Langmuir 1994,10, 3550-3554
3550
Sterically Stabilized Colloidal Particles as Model Hard Spheres S. M. Underwood, J. R. Taylor, and W. van Megen* Department of Applied Physics, Royal Melbourne Institute of Technology, Melbourne 3000, Australia Received March 15, 1994. I n Final Form: June 16,1994@ In this paper we show that the same freezing-melting miscibility gap can be identified for suspensions of spherical polymer particles of radii 138-440 nm sterically stabilized in all cases by the same coating of a different polymer. Diffusion coefficients and sedimentation velocities of the colloidal fluid at the freezing concentration scale with that power of the particle radius expected for hard spheres. These observations demonstrate that these suspensions constitute a good experimental representation of the ideal hard-sphere system. 1. Introduction
Recognitionof the importance of the hard-sphere system as a reference system for condensed states of matter and for concentrated suspensions generally has led to extensive use of sterically stabilized colloidal particles, where the interactions approximate those of hard Common choices for model hard-sphere systems include particles of poly(methy1 methacrylate) (PMMA)2t3and silica4 coated with relatively thin surface layers of macromolecular material and suspended in various organic liquids. Variation of the optical contrast, achieved by adjusting the refractive index of the continuous liquid phase, allows multiply scattered light to be minimized and also allows the independent measurement of the coherent and incoherent intermediate scattering function~.~ A reliable and universal definition of the suspension concentration is of vital importance, particularly a t high particle concentrations. As a first approximation the volume fraction, 5, can be calculated from the known weight composition of the the sample using literature values for the densities ofthe components. However, this measure of the particle concentration does not include the solvated adsorbed layer for which neither the exact thickness nor density is generally known. Possible solvent imbibition by the particles is a further complication. Consequently, 5 is only a n estimate of the volume fraction of the dry particle cores. These unknown factors, in particular, the contribution of the solvated adsorbed layer, can be dealt with by expressing the sample concentrations as effective hard-sphere volume fractions, p,. Typically, either one of two procedures is selected to achieve this. The first equates the measured intrinsic viscosity with that given by the Einstein expression for a suspension of hard sphere^.^,' The second identifies the measured Abstract published in Advance ACS Abstracts, September 1, 1994. (1)Pusey, P. N.; van Megen, W.Nature 1986,320, 340. (2) Pusey, P. N. In Liquids, Freezing and the Glass Transition; Hansen, J. P., Levesque, D., Zinn-Justin, J., Eds.; Elsevier Science Publishers B. V.: Amsterdam, 1991. (3) Bartlett, P.; van Megen, W. In Granular Materials; Mehta, A., Ed.; Springer-Verlag: Berlin, 1993. ( 4 ) Smits, C.;van Duijneveldt,J. S.; Dhont, J. K. G.; Lekkerkerker, H. N. W.; Briels, W. J. Phase Transitions 1990,21, 157. (5) van Megen, W.;Underwood, S.M. Langmuir 1990, 6 , 35. (6) de Kruif, C.G.; van Iersel, E. M. F.; Vrij, A.; Russel, W. B. J . Chem. Phys. 1986,83, 4717. (7) Jones, D. A. R.; Leary, B.; Boger, D. V. J . Colloid Interface Sci. 1991, 147, 479. @
suspension freezing concentration with that known for the hard-sphere system.lr8 Despite the significant number of studies on suspensions of hard-sphere-like particles, the effect of deviations of the actual interactions from those of hard spheres on the suspension properties, such as the phase behavior, particle dynamics, and crystallization kinetics, is not well established. Whether the rheological studies of one “hardsphere” suspension and the dynamic light scattering studies of another apply to the same or comparable thermodynamic states is a n issue that appears not to have been addressed. There are significant differences, for example, between published estimates of the effective hard-sphere volume fraction where the kinetic glass transition occurs (p, * 0.57) and particle diffusion ceasesgJo and that obtained (p, % 0.64)for the volume fraction where the low-shear viscosity diverge~.~,’JlIf the measurements of particle diffusion and viscosity apply to the metastable amorphous colloidal fluid of effectively hard spherical particles, then the cessation of diffusion and the divergence of the low-shear viscosity should occur at the same volume fraction. I t is not clear whether these differences in concentration arise from experimental errors, such as those incurred by implementing the above definitions of the hard-sphere volume fractions, from differences in the particle size distributions, or from deviations from hardsphere behavior due to deformation of the adsorbed layers.12 The latter are more likely to occur a t very high concentrations, particularly for suspensions under shear. In this paper we address some of these questions by measuring the phase behavior and also diffusion coefficients of several PMMA latices with particle radii, R , from 138to 440 nm with comparable polydispersities and the same stabilizing coating. Our results show the following: (i) The effective hard-sphere freezing and melting volume fractions are independent of the relative thickness, 0.l.R to 0.03R,of the stabilizing coating. (ii) Effective hard-sphere volume fractions deduced from measurement of intrinsic viscosity are consistent with those obtained from the phase behavior. (iii) Collective diffusion coefficients and sedimentation velocities, measured on the colloidal fluid at freezing, scale with R and R2,respectively. (iv) Unexpectedly, a suspension with a polydispersity of 0.1 still shows the freezing-melting (8) Paulin, S. E.; Ackerson, B. J. Phys. Rev. Lett. 1990, 64, 2663. (9) van Megen, W.;Pusey, P. N. Phys. Rev. A 1991,43, 5429. (10)van Megen, W.; Underwood, S. M. Phys. Rev. Lett. 1993, 70, 2766. (11) Marshall, L.; Zukoski, C. F. J.Phys. Chem. 1990,94, 1164. ( 1 2 ) Mewis, J.; Frith, W. J.; Strivens, T. A.; Russel, W. B.AlCHE J . 1989,35, 415.
0743-746319412410-3550$04.50/00 1994 American Chemical Society
Sterically Stabilized Colloidal Particles
Langmuir, Vol. 10, No. 10, 1994 3551
Table 1 series designation
E D F G
Rh (nm)
R (nm)
u (DLS)
443f8 340f5 205f3 144f3
440f8 337f5 201f3 138f3
0.04 0.04 0.06 0.09
u (EM)
0.06 0.11
Rhs (nm) 455 f 4 336 f 3 201 f 2 138f1
a Radii: Rh is the hydrodynamic radius measured by DLS; R is the number-average hydrodynamic radius calculated from Rh (see text);Rhsis the effective hard-sphere radius. The solvent is decalid CS2 for all listed series; 29D series (in decalin, see Table 2 and text) was prepared from the same latex as the F series.
transition predicted for monosized hard spheres. We speculate on a mechanism to explain this observation. The Experimental Section of this paper covers descriptions of the sample preparation and characterization, phase behavior, viscosity measurements, sedimentation equilibrium, collective diffusion coefficients, and sedimentation velocities of the coexisting colloidal fluid and crystal. A discussion of the results is presented in section 3 and concluding remarks are presented in section 4. 2. Experimental Section 2.1. Sample Preparation and Characterization. The synthesis of PMMA particles stabilized by poly(l2hydroxystearic acid) (PHSA) in hydrocarbon media has been reported in detail e1~ewhere.l~ For the present study we prepared four ofthese latices with particles of different size but coated with the same PHSA stabilizer. Therefore in a given solvent (mixture) different sized particles have adsorbed layers of the same nominal thickness. The particle hydrodynamic radii, Rh, determined by dynamic light scattering (DLS) and polydispersities, or coefficient of variation of the particle size distribution (a), determined by both DLS and electron microscopy (EM), are listed in Table 1. Note that the first three latices listed have similar u but the last latex (used for the G series) has a broader size distribution. A further distinguishing feature of this broad particle size distribution, determined by EM, is its relatively large negative skewness of about -1.0, compared with -0.6 for the other latex ( F series) analyzed by this technique. For each latex the number-averaged hydrodynamic radius,_R_ , calculated from the intensity weighted radius, Rh = R61R5,obtained by DLS,14 is also tabulated. After determination of the weight fraction of polymer of a given stock suspension in decalin, about eight samples were prepared in 1cm path-length cuvettes and optically matched to turbidity values, t,of about 0.2 cm-' by the addition of carbon disulfide (CS2). Although lower turbidities can be achieved (see below), the value t % 0.2 cm-l ensures adequate single but negligible multiple scattering. The samples were then left for 24 h, to allow any solvent imbibition by the particles, before concentrating by centrifugation and removal of a weighed amount of supernatant. This was followed by gentle tumbling of the samples to redisperse the compacted sediments, after which they were left undisturbed for observation. Each series spanned the concentration range between the anticipated freezing and melting concentrations. One latex was also studied in decalin without optical matching, i.e. no added CS2 (29D series). The issue of CS2 imbibition by PMMA particles has been alluded to in several works but the kinetics has been studied in detail only by Ottewill and Livsey.15 These (13)Antl, L.; Goodwin, J. W.; Hill, R. D.; Ottewill, R. H.; Owens, S. M.; Papworth, S.; Waters, J. W. Colloids Surf. 1986,17, 67. (14)Pusey, P. N.; van Megen, W. J. Chem. Phys. 1984,80,3513. (15)Ottewill, R. H.; Livsey, I. Polymer 1987,28,109.
authors found that the overall imbibition process could be described by a relatively fast initial solvent uptake by the particles followed by a much slower relaxation of the polymer segments within the particles. Their finding that the total time scale of these processes was several hours is consistent with our own observations. In order to quantify the CS2 absorption, the weight of CS2 required for minimum turbidity was measured as a function of PMMA concentration. From this we estimated that the particles imbibe CS2 to approximately 6% their weight in optically matched suspensions. For minimum sample turbidity, typically 0.01 em-', the weight fraction of CS2 in the solvent was found to be about 0.30. 2.2. Phase Transition Studies. Crystallization by homogeneous nucleation observed in these samples was similar to that described in several earlier w o r k ~ . l - ~ The procedure used to determine the equilibrium freezing and melting concentrations followed that introduced by Paulin and Ackerson8 with two refinements. First, the cuvettes were sealed in a manner that minimized solvent evaporation (particularly CS2 in our case) and obviated the necessity to weigh the samples during the study, thus avoiding potential perturbation of the crystallization process. Weights measured a t the commencement and termination of each study showed that total weight loss over a period of a year was, a t worst, only 0.5% of the sample weight resulting in a n increase in volume fraction of less than 1%. Second, instead of measuring the crystal fractions using a cathetometer, the sets of samples were photographed; a white light source and camera were placed to observe the strongest Bragg reflections and give optimum distinction between colloidal crystal and fluid phases. Depending on the particle size and resulting sedimentation rate, photographs were taken at intervals that varied from about 10 h (E series, R = 440 nm) to several days (G series, R = 138 nm). Each series was photographically monitored until the crystals had formed and settled and the particles in the remaining colloidal fluid had settled to a gravitationally compacted crystal. At the completion of these processes the samples contained varying amounts of crystal, a thin diffuse layer of colloidal fluid, in which sedimentation equilibrium had apparently been attained, and clear supernatant. The required total observation period ranged from about 2 weeks (E series) to 8 months (G series). Analysis of the photographs yielded, for each sample, a plot of the fluid and crystal heights, Hf and H, respectively, as functions of time. Typical results are shown in Figure 1. The initial nonlinear sections are a consequence of the curved meniscus of the sample-air interface and competition between settling crystallites and the slower settling of the particles. The constant difference between Hf and H, a t long times (2450 h in Figure 1)provides a n estimate of the extent of the diffuse layer in which sedimentation equilibrium is reached. Following Paulin and Ackersona the gravity free fluid and crystal heights were obtained by extrapolating the linear sections of these plots (between about 100 and 400 h in Figure 1)to zero time. After allowing for the diffuse layer (see below), which is included in the initial measurement of Hf,we constructed plots, such as that in Figure 2, of percent crystal (by volume) versus core volume fraction. Freezing and melting core volume fractions, Ef and Em, were obtained by extrapolating the line of best fit to zero and 100% crystal, respectively. Multiplication of the core volume fractions, E, by the factor f = qd&, where ipf = 0.494 is the volume fraction a t which the perfect hardsphere fluid freezes,16 for each of the samples in a particular sequence gives their corresponding effective
3552 Langmuir, Vol. 10, No. 10, 1994
Underwood et al. 1.4
1
r
c v)
0 0
2> 1
:.
m
.-> -O
12
.I-
? I
0
200
400
800
600
time (hours) Figure 1. Heights of fluid phase (Hf, 0 )and crystal phase (Hc, m) as a function of time for a typical sample from the D series. 100
80
-0
,
I
/
i
c
F 60: 0
.I-
5
2m
40-
n
-
P
2ol 0 0.38
0.39
0.40
0.41
0.42
0.43
0.44
l Figure 2. Percent crystal by volume for the D series, corrected for effects of gravity and sedimentation equilibrium layer (see text) versus core particle volume fraction, 5. Table 2
E D F G 29D
1.22f 0.01 1.27f 0.01 1.31f 0.01 1.35f 0.01 1.25 ?c 0.01
0.544f 0.003 0.547 f 0.003 0.539f 0.007 0.548f 0.01 0.544f 0.003
0.11f 0.01 0.10f 0.01 0.12f 0.01 0.13f 0.01
0.018f 0.002 0.015f 0.002 0.016f 0.002 0.016 0.002
*
a f is the scaling factor required to convert volume fractions (6) calculated from dry weight fractions to effectivehard sphere volume fractions (q);q m is the experimental effective hard-spherevolume fraction at melting. DJDo and DdDo are the normalized short and long time diffusion coefficients, respectively, measured at the position, qm,of the main peak in the static structure factor in the fluid phase at the freezing volume fraction.
hard-sphere volume fractions, q. This procedure was followed for each of five series of samples and the corresponding factors f and ex erimental melting volume (=Emf) of the coloidal crystal are listed in fractions Table 2. e quantities q m are to be compared with the value 0.545 for the melting volume fraction of the hardsphere crystal.16 The relatively large uncertainty for the G series is due to the error in estimating the magnitude of the diffise sedimentation equilibrium layer (see below).
1
0
Figure 3. Relative viscosities versus core particle volume fraction, 5. The solid line is the quadratic line of best fit (see text).
As discussed earlier, the weight fraction of polymer in a stock suspension used to calculate E includes only the collapsed adsorbed layer, whereas a t least in principle q includes the solvated layer. However, due in part to the imbibition of solvent by the particles, the factor f = q ~ / g applies only to a particular combination of stock latex composition, solvents, and procedure for dry weight analysis; we found that drying a weighed aliquot of stock latex under vacuum for 48 h at 140 "C provided the most reproducible dry weight fractions. Consequently, the quantity 6 = cf/" - 1)R,which one might interpret as a n adsorbed layer thickness, can be caused to vary from about 12 to 30 nm for a given latex through changes in these conditions. Furthermore, the polymer "cores"are not pure PMMA13but include up to 4% copolymerized methacrylic acid and they may also contain excess stabilizer trapped during polymerization. Thus the particle density even in the dry state cannot be assumed to be that ofbulk PMMA (1.19 g ~ m - ~ used ) in these analyses. For particles suspended in the solvent mixture the discrepancy is likely to be greater. The factor f incorporates all these uncertainties. Although 6 is not entirely inconsistent with the extended chain length of above five ester linkages reported for a similar PHSA ~tabilizer,~' it should not be interpreted as more than an indication of the adsorbed layer thickness. Note that the difference, about 5%, in the factor f for unmatched samples (29D series of particles in decalin alone) and matched samples (Fseries ofparticles in decalin and CS2) is consistent with the weight fraction of CS2 imbibed by the particles, discussed earlier. 2.3. Intrinsic Viscosities. Viscosities, 17, were measured with a Cannon Fenske Viscometer a t 20 f 0.01 "C using a k e d volume of dilute suspensions ( 5 from 0.01 to 0.1) of the 201-nm latex particles dispersed in decalin. Relative viscosities, vr = 17/70, where 170 is the viscosity of the solvent (decalin),were calculated and they are plotted as a function of 5 in Figure 3. The intrinsic viscosity of the suspension
was obtained by subjecting these data to quadratic regression. For a suspension ofhard spheres the intrinsic (16)Hoover, W. G.;Ree, F. H. J. Chem. Phys. 1988,49, 3609. (17)Cairns, R. J.R.; Ottewill, R. H.; Osmond, D. W. J.;Wagstaff, I. J. Colloid Interface Sei. 1976,54, 45.
Sterically Stabilized Colloidal Particles 6
T' E
U
I
I" = 2
Langmuir, VoE. 10, No. 10, 1994 3553 diffuse layer dH and hence the equivalent amount, or height dHf, of colloidal fluid at qfin the diffise layer. The small correction, dH - dHf,significant only for the smaller particles (F and G series), was then subtracted from the measuredHfto give the final fluid height used to determine the fraction of colloidal crystal shown in Figure 2. This assumes that sedimentation equilibrium is reached a t the interface of the colloidal fluid and supernatant while the colloidal fluid is still sedimenting, i.e. over the linear sections of Hf and H, versus time of Figure 1. 2.5. Diffusion Coefficients. Short and long time collective diffusion coefficients, D,(qm) and DL(qm), respectively, were obtained from the initial and long-time gradients of the intermediate scattering functions measured, by dynamic light s ~ a t t e r i n g ,a~t !the ~ position, qm, of the main peak of the static structure factor of the colloidal fluid to the freezing volume fraction, qf. This approach ensures that the measurements made on latices of different sized particles apply to the same thermodynamic state and the same (relative) spatial scale. The procedure for locating qmis described in ref 9. From this quantity we also calculate the effective hard-sphere radius, Rhs, shown in Table 1, using the theoretical value q&hs = 3.47 for the position of the primary structure factor maximum of the hard-sphere fluid a t freezing.lg One sees that, within the experimental uncertainties of about 1%,the effective hard-sphere particle radii are consistent with the hydrodynamic radii. It was clearly not possible to perform these light scattering measurements on the very turbid samples of the 29D series. The quantities Ds(qm)lDoand DL(qm)lDO,where DOis the free particle diffusion coefficient, are listed in Table 2. Within the estimated errors of about lo%,the normalized diffision coefficients are the same for suspensions of particles of different radii. 2.6. Sedimentation Velocities. Sedimentation velocities, Vf and V,, for the coexisting colloidal fluid and crystal were obtained from the linear sections of the height versus time plots such as those shown in Figure 1. Vf is simply given by dHddt and, as shown by Paulin and Ackerson8
The last result accounts for the increase in H , due to the conversion of the settling colloidal fluid into sedimentary colloidal crystal as well as the decrease in H , due to gravitational compacting of the crystal phase itself. From Figure 5 one sees that both Vf and V, scale, as expected for hard spheres, as R2.
the distance over which the concentration decreases from qfto qde. Here mb = (4nR313)(ep- el). The samples were monitored until sedimentation equilibrium, indicated by a constant difference, dH = Hf- H,, between heights of the fluid and crystal (see Figure l), was attained. dH was measured for each series of samples and is plotted as a function of R-3 in Figure 4. The error bars reflect the uncertainty in this quantity which is most significant for the smaller particles. From Figure 4 one sees that the measured sedimentation equilibrium heights are roughly consistent with the R-3 dependence, expected for hard spheres, and also with the calculated Mqde). We exploit the fortuitous agreement between dH and h(qf/e)by calculating the average volume fraction of the (18) Carnahan, N.F.;Starling, K. E. J. Chem. Phys. 1969,51,635.
3. Discussion The results ofthe phase equilibrium study, summarized in Table 2, indicate that, once the freezing concentration is identified with the theoreticalvalue qf=0.494, we obtain experimental melting volume fractions that are consistent with the value q,,, = 0.545 f 0.002 for the perfect hardsphere solid. Significantly, there is no systematic dependence of the experimental q m on the particle size. qm is also independent ofthe refractive index of the continuous phase, in so far that it is unaffected by the addition of CS2, seen by comparing the values for the F series and 29D series. The difference between freezing and melting concentrations is sensitive to the form of the interparticle potential. Therefore, our results indicate that, at least in the thermodynamic sense, suspensions of PMMA particles stabilized by PHSA behave as hard spheres. Moreover, for the range of particle radii examined here, this (19) Verlet, L.;Weis, J. J. Phys. Rev. A 1972,5,939.
3554 Langmuir, Vol. 10, No. 10,1994
R * ( ~ Io - ~ ~ ~ ) Figure 6. Sedimentation velocities for the fluid, Vf (A),and crystal phases, V, (A), as a function ofR2,with lines of best fit.
description is independent of the relative thickness of the adsorbed steric barrier and solvent composition (i.e. addition of CS2). In all samples sedimentation equilibrium gives rise to a diffuse layer of gradually decreasing concentration from q = 0.494to infinite dilution in the transparent supernatant. The magnitude of this diffuse layer was found to be consistent with the R-3 dependence, expected for hard spheres. In previously published studies of the phase behavior of PMMA suspensions,1p8 the particles were sufficiently large (R L 300 nm) that the sedimentation equilibrium height was negligible. The sedimentation velocities of the coexisting colloidal fluid and crystal scale as R2. However, one can see that the linear least-squares fit to the data, in Figure 5, does not extrapolate exactly to the origin. It is not possible to specify whether this is due to genuine random experimental errors or a systematic reduction in the sedimentation velocities resulting from convection within the samples. We have used the measured intrinsic viscosity and the freezing concentration to determine factors, f ’ and f, respectively, that convert the core volume fraction into the effective hard-sphere volume fraction. Results from the two procedures are compatable but the random errors in f ’ are very large. It is conceivable, therefore, that the unphysically high effective hard-sphere volume fractions deduced from intrinsic viscosity measurements in other ~ o r k ~ are * ~ aJ consequence l of random errors comparable to ours or, possibly, systematic errors caused by the limitation of analyses to linear regression. We now consider the observed crystallization in the suspensions of the smallest particles (G series). Arguments based on the Lindemann melting criterion suggest that the polydispersity (a = 0.1)of these particles is equal to the critical value where crystallization is no longer possible,20i.e. where the distortion of the putative crystal is so severe that Bragg reflections do not occur. (Density functional theories21s22predict a lower value, u 5 0.065, (20) Pusey, P. N. J. Phys. (Paris) 1967,48, 709. (21) Barrat, J. L.; Hansen, J.-P. J.Phys. (Paris) 1986, 47, 1547. (22) McRae, R.; Haymet, A. D. J. J. Chem. Phys. 1988, 88, 1114.
Underwood et al. for this critical polydispersity.) However, we find that homogeneously nucleated crystallization, although much slower than for latices of lower polydispersity, occurs for volume fractions up to the glass transition (qg= 0.57).9J0r23 We also find no evidence of the predicted narrowing of the miscibility gap (qm - qf)as a result of polydisper~ity.~~ To explain this contradiction between our observations and predictions, we speculate that rather than forming a substitutionally disordered solid, so distorted thht it cannot Bragg reflect, these polydisperse suspensions fractionate during the nucleation and growth stages. This segregation may well be facilitated kinetically by the significant negative skewness (=-l.O) ofthe particle size distribution of this latex; one can visualize small particles being readily excluded from the advancing crystal fluid interface. A corollary to this argument is that crystallization will not occu in a suspension in which the particle size distribution is of comparable width but with zero or positive skewness. Our experimental work on this aspect is ongoing and we expect to give a more detailed account on these issues in a future publication. A study, in several respects similar in spirit to the above, has been published by Smits et al.4 By comparing several nonaqueous suspensions of silica particles coated with different adsorbents of different thicknesses, these authors concluded that the crystallization rate increases significantly as the range of the interparticle repulsion is increased. Homogeneously nucleated crystallization and separation into coexisting fluid and crystal, consistent with that expected for hard spheres, was only observed when particles were coated with PHSA. In some cases it appears that, as a result of Coulomb repulsion between charges on the particle surfaces, the miscibility gap, q,,, - qf,was significantly reduced. In other cases, possibly due to interparticle attraction, gravitational settling preceded crystallization. 4. Conclusion We have compared some properties of a range of suspensions of PMMA particles of different radii coated with the same PHSA stabilizer. Our results show that the effective hard-sphere freezing and melting volume fractions are independent of the relative thickness of the stabilizing coating. Sedimentation equilibrium heights, diffusion coefficients and sedimentation velocities are shown to scale with the particle radius in a manner expected for hard spheres. (This behavior is consistent with the scaling of shear viscosities observed for similar PMMA suspensions over a comparable range of particle sizes,25although for smaller particles under shear there is evidence of compression of the adsorbed layers.) Interestingly, a suspension with a polydispersity of 0.1 was found to have the same freezing-melting transition as those of much lower polydispersity, although, according to theoretical predictions and simple packing arguments, the suspension should not crystallize a t all.
Acknowledgment. This research is supported by the Australian Research Council. We thank Phil Francis for his assistance and the Department of Chemical Engineering, University of Melbourne for use of the viscometer. We are also grateful to Peter Pusey for several valuable suggestions. (23) van Megen, W.; Underwood, S.M. Nature 1993,362, 616. (24) Dickinson, E.; Parker, R. J.Phys. Lett. (Paris) 1985,46, L229. (25) d’Haene, P. PhD thesis, Katholieke Universiteit Leuven, 1992.
