J . Chem. SOC., Faraday Trans. I , 1984,80, 13-27 Small-angle Neutron-scattering Study of Microemulsions Stabilised by Aerosol-OT Part 1 .-Solvent and Concentration Variation BY BRIAN H. ROBINSON,*? CHRISTO TOPRAKCIOGLU~~ AND JOHN C. DORE$ ?Chemistry and $Physics Laboratories, University of Kent, Canterbury CT2 7NR AND PIERRE CHIEUX Institut Laue-Langevin, Grenoble, France Received 1st November, 1982 Small-angle neutron-scattering (SANS) measurements have been made for a series of aerosol-OT (A0T)-stabilised water-in-oil microemulsions. The intensity pattern has been used to extract a value for the radius of the water core, rw, using D,O to provide the required contrast profile. In heptane the radii are found to follow an approximately linear relationship with respect to the [D,O]/[AOT] concentration ratio, R.At 20 OC, and R = 20, the structure of the water-droplet system is dependent on the hydrocarbon chain length of the oil medium. The experimental SANS patterns show increasing discrepancies with a fitted function based on monodisperse spheres as the length of the alkane chain is increased from n-heptane to n-dodecane. This effect is attributed to polydispersity and indicates that the droplet-size distribution within these microemulsion systems is much larger than had previously been thought. In this paper we report a small-angle neutron-scattering (SANS) study of water-in-oil microemulsions stabilised by aerosol-OT (AOT) as surfactant. This system was chosen for detailed study since no cosurfactant is required to form a stable microemulsion and the behaviour is therefore considerably simplified.Structural studies of four- component microemulsions pose various problems which have not as yet been fully resolved. On adding water to a solution of AOT in a hydrocarbon (e.g. heptane), the microemulsion is formed spontaneously after being shaken for a few seconds. It is generally accepted that the system is thermodynamically stable and the equilibrium sizes are established rapidly following dispersion. However, evidence has recently been obtained which suggests that water-in-oil microemulsions formed by the system AOT/water/decane can separate into two phases on standing for a few m0nths.l have clearly indicated the existence of discrete water droplets dispersed in a continuous oil phase for the AOT systems.The effect of temperature was studied by Zulauf and Eicke.2 These results taken together show unambiguously that the size of the water droplets increases with R (or coo), the molar ratio of water to AOT in the system. For R < 20 the size did not vary significantly over a fairly wide temperature range. Neutron scattering offers several advantages over other techniques since deuteration of the various chemical constituents of the system enables different structural parameters to be obtained. In the present paper the main emphasis of the investigation concerns the size of the water core. This is obtained by using fully deuterated water Previous studies by photon correlation spectroscopy 2-4 and 1314 SANS FOR AOT MICROEMULSIONS in a hydrogenated surfactant and solvent medium.The information is complementary to that obtained in PCS experiments in which the overall droplet size, as characterised by the hydrodynamic radius, is determined from the translational diffusion coefficient. Further considerations related to polydispersity and surfactant-partitioning effects are also considered and related to the detailed behaviour of the SANS intensity profile. This work builds upon the pioneering neutron study of AOT-based microemulsions reported by Cabos and de Lord.5 THEORY OF THE SANS METHOD Small-angle scattering occurs when a beam of neutrons passes through a sample containing regions of different mean coherent scattering lengths. The intensity profile for the scattered neutrons depends on the size distribution of these regions as well as the difference (or contrast) in their relative scattering amplitudes, which are determined by the nature of the nuclei contained in the regions concerned.The coherent neutron scattering lengths of hydrogen and deuterium are -3.74 and 6.67 fm, respectively.' These values are significantly different, so that selective deuteration of the various components of the system conveniently adjusts the contrast profile to provide the required information. The scattered intensity for a system consisting of an infinitely dilute solution of non-interacting, monodisperse spherical particles of radius r may be expressed as where p and ps are the mean scattering lengths per unit volume of the particle and solvent, respectively, V is the volume of the particle, n is the number of particles per unit volume and Q is the scattering vector, whose magnitude is given by 4n Q = sin 8/2 where 8 denotes the angle of the scattered beam with respect to an incident beam of wavelength A.The function I@, r ) is essentially the form factor representing the interference pattern for scattering from a single droplet. In a real system of finite concentration there is also interference between droplets, so that the observed intensity distribution is given by (3) where S(Q) is the structure factor for the droplet distribution. At sufficiently low concentrations S(Q) approximates to unity over the Q-range of interesta and the intensity pattern may be used directly to determine the structural parameters of the droplet.In practice, however, this is not necessarily the case for all low-concentration dispersion^.^ In the case of charge-stabilised dispersions, such as aqueous micelles, the interaction between different aggregates can be represented by electrostatic forces with a screened Coulomb potential so that some estimate can be made of S(Q). For microemulsions there is a much weaker electrostatic interaction due to charge cancellation of counter-ions in the region of the surfactant head-group at the water interface. Under these circumstances there is less overall structure in the particle distribution and the system can be represented in first order as a system of hard spheres with negligible long-range interaction. The second virial coefficient, B,, is thought to have a smallB.H. ROBINSON, C. TOPRAKCIOGLU, J. C. DORE AND P. CHIEUX 15 negative value,l0 which indicates a residual attractive force. In the present measurements a low droplet concentration (typically ca. 4% by volume) is used and the system is not normally close to the phase boundary defining the stability of the microemulsion (fig. I). Under these conditions the structure factor will approximate to unity for the @range of interest. 100- R 50 - 0 70 0 70 0 70 T/" C Fig. 1. Phase behaviour of the systems : (a) water/AOT/heptane, (b) water/AOT/decane and (c) water/AOT/dodecane, with [AOT] = 0.1 mol dm-3 in each case. The area under each curve corresponds to the 'clear' microemulsion region. In the case of heptane, in the upper part of the 'clear' region where the two phase boundaries converge, the system acquires a light blue appearance due to Tyndall scattering.EXPERIMENTAL MATERIALS The AOT (fig. 2) was obtained from Sigma Chemicals. The samples obtained from this source were found to be consistently pure and contained minimum amounts of carboxylic acid and alcohol, which form as a result of self-hydrolysis of the ester AOT.ll The heavy water (D,O) was obtained from Prochem (B.O.C.) and B.D.H. and had a quoted isotopic purity of > 99.8%. The microemulsion samples were prepared by adding D,O (with the aid of a microsyringe) to a solution of AOT in one of the various hydrocarbons used and shaking the mixture vigorously for a few seconds. The hydrocarbon solvents used in the present experiments (hexane, heptane, octane, decane and dodecane) were Fisons or B.D.H.materials and were distilled before use. APPARATUS Neutron-scattering measurements were performed on the D 17 small-angle diffractometer at the Institut Laue-Langevin (ILL), Grenoble, and also on the Pluto small-angle instrument at A.E.R.E., Harwell. The wavelength of the monochromatic neutron beam was varied in the range 6-9 A and corresponded to scattering vectors, Q, of 0.02-0.20 A-l for most measurements. The samples were contained in stoppered quartz cells and had a neutron path length of 1 mm. The temperature was maintained constant to kO.1 K throughout each run. The small-angle intensity profile was recorded using a two-dimensional multidetector. The separate channels were grouped to provide a radial summation and the usual corrections applied to the data in order to produce a SANS intensity profile, lob&?).16 SANS FOR AOT MICROEMULSIONS o .1 P 0 ter AOT n - h e / ptane 0 Na \ \ Fig. 2. Structure of the AOT molecule and the spherical microemulsion droplet with the corresponding contrast in the mean coherent-scattering length per unit volume, p , which is given in units of 10-12 cm 81-3. RESULTS PHASE BEHAVIOUR OF MICROEMULSION SYSTEMS Since the dimensions of water-in-oil microemulsion droplets are typically in the range 20-100 A these systems have a visually clear appearance. If at a given temperature increasingly larger amounts of water are added to a solution of AOT in heptane (or other hydrocarbon), the surfactant concentration being kept constant, a stage is eventually reached at which the system undergoes what appears to be a phase transformation, and becomes visually turbid.Denoting the molar ratio of water to surfactant by R, it is possible to construct phase diagrams by measuring the value of R at which the transition occurs as a function of temperature. The phase behaviour of water/AOT/heptane is shown in fig. 1 ; when D,O is substituted for H,O the phase diagram shifts to higher temperature by ca. 6 K.12 The transition involves the formation of two phases separated by a well defined interface. The nature of the two phases has yet to be fully characterized, and the behaviour does not appear to correspond to simple separation of the oil and water components. The results reported in this investigation correspond to the clear microemulsion region of the phase diagram for all the systems studied; data for conditions close to the transition region will be reported in a later paper.SANS MEASUREMENTS The constant profile for a typical AOT-stabilised microemulsion droplet is shown in fig. 2, for a water core consisting of D,O surrounded by hydrogenated surfactantB. H. ROBINSON, C. TOPRAKCIOGLU, J. C. DORE AND P. CHIEUX 17 1.00 0.75 h 2 + 0.50 0.25 0.05 0.10 0.15 0 -75 h 0.50 O a Z 5 i 1 0.05 0.10 0.15 QI A I I I 0.05 0.10 0.15 i -\. 1 L I I I I I I I 0.05 0.10 0.15 0.05 0.10 0.15 0.25 QIA Fig. 3. Fit to the observed SANS intensity pattern for R = 20 microemulsions in various solvent media: (a) hexane, (b) heptane, (c) octane, (d) decane and (e) dodecane. The solid line is the fit using the monodisperse formalism [eqn (l)].and solvent. The surfactant head-groups are hydrated and protrude into the aqueous region, so that the overall profile includes the sulphosuccinate portion of the AOT molecule in addition to the water core. The hydrocarbon chains of the surfactant are well matched to the n-alkane, which constitutes the continuous phase. The SANS measurement is therefore dependent on the radius, rw, which includes the volume occupied by the head-group. If it is assumed that the interface region is relatively sharp, the observed SANS intensity may be characterised by eqn (1) and used to extract a value for the radius parameter, rw. Typical results for various hydrocarbon media with R = 20 are shown in fig. 3. The curves represent least-squares fits based on the intensity function for monodisperse18 SANS FOR AOT MICROEMULSIONS Table 1.The core radius, rw, obtained from the fit to the SANS intensity profile for various hydrocarbon mediaa ~~ ~ hydrocarbon medium rw/A n-hexane 33.1 & 1 .O n-heptane 35.8 k 1 .O n-octane 36.8 & 1 .O n-decane 34.4 & 1 .o n-dodecane > 50 a [AOT] = 0.1 mol dm-3; R = 20; T = 293f 1 K. spherical particles [eqn (l)]. The value of rw for each system is shown in table 1, and the quoted error incorporates the variation for several runs; the calculated statistical error from a single observation is much smaller. With the exception of dodecane, both the shape of the intensity pattern and the extracted value for rw remain virtually independent of the hydrocarbon solvent. Although the particle size in dodecane appears to be considerably larger than in the other solvents, it is clear that eqn (1) fails to produce an adequate fit to the experimental data.Closer examination of the results for hexane to decane reveals small but significant systematic deviations in the higher region of Q, these being most pronounced in the case of decane. In particular the minimum predicted at Q x 4.49/r [eqn (111 is never actually observed in the experimental data. The absence of this minimum can be attributed to various factors, although the effects due to polydispersity are expected to dominate; these features are discussed more fully later in the paper. SIZE DEPENDENCE ON R It is possible to predict an approximate relationship between rw and R on the basis of certain simplifying assumptions.13 The amount of water in a microemulsion system consisting of monodisperse spheres is equal to jn(rw - ?-k)3 pw n, where n is the number of droplets, rk is the effective size of the head-group protruding into the aqueous phase and pw is the density (in moles per unit volume) of water in the droplet.If all of the surfactant present in the system is located at the interface, the total amount of surfactant is equal to 47t(rw - rk)2 a, n, where a, is the density (in moles per unit area) of the surfactant at the interface. The molar ratio, R, of water to surfactant is then given by R = in(rw - r,)3 pw n/4n(rw - rk)2 a, n which establishes a linear relationship between R and rw provided that pw and a, are independent of droplet size. Experimental results2v seem to support this simple relationship quite well for AOT systems, and this has contributed to the view that the systematic variation of R has a well defined effect on the structure of the droplets.There is now increasing evidenceff to suggest that the assumptions are probably not justified and that the linear relationship is a result of various competing factors which conceal the underlying complexity of the structural changes.B. H. ROBINSON, C. TOPRAKCIOGLU, J. C. DORE AND P. CHIEUX 19 I 50 I i 10 20 30 40 R = [ DzO] /[ AOT] Fig. 4. Radius parameter, rw, as a function of R for water/AOT/heptane at 293 k 1 K. ([AOT] = 0.1 mol dm-3). The results of the present measurements for water/AOT/heptane are shown in fig.4 for the range 5 < R < 40 and are qualitatively similar to those given by micro- emulsions prepared with different hydrocarbon media ; the error bars again include variations from several different runs and therefore incorporate any uncertainties arising from sample preparations. Note that at low values of R the errors are mainly associated with sample preparation as well as unfavourable statistics (due to low count rates) while at high R there are increasingly larger uncertainties arising from the least-squares fits (due to the fact that the droplets are larger, and consequently most of the intensity profile falls outside the accessible Q-range). Tests were made at R = 20 with samples that had been freshly prepared before the run and other microemulsion samples that had been measured in an earlier experiment, two months previously.The two data sets were in good agreement and did not indicate any change in droplet size. The results are also in satisfactory agreement with those reported by Cabos and de Lord,5 although their radius values were extracted from a simpler analysis based on Guinier-radius formalism and therefore restricted to a much smaller Q-range . The overall features are therefore well represented by the linear relationship between rw and R which can be used to predict approximate values that are in accordance with a comprehensive photon correlation spectroscopic study of AOT-stabilised micro- emulsions in iso-octane.2 For larger R values (> 40) the available data suggest that there are significant deviations from this simple relationship which are probably due to the partitioning of the surfactant between the interface and the oil medium." Furthermore the current analysis is based on monodisperse size of the droplets and there is increasing evidence (see next section) to indicate that the spread in droplet size is much greater than has previously been thought.If this is the case there is no reason to expect a linear relation between rw and R, and it may be dangerous to assign too much significance to the value for the effective head-group area of the AOT molecule at the interface which may be extracted from the data. 2 FAR 120 SANS FOR AOT MICROEMULSIONS TREATMENT OF POLYDISPERSITY The results of the previous section show that radius values can be readily extracted from the observations using a simple formulation based on the assumption of a monodisperse distribution. However, a close examination of the experimental data shows that there are systematic differences particularly in the region where there is a minimum in the fitting function.This indicates that the detailed shape of the SANS pattern contains further information about the contrast profile in the sample material. For spherically symmetric particles the intensity can be more generally expressed as where QA(Q) = s" rp(r) sin Qr dr 0 and p(r) is the contrast profile for the particle. A constant value for p ( r ) with a sharp cut-off at the interface leads directly to eqn (1). In a real system it would be expected that the interface region will not be sharp and the presence of the head-group and associated counter-ions may alter the shape of the profile.However, the resulting A(Q) function will still have an oscillatory form, so that I(Q) will always have a zero at points where the curve crosses the axis. Hence the introduction of a diffuse interface is unlikely to explain the deviations from eqn (1). The introduction of elliptical distortion for the droplet shape changes this situation slightly but does not have a large effect on the results unless large eccentricities are used. The uncertainty in neutron wavelength associated with the velocity selectors used in our experiments was of the order of f 10%. This can be expected to cause a certain amount of flattening of the intensity profile in the higher region of Q, but it cannot account for the systematic changes observed in the profile as a function of the hydrocarbon medium.It is clear from the experimental data that there is no zero-scattering region [i.e. I(Q) = 01 in the high-Q part of the SANS intensity pattern. This immediately implies a variation in the sizes of the particles such that the minimum occurs at different Q values for different particle sizes and is smeared out in the total pattern. The observed intensity therefore becomes a weighted mean of independent contributions from the different sizes, i.e. where w(r) is an appropriate weighting function. In order to study the effects of droplet-size variation on the SANS profile, some assumptions must be made about the shape of the distribution which defines the weighting function w(r).Unfortunately there are no theoretical predictions based on statistical-mechanical principles, and it has been necessary to choose three simple models based on empirical distribution functions to study the effects on the SANS pattern. The mathematical basis of the three models is given in the Appendix for symmetric, triangular and concave shapes of the radius distribution function, p(r). An index of polydispersity, A, is also introduced which is linearly related to o/f where o is the spread and F is the mean for the distribution. The effect of polydispersity on the shape of the intensity pattern is illustrated in fig. 5 for the triangular distribution [eqn (6)]. As the index of polydispersity, A7 is increased, the minimum in I(Q) for the monodisperse system [curve (a)] becomes lessB.H. ROBINSON, C. TOPRAKCIOGLU, J. C. DORE AND P. CHIEUX 21 1 .oo 0.75 0.50 0.2 5 0.10 0.15 0.20 0.05 0.10 0.15 0.20 QIA Fig. 5. Effect of increasing polydispersity on the SANS intensity pattern using a triangular distribution (model 2). Curves are shown for fixed P and for different values of I : (a) 0, (b) 0.2, (c) 0.4 and ( d ) 0.6. The region of the minimum is shown on a larger scale. pronounced [curve (b)] and is eventually smoothed to give a monotonically decreasing curve [curves (c) and (41. There is also a change in the value of I(O), which eventually increases with il. It is immediately apparent that the experimental data are more closely related in general shape to curves (c) and (d) than to curves (a) and (b).The overall behaviour is similar for the other two models [eqn (5) and (7)] and this suggests that the level of polydispersity in AOT microemulsions is much larger than is normally assumed. EVALUATION OF POLYDISPERSITY MODELS Although the three models of polydispersity have been introduced in an empirical way in the absence of any detailed theoretical formulation, it is convenient to compare the results directly with the experimental data in order to investigate the most likely shape of the distribution function, p(r). This has been carried out by means of a x2 fit for the data on water/AOT/decane at R = 20, using il as an adjustable parameter. As expected, there is a substantial improvement in the fit as shown by the x2((a) dependence given in fig. 6.The results are found to be strongly model-dependent and the best fit is obtained with the concave parabolic distribution. In all cases the polydispersity index is large and it is therefore clear that there are substantial departures from the monodisperse condition (13. = 0). The results are summarised in table 2 and the improvement to the fit is shown in fig. 7. 2-222 10 x / \ \ , 0.2 0.L 0.6 0.8 1.0 h Fig. 6. Variation of x2 for water/AOT/decane (R = 20) using different models of polydispersity (see Appendix for details). The arrows indicate the optimum values of ,I corresponding to the fits shown in fig. 7. Table 2. Parameter values obtained from the fit to Iobs(Q) for D,O/AOT/decane" 0 monodisperse [eqn (l)] 0 123 - symmetric (eqn (5)] 0.2 66 31 31 3 0.09 triangular [eqn (6)] 0.8 15 25 19 9.6 0.50 concave [eqn (7)] 0.6 9 32 22 7.5 0.34 - - a [AOT] = 0.1 mol dm-3, R = 20, using different models for polydispersity where rm = (rl + r2)/2.DISCUSSION The present SANS study of AOT-stabilised microemulsions suggests that the structural composition of the droplets is not as simple as had been previously thought. It seems clear that polydispersity is strongly linked to the dynamics of droplet collisions. This implies a continuous process in which two droplets collide and formB. H. ROBINSON, C. TOPRAKCIOGLU, J. C. DORE AND P. CHIEUX 23 0.75 0.50 0.2 5 0.05 0.10 0.15 0.05 0.10 0.15 0.20 Q/A Fig. 7. Experimental data and optimised fits corresponding to different models of polydispersity for water/AOT/decane ( R = 20); [AOg = 0.1 mol dm-s: (a) monodisperse, R = 0, (6) sym- metric (model l), il = 0.2, (c) triangular (model 2), 1 = 0.8, ( d ) concave (model 3), il = 0.6.an aggregate which subsequently disintegrates. l1 Reaction kinetics can give a valuable insight into these processe~.~*~ l5 There is some evidence4? l1, l6 to suggest that for a given value of R, a certain fraction of the surfactant molecules reside in the oil and continuous phase in dynamic equilibrium with those associated with the interface. Since droplet collisions leading to aggregation are expected to result in the expulsion of some surfactant molecules from the interface in view of the reduced surface-to-volume ratio of the aggregate, surfactant partitioning may be linked to po1ydispersity.l' Calculations based on the droplet-size distribution models considered here show that the total interfacial area of a microemulsion system is a decreasing function of polydispersity. Several lines of further investigation will be required to elaborate the findings of the present measurements.For low-R values the AOT/heptane system appears to follow24 SANS FOR AOT MICROEMULSIONS the idealised behaviour reasonably well, and this can be readily understood by reference to the photon correlation spectroscopy results for AOT/iso-octane obtained by Zulauf and Eicke.2 The deviations from ideal behaviour are most pronounced for systems close to the phase-transition region. From fig. 1 it is clear that at R = 20 and T = 293 K the higher-temperature phase boundary is progressively approached as the solvent hydrocarbon chain length is increased in the order heptane, decane, dodecane. The decane system can be analysed by the polydispersity approach discussed in this paper, but in the dodecane system [fig.3(a)] critical phenomenal7-lg are already becoming apparent in the low-Q region. In this connection it is also of interest to extend the study to R values > 40. As the droplets increase in size it becomes necessary to cover a much lower Q-range, typically down to 5 x A-l, which is accessible using the D11 instrument at ILL. Another method of approaching the transition region is by temperature variation. It is already known that there are small droplets present even when the system is visibly turbid,20 but it is not known for how long these persist.In addition, the investigation of structural properties over lengthy time periods to check for possible changes is now an important aspect of both the stability and polydispersity features. Unfortunately the beam schedules for neutron experiments do not make these time-dependent measurements very easy, but it would now seem desirable to extend these studies, particularly for conditions close to the transition. Another aspect of the neutron-scattering technique has not yet been utilised in the work presented here. This concerns the ability to vary the contrast profile H/D substitution in both the water and solvent media. In addition to providing information on the overall droplet size, including the surfactant thickness, it can be shown that the shape of the SANS profile is very sensitive to polydispersity effects as well as shape variation and solvent penetration into the surfactant coat.The neutron measurements are therefore capable of providing a more detailed picture than has been achieved in this initial work. It may also be possible to refine the method of fitting the data. It is feasible that more complex forms for the distribution could give a better fit, and there is some indication that a bimodal distribution may be more representative for the system. Other studies4 also support this viewpoint, but our present results do not justify such a treatment. APPENDIX: MODELS FOR POLYDISPERSITY In order to investigate the effects of polydispersity it is convenient to define a radius dis- tribution function, p(r), which will determine the weighting function w(r) of eqn (4).The form of this function is not known a priori and therefore several simple relationships may be chosen to illustrate the sensitivity of lobs(Q) to the parameters defining the distribution. The integral runs over all possible values of r (i.e. 0-co) but in practice there must be a range which is defined by the physical constraints of the system. In order to provide a mathematically convenient formalism which is correctly normalised, it is useful to consider models with a well defined range of r values from rl to r2. Three models have been chosen for p(r) where p(r) dr is the probability of finding a droplet of radius between r and r+dr. The peak probability is represented by po and the mean value is with a variance, 0, given by where the integrals are evaluated between the limits rl and r2.B.H. ROBINSON, C. TOPRAKCIOGLU, J. C. DORE AND P. CHIEUX 25 It is also useful to introduce a dimensionless parameter A as an index of polydispersity. It is defined as a ratio of the spread to the mid-point of the range, i.e. r2-r1 A=- r2 + rl and is linearly related to the variance for each model distribution. ( a ) Model 1 : symmetric j p " r - (r - r2) , r , < r < r , otherwise. This has the form of an inverted parabola with a mean value of P k - 1 = (r-r2I2 and a variance such that (b) Model 2 : triangular ( 5 ) I 0 otherwise. This has a linear distribution with a maximum at the lowest value of the range. The mean value and variance are - r2+2r1 r = - 3 (c) Model 3: concave I 0 otherwise.This gives a more strongly peaked distribution at lower r values in the range. It is therefore asymmetric with a parabolic shape. The mean value and variance are - r2+3r, r=------- 4 0 = (r2 -rl). All the curves are modelled as simple power-law relationships and are schematically illustrated in fig. 8. Since the droplet volume is also of interest the corresponding distribution, A V ) , is shown where p(r)dr =f(V)dV and A V ) = -. 4nr226 SANS FOR AOT MICROEMULSIONS m o d e l 1 I m o d e l 2 30 60 X f IX 1: I -j( I 30 50 i "!$& i- Fig. 8. Shapes of the p ( r ) andfTV) distribution functions for the three models of polydispersity (see text for details). The total volume of water in the system may be expressed as 7 2 V, = n jn r2p(r) dr 7-1 where n is the number of droplets.This provides a relationship between po, V, and n for each model and shows that p o may be treated as a scaling parameter. For a fixed volume of water, V,, the number of droplets will be dependent on the shape of the p(r) distribution function. The observed SANS pattern [eqn (4)] can therefore be expressed as [l ~ ( r ) W , r ) dr W ) d r L s ( Q ) = V, ,.,.* rl where V(r) = %nr3. For a case where the total water volume is V,. Strictly the volume V, appropriate to the present experimental studies should also include a contribution from the surfactant head-groups but this is a minor correction in the present treatment. Each model therefore defines a function which represents the SANS profile and is dependent on the chosen range given by the parameters rl and rz.A value for A can readily be obtained by comparing the intensity pattern for the model distribution with the observed data, Zobs(Q). The results are presented in the main text.B. H. ROBINSON, C. TOPRAKCIOGLU, J. C. DORE AND P. CHIEUX 27 The work was carried out at A.E.R.E., Harwell and Institut Laue-Langevin, Grenoble. We thank Vic Rainey (A.E.R.E.) for assistance during the experiments and the Neutron Beam Research Committee of the S.E.R.C. for financial support. 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