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Langmuir 1996, 12, 3876-3880
Mixing of Alkanes with Surfactant Monolayers in Microemulsions Julian Eastoe,* Karen J. Hetherington, Donal Sharpe, and Jinfeng Dong School of Chemistry, University of Bristol, Bristol BS8 1TS, U.K.
Richard K. Heenan ISIS Facility, Rutherford Appleton Laboratory, Chilton, OXON OX11 0QX, U.K.
David Steytler School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, U.K. Received February 27, 1996. In Final Form: May 31, 1996X The internal structure of water-in-oil microemulsion droplets has been studied by small-angle neutron scattering (SANS). Dichained surfactants with different tail structures were investigated: synthetic phosphatidylcholines, dialkyldimethylammonium bromides, and Aerosol-OT. The oil was cyclohexane, but for AOT n-heptane was used. The data from core-shell-drop neutron contrast series were globally analyzed to investigate the possibilities that water and/or oil penetrates the curved surfactant film. Three different models for the interfacial structure were tested: sharp-step as well as linear or exponential changes in neutron-scattering length density. Penetration of water was not detected. For phosphatidylcholine and n-octadecyl-n-dodecyldimethylammonium bromide layers, the SANS data were most consistent with a volume fraction of cyclohexane ΦC6D12 between about 0.05 and 0.20. However, for DDAB and AOT there was no clear evidence for any significant oil mixing. These results indicate that the extent of alkane penetration into such negatively curved monolayers depends somewhat on the surfactant alkyl chain structure.
Introduction The understanding of surfactant monolayers at airwater interfaces has been improved by new advances in neutron reflectometry and computational methods (e.g. refs 1 and 2). However, less detail is known about surfactants at oil-water interfaces. An unresolved question is, “To what extent do water and oil penetrate the surfactant film?” Since microemulsions are thermodynamically stable, they are ideal systems for studying these effects. Small-angle neutron scattering (SANS) is a versatile method for investigating structure in water-surfactantoil mixtures, as different components can be highlighted using contrast variation. Strey et al. analyzed SANS data from nonionic surfactant films in bicontinuous C12E53 and droplet C10E44 microemulsions assuming a Gaussian distribution of solvent across the interface. In this model the extent of penetration of water and oil is identical, and this may not strictly be the case, especially at highly curved interfaces. We have recently used SANS to study waterin-oil (w/o) microemulsion droplets formed by n-alkyl-ndodecyldimethylammonium bromide (Cn-C12) surfactants.5 The C12-C12 compound is commonly known as DDAB. The SANS data from a contrast variation series were globally analyzed using a sharp step model, which * To whom correspondence should be addressed. E-mail: julian.
[email protected]. Telephone: U.K. +117-9289000 ext 4726. Fax: U.K. +117-9250612. X Abstract published in Advance ACS Abstracts, July 15, 1996. (1) (a) Lu, J. R.; Marrocco, A.; Su, T. J.; Thomas, R. K.; Penfold, J. J. Colloid Interface Sci. 1993, 158, 303. (b) Lu, J. R.; Thomas, R. K.; Aveyard, R.; Binks, B. P.; Cooper, P.; Fletcher, P. D. I.; Sokolowski, A.; Penfold, J. J. Phys. Chem. 1992, 96, 1097. (2) Smit, B.; Hilbers, P. A. J.; Esselink, K.; Rupert, L. A. M.; van Os, N. M.; Schlijper, A. G. Nature 1990, 348, 624. (3) Strey, R.; Winkler, J.; Magid, L. J. Phys. Chem. 1991, 95, 7502. (4) Gradzielski, M.; Langevin, D.; Magid, L.; Strey, R. J. Phys. Chem. 1995, 99, 13232.
S0743-7463(96)00178-3 CCC: $12.00
allows for different amounts of water and oil to mix homogeneously into the film. No evidence was found for water in the layers; however, the apparent volume fraction of cyclohexane ΦC6D12 increased from zero for C12-C12 to about 0.08 for the asymmetric chain C18-C12. Here this approach has been extended to linear and/or exponential mixing profiles and also dichain surfactants with different structures. The presence of carbon-carbon double bonds, or chain asymmetry, in the surfactant chains is expected to increase disorder within the layer and so enhance oil penetration. It was found that the sharpstep model is most consistent with the data. Furthermore, for synthetic di-C18:1 phosphatidylcholine (PC) cis or trans isomers, or the cationic C18-C12, the value of ΦC6D12 lies in the range 0.05-0.20. It is difficult to differentiate the fits for homogeneous and linear models, except by also considering certain fitted values. For the saturated alkyl chain surfactants with two equal chains, C12-C12 (DDAB) and Aerosol-OT, no clear evidence was found for any significant alkane mixing (cyclohexane and n-heptane, respectively). The formation of w/o systems, using PC’s similar to those described here, has been reported previously (e.g. refs 6-8), but there have been no detailed studies of the structure of monolayers in microemulsions. (5) (a) Eastoe, J.; Dong, J.; Hetherington, K. J.; Steytler, D. C.; Heenan, R. K. J. Chem. Soc., Faraday Trans. 1996, 92, 65. (b) Eastoe, J.; Heenan, R. K. J. Chem. Soc., Faraday Trans. 1994, 90, 487. (6) (a) Schurtenberger, P.; Scartazzini, R.; Magid, L. J.; Leser, M. E.; Luisi, P.-L. J. Phys. Chem. 1990, 94, 3695. (b) Peng, Q.; Luisi, P.-L. Eur. J. Biochem. 1990, 188, 471. (c) Schurtenberger, P.; Peng, Q.; Leser, M. E.; Luisi, P.-L. J. Colloid Interface Sci. 1993, 156, 43. (7) (a) Shinoda, K.; Araki, M.; Sadaghiani, A.; Khan, A.; Lindman, B. J. Phys. Chem. 1991, 95, 989. (b) Shinoda, K.; Shibata, Y.; Lindman, B. Langmuir 1993, 9, 1254. (8) (a) Kahlweit, M.; Busse, G.; Faulhaber, B. Langmuir 1995, 11, 1576. (b) Kahlweit, M.; Busse, G.; Faulhaber, B.; Eibl, H. Langmuir 1995, 11, 4185.
© 1996 American Chemical Society
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Table 1. Scattering Length Densities Used in the Analysis of SANS Dataa 10-10F/cm-2 C6H12 C6D12 C7H16 C7D16 H2O D2O cis- and trans-PC C12-C12 DDAB C18-C12 AOT
-0.28 +6.67 -0.55 +6.30 -0.56 +6.40 +0.30 -0.44 -0.46 +0.58
a For solvents the mass densities were taken as the literature values at 25 °C, and for surfactants 1.0 g cm-3 was assumed.
Figure 1. Schematic scattering length density profiles F(r) used to fit SANS data from droplet microemulsions. This example shows the shell contrast (D/H/D). For the sharp model (a) ‚‚‚ represents without and s with solvent mixing.
Core-Shell-Drop Model Experimental Section The synthetic phosphatidylcholines 1,2-dioleoyl-sn-glycero3-phosphocholine (cis-PC) and 1,2-dielaidoyl-sn-glycero-3-phosphocholine (trans-PC) were obtained from Avanti Polar Lipids. These two isomers (MW ) 786.1) have either a cis or trans CdC bond on average halfway along the C18 chains. The phosphocholines were stored at -20 °C and used without further purification. The di-n-dodecyldimethyl- and n-octadecyl-ndodecyldimethylammonium bromides (C12-C12 and C18-C12, respectively) were prepared, purified, and analyzed as described previously.5 Aerosol-OT (AOT, Sigma) was used as received. These surfactants were stored over refreshed P2O5 in a dessicator cabinet until used. 1H-NMR (Jeol GX270, using CDCl3 as solvent) and CHNBr elemental analysis were used to confirm the structure and purity of the surfactants. Cyclohexane-d12 and D2O, from CDN Isotopes, and n-heptane-d16, from MSD, were used as received. The deuterated chemicals contain >99% D atom. Cyclohexane-h12 and n-heptane-h16 were obtained from Aldrich and purified by column chromatography over alumina before use. H2O was doubly distilled. The microemulsion phase equilibria were determined by visual inspection of samples made up in stoppered 5 mL volumetric flasks, thermostated to (0.1 °C. The composition parameter w is given by [water]/[surfactant]. The Winsor II systems consisted of a w/o microemulsion, at a maximum value wmax, in equilibrium with excess water. At 25 °C in cyclohexane the values of wmax were 20.0, 22.0, 12.0, and 10.0 (to (1) for the cis-PC, trans-PC, C12-C12 (DDAB), and C18-C12 surfactants. For AOT at w ) 40.0 the Winsor II solubilization phase boundary occurs at 12.0 °C.9 The SANS experiments used the LOQ instrument at ISIS, U.K. The procedures for data normalization, reduction, and background subtraction were all carried out as described previously.5,10 The measurements determine the absolute scattering probability I(Q) (cm-1) as a function of momentum transfer Q ) (4π/λ) sin(θ/2) with λ the incident neutron wavelength (2.210 Å) and θ the scattering angle ( 0.04 Å-1 (ref 11)) is also included in the modeling. (9) Howe, A. M.; Fletcher, P. D. I.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1 1987, 83, 985. (10) (a) Eastoe, J. In New Physico-Chemical Techniques for the Characterisation of Complex Food Systems; Dickinson, E., Ed.; Blackie: Glasgow, 1995; Chapter 1, pp 268-294. (b) Heenan, R. K.; King, S. M. In Proceedings of the International Seminar on Structural Investigations at Pulsed Neutron Scources, Dubna, Russia, Sept 1992; Publication E3-93-96; JINR: Dubna, 1993. (c) Heenan, R. K. FISH Data Analysis Program. Rutherford Appleton Laboratory Report RAL-89-129; 1989. (11) Cummins, P. G.; Staples, E.; Penfold, J. J. Phys. Chem. 1990, 94, 3740.
For polydisperse spherical particles at volume fraction φ, radius Ri, volume V, and coherent scattering length density Fp, dispersed in a medium of Fm, the normalized SANS intensity I(Q) (cm-1) may be written
∑i P(Q,Ri) X(Ri)]S(Q,Rhs,φhs)
I(Q) ) φV(Fp - Fm)2[
(1)
P(Q,Ri) is the single-particle form factor. The Schultz distribution X(Ri)12 defines the polydispersity using an average radius Rav and RMS deviation σ ) Rav/(Z + 1)1/2 with Z a width parameter. S(Q,Rhs,φhs) is the structure factor, and here a hard-sphere model modified for polydispersity12 was used. The scattering law P(QR)cs for spherical particles composed of a core plus multiple shells, with sharp interfaces, has been given by Ottewill et al.13 Owing to the difference in scattering length density F between hydrogen- and deuterium-containing materials, contrast variation SANS experiments can resolve different domains. In this way scattering from the water core, surfactant shell, or overall droplet can be selected using either D-water/H-surfactant/H-oil (D/H/H), D-water/Hsurfactant/D-oil (D/H/D), or H-water/H-surfactant/D-oil (H/H/D). In principle this core-shell-drop (CSD) contrast series contains information on the core radius Rc, the effective head group and tail thicknesses in the film (h and t), and the overall droplet radius Rdrop ()Rc + h + t). These three data sets are complementary, and any analysis of them must be consistent; this can be ensured by analyzing the files individually as well as simultaneously.5 In the latter approach the model is highly constrained, and any change in the parameters for a particular set must be reflected in the other two. This is expected to yield the most representative structural parameters, a point demonstrated by this work. Mixing of water and/or oil solvent into the surfactant film may also contribute to the scattering, and the effects may be manifest in each of the relevant CSD data sets. The analysis below allowed three different possibilities for the distribution of scattering length density to be modeled in separate head and tail group regions: (a) sharp interfaces with homogenous mixing, (b) linear mixing, and (c) an exponential profile. Model (a) can be readily modified for the case of no solvent mixing. Schematic representations of the final fitted profiles are shown in Figure 1, and details of these models are given in the Appendix. (The lack of evidence for a discrete head group region is discussed below.) Trial calculations, and the limitations of this approach for sharp interfaces (a), have been described in detail previously.5 Here, the three (12) (a) Kotlarchyk, M.; Chen, S.-H.; Huang, J. S.; Kim, M. W. Phys. Rev. A 1984, 29, 2054. (b) Hayter, J. B.; Penfold, J. Colloid Polym. Sci. 1983, 261, 1022. (13) Markovic, I.; Ottewill, R. H.; Cebula, D. J.; Field, I.; Marsh, J. Colloid Polym. Sci. 1984, 262, 648.
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models are compared for five different surfactants, in order to establish which represents the best description of the interfaces. In the modeling a number of parameters were fixed: for S(Q,Rhs,φhs), φhs ) φdrop and Rhs ) Rdropav, and the F values for solvents were fixed as well (Table 1). For any given set of structural parameters the core volume fraction φc defines the absolute scattering intensity,5 and (10% of the known value is allowed in the modeling. Therefore Rcav, σ/Rcav, h, and t, with the addition of Fhead and Ftail for the sharp interface model, may be fixed or adjusted both individually and collectively. Results and Discussion 1. Unsaturated Chain Phosphocholines. Sharp and Linear Interface Models. Figure 2 shows example core, shell, and drop contrast data for cis-PC Winsor II microemulsions. Similar results were obtained for the trans-PC surfactant. The fitted functions for sharp interfaces, both with and without mixing, and also a linear F(r) profile are indicated in Figure 2. Values of the fitted parameters are given in Table 2. The uncertainties in I(Q) are greatest for the lower intensity core data, but for shell and drop contrasts the error bars mostly lie within the data points. The exponential profile did not give good fits to any of the five different microemulsions. In particular, the characteristic oscillations in the shell contrast fitted functions are less well defined as compared to those in the measured profiles. The possibility that both water and oil penetrate the layers was thoroughly investigated using all three types of F(r) function. No clear evidence for any significant hydration of the surfactants was found. Therefore, in the final fits presented here a distinct shell for the surfactant head group is not included, and there are only small differences in the calculated I(Q) profiles for the core contrast case. However, the modeling does suggest that there is some mixing of cyclohexane into the layer. This effect will be most significant in the shell and drop data. In shell contrast, solvent mixing will reduce the total scattering intensity across the entire Q-range.5 If there were no solvent penetration, the film scattering length density Ffilm would be given by that of the surfactant only Fsurf (in Table 1). Making this constraint in the calculations results in a higher intensity than is observed experimentally, and this is shown in Figure 2b and c. On a linear I(Q) vs Q representation at low Q < 0.06 Å-1 the differences are obvious. For example, at Q ∼ 0.02 Å-1 the calculation with Ffilm ) Fsurf gives an intensity of 36 cm-1 as compared with the measured 25 cm-1. Even allowing for the maximum uncertainties in both the intensity calibration and the volume fraction φc still leaves a discrepancy of ∼6 cm-1. When solvent mixing is allowed for, the calculated and measured intensities now agree to better than 1 cm-1. Which is the best model for the film, a sharp or linear scattering length density profile? Firstly, it is important to note that the fitted values for Rcav and σ/Rcav are very similar for the two cases (Table 2). Looking at the I(Q) profiles only, it is very difficult to tell these two possibilities apart, although for the drop contrast the sharp model appears marginally better in the region of the oscillation at high Q. Figure 3 shows a plot of I(Q)Q2 vs Q for the shell data, and this serves to accentuate the high Q-range, 0.06 < Q < 0.20 Å-1, where the signal is most sensitive to the nature of the interface. As expected, the predicted intensity is consistently too high if no mixing is assumed, and including the mixing improves the fit. Visually the sharp model appears marginally better, and also the residuals are slightly lower (e.g. SWSE ) 7 × 103 vs 9 × 103 for the linear profile). As can be seen in Table 2 the
Figure 2. Simultaneous analysis of SANS data from a microemulsion contrast series. The surfactant is cis-PC at 0.05 mol dm-3, the water content wmax ) 20.0, and cyclohexane is the continuous phase. T ) 25 °C. Error bars are shown. (a, top) core O (D/H/H); (b, center) shell b (D/H/D); (c, bottom) drop 0 (H/H/D). Fitted interfacial profile: (s) sharp with cyclohexane mixing; (‚‚‚) sharp without mixing; (- - -) linear.
apparent layer thicknesses for the sharp (ts) and linear (tlin) interface agree well for the trans-PC but not for the cis-PC isomer. Using the linear model for both surfactants, the apparent volume fractions ΦC6D12 given in Table 2 are approximately 0.60, and this does seem rather high. On balance the sharp interface description fits slightly better, and it also gives a more physically reasonable value for
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Table 2. Values Obtained from a Simultaneous Analysis of Contrast Variation SANS Data of Water-in-Oil Microemulsionsa sharp interface cis-PC trans-PC C12-C12 (DDAB) C18-C12 AOT
linear gradient
w
Rcav/Å
σ/Rcav
ts/Å
Φoil
Rcav/Å
σ/Rcav
tlin/Å
Φoil
20 22 12 10 40
30.0 31.6 19.7 18.6 69.7
0.21 0.24 0.25 0.18 0.19
15.8 26.1 11.0 13.5 7.4
0.10 0.20 0 0.08 0
29.5 30.9 18.7 19.3 62.9
0.21 0.23 0.20 0.18 0.20
25.6 24.0 19.1 19.3 15.4
0.60 0.59 0.61 0.61 0.53
a The parameters are as described in the text. Cyclohexane is the oil for the phosphatidylcholine and cationic surfactants, whilst for AOT n-heptane was used. Φoil is the apparent volume fraction of oil in the surfactant film. Concentrations: phosphatidylcholines, 0.05 mol dm-3; cationic surfactants and AOT, 0.10 mol dm-3. Temperature ) 25 °C.
Figure 3. Comparison of different models for the cis-PC film with shell contrast data. [cis-PC] ) 0.05 mol dm-3, wmax ) 20.0, T ) 25 °C. Error bars are shown. Fitted interfacial profile: (s) sharp with cyclohexane mixing; (‚‚‚) sharp without mixing; (- - -) linear.
Φoil. The effects of concentration and surfactant isomer type are now discussed with respect to this model. Droplet Concentration Variation. The effects of droplet concentration on the fitted parameters were carefully checked for all of the microemulsion systems. The data were analyzed with the sharp interface model, both individually and simultaneously. As an example Figure 4 shows fitted values for Rcav, σ/Rcav, and ts, obtained for the cis-PC Winsor II system at three different core volume fractions. The agreement is typical of all the other systems that were studied. A constant value for Rcav is expected for droplets in a Winsor II system at a maximum water loading wmax, and the polydispersities are typical of many other microemulsions.5 Note that the simultaneous analysis gives the most consistent parameter values, particularly for the penetration of solvent. cis- vs trans-PC Films. The length of a di-C18:1-PC molecule is approximately 26 Å.14 Owing to rotational isomerism a difference in the film thicknesses is to be expected, and the trans layer should be the thicker one. The values are given in Table 2 and for cis-PC ts is approximately 16 Å, whereas for the trans-PC it is ∼26 Å. Although there is poor agreement in the ΦC6D12 values for the cis-PC obtained by the individual and simultaneous methods, it is clear that there is some solvent penetration (Figure 4). The volume fractions ΦC6D12 given in Table 2 also indicate that the extent of this penetration appears to depend on the surfactant chain configuration. Presumably, owing to the more “linear” chain conformation, (14) Small, D. M. In The Physical Chemistry of LipidssThe Handbook of Lipid Research; Plenum Press: New York and London, 1986; Vol. 4.
Figure 4. Fitted parameters obtained at different core volume fractions φc for the cis-PC microemulsions at wmax ) 20.0 and 25 °C using the sharp interface model. The CSD data sets were analyzed simultaneously (b) and individually (O).
the trans isomer forms a thicker shell which is able to accept more oil than the thinner film formed by the cis isomer. 2. Saturated Chain Ionic Surfactants. For the asymmetric chain C18-C12 surfactant the best fits were obtained when solvent mixing was included. Figure 5 shows the shell data plotted as I(Q)Q2 vs Q, along with fits to the sharp and linear film profiles. In this case it is clear that the sharp model is the best description, and it also gives a SWSE half of that obtained for a linear F(r) function. The analysis is consistent with a volume fraction ΦC6D12 ) 0.08 (Table 2). When the two alkyl chains are saturated and of equal length, as in C12-C12 (DDAB) and AOT, the contrast series data were best modeled using sharp interfaces but with no solvent mixing. The fitted parameters are given in Table 2. The values obtained for the core radii Rcav agree well with other SANS work (e.g. refs 5 and 15). When the linear gradient model was used, the film thickness was greater than the surfactant molecular length, and the residuals were also poor. For C12-C12 SWSE was two (15) Eastoe, J.; Young, W. K.; Robinson, B. H.; Steytler, D. C. J. Chem. Soc., Faraday Trans. 1990, 86, 2883.
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thanked for provision of a studentship to K.J.H., funds for travel, subsistence, and consumables for the SANS experiments. A Sino-British Fellowship, and grant for consumables, was awarded to J.D. Prof. Brian H. Robinson (UEA) and Dr. Paul Fletcher (Hull) are thanked for stimulating discussions. Appendix The single particle form factor P(Q) is F(Q)2, where for a spherically symmetric particle of scattering length density F(r) at radius r
∫0∞r2F(r)
F(Q) ) 4π
Figure 5. Comparison of different models for the C18-C12 film with shell contrast data. [C18-C12] ) 0.10 mol dm-3; wmax ) 10.0; T ) 25 °C. Error bars are shown. Fitted interfacial profile: (s) sharp with cyclohexane mixing; (‚‚‚) sharp without mixing; (- - -) linear.
times, and for AOT nine times, higher for the linear profile than with the sharp model. This, taken together with the high values of Φoil that are found with the linear F(r), suggests that the sharp model is best. The simultaneous analysis method was also tested with contrast data from nonionic C12E5-stabilized w/o and o/w microemulsions.16 In all cases inconsistent values for the fitted parameters and poor fits were obtained. Further work is required to clarify whether structural fluctuations and/or subtle effects of solvent deuteration are the causes of these discrepancies. Conclusions A simultaneous analysis of SANS contrast series data can be used to investigate the internal structure of w/o microemulsion droplets. Different models for the interfacial structure, allowing for water and oil penetration into the film, have been applied to a range of different dichain surfactants. In all cases an exponential scattering length density profile F(r) for the film is incompatible with the data. Furthermore there is no evidence for any significant penetration of water into the layers. When either CdC bonds or chain asymmetry are introduced into the surfactant tails, the analysis is consistent with a mixing of alkane into the layer. In these situations it is difficult to tell from the fitted functions alone if the solvent penetration is homogeneous or varies linearly across the film. This may be due to the inherent polydispersity and dynamic nature of the microemulsion droplets, which tends to smear out any of these details. Despite these problems, the sharp model seems to be the most physically reasonable one, since the values for the volume fraction Φoil are more realistic. For the unsaturated chain phosphatidylcholines, and asymmetric chain alkylammonium bromides, the apparent volume fraction of cyclohexane in the layer is in the region 0.05-0.20. With the more common surfactants DDAB and AOT, both having symmetric saturated tail groups, the analysis infers that there is no significant penetration of alkane into the monolayers. Acknowledgment. A BBSRC research grant was awarded to study the phosphocholine systems. D.S. thanks BBSRC for a postdoctoral fellowship. EPSRC is (16) Binks, B. P.; Fletcher, P. D. I.; Eastoe, J.; Heenan, R. K. Unpublished work.
sin(Qr) dr Qr
(A1)
When F(r) takes a complex form, as for the cases in Figure 1, this Fourier integral may be split into a sum of terms for each step or feature. For a vertical step of upward of ∆F at radius a, as in the sharp interface model:
F(Q) ) -4π∆F(sin(Qa) - Qa cos(Qa))/Q3 (A2) The term 4π/Q3 is often replaced by 3V/(Qa)3, where V is the volume of the sphere. The sharp interface in Figure 1a requires a sum of two such terms at r ) a and (a + ts).13 If instead F(r) increases linearly from zero to ∆F between radii a and b ) (a + tlin), as in the linear gradient model of Figure 1b, then:
F(Q) ) 4π∆F{2(cos(Qa) - cos(Qb)) + Qa sin(Qa) - Qb sin(Qb)}/(Q4(b - a)) (A3) For the same shell volume the equivalent thickness ts for the sharp interface model is given by the solution of (a + ts)3 ) {(a + tlin)4 - a4}/(4tlin). Summation of eq 1, for I(Q), over polydispersity X(Ri) for the core radius was treated numerically by a 64 point Gaussian quadrature. To model a more diffuse boundary, Gradzielski et al.4 used a symmetrical Gaussian distribution for F(r), which, after approximating the lower integration limit of eq A1 to -∞, gives an analytic equation for F(Q). Unfortunately this is only realistic for a well matched “shell” contrast, and, since the integral for a “half Gaussian” appears not to be analytic, may not easily be applied to core or droplet contrasts except by numerical integration. A possible alternative is an exponential scattering density profile (Figure 1c). If F(r) decreases from a value ∆F at r ) a to zero at infinity as F(r) ) ∆F exp{-(r - a)/L}, then
F(Q) )
(
4π∆F Qa(hQL sin(Qa) - cos(Qa)) + Q (Q2L2 + 1) (1 + 3Q2L2) sin Qa + 2hQ3L3 cos(Qa) (A4) (Q2L2 + 1) 3
)
may be derived where the constant h ) +1. For the purposes of this work a finite shell thickness of 3.5L was assumed by setting F(r) to zero above (a + 3.5L). This was achieved by subtracting a further term (eq A4) for the exponential shell from (a + 3.5L) to infinity, and a term (eq A2) for the small vertical step was introduced. For completeness it is noted that if F(r) increases from zero at minus infinity to ∆F at r ) a according to F(r) ) ∆F exp{-(a - r)/L}, then h ) -1 in eq A4. Again by adding appropriate terms it is possible to truncate F(r) at say r ) (a - 3.5L) or at r ) 0. LA960178S
