6024
J. Phys. Chem. 1993,97, 6024-6033
Structure of an Octadecyltrimethylammonium Bromide Layer at the Air/Water Interface Determined by Neutron Reflection: Systematic Errors in Reflectivity Measurements J. R. Lu, E. A. Simister, and R. K. Thomas' Physical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, United Kingdom
J. Penfold Rutherford-Appleton Laboratory, Chilton, Didcot, Oxon OX1 1 O R A , United Kingdom Received: January 19, 1993
The adsorption of octadecyltrimethylammonium bromide (ClgTAB) a t the air/water interface has been studied by surface tension and neutron reflection measurements. The critical micelle concentration (cmc) was found to be 3.1 X 10-4 M a t 33 OC and the area per molecule a t the cmc 44 f 2 A2. The results for the area per molecule from the two methods agreed. The structure of the CISTAB layer was determined a t the cmc by measuring the neutron reflectivity profiles of six isotopic compositions. Assuming the distributions of chain and head-group regions to be Gaussian in shape, their widths were found to be respectively 17 f 1 and 13 f 3 A and, taking the depletion layer of water to be uniform, the width of the depleted water region was 10.5 f 1 A. The separations of the three distributions, chain, head, and solvent, denoted by aa, 6hs, and a&, were determined directly and found to be respectively 9 f 0.75, 9 f 1.5, and 1 f 1 A. The method of analysis of reflection data has been improved and possible sources of systematic error have been fully analyzed. Application of the improved analysis to data from C14TAB a t the same surface coverage leads to values of 7 f 0.75,6 f 1.5, and 1 f 1 A, respectively for the corresponding three separations, 6, in C14TAB. The separation of the distributions for the two surfactants and the widths of head-group and solvent distributions are found to be in good agreement with a recent computer simulation. The width of the chain distribution, on the other hand, is found to be larger than from the simulation.
Introduction Neutron reflection has recently emerged as a technique able to give hitherto inaccessible information about the structure of layers of surfactants adsorbed at the airlwater interface and in equilibrium with bulk solution. The type of information that can be obtained includes the surface excess, the mean thickness of the chain and head-group regions of the layer, and the extent of penetration of water into the surfactant layer. To advance our understanding of the factors that determine the structure of an adsorbed surfactant layer, it is necessary to determine the effects on the structure of the layer of changing different parameters such as, for example, the chain length, the type of head group, and the temperature. To this end we are using neutron reflection to make a systematic study of the series of cationic surfactants C,TAX, where C, denotes a hydrocarbon chain of 10-1 8 carbon atoms and X denotes an anion. In this paper we present results for octadecyltrimethylammonium bromide (C18TAB). Since there have been significant advances in the sensitivity of the neutron reflection technique and in the method of analyzing reflectivity data since the first measurements on CloTAB,l we also use the measurements on CISTABboth to demonstrate some improvements in the analysis procedure and to make a critical assessment of possible systematic errors in the neutron reflection experiment. Experimental Details Four isotopic species of octadecyltrimethylammonium bromide were used in the experiments, C18D37N(CD&Br, C18D37N(CH3)3Br, C I S H ~ ~ N ( C D ~and ) ~ C1&37N(CH3)3Br, B~, which we refer to as dCISdTAB, dClshTAB, hClsdTAB, and hClshTAB, respectively. The method of preparation and purification was identical with that used for C14TAB and described by Simister
* Address correspondence t o this author.
et al.z.3 The raw materials used for the preparations were C18D37Br and N(CD3)3 from Merck, Sharp & Dohme and Cl8H370H from Larodan. The latter was converted into C1~H37Br by using standard procedures.4 At least two and sometimes as many as five recrystallizations of the crude surfactant were usually necessary to obtain satisfactory purity as assessed from surface tension measurements. High-purity water was used throughout (Elga Ultrapure system) and the methods of cleaning the glassware and Teflon troughs for the neutron experiment were as described in ref 1. All the experiments were performed at 33 OC to avoid any problems with the sample crystallizing out of the solution. The neutron reflection measurements were made on the reflectometer CRISP at the Rutherford-Appleton Laboratory (Didcot, U.K.). The procedure for making the measurements has been described previo~sly.1~~ The measurements were all madeat a fixed incident angleof 1Soand theintensities calibrated with respect to D20. A flat background determined by extrapolation to high values of the momentum transfer, K ( K = ( 4 sin ~ @ / A where 0 is the glancing angle of incidence), was subtracted. This has been shown to be a valid procedure provided that there is no small-angle scattering from the bulk s ~ l u t i o n .Since ~ this only starts to become a problem at concentrations of the order of 1%by volume, well above the concentrationsused in the present experiments, the procedure is valid for the experiments described here. Surface tension measurements were made by using a Kruss K10 maximum pull tensiometer using a Pt du Nouy ring. Results Figure l a shows the surface tension ( 7 )plotted against the log of the concentration for fully deuterated C I ~ T A Bin HzO. The critical micelle concentration (cmc) is found from the break in the plot to be (3.1 f 0.2) X 10-4 M, in good agreement with the literature value of 3 X 10-4 M,6 determined by streaming current measurements on the protonated material. It is interesting to
0022-3654/93/2091-6024S04.00/00 1993 American Chemical Society
Structure of a CISTABLayer
The Journal of Physical Chemistry, Vol. 97, No. 22, 1993 6025 1o
-~
0
2 0
60[
7
55
X
0
'ij 50
0
C
a,
c
45
40
t
0 0
I
-100
0 0
0
-95
-90
-85
-80
0
0
I
10-6
-7.5
4
I
In [C I MI
0.05
0.10
0.15
0.20
0.25
0.3C
Figure 2. Neutron reflectivity profiles of chain deuterated dClahTAB in null reflecting water at different concentrations: (0) 3.0 X M, (A) 1.0 X lo4 M, (+) 3.0 X M, and (0)6 X 10" M. T = 306 K. The incoherent background has been subtracted.
TABLE I: Scattering Lengths and Volumes of Constituent Parts of CISTABand C14TABs
I
I
I
I
45
50
55
60
I
I
65
70
extended length/A
scattering length/lk5 A
24.3 24.3 19.2 19.2
363 (98.6%D) -18.7 283.8 (98.6%D) -15.4 95.6 (99%D) 2.5 19.14 -1.68
AIA2 Figure 1. (a, Top) Surface tension of octadecyltrimethylammonium bromide as a function of ln[concentration] and (b, Bottom) the surface pressure-area isotherm deduced from (a).
observe that these surface tension measurements are, to the authors' knowledge, the first in the literature for CISTAB,which is probably because this series of cationic surfactants has always presented particular difficulties of measurement. We comment on this in the Discussion. The surface excess was determined by fitting a second-order polynomial to the points in Figure l a below the cmc and substituting the slope into the Gibbs equation. Fibure 1b shows the pressure/area isotherm deduced from the surface tension measurements, from which it can be seen clearly that an adsorption plateau is not reached at the cmc, implying that the surface excess continues to increase above the cmc. This reinforces our earlier conclusion that a straight line fit to the y / l n c plot should never be used to determine the surface excess of a surfactant at the cmc.2 In practice, we have found that, for the limited number of surfactants we have so far studied by reflection, the plateau occurs above the cmc for charged surfactants and at about the cmc for nonionic surfactants.8 This is qualitatively consistent with the effect on surface excess of increasing the ionic strength. The surface excess from the surface tension measurements is compared with the neutron reflectivity excess in Figure 3. Figure 2 shows the neutron reflectivity at different concentrations of dClshTAB in null reflecting water. After subtraction of the incoherent flat background (approximately 6 X 1W at this isotopic composition) the reflectivity results entirely from the surfactant chain. Such reflectivity profiles are easily fitted by using a model of a single uniform layer and the known scattering lengths of the constituent parts of the surfactant (Table I) to give the surface excess with an accuracy of better than 5% at surface excesses in the region (3-4) X 1WO mol cm-2. The value of the
Volumes and extended lengths are from ref 13 and scattering lengths from ref 14.
TABLE II: Parameters for C18TAB Determined from Neutron Reflection Using a Single Uniform Layer Model c/104 M
0.06 0.3 0.6 1 .o 2.0 3.0
15f2 16.5 f 2 18.5 f 1 18.5 f 1 19f 1 18.5 f 1
A/&
I?/iO-'O mol cm-2
83 f 3 65 f 3 58 f 2 54 f 2 47 f 2 44 f 2
2.00 2.55 2.85 3.05 3.50 3.75
surface excess found by this means has been shown to be independentof the model used to fit the reflectivity.2 The neutron surface excess is compared with the surface tension surface excess in Figure 3 and the agreement between the two is within the experimental error. The error in the neutron surface excess increases as the surface excess decreases but always remains smaller than the error in the correspondingsurface tension excess. The fitting procedure also gives a measure of the mean thickness of the layer and values of the derived thickness and area per molecule are given in Table 11. The combination of data at different isotopiccompositions can be used to determine the structure of the layer in more detail. We have done this at the cmc, where we have measured the reflectivity from seven different isotopic compositions. These compositions were dClsdTAB, dClshTAB, and nClsdTAB in null reflecting water (Figure 4) and dClsdTAB, dClshTAB, nClshTAB, and hClsdTAB in D20, of which the first three are shown in Figure 5 . nC18hTABdenotes a mixture of chain-deuterated and chainprotonated material in a ratio that makes the m a n scattering length of the chains zero (null scattering). The data were fitted in two ways. In the first we used a model of two uniform layers
Lu et al.
6026 The Journal of Physical Chemistry, Vol. 97, No. 22, 1993
3 75 -
4 00
0
T
'.
Fi P
350-
-
325-
E
300-
P
d
0
n
0
0 2757
:
rll
I
2 00
05
10
15
20
25
30
c x 104i M Figure 3. Surface excess against concentration from neutron reflectivity (A) and surface tension (0).
0.05
0 15
0.10
0.20
0.25
0.30
KIA-' Figure 5. Observed and calculated reflectivity profiles for three isotopes of C18TAB in D2O after subtraction of a flat background: (0)dCl8dTAB, (A)dC18HTAB, and (+) nCl8hTAB. The continuous lines are calculated by using the parameters in Table 111.
TABLE III: Parameters for Calculated hordes of Figures 4 and 5 A f ff t f pcx fcf PhX fhf
005
010
015
020
025
C
Figure 4. Observed and calculated reflectivity profiles for three isotopes of ClXTAB in null reflecting water after subtraction of a flat background: (0)dClxdTAB,(A) dC18hTAB, and (+) nCl8dTAB. The continuous lines are calculated by using the parameters in Table 111.
similar to that used by Simister et al.3 The two layers areassumed to be a chain-only region containing a fraction of chains, (1 -f ), and a region containing all the head groups and counterions and a proportion of chains, f, with the remaining space filled with water. The equations giving the scattering length densities and thicknesses of the two layers are
7h
7,
= (1 - f ) l c c
=
uh fu, A
+ + nu,
where T~ and 7 h are the thicknesses, b, and bh are the scattering lengths, I, is the fully extended length of the chain, n is the number of water molecules in the head-group layer, and uc, uh, and u, are the molecular volumes of chain, head, and water, respectively (Table I). In the paper by Simister et al. the second of the set
species
21.42
0.1 0.1 1061A-2 I / A 106JA-2
dC,sdTAB/nrw dCl8hTABJnrw "O"ClsdTAB/nrw dCIgdTAB/D*O dClghTAB/D20 "O"C18hTAB/D20
45 43.5 48 44 43.5 44
0.3 0.7 0.3 0.7
4.75 4.9
12 12
0.3 0.7 0.3 0.7
4.85 4.9
12 12
4.35
118,
1.3 6.95
10.5 10 14f 3 10.5
5.2 2.55
10 10.5
2.6
of eqs (1) was different and the comment was made that inclusion of the degree of extension e in the second equation altered (increased) the values off and t obtained in the final fit. Since then we have found that the direct method of analysis to be described below gives results closer to this second procedure and we have therefore used it here. The version of the second equation given hereis moreeasily interpreted. It makes a difference, which should be taken into account when comparing the results of Table I11 with those given in ref 3. Approximately,f increases by about 0.1 in the present model over the values in ref 3. There is a choice of the four independent parameters for fitting the set of isotopic reflectivity profiles, which is normally taken to bef, c, A, and n. The fitted curves for a single structure to the set of six profiles are shown as continuous lines in Figures 4 and 5 and the parameters derived for the layer are listed in Table 111. We comment on the values of these parameters in the Discussion. The consistency of the fit of a single structure to the data from seven very different scattering situations is excellent and indicates that the derived structure is unique within the errors and assumptions of the model. The second method of analyzing the data is more direct. We have described it p r e v i ~ u s l ybut ~ ~ here ~ ~ * introduce ~ some improvements, some of which have already been described in other publications but not yet applied to the more complex situation of chains, heads, and solvent. In particular we will show that the more accurate analysis leads to some clear and interesting differences between C14TAB and C18TAB. For the surfactant layer the main features of interest are the relative positions of chains, heads, and water and the widths of their distributions normal to the interface. A simple description of the structure of the air/solution interface can then be made in termsofthedistributionsof chainsc, heads h, and water(so1vent) s. In terms of these three labels the scattering length density can
The Journal of Physical Chemistry, Vol. 97, No. 22, 1993 6021
Structure of a CI8TABLayer
be written P(z) = bcnc(z)
+ 6hnh(z) + bsns(z)
(2) where ni is the number density profile of species i and bi is its scattering length. The kinematic approximation for the reflectivity R ( K may ) be written in terms of the partial structure factors hii
where hij are the partial structure factors given by
hji = hij(K) = Re(nj(K)nj*(K))
(4)
The n i ( ~are ) the one-dimensional Fourier transform of ni(z),the average number density profile of atom, or group, i in the direction normal to the interface, ni(K) =
s-1exp(-iKz)ni(z) dz
An alternative expression, equivalent to (3), can be written in terms of dn/dz = n(l) and is
In principle it is possible to Fourier transform I n i ( ~ )to l ~ obtain the YPatterson” function for the number density,
P(z) = s-lni(u)nj(u - z ) du
(7)
where P(z) represents the correlation of the number density with itself. A set of equations similar to (3)-(7) can be written in terms of the differential of n(z), denoted n(I)(z),and its Fourier transform, h(l)(K). Because the Fourier transforms are one dimensional the two sets of equations are equivalent and the relation between h(K) and h(l)(K)is K2h(K) = h ( ’ ) ( K )
(8)
R ( K )and h ( ~decay ) rapidly with K and, although it is simpler to
discuss reflectivities in terms of the partial structure factors hii, it is more convenient to plot and fit data in terms of K2h(K). The six isotopic compositions, whose reflectivity we have measured at the cmc, correspond to different values of the 6, in eq 3 (see Table I) and the resulting simultaneous equations may be solved for the partial structure factors. The set of six partial structure factors so determined is shown in Figures 6 and 7. The reflectivity given by eq 3 is approximate and without further correction will lead to errors in the partial structure factors, particularly that of the solvent. This has been discussed in detail by Lu et a1.8 who haveshown that an equation derived by Crowleyl can be used to convert the experimental data into a reflectivity for which eq 3 holds almost exactly. The equation is
Figure 6. Self partial structure factors for octadecyltrimethylammonium bromide at the air/water interface: (a) chain/chain, K 2 h c c , (b) head/ head, K Z h h h , and (c) water/water, K 2 h s s . Continuous lines in (a) and (b) are calculated for Gaussian distributions and that in (c) is calculated for a uniform layer. The parameters used are those of Table IV.
by a uniform layer, any difference between the two fits being within the margin of error. We therefore fit the three self partial structure factors using both models to obtain a mean width of each distribution, the values of which are given in Table IV. The resulting fitted cuwes are shown as continuous lines in Figure 6, using Gaussian distributions for chain and lead distributions and a uniform layer for the water (see ref 3 for the formula). The Gaussian distribution for the chains is defined by
n = n, exp(-4z2/u,2)
(10) where the full width is u, at l / e of the height. The corresponding partial structure factor is
The surface excess for the distribution (10) is u,n,a’/2
rc= I/A = 2 .
(9)
There is considerable uncertainty in fitting the head-group distribution. However, if the constraint is introduced that the number of heads must be equal to the number of chains, then eq 12 gives
where Rr and Rk are the exact and kinematical reflectivities for a perfectly smooth surface between the two bulk phases and K, is the momentum transfer at which total reflection would occur for the two bulk phases. The hij of Figures 6 and 7 have been derived by using this modification. We first interpret the self partial structure factors hii. The data do not extend to high enough K for it to be worthwhile carrying out a Fourier transform. This is illustrated very simply by the fact that h,, in Figure 6 is equally well fitted by Gaussian and
uh = ucnc/nh (13) The relatively high accuracy of the determination of uc and n,, which results from a much greater signal from the larger number of scattering species in the chain, can then be used to obtain a value for the width of the head distribution, which has a final accuracy of about 25%. This is an improvement on what could be obtained by fitting Uh and nh independently. The self partial structure factors, hii,contain information about the distribution of each labeled component but not about the
6028 The Journal of Physical Chemistry, Vol. 97, No. 22, 1993
Lu et al. in comparing calculated and observed data and is also confusing in presentation. Since we obtain good fits of Gaussians to the distributions of chains and heads, it is reasonable to replace the observed values of h, and hhh with the fitted Gaussians. In the case where the two partial structure factors are exactly Gaussian, the following expression for hch can be derived from eqs 11, 12, 15, and 16:
33 - 3
k/A
‘
Figure7. Cross partial structure factors for octadecyltrimethylammonium bromide at the air/water interface: (a) chain/head, K’hch, (b) chain/ water, K2hcs,and (c) head/water, K2hhs. The continuous lines are calculated by using Gaussian distributions for chains and heads and a uniform layer for water and eqs 15 and 16. The parameters used are those of Table IV.
relative positions of the components. The information about the relative positions is contained in the cross partial structure factors, h,, and these may give important structural information without being Fourier transformed. It can be shown that the cross term between two distributions centered at 6, and 6, is
h,,M = Re(n,(K)n,*(K) exp[-ir(b, - 6,)l)
(14) It may often be the case that the distributions are either perfectly even about their centers or perfectly odd. For example, both chain and head have distributions that are zero a t large positive and negative values of z and are therefore predominantly even functions, whereas the solvent density is zero at large negative z but has its bulk value at large positive z and is therefore predominantly an odd function. When nc(z)and ndz) are exactly even about their centers and n,(z) is exactly odd, eq 14 gives the following results
h,, = *(hcchss)”2sin ~6,,
(15)
and h,h = (hC,.hhh)”* cos K6ch
(16) If the positive roots of hc, and h,, are taken, the negative sign applies in eq 15. However, the formula is more correctly written
h,, = ncns sin KB
(17) which leads to a phase uncertainty in deriving (1 5) from (14) and is the cause of the ambiguity of sign. However, it is usually possible to determine the sign from other knowledge of the physical situation at the interface. Previously we have used eqs 15 and 16 directly to determine the distance between the different regions of the interface. Since there are statistical errors in both sets of data this leads to errors
Since A, a,, and Uh are already known, 6ch can be determined directly from hch. The result of fitting eq 18 to hch is shown in Figure 7a and gives a value of 6ch = 9 f 1.5 A. In a similar fashion we use the fitted single uniform layer expression for h,, in fitting eq 15 to the experimental partial structure factors hcs and hhs (Figure 7b,c) to obtain values of 6,, = 9 f 0.75 A and 6hs = 1 1 A. The large relative uncertainty in 6hs is clear from the statistical errors in Figure 7c. The set of parameters derived for the layer is given in Table IV. In the discussion we compare the structures of CtgTAB and CI4TABa t the same surface coverage. Since our paper on C14TAB there have been a number of improvements in the analysis of the data and we use these to reanalyze the Cl4TAB data at 4.5 mM, for which the area per molecule A is close to the value for C16TABat the cmc. This reanalysis makes a small change in one of the six parameters obtained in the earlier paper. Two of the improvements have been outlined above, the use of eq 9 to make the adjustment between observed and kinematic reflectivity and the use of the fitted Gaussians rather than actual experimental h,,, hhh, and h,, when using eqs 15 and 16. There are two further improvements, one of which results from introducing a constraint equivalent to eq 13 but in a less trivial fashion and the other of which gives a clear means of refining the calibration of runs with DzO as solvent. We discuss both in more detail in the following section and simply note here that in calculating hch we first normalize each of the reflectivities from dCI8dTABand dC18hTABin null reflecting water to the same coverage before applying the kinematic analysis. We have followed this procedure for ClgTAB and have reanalyzed the CI4TABat 4.5 mM using all the corrections above. The original and revised parameters for C14TAB are given in Table IV. The main effect is to change 6ch. Figure 8 shows the old and new fits for h,h for CI4TAB. The other partial structure factors and their fits are hardly changed by the new procedure.
Errors in the Determination of Partial Structure Factors Since the main purpose of the present work is to determine the effect of chain length on the structure of the surfactant layer structure, it is important to establish the accuracy with which such changes may be measured. Although the use of partial structure factors to analyze the structure of the layer gives a clear picture of what exactly can be determined in a reflection experiment, it is not at all easy to track the propagation of systematic errors. The statistical errors are easily determined and have already been shown in Figures 6 and 7. Here we concentrate on the systematic errors. On the basis of our experience with the present and other surfactant solutions, we classify the systematic errors that might arise in the determination of layer structure from reflectivity into four types: (i) errors in the scaling of the reflectivity arising from errors in calibration and alignment of the sample in the beam; (ii) incorrect background subtraction; (iii) errors in the absolute level of the reflectivity resulting from contamination of the surface, incorrect determination of the isotopic composition, or isotopic dependence of the coverage a t the surface; (iv) errors in the structure consequent on contamination or isotopic effects. It might be thought somewhat artificial to separate (iii) and (iv). However, the reason for doing so is illustrated by considering the differences between soluble and insoluble monolayers. When
The Journal of Physical Chemistry, Vol. 97, No. 22, 1993 6029
Structure of a C18TABLayer
TABLE n7: Structural Parameters (A)from the Partial Form Factors# model
a,(chains)
ah(heads)
?(water)
19h 1 17f 1
14h 3 13f3
10.5 10.5
18.5 f 1 16* 1
13f3 12f3
10 10
bch
f 1.5
d,, f 0.75
dhs f
1
C , ~ T A B(3 x 10-4 M) none uniform layer Gaussian/uniform layer C,~TAB (4.5x 10-3M) none uniform layer Gaussian/uniform layer
9
9
1
9
9
1
6(5)
7(8)
1(2)
6
7
1
Figures in parentheses are the values obtained in ref 3.
TABLE V Parameter Determining the Background Subtraction Error in Eq 19 accN LYhh
= 1.5 x 10-2
1.5 X 10-4 1.5 x
ass
(Ych I!
3.3 x
about 5% and actual values of the backgrounds in our measurements are currently about 2 X 10-6 for samples with a D20 substrate and 6 X 10-6 for a null reflecting water substrate. The difference results from the different incoherent scattering cross sections of H20 and D2O. These errors are small relative to the reflected signal for values of K below about 0.15 A-l but become important at higher values of K or when the signal from the layer is especially low. They may also be significant in a partial structure factor, which depends on small differences between two reflectivities. Weconsider backgrounderrors in the self partialstructure factors and their cross terms separately. Ideally h,,, hhh, and h,, are determined in single measurements from the reflectivity profiles of dC18nTAB/nnv,nClsdTAB/nrw, and nC18nTAB/D20, respectively, where n denotes null reflecting relativetoair,i.e., thesamescattering lengthdensity asair. Though these were not the exact combinations used, they are close enough to be a good approximation. For the corresponding reflectivities we use the notation R,,,, Rnhn, and R,,,, respectively. The value of a in eq 19 is then easily determined from relations of the form
0.05
0.10
0.15
0.20
0.25
0.30
dA-1 Figure8. Observed K2hchfor C14TAB(points) showing the calculated fit (a, top) using 16 from ref 3 and (b, bottom) the fit using the improved procedure described in the text.
an insoluble monolayer is spread on a surface the natural comparison to make between two isotopic species in a reflection experiment is a t constant area, the area per molecule being controlled simply by the amount spread on the surface. There is then no error of type (iii). The manifestation of any contamination or isotope effect will then be a different surface pressure between the two isotopic species a t the same coverage, which may lead to errors of type (iv). For soluble surfactants it is not possible to control the area per molecule directly and errors of type (iii) are relatively common. On the other hand it seems that for these more looselypacked layers the structure only changes slowly with coverages so that errors of type (iv) are relatively insignificant. The choice of background for subtraction has been discussed elsewhere.5 If we assume that the correct background does not vary with K , then it follows from eq 2 that the error that arises in a partial structure factor, KZhcc,will be of the form Ab = a K 4 (19) where a is a constant directly related to the error in the chosen level of the background. The error in the background is typically
where A is the error. Using the values of b,, bh, and b, in Table I and the errors in the backgrounds given above we obtain the values of aii for the three hii given in Table V. The results of these errors are shown as shaded bands in Figure 9. Relatively, the largest error is in hhh, but in ail three cases the systematic background error is less than the statistical errors shown in Figures 6 and 7. The cross terms in the partial structure factors are obtained by taking differences between reflectivities and then the influence of a poor background subtraction is more subtle. The actual value determined for the background is itself subject to a random error, although it becomes systematic once incorporated into a single profile. When evaluating the errors from differences between reflectivity profiles we may therefore use the standard formula for calculating the combination of statistical errors, i.e., lRlA1l+ 1 w 2 1
2(R, - R2)
(21)
We first consider hch, which is determined from K4
A ( K ~ ~= , ,) A(Rchn - Rcnn - Rnhn) 32r2bcbh
(22)
To estimate the error we need to know the values of the three reflectivities. For null reflecting water each of the three reflectivities is approximately proportional to the square of the appropriate scattering length. With this approximation, eq 2 1
6030 The Journal of Physical Chemistry, Vol. 97, No. 22, 1993
Lu et al.
I
00s
KA’
0.m
0.5 K/A’
0.20
azs
I
Figure 9. Effects of errors in the level of subtracted background in the self partial structure factors. The band shows the two limits of the error and the scale is the same as for Figure 6.
Figure 10. Effects of a 5% error in the level of subtracted background in the cross partial structure factors. The band shows the two limits of the error and the scale is the same as for Figure 7.
gives
All errors of the types (i) and (iii) will have the same consequence, that the reflectivity will be abnormally high or low over its whole range, changes in the shape of the reflectivity being covered by the errors of type (iv). We therefore combine all the possibilities(i) and (iii) into a fractional error A. It is convenient, however, to consider errors in the null reflecting water profiles separately from those in DzO. For the three isotopic combinations on null reflecting water, the reflectivity R(K)of eq 1 simplifies to
A(Rchn
- Rcnn - Rnhn) =
(bf
+ bcbh+ bi) 4bcbh
4
(23)
where An is the background error for null reflecting water. Using eqs 22 and 23 and the values of 6 in Table I leads to the value in Table V and the band of error shown in Figure loa. of The errors in the remaining two cross terms are more complicated because the reflectivities depend in more detail on the various hip Thus the error in h, is given by
R(K)= , y { b f h ,
and, using eq 21, A(Rcnd
- Rnnd - Rcnn) (bth,
+ 2bCbsha+ 2b:h,)&/2 + b:h,4
(25) 4bcbsha Equation 23 is the special case of (25) when the background errors for null reflecting water and D20 are the same and when the hip are all taken to be unity. The error arising from eq 25 can only be calculated once the functions h, have been determined by experiment and it will not follow such a simple pattem as eq 19. Taking functional forms that fit the experimental data for the h , e.g., eqs 11 and 15, we calculate the error bands for h, shown in Figure 1Oc. The result of a similar calculation for hhs is shown in Figure lob. It is clear from Figures 9 and 10 that the effects of incorrect background subtraction on the self partial structure factors are comparable with the easily evaluated statistical errors but that errors in the cross partial structure factors are more complicated and are specific to the system being studied.
+ bib,, + 2bcb,hc,)
(26)
Since h, and hhh are determined directly in a single measurement, a fractional error A in the reflectivity carries over directly into h: and hi,, where the denotes the apparent partial structure factor. The relative errors in these two quantities are then the same as in the reflectivity itself, 17%. h:h is determined by subtraction from the reflectivity &nn by using
*
hrh
=a
K2
R c h n
- Rcnn - Rnhn)/2bcbh
(27)
The maximum error in hch occurs when the relative errors A in the three reflectivities combine unfavorably, i.e.
giving a fractional error
&h
in hch of
+ Rcnn + Rnhn)/(Rchn Rcnn - Rnhn) (29) Making the approximationthat the reflectivities in null reflecting water are proportional to the square of the appropriatescattering A(Rchn
The Journal of Physical Chemistry, Vol. 97, No. 22, 1993 6031
Structure of a C18TABLayer
0.1s 0.20 0.25 a30 K /K’ Figure 11. Effects of reinforcing 5% systematic errors in each of Rchn and R,,, in hch (see eq 33). The continuous line is the smoothed observed partial structure factor and the scale is the same as Figure 7a. ’
0.05
0.10
length, as before, we obtain
Ach E A{(bc
+ bh)’ + b,Z + bi)/((bc+ bh)2 - b: - bi) = A(b: + 6; + b,bh)/b,bh
(30)
For C I ~ T A bc/bh B = 3.5 (Table I) making &, 5A a t low K . Thus, even systematic errors of the order of 5% can lead to serious errors in hch. However, an examinationof the consequences shows that such errorscan be greatlyredudif not eliminated altogether. Since Rchnand R,,, are much larger than&, (the approximate ratio is 1:0.6:0.05) the error in hch arises almost entirely from the errors in Rchnand R,,,, the contribution of Rnhnbeing negligible. If Rchnand R,,, are found to correspond to slightly different coverages, which is easily checked by using the single uniform layer model to fit the data, then, neglecting the contribution of Rnhn,the resultant hh: consists of the true hch plus a fraction of hcc. From eqs 27 and 28
Rnhn. This is because of a combination of two factors, the low signal and the limited K range of the measurement. We have demonstrated that the use of the single layer model to fit the data gives a reliable value of the surface coverage. However, this is only true if the lowest value of K a t which the reflectivity is measured gives a relatively large signal. If not, then the roughness of the interface becomes an adjustable parameter, which can affect the derived surface coverage if it is larger than about 5 A. Given this additional uncertainty and given that Rnhndoes not contribute significantly to the errors in hch, we prefer to leave this reflectivity alone. It is also clear that the correction described above would be invalid if errors of types (i) and (iii) were accompanied by an error of type (iv). However, provided that the original difference is within certain limits, the structural changes may be insignificant. Thus analysis of a wide range of data for C12E3shows that, at values of A above about 35 A2, a 10% change in the reflectivity corresponds to a 5% change in coverage and a 3% change in the structural parameters. Since the largest acceptable systematic error one might envisage in the reflectivity is of the order of lo%, the procedure outlined above will not be invalidated by more than the intrinsic errors in the analysis. Consideration of the errors in reflectivity from D2O solutions leads to quite different corrections. The chain-solvent correlation is given most simply by the combination of the following three runs: dCdTAB/nrw, dCnTAB/D20, and nCnTAB/D20, i.e., by R,,,, Rend, and Rnnd. This combination is seldom exactly realized in practice but is sufficient for considering the possible systematic errors. The combinations are such that h,, and h,, are determined from R,,, and Rnnd,respectively, by equations similar to (26) and any relative error A passes directly over into the partial structure factors. Rnndis only indirectly sensitive to the surfactant coverage. For a value of A of about 45 A2, a model calculation shows that an error of 5% in the surfactant coverage leads to an error of less than 1% in h,,. Thus the only important errors are those of type (i) and/or incorrect isotopic composition. The simplest possible model of h,, is
~ ~ h=, nf, exp(-K2a2)
hch A(bch:c/bh + bhhi,/2bc + h:h) (31) which, for the particular case of C I ~ T A Bgiven , the ratio bc/bh = 3.5, becomes
hch f A(3.5hfc + hlh/7
+ h:h)
(32) The three partial structure factors are comparable and so the dominant error is the inclusion of a significant fraction of h:c into We assume that this is the only error and then h:h
*
h:h h,h 3.5Ah;, (33) An examination of hch and h,, (Figure 6 ) shows that such an error will have a disastrous effect on the determination of bch. Figure 11 shows the effects of such an unfavorable combination of a 5% systematic error in each of Rchn and Rcnn. If there are good reasons for supposing that there is no isotope effect on the system, for example, as demonstrated by surface tension measurements, it is possible to eliminate the systematic errors of types (i) and (iii) from the calculation of hch as follows. If the two reflectivities Rchnand R,,, are normalized to the same average coverage, they will finish with identical systematicerrors, which will cancel out when eq 28 is applied, Le., the factor 3.5 in eq 33 becomes 0 and the awkward mixing in of hf, is avoided. One way of looking at this correction is that it is the most effective way of introducing the stoichiometricconstraint that the number of chains equals the number of heads. It could be argued that such a constraint could be applied to all three reflectivities in eq 28. This will depend on the system and for our measurements we feel that it would not be safe to extend the correction to include
(34) wheren, is the number density of the solvent and nt is the limiting value of K Z ~ at ~ ,low K . This limit is reached for all models of the solvent structure and for water it has the value 1.1 X 10-3 A+. If there are any errors of type (i) or of isotopic composition of the solvent, ~ ~ hwill , , extrapolate to an incorrect value of the limit. However, given the earlier argument that Rnndis only very weakly affected by changes in surfactant concentration and such effects can be neglected, any scaling error in h,, can be eliminated totally by multiplying Rnndby a factor that ensures that thecorrect limit is reached. Such a correction, which merely incorporates the known bulk density of the solvent into the analysis, has only recently been made possible by the introduction of eq 9. Previously, the kinematic approximation failed at values of K where h,, started to approach its limiting value. The determination of systematic errors in the two cross partial structure factors h,, and hhs is somewhat more complex because these cross terms depend on reflectivity differences. By the same argument as for h,,, all errors of type (i) and effects of incorrect isotopic composition of the solvent can be eliminated by scaling each reflectivity to give the correct limiting density of the solvent as K tends to zero. This is done by using the following treatment (35) where h* is an apparent solvent structure factor. This may not be easy if measurements have not been extended to low enough K where effects of the surface layer on the reflectivity become insignificant. However, this problem may be avoided either by extending the measurements or by using an approximate model to assess the layer contribution and enable an extrapolation to
Lu et al.
6032 The Journal of Physical Chemistry, Vol. 97, No. 22, 1993
KA’
be made. Any errors that remain depend only on errors in the surface coverage and these, in turn, assuming that surface tension measurements have established that there is no isotope effect, can only be caused by surface contamination. The possible effects of surface contamination on the two cross terms h,, and hhs are determined respectively from
0
(37) Simple model calculations show that Rend is the only one of the four DzO reflectivities in eqs 36 and 37 to be significantly sensitive to surfactant coverage and therefore to contamination of the surface. Thus there are two possible sources of error in h,,, from Rend and from R,,,. Neglecting for the moment any error in R,,, a relative error A in Rend resulting from surface contamination is carried over directly into h,. It is therefore necessary to consider what types of surface contamination are likely because each one will have a different effect on Rend. There are four possibilities: (i) Contamination by electolyte. This will increase the surface coverage of surfactant if it is charged and have little effect otherwise (at low electrolyte concentrations). (ii) Contamination of theoriginal sample by highly surface active deuterated precursor (in the present case octadecyl bromide). This will increase the apparent surface coverage in the null reflecting water measurement. (iii) Contamination of the original sample by highly surface active protonated material. This is unlikely but would decrease the apparent surface coverage in the null reflecting water measurement. (iv) Contamination of the solution by residual protonated surfactant from cleaning procedures. This will decrease the apparent surface coverage in the null reflecting water measurement. The most probable types of contamination are (ii) and (iv), but (ii) would have effects on measurements in both null reflecting water and D20. Thus, provided Rend is measured first and found to be consistent with isotherm measurements, either neutron or surface tension, there will be no such error when it comes to the D20measurement. Thus (iv) is the only remaining source of error. A reasonable upper limit on this contamination, based on our experience of handling a range of surfactant solutions, is 5% for the CTABs (it would be much less for more surface active species such as the alkyl ethylene glycols). The results of a model calculation indicate that at the surface concentrations of the present experiment 5% contamination of the surfactant layer by residual protonated surfactant during the measurement of Rend would lead to a final error in 6,, of about 8%. There is also a contribution to h,, from errors in R,,,. These have already been discussed above and again should be less than 5%. This introduces a further significant error into h,,, but this is partially cancelled when h,, is introduced into the calculation of ~5,~. The effects of both errors are shown in Figure 12 and the final total systematic error in 6,, is a maximum of 10%. A comparable analysis of 6hs indicates that systematic errors are outweighed by statistical and background errors, so that 6,, is the one parameter in the whole set where systematic errors may outweigh statistical and background errors. However, it should be emphasized that the combination of errors leading to a 10% overestimate of 6,, is a very unlikely one and the contamination error can only have the effect of increasing the apparent 6,,. The overall effect of including all the errors, statistical and systematic, is that the background and statistical errors cause considerable uncertainty in measurements involving the trimethylammonium head group. This is clearly a consequence of its small scattering length and therefore low contribution to the reflectivity. On the other hand, provided great care is taken to avoid contamination errors, the determination of all the dimensions
Figure 12. Effects of a 5% contaminationerror in Rend (dashed h e ) and an additional 5% error in h,, (dotted line) on hcs. The smoothed experimental partial structure factor is the continuous line. The values of b,, required to fit the threecurves are 9.0 (continuousline), 9.7 (dashed line), and 9.9 (dotted line) A.
involving the chains and particularly the useful 6,, should be accurate to better than 10%. The six partial structure factors evaluated here overdetermine the structure as defined. Thus, it is not necessary to determine 6ch, 6,,, and 6hs. Agreement of the three values, Le., (6ch - 6,) equaling 6hs within error, indicates the absence of many of the systematic errors considered. Thus it is easy to show that an anomalously high value of h,, leads to an increase in 6, and a decrease in dch, but has no effect on ah,, and would therefore make the threevalues inconsistent. In the particular caseof the CTABs it would have been possible to determine the structure with only five partial structure factors, leaving out hhs by not measuring Rnhd. However, this can only be done if the head-group scattering length can be adjusted to zero so that it makes no contribution to the reflectivity of other isotopic species. Of the common surfactants this is only possible with the CTAX series.
Discussion We first use the results of Table IV (direct fitting) to compare the structure of the two surfactants. The fully extended lengths 1, of the chains of C14TABand C18TABare respectively 19.2and 23.8 A. At 4.5 mM the C14TAB thickness is slightly less than 1, and the C 18TABis substantially less than I, being little different from that of C14TAB. This trend is also extended to CloTAB, where the chain thickness was found to be significantly greater than I,, at a lower surface excess. The CISTABchain region is correspondingly moredense than that of C14TAB. It is not possible to give absolute values of the density of the chain layer because they are model dependent. Taking the maximum density in the Gaussian distribution (eq lo), the maximum number densities are approximately the same as 1.5 X A-3. The corresponding number densities of octadecane and tetradecane are 1.8 and 2.3 X A-3. The maximum density in the Cl8TAB chain layer is therefore nearly 85% of its corresponding liquid alkane but for C14TABis only 65%. The penetration of the chain by water is most directly assessed from the thicknesses of chain and water layers and the separation 6,,. As shown in ref 10 an estimate of the fraction of chain penetrated by solvent, 4, is given by
Taking values from Table IV, we find that the ClsTAB and (214TAB chains are respectively about 30 and 40% immersed. The model-fitted parameters in Table I11give the same result, although as noted earlier the model fitting can give a different result. The difference between the two surfactants is probably what one would
Structure of a ClgTAB Layer
I
The Journal of Physical Chemistry, Vol. 97, No. 22, 1993 6033 Error
Solvent
Distance/A Figure 13. Experimental distributions of chains, heads, and solvent for ClgTAB (dashed line) and C14TAB (dotted line) compared with those from computer simulation of C16TACI (continuous line). The reference point has been taken to be the center of the chain distribution. The simulated distributions, from ref 12, have been smoothed and the peak heights of chains and heads have been scaled to place emphasis on the shapes of the distributions. The experimental water distribution is here described by a tanh function with a width parameter of 6 R,3 which is equivalent to the uniform layer of 10.5 A in Table IV.
expect given that the alkyl chains become more like liquid alkane as the chain length increases. The relative position of the head-group and chain distributions is less easily understood. If the chain were fully extended and aligned normal to the surface the chain head separation would be ( I , 11,)/2,Le., about 11 and 13 A for C14TAB and C18TAB to be compared with the measured values of 7 and 9 A. There are three mechanisms that would account for the discrepancy, a high incidence of trans-gauche conformations in the chains, an average tilting of the chains, and either static or thermal disorder in the direction normal to the surface. We discuss this below in relation to recent computer simulationsof C16TAC1.12The width of the head-group region certainly indicates that the third of these mechanisms is important. The thicknesses of both headgroup distributions are significantly larger than would be expected if the molecules lay on a single surface plane and, although there is considerable experimental uncertainty about the head-group thicknesses because of the weakness of the signal from the deuterated head, they are consistent with the third mechanism. Further confirmation of this come from the penetration of the chain region by the head groups. The application of eq 37 gives values of 35 and 50% for head-group penetration for CI8TAB and C I ~ T A Brespectively. , It is surprising that this number is larger than the value of the water penetration and we have no explanation for it. However, it may simply be that it is not appropriate to apply such a simple equation as eq 38. For example, the shapes of the different distributions may not be as idealized as we have supposed. We have argued elsewhere that assumptions about the shapes cause little error in the determination of the 6 values and this is certainly borne out by the self-consistency of the three 6 values for both surfactants. However, deviations from the assumed shapes may well make the subsequent interpretation unreliable. There has been a recent computer simulation by Biicker et al. of C16TAClat an area per molecule (45 %L2) similar to that at which we have studied C18TABand C14TAB.12 In Figure 13 we compare their distributions for chains, heads, and solvent with those of our two surfactants. We have taken as a reference point the midpoint of the chain distribution and we have converted their separate methylene and methyl group distributions into a single chain distribution. We have also altered the limit of their
+
simulation for the solvent for greater clarity. The most reliable parameters from our measurements are the separations of the three distributions and it is clear that for both solvent and head group our values for the shorter C14TAB and the longer &TAB straddle Biicker et al.'s values for C16TACl. Also in good agreement with the computer simulations are the widths of headgroup and solvent distributions, which are approximately independent of surfactant chain length. The main disagreement is that the experimental width of the chain distribution is wider than the simulation for both C I ~ T A Bwhich , is not surprising, and C14TAB, which is surprising. The latter disagreement indicates that some refinement in the effective hydrocarbon interactions used in the computer simulation will be necessary, but nevertheless the agreement between experiment and simulation is remarkably good. An especially interesting result from the experimental point of view is that the simulation indicates that the assumption of evenness and oddness of the distributions is a good one. The small deviation from evenness in the chain distribution from the simulation falls well below the level a t which corrections would be necessary in the a n a l y ~ i s .The ~ simulation alsojustifies our conclusions concerning the disorder in the chains. A secondary conclusion from this work concerns the apparent difficulty in measuring and interpreting surface tension measurements on cationic surfactants. There are few published measurements and this seems to be because of partial wetting leading to problems of reproducibility, which then discourages publication. In this particular case we have had no particular difficulty and, provided that B polynomial is used to fit the Gibbs equation to the data, there is reasonable agreement with the neutron reflection measurement. We have shown in our earlier paper on C14TAB that these materials are not easy to purify and our conclusion is that it has mainly been impurities that have prevented accurate surface tension measurements. This is supported by the fact that it is usually easier to obtain good surface tension results with the deuterated surfactants. The cost of the deuterated raw materials is very high and it seems that some of this cost is reflected in a higher purity.
Acknowledgment. We thank the Science and Engineering Research Council for support. E.A.S. also thanks Unilever Research, Port Sunlight, for studentship. References and Notes (1) Lee, E. M.; Thomas, R. K.; Penfold, J.; Ward, R. C. J. Phys. Chem. 1989, 93, 381.
(2) Simister, E. A.; Thomas, R. K.; Penfold, J.; Aveyard, R.;Binks, B. P.; Cooper, P.; Fletcher, P. D. I.; Lu, J. R.; Sokolowski, A. J . Phys. Chem. 1992, 96, 1383. (3) Simister, E. A,; Lee, E. M.; Thomas, R. K.; Penfold, J. J. Phys. Chem. 1992, 96, 1373. (4) Furniss, B. S.; Hannford, A. J.; Smith, P. W. G.; Tatchell, A. R. Vogel's Textbook of Practical Organic Chemistry, 5th ed.; Longman: Essex, U.K., 1989. ( 5 ) Penfold, J.; Lee, E. M.; Thomas, R . K. Mol. Phys. 1989, 68, 33. (6) Cardwell, P. H. J. Colloid Interjace Sci. 1966, 22, 430. (7) Deleted in proof. (8) Lu, J. R.;Lee, E. M.; Thomas, R. K.; Penfold, J.; Flitsch, S. L. Langmuir, in press. (9) Simister, E. A.; Lee, E. M.;Thomas, R. K.; Penfold, J. Macromol. Rep. 1992, A29, 155. (10) Lu, J. R.;Simister, E. A,; Lee, E. M.; Thomas, R. K.; Rennie, A. R.; Penfold, J. Langmuir 1992, 8, 1837. (1 1) Crowley, T. L. Phys. A , in press. (12) Bocker, J.; Shlenkrich, M.; Bopp, P.; Brickmann, J. J . Phys. Chem. 1992, 96, 9915. (13) Tanford, C. J. J. Phys. Chem. 1972, 76, 3020. (14) Willis, B. T. M., Ed. Chemical Applications of Thermal Neutron Scattering, Clarendon Press: Oxford, 1973.
