4638
J. Phys. Chem. B 1999, 103, 4638-4648
The Structure and Composition of Ethyl Hexadecanoate Layers Spread on Aqueous Solutions of Hexaethylene Glycol Monododecyl Ether J. R. Lu* and T. J. Su Department of Chemistry, UniVersity of Surrey, Guildford GU2 5XH, U.K.
M. J. Lawrence, D. J. Barlow, W. Warisnoicharoen, and T. Zuberi King’s College London, Manresa Road, London SW3 6 LX, U.K. ReceiVed: July 29, 1998; In Final Form: December 3, 1998
Specular neutron reflection combined with deuterium labeling has been used to study the structure and composition of a mixed layer of oil and surfactant at the air/water interface, formed by spreading ethyl hexadecanoate from small lenses placed on the surface of hexaethylene glycol monododecyl ether (C12E6). The area per oil molecule was found to be 128 ( 10 Å2 in the mixed monolayer and that of C12E6 to be 54 ( 3 Å2, identical to the value obtained in the absence of the oil. The thicknesses for the dodecyl chain layer and ethoxylate layer were 18.5 ( 2 Å and 20 ( 2 Å, assuming Gaussian distributions, as compared with the respective values of 16 ( 2 Å and 16.5 ( 2 Å without the oil, suggesting that oil mixing has broadened the distributions of the two fragments. The extent of penetration of the ester oil into the surfactant layer was described by the relative distances between the centers of the distributions. The use of alkyl chain deuterated and ethoxylate headgroup deuterated surfactants allowed separate determinations of the relative distances between the centers of the distributions of the dodecyl chains and of the oil (δco) and between the ethoxylate headgroups and the oil (δeo). The value of δco was found to be 0.5 ( 1 Å and that of δeo to be 10.5 ( 1 Å, showing that while the center of the ethoxylate headgroups was further into aqueous solution the ester oil layer was almost entirely overlapped with the distribution of the dodecyl chain layer. On the basis of comparisons made with the results from other systems involving the spreading of dodecane and dodecanol onto the layers of ionic surfactants and nonionic alkyl ethoxylates, we attribute the deep insertion of the ester oil into the C12E6 layer to the combined effect of the affinity of the ester groups toward water and the affinity of the oil toward the ethoxylate groups.
Introduction The use of long chain alcohols and esters in the formulation of detergent products has been a well-known practice. The organic additives are usually thought to intermix with the surfactant monolayers surrounding the aggregates, resulting in the modification of the aggregational properties. The role played by these organic additives has been demonstrated by many studies using model surfactant systems. For example, addition of n-pentanol into sodium dodecyl sulfate (SDS) solution increases the size of the droplets, leading to the formation of microemulsions.1 Pentanol in this case works as a cosurfactant and its mixing into surfactant monolayers alters the spontaneous curvature and bending elasticity of the monolayers at the oil/ water interface. The extent of mixing and penetration of a cosurfactant into a surfactant layer is considered to be a dominant factor in determining the physical properties of microemulsion systems. Surface tension measurement has been used to elucidate various aspects of microemulsion behavior, e.g., the extent of solubilization and phase inversion, but it does not offer any direct information about the mixed interface. In fact, little is known about the structural composition of the interfacial layer at the oil/water interface, especially in the presence of cosurfactant. Although measurements at the oil/water interface are directly relevant to microemulsion systems, few techniques are capable * Corresponding author.
of probing such structural information with acceptable sensitivity. In contrast, similar measurements can be reliably made at the air/water interface although certain differences exist between the air/water and oil/water interfaces, e.g., the amplitude of the capillary waves and the distribution of the alkyl chain region due to the presence of air and bulk oil. We have recently demonstrated that neutron reflection at the air/water interface can reveal valuable information about the surface excess of oil additives and the extent of penetration of oils into different surfactant layers.2 Several oil/surfactant systems have been studied, and the results show that dodecane is partly mixed with the cationic surfactant layer formed by dodecyl trimethylammonium bromide (C12TAB)2 and the cross distance between the centers of the dodecane and dodecyl chain distributions is about 6.5 Å. In contrast, the cross distance between the centers of the dodecanol and SDS distributions is only 3.5 Å, indicating that the dodecanol layer is further inserted into the SDS layer.3 The increased penetration of the dodecanol is attributed to the affinity of the OH groups to aqueous solution which drives the alcohol molecules further into the surfactant layer. In addition, we have found that the surface excess of dodecanol is much higher than that of dodecane. Because the area per surfactant molecule in the two systems is almost the same, these results suggest that the dodecanol/SDS mixed layer is much more densely packed than that of the dodecane/C12TAB system. Very recently, we have also studied the spreading of dodecane onto the surface of pentaethylene glycol monododecyl ether (C12E5)4 and found
10.1021/jp9832239 CCC: $18.00 © 1999 American Chemical Society Published on Web 05/13/1999
Ethyl Hexadecanoate Layers that the cross distance between the centers of the dodecane and surfactant chain distributions is only about 3 Å, comparable to the value found for the dodecanol/SDS system. This high degree of penetration has been attributed to the attractive interaction between the oil and the ethoxylate headgroups. To further explore the interaction between ethoxylate headgroups and hydrophobic fragments within the mixed layer, we have extended this work to the study of the solubilization of a polar oil into a nonionic surfactant monolayer at the air/water interface using neutron reflection. Alkyl esters have been used as oil in this work partly because their polarity is similar to the dodecanol studied previously and partly because they have been used as model cosurfactants in our previous work on the solubilization of model steroids into nonionic surfactant solutions.5,6 Because many drug molecules contain large nonpolar and small polar moieties, their solubility in water is very limited. Aggregated surfactant solution creates a large interfacial area around the surface of aggregates which is amphiphilic in nature and which is ideal for drug solubilization. Our previous studies have shown that addition of alkyl esters can substantially affect the amount of drug solubilized.5,6 For a series of alkyl esters studied, the extent of solubilization was found to be dependent on the alkyl chain length, among many other factors, e.g., the ratio of surfactant to cosurfactant, temperature, and salt. The neutron measurements at the air/water interface will thus offer information on the effect of alkyl chain length on the structural composition of the mixed layer. In the first part of the study we report the characterization of the mixed layer formed by C15H31COOC2H5/C12E6. Experimental Section Two isotopic species of the surfactant were synthesized, C12D25(OC2H4)6OH and C12H25(OC2D4)6OH, abbreviated as dC12hE6 and hC12dE6 respectively. dC12hE6 was prepared by the standard Williamson synthesis from deuterated dodecyl bromide (Merck, Sharp and Dohme, 98%D), an equimolar amount of sodium tert-butoxide, and a 5-fold molar excess of hexaethylene glycol (Fluka, 99%). Hexaethylene glycol was dried overnight with stirring at 40 °C before sodium butoxide was added. When all of the solid powder was dissolved, the reaction flask was connected to a vacuum line to remove the released tert-butyl alcohol. Deuterated dodecyl bromide was then added to the flask. The mixture was stirred and heated to 120 °C for half an hour before leaving it for a further 2 h under stirring. About 10 mL of water was subsequently added to the mixture, followed by addition of HCl to adjust the solution pH to be acidic. The solution was then extracted with ether a few times, and evaporation of the solvent left a light yellow oil which was purified by flash chromatography on a silica column. Head deuterated hC12dE6 was synthesized by reacting ethylene oxided4 (Merck, Sharp and Dohme, 96%D) with dodecanol (Fluka, 98%), and the mixed samples with different headgroup length were separated by flash chromatography. Fully hydrogenated surfactant, hC12hE6, was purchased from Fluka (99%) and was purified before use. The fully hydrogenated and partially deuterated ethyl hexadecanoates were prepared by esterifying the protonated (Aldrich, 99%) or deuterated (MSD, 99%D) hexadecanoic acids with fully hydrogenated (BDH, 96%) or partially deuterated (MSD, 95%) ethanol (CH3CD2OH, 99%D), under standard carbonyldiimidazole coupling conditions. After refluxing the solution overnight, the solvent was removed and the residue dissolved in ether and washed sequentially with 2 M HCl, aqueous NaHCO3, and water. It was then dried, concentrated, and subsequently purified on a silica column to yield the desired esters.
J. Phys. Chem. B, Vol. 103, No. 22, 1999 4639 The neutron reflection measurements were made on the white beam reflectometer SURF at the Rutherford-Appleton Laboratory, ISIS, Didcot, U.K.7 The procedure for performing the measurements has been described previously.8 Surfactant solutions were poured into a Teflon trough to give a positive liquid meniscus. One small ester oil drop was then carefully placed on the corner of the trough in contact with the aqueous solution. The solubility of C12E6 in the ester oil was estimated to be about 1.5%, and the oil was saturated with the surfactant before it was placed on the surface of the surfactant solution. This was done to avoid the strong partitioning of surfactant molecules into excess oil lenses. It should be noted that while a slight excess of oil on the solution surface is necessary to maintain the saturated oil monolayer, its effect on the neutron reflectivity is negligible as discussed elsewhere.2,9 To make sure that the sample can was saturated with oil, four more oil drops were placed on filter papers standing on the edge of the trough. The lid was then closed to avoid evaporation, and the sample carrier was mounted on an antivibration bench. Beam intensities were calibrated with respect to D2O. A flat background determined by extrapolation to the high values of momentum transfer, κ (κ ) (4π sin θ)/λ, where λ is the neutron wavelength and θ is the glancing angle of incidence), was subtracted. The background in D2O was typically around 2 × 10-6 and that in H2O was around 6 × 10-6. This means that when the specular reflectivity is close or below the background level it will suffer from the uncertainty in the background subtraction. All of the experiments were performed at the critical micellar concentration (cmc ) 8 ( 0.5 × 10-5 M) for C12E6 and at 25 °C. High purity water (UHQ) was used for all of the measurements, and all of the glassware and Teflon troughs for the reflection measurements were cleaned using alkaline detergent (Decon 90) followed by repeated washing in ultrapure water. Theory Neutrons are reflected from a surface in the same way as light. The standard theory of reflection of electromagnetic radiation can be applied to interpret neutron reflectivity profiles.10 The reflectivity, defined as the ratio of the intensity of the reflected beam to that of the incoming one, depends on the neutron refractive index variation perpendicular to the interface. Neutrons have two distinct advantages over light. First, the wavelength of neutrons is short, typically a few angstroms. This allows neutron reflection to probe structural information at dimensions close to the molecular level. Second, neutron reflectivity is related to the scattering length or scattering amplitude which varies from isotope to isotope. More than one reflectivity profile can be measured for the same chemical system under different isotopic compositions. Isotopic substitution can thus help to improve the reliability of interfacial profiles derived from neutron reflection measurements. The relationship between neutron reflectivity R(κ) and the scattering length density distribution F(z) is approximately expressed as:11,12
R(κ) )
16π2 |Fˆ (κ)|2 2 κ
(1)
where Fˆ (κ) is the one-dimensional Fourier transform of the scattering length density across the interface
Fˆ (κ) )
∫-∞∞ exp(-iκz)F(z) dz
(2)
The scattering length density is a function of chemical composi-
4640 J. Phys. Chem. B, Vol. 103, No. 22, 1999
Lu et al.
tion across the interface and is given as
F)
∑nibi
(3)
where ni is the number density of the ith element, i.e., the number of atoms per unit volume, and bi is the corresponding scattering length. Equations 1-3 establish a direct relation between neutron reflectivity and interfacial composition. We have described previously11 that at the air/solution interface eq 1 becomes exact if the scattering length density of the water is matched to air, i.e., null reflecting water (nrw). In the case of adsorption from D2O whose scattering length density is nonzero, an equation developed by Crowley12 can be used to scale the measured reflectivity so that the quantitative relation given in eq 1 still holds. The key behind the manipulation of scattering length densities for surfactants or for water is the substitution of hydrogen by deuterium. Because the scattering lengths of D and H are of opposite sign, the scattering length density of water can be varied over a wide range and can be used to highlight an adsorbed layer of surfactant in different ways. For example, null reflecting water is made of approximately one mole of D2O and 11 mol of H2O. When a layer of deuterated surfactant is adsorbed at the air/water interface from this mixed water, the specular reflectivity arises almost entirely from the adsorbed surfactant layer. The area per surfactant molecule, A, is given by
A)
∑mibi Fτ
(4)
1 NAA
TABLE 1: Structural Parameters Obtained from Fitting a Uniform Layer Model to the Mixed Layer on Null Reflecting Water isotope
where mi is the number of atoms of species i and τ is the thickness of the layer. The surface excess Γ is related to A by
Γ)
Figure 1. Neutron reflectivity profiles plotted against κ for the mixed layers of dC12hE6/h-oil/nrw (+), hC12hE6/d-oil/nrw (b), and dC12hE6/ d-oil/nrw (4) at the air/water interface. The continuous lines were calculated using a uniform layer model, and the fitted structural parameters are tabulated in Table 1.
(5)
where NA is Avogadro’s constant. The procedure for obtaining structural information from experimental reflectivity profiles is usually through model fitting. The fitting usually proceeds with the assumption of a structural model, followed by calculation of its corresponding reflectivity using the optical matrix formulas.13 The calculated reflectivity is then compared with the measured one, and the structural parameters are modified in a least-squares iteration. The parameters used in the calculation are the thicknesses of the layers, τi, and their corresponding scattering length densities, Fi. Because the scattering length density of a given layer varies with isotopic composition, the fitting of a set of reflectivity profiles for different isotopic compositions greatly increases the certainty of the structural interpretation. In fitting the reflectivity profiles with uniform layer models, it is usually found that, although a single layer model describing the surfactant distribution is not always appropriate, the value of A is approximately independent of the model chosen for the interface. Results and Discussion Neutron reflectivity profiles were first measured for the mixed layers in null reflecting water to determine the surface composition. The three reflectivity profiles involving chain deuterated surfactant and deuterated oil are shown in Figure 1. The sensitivity of the neutron reflectivity to the variation of isotopic labeling can be seen from these profiles because in each case the signal arises primarily from the deuterated fragments in the adsorbed layer, as outlined previously. The level of the reflectivity is dominated by the adsorbed amount of the
dC12hE6/h-oil hC12dE6/h-oil hC12hE6/d-oil dC12hE6/d-oil hC12dE6/d-oil
Aa ( 3 Å2 (56)a
55 53.5 (57) 54 54 54
Ao ( 10 Å2
τ(2Å
σ(2Å
128 128 132 133 130
22 (19) 22 (20.5) 21.5 23 27
20 (16) 19 (16.5) 18 20.5 24
a
The numbers in the parentheses were from ref 20 for the pure C12E6 monolayer at the cmc.
deuterated species, and the slope is determined by its thickness. Hence, the level of the reflectivity from dC12hE6/d-oil/nrw is the highest among the three reflectivity curves, as the signal results from the combined contribution of the deuterated oil and the chain deuterated surfactant. The closeness of the slopes of the lines for dC12hE6/h-oil/nrw and d-oil/hC12hE6/nrw suggests that the thicknesses for the two layers are similar, showing that the thickness of the deuterated dodecyl chain layer is close to that of the oil. The extent of insertion of the oil layer into the surfactant layer is reflected in the slope of the reflectivity profile for dC12hE6/d-oil/nrw. The steeper the slope the greater the thickness and thus the further apart the centers of the two distributions. That the slope for the dC12hE6/d-oil/nrw profile is just slightly higher than those of the other two curves suggests that the centers of the two distributions are very close, indicating that the oil is well mixed with the layer of dodecyl chains. Because scattering length for the hydrogenated oil is -3 × 10 -5 Å -2 and is close to zero, the contribution from the hydrogenated oil to the total scattering length of the layer is negligible. Information about the distribution of surfactant in the presence of oil can thus be obtained directly by fitting the reflectivity profile for dC12hE6/h-oil/nrw. The continuous line in Figure 1 through the measured profile of dC12hE6/h-oil/nrw is the best fit, assuming a uniform layer distribution for the surfactant layer with the resultant structural parameters listed in Table 1. The physical constants for different fragments including scattering lengths and volumes used in the calculation are tabulated in Table 2.
Ethyl Hexadecanoate Layers
J. Phys. Chem. B, Vol. 103, No. 22, 1999 4641
TABLE 2: Physical Constants Used in Data Analysisa unit
volume [Å3]
extended length [Å]
scattering length × 10-5 [Å]
C15D31COOCD2CH3 C15H31COOC2H5 C12D25 C12H25 (C2D4O)6OH (C2H4O)6OH D2O H2O
510 510 350 350 395 395 30 30
26 26 16.7 16.7 23 23
338 (99%D) -3 241 (98%D) -14 267 (96%D) 27 19 -1.7
a The volumes and fully extended lengths were obtained from refs 25 and 26 and the scattering lengths from ref 27.
The thickness of the layer for the deuterated surfactant chain was found to be 22 ( 2 Å, and the area per molecule for the surfactant was calculated to be 55 ( 3 Å2 using eq 4. In comparison with the structural distribution of C12E6 at its cmc addition of oil has increased the layer thickness by 3 Å. Because the thickness under this contrast is mainly a measure of the dodecyl chain projection onto the surface normal, the result indicates that the distribution of the alkyl chain is broadened in the presence of oil. However, in contrast with this increased thickness, the interfacial area of the surfactant remains unchanged. A similar observation was recorded from the case of dodecane spread on the surface of C12E5 solution.4 The surface excess of the oil in the mixed layer can be estimated in a similar manner from the uniform layer model fitting to the data for d-oil/hC12hE6/nrw by neglecting the contribution from the fully hydrogenated surfactant. This gives a thickness of 22 ( 2 Å for the oil layer and an area per molecule of 128 ( 10 Å2. It should be noted, however, that because the scattering length of hC12hE6 is not so small that it can be completely neglected, the calculation described above may lead to some degree of error. The surface coverage of oil can be determined more accurately by taking into account the contribution from the surfactant layer using the expression
F)
bo + baAo/Aa Aoτ
(6)
where the subscript “o” denotes oil and “a” denotes surfactant. Because the surfactant area (Aa) is known from the fitting to dC12hE6/h-oil/nrw, and F and τ for d-oil/hC12hE6/nrw are also known from the uniform layer fitting, input of these parameters into eq 6 gives an area per oil molecule (Ao) of 130 ( 10 Å2, as compared with the value of 128 ( 10 Å2 obtained by neglecting the contribution from hC12hE6, a difference which is well within the experimental error. The consistency in the interfacial composition between the three measurements shown in Figure 1 can be further examined by comparing the areas per molecule for the oil and the surfactant with those calculated from the third set of data, obtained for dC12hE6/d-oil/ nrw. Input of the fitted scattering length density and layer thickness calculated from the dC12hE6/d-oil/nrw data into eq 6 gave the values of 131 ( 10 Å2 for Ao assuming that Aa is 54 ( 3 Å2, in good agreement with the values calculated from the other two contrasts. That the thickness of the mixed layer of dC12hE6/d-oil is only about 1 Å thicker than the values obtained from the other two contrasts again confirms that the oil is well inserted into the dodecyl chain region of the surfactant monolayer. The effect of oil addition on the distribution of the ethoxylate headgroups can be examined by using the head deuterated surfactant. Figure 2 shows the reflectivity profiles involving the
Figure 2. Neutron reflectivity profiles plotted against κ for the mixed layers of hC12dE6/h-oil/nrw (+), hC12hE6/d-oil/nrw (b), and hC12dE6/ d-oil/nrw (4) at the air/water interface. The continuous lines were calculated using a uniform layer model, and the fitted structural parameters are tabulated in Table 1.
head deuterated surfactant in nrw. The profile for d-oil/ hC12hE6/nrw is also shown for comparison. It can be seen that while the slopes for the two lower curves are almost equal, that of hC12dE6/d-oil/nrw is clearly greater, indicating a greater thickness of the mixed layer under this contrast. This increase is due to the further distance between the center of the distribution of the headgroups and that of the oil. The actual layer thickness was found to be 27 ( 2 Å, and the areas per molecule for the oil and surfactant are also in good agreement with the values calculated from the data involving the chain deuterated surfactant. Similar measurements have been recorded using a subphase of D2O, and the analysis of these profiles provide information about the extent of mixing of the oil/surfactant layer with water. Figure 3 shows the measured reflectivity profiles in D2O for differently labeled surfactants and oils. We first attempted uniform layer model fitting to these profiles and found that the model was inadequate for fitting these reflectivity profiles. Poor fitting mainly arises from the large difference in scattering length density between the alkyl chain part in air and the ethoxylate headgroup immersed in water. However, we have shown in our previous work on the mixing of dodecane into cationic surfactant layers2 that good fits can be obtained by dividing the interfacial layer into two layers, a chain-only region containing a fraction of dodecyl chains (1 - fc) and the oil, and a region containing the surfactant headgroups and a proportion of dodecyl chains, fc, with the remaining space filled with water. In analyzing the current set of data, it has been found that this model could not produce satisfactory fits to all of the measured profiles simultaneously without also allowing for a fraction of the headgroups, fh, mixed into the dodecyl chain region. Attempts were also made to allow for a fraction of the polar oil to be mixed with the water layer, bearing in mind that the oil has a polar ester group, but it was found that the model was insensitive to the possible association of the ester group with water because its scattering length was small, being determined primarily by the hydrogenated ethyl group. The equations used in calculating the scattering length densities and thicknesses of the two layers were finally taken as
4642 J. Phys. Chem. B, Vol. 103, No. 22, 1999
Lu et al.
τc ) (1 - fc)lcc + fhlhh
(7)
above the dividing line, equivalent to about 1-1.5 ethoxylate units. From the calculation of the scattering length densities required to fit the aqueous layers, the number of water molecules associated with each headgroup was found to be 12 ( 2. The assumption of the dividing line in the model described above is physically unrealistic and may lead to errors in the extent of mixing. The main reason for presenting this model is that direct comparison can be made with the existing literature work where model fitting of this type has been widely adopted. A similar two layer model has been used in describing the adsorbed monolayers of C12Em, where m ) 2, 3, 4, 5, 6, 8, and 12.14 The extent of the alkyl chain mixing with water was found to be usually between 25 and 40% at their cmcs. In comparison with the results from this work, it appears that the penetration of oil has reduced the overlap between the chain and water by some 10%. The more direct and less model dependent way of determining the structure of the mixed layer is to analyze the set of reflectivity profiles obtained from different isotopic labeling simultaneously, in terms of partial structural factors of the different components in the layer. For mixed C12E6/oil layers, the main features of interest are the widths of the distributions of oil (o), water (w), surfactant chain (c) and surfactant head (h) normal to the interface, and their relative locations. The scattering length density can be written in terms of these components11
τh ) (1 - fh)lhh + lcfcc
(8)
F(κ) ) bcnc(z) + bhnh(z) + bono(z) + bwnw(z)
Figure 3. Neutron reflectivity profiles plotted against κ for the mixed layers of dC12hE6/h-oil/D2O (+), hC12hE6/d-oil/D2O (b), hC12dE6/hoil/D2O (]), and hC12hE6/h-oil/D2O (4) at the air/water interface. The continuous lines were calculated using a two layer model, and the fitted structural parameters are tabulated in Table 3.
bc(1 - fc) + fhbh + bo Aτc
(9)
bh(1 - fh) + fcbc + nbw Aτc
(10)
Fc ) Fh )
(11)
Substituting eq 11 into eq 2 and eq 1 gives
16π2 2 [bc hcc(κ) + bh2hhh(κ) + bo2hoo(κ) + κ2 bw2hww(κ) + 2bcbhhch(κ) + 2bcbohco(κ) + 2bcbwhcw(κ) + 2bhbohho(κ) + 2bhbwhhw(κ) + 2bobwhow(κ)] (12)
R(κ) )
where τc and τh are the layer thicknesses for the alkyl chain layer and ethoxylate headgroup layer, lc and lh are the fully extended lengths of chain and head, c and h are the degrees of the extension of the chains and heads, and n is the number of water molecules associated with each ethoxylate headgroup. The two layer model can produce satisfactory fits simultaneously to all of the nine measured reflectivity profiles including the five recorded using a subphase of nrw. The solid lines in Figure 3 are the resultant fittings to the measured reflectivity profiles in D2O and the structural parameters obtained are listed in Table 3. It can be seen from Table 3 that while the thicknesses for the headgroup layers are constant at 13 ( 3 Å, those for the alkyl chain layers tend to vary in the range 10-15 Å. Such variation reflects the sensitivity of the individual profiles to the labeling of the alkyl chain or oil. For example, while the reflectivity for dC12hE6/h-oil/D2O shows a good sensitivity to the thickness of the alkyl chain region, that for hC12hE6/h-oil/ D2O has little signal from the fully hydrogenated material in the chain region and the fitting is only sensitive to the region associated with water. For the systems containing alkyl chain deuterated surfactant and deuterated oil, there are small differences between the measured and calculated reflectivities over the high κ range. This cannot be viewed as an inappropriateness of the model because over this κ range the reflectivity is around 10-6 and is strongly affected by the accuracy of the background subtracted. The results shown in Table 3 suggest that there is on average some 10-15% chain mixed with water, equivalent to about one methylene group immersed in the aqueous region. Equally, there is some 20% ethoxylate group mixed with the dodecyl chain
where hii(κ) and hij(κ) are the partial structure factors. The relationship between the partial structure factors and number density distribution functions are
hii(κ) ) |nˆ i(κ)|2
(13)
hij(κ) ) Re |nˆ i(κ)nˆ j(κ)|
(14)
where nˆ i(κ) and nˆ j(κ) are the one-dimensional Fourier transforms of the appropriate number density and can be obtained from an equation similar to eq 2. If the number density distributions corresponding to the two self-structure factors hii and hjj are symmetrical, the relative distance between the centers of the two distributions can be determined through the cross term in the structure factor
hij(κ) ) (xhiihjj cos(κδij)
(15)
where δij is the distance between the centers of the two distributions. If one distribution is even and the other odd, then
hij(κ) ) (xhiihjj sin(κδij)
(16)
The distributions of oil, surfactant and its fragments across the interface approximate closely to these two limits. Thus equations 15 and 16 offer a route for determining the separation between pairs of distributions without either Fourier transformation or specific assumptions about the type of distribution apart from its being even or odd.
Ethyl Hexadecanoate Layers
J. Phys. Chem. B, Vol. 103, No. 22, 1999 4643
TABLE 3: Structural Parameters Obtained from the Two-Layer Model Fitting
a
contrast
A a ( 3 Å2
Ao ( 10 Å2
τc ( 2 Å
τh ( 2 Å
fc ( 0.05
fe ( 0.05
dC12hE6/h-oil/nrw hC12hE6/d-oil/nrw dC12hE6/d-oil/nrw hC12dE6/h-oil/nrw hC12dE6/d-oil/nrw dC12hE6/h-oil/D2O hC12hE6/d-oil/D2O hC12dE6/h-oil/D2O hC12hE6/h-oil/D2O
54 54a 54 54 55 54.5 54a 54 54a
126a 124 126 126a 126 128 126 126a 130a
13 14 13.5 13a 14 12 15 10a 10a
13a 13a 13 13.5 13 13 13.5 13 13
0.15 0.15a 0.15 0.15a 0.15 0.1 0.1a 0.1a 0.15a
0.2a 0.2a 0.2 0.25 0.2 0.2a 0.2a 0.2 0.2a
Denotes insensitive parameters in the model fitting.
The choice of the optimum route for deriving number density distributions from self-partial structure factors is less straightforward. We have previously explained11 that number density distributions can in principle be obtained directly by Fourier transformation, but in practice the errors are too large because of the limited range of momentum transfer over which the reflectivity profiles were measured. We have also demonstrated that the use of suitable analytic expressions for the partial structure factors is a more convenient approach. A simple and appropriate model to represent the distribution of an interfacial component is a Gaussian distribution11
ni(z) ) no exp(-4z2/σ2)
(17)
where no is the number density at the center of the distribution, and σ is the full width at the height of no/e. Substitution of eq 17 into eq 2 and 13 gives
hii(κ) ) Γ2 exp(-σ2κ2/8)
(18)
where Γ ) xπσno/2. The distributions of the alkyl chain, the headgroups, and the oil are each fairly symmetrical and are expected to be quite well described by eq 18. For the water distribution across the interfacial region, a tanh function has been found to be suitable
hww(κ) ) nwo2(πξ/2)2csch2(πξκ/2)
(19)
where nwo is the number density of bulk water and ξ is the width parameter of the tanh function. Although the ten partial structure factors in eq 12 can be obtained by solving the combined equations measured under different labeling compositions, the fact that there were only nine reflectivity profiles measured here imposes some degree of difficulty in solving these equations without making an assumption about the distribution for one of the structure factors. Furthermore, a major problem related to the combined resolution of ten equations is the possible propagation of errors which may cause systematic deviations in some of the partial structure factors. We have shown that the most serious systematic error comes from small differences in the surface coverage.15 These errors tend to cause distortions to the true shape of the crossterm partial structure factors hij. The reflectivity profiles shown in Figures 1-3 were all fitted with the fixed values of Aa ) 54 Å2 and Ao ) 128 Å2. That all of the calculated profiles fit the measured ones well without any need for further adjustment suggests that the current set of reflectivity profiles is of sufficient consistency. Nevertheless, it is still useful to obtain structure factors in a more direct route before any sophisticated method is attempted. We have shown previously15 that with some minor assumptions structure factors can be obtained more straightforwardly by combining reflectivity profiles in smaller groups. Partial
structure factors for hcc, hoo, and hco can be obtained by combining dC12hE6/h-oil/nrw (1), d-oil/hC12hE6/nrw (2), and dC12hE6/d-oil/nrw (3), where each of the measurements offers the most reliable determination of the three partial structure factors, respectively. In this case, eq 12 can be written as
R3(κ) )
R1(κ) )
16π2 2 b′c h′cc(κ) κ2
R2(κ) )
16π2 2 bo hoo(κ) κ2
16π2 2 [b′c h′cc(κ) + bo2hoo(κ) + 2b′cbohco(κ) 2 κ
(20)
where b′c is the scattering length of the deuterated chain containing the contribution from the hydrogenated headgroup and h′cc is the corresponding structure factor. We have shown previously16 that the contribution from the hydrogenated headgroup will slightly increase the calculated thickness of the dodecyl chain layer, but its effect on the cross-term partial structure factor is usually well within the quoted error of (1 Å. The fractional contribution from the hydrogenated parts can be estimated from equations 5 and 20 where the thickness of the interfacial layer is approximately proportional to the total scattering length. An increment of 10% in the scattering length for the hydrogenated headgroup will result in an equivalent increase in the calculated thickness of the alkyl chain layer. This estimate can be justified when the exact thickness of the dodecyl chain layer is obtained using an alternative analysis procedure. Likewise, by combining hC12dE6/h-oil/nrw, d-oil/hC12hE6/ nrw, and hC12dE6 /d-oil/nrw h′ee and heo can be obtained using a set of equations similar to eq 20. The resultant self-term partial structure factors for the alkyl chains, the headgroups, and the oil are shown in Figure 4 together with the best fits to the structure factors using eq 18. The cross-term partial structure factor between the alkyl chain and the oil, hco, and that between the oil and the headgroup, heo, are shown in Figure 5 together with the best fits using eq 15. The cross-term partial structure factors between oil and surfactant fragments and water, how, hcw, and hew, are shown in Figure 6 together with the best fits using eq 16. All of the structural parameters obtained from the calculations are summarized in Table 4. It should be noted that all of the layer thicknesses were calculated assuming Gaussian distributions, and these values are about 90% of the values obtained under the uniform layer models. Such systematic differences are entirely due to the assumptions of the models, and we have shown previously that the Gaussian distribution is a more realistic representation for soluble surfactant layers at the air/water interface, where thermal roughness and structure disorder tend to smooth the distribution of the surface layer.17
4644 J. Phys. Chem. B, Vol. 103, No. 22, 1999
Figure 4. Plots of self-term partial structure factors versus κ for (a) h′cc, (b) h′ee, and (c) hoo. The continuous lines are the best fits using eq 18 with σ′c ) 20 ( 2 Å, σ′e ) 18.5 ( 2 Å and σo ) 18 ( 2 Å.
Lu et al.
Figure 6. Plots of cross-term partial structure factors versus κ for (a) hcw, (b) how, and (c) hew. The continuous lines are the best fits using eq 16 with δcw ) 12 ( 1 Å, δow ) 12 ( 1 Å, and δew ) 0.5 ( 1 Å.
TABLE 4: Comparison of the Structural Parameters Obtained from the Direct Route of Analysis, the Least-Squares Fitting to All of the Isotopic Data with Those in the Absence of Oil Aa ( 3 Å2 Ao ( 10 Å2 σc ( 2 Å σe ( 2 Å σo ( 2 Å ξw ( 2 Å δco ( 1 Å δcw ( 1 Å δce ( 1 Å δow ( 1 Å δew ( 1 Å δeo ( 1 Å
direct route
least-squares fit
54 128 20 (18)a 18.5 (16) 18 (16) 8.5 0.5 12 11 12 0.5 10.5
54 128 18.5 (16) 20 (18) 17.5 (15) 10 1 12.5 12 11.5 0.5 11
without oil 54 16 (13.5) 16.5 (14) 8 10 9 1
a The numbers in parentheses were the thicknesses after removal of roughness using eq 22.
Figure 5. Plots of cross-term partial structure factors versus κ for (a) hco and (b) heo. The continuous lines are the best fits using eq 15 with δco ) 0.5 ( 1 Å and δeo ) 10.5 ( 1 Å.
The structural parameters obtained from the direct analysis can be compared with those obtained following the more
systematic route based on the least-squares routine. This was done by the simultaneous fitting to the nine reflectivity profiles based on the relation given in eq 12. Equation 18 was used to represent the distributions for the alkyl chains, the headgroups, and the polar oil, and eq 19 was used for describing the distribution of the water. Equations 15 and 16 were used to describe the relations between these distributions. The fits were optimized against the residue differences between the measured and the calculated reflectivities, and the deviation in the total volume fraction distribution was constrained to within ( 10%.
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These equations lead to a total of four width parameters, σc, σh, σo, ζ, and six cross distances, δco, δce, δeo, δcw, δow, δew. The simple geometrical relations between these cross distances can be expressed as
δow ) δco + δcw ) δoe + δew
(21)
This set of equations leaves only three genuine variables among the six cross distances, giving a total of seven independent parameters. Because the number of reflectivity profiles exceeds the number of independent variables, the structural parameters have thus been overdetermined. In performing the least-squares fitting, the area per surfactant was fixed at 54 Å2 and that for the oil at 128 Å2. Different sets of starting values were used to test the fitting, and it was observed that as long as the starting value for each variable was close to that obtained from the direct analysis (typically within (3 Å), the final results only varied by (1 Å. However, varying the initial values further from the true ones resulted in an erratic scattering of the refined values. Comparison between the optimized parameters from the leastsquares fitting and those from the direct analysis is given in Table 4 where it can be seen that the difference between the cross distances is about 1 Å, indicating that the assumptions made in the direct analysis did not affect the determination of the cross distances between different fragments. There are however some apparent differences in the thicknesses of the dodecyl chain region and of the headgroup region. These are obviously caused by the inclusion of the scattering lengths from the hydrogenated fragments in the direct analysis. Neglect of the scattering contributions from the hydrogenated portions leads to a decrease of the alkyl chain layer thickness by 1.5 Å and an increase of 1.5 Å to the headgroup layer thickness, a trend entirely consistent with the estimate made earlier. That the width for the oil is identical within the experimental error suggests that the assumption about the scattering lengths for the surfactant fragments does not affect the oil distribution. The agreement between the values obtained for the width of the water distribution is less good, and the difference is some 2 Å. It was found that ζ was sensitive to the constraint imposed on the variation of the total volume fraction distribution. As described above, the results given in Table 4 were obtained with a deviation of the total volume fraction less than 10%. If this was set to a broader range, the fit to the reflectivity of hC12hE6/ h-oil/D2O would be improved and ζ would approach a value of 8.5 Å. This observation may suggest that the discrepancy in ζ in the simultaneous fitting is either a consequence of error accumulation or the inappropriateness of the tanh function for the water distribution as discussed elsewhere.18 The effect of oil addition on the distributions of the dodecyl chain and ethoxylate headgroup can be seen from Figure 7 where hcc and hee in the presence and absence of oil are plotted together. The continuous lines were calculated assuming Gaussian distributions for the two fragments, and the results show that addition of oil resulted in the broadening of the chain distribution as well as of the head distribution. The structural parameters for the pure surfactant layer are also listed in Table 4 for comparison. The thickness of the chain distribution assuming a Gaussian model increases from 16 to 18.5 Å. Because the fully extended length of the dodecyl chain is 16.7 Å, this result suggests that the actual chain distribution in the presence of oil is wider than the fully extended one. Some contribution to the broadening of the layer arises from capillary wave roughness, for which an estimate can be made from the known surface tension and the known surface roughness of pure water.19 Because roughness is inversely proportional to the square root
Figure 7. Comparison of self-term partial structure factors of C12E6 with (b) and without (+) ester oil, (a) hcc and (b) hee. The values of σc and σe are given in Table 4.
of the surface tension and with the known value of the water roughness of 2.8 Å at the surface tension of 72 mNm-1, the roughness at any other surface tension can be calculated. The total thickness, σ, and the roughness, $, are related according to the following relation
σ2 ) l2 + $2
(22)
where l is the layer thickness after roughness removal. The values of l were found to be 16 and 17.5 Å, as compared with the respective values of 13 and 14 Å for the dodecyl chain layer and ethoxylate group layer in the absence of oil. The increase in layer thicknesses results from the change in the projection of the chains onto the surface normal direction. The extent of the change can be quantified in terms of a mean tilt angle of the chains or heads and, in the case of the alkyl chain, this can be expressed as
〈cos2 θ〉 )
l2 16.72
(23)
which gives a value of 40° for the alkyl chain layer with respect to the surface normal in the absence of oil and 15° in the presence of the oil, suggesting that the alkyl chain is almost vertical when oil is present. It should be stressed that the treatment described above is very approximate because several assumptions are made in the use of eq 23 as already discussed elsewhere.17,20 Given that the cross distance between the oil distribution and that of the dodecyl chain distribution is virtually zero (within an error of (1 Å) and that the width of the two distributions is comparable, it can be said that the alignment of the alkyl chains with respect to the surface normal is a direct consequence of the interpenetration of the dodecyl chain region by the oil. It is of interest to discuss the distribution of the ethoxylate headgroups at the air/water interface. The distribution of the headgroup resembles the anchored hydrophilic chain distribution at the solid/solution interface except that there is an extra
4646 J. Phys. Chem. B, Vol. 103, No. 22, 1999
Lu et al.
Figure 8. Comparison of cross-term partial structure factors of C12E6 with (b) and without (+) ester oil, (a) hcw and (b) hew. The cross distances are given in Table 4.
Figure 9. Volume fraction distributions normal to the interface for the C12E6 layer with (a) and without (b) ester oil. The center of dodecyl chain distribution was chosen as reference point. The alkyl chain distribution is shown as a continuous line, the ethoxylate headgroup and the oil distributions as dashed lines, the water distribution as a dotted line, and the total volume fraction as a dash-dotted line.
contribution from the thermal roughness at the air/water interface which serves to broaden the headgroup distribution. Sarmoria et al.21 have used the rotational isomeric state model to analyze
Figure 10. Volume fraction distributions for (a) dodecane/C12TAB/ water, (b) dodecanol/SDS/water, (c) dodecane/C12E5/water, and (d) ethyl hexadecanoic ester/C12E6/water. The center of dodecyl chain distribution was chosen as reference point. The alkyl chain distribution is shown as a continuous line, the ethoxylate headgroup and oil as dashed lines, the water distribution as a dotted line, and the total volume fraction as a dash-dotted line. In the case of dodecane/C12E5/water, σe was taken to be 18 Å and δce ) 11 Å.
how polymer-like behavior might develop with chain length for ethylene oxide groups in aqueous media and concluded that polymer behavior is more or less developed for a free chain at m = 4 and for an anchored chain at m = 10, where m is the number of ethoxylate units in the chain. We have shown that for the C12Em series at the air/water interface polymer-like behavior is already developed at m g 4,22 and at m ) 6 the
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Figure 11. Schematic representation of the C12E6 layer without (a) and with (b) the ester oil.
thickness of 14 Å for the hexaethylene glycol groups is virtually identical to the value predicted by the polymer model. However, the presence of oil has increased the headgroup layer thickness by a further 4 Å, and the headgroup layer is even thicker than the end-to-end distance for the corresponding free chain polymer predicted by Sarmoria et al.21 This broadening is clearly caused by the insertion of oil molecules into the surfactant layer. The effect of oil addition on the extent of broadening of the surfactant layer can also be seen from the two main cross-term partial structure factors, hcw and hew, and the comparison is made in Figure 8. The cross-distance between the centers of the alkyl chain and water distributions is 12.5 ( 1 Å, as compared with 10 ( 1 Å in the absence of oil, indicating that the alkyl chain is further away from the surface of the water. Moreover, the close similarity between the two distributions of hew suggests that the relative position between the distribution of the headgroup and that of water is unchanged. The relative location and the extent of intermixing between different fragments can be better visualized by plotting the volume fraction distributions for the whole interfacial system, and this is shown in Figure 9. Gaussian distributions have been used for oil, the alkyl chain, and headgroup of the surfactant and a tanh distribution for water. The overlapping of the alkyl chain distribution with that of the oil results in a very high density for the alkyl chain region. The volume fraction over the chain region amounts to over 90%, indicating that the alkyl chain region is almost liquid-like. For comparison, the volume fraction distributions for the fragments of the pure surfactant at the cmc are also shown in Figure 9. The penetration of the oil has broadened the distributions for the alkyl chain, the headgroup, and the whole surfactant layer. Figure 9 also shows that the overlapped region between the alkyl chain and the headgroup in the presence of oil is reduced, showing the lesser extent of mixing between the two fragments. It can also be seen from Figure 9 that when the bulk solution is approached the total volume fraction exceeds unity, showing an impossible amount of the material in this region. This is clearly indicative of the errors involved in the data analysis. In several previous publications involving similar interfacial systems,9,11,15,23 the possible sources of errors contributing to the deviation of the
total volume fraction have been systematically analyzed. Because uncertainty arising from these errors affects the reliability of the conclusion in this work, we briefly repeat the outline in the following. The errors come from the use of models in the extraction of the structural parameters and the uncertainty in the estimates of the volumes of the surfactant components used in the calculation. The main error in the modeling arises from the inappropriateness of the models used for describing the fragment distributions, that is, Gaussian distributions for the alkyl chains, the ethoxylate heads, and the oil, and the tanh function for the water distribution. Although the combined reflectivity profiles offer reliable measurements on the relative locations between the centers of different components across the interface, they are relatively insensitive to the exact shape of the distributions. The least sensitive is the distribution of water across the interface, and this is likely to be the main source of error to the total volume fraction distribution. Although a tanh profile is far more realistic than the uniform layer model, the true water density distribution may not follow the tanh function exactly. Deviation between the calculated and measured distributions is expected to occur in some part of the interfacial region. This is particularly possible when the headgroup is large, as in the nonionic surfactant systems. The broad distribution of the surfactant headgroup disrupts the water distribution across the interface, which can lead to a deviation of up to 10% in the localized region. In comparison, the errors from the components within the surfactant layers are less. The main misrepresentation in using the Gaussian distributions for the alkyl chains and the ethoxylate heads is also the shape of the profiles. Gaussian distributions produce symmetrical profiles, but a recent computer simulation by Bocker et al.24 has shown that the distribution of the alkyl chain in a cationic surfactant monolayer is slightly unsymmetrical. The penetration of the ester oil may cause some further distortion of the density profiles within the surfactant layer away from the symmetrical distributions. The accumulation of errors of this type is less significant than that arising from the water distribution, but can still result in the error of a few percent in the total volume fraction distribution. Because the measurements in this work were focused on the structure and
4648 J. Phys. Chem. B, Vol. 103, No. 22, 1999 composition of the interfacial layers, the potentially greater error from the water density profile is less undermining to the interpretation of the results. It can be argued that more realistic models should be used, but the more realistic the model is the more parameters are usually involved in the fitting. As already indicated previously, the main constraint is that the measured reflectivity profiles are not sensitive enough to test the appropriateness of any more elaborate models. It can be said, however, that the models used in the direct route of analysis in this work are far more realistic than the uniform layer models which have been widely used in the literature. The deviation in Figure 9 is under 10% and is consistent with the limit set out in the simultaneous analysis described previously, suggesting that the results are reasonably reliable. It should be emphasized that although the uncertainty about the exact shape of the distributions affects the total volume fraction distribution, it has almost no effect on the cross-distances between the centers of the distributions. The spreading of oil onto the surface of different surfactants has been studied by neutron reflection,2-4 and it is interesting to compare the extent of oil penetration into these different surfactant layers. Figure 10 compares the volume fraction distributions for dodecane/C12TAB2, dodecanol/SDS3, dodecane/ C12E54, and C12E6/hexadecanoate, all at the air/water interface. Although the difference in the alkyl chain length of the oils means that the entropy of mixing is different, it is useful to compare the actual packing density in the mixed layers. The largest cross-distance between the center of the oil and that of the dodecyl chain distribution is seen in the cationic surfactant system, with δco ) 6.5 Å. This indicates that the oil here penetrates only into the outer part of the surfactant chain. Despite the fact that the fully extended length of ethyl hexadecanoate is much greater than that of dodecane the actual width of the dodecane layer spread on C12TAB is close to that found for the ester spread on C12E6. This means that if the surface coverage of the oils is the same, the ester layer would have some 35% higher packing density because of its greater molecular volume. However, this difference is almost exactly compensated by the opposite contribution from the oil surface coverage, resulting in almost identical volume fraction distribution profiles for the two oils. While the area for both surfactants is 54 Å2, the area for dodecane is 80 Å2 as compared with 130 Å2 for the ester oil. However, a distance of 6.5 Å from the center of the alkyl chain distribution gives a broad but less dense hydrocarbon chain distribution in the dodecane/C12TAB layer. The almost zero cross-distance between the centers of the ester and C12E6 distributions indicates that the overall packing density in the ester/C12E6 layer is much greater. Thus, the two mixed layers are very different in physical character. In comparison with the dodecanol/SDS mixed layer it can be said that the deep insertion of the ester oil into the nonionic surfactant layer is due to the polar ester group. The small polar ester groups drive the oil molecules to be as close to the water surface as possible. However, despite the shorter cross-distance of 3 Å between the centers of the dodecanol and SDS distributions, it is still 3 Å further away from the complete overlapping of the two centers in the ester/C12E6 mixed layer. This additional insertion can be explained by the feature observed in the dodecane/C12E5 system. The cross-distance between the centers of the dodecane and dodecyl chain distributions is also about 3 Å, identical to that found in dodecanol/SDS system. This was attributed to the weak attractive interaction between the ethoxylate groups and the dodecane, which would have an equivalent effect to the OH/ water attraction in the dodecanol/SDS system.
Lu et al. The structure of the C12E6 monolayer before and after addition of ester oil is summarized schematically in Figure 11. Addition of C15H31COOC2H5 leads to the broadening of the distributions for the dodecyl chain, the hexaethylene glycol chain, and the whole surfactant layer. The insertion of the ester oil molecules forces the alkyl chains and the ethoxylate headgroups to become more vertically oriented. That the center of the headgroup distribution is almost identical to that of water suggests that although the distribution of the headgroup is broadened in the presence of oil, its relative location with respect to water is unchanged. Acknowledgment. We thank the Engineering and Physical Sciences Research Council (EPSRC) and the Biotechnology and Biological Sciences Research Council (BBSRC) for support. We also thank Dr. Sean Langridge at the ISIS neutron facilities for technical support. W.W. thanks the Thai Government for the studentship. References and Notes (1) Verhoeckx, G. J.; Bruyn, P. I.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1987, 119, 409. (2) Lu, J. R.; Thomas, R. K.; Binks, B. P.; Fletcher, P. D. I.; Penfold, J. J. Phys. Chem. 1995, 99, 4113. (3) Lu, J. R.; Purcell, I. P.; Lee, E. M.; Simister, E. A.; Thomas, R. K.; Rennie, A. R.; Penfold, J. J. Colloid Interface Sci. 1995, 174, 441. (4) Lu, J. R.; Li, Z. X.; Thomas, R. K.; Binks, B. P.; Crichton, D.; Fletcher, P. D. I.; McNab, J. R.; Penfold, J. J. Phys. Chem. 1998, 102, 5785. (5) Malcolmson, C.; Lawrence, M. J. J. Pharm. Pharmacol. 1993, 45, 141. (6) Malcolmson, C.; Sidhu A.; Sacra C.; Kantaria S.; Lawrence, M. J. J. Pharm. Sci. 1998, 87, 109. (7) Penfold, J.; Richardson, R. M.; Zarbakhsh, A.; Webster, J. W. P.; Bucknall, D. G.; Rennie, A. R.; Jones, R. A. L.; Cosgrove, T.; Thomas, R. K.; Higgins, J. S.; Fletcher, P. D. I.; Dickinson, E. J.; Roser, S. J.; McLure, I. A.; Hillman, A. R.; Richards, R. W.; Staples, E. J.; Burgess, A. N.; Simister, E. A.; White, J. W. J. Chem. Soc., Faraday Trans. 1997, 93, 3899. (8) Lee, E. M.; Thomas, R. K.; Penfold, J.; Ward, R. C. J. Phys. Chem. 1989, 93, 381. (9) 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, 10971. (10) Born, M.; Wolf, E. Principles of Optics; Pergamon: Oxford, 1970. (11) Lu, J. R.; Lee, E. M.; Thomas, R. K. Acta Crystallogr. 1996, A52, 11. (12) Crowley, T. L. Physica 1993, A195, 354. (13) Lu, J. R.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys.: Condens. Matter. 1994, 6, A403. (14) Lu, J. R.; Su, T. J.; Li, Z. X.; Thomas, R. K.; Staples, E. J.; Tucker, I.; Penfold, J. J. Phys. Chem. B 1997, 101, 10332. (15) Lu, J. R.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys. Chem. 1993, 97, 6024. (16) Lu, J. R.; Simister, E. A.; Lee, E. M.; Thomas, R. K.; Rennie, A. R.; Penfold, J. Langmuir 1992, 8, 1837. (17) Lu, J. R.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys.: Condens. Matter 1994, 6, A403. (18) Lu, J. R.; Li, Z. X.; Smallwood, J.; Thomas, R. K.; Penfold, J. J. Phys. Chem. 1995, 99, 8233. (19) Schwartz, D. K.; Schlossman, M. L.; Kawamoto, E. H.; Kellogg, G. J.; Pershan, P. S.; Ocko, B. M. Phys. ReV. 1990, A41, 5687. (20) Lu, J. R.; Li, Z. X.; Thomas, R. K.; Staples, E. J.; Tucker, I.; Penfold, J. J. Phys. Chem. 1993, 97, 8012. (21) Sarmoria, C.; Blankschtein, D. J. Phys. Chem. 1992, 96, 1978. (22) Lu, J. R.; Thomas, R. K. J. Chem. Soc., Faraday Trans. 1998, 94, 995. (23) Lu, J. R.; Hromadova, M.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys. Chem. 1994, 98, 11519. (24) Bocker, J.; Shlenkrich, M.; Bopp, P.; Brickmann, J. J. Phys. Chem. 1992, 96, 9915. (25) Tanford, C. J. J. Phys. Chem. 1972, 76, 3020. (26) Takahashi, Y.; Sumita, I.; Tadokoro, H. J. Polym. Sci. 1973, 11, 2113. (27) Sears, V. F. Neutron News 1992, 3, 26.
