GENERAL DISCUSSION Dr K. Schatzel (University ojKiel, Federal Republic ofGerrnany) said: Just like other methods of Laplace inversion, the maximum entropy method definitely requires data measured over a very large span of lag-times, if reasonable resolution and reproducibility are to be achieved in the calculated distribution of relaxation times. PCS data in the paper were in fact covering up to 7 decades of lag-time. Such large ranges cannot be economically covered by traditional digital correlators with a linear channel arrangement. The data given in the paper were obtained by running a series of correlation measurements using different lag-times. Besides being a time consuming procedure, this scheme has two major shortcomings. First, there is a need for ‘matching’ adjacent correlation functions, i.e. the introduction of more or less arbitrary multiplicative constants needed to correct for statistical signal fluctuations on long time scales and, if not a great amount of overlap is provided between adjacent correlograms, to compensate the triangular weighting distortions due to finite channel width.’ Secondly, the limited accuracy of traditional correlators (typically 4 bit) requires prescaling of the photon count signal at large sampling times. This procedure introduces a significant amount of quantization noise2 in the far tail of the correlation function, and can only be overcome by the use of extremely large total measurement times. While the use of single clipping3 would relax this second problem, it would make the matching problem considerably more difficult and restrict the method to signals with Gaussian complex amplitude statistics.A novel correlator architecture was required to overcome these problems of uneven channel spacing, matching and excess quantization noise. Combining a new fully parallel processor design, which provides a significant increase in computational speed, with data buffering and a facility to contract adjacent samples,’ I was able to design a new generation of correlator~,~ which can cover more than 7 decades in lag-time (e.g. 1 p s to ca. 60 s) in one single truly real-time experiment. The whole lag-time range is evenly covered by 192 channels with both the sample time and the lag-time increment doubling every 8 channels (resulting in 23 different simultaneous sample times).The channel width increases from 4 x 4 bit to 16 x 16 bit processing, and a new normalization scheme considerably reduces noise (and hence total measurement time) for very large lag-time channels.’ The ability to measure photon correlation functions far beyond their final decay implies new demands on inversion programs. The common scheme of reducing second- order correlation functions to first-order by baseline subtraction and taking a square root cannot handle data points properly, which fall below the baseline due to residual noise. This problem seems to be overcome with some elegance by the maximum entropy method’s ability to use second-order correlation data directly. Another potential problem results from the dominance of signal fluctuations rather than photon noise in large lag-time As a result, we obtain highly correlated noise in adjacent channels and a non-diagonal covariance matrix for the residuals. A modification of the simple chi-squared test used in the maximum entropy program’ seems to be appropriate.1 K. Schiitzel, Inst. Phys. Con$ Ser., 1985, 77, 175. 2 K. Schatzel, Appl. Phys., 1980, 22, 251. 3 E. Jakeman, .I. Phys. A, 1969, 3, 201. 4 ALV-3000 User’s Manual, (ALV Laser GmbH, D-6070 Langen, F.R.G., 1985). 5 K. Schatzel and M. Drewel, J. Mod. Opt., 1987, in press. 6 E. Jakeman, E. R. Pike and S. Swain, J. Phys. A, 1971, 4, 517. 7 K. Schatzel, Opt. Ada, 1983, 30, 155. 8 K. Schatzel, Appl. Phys. B, 1987, 42, 193. 9 A. K. Livesey, P. Licinio and M. Delaye, J. Chem. Phys., 1986, 84, 5102.317318 Genera 1 Discussion Dr A. K. Livesey (DAMTP, Cambridge) replied: We certainly agree that such a correlator would significantly reduce the measurement time involved in measuring 7 decades. Indeed, simply having 192 channels instead of our 48 would cut the measure- ment time by a factor of 4. The matching of our successive time windows has a firm justification. We adjusted these (inaccurately known) experimental parameters until the highest entropy curve was found. If, as a result of using your correlator, these parameters are accurately calibrated between the banks of 8 channels then this restriction on their values can only improve the reconstruction. I would like to stress the importance of maximum entropy’s ability to handle the second-order correlation data directly.This enables the measurements to be taken out to very long decay times when the measured signal (after background subtraction) can go negative owing to the noise in the background. Since maximum entropy can handle such data correctly, weak long delay signals (perhaps due to a few clusters or impurities) can be correctly determined and will not bias the shape or estimate of polydispersity of the major peak. The program is quite capable (in theory at least) of increasing the sophistication of the test of acceptability of a spectrum by including the off-diagonal terms in the covariant matrix. Currently these are set ‘unmeasured’ ( i e . with infinite errors) and not set to zero. Thus our current estimates of the spectra will have less structure than could possibly be extracted.It is therefore ‘safe’. In our particular case, of course, the 15 or so separate sets of lag times used were measured at different times, so that the correlation between these groups would necessarily be zero and only the near-diagonal terms would be relevant. Prof. V. Degiorgio (Pavia, Italy) asked: Concerning the examples discussed in fig. 2 and 3 of your paper, what is the minimum accuracy required on a single channel of the measured autocorrelation function in order to extract the information about the position and relative amplitude of the three peaks? Dr Livesey replied: Typically I estimate that ca. lo6 counts or more are needed in about a hundred channels spread out in log(measurement time) to resolve three well separated peaks.This gives a signal-to-noise ratio running from 500: 1 to 1 to 10 (or less). However, at this poor signal-to-noise ratio (for QLS measurements) there would probably be a certain overlap of the peaks and the user will have to set his own limits on the integration under the peaks. Such overlaps are still apparent in our fig. 2 and 3 taken with ca. 2 x lo7 counts in 200-400 channels (depending on the temporal range of the d-ta). Nevertheless, the MEM graphs also give the number of distinguishable peaks, their width, size, asymmetry, whether there are any shoulders etc. directly and visually without recourse to any models. Prof. M. Fixman (Colorado State University) asked: Is your method a candidate for a general ‘black box’ Laplace inversion scheme, at least for the recovery of positive functions? Is it possible to state general criteria for such a ‘library’ routine that would permit a competitive test of the different methods? Dr Livesey replied: The maximum entropy programs run entirely automatically without any user intervention except to set the required background level (although even this could be automated). The technique is not strictly limited to positive functions since it can recover the difference between two positive functions.This has been successfully applied in chemical n.m.r. spectrum analysis.’General Discussion 319 To run the routine we require data in the form of triples; autocorrelator delay, measured value and estimated standard deviation of random noise on the measurement. We have analysed Laplace transform data ranging from only 15 points with 10% accuracy to over 500 points with better than 0.1% accuracy.We would be glad to take part in any competitive test of analysis methods and hope the test includes the analysis of data from complex spectra containing many, asymmetric broad peaks, as well as simpler unimodal spectra. 1 E. D. Laue, J. Skilling and J. Staunton, J. M a p . Reson., 1985, 63, 418. Dr P. N. Pusey (R.S.R.E., Malvern) remarked: A measure of the resolution of an inversion technique is its 'response' to an ideal signal. In your case, if you construct numerically a single-exponential correlation function with noise (corresponding to an experiment of finite duration on a monodisperse sample) what is the width of the peak in Laplace space returned by the maximum entropy method? With careful technique it is possible, when studying, say, nearly monodisperse polystyrene spheres, to obtain experimental values for the normalized second cumulant Q in the range 0.01 +0.01.Since for narrow distributions Q =: ( T ~ , where (T is the normalized standard deviation of the distribution (see my comment on Dr van de Ven's paper), these values correspond to an apparent size distribution having half-width at half-height of ca. 10%. Does maximum entropy achieve this when an appropriate noise level is chosen? Dr Livesey replied: A delta function can be recovered as a single peak with half-width at half-height of 10%, if 48 channels (linearly spaced in time) are measured to contain between 1 x 10' ( t - 0 ) and 5 x lo6 (t=m) counts.Dr D. A. Weitz (Exxon , Annandale, NJ) addressed Dr Horne: You are no doubt aware that the power-law dependence of scattered light intensity on wavevector, which yields the fractal dimension as its exponent, holds only over a restricted range of that wavevector, q. When q is small, intensity becomes independent of q. Turbidity encom- passes the results of scattering over all, or nearly all angles, some very small, and therefore I wonder if your results will not be influenced by part of your scattering coming from this low-q regime and not all of it taking place in the power-law-dependent range. I would also like to ask you what effect cluster polydispersity has on the measurement of the fractal dimension using your turbidity technique.Aggregation processes often produce cluster mass distributions which are power-law in form, N ( M ) 0~ M-'. A measurement of dt using the k dependence of the scattering intensity is insensitive to polydispersity provided 7 < 2. Does the same limit hold for this turbidity measurement? Further, how sensitive is the measure of df to the choice of the lower cut-off angle included in your calculation of the scattering when the clusters have a power-law p o 1 y di s per s it y '? Dr Horne replied: I am aware of the restricted range of validity of the power-law dependence equation you cite. My paper includes a series of calculations of the turbidity/wavelength exponent using for the structure factor of the cluster the equation proposed by Teixeira [ref. (13) and eqn (S)].Qualitatively this equation gives the expected behaviour you cite, tending to unity at high y, power-law dependence at intermediate q. and, as Teixeira demonstrates, tending towards the Ornstein-Zernicke result at low 9 when D = 2. The full equation, giving structure factor as a function of monomer radius, a, cluster size, (, and fractal dimension, D, was used in my calculations of turbidity. Fig. 1 of the paper shows the behaviour of the dissipation factor/wavelength exponent as the cluster size increases, demonstrating that its asymptotic value for large clusters is the fractal dimension. Experimentally I follow this exponent as a function of reaction time, finding that it too reaches an asymptotic value (fig. 3 and 6) by which time the aggregates are sedimenting.I am therefore of the opinion that the wavelength320 Genera 1 Discussion exponents measured are for clusters large enough for the product q,$ to be in the range where power-law dependence is expected at all angles and that the experiments provide a value for the cluster fractal dimension. As to the influence of cluster size polydispersity on the value of the measured fractal dimensions, I think it would be quite small. Looking at fig. 1, once again, if we consider the plot for a fractal dimension D = 2, the exponent, p, is already > 1.9 for clusters > 1250 nm in radius, the limiting value of ,$+ being 2. It is my intention to carry out further calculations, involving realistic cluster size distribution functions, to verify these speculations.Dr Rarity then added: In the limit of small monomer particles and large aggregates the turbidimetry technique you have used measures the same moment of the aggregate size distribution as in the conventional measurement of light scattering us. scattering vector. Martin’ has studied the effects of polydispersity on measured fractal dimension for this case. He concludes that within this range of polydispersity expected from non-gelling reaction-limited aggregation the measured dimension is the fractal dimension. 1 J. E. Martin, J. Appl. Crystalfogr., 1986, 19, 25. Prof. M. Fixman (Colorado State University) addressed Prof. Fuller: Is it possible to specify which aspect(s) of the behaviour of non-Newtonian fluids are illuminated by experiments of this kind? The different aspects that I have in mind might be classified as follows.( a ) Fluid motion: ( 1 ) non-locality in time (memory functions, (2) non-locality in space (or higher spatial derivatives) and (3) non-linearity. ( b ) Particle-fluid interaction: (4) boundary conditions at surface of particle. Does a non-slip boundary condition suffice to determine a solution? Is it physically correct? ( c ) Mathematical solutions for a given model: (5) are there notable approximations made in solving the problem, the accuracy of which is tested? There then followed informal discussion on the analogue of Stokes’ Law for a second-order fluid, a problem which had been addressed by L. G. Leal. Prof G. G . Fuller (Stanford University) said: The points raised by Prof. Fixman and the problems of Stokes’ law for a second-order fluid are closely related and we will respond to them simultaneously.The objective of this paper was to exaiiiiile the respmse of a colloidal particle to flowing, non-Newtonian suspending fluids, and the materials were chosen in order to access the influence of Brownian motion and to mimic, to the best degree possible, the rheological properties of the second-order fluid constitutive model employed in the only available theories. The class of Boger fluids is currently thought to offer the closest approximations to second order fluid behaviour but even these systems will fail in several important ways, the principal difference being that the polymeric Boger fluids have finite relaxation times whereas an ideal second order liquid is modelled to respond instantaneously. For this reason, non-linear constitutive models such as the Oldroyd B model would be more appropriate.With respect to promoting the influence of Brownian motion, the appropriate dimensionless group which must be kept in the vicinity of unity is [from ref. (18) in our paper] +,G2/2r~77nr where I), is the first normal stress difference coefficient, D, is the rotary diffusivity and all other symbols are defined in our paper. This criterion was met for all three particles used in our study and the principal conclusion obtained by Cohen et al. that the effect of Brownian motion is to induce alignment in the flow direction at lower shear rates was verified in our measurements. This point is most clearly brought out in fig.9(c) where the ordinate passes from positve values at low shear rates to negative values at high shear rates.Genera l Discussion 321 The available theory, however, cannot be expected to capture the full response of the suspensions which were studied, and memory effects in the fluid are readily apparent in the data following the cessation of flow. In fig. 7, for example, examination of the response of the average particle orientation angle reveals that the particles rotate significantly after the flow is stopped whereas the theory predicts this angle should remain constant as the particles disorient. The timescale for this rotation is proportional to the relaxation time of the fluid and not the rotational diffusion of the particles. These two timescales are separated by ca.2 orders of magnitude. Indeed, this dimensionless group D,T, where T = +,/a7 is the fluid relaxation time, specifies whether or not the rotational diffusivity can be considered to be independent of time in these systems. It would appear that the separation of timescales was sufficient in our experiments to make this assumption. The question of the choice of boundarj conditions at the interface of a coiioidai particle and a polymeric liquid are certainly important to the problem being addressed in our paper. However, the answer to this question would more easily be obtained by experiments which are designed to look specifically at the friction fact of a particle residing in polymeric liquids. We are currently carrying out such an investigation by measuring the rotational diffusivity using electric dichroism techniques.In the experi- ments discussed in the present paper, the length scales of the particles (ca. 0.5 pm) and the polymer chains (ca. 0.1 pm) are not dissimilar, and there may be situations where non-slip boundary conditions are not appropriate. However, at the low polymer con- centrations employed here, we suspect that this is not the case. Our data do indicate, however, that the rotational diffusivities were altered upon the addition of the small amount of polyisobutylene, even though this addition did not significantly increase the shear viscosity. This effect is most clearly brought out by comparing the relaxation of the dichroism for particle A in fig. 5 ( b ) (Newtonian case) and fig.7( a ) (the Bsger fluid). The particles clearly relax more quickly in the Boger fluid. Dr J. Penfold ( Rutherford-Appleton Laboratory ) said: During his Lennard-Jones lecture Prof. Oppenheim spoke of the complexities of Brownian motion for spherical particles, and he alluded to the greater difficulties of treating anisotropically shaped particles. I would like to draw your attention to some experimental work on anisotropi- cally shaped micelles. Other groups’ and myself in collaboration with John Hayter, Oak Ridge’ and Ed Stables and Phil Cummins, Unilever3 have made investigations by small-angle neutron scattering of rod-like micelies aligned by shear flow. Experimentally the situation is simpler than the situation so well described by Prof. Fuller for non- Newtonian fluids, alignment is in the flow direction and is dependent on the ratio of the shear gradient, G, to the randomising factor, the rotational Brownian diffusion, 0,.Much of the information obtainable from such measurements is not particularly relevant to this meeting, such as particle size, size distribution, structure, conformation, and flexibility; and I will emphasise only aspects of interest to a meeting on Brownian motion. Fig. 1 shows some representative data for a 0.04 mol dm-3 dodecyldimethyl ammonium chloride-4 mol dm-3 sodium chloride micellar solution at different shear gradients; and the alignment of the rod-like micelles in the flow direction ( Qil) is clearly demonstrated. For dilute non-interacting systems we can, using existing theories,2 obtain an ‘effective’ rod length from the shear dependence of the anisotropy.At higher shear gradients (2 10 000 s-’) polydispersity and the onset of turbulence perturb the interpreta- tion. Although in terms of interactions the systems investigated by us were dilute, in the language of Doi and Edwards4 they are in the entangled regime where C CK: 1/ L3, with C the micelle number density and L the rod length. At low values of shear gradient hindered rotational diffusion, due to entanglements, is observed. This is illustrated clearly in fig. 2, where the effective rod length increases dramatically at low shear322 General Discussion e ( i i ) - - 01, Fig. 1. Intensity contours for scattering from 0.04 mol dm-? dodecyldimethyl ammonium chloride- 4 mol dm-3 sodium chloride-deuterium oxide at T = 298 K.The Q range is 10.6 nm-' in both the QII and Q, directions. The shear gradient, G, is ( i ) 250, (ii) 5000 and (iii) 7500 s-'. gradients. At higher shear gradients [in this case ca. (5-10 000) x lo3 s-I] the rods are sufficiently well aligned that hindered rotation is not present. Fig. 3 shows the shear dependence of the effective diffusion coefficient, D,, obtained from such assumptions. The plateau at 5000 s-l corresponds to dilute free rotational diffusion. The form of the dependence is similar to that reported by Keller et al.' for polymer solutions. Evidence of rod flexibility in some systems has been observed3 and will further complicate the shear dependence of the rotational diffusion. We are not, at present, in a position to offer any theoretical interpretation to our data, but nevertheless feel it contains important information with regard to the rotational diffusion of anisotropically shaped particles.1 H. Thurn, J. Kalus and H. Hoffman, J . Phj~s. Clrem., 1984, 80, 3440. 2 J . 3. Hayter and J. Penfold, J. Phyx Chein., 1984, 88, 4589.General Discussion 3 23 0 0 0 0 i 0 L I 1 1 I 1 I I 0 2 L 6 8 1 0 1 2 l L shear gradient, G/ lo3 s-' Fig. 2. Effective rod length as a function of shear gradient for 0.03 mol dm-3 cetyltrimethyl ammonium bromide-0.4 mol dm-3 potassium bromide-deuterium oxide at T = 313 K. .' 800 8 I I I I I I I 1 0 2 L 6 8 10 1 2 - 1 1 , shear gradient, G / lo3 s-' Fig. 3. Effective rotational diffusion coefficient as a function of shear gradient for 0.03 mol dm-3 cetyltrimethyl ammonium bromide-0.4 MOI dm-3 potassium bromide-deuterium oxide at T = 313 K. 3 P.G. Cummins, J. B. Hayter, J. Penfold and E. Staples, J. Chem. SOC., Faraday Trans. I , in press. 4 M. Doi and S. F. Edwards, J. Chem. Soc., Faraday Trans., I , 1978, 74, 560. 5 J. A. Odell, A. Keller and D. T. Atkins, Macromolecules, 1985, 18, 1443.324 General Discussion Dr W. van Megen (R.M.I. T., Melbourne, Australia) addressed Prof. Degiorgio: In the interpretation of, what I assume, as small-k diffusion constants (fig. 5 of your paper), do you think that Oseen level hydrodynamics is sufficiently accurate? Approximate calculations of the k-dependent diffusion constant D( k) (as obtained from the initial decay of the intensity autocorrelation function) for dilute (4 5 0.1) charge stabilized dispersions, suggest that the contribution of hydrodynamic interactions to this quantity is small at intermediate and large k.However, at small k, D( k ) is very sensitive to both direct and hydrodynamic interactions.’ 1 1. Snook and W. van Megen, J. Colloid Interface Sci., 1984, 100, 194. Prof. Degiorgio replied: Our results agree completely with your remark that, at small k, D( k) should be very sensitive to hydrodynamic interactions notwithstanding the fact that particles are, on the average, quite far apart one from the other in our system. We have only a qualitative justification for the fact that our calculations take into account hydrodynamic interactions at the Oseen level: since g ( r ) is different from 1 in a range of distances r which is much larger than the particle size a, and since the terms which appear beyond the Oseen level are of the order ( a l r ) ” with n = 2 or larger, we suppose that in our case tne Oseen level is sufficiently ac- -u rat e.Dr P. N. Pusey (R.S.R.E., Malvern) then asked: Quite plausibly you attribute the slow decays in fig. 4 to polydispersity. Is it possible that the disagreement between experiment and theory seen at low ionic strength in fig. 1 reflects incoherent scattering also caused by polydispersity? Prof. Degiorgio replied: If we analyse our correlation functions in terms of a superposition of two exponentials, A , exp (- t / 7’) + A2 exp (- t / T2) with r2 >> T ~ , the observed ratio A , / A , is too small to explain the disagreement between theory and experiment.This would indicate that the cause for the disagreement is not polydispersity alone. On the other hand, the dependence of A 2 / A l on the salt concentration is not in agreement with the decoupling approximation, so that our conclusion should be taken with some caution. Dr 0. A. Nehme (Cavendish Laboratory, Cambridge) said: I have a cautionary tale for the dynamic-light-scattering experimentalists, following Dr Pusey’s comment on polydispersity values deduced from light scattering data. There are evidently many sources of noise that produce non-exponential, non-random contributions to the correlation function of supposedly nearly monodisperse systems. This will give an abnormally high value for the polydispersity. The origins of the non-random ‘noise’ could be mains electrical supply problems, thermal fluctuations and at long sample times, mechanical vibrations.We have been able to track down the non-random ‘noise’ and separate from the exponential decay by using the maximum entropy data analysis, and by thorough examination of the residuals. It is hoped that non-random ‘noise’ could be minimised experimentally, although it is advisable to examine the residuals of the fit before trying to obtain quantitative values for polydispersity. Dr Pusey commented: I do not think it is necessary to resort to sophisticated methods of data analysis such as maximum entropy to detect many of the artifacts which can arise in a PCS experiment. As mentioned in my comment on Dr Livesey’s paper, it is possible, although not easy, to take a carefully prepared nearly monodisperse suspension of particles and to obtain essentially a single-exponential correlation function (to withinGeneral Discussion 325 experimental error).It seems much better to discover and remedy experimental di ficul- ties during such a test of equipment and technique than to attempt to allow for it in subsequent analysis of data obtained from a more complicated system. Dr A. K. Livesey added: Maximum entropy itself cannot correct for these systematic effects often seen in poor experimental measurements, unless some correct model of their origin is supplied. However, since the data analysis is completely general and makes no assumptions about the shape of the correlation times, then any non-random residuals which appear during the data-analysis must have arisen from some experimental artifact.It is this confidence that the problem lies in the experiment and not the fitting procedure which enables the research worker to pursue subtle experimental artifacts and eliminate them at source. The experimental papers in this Discussion have demonstrated how well (or other- wise) they have fitted their data by displaying a graph of their calculated data superim- posed on their experimental data. Given the large dynamic range of the data, and the high precision to which QLS scatterers work, I suggest that such a representation cannot adequately display the small differences between these curves. I recommend that users wishing to report an 'excellent' fit to their data should publish a graph of their normalised residuals where Dy''d is the kth calculated datum point, Dibsd is the kth observed datum point and 0; is the (estimated) standard deviation of the kth measurement.These graphs should appear random (further statistical tests could be applied to check this visual appearance if wished) and have an average deviation from zero of ca. *1 with very few outliers lying beyond *3. Dr A. Philipse (Utrecht, The Netherlands) said to Prof. Degiorgio: You state that the non-exponentiality of the correlation functions at low ionic strength is due to polydispersity in micelle size. However, if K - ' is large V ( r ) [eqn (3)] only weakly depends on the micelle size so the influence of this polydispersity will be small. At high salt concentration V ( r ) is more sensitive to the micelle size.Therefore, I would expect the variance V in fig. 6 to slope upwards at high ionic strength instead of downwards. Since a weakly screened V ( r ) depends strongly on the particle charge Q, V in fig. 6 probably reflects polydispersity in Q. Its effect will be smaller if the charge is sufficiently screened, in accordance with your data. This polydispersity also explains why the HNC fit of S ( K ) fails at low ionic strength, whereas at high ionic strength a fit is possible for one value of Q. [In addition, if K - ~ increases the S ( K ) peak shifts to higher K-values, away from the value of K in your experiment. So (any) polydispersity will manifest intself more clearly.] Thus, it seems to me that charge polydispersity explains as much as your suggestion that Q changes at low salt concentration.Prof. Degiorgio replied: The expression of the correlation function for the case of arbitrary polydispersities and interactions is very complicated [see ref. (2) and (22) of our paper], but it seems to me that what is required in order to give a non-exponential correlation function is a polydispersity in scattering amplitude rather than in the electric charge. However, as your remark implicitly suggests, a polydispersity in aggregation number implies a polydispersity in the micellar charge. As a consequence, this would lead to a strong correlation between concentration fluctuations and number fluctuations, and would not allow one, in agreement with our observations, to consider the correlation function as the sum of two independent modes controlled by collective diffusion and self-diffusion, respectively.326 General Discussion 1 I 1 " 0 15 Fig.4. Scattered neutron intensity for a 3 mmol cmP3 GMl ganglioside solution in D20 with no salt (0) and with 100 mol dm-3 NaCl. (W). Prof. V. Degiorgio (Pavia, Italy) said: We have recently performed small-angle neutron-scattering measurements on the same solutions studied by light scattering by using the spectrometer D17 at ILL, Grenoble. I show in fig. 4 the intensity of scattered neutrons as a function of k. The high-salt measurement gives essentially the form factor of the globular micelle, whereas the no-salt curve gives the product between form and structure factor.I mention here two features of our neutron data: (i) the two curves are perfectly superposed at high k, thus confirming that the size of the ganglioside micelle is very insensitive to salt concentration; (ii) the no-salt curve presents a marked peak at a value of k which is given approximately by 2v/(average interparticle distance), i.e. the micellar solution begins to form a structure reminiscent of that of colloidal crystals. Dr A. R. Rennie (I.L.L., Grenoble, France) then commented on the small-angle neutron scattering data presented by Prof. Degiorgio in the discussion in fig. 4 of this section. These data confirm the view that the ganglioside micelles are extremely monodis- perse. A least-squares fit to his high salt data gives a polydispersity of <2%.Dr Pusey then said: You mentioned that your analysis of the neutron-scattering data of Cant6 et al. yielded a very small polydispersity for their micelles. How was this analysis performed? If we consider their fig. 6 and take the data at high ionic strength to apply to a non-interacting system, the value of second cumulant V== 0.06 implies a polydispersity (standard deviation/mean) of cr = V1'2 == 0.25. How does this compare with your findings? Prof. Degiorgio answered: We believe that some of the observed polydispersity is due to some disturbance which is difficult to eliminate because the signal is very low: the intensity scattered from the solution of 1 mmol drn-3 GMl with no salt is only a few times that scattered by pure water! We were mainly interested in the trend of the variance u as a function of the salt concentration.Dr M. La1 (Unilever Research, Port Sunlight Laboratory) turned to Prof. Prieve: The disagreement between the calculated and measured potential-energy profiles, presentedGeneral Discussion 327 in fig. 2 of the paper, has been attributed by the authors to the fact that the particle-wall van der Waals interaction has not been taken into account. Surely, it is a simple matter to include this component of the interaction (assuming that the Hamaker constant for the system is known) in their eqn (6) and see whether better agreement is achieved. Can the experimental technique used by the authors be adapted to study the dynamical behaviour of surface-confined particles? Prof. D. C. Prieve (Carnegie-Mellon University) replied: We have estimated the van der Waals potential using Lifshitz theory.This estimate suggests that the van der Waals contribution cannot be neglected at the separation distances expected in our experiments, despite significant weakening of the van der Waals force by retardation as well as Debye screening. A more quantitative comparison of theory and experiment must await evaluation of the parameters appearing in the theory by independent experiments. In more recent experiments, histograms of scattering intensity have also been obtained for particles which appear to have stuck to the plate. As expected, they are much brighter than free particles. Although the intensity of stuck particles also fluctuates, the fluctuations represent a smaller percentage of the mean than for free particles. Prof.B. U. Felderhof ( R . W. T.H., Aachen, Federal Republic of Germany) addressed Prof. Ackerson: I would like to remark that the pair correlation function may be calculated exactly, not only for a one-dimensional system of hard rods, but more generally for one-dimensional fluids with nearest-neighbour interactions. This was first shown by Salsburg et al,' See also the article by Fisher and Widom.' 1 2. W. Salsburg, R. W. Zwanzig and J. G. Kirkwood, 1. Chem. Phys., 1953, 21, 1098. 2 M. E. Fisher and B. Widom, J. Chem. Phys., 1969, 50, 3756. Prof. Ackerson replied: The calculation may be difficult to carry out explicitly. Prof. Felderhof continued: That may be. I would also like to ask you whether it would be possible to measure the velocity of a colloidal sphere driven by radiation pressure.For known friction coefficient that would allow one to find the force. This may be of importance in connection with the longstanding controversy on the form of the Maxwell stress tensor in a liquid. Prof. Ackerson: Such a measurement would be possible. Prof. Fixman interjected: Which controversy are you referring to? Prof. Felderhof answered: In the early days of special relativity different forms have been proposed for the form of the electromagnetic stress tensor in a medium. Different proposals have been made by Minkowski, Abraham, Einstein and Laub and by others. The controversy has been discussed in great detail by S. R. de Groot and L. G. Suttorp in their book Foundations of Electrodynamics.They come to the conclusion that the issue cannot be decided on macroscopic grounds and that a microscopic derivation is needed. They gave such a derivation and came to a new form of the electromagnetic stress tensor. Prof. Fixman: For such matters I refer to the book by Landau and Lifshitz! Prof. Felderhof: There you will find a particular form derived on the basis of macroscopic arguments, but it is not the only possible one.328 General Discussion Prof. Fixman: Does it. make any difference for non-relativistic effects? Prof. Feiderhof: It does. The choice of the stress tensor and of the electromagnetic momentum density determines the electromagnetic force density exerted on a fluid element. Prof. Aekerson then took up this point: I assume the dispute to which you refer is whether the form of the momentum density for electromagnetic waves in a material medium is ( D x B)/47rc (Minkowski form) or ( E x H)/47rc (Abraham form).Presum- ably one could measure the Doppler shift of light scattered by a particle which is being pushed by the radiation pressure forces of an intense laser beam. From the measured velocity and known drag coefficient a force is determined, and the two theoretical predictions may be tested. Alternatively, one may test the two forms by measuring the degree of alignment of particles in a given magnitude intensity fringe pattern (by scattering from the induced diffraction grating) as a function of solvent refractive index. However, there are many experimental difficulties to overcome, especially for measure- ments which be can be compared from sample to sample. So I do not know if the dispute can be resolved easily.Gordon' has given a detailed discussion of this problem with reference to colloids and fluid surfaces. He favours the Abraham form but states that 'laboratory experiments designed to demonstrate the nature of the true electromag- netic momentum in dielectric media may not be feasible'. 1 Gordon, Phps. Rev. A, 1973, 8, 14. Prof. J. A. McCammon asked: Tn the two-dimensional colloid with solid-like order, the particles are trapped in potential wells that arise from interactions among the particles as well as between the particles and the radiation field. Thermal displacement fluctu- ations away from the minimum-energy positions should affect the scattering intensities; cf: the Debye-Waller factors in other diffraction experiments.Is it possible to extract information on the interparticle potentials by analysing such intensity variations? Prof. Ackerson replied: In principle, it is possible. However, to do thermal diffuse measurments will require a change in our scattering technique. Presently we are in a self-scattering mode. Both incident beams which produce the intensity fringes also scatter from the particles. The observed scattering pattern is a coherent mixture of the scattering patterns produced by each beam. A region which is bright with either beam alone (for a given particle configuration) may be dark with both beams on, because the scattering amplitudes from each beam are out of phase at that point.One can utilize this effect to make optical logic gates with the desired binary output properties or to resolve the phase problem inherent in standard scattering measurements. However, to measure thermal diffuse scattering one should filter out the self-scattered light from the input beams and probe with another beam of a different wavelength. Prof. J. F. Brady (CaItech, Pasadena) remarked: You have worked with two- dimensional colloidal systems. Can you extend this work to three dimensions? Prof. Ackerson answered: We have found it relatively easy to work with two- dimensional systems, both experimentally and in interpreting the results. In three dimensions one begins to have difficulties with the light scattered by the particles, and the fringes degrade in quality as the sample becomes thick.There may be some geometries where the scattered light will reinforce the interference pattern, but we have not followed this up. On the other hand, we have observed (microscopically) the three-dimensional ordering of particles near a wall with applied intensity fringes. The ordering and particle motion was interesting but restricted to a few layers of particles. I think these restricted three-dimensional systems are worth studying further.General Discussion 329 Dr R. B. Jones (Queen Mary College, London) said: I would like to ask if the radiation pressure effect could be used in an optically polydisperse suspension to separate the more optically active particles from the less active ones. If such a procedure were possible it offers a mechanism for particle sorting and could serve as another means of establishing the initial conditions for an exchange or tracer-diff usion experiment. Prof. Ackerson replied to Dr Jones: The radiation pressure forces depend on the difference in index of refraction of the particle and the solvent as well as the particle size. A focussed beam in a suspension will preferentially draw in the larger and more optically dense particles (assuming the solvent refractive index is less than that of all the particles). Because our particles are optically homogeneous and size monodisperse, we do not seem to induce preferential motion in certain particles. However, it may be interesting to seed a suspension with a trace amount of larger or more optically dense particles, as you suggest, then use radiation pressure to form a grating of these particles. By monitoring the time decay of the grating, when the fringes are eliminated, the low-k long-time self- or exchange-diffusion constant could be measured. These would be measurements on a different scale in length and time from those discussed in the Bristol and Utrecht papers at this Discussion. Prof. N. Sheppard (University of East Anglia) asked: Can radiation pressure be used to move single colloidal particles? Prof. Ackerson replied: Yes, we have trapped single particles near the cell wall and translated them about by moving the laser beam. One should also be able to scan a beam throughout the sample, stopping briefly in certain selected positions. Particles will tend to accumulate at the selected positions. In this way any pattern (including a Penrose tiling) of colloidal particles may be achieved. Prof. E. R. Smith (LaTrobe Uniuersity, Australia) then asked: In your one-dimensional colloidal hard-rod system, do particles which are not nearest neighbours interact? If not, then we can calculate exactly the properties of the system in terms of the Laplace transform of the Mayer $function for the pair potential. This will allow calculation of potential parameters from experimental properties of the system. Prof. Ackerson: I am very interested in an potential for another problem that I am involved with. Prof. Smith: I’ll send you the answer!!