J. Phys. Chem. 1992,96, 1373-1382
1373
Structure of a Tetradecyltrimethylammonium Bromide Layer at the Air/Water Interface Determined by Neutron Reflection E. A. Simister, E. M. Lee, R . I(.Thomas,* Physical Chemistry Laboratory. South Parks Road, Oxford, OX1 3QZ UK
and J. Penfold Rutherford-Appleton Laboratory, Chilton, Didcot, Oxon, OX1 I ORA UK (Received: July 3, 1991)
We have determined the structure of a layer of tetradecyltrimethylammonium bromide (C14TAB)adsorbed from solution at the air/water interface using neutron specular reflection data from several isotopic compositions. Two different methods of analysis of the data have been used, one fitting a single structural model to the set of reflectivity profies at a given concentration, and the other using an approximatebut more direct method. The relative locations of the chain, head, and water distributions across the interface have been determined directly. For the saturated monolayer Gust below the critical micelle concentration (cmc)) the mean center-to-center distribution of chains and heads is found to be 7 f 0.5 A, that between chains and water to be 7 f 0.5 A, and that between heads and water 1 f 0.5 A. The mean thickness of the different regions is model dependent. The chain region is found to have a thickness of 17.5 f 1 A for a uniform layer model and 16 f 1 A for a Gaussian distribution (at l / e of the height), both less than the fully extended chain length. The corresponding values for the heads are 7 f 3 (uniform layer) and 6 f 3 A. At about one-third the cmc the chain region becomes thinner (Gaussian width = 12 f 2 A) and the mean separation of chains and water decreases to 6 A. Above the cmc the surfactant is more closely packed and the thickness of the head-group distribution increases to 12 f 3 A (Gaussian profile). This is attributed to a "roughening" of the head-group part of the layer.
htroduction In this paper we apply the recently developed technique of neutron reflection to the study of the structure of layer of the surfactant tetradecyltrimethylammonium bromide (C14TAB) adsorbed at the air/water interface. An important feature of neutron reflection is the use of different isotopic species to establish a structure. This often depends on the assumption that isotopic substitution does not affect the chemical and physical properties of a substance. We have already established in a previous paper that the interfacial properties of C14TABare not affected by deuterium substitution, and we have compared some of the results obtained from neutron reflection with surface tension measurements.' We now use isotopic substitution to determine the structure of the layer, and we also analyze in some detail how best to determine structure from reflection measurements.
Experimental Details Four isotopic species of tetradecyltrimethylammoniumbromide were used in the experiments, CI4D2&J(CD3),Br, C14DB(CH3),Br, C14H29N(CD3)3Br, and C14H29N(CH3)3Brr which we refer to as dC ,,dTAB, dC I &TAB, hC ,PTAB, and hC ,hTAB respectively. They were the same samples described in ref 1, where a full description of the preparation, purification, and characterization is given. High-purity water was used throughout (Elga Ultrapure system) and the methods of cleaning the glassware and Teflon troughs for the neutron experiment are also described elsewhere? 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 previou~ly.~.~ The measurements were all made at a fixed incident angle of 1.5O, and the intensities calibrated with respect to D20. One of the limitations of reflection measurements is the presence of an incoherent background. There are a number of ways of determining the background, by off specular measurements," direct mea~urement,~ or by extrapolation (1) Simister, E. A.; Thomas, R. K.;Penfold, J.; Aveyard, R.; Binks, B. P.; Cooper, P.; Fletcher, P. D. I.; Lu, J. R.; Sokolowski, A., in press. (2) Lee, E. M.; Thomas, R. K.; Penfold, J.; Ward, R. C. J . Phys. Chem. 1989,93,381. (3) Penfold, J.; Le+, E. M.; Thomas, R. K. Mol. Phys. 1989,68, 33. (4) Rennie, A. R.; Crawford, R. J.; Lee, E. M.;Thomas, R. K.;Crowley, T. L.; Roberts, S.; Qureshi, M.S.;Richards, R. W. Macromolecules 1989, 22, 3466.
to high values of the momentum transfer, K . Provided that there is no small-angle scattering from the bulk solution the background is found to be flat in K except at very small values of K where the subtraction is, in any case, unimportant. For surfactants at concentrationsbelow the critical micelle concentration (cmc) the background is flat and can easily be determined accurately. All the results shown here had a flat background subtracted.
Neutron Reflection Specular reflection of neutrons gives information about inhomogeneities normal to an interface.5 The effectiveness of the method for surfactants depends on being able to combine reflectivity profiles from solutions of the same chemical but different isotopic composition. The reflectivity profiles can be analyzed in two ways. In the most commonly used method a structural model is assumed for the interface and the reflectivity calculated exactly using the optical matrix method.6 The structural parameters are the thicknesses of the component layers and their scattering length densities, which depend on the number densities of each atomic species and their known scattering lengths. When using sets of isotopic data, it is necessary to establish how the scattering length density of a given layer varies with isotopic substitution and to attempt to optimize the structure to fit simultaneously the reflectivity profiles from the different isotopic species. This may become a difficult exercise if the structure of the interface is at all complicated, and there is always the risk that the final structure is not a unique solution. An alternative approach has been outlined by Crowley et al.' In the kinematic approximation the reflectivity is given either by
or by
where K is the momentum transfer normal to the interface ( K = 47r sin ell),P(K) is the onedimensional Fourier transform of p(z), ( 5 ) Penfold, J.; Thomas, R. K. J . Phys. Condens. Marrer 1990, 2, 1369. (6) Born, M.; Wolf. E. Principles of Oprics, 5th ed.;Pergamon: Oxford,
1975. (7) Crowley, T. L.; Lee, E. M.; Simister, E. A.; Thomas, R. K. Physica B 1991,8173,143.
0022-3654/92/2096-1373%03.00/00 1992 American Chemical Society
1374 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
Simister et al.
and ;(')(K) is the Fourier transform of dp/dz = p(')(z): P(K)= S_:exp(-iKz)p(z)
dz
P(')(K)= S_Iexp(-irz)p(')(t) dz
(3) (4)
p(z) is the mean scattering length density at level z in the interface.
Equations 1 and 2 are approximate and hold only when K is significantly greater than its value at the critical angle of total reflection. For all the reflectivity profiles discussed in this paper, (1) and ( 2 ) are good approximations. The simplest description of the structure of the air/solution interface is in terms of the distributions of three group, the head group, h, of the surfactant, its chain, c, and the solvent, s. We can write the scattering length density profile across the interface in terms of the number densities of these three groups: p(z) = bcnc(z) + bhnb(z) + ~ S ~ S ( ~ ) (5) where b, and n, are the scattering lengths and number densities of the different groups. Substituting ( 5 ) into (1) gives 16u2 R(K) = -(b:h, K2
+ bh2hhh + b?h, + 2bcbhh& + 2bcb~ha+ 26hb,hhs) (6)
where h, depends on K and is given by h,,(~)= lfi,(~)V = Re lfih4fi;(~)I (7) where fir(.) is given by an equation corresponding with (3) except that it is in terms of the number rather than the scattering length density. An equivalent expression, in terms of hjf)(~), would be obtained by substituting the derivative of ( 5 ) into ( 2 ) . As can be seen from eqs 1 and 2
hljl)(~)= K ~ ~ , , ( K )
(8)
A possible procedure for determining the structure of the interface directly is then to measure six reflectivity profiles using isotopic substitution to obtain different values of bo 4, and b,, from which the six different h, in (6) can be obtained. We shall refer to these functions as the partial structure factors of the surface layer. Each one of the structure factors may be Fourier transformed directly to give P,,(z) and P,,(z), where PJz) = j m -_n , ( u )n,(u - z) du
(9)
The functions P are similar to the Patterson function used by crystallographers, and they will usually be sufficient to defme the structure of the layer unambiguously.' It should be noted that the structure can only be defined in terms of the labeled groups. For example, to determine any differences in the distribution of methylene groups at either end of a surfactant alW chain, it would be necessary to make the appropriate isotop. "dbstitutions to determine the additional four Patterson functions. A particularly simple example of this method is when the isotopic composition of the water is adjusted so that its scattering length density matches that of air (null reflecting water (nrw)). If p of the head group is also matched to air (approximately achieved by using the protonated head group) but p of the chain is not (achieved by deuteration), all terms except the first one in eq 6 vanish. Multiplication of this reflectivity profile by 2/16&: or ~ ~ / 1 6 ? r %gives 2 h, or which may be transformed to give respectively either P,(z) or Pd)(z),from which it is straightforward to determine the chain distribution. &(Z) and Ps(z) can similarly be determiid directly in separate experiments. In the kinematic approximation multiplication of R(K)by K ~ 16rZb,2 / is equivalent to division by R&K)where R,(K) is the reflectivity of the perfectly sharp interface. The limitation of this second method of analysis is that data can be obtained only over a limited range of K, and the practical problems of Fourier transformation are such that it will seldom be worth doing. However, the form factors h ( ~offer ) the possibility of an intermediate stage of analysis, which is valuable for
lo4
1... .
0
A t
0
A
t
€
i
i'
lo4
0.100
0.150
8
L
T
P 0.200
0.250
K/k' Figure 1. Neutron reflectivity profdcs of fully deuterated C14TABin null reflecting water at different concentrations: (0)4.5 X lo+ M, (A) 3.0 X lO-'M, (+) 1.0 X M, ( 0 )3 X l V M , and ( 0 ) 1 X lo4 M. T = 298 K. The incoherent background has been subtracted.
assessing the uniqueness and resolution limitations of any assumed structural model. Equation 6 also gives a means for reducing a whole set of isotopic data to the minimum number of profiles required to determine the structure of the layer. For C14TABwe have measured the reflectivity profiles of 11 different isotopic compositions (three labels) at one concentration in order to assess this alternative method of analysis.
ReNlk3 Figure 1 shows the variation of reflectivity with surfactant concentration for dC14dTABin nrw. The measurements cover a range of concentration from about 0.3 cmc (the cmc = 3.7 X M) to above the cmc at 1.5 cmc. In nrw the general level of the reflectivity depends on the surface excess and the simplest result obtainable from the set of profiles in Figure 1 is the adsorption isotherm. This has been fully discussed in ref 1 where we have given the equivalent results for dCl&TAB and compared the results for both isotopes with surface tension measurements. The simplest model for fitting the reflectivity is to assume that the structure is a uniform layer. Such a model fits the data in nrw very well and gives values for the mean thickness of the layer at different concentrations. The values obtained from this simple analysis are given in Table I for dC14dTABand dCl&TAB. The mean thickness of the layer decreases by up to 20% for the fully deuterated species and by about 10% for the partially deuterated species over the concentration range measured for a change in the coverage of a factor of three. The sensitivity of the profiles to the fitted thickness is shown in Figure 2, which shows x2 plots at four concentrations when both T and r are allowed to vary. We have given an error of 10%in T in Table I, but the x2 plots indicate that that may be a bit pessimistic. The relatively small change in the thickness of the layer between the two isotopic species is a little surprising. At 5% this is smaller than the individual error but doea seem to be systematic. Figure 3 compares the reflectivity of three isotopic species, dCI4dTAB,dC,&TAB, and hCI4dTAB,at the same concentration in nrw (3 X lC3M)just below the cmc. The signals in these three reflectivity profiles result respectively from the whole molecule, the chains only, and the heads only, because to a good approximation the protonated group are matched to the nrw. The low level of the reflectivity of the hC14dTABprofile makes the errors in fitting a uniform layer model rather large, but this profile plays an important role in establishing the detailed structure of the layer. Figure 4 compares the reflectivity of all four isotopic species at 3 X M in D20with D20on its own. Bemuse of the high scattering length density of D20the overall reflectivity is much
The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 1375
Tetradecyltrimethylammonium Bromide Layer. 1
0
0 D
O 6
0 0
0
A
o D
o 0
0
00
0
0
0 0 0
A
N
0
X
0 0
on
0.20
0.15
A
8
0
025
I 3
4’ F i i 3. Neutron reflectivity profdcs of dC14dTAB(0),dCI4hTAB(A), and hCI4dTAB(0)in null reflecting water at 3.0 X l V 3 M and T = 298
K. The incoherent background has been subtracted.
14
18
26
22
wh
Figure 2. x2 plots for the fit of a single uniform layer to the data of
Figure 1. The fits are plotted as a function of T but were done simultaneously for both 7 and pI. The concentrations of C14TABare (0)4.5 X 10-3 M, (X) 3.0 X lo-’ M, (A) 1.0 X M, and (+) 3 X lo4 M. TABLE I: Parameters Determined from Neutron Reflection Using a Single Uniform Layer Model r/ 10-10 e l lo-’ M pJ 1 O4 A-2 71.4 A I A ~ molcm-2
dCI4dTAB 4.5 3.0 1 .o 0.3 0.1
4.7 4.1 3.3 2.4 1.5
4.5 3.7 3.5 3.0 2.5 1.9
3.5 3.5 3.4 3.1 3.3 2.8 2.8 2.3
20 20 18 16 16
*2
*
44 2 48 67 3 100 f 7 160* 20
3.8 3.5 2.5 1.7 1.1
0.2
dCl4hTAB
1 .o
0.3
19 19 19 19 17 17 16 17
44 45 46 49 50 60 64 74
3.8 3.7 3.6 3.4 3.3 2.8 2.6 2.3
higher than for the nnv profdes in Figure 3. dCI4dTABenhances the reflectivity at the lowest value of K but all four depress the reflectivity of D20 at the highest value of K . The greatest effect on the reflectivity of D20 is caused by dC&TAB. It can be shown for simple models that the depression of the D 2 0 reflectivity is maximized when the scattering length density of the layer is half that of D2OS7In principle this could be brought about by a chain layer of intermediate scattering length density or a head group layer of intermediate scattering length density. That none of the other three isotopes gives such a large effect shows that there is not total separation of head and chain groups. It might at first seem that little is to be gained from a measurement where the surfactant has hardly any effect. However, it is largely this set of results that makes it possible to draw conclusions about the structure within the layer. As discussed in the previous section the contribution of the surfactant layer to the reflectivity is emphasised by plotting the data in the form of R / R , against K . Some of the D 2 0 data of
0~100
0150 I(
om
0250
/A
Figure 4. Neutron reflectivity profiles of the four isotopic species of C14TABat 3 X M in D20,(a) dC14dTAB(0)and dC,,hTAB ( 0 ) and (b) hC14dTAB( 0 )and hC14hTAB(0).In both cases the continuous line is the observed D20profile.
Figure 4 are replotted in this form in Figure 5. D20itself is again included for comparison. If the surface of D20were perfectly flat it would be a horizontal line in Figure 5 . The slight fall off in R / R , is caused by the small roughness of its surface (2.8 A). This has been attributed to capillary waves? although in the specular reflection experiment as done here no distinction can be made between diffuseness and capillary waves. Since the fully protonated CI4TABis almost exactly matched to air the R / R , (8) Schwartz, D. K.;Schlossman, M.L.; Kawamoto, E.H.; Kellogg. G. J.; Pershan, P. S.;Ocko, B. M. Phys. Rev. 1990, A l l , 5687.
1376 The Journal of Physical Chemistry, Vol. 96, No. 3, I992
+
Simister et al.
+
0
0
0 0
3
A
a
0.n
0.10
0.20
010
020
015
KA’
025
$30
KIA.’
Figure 5. Neutron reflectivity profdes of dC14dTAB(0) and dCl,hTAB (A) at 3 X IO-.’ M in D20, multiplied by K4. The observed D20profile (+) is included for comparison.
Figure 7. Effect of water in the surfactant layer. The data (points) are for 3 X IO-) M hCI4hTABin D20.The calculated lines are for any remaining space in just the head-group region filled with water (upper line), and the whole layer filled with water.
layer relative to that of water changes through unity. When it is less than 1 the reflectivity of the solvent is depressed, and when it is greater than 1 the reflectivity is enhanced. In principle this could be used as a method to determine the exact scattering length density of the layer but this would not be valid if the distribution of material through the interface were not uniform, as is probably the case for CI4TAB.
+ 0
10.6
I
0
X 0
c
9 X
8 +
I
X
I
( o o o o o 0.. 0
u o o o
03-
o
o
o
o
O
0
Interpretation (a) Interpretation Using a Structural Model and the Optical Matrix Method. Just as we have found for other surfactant layer^,^.^ it is impossible to fit a single structure consisting of a uniform layer to all the isotopic data. It is possible to fit the nrw data to such a model but totally impossible to fit the D 2 0 data, even approximately. The next simplest and most obvious model is to divide the interface into two uniform layers, one predominantly consisting of alkyl chains, and the other predominantly head groups, with any available empty space being partially filled with water. If the segregation of chains and head groups is taken to be total, this model also fails to fit the whole set of reflectivities of the different isotopic species. Examples are shown in Figure 7 for 3 X M hC14hTABin D20with (a) just the head-group region filled with water and (b) all the available space in the layer filled with water. We have used the two-layer model successfully on other surfactants by constraining the water to remain in the head-group region but allowing a certain proportion of alkyl chains also to enter the head-group region. The adjustable parameters of such a model are then the area per molecule, A, the proportion of alkyl chain in the head group region,f, and the degree of extension of the alkyl chain, c. In terms of these adjustable parameters the thickness 7 and scattering length densities p are as follows: Tc = (1 - j ) l , e Th
Pc
Ph
=
= 1,
=
+ I,-f
(1 - A b , 7
f b c -k
bh + ribs 7hA
(9a)
where 1, and lh are the lengths of chain (fully extended) and head group, b,, bh,and bs are the known scattering lengths of chain, head, and solvent, respectively, and n is the number of water molecules per surfactant molecule. I, was taken to be 19.2 A? and estimated to be 5 A. n is determind by the available free ( 9 ) Tanford, C. J. J. Phys. Chem. 1972, 76, 3020.
The Journal of Physical Chemistry, Vol. %, No. 3, 1992 1377
Tetradecyltrimethylammonium Bromide Layer. 1 TABLE II: Parameters for Calculated Profiles of Figure 7 species A/A' f dC,.dTAB/nrw 47.8 0.25 dC;;hTAB)nrw 48.5 0.28 hC,,dTAB/nrw 51.9 0.26 dCl4dTAB/D20 47.9 0.26 dCI4hTAB/D20 46.0 0.23 hCI4dTAB/D20 47.8 0.32 hCI4hTAB/DzO 41.3 0.25
t
0.68 0.76 0.73 0.64 0.71 0.67 0.67
pc
%/A
x 106/A-2 4.63 4.05 -0.2 4.9 4.6 -0.25 -0.25
9.7 10.5 10.4 9.1 10.5 8.8 9.7
Ph X
106/A-' 3.6 1.6 1.8 6.8 4.6 5.0 3.2
%/A 9.8 10.4 10.0 10.0 9.4 11.1 9.8
TABLE Ilk Mean Parameters of the Two-Layer Structure at Different Concentrations ~
~~
?CIA
%/A
4.5 X lo-' 43 f 3 0.3 f 0.1 0.75 f 0.1 10 1 3.0 X lo-) 48 f 2 0.3 f 0.1 0.75 f 0.1 10 f 1 1.0 X lo-) 62 f 4 0.25 f 0.1 0.65 f 0.1 9 i 1
10 i 1 10 i 1 10 f 1
concn/M
A/A~
f
6
*
TABLE IV: Sensitivity of Structural Parameters to Different Reflectivitv Profiles dC,,dTAB/nrw A,' t, f dC14hTAB/D20 A, t, f" dC,,hTAB/nrw A," c, f hC14dTAB/D20 f hC,,dTAB/nrw A hC14hTAB/D20 f dC14dTAB/D20 A, t, f" ~~
a
~
Most sensitive parameter.
volume in the head-group region taking the molecular volume of water to be 30 ASand that of the head group to be 141 A3.lo In principle the reflectivity profiles at a given concentration should be fitted with single values of A , f , and e. However, to use a least-squares fitting routine most effectively, we fitted each profile independently allowing A,J and e to float within narrow limits.This also prevents any systematic error in one profile having too dominant an effect on the derived parameters. The resulting fits to the seven profiles for 3 X M C14TABin nrw (three isotopic species) and DzO(four isotopic species) are shown in Figure 8 and the actual parameters for each calculation tabulated in Table 11. At this concentration we can therefore fit the whole set of data with A = 48 f 2 A2,f = 0.27 f 0.05,and e = 0.70 f 0.05.The same procedure was used to analyse the set of profiles at 4.5 X lo-' and 1 X M. The mean values of the structural parameters are given in Table 111. The errors quoted above for A , f , and t are based on the least-squares fitting routine. However, the two-layer model may be parametrized differently without substantially changing the physical model. For example, inclusion of t in the expression for Th above, which would allow the chain to be flexible in the head-group region as well as in the chain region, gives higher values off and t. Table I11 takes account of this in the final estimates of the errors. Another limitation of the model used above is the assumption that the layers are uniform. Thus a physically more realistic description of the chain layer would be a distribution of scattering length density normal to the surface falling from a liquidlike value next to the head-group region to zero next to air. This would introduce considerable extra complexity into a fitting procedure of the kind described above. We return to the question when we discuss the alternative method of analyzing the reflectivity data below. The profiles of the various isotopic species have quite different sensitivities to the adjustable parameters. For example, hC,,hTAB in DzOis totally insensitive to the values of A but very sensitive to the value off, and dC14dTABin nrw is very sensitive to A but not very sensitive to$ We summarize the different sensitivities in Table IV,but in deriving the values of the parameters given in Table I11 we have given equal weighting to the result from each profile. We return to a more quantitative assessment in the next section. (b) A More Direct Method of Analysis Using the Kinematic Approximation. The basis for an alternative analysis of the data (10)
Immirzi, A.; Perini. B. Acra Crystallog. 1977, A33, 216.
h
io
Figure 8. All the fits to the CllTAB solutions at 3 X lo-' M. (a) The three deuterated C14TABspecies in nrw. (b) The four isotopic species in D20.
has been given above. Making use of eq 6 and measurements of reflectivity profiles at 11 different contrasts, we have determined, by solving the appropriate simultaneous equations, the six partial structure factors, three hf;)and three hg), appropriate to the three labels in the system. These structure factors are shown h Figures 9 and 10 for the concentration 3 X lW3 M. At high K the errors arise from statistical errors amplified by the background subtraction. At low K the statistical errors are insignificant but systematic errors may arise from errors in K , alignment errors, and small differences between the samples, becausethe reflectivity is changing very rapidly with K in this region. We have not shown the systematic errors, which we estimate to be f796, in Figures 9 and 10. The most obvious advantage of the use of eq 6 is that it reduces the set of scattering data from the system as a whole to a set of functions dependent only on the structure at the interface, the scattering characteristics of the problem and any contributions from the bulk having been eliminated. This procedure is unique to the neutron experiment. It is immediately clear from the experimentally determined structure factors that they have not been determined over a sufficiently wide range of momentum transfer for it to be possible to Fourier transform them directly to obtain the structure. However, the fitting of model structures to these functions is straightforwardand usually unambiguous except for the resolution limitation imposed by the maximum value of K . Furthermore, some structural features of the layer can be deduced without any
1378 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
Simister et al. 4x10“/
3~lO’~-
2x106-
lod -
5
-2.4~10’~ /@ *ab’--0
-.-.-:no
0
005
0.10
0.15
0 20
K/k’ Figure 9. Structure factors hii of the CIITAB layer at a concentration of 3 X M: (a) chain/chain; (b) water/water; (c) head/head. The calculated curves are for the model given in Table 111 (dashed line) the best fit of uniform layers using the parameters of Table V (dotted line) and the best fit of Gaussian and tanh distributionsusing the parameters of Table V (continuous line). recourse to Fourier transformation. We start by testing the model we used in the previous section to fit the actual reflectivity profiles. For a layer of uniform composition given by -r/2 < z < r / 2 ni(z) = nil
=0 all other z (10) where T is the thickness of the layer and nil its number density, f i i ( K ) and hii(K) are easily shown’ to be given by
KZhji(K) 441’ Sinz ( K T / 2 ) and the surface excess (in molecules A-2) is
(12)
ri =
-5~10’~
005
0.10
K / P
0.15
0.20
Figure 10. Structure factors hi. of the CI4TABlayer at a concentration of 3 X M: (a) chain/head; (b) chain/water; (c) head/water. The calculated curves are for the model given in Table 111 (dashed line) the best fit of uniform layers using the parameters of Table V (dotted line) and the best fit of Gaussian and tanh distributionsusing the parameters of Table V (continuous line).
In the model we used for the chain distribution in the previous section we had two uniform layers. For such a distribution given by -7
