (ref. 5) based on this pair interaction has only been partly experimentally verified (ref. 8) and is limited to the zero angle regime in light scattering ("forward" scattering)_
The aim of this work is: 1) to investigate the angle or wave vector (see next section) dependence of PCS of hydrodynamically interacting systems; 2) to study the concentration dependence of the friction coefficient; and 3) to explain the nutually related results in terms of particle pair interactionsTHEORY III. In light scattering experiments the scattered light is observed at an angle d with respect to the incident direction of radiation of wavelength 1 in the scattering medium. The dependence of the scattering on A and 2 is expressed by the wave vector K with length: K = (47/l) sin (s/2).
(1)
In PCS experiments the temporal autocorrelation function p(K,r) is observed (refg), which is defined by:
163
:(#,:)=
I(K, t + T)>
9
(2)
where I(K,t) is the K and time-dependent scattered intensity_ The scattered intensity is supposed to be a stationary quantity_ From r(K,r) one can, apart from a proportionality constant, derive the dynamic structure factor S(K;:) (ref. 9) which in fact is the normalized temporal autocorrelation function of the scattered field. For a system of monodisperse, identical and spherical particles suspended in a structureless solvent S(K,:) is given by (ref. 10, II): S(K,r) = S(K) exp [- K2De(K)y\ _
(3)
Here S(K) is the so called static structure factor which can be obtained from the time-averaged scattered intensity. S(K) reflects the influence on the scatterinS of static particle interactions, which may induce local structure in the suspension (ref. 12). D,(K) stands for the effective diffusion coefficient of the suspended particles_ D,(K) depends on the concentration and can be a function of K. To first order in the hydrodynamic volume fraction : of the particles, De(K) can be written as: De(K) = Do [l+:
'S,(K)+HI+H2(K):]
,
(4)
Where Do is the diffusion coefficient at infinite dilution_ The term SI(K) reflects the influence of the static interactions on De(K), whereas HI and HZ(K) contain the hydrodynamic contribution to the dependence of the effective diffusion coefficient on K and the concentration. Recently one of us (ref. 11) has given explicit expressions for SI, HI and H2 based on a theory of Felderhof (ref. 7): SI(K) = i3/(Ka)3/"x sin x Cl-g(x/K)> dx
(5)
;
0 <”
H1 = (27/8) a3
Tg(x)/x4?dx- (15/4)a Ig(x)/x2: dx I I 0
;
(6)
0
H2 = C(9/2)/(Ka)2; (sin x + cos x/s - sin x/x2) :g(x/K)-1:dx / 0
4 (3 sin x/x - 3 cos x/*3 -
+3
sin
x/x2)g(x/K) dx
/
0 +
(225/4)(Ka)4 / 0
(sin x/x6+2
cos
x/x7-
2 sin x/x8)g(x/K) dx,
(7)
164 where a is the radius of the particle and g(x) is the zero order radial distribution function (ref. 12). It is worthwhile to mention the relation between S(K) and SI(K) (ref. 11): S(K) = I - rSl(K) , which is valid to the first order in 2. In sedimentation experiments the sedimentation constant s is observed (ref. 13)_ Expressions for the concentration dependence of s have been derived by Batchelor (ref_ 5) and by Felderhof (ref. 7)_ Using our notation one has: s = so[L+,fHI+H2(K=0)l]
=
(9)
where s
is the sedimentation constant at infinite dilution_ Consgdering the expressions (Eqs. 4, 8 and 9), one observes how closely the concentration dependence of time-averaged light scattering, PCS and sedimentation experiments are related, especially in the limit of zero K. In fact at K=O the relation between De(K=il)/DO, s/so and S(K=O) is simply the generalized StokesEinstein relation: D,(K=O)
(10) DO
which can be easily verified from the relation between the sedimentation constant, sedimentation velocity and friction coefficient (ref_ 5) and from the relation between S(K=O) and the osmotic compressibility (ref_ 14). For KPO one can find more detailed information about hydrodynamic interactions from PCS experiments in combination with sedimentation and static light scattering results. This possibility is the subject of this and future investigations. It should be mentioned here that the application of the expressions given above in the experimental situation is not always simple because of uncertainties in the relation between volume fraction and particle concentration (ref. 8). The hydrodynamic volume fraction of suspended particles can be quite different from the volume fraction calculated on the basis of the density and the weight concentration of these particles, especially if swelling occurs of the particles when they are brought into the solvent EXPERIMENTS IV_ The PMMA latex particles were prepared by emulsion polymerization in water (ref_ 15)_ Samples 6. 7 and 8 were prepared with sodium dodecylsulfate as the emulsifier, whereas sample 5 was prepared without emulsifier. In all cases ethylene glycol dimethacrylate (5 weight percent of monomer) was added to the
165 reaction mixture to build a three-dimensional network inside the particle_ For further details, see the work of Nieuwenhuis et al. (ref. 15), who used the same sample indication as we do. After preparation in water, the dispersions were dialyzed and ion-exchanged to remove residual monomer and ionic impurities_ The dried latex was easily dispersible in benzene and some other apolar solvents. The samples were sedimented 3 or 4 times to remove the linear polymer_ Upon dispersing the particles in benzene, the Pf+lAparticles swelled with respect to their dimensions in water. In water, the particles were compact spheres stabilized by an electric double layer. In benzene, however, they were swollen microgels sterically stabilized by polymer threads sticking out of the particles into the solvent_ The PCS experiments were performed with an Ar ion laser operating at 514.5 nn at low power. The laser beam was focused in a sample cell placed in a sample holder at a temperature of 25.0 + O.l°C. The scattered light was detected by a photomultiplier at scattering angles between 15' and 135'_ The photon pulses from the photomultiplier were processed in a Malvern (type K 7025) photon correlator to obtain the autocorrelation function :(K,T) of the scattered intensity. By the standard method of cumulants (ref- 16), one can find from :(K,:) the value of the effective diffusion constant D,(K) as a function of K and the particle (weight) concentration_ Except for the highest concentration, ,:(K.T)is a single exponential_ In Figure I the results for latex 5 are shown. The values of D,(K=O) are found from a linear extrapolation to K=O
1
0
0.5
I
1.0
1
7.5
of the K* dependence of D,(K) for
1
1
2.0
2.5 K
-10 7
i
(z-n-‘1
Fig_ 1. The effective diffusion coefficient as a function of the scattering wave vector for latex 5. Lines are drawn to connect the appropriate data points-
166
values of K smaller than 1x10 the same K regime.
7
m
-I
_
Fig. 2 shows the results for latex 7 in
The results for latex 6 have been published before (ref. 4)
and look very similar to those of latex 7.
Also, latex 8 shows a similar K-
dependence of the effective diffusion constant as latex 7_
In Figure 1 one
clearly observes that for large K values and high particle concentration value of D,(K) is smaller than the corresponding theoretically
value of Do.
(ref- 10, II) expected and now also experimentally
-
the
This behavior is observed.
ncm-‘1
Fig_ 2. The effective diffusion coefficient versus wave vector of latex 7. From the PCS experiments one can find Do from experiments at very low concentration.
The results are given in Table 1.
Applying the Stokes-Einstein
relation
enzene) of the particles in benzene radius a to Do, one finds the hydrodynamic h (b (see also Table 1). The accuracy of D (and thus of the radius) is typically + 5%. The hydrodynamic radii in water have been found earlier (ref. 15) and for convenience are also added to Table 1 (accuracy 5 5%)_ The dependence of D,(K=O)
on the particle concentration can also be found from
the PCS experiments by extrapolating coefficient to K=O latex 5 (at K=O
the K dependence of the effective diffusion
at a given concentration.
and K=23.6
x lo6 m-l)_
Figure 3 shows the results for
Figure 4 shows the ratio De/Do as a
function of the concentration c for latexes 6, 7 and 8.
The experimental data
in Figures 3 and 4 are fitted according to: D,(K=O)
(11)
= Do(lfoDc)
by means of a least squares fit. The results for uD are given in Table I_ in Table 1 the slope of De(K=23.6x106 is given and indicated by aD(K)_
Also
m-')/Do as a function of c for latex 5
It should be noted that Do. as found from the
167
TABLE 1 5ummary of experimental the text. Latex Sample
data. The meaning of the various symbols is explained in
Frepared without emulsifier
Do/1D-12 (m2 S-l)
1.73
Prepared with sodium dodecyl sulfate as the emulsifier -_2.9 3.7 2.5
ah(benzene)
210
120
97
139
106
56
41
75
Ci D(cm3 g-l)
1421
24L3
3223
3.8t1.2
aD(K) (cm3 g -1)
-17+1 -
So/lo-1O (5)
3.250-2
l-23+0-02
0_40~0_01
3-2~0-2
(cm3 g-I)
-(56+6)
-(45+3)
-(67+4)
-(20+3)
Y/lo8 (g mol-l)
16
3.5
0.91
11
qh (cm3 g-l)
15
12
25
6.4
1.0~0.2
2.1+0.4
l-3+0.2 -
l-4+0.2
-3.6&O-5
-2.620.4
-3.3+0_5
ah(water)
'-%
(nm)
(rim)
"D SD(K)
-1_2+0.2
s
-3_6+0_5
S
2 2-12 2 -1 De/10 m s
1.7,
P
LATEX
C1.8-
0
I
0.01
5
I
0.02 C (gcm31
Fig_ 3. The effective diffusion coefficient of latex 5 as a function of the weight concentration for two different values of the wave vector-
168
LATEX
0.8 -
IN
BENZENE
0.L -
0 0
1
Fig. 4. Values of De/Do at scattering wave vector zero for latexes 6, 7 and 8 as functions of weight concentrations. extrapolations of De(K=23.6x10
of D (K=O) as a function of c and as found from the extrapolation 6 e-l m ) as a function of c to c=O are the same within experimental
accuracy_ Sedimentation measurements were carried out in a Beckman Spinco model E analytical ultracentrifuge.
The measured sedimentation
constants s are given
ures 5 and 6 as functions of the weight concentration; constant
at
infinite
dilution.
s
in Fig-
is the sedimentation
The observed data are fi&ed
to the relation:
s = so (1
(12)
The resulting values for so and c~s are given in Table 1.
LATEX
Fig. 5. Sedimentation coefficient of latex drawn curve connects data points.
5
5 versus weight
concentrrtion.
The
LATEX
IN
, 1
BENZENE
I3
I.
. D
Ol
1
0
-
$0-2gcd3
Fig. 6. Values of the relative sedimentation as functions of the weight concentrations. V.
169
coefficient for latexes 6, 7 and 8
DISCUSSION The value of aD (see Table 1) found from the PCS experiments
a71 latexes. is
that
The physical
meaning
of
the
effective
diffusion
coefficient
in this K limit the diffusion coefficient observed from
diffusion coefficient
(ref. II).
is positive for PCS
at K= 0
is the mutual
Positive values of fL mean that the mutual D in the concentration range
diffusion coefficient increases with concentration observed.
This phenomenon must be attributed to a dominating repulsive inter-
action between the latex spheres since the effect of repulsions is to push the particles away from each other faster than by thermal agitation alone (the driving force for diffusion at infinite dilution)_
Although the hydrodynamic
friction
also increases with concentration, which results in slowing down the separation speed between the particles, this increased friction cannot reverse the sign of the relative motion induced by the interaction forces_ crease in the mutual For values
of
diffusion
Kah’>l
diffusion coefficient.
Thus one expects an in-
coefficient.
one expects to observe that D,(K) approaches the selfIn this case the influence of direct interactions is
averaged out: the term SI(K) of Eq. 5 becomes zero. But still the hydrodynamic friction increases with increasing particle concentration. Therefore, one now expects a decrease in the effective diffusion coefficient. clearly observed with latex 5 in Figures 1 and 3.
This behavior is
The radii of the other latexes
are too small to observe the expected phenomenon of self-diffusion
in the available
K rangeThe increase of the hydrodynamic
friction in the
case
of
self-diffusion
as
observed from PCS is due to the contribution H1 of Eq_ 6, since H1 is independent of K and HZ(K) becomes zero for sufficiently high K values. hydrodynamic friction as observed from sedimentation contains both HI and Hp(K=O).
The increase in the
experiments
Clearly both HI and H2(K=O)
(Figures 5 and 6)
are negative, which
agrees with the idea of increasing hydrodynamic drag with increasing particle concentration.
To compare the experiments with theoretical predictions, one has to convert the weight concentration c of the particles into the hydrodynamic volume fraction .t. Both quantities are t-elatedby (ref. 8): (13) where N is Avogadro's number and M is the molar mass of the latex. The value or F?can be found from the Svedberg equation: II= soRTIDo(l-&)1-I
,
(14)
whet-eR is the gas constant, V a partial specific volume defined in Ref. 3 and : is the solvent density. The molar masses for latexes 6, 7 and 8 are taken from Ref. 3, whereas the result for lat-x 5 is found from so and Do from Table 1 and from c and r values used in Ref. 3. The results for M are given in Table 1. The accuracy in M is about lo:,. The value of (I,, (Eq. 13) can be calculated now directly and is also given in Table 1. The experimental data as functions of c can now be converted into function of :. Therefore, we writer D,(K=O) = Do (I+:',&
D,(K) = K. (I+ZD(K););
;
(15)
(16) (17)
From the numerical values of qh, :tD..-,D(K) and ccs,the corresponding values of SD' eD(K) and ss are calculated and given in Table 1. The results for the diffusion experiments show that latexes 5, 7 and 8 have about the same value of SD. whereas latex 6 shows a somewhat higher value. In the case of sedimentation, the concentration effects of latex 7 show a Small deviation from the other latexes. The results for latex 5 are especially suited to further consideration. It is a reasonable assumption that the K value at which 6,,(K)is evaluated can be taken as the limit K-x- in view of the accuracy of s,(K), which means that HI = sD(K) = -1.2. From this result, it immediately follows that H2(K=O) = es- ED(K). Comparison of Eqs. 4, 16 and 17 shows that S1(K=O) = sD(K=O)- ss_ The experimental values for HI, H2(K=O) and SI(K=O) are given in Table 2. In the same table the results for hard sphere interacting particles are also given (ref. 7, 11). One clearly observes the large deviations between latex 5 and hard sphere behavior_ The differences are most pronounced for H, and S_. L I
171
TABLE 2 Interaction characteristics. See Eqs. 4, 8 and 9. --________ Hl Hard sphere
Latex 6 Latex 7 Latex 8
_H2(K=0) -2.4+0_5 -4.71
-1.2+0.2 -1.73
Latex 5
--3.6+0.5 -2.6+0.4 -3.3+0.5 ---__-------__._____.____-.-
-_ ----_SI(K=O) --__-~---.-4.5+0.5 _ 8
HI+H2(K=O) ~~~~~-_-__--~-~~~_~_~_~~~ 5.7 +0.6 3.9+0.5 4.7-to.5 _ _.--_- ._._ -_-
-.__ _._--._
In Table 2 the experimental values of HI+H2(K=O) = Gs and SI for latexes G, 7 and 8 are also given. These latexes also show lower values of SI(K=O) and less negative values for HI+H2(K=0) compared to hard spheres. Not only the K= 0 and K=asalimits deviate from hard sphere behavior, but the overall shape of the K dependence of O,(K) also deviates very clearly from the hard sphere predictions. To illustrate this phenomenon, we have campared the experimental results of D,(K) for one concentration of latex 5 with the hat-dsphere K dependence (Figure 7). To this purpose we have adjusted the theoretical hard sphere curve shape with the experimental curve by letting them coincide at the two points K=O and K=Ko, where K, is defined by De(Ko) =Do. If instead of the coincidence at K=O the volume fraction as a fitting parameter is used, then the theoretical curve can be found from the drawn curve by the operation iDe(K)/Do-i: x 1.56 f 1. It is interesting to consider the K. value in more detail. For hard spheres one observes from computer calculations that KoaHS= 1.08, where aHS means the hard sphere radius. If aHS is determined from the experiments (assuming thus that the latex particles are hard spheres), we find values of aHS always smaller than the hydrodynamic radii of the particles. The ratio aHS/ah decreases Somewhat with increasing volume fraction. The ratio is about 0.7 for latexes 5, 7 and 8 and about 0.6 for latex 6.
0.6 0.7
C6-
* *
‘-
0
OS
1.0 -e
15 20 Kvt-r~l
25x107
Fig. 7. Comp.arisonbetween hard sphere and experimental K-dependence for one concentration of latex 5.
From the foregoing it is very clear that none of the latexes behaves as a hard sphere and that there are small differences between the different factors than hard sphere model potentials have to be considered. to consider as the next step a hard sphere core potential with an repulsive disturbance. In view of the microgel type structure of we have first tried a model potential of the triangle type:
latexes. Other It is customary to attractive or the particles,
=mxcazZaHS U(x)/kBT
= .(,1-x/,)/(;:-I), c .: < <:c = 0 x,;:o : 2 ah
(18)
9
where U(X) iS the pair potential between tbJ0 particles with CenterS a distance X apart; k8 is Boltzmann's constant; T the temperature_ :Iis taken larger than 1 ant +cis the hard core diameter of the particles. The relation between U(x) and g(x) is: g(x) = exp I-U(x)/k8Tl _
: 14)
ihe physical meaning of the potential (Eq. I8) is that the particles can be pressed together up to the hard core. But the energy of repulsion increases linearly with the decrease in the particles' center separation x in the interval (c,:G)_ Thus E in Eq. I8 is positive_ The use of this potential VJaS also suggested by the value of SI(K=O) in Table 2_ SI being smaller than 8 means that the repulsion is less than the hard sphere type: aHS
with E>O. does not give a good description of the experiments. However, the same tendency as with the triangle potential was observed: a relatively small hard core is needed to make HI less negative than -1.73. We also considered hard sphere core repulsion with an attractive disturbance. The physical meaning of this assumption is the existence of a London-van der Waals attraction between particles together with an excluded volume repul-sion. If the
173
hard sphere repulsion starts at particle separation ~12
ah and the attraction
exists beyond that separation, no agreement between theory and experiment can be obtained. The exact shape of the attraction as a function of the separation is not important here. Although the experimental value of SI(K=O) can be obtained easily from such a pair potential, it appears that the value of HI is always more negative than without attraction, where it has the pure hard sphere value of -1.73. Vi. CONCLUSIONS The concentration dependence of sedimentation and PCS experiments contains detailed information about the interactions between latex particles. As far as the K-dependence of the effective diffusion coefficient is concerned, the results are in qualitative agreement with the theoretical predictions about the mutualand self-diffusion concepts. Going from small to large values of the scattering wave vector K, one observes in the effective diffusion coefficient a gradual transition from mutual- to self-diffusion. The K-dependence contains detailed information about the particle interactions. These interactions cannot be described by triangle or square step repulsions with a hard sphere core inside the particle. Neither can the interaction be described by an excluded volume repulsion in combination with an attraction starting at particle separations larger than the hydrodynamic particle diameter. A method to find the detailed interaction potential is the subject of further investigationThe combination of sedimentation with PCS experiments enables one to distinguish between two terms containing the hydrodynamic interactions: HI and H2(K=O). The absolute value of HI is not very sensitive to the type of interaction, contrary to the value of ti2(K=O). ACKNOWLEDGEMENTS VII. The authors are very indebted to H. Mos and C_ Pathmamanoharan for their assistance in the latex preparation and in the PCS experiments. VIII. REFERENCES 1 R-H. Ottewill, Progr. Colloid Polymer Sci., 59(1976)14. 2 K. L. Hittal, ed., Micellization, Solubilization and Micro-emulsions, Vol. 2, Plenum Press, Nev~York.1977_ 3 E-A. Nieuwenhuis, C. Pathmamanoharan and A. Vrij, J. Colloid Interface Sci., 81(1981)196. 4 H-M. Fijnaut, C. Pathmamanoharan, E.A. Nieuwenhuis and A. Vrij, Chem. Phys. Lett., 59(1978)351. 5 G.K. Batchelor, J. Fluid Mech., 52(1972)245_ 6 G.K. Batchelor, J. Fluid Mech., 74(1976)1_ 7 B.U. Felderhof, J. Phys. A: Math. Gen., 11(1978)929. 8 M.M. Kops-Werkhoven and H.M. Fijnaut, J. Chem. Phys-, 74(1981)1618. 9 B-J. Berne and R. Pecora, Dynamic Light Scattering, Wiley, New York, 197610 W-B. Russel and A-B. Glendinning, J. Chem. Phys., 74(1981)948. 11 H.M. Fijnaut, J_ Chen. Phys., 74(1981)6857. 12 J-P. Hansen and I_!!_McDonald, Theory of Simple Liquids, Academic Press, London, 1976.
174
13 H. Fujita, Foundation of Cltracentrifuga? Analysis, Wiley, New York, 1975. 14 A_ Vrij, E.A. Nieuwenhuis, H-M. Fijnaut and W.G.t-l_ Agterof, Disc. Faraday Sot_, 65(1978)101. 15 E-A_ Nieuwenhuis, C. Pathmamonoharan and A. Vrij., Progr. Colloid Polym. Sci., 67(1980)85_ 16 P-N. Pusey, in Photc Correlation and Light Beating Spectroscopy, H.Z. Cummins and E-P,.Pike, eds., p_ 387, Plenum, New York, 1973.
161
in The Netherlands
PHDTCN CORRELATION AND SEDIllENTATION EXPERIMENTS ON INTERACTING PMMA PARTICLES IN BENZENE
H.Fl. FIJNAUT, J.K.G. DHONT and E-A. NIEUWENHUIS Van't Hoff Laboratory for Physical and Colloid Chemistry, Transitorium
3, Padualaan
3, 3584 CH Utrecht, The Netherlands
CONTENTS I.
AGSTRACT
II.
INTRODUCTION
. . . .. ... .. . . . . .. . ..._________________________________________161
III_
THEORY
IV.
EXPERIMENTS
V.
DISCUSSION
.. . . . .._.___......._____________._________________~______~6l
.____...____.._..__._________.______.___...____.___.__._.._._._16~ . . ..._........._....__.______~___~~___~~~__.~~____________16~ . ...___....__..____________________________________.___.___l69
VI.
CONCLUSIONS
VII.
ACKNOWLEDGEMENTS
__..___._____._._____________._.___.........................173
VIII.
REFERENCES
I.
ABSTRACT
__...__.______._.._.____~___~~___~~~___~____.~_______173
. .. .. . .. . . . .. . .._...~___.__~_.~_~~.~._~.~~~~~~~.~~~~_~~_.__173
Photon correlation
(PCS) experiments on interacting
PMHA (polymethylmethacrylate)
latexes show the existence of static (thermodynamic) and hydrodynamic The interactions are observed through the measurement
interactions.
of an effective diffusion
coefficient as a function of concentration and scattering wave vector.
For the
largest particles the wave vector dependence of the effective diffusion coefficient confirms theoretical predictions that from PCS experiments mutual and self-diffusion coefficients can be observed_
The combination of PCS experiments with sedimentation
experiments enables the separation of hydrodynamic and static interactions.
The
hydrodynamic interactions contain two physical significant terms which can be deduced separately only by the combination of PCS experiments at large wave vectors with the sedimentation experiments._ The combined experimental data show that the interaction is too complicated to be described by simple model potentials of hard sphere repulsion with either an attractive or a repulsive disturbance. II_
INTRODUCTION Nowadays the study of colloidal interactions is subject to a large number of
theoretical and experimental
investigations.
Part of the work is concerned with
colloidal suspensions as, for example, electrically OOOl-8686/82/0000-00001$03.500
charged particles in water
1982Elsevier ScientificPublishingCompany
(ref. 11, microemulsion systems (ref. 2) and sterically stabilized particles in non-polar solvents (ref. 1, 3). These systems are, among other things, intensively studied by means of light scattering techniques and, to a somewhat lesser extent, by means of ultra centrifugal sedimentation. The ligiitscattering methods include time-averaged intensity, photoncorrelationspectroscopy (PCS) and forced Rayleigh scattering. Time-averaged light scattering gives information about particle interactions and possible (static) structures as a result of these interactions. PCS and forced Rayleigh scattering give information about interaction effects on the particle dynamics_ Sedimentation experiments can give the concentration dependence of the friction coefficient of the suspended particles_ In this paper we will deal with PCS and sedimentation experiments on polymethylaethacrylate (PUMA) latexes dispersed in benzene. The investigations are concerned +lithsuspensions of latex volume fractions up to 25 volume percent. Some of the latex samples have been studied earlier by means of time-averaged light scattering (ref- 3, 4). The concentration regime studied is such that, apart from static interactions, hydrodynamic interactions also play an important role in the particle dynamics. This study, therefore, is intended to increase our understanding of the hydrodynamic interactions from an experimental point of view_ Theories about hydrodynamic interactions, resulting in concentration dependent friction and viscosity, appear to be very complicated_ Nowadays only pair interfor PCS actions are reasonably well understood (ref. 5, 6, 7). The predictions (ref. 7) and friction (ref. 5) based on this pair interaction has only been partly experimentally verified (ref. 8) and is limited to the zero angle regime in light scattering ("forward" scattering)_
The aim of this work is: 1) to investigate the angle or wave vector (see next section) dependence of PCS of hydrodynamically interacting systems; 2) to study the concentration dependence of the friction coefficient; and 3) to explain the nutually related results in terms of particle pair interactionsTHEORY III. In light scattering experiments the scattered light is observed at an angle d with respect to the incident direction of radiation of wavelength 1 in the scattering medium. The dependence of the scattering on A and 2 is expressed by the wave vector K with length: K = (47/l) sin (s/2).
(1)
In PCS experiments the temporal autocorrelation function p(K,r) is observed (refg), which is defined by:
163
:(#,:)=
I(K, t + T)>
9
(2)
where I(K,t) is the K and time-dependent scattered intensity_ The scattered intensity is supposed to be a stationary quantity_ From r(K,r) one can, apart from a proportionality constant, derive the dynamic structure factor S(K;:) (ref. 9) which in fact is the normalized temporal autocorrelation function of the scattered field. For a system of monodisperse, identical and spherical particles suspended in a structureless solvent S(K,:) is given by (ref. 10, II): S(K,r) = S(K) exp [- K2De(K)y\ _
(3)
Here S(K) is the so called static structure factor which can be obtained from the time-averaged scattered intensity. S(K) reflects the influence on the scatterinS of static particle interactions, which may induce local structure in the suspension (ref. 12). D,(K) stands for the effective diffusion coefficient of the suspended particles_ D,(K) depends on the concentration and can be a function of K. To first order in the hydrodynamic volume fraction : of the particles, De(K) can be written as: De(K) = Do [l+:
'S,(K)+HI+H2(K):]
,
(4)
Where Do is the diffusion coefficient at infinite dilution_ The term SI(K) reflects the influence of the static interactions on De(K), whereas HI and HZ(K) contain the hydrodynamic contribution to the dependence of the effective diffusion coefficient on K and the concentration. Recently one of us (ref. 11) has given explicit expressions for SI, HI and H2 based on a theory of Felderhof (ref. 7): SI(K) = i3/(Ka)3/"x sin x Cl-g(x/K)> dx
(5)
;
0 <”
H1 = (27/8) a3
Tg(x)/x4?dx- (15/4)a Ig(x)/x2: dx I I 0
;
(6)
0
H2 = C(9/2)/(Ka)2; (sin x + cos x/s - sin x/x2) :g(x/K)-1:dx / 0
4 (3 sin x/x - 3 cos x/*3 -
+3
sin
x/x2)g(x/K) dx
/
0 +
(225/4)(Ka)4 / 0
(sin x/x6+2
cos
x/x7-
2 sin x/x8)g(x/K) dx,
(7)
164 where a is the radius of the particle and g(x) is the zero order radial distribution function (ref. 12). It is worthwhile to mention the relation between S(K) and SI(K) (ref. 11): S(K) = I - rSl(K) , which is valid to the first order in 2. In sedimentation experiments the sedimentation constant s is observed (ref. 13)_ Expressions for the concentration dependence of s have been derived by Batchelor (ref_ 5) and by Felderhof (ref. 7)_ Using our notation one has: s = so[L+,fHI+H2(K=0)l]
=
(9)
where s
is the sedimentation constant at infinite dilution_ Consgdering the expressions (Eqs. 4, 8 and 9), one observes how closely the concentration dependence of time-averaged light scattering, PCS and sedimentation experiments are related, especially in the limit of zero K. In fact at K=O the relation between De(K=il)/DO, s/so and S(K=O) is simply the generalized StokesEinstein relation: D,(K=O)
(10) DO
which can be easily verified from the relation between the sedimentation constant, sedimentation velocity and friction coefficient (ref_ 5) and from the relation between S(K=O) and the osmotic compressibility (ref_ 14). For KPO one can find more detailed information about hydrodynamic interactions from PCS experiments in combination with sedimentation and static light scattering results. This possibility is the subject of this and future investigations. It should be mentioned here that the application of the expressions given above in the experimental situation is not always simple because of uncertainties in the relation between volume fraction and particle concentration (ref. 8). The hydrodynamic volume fraction of suspended particles can be quite different from the volume fraction calculated on the basis of the density and the weight concentration of these particles, especially if swelling occurs of the particles when they are brought into the solvent EXPERIMENTS IV_ The PMMA latex particles were prepared by emulsion polymerization in water (ref_ 15)_ Samples 6. 7 and 8 were prepared with sodium dodecylsulfate as the emulsifier, whereas sample 5 was prepared without emulsifier. In all cases ethylene glycol dimethacrylate (5 weight percent of monomer) was added to the
165 reaction mixture to build a three-dimensional network inside the particle_ For further details, see the work of Nieuwenhuis et al. (ref. 15), who used the same sample indication as we do. After preparation in water, the dispersions were dialyzed and ion-exchanged to remove residual monomer and ionic impurities_ The dried latex was easily dispersible in benzene and some other apolar solvents. The samples were sedimented 3 or 4 times to remove the linear polymer_ Upon dispersing the particles in benzene, the Pf+lAparticles swelled with respect to their dimensions in water. In water, the particles were compact spheres stabilized by an electric double layer. In benzene, however, they were swollen microgels sterically stabilized by polymer threads sticking out of the particles into the solvent_ The PCS experiments were performed with an Ar ion laser operating at 514.5 nn at low power. The laser beam was focused in a sample cell placed in a sample holder at a temperature of 25.0 + O.l°C. The scattered light was detected by a photomultiplier at scattering angles between 15' and 135'_ The photon pulses from the photomultiplier were processed in a Malvern (type K 7025) photon correlator to obtain the autocorrelation function :(K,T) of the scattered intensity. By the standard method of cumulants (ref- 16), one can find from :(K,:) the value of the effective diffusion constant D,(K) as a function of K and the particle (weight) concentration_ Except for the highest concentration, ,:(K.T)is a single exponential_ In Figure I the results for latex 5 are shown. The values of D,(K=O) are found from a linear extrapolation to K=O
1
0
0.5
I
1.0
1
7.5
of the K* dependence of D,(K) for
1
1
2.0
2.5 K
-10 7
i
(z-n-‘1
Fig_ 1. The effective diffusion coefficient as a function of the scattering wave vector for latex 5. Lines are drawn to connect the appropriate data points-
166
values of K smaller than 1x10 the same K regime.
7
m
-I
_
Fig. 2 shows the results for latex 7 in
The results for latex 6 have been published before (ref. 4)
and look very similar to those of latex 7.
Also, latex 8 shows a similar K-
dependence of the effective diffusion constant as latex 7_
In Figure 1 one
clearly observes that for large K values and high particle concentration value of D,(K) is smaller than the corresponding theoretically
value of Do.
(ref- 10, II) expected and now also experimentally
-
the
This behavior is observed.
ncm-‘1
Fig_ 2. The effective diffusion coefficient versus wave vector of latex 7. From the PCS experiments one can find Do from experiments at very low concentration.
The results are given in Table 1.
Applying the Stokes-Einstein
relation
enzene) of the particles in benzene radius a to Do, one finds the hydrodynamic h (b (see also Table 1). The accuracy of D (and thus of the radius) is typically + 5%. The hydrodynamic radii in water have been found earlier (ref. 15) and for convenience are also added to Table 1 (accuracy 5 5%)_ The dependence of D,(K=O)
on the particle concentration can also be found from
the PCS experiments by extrapolating coefficient to K=O latex 5 (at K=O
the K dependence of the effective diffusion
at a given concentration.
and K=23.6
x lo6 m-l)_
Figure 3 shows the results for
Figure 4 shows the ratio De/Do as a
function of the concentration c for latexes 6, 7 and 8.
The experimental data
in Figures 3 and 4 are fitted according to: D,(K=O)
(11)
= Do(lfoDc)
by means of a least squares fit. The results for uD are given in Table I_ in Table 1 the slope of De(K=23.6x106 is given and indicated by aD(K)_
Also
m-')/Do as a function of c for latex 5
It should be noted that Do. as found from the
167
TABLE 1 5ummary of experimental the text. Latex Sample
data. The meaning of the various symbols is explained in
Frepared without emulsifier
Do/1D-12 (m2 S-l)
1.73
Prepared with sodium dodecyl sulfate as the emulsifier -_2.9 3.7 2.5
ah(benzene)
210
120
97
139
106
56
41
75
Ci D(cm3 g-l)
1421
24L3
3223
3.8t1.2
aD(K) (cm3 g -1)
-17+1 -
So/lo-1O (5)
3.250-2
l-23+0-02
0_40~0_01
3-2~0-2
(cm3 g-I)
-(56+6)
-(45+3)
-(67+4)
-(20+3)
Y/lo8 (g mol-l)
16
3.5
0.91
11
qh (cm3 g-l)
15
12
25
6.4
1.0~0.2
2.1+0.4
l-3+0.2 -
l-4+0.2
-3.6&O-5
-2.620.4
-3.3+0_5
ah(water)
'-%
(nm)
(rim)
"D SD(K)
-1_2+0.2
s
-3_6+0_5
S
2 2-12 2 -1 De/10 m s
1.7,
P
LATEX
C1.8-
0
I
0.01
5
I
0.02 C (gcm31
Fig_ 3. The effective diffusion coefficient of latex 5 as a function of the weight concentration for two different values of the wave vector-
168
LATEX
0.8 -
IN
BENZENE
0.L -
0 0
1
Fig. 4. Values of De/Do at scattering wave vector zero for latexes 6, 7 and 8 as functions of weight concentrations. extrapolations of De(K=23.6x10
of D (K=O) as a function of c and as found from the extrapolation 6 e-l m ) as a function of c to c=O are the same within experimental
accuracy_ Sedimentation measurements were carried out in a Beckman Spinco model E analytical ultracentrifuge.
The measured sedimentation
constants s are given
ures 5 and 6 as functions of the weight concentration; constant
at
infinite
dilution.
s
in Fig-
is the sedimentation
The observed data are fi&ed
to the relation:
s = so (1
(12)
The resulting values for so and c~s are given in Table 1.
LATEX
Fig. 5. Sedimentation coefficient of latex drawn curve connects data points.
5
5 versus weight
concentrrtion.
The
LATEX
IN
, 1
BENZENE
I3
I.
. D
Ol
1
0
-
$0-2gcd3
Fig. 6. Values of the relative sedimentation as functions of the weight concentrations. V.
169
coefficient for latexes 6, 7 and 8
DISCUSSION The value of aD (see Table 1) found from the PCS experiments
a71 latexes. is
that
The physical
meaning
of
the
effective
diffusion
coefficient
in this K limit the diffusion coefficient observed from
diffusion coefficient
(ref. II).
is positive for PCS
at K= 0
is the mutual
Positive values of fL mean that the mutual D in the concentration range
diffusion coefficient increases with concentration observed.
This phenomenon must be attributed to a dominating repulsive inter-
action between the latex spheres since the effect of repulsions is to push the particles away from each other faster than by thermal agitation alone (the driving force for diffusion at infinite dilution)_
Although the hydrodynamic
friction
also increases with concentration, which results in slowing down the separation speed between the particles, this increased friction cannot reverse the sign of the relative motion induced by the interaction forces_ crease in the mutual For values
of
diffusion
Kah’>l
diffusion coefficient.
Thus one expects an in-
coefficient.
one expects to observe that D,(K) approaches the selfIn this case the influence of direct interactions is
averaged out: the term SI(K) of Eq. 5 becomes zero. But still the hydrodynamic friction increases with increasing particle concentration. Therefore, one now expects a decrease in the effective diffusion coefficient. clearly observed with latex 5 in Figures 1 and 3.
This behavior is
The radii of the other latexes
are too small to observe the expected phenomenon of self-diffusion
in the available
K rangeThe increase of the hydrodynamic
friction in the
case
of
self-diffusion
as
observed from PCS is due to the contribution H1 of Eq_ 6, since H1 is independent of K and HZ(K) becomes zero for sufficiently high K values. hydrodynamic friction as observed from sedimentation contains both HI and Hp(K=O).
The increase in the
experiments
Clearly both HI and H2(K=O)
(Figures 5 and 6)
are negative, which
agrees with the idea of increasing hydrodynamic drag with increasing particle concentration.
To compare the experiments with theoretical predictions, one has to convert the weight concentration c of the particles into the hydrodynamic volume fraction .t. Both quantities are t-elatedby (ref. 8): (13) where N is Avogadro's number and M is the molar mass of the latex. The value or F?can be found from the Svedberg equation: II= soRTIDo(l-&)1-I
,
(14)
whet-eR is the gas constant, V a partial specific volume defined in Ref. 3 and : is the solvent density. The molar masses for latexes 6, 7 and 8 are taken from Ref. 3, whereas the result for lat-x 5 is found from so and Do from Table 1 and from c and r values used in Ref. 3. The results for M are given in Table 1. The accuracy in M is about lo:,. The value of (I,, (Eq. 13) can be calculated now directly and is also given in Table 1. The experimental data as functions of c can now be converted into function of :. Therefore, we writer D,(K=O) = Do (I+:',&
D,(K) = K. (I+ZD(K););
;
(15)
(16) (17)
From the numerical values of qh, :tD..-,D(K) and ccs,the corresponding values of SD' eD(K) and ss are calculated and given in Table 1. The results for the diffusion experiments show that latexes 5, 7 and 8 have about the same value of SD. whereas latex 6 shows a somewhat higher value. In the case of sedimentation, the concentration effects of latex 7 show a Small deviation from the other latexes. The results for latex 5 are especially suited to further consideration. It is a reasonable assumption that the K value at which 6,,(K)is evaluated can be taken as the limit K-x- in view of the accuracy of s,(K), which means that HI = sD(K) = -1.2. From this result, it immediately follows that H2(K=O) = es- ED(K). Comparison of Eqs. 4, 16 and 17 shows that S1(K=O) = sD(K=O)- ss_ The experimental values for HI, H2(K=O) and SI(K=O) are given in Table 2. In the same table the results for hard sphere interacting particles are also given (ref. 7, 11). One clearly observes the large deviations between latex 5 and hard sphere behavior_ The differences are most pronounced for H, and S_. L I
171
TABLE 2 Interaction characteristics. See Eqs. 4, 8 and 9. --________ Hl Hard sphere
Latex 6 Latex 7 Latex 8
_H2(K=0) -2.4+0_5 -4.71
-1.2+0.2 -1.73
Latex 5
--3.6+0.5 -2.6+0.4 -3.3+0.5 ---__-------__._____.____-.-
-_ ----_SI(K=O) --__-~---.-4.5+0.5 _ 8
HI+H2(K=O) ~~~~~-_-__--~-~~~_~_~_~~~ 5.7 +0.6 3.9+0.5 4.7-to.5 _ _.--_- ._._ -_-
-.__ _._--._
In Table 2 the experimental values of HI+H2(K=O) = Gs and SI for latexes G, 7 and 8 are also given. These latexes also show lower values of SI(K=O) and less negative values for HI+H2(K=0) compared to hard spheres. Not only the K= 0 and K=asalimits deviate from hard sphere behavior, but the overall shape of the K dependence of O,(K) also deviates very clearly from the hard sphere predictions. To illustrate this phenomenon, we have campared the experimental results of D,(K) for one concentration of latex 5 with the hat-dsphere K dependence (Figure 7). To this purpose we have adjusted the theoretical hard sphere curve shape with the experimental curve by letting them coincide at the two points K=O and K=Ko, where K, is defined by De(Ko) =Do. If instead of the coincidence at K=O the volume fraction as a fitting parameter is used, then the theoretical curve can be found from the drawn curve by the operation iDe(K)/Do-i: x 1.56 f 1. It is interesting to consider the K. value in more detail. For hard spheres one observes from computer calculations that KoaHS= 1.08, where aHS means the hard sphere radius. If aHS is determined from the experiments (assuming thus that the latex particles are hard spheres), we find values of aHS always smaller than the hydrodynamic radii of the particles. The ratio aHS/ah decreases Somewhat with increasing volume fraction. The ratio is about 0.7 for latexes 5, 7 and 8 and about 0.6 for latex 6.
0.6 0.7
C6-
* *
‘-
0
OS
1.0 -e
15 20 Kvt-r~l
25x107
Fig. 7. Comp.arisonbetween hard sphere and experimental K-dependence for one concentration of latex 5.
From the foregoing it is very clear that none of the latexes behaves as a hard sphere and that there are small differences between the different factors than hard sphere model potentials have to be considered. to consider as the next step a hard sphere core potential with an repulsive disturbance. In view of the microgel type structure of we have first tried a model potential of the triangle type:
latexes. Other It is customary to attractive or the particles,
=mxcazZaHS U(x)/kBT
= .(,1-x/,)/(;:-I), c .: < <:c = 0 x,;:o : 2 ah
(18)
9
where U(X) iS the pair potential between tbJ0 particles with CenterS a distance X apart; k8 is Boltzmann's constant; T the temperature_ :Iis taken larger than 1 ant +cis the hard core diameter of the particles. The relation between U(x) and g(x) is: g(x) = exp I-U(x)/k8Tl _
: 14)
ihe physical meaning of the potential (Eq. I8) is that the particles can be pressed together up to the hard core. But the energy of repulsion increases linearly with the decrease in the particles' center separation x in the interval (c,:G)_ Thus E in Eq. I8 is positive_ The use of this potential VJaS also suggested by the value of SI(K=O) in Table 2_ SI being smaller than 8 means that the repulsion is less than the hard sphere type: aHS
with E>O. does not give a good description of the experiments. However, the same tendency as with the triangle potential was observed: a relatively small hard core is needed to make HI less negative than -1.73. We also considered hard sphere core repulsion with an attractive disturbance. The physical meaning of this assumption is the existence of a London-van der Waals attraction between particles together with an excluded volume repul-sion. If the
173
hard sphere repulsion starts at particle separation ~12
ah and the attraction
exists beyond that separation, no agreement between theory and experiment can be obtained. The exact shape of the attraction as a function of the separation is not important here. Although the experimental value of SI(K=O) can be obtained easily from such a pair potential, it appears that the value of HI is always more negative than without attraction, where it has the pure hard sphere value of -1.73. Vi. CONCLUSIONS The concentration dependence of sedimentation and PCS experiments contains detailed information about the interactions between latex particles. As far as the K-dependence of the effective diffusion coefficient is concerned, the results are in qualitative agreement with the theoretical predictions about the mutualand self-diffusion concepts. Going from small to large values of the scattering wave vector K, one observes in the effective diffusion coefficient a gradual transition from mutual- to self-diffusion. The K-dependence contains detailed information about the particle interactions. These interactions cannot be described by triangle or square step repulsions with a hard sphere core inside the particle. Neither can the interaction be described by an excluded volume repulsion in combination with an attraction starting at particle separations larger than the hydrodynamic particle diameter. A method to find the detailed interaction potential is the subject of further investigationThe combination of sedimentation with PCS experiments enables one to distinguish between two terms containing the hydrodynamic interactions: HI and H2(K=O). The absolute value of HI is not very sensitive to the type of interaction, contrary to the value of ti2(K=O). ACKNOWLEDGEMENTS VII. The authors are very indebted to H. Mos and C_ Pathmamanoharan for their assistance in the latex preparation and in the PCS experiments. VIII. REFERENCES 1 R-H. Ottewill, Progr. Colloid Polymer Sci., 59(1976)14. 2 K. L. Hittal, ed., Micellization, Solubilization and Micro-emulsions, Vol. 2, Plenum Press, Nev~York.1977_ 3 E-A. Nieuwenhuis, C. Pathmamanoharan and A. Vrij, J. Colloid Interface Sci., 81(1981)196. 4 H-M. Fijnaut, C. Pathmamanoharan, E.A. Nieuwenhuis and A. Vrij, Chem. Phys. Lett., 59(1978)351. 5 G.K. Batchelor, J. Fluid Mech., 52(1972)245_ 6 G.K. Batchelor, J. Fluid Mech., 74(1976)1_ 7 B.U. Felderhof, J. Phys. A: Math. Gen., 11(1978)929. 8 M.M. Kops-Werkhoven and H.M. Fijnaut, J. Chem. Phys-, 74(1981)1618. 9 B-J. Berne and R. Pecora, Dynamic Light Scattering, Wiley, New York, 197610 W-B. Russel and A-B. Glendinning, J. Chem. Phys., 74(1981)948. 11 H.M. Fijnaut, J_ Chen. Phys., 74(1981)6857. 12 J-P. Hansen and I_!!_McDonald, Theory of Simple Liquids, Academic Press, London, 1976.
174
13 H. Fujita, Foundation of Cltracentrifuga? Analysis, Wiley, New York, 1975. 14 A_ Vrij, E.A. Nieuwenhuis, H-M. Fijnaut and W.G.t-l_ Agterof, Disc. Faraday Sot_, 65(1978)101. 15 E-A_ Nieuwenhuis, C. Pathmamonoharan and A. Vrij., Progr. Colloid Polym. Sci., 67(1980)85_ 16 P-N. Pusey, in Photc Correlation and Light Beating Spectroscopy, H.Z. Cummins and E-P,.Pike, eds., p_ 387, Plenum, New York, 1973.