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Orientational ordering of surfactant monolayers adsorbed at the air—water interface. Structural model and fit to neutron reflectivity data
11 March 1994
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 219 ( 1994) 3 1O-3 18
Orientational ordering of surfactant monolayers adsorbed at the air-water interface. Structural model and fit to neutron reflectivity data A.R. Denton, C.G. Gray, D.E. Sullivan Department of Physics, University of Guelph, Guelph. Ontario, Canada Nl G 2 WI Guelph- Waterloo Program for Graduate Work in Physics, University of Guelph, Guelph, Ontario, Canada NIG 2 WI
Received 13 December 1993
Abstract A new model for the structure of surfactant monolayers at interfaces is proposed and its utility demonstrated by application to data from a recent neutron reflectometry experinient. From a least-squares tit of the model to the reflectivity of a &TAB monolayer adsorbed at the air-water interface, the polar tilt angle of the hydrocarbon chains is extracted as a fitting parameter. Implications for the nature of orientational ordering in adsorbed surfactant monolayers are noted and discussed.
1. Introduction Interest in the thermodynamic and structural properties of monomolecular layers (monolayers) at interfaces dates back at least to the origins of the atomic theory of matter in the last century. Pioneering investigations of Rayleigh [ 1 ] and others into the surface tension of oils spread on water revealed the existence of a lower limit to the thickness of an oil layer, which was interpreted as the size of an oil molecule. While of great practical value for their potential to alter surface and interfacial properties of materials, monolayers are also of fundamental interest as models of quasi-two-dimensional systems in which the interplay between two- and three-dimensional behaviour may be studied. Surfactants comprise a class of chain-like amphiphilic molecules that readily form monolayers at interfaces between water and other fluids (e.g., oil, air, etc. ) . One end of each molecule (the hydrophilic head group) is usually narrowly confined to the plane of
the interface, while the other end (the hydrophobic chain) is relatively free to rotate out of the plane. It is convenient to define two broad classes of surfactant monolayers. In the first class are those, commonly known as Langmuir monolayers, composed of relatively long and usually densely packed molecules that are essentially insoluble in water. The coverage (area/molecule) of such a monolayer is entirely determined by the surface area of the water and the quantity of surfactant deposited, and is typically in the range 18-25 AZ/molecule. In the second class, our primary concern here, are monolayers whose constituent molecules are shorter (i.e. whose chains contain fewer CH2 groups) and hence are soluble to some extent in water. Such monolayers form by surface adsorption of some fraction of surfactant molecules from the bulk solution, the bulk concentration of surfactant governing the coverage, which typically exceeds 40 AZ/molecule. Structural characterization of interfaces has advanced rapidly over the past decade, largely due to
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. AlI rights reserved SSDIOOO9-2614(94)00082-2
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
the development of sensitive new experimental techniques for probing interfacial structure on a molecular scale #‘. One new technique that is proving especially valuable is neutron reflectometry #2.In a typical experiment, thermal neutrons are directed at glancing incident angle onto a sample surface and specularly reflected into a detector that measures the reflectivity R(K) - the ratio of reflected to incident neutrons - as a function of the transferred wavevector K normal to the surface. Compared with other techniques, such as ellipsometry, second-harmonic generation, or fluorescence microscopy, the two principal virtues of neutron reflectometry are first, the direct relation between the measured reflectivity and the density profile of the interface, and second, the possibility of exploiting selective isotopic substitution to effectively contrast specific components (layers) of the interface. X-ray reflectometry shares the first virtue, but does not allow for any practical means of contrast variation. On the other hand, X-ray intensities emerging from modem synchrotron sources far exceed the neutron intensities achievable with the current generation of reactor and spallation sources. Numerous previous studies [ 2 ] - by experiment, simulation, and theory - have revealed ample evidence of orientational ordering in Langmuir monolayers. It is clear that in such dense monolayers geometric packing constraints, due to short-range repulsive chain-chain interactions, are the chief cause of ordering. Until recently, issues concerning orientational ordering in adsorbed monolayers have received comparatively little attention. In particular, the specific interactions and the mechanism responsible for ordering in the lower surface density regime have not yet been identified. In a series of recent neutron reflectometry experiments, however, Thomas and co-workers [4-l 31 have begun to explore such issues by exploiting isotopic substitution to measure layer thicknesses, separations, and other structural properties of ionic and nonionic surfactant monolayers adsorbed at the airwater interface. In analysing their reflectivity data, these authors have adopted a structural model of the monolayer in which the head groups and chains of ’ For recent reviews of Langmuir monolayers, see ref. [ 21. ’ For a recent review of the theory and applications of neutron reflectometry, see ref. [ 31.
311
the surfactant molecules form two sublayers. Aside from qualitative inferences from the measured thicknesses, the initial studies of simple ionic surfactants did not directly address the nature of orientational ordering of the chains. In more recent studies of the C12Emseries of nonionic surfactants [ 12,13 1, however, more quantitative analyses of chain tilt angles have been attempted. Motivated by these experiments, the purpose of the present Letter is first, to propose a more detailed structural model of smfactant monolayers than has been used to date, and second, to demonstrate the utility of the model by fitting it to neutron reflectivity data for a simple ionic surfactant and extracting as fitting parameters tilt angles of the hydrocarbon chains. The remainder of the Letter is organized as follows. In section 2 we begin by briefly reviewing the theory of neutron reflectometry, including the kinematic (or first Born) approximation. In section 3 we describe a new structural model for surfactant chains, which, together with the kinematic approximation may be used to calculate neutron reflectivities. In section 4 we describe the application of the model to recent reflectivity data for the ionic surfactant C14TAB adsorbed at the air-water interface, and in section 5 discuss the physical interpretation of the results. Finally, in section 6 we close with a summary and conclusions.
2. Theory of neutron reflectometry 2.1. Basic principles The specular reflection of neutrons of wavelength L at a planar interface can be shown to obey the same basic laws that govern reflection of electromagnetic radiation polarized perpendicular to the plane of incidence [ 14- 16 1. The reflectivity is thus directly related to the neutron refractive index profile q(z) normal to the interface (in the z= 0 plane), which in turn is related to the neutron scattering length density profile p(z) of the interface via [ 171 V(Z) = 1- (12/21C)P(Z) ,
(1)
where we neglect a usually small imaginary term corresponding to the neutron absorption cross section. The scattering length density is determined by the co-
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A.R. Denton et al. /Chemical PhysicsLetters 219 (1994) 310-318
herent scattering lengths bj and number density profiles nj( z) of the various components, according to P(z) = C bini j
(2)
>
where j labels the components. Because the neutron refractive index is very close to yet smaller than unity, total external reflection occurs for incident angles 0, smaller than a critical angle 6,=&/m. For 0i> 0,, reflection is only partial, but may still be considerable for sufficiently small 0,. A standard method of calculating interfacial reflectivity profiles is the optical matrix method of multilayer film optics [ 18 1. It proceeds by modelling the interface as a series of discrete layers, each layer characterized by a different refractive index and a corresponding 2 x 2 reflectivity matrix. The net reflectivity is then obtained from the elements of the product of the single-layer matrices. For idealized sharp interfaces the method is exact. For real diffuse interfaces it is only approximate, but in principle may be systematically improved by finer discretization of the interface. In practice, however, the method becomes unwieldy when more than a few layers are required. In many of the original neutron reflectometry experiments, the data analysis was conducted by means of the optical matrix method, with interfacial models consisting of one, two, or at most three layers (see, e.g., refs. [ 4-6 ] ). More recent studies have made use of an alternative method based on the kinematic approximation, which we now briefly review. 2.2. The kinematic approximation In the kinematic approximation the reflectivity is directly related to the Fourier transform of the scattering length density profile normal to the interface p(z), according to [ 19,201 R(K)=
$
M(K) I2 7
R(K)=
$
ljs(‘) (K) 12 )
(da)
where m /5(‘)(K) =
I
dze’“p’(z)
,
(4b)
-ca
andp’(z)=@/dz. Practical application of Eqs. (3 ) or (4) evidently requires an explicit model for p( z). In the case of an air-surfactant-solvent interface it is natural and convenient to first separate p(z) into its three principal contributions, P(Z) =P,(z) +ph(z) +PC(z) ,
(5)
where pJ( z) , ph (z) , and pC(z) are the scattering length density profiles of the solvent, head groups, and chains, respectively. By regarding the solvent molecules and the head groups as discrete scattering centres, the first two terms in Eq. (5) may be factored into the product of average scattering length and number density, such that P(Z) =&n,(z) +hn,(z)
+p,(z) ,
(6)
where b, and & are the average scattering lengths, and n,(z) and nh(z) the number density profiles, of the solvent and head groups, respectively. Previous workers [ 4- 13 ] have applied the same simple factorization to the chains as well, modelling them as discrete scattering centres, despite their relatively complex structure. Although such a model may be adequate for extracting such coarse structural properties as layer thicknesses, it is less sensitive to more detailed properties related to chain conformations and orientational ordering. With the latter properties in mind, we now propose a more realistic and explicit treatment of the chain scattering length density.
(3a)
3. Structural model of adsorbed smfactant where
monolayers
03 p(K) = I dze’“p(z)
.
(3b)
--oo
An alternative (and equivalent) expression in terms of the gradient of the scattering length density is
In modelling the scattering length density of the hydrocarbon chains one could, of course, go so far as to treat each nucleus as a separate scattering centre. Such detail, however, is usually not warranted by the current precision of experimental data, and so in-
A. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
stead we simply approximate each methylene group ( CH2 or CD2) - as well as the terminal methyl group (CH3 or CDS) - as a separate scattering centre and model each chain as a string of N discrete scattering centres localized on a flexible backbone, as depicted in Fig. 1a. Neighbouring sites are connected by bonds offixed length x but arbitrary orientations. Thus, although for simplicity vibrational degrees of freedom are ignored, orientational degrees of freedom are included. Vibrational degrees of freedom may of course be included in the model, but at the cost of additional
313
complexity. The orientational state or conformation of a chain is sufficiently specified by the set of polar angles {O}=(f3i,..., O,), or equivalently their cosines {u}= (U,,..., uN) , where ui = cos 0,. Although a complete specification of chain conformation would require both polar and azimuthal bond angles, only the polar angles are relevant here, since specular reflectometry probes structure only in the direction normal to the plane of the interface. For application to diffuse (nonspecular) reflectometry, which probes inplane structure, the model may be generalized to include azimuthal orientations as well. Factoring the scattering length density of each chain site into the product of average scattering length and number density, the scattering length density of a single chain is expressed by PC(~)= i
bj4tz)
3
j=l
(7)
where bj is the average scattering length, and n’, (z) the number density, of thejth group on the chain. If we now suppose the number density of each group on the chain to have the same form as that of the corresponding head group number density, aside from a separation in the z direction, then n’, (z) may be simply related to nh(z) via tZ’,(Z)=?&(Z+Zj)
)
(8)
where Zj=X(U,
b
Fig. 1. Schematic depiction of structural model of the interfacial density profile for a surfactant monolayer at the air-water interface. (a) Arbitrary chain conformation. (b) Linear chain approximation. The independent parameters of the model are ( 1) the interfacial width Cof the air-solvent (water) density profile; (2) the Gaussian width u,, of the head group layer distribution; (3) the separation S, between the centres of the head group and solvent distributions; and (4) the polar angles Orof the chain bonds in model (a) or simply the chain tilt angle 8, in model (b). A possible additional parameter (not shown) is the width of the polar angle distribution.
+***+Uj)
(9)
is the separation in the z direction between the head group and thejth group on the chain. Denoting the distribution of chain conformations by P( {u}), the average chain scattering length density may be formally expressed as an average over all possible conformations, 1
1 p,(z)=
I
0
X
du,...
s
dUNP({U})
0
5 bjnh(Z+Zj) .
j=l
(10)
The corresponding Fourier transform may be written in the compact form
314
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318 1
PC(K)=
(iI bjP(KZj))h(K) y
(lla)
P(KZj)= s
due-‘“‘“p(u).
(14)
0
where CO i&(K)=
dze'"n,(z)
(lib)
I -co
In the simplest, and somewhat idealized, case of a fixed tilt angle $, whose cosine we denote by uo, we havep(u) =d(u-uo), and thus p”(KZ,) = exp ( - iKjXU, ) .
(15)
Substituting Eq. ( 15) into Eq. ( 12), the reflectivity for the case of linear chains with fixed tilt angle is then given simply by
and
R(K)=
167~’
7
b,&(K)
(llc)
x P({ul) *
2
Combining Eqs. (3), (6)) and ( 11), our model for the reflectivity may now be expressed in the explicit form R(K)+
b,%(K) 2
+
bh
+jil
bjP(KZj)
h(K) >
.
(12)
I
Together with parametrized forms for the distributions n,(z), nh(z), and P({u}), Eq. (12) constitutes a quite general model, which in principle may be applied to neutron reflectivity data for any air-surfactant-solvent interface. In practice, it may in some cases be a reasonable approximation to assume linear chains (see Fig. 1b ) . For instance, in Langrnuir monolayers, where the chains are relatively densely packed, this is usually quite a valid assumption. In the more loosely packed adsorbed monolayers the chain conformations are more likely to deviate from linearity due to the presence of gauche defects. Nevertheless, for a sufficiently low defect frequency the conformations may be considered essentially linear. The orientational distribution function then reduces to P((U))=P(U,)@U,--%) x&u,-U,)...&U,-u,)
)
(13)
wherep( U) is a simple tilt angle distribution function that by itself completely characterizes the conformation, and from Eqs. (9) and ( 11 c),
+
bh+ g b,exp(-iKjxz.&) j=l
A,,(K) >
.
(16)
I
Compared with the model of Thomas et al., our model differs in essentially two respects. First, by including a more detailed representation of the chain layer scattering length density, the model is sufflciently general that it may be applied to systems of chains with arbitrary conformations and isotopic compositions. Second, since the static or geometrical thickness of the chain layer is now decoupled from the overall chain layer thickness, which includes a significant dynamic contribution from capillary wave roughness, orientational ordering of the chains may be studied more directly. In section 4 we proceed to apply the model to neutron reflectivity data for a surfactant monolayer at the air-water interface and attempt to extract chain tilt angles.
4. Application to a surfactant monolayer at the airwater interface To illustrate the model described in section 3 we now apply it to a reanalysis of data from a recent neutron reflectometry study [ 71 of a monolayer of the ionic surfactant tetradecyltrimethylammonium bromide ( Ci4TAB) adsorbed at the air-water interface. (The full chemical formula of the protonated species is C14H29N(CH3)3Br.) In the original experiment, reflectivity profiles were measured for two bulk surfactant concentrations and for a variety of isotopic compositions. The concentrations were c= 3 mM and
315
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
c=4.5 mM, corresponding to areas/molecule of roughly 48 and 43 A*, respectively. (For comparison, the critical micelle concentration for this system is c= 3.7 mM. ) The four isotopic compositions that we consider are C&TAB-d (fully deuterated chain and head group) and C&-TAB-h (fully deuterated chain, protonated head group ) , each adsorbed either from null reflecting water [ 3 ] (nrw ) or DzO. Reflectivities were also measured for protonated-chain species, but since the signals are considerably weaker in these cases, and the statistical errors correspondingly greater, we have chosen not to include these data in our analysis. Following Simister et al. [ 7 1, we model the head group and solvent number densities as normalized Gaussian and hyperbolic tangent profiles, respectively, their centres separated by a distance &,. Taking the centres of the head group and solvent profiles to be at z= -&, and z= 0, respectively, we thus assume nh(Z)
=
where A is the area/molecule and ch the full width at (l/e) x (maximum), and n,(z) =InsO
11+taWz/C) 1 ,
(18)
where nso (~0.0332 A-’ for water) is the bulk solvent density and [the characteristic width of the profile. The corresponding Fourier transforms, required for the kinematic approximation, are Ah(7c)= f exp( - hrc’at) exp( -iK&)
head group layer width ah, the head group layer-solvent (water) separation &, and the chain layer width. The numerical value and physical interpretation of the latter parameter, however, are somewhat imprecise due to the approximate treatment of the chain layer. The purpose of the present reanalysis, then, is simply to refine the characterization of the chain layer structure by means of our more detailed model. In applying the model, we assume for simplicity a bond length X= 1.265 A, corresponding to fully extended chains [ 2 1 ] (no defects). The validity of this assumption is discussed below. The required scattering lengths may be obtained from standard tables (e.g., ref. [ 17 ] ) and are given in Table 1. Although a complete implementation of the model would involve independently varying the four parameters c, 4, &,, and uo, we have chosen here to simply f= the first three, which characterize the air-solvent interface and the head group layer, at those values previously determined [ 22,231 (see Table 2) and to vary only the tilt angle parameter uo. For each of the two concentrations, we fit the reflectivity profiles of all four isotopic species by a single tilt angle parameter. Plots of the resulting least-squares error x2 versus uo, shown in Fig. 2, are seen to exhibit well-defined minima corresponding to optimal values of uo, or equiv-
,
Table 1 Average scattering lengths used in fitting the structural model to the neutron reflectivity data for C1,TAB
(19)
and fi,(rc) = tilr: m, csch( hrt C;K).
(20)
It may be shown that if the head group number density is any even function of z, then for KZ<<1 its Fourier transform is well approximated by Eq. ( 19 ) . The initial analysis performed by Simister et al. [ 71 made use of a model very similar to ours with regard to the solvent and head group layer densities, but less detailed in its treatment of the chain layer. A leastsquares tit to the reflectivity data produced optimal values for various structural parameters of the interface, including the air-water interfacial width c, the
Unit
Scattering length (lo-4A)
N(CD&Br N(CW3Br CD2 CD3 Dz0 nrw
9.618 0.249 1.999 2.666 1.914 0
Table 2 Structural parameters for (&TAB. The values of C, 4, and S, were obtained from refs. [ 22,231 C
c
4
s,
(mW
(A)
(A)
(A)
3.0 4.5
5.2kO.S 5.9kO.5
lOf2 12+2
If1 2+1
a3
& (de&
0.71 0.67
45 48
316
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
Fig. 2. Least-squares error x2 versus tilt angle parameter u,, for a fit of the structural model to neutron reflectivity data for a C14TAB monolayer adsorbed at the air-water interface. The solid and dashed curves correspond, respectively, to bulk surfactant concentrations of c= 3 mM and ~~4.5 mM.
alently the tilt angle $. The latter are calculated using 0, = cos- ’ u. and are included in Table 2. The experimental reflectivity profiles themselves, and our best numerical fits to them, are shown together in Fig. 3. Although we have considered scaling the reflectivity data according to the kinematic correction formula of Crowley (241, given the statistical errors in the data and the approximations underlying our model, we find that the effects of scaling are in this case quite insignificant.
5. Discussion The accuracy with which we are able to tit the reflectivity data would seem to lend support to our assumption of linear conformations with a fixed tilt angle. As a check on that assumption though, we have also explored other tilt angle distributions (e.g., a twoparameter Gaussian). In general, the accuracy of the fit is quite insensitive to the functional form of the distribution function, and the width of the distribution tends to be rather narrow (e.g., 2-4 A for a truncated Gaussian). It should be noted, however, that in a recent molecular dynamics simulation, Backer et al. [ 25 ] computed a relatively broad tilt angle distribution for a similar system, namely a Cr6TAC monolayer at the air-water interface. The basic form of the tilt angle distribution - or, more generally, the ori-
entational distribution function - and its dependence on such factors as surface density and chain length are interesting issues for future study. Although a model of surfactant chains with strictly linear conformations and fixed tilt angles must certainly be an idealization, nevertheless, the fact that the C&TAB data may be fit at all with such a model - let alone with such accuracy-strongly suggests the possibility of some orientational ordering of the chains in the monolayer. Quantitative results for the tilt angle are subject to some uncertainty due to an unknown frequency of gauche defects in the chains, each defect acting to shorten a chain by roughly 1 A. In the simulation of Biicker et al. the defect frequency was found to be about 200/bfor Ci6TAC. If the same frequency were present in the case of C,,TAB, then the average bond length would be reduced to xx 1.065 and the average tilt angle range to roughly 33”-37”, illustrating the magnitude of uncertainty in our tilt angle determinations. In any case, it is clear that the chains are significantly tilted with respect to both the vertical and the horizontal. It is interesting to compare this conclusion with a similar conclusion reached in recent studies of the &E,,, series of nonionic surfactants. From analyses of neutron reflectivity data for Cr2E3 [ 121 and Cr2E6 [ 131, Lu et al. conclude that the alkyl chains are significantly tilted away from the normal over a range of concentrations, and that the ethylene glycol chain orientation varies strongly with concentration, with the chains oriented close to vertical at high concentration (Ax 36 A’), but close to horizontal at substantially lower concentrations (A~89 A’). Although the molecular structures of these nonionic surfactants are rather different from those of the ionic Ci4TAB surfactant, nevertheless the occurrence of sizeable chain tilt angles away from the normal appears to be a feature common to both systems. Finally, we note that the small difference in coverage of only about 10% between the two bulk concentrations is unfortunately too small to permit any conclusions regarding dependence of tilt angle on coverage. This dependence, however, is an interesting issue to which the model may be applied in future.
311
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
-6 0
I 0.05
01K ',&';i
'0'2''
0.25 '
&TAB
&TAB on m-w c=4.5 mM
on D,O
-6 -6
0
,,,"'~,,',,,"""""' 0.05
0.2
0.25
Fig. 3. Reflectivity R(K) versus wavevector transfer K normal to the interface for the surfactant C14TABadsorbed at the air-water interface, for two different bulk concentrations and four different isotopic compositions. The symbols are the data from ref. [ 71, and the curves are fits to the data of our structural model (Fig. lb). The solid curves and solid symbols correspond to the isotopic composition C&TAB-h, the &shed curves and open symbols to Cr.,-d-TAB-h. 6. Summary and conclusions
In summary, we have proposed a new model for the structure of surfactant monolayers at interfaces which, in conjunction with the kinematic (or first Born) approximation, may be used to calculate neutron reflectivities. Compared with previous models, the new model is distinguished by a more realistic and detailed representation of the chain scattering length density. As a practical demonstration, we have applied the model to a reanalysis of data from a recent neutron reflectometry study of a &TAB monolayer adsorbed at the air-water interface. Assuming, for simplicity, fully extended linear chains with fned tilt angles, a least-squares fit of the model to reflectivity data for two different bulk concentrations and four differ-
ent isotopic species yields chain tilt angles in the range 45”-48”. Although considerable uncertainty must be attached to the tilt angle and its distribution due to an unknown frequency of gauche defects in the chains, it is at least qualitatively clear that the chains are significantly tilted away from the normal to the interface and quite possibly orientationally ordered. It is hoped that future refinements and applications of the model to neutron reflectivity data for a range of adsorbed surfactant monolayers, with various coverages and isotopic compositions, will aid in illuminating fundamental issues concerning the nature of orientational ordering of monolayers at interfaces.
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A.R. Denton et al. / ChemicalPhysics Letters 219 (1994) 310-318
Acknowledgement We are grateful to Professor P.A. Egelstaff, Dr. J. Penfold, Dr. J.R. Lu, and Professor R.K. Thomas for helpful discussions, and especially to Dr. Lu and Professor Thomas for kindly providing us with the neutron reflectivity data from ref. [ 7 1. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). One of us (ARD) gratefully acknowledges support through an NSERC Postdoctoral Fellowship during the early stages of the work.
References [ 1 ] Rayleigh, Proc. Roy. Sot. 47 (1890) 364; Phil. Mag. 48 (1899) 321. [2] C.M. Knobler and R.C. Desai, Ann. Rev. Phys. Chem. 43 (1992) 207; H.M. McConnell, Ann. Rev. Phys. Chem. 42 ( 199 1) 171; C.M. Knobler, Advan. Chem. Phys. 77 (1990) 397; H. Miihwald, Ann. Rev. Phys. Chem. 41 (1990) 441; J.D. Swalen, Ann. Rev. Mater. Sci. 21 (1991) 373; V. Vogel and Y.R. Shen, Ann. Rev. Mater. Sci. 2 1 ( 1991) 515. [3] J. Penfold and R.K. Thomas, J. Phys. Condens. Matter 2 (1990) 1369. [ 41 E.M. Lee, R.K. Thomas, J. Penfold and R.C. Ward, J. Phys. Chem. 93 (1989) 381. [ 5] J.B. Hayter, R.R. Highfield, B.J. Pullman, R.K. Thomas, AI. McMullen and J. Penfold, J. Chem. Sot. Faraday Trans. 177 (1981) 1437.
[6] J.E. Bradley, E.M. Lee, R.K. Thomas, A.J. Willatt, J. Penfold, R.C. Ward, D.P. Gregory and W. Waschkowski, Langmuir 4 (1988) 821. [7] E.A. Simister, E.M. Lee, R.K. Thomas and J. Penfold, J. Phys. Chem. 96 (1992) 1373. [ 81 J.R. Lu, E.A. Simister, E.M. Lee, R.K. Thomas, A.R. Rennie and J. Penfold, Langmuir 8 (1992) 1837. [9] J.R. Lu, E.M. Lee, R.K. Thomas, J. Penfold and S.L. Flitsch, Langmuir 9 (1993) 1352. [ lo] E. Staples, L. Thompson, I. Tucker, J. Penfold, R.K. Thomas and J.R. Lu, Langmuir 9 (1993) 1651. [ 111 J.R. Lu, 2.X. Li, T.J. Su, R.K. Thomas and J. Penfold, Langmuir 9 (1993) 2408. [ 121 J.R. Lu, M. Hromadova, R.K. Thomas and J. Penfold, Langmuir 9 (1993) 2417. [ 131 J.R. Lu, Z.X. Li, R.K. Thomas, E.J. Staples, I. Tucker and J. Penfold, J. Phys. Chem. 97 (1993) 8012. [ 141 M.L. Goldberger and F. Seitz, Phys. Rev. 71 (1947) 294. [ 151 J. Lekner, Theory of reflection (Martinus Nijhoff, Dordrecht, 1987). [ 161 V.F. Sears, Neutron optics (Oxford Univ. Press, Oxford, 1989). [ 171 G.E. Bacon, Neutron diffraction, 3rd Ed. (Clarendon Press, Oxford, 1975). [ 181 M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford, 1970). [ 191 T.L. Crowley, E.M. Lee, E.A. Simister, R.K. Thomas, J. Penfold and A.R. Rennie, Colloids Surfaces 52 ( 1990) 85. [20] J. Ah-Nielsen, Z. Physik B 61 (1985) 411. [ 2 1 ] C. Tanford, J. Phys. Chem. 76 ( 1972) 3020. [ 22 ] J.R. Lu, private communication. [23] J.R. Lu, E.A. Simister, R.K. Thomas and J. Penfold, J. Phys. Chem. 97 (1993) 6024. [24] T.L. Crowley, Physica A 195 (1993) 354. [ 25 ] J. Biscker, M. Schlenkrich, P. Bopp and J. Brickman, J. Phys. Chem. 96 (1992) 9915.
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 219 ( 1994) 3 1O-3 18
Orientational ordering of surfactant monolayers adsorbed at the air-water interface. Structural model and fit to neutron reflectivity data A.R. Denton, C.G. Gray, D.E. Sullivan Department of Physics, University of Guelph, Guelph. Ontario, Canada Nl G 2 WI Guelph- Waterloo Program for Graduate Work in Physics, University of Guelph, Guelph, Ontario, Canada NIG 2 WI
Received 13 December 1993
Abstract A new model for the structure of surfactant monolayers at interfaces is proposed and its utility demonstrated by application to data from a recent neutron reflectometry experinient. From a least-squares tit of the model to the reflectivity of a &TAB monolayer adsorbed at the air-water interface, the polar tilt angle of the hydrocarbon chains is extracted as a fitting parameter. Implications for the nature of orientational ordering in adsorbed surfactant monolayers are noted and discussed.
1. Introduction Interest in the thermodynamic and structural properties of monomolecular layers (monolayers) at interfaces dates back at least to the origins of the atomic theory of matter in the last century. Pioneering investigations of Rayleigh [ 1 ] and others into the surface tension of oils spread on water revealed the existence of a lower limit to the thickness of an oil layer, which was interpreted as the size of an oil molecule. While of great practical value for their potential to alter surface and interfacial properties of materials, monolayers are also of fundamental interest as models of quasi-two-dimensional systems in which the interplay between two- and three-dimensional behaviour may be studied. Surfactants comprise a class of chain-like amphiphilic molecules that readily form monolayers at interfaces between water and other fluids (e.g., oil, air, etc. ) . One end of each molecule (the hydrophilic head group) is usually narrowly confined to the plane of
the interface, while the other end (the hydrophobic chain) is relatively free to rotate out of the plane. It is convenient to define two broad classes of surfactant monolayers. In the first class are those, commonly known as Langmuir monolayers, composed of relatively long and usually densely packed molecules that are essentially insoluble in water. The coverage (area/molecule) of such a monolayer is entirely determined by the surface area of the water and the quantity of surfactant deposited, and is typically in the range 18-25 AZ/molecule. In the second class, our primary concern here, are monolayers whose constituent molecules are shorter (i.e. whose chains contain fewer CH2 groups) and hence are soluble to some extent in water. Such monolayers form by surface adsorption of some fraction of surfactant molecules from the bulk solution, the bulk concentration of surfactant governing the coverage, which typically exceeds 40 AZ/molecule. Structural characterization of interfaces has advanced rapidly over the past decade, largely due to
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. AlI rights reserved SSDIOOO9-2614(94)00082-2
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
the development of sensitive new experimental techniques for probing interfacial structure on a molecular scale #‘. One new technique that is proving especially valuable is neutron reflectometry #2.In a typical experiment, thermal neutrons are directed at glancing incident angle onto a sample surface and specularly reflected into a detector that measures the reflectivity R(K) - the ratio of reflected to incident neutrons - as a function of the transferred wavevector K normal to the surface. Compared with other techniques, such as ellipsometry, second-harmonic generation, or fluorescence microscopy, the two principal virtues of neutron reflectometry are first, the direct relation between the measured reflectivity and the density profile of the interface, and second, the possibility of exploiting selective isotopic substitution to effectively contrast specific components (layers) of the interface. X-ray reflectometry shares the first virtue, but does not allow for any practical means of contrast variation. On the other hand, X-ray intensities emerging from modem synchrotron sources far exceed the neutron intensities achievable with the current generation of reactor and spallation sources. Numerous previous studies [ 2 ] - by experiment, simulation, and theory - have revealed ample evidence of orientational ordering in Langmuir monolayers. It is clear that in such dense monolayers geometric packing constraints, due to short-range repulsive chain-chain interactions, are the chief cause of ordering. Until recently, issues concerning orientational ordering in adsorbed monolayers have received comparatively little attention. In particular, the specific interactions and the mechanism responsible for ordering in the lower surface density regime have not yet been identified. In a series of recent neutron reflectometry experiments, however, Thomas and co-workers [4-l 31 have begun to explore such issues by exploiting isotopic substitution to measure layer thicknesses, separations, and other structural properties of ionic and nonionic surfactant monolayers adsorbed at the airwater interface. In analysing their reflectivity data, these authors have adopted a structural model of the monolayer in which the head groups and chains of ’ For recent reviews of Langmuir monolayers, see ref. [ 21. ’ For a recent review of the theory and applications of neutron reflectometry, see ref. [ 31.
311
the surfactant molecules form two sublayers. Aside from qualitative inferences from the measured thicknesses, the initial studies of simple ionic surfactants did not directly address the nature of orientational ordering of the chains. In more recent studies of the C12Emseries of nonionic surfactants [ 12,13 1, however, more quantitative analyses of chain tilt angles have been attempted. Motivated by these experiments, the purpose of the present Letter is first, to propose a more detailed structural model of smfactant monolayers than has been used to date, and second, to demonstrate the utility of the model by fitting it to neutron reflectivity data for a simple ionic surfactant and extracting as fitting parameters tilt angles of the hydrocarbon chains. The remainder of the Letter is organized as follows. In section 2 we begin by briefly reviewing the theory of neutron reflectometry, including the kinematic (or first Born) approximation. In section 3 we describe a new structural model for surfactant chains, which, together with the kinematic approximation may be used to calculate neutron reflectivities. In section 4 we describe the application of the model to recent reflectivity data for the ionic surfactant C14TAB adsorbed at the air-water interface, and in section 5 discuss the physical interpretation of the results. Finally, in section 6 we close with a summary and conclusions.
2. Theory of neutron reflectometry 2.1. Basic principles The specular reflection of neutrons of wavelength L at a planar interface can be shown to obey the same basic laws that govern reflection of electromagnetic radiation polarized perpendicular to the plane of incidence [ 14- 16 1. The reflectivity is thus directly related to the neutron refractive index profile q(z) normal to the interface (in the z= 0 plane), which in turn is related to the neutron scattering length density profile p(z) of the interface via [ 171 V(Z) = 1- (12/21C)P(Z) ,
(1)
where we neglect a usually small imaginary term corresponding to the neutron absorption cross section. The scattering length density is determined by the co-
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A.R. Denton et al. /Chemical PhysicsLetters 219 (1994) 310-318
herent scattering lengths bj and number density profiles nj( z) of the various components, according to P(z) = C bini j
(2)
>
where j labels the components. Because the neutron refractive index is very close to yet smaller than unity, total external reflection occurs for incident angles 0, smaller than a critical angle 6,=&/m. For 0i> 0,, reflection is only partial, but may still be considerable for sufficiently small 0,. A standard method of calculating interfacial reflectivity profiles is the optical matrix method of multilayer film optics [ 18 1. It proceeds by modelling the interface as a series of discrete layers, each layer characterized by a different refractive index and a corresponding 2 x 2 reflectivity matrix. The net reflectivity is then obtained from the elements of the product of the single-layer matrices. For idealized sharp interfaces the method is exact. For real diffuse interfaces it is only approximate, but in principle may be systematically improved by finer discretization of the interface. In practice, however, the method becomes unwieldy when more than a few layers are required. In many of the original neutron reflectometry experiments, the data analysis was conducted by means of the optical matrix method, with interfacial models consisting of one, two, or at most three layers (see, e.g., refs. [ 4-6 ] ). More recent studies have made use of an alternative method based on the kinematic approximation, which we now briefly review. 2.2. The kinematic approximation In the kinematic approximation the reflectivity is directly related to the Fourier transform of the scattering length density profile normal to the interface p(z), according to [ 19,201 R(K)=
$
M(K) I2 7
R(K)=
$
ljs(‘) (K) 12 )
(da)
where m /5(‘)(K) =
I
dze’“p’(z)
,
(4b)
-ca
andp’(z)=@/dz. Practical application of Eqs. (3 ) or (4) evidently requires an explicit model for p( z). In the case of an air-surfactant-solvent interface it is natural and convenient to first separate p(z) into its three principal contributions, P(Z) =P,(z) +ph(z) +PC(z) ,
(5)
where pJ( z) , ph (z) , and pC(z) are the scattering length density profiles of the solvent, head groups, and chains, respectively. By regarding the solvent molecules and the head groups as discrete scattering centres, the first two terms in Eq. (5) may be factored into the product of average scattering length and number density, such that P(Z) =&n,(z) +hn,(z)
+p,(z) ,
(6)
where b, and & are the average scattering lengths, and n,(z) and nh(z) the number density profiles, of the solvent and head groups, respectively. Previous workers [ 4- 13 ] have applied the same simple factorization to the chains as well, modelling them as discrete scattering centres, despite their relatively complex structure. Although such a model may be adequate for extracting such coarse structural properties as layer thicknesses, it is less sensitive to more detailed properties related to chain conformations and orientational ordering. With the latter properties in mind, we now propose a more realistic and explicit treatment of the chain scattering length density.
(3a)
3. Structural model of adsorbed smfactant where
monolayers
03 p(K) = I dze’“p(z)
.
(3b)
--oo
An alternative (and equivalent) expression in terms of the gradient of the scattering length density is
In modelling the scattering length density of the hydrocarbon chains one could, of course, go so far as to treat each nucleus as a separate scattering centre. Such detail, however, is usually not warranted by the current precision of experimental data, and so in-
A. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
stead we simply approximate each methylene group ( CH2 or CD2) - as well as the terminal methyl group (CH3 or CDS) - as a separate scattering centre and model each chain as a string of N discrete scattering centres localized on a flexible backbone, as depicted in Fig. 1a. Neighbouring sites are connected by bonds offixed length x but arbitrary orientations. Thus, although for simplicity vibrational degrees of freedom are ignored, orientational degrees of freedom are included. Vibrational degrees of freedom may of course be included in the model, but at the cost of additional
313
complexity. The orientational state or conformation of a chain is sufficiently specified by the set of polar angles {O}=(f3i,..., O,), or equivalently their cosines {u}= (U,,..., uN) , where ui = cos 0,. Although a complete specification of chain conformation would require both polar and azimuthal bond angles, only the polar angles are relevant here, since specular reflectometry probes structure only in the direction normal to the plane of the interface. For application to diffuse (nonspecular) reflectometry, which probes inplane structure, the model may be generalized to include azimuthal orientations as well. Factoring the scattering length density of each chain site into the product of average scattering length and number density, the scattering length density of a single chain is expressed by PC(~)= i
bj4tz)
3
j=l
(7)
where bj is the average scattering length, and n’, (z) the number density, of thejth group on the chain. If we now suppose the number density of each group on the chain to have the same form as that of the corresponding head group number density, aside from a separation in the z direction, then n’, (z) may be simply related to nh(z) via tZ’,(Z)=?&(Z+Zj)
)
(8)
where Zj=X(U,
b
Fig. 1. Schematic depiction of structural model of the interfacial density profile for a surfactant monolayer at the air-water interface. (a) Arbitrary chain conformation. (b) Linear chain approximation. The independent parameters of the model are ( 1) the interfacial width Cof the air-solvent (water) density profile; (2) the Gaussian width u,, of the head group layer distribution; (3) the separation S, between the centres of the head group and solvent distributions; and (4) the polar angles Orof the chain bonds in model (a) or simply the chain tilt angle 8, in model (b). A possible additional parameter (not shown) is the width of the polar angle distribution.
+***+Uj)
(9)
is the separation in the z direction between the head group and thejth group on the chain. Denoting the distribution of chain conformations by P( {u}), the average chain scattering length density may be formally expressed as an average over all possible conformations, 1
1 p,(z)=
I
0
X
du,...
s
dUNP({U})
0
5 bjnh(Z+Zj) .
j=l
(10)
The corresponding Fourier transform may be written in the compact form
314
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318 1
PC(K)=
(iI bjP(KZj))h(K) y
(lla)
P(KZj)= s
due-‘“‘“p(u).
(14)
0
where CO i&(K)=
dze'"n,(z)
(lib)
I -co
In the simplest, and somewhat idealized, case of a fixed tilt angle $, whose cosine we denote by uo, we havep(u) =d(u-uo), and thus p”(KZ,) = exp ( - iKjXU, ) .
(15)
Substituting Eq. ( 15) into Eq. ( 12), the reflectivity for the case of linear chains with fixed tilt angle is then given simply by
and
R(K)=
167~’
7
b,&(K)
(llc)
x P({ul) *
2
Combining Eqs. (3), (6)) and ( 11), our model for the reflectivity may now be expressed in the explicit form R(K)+
b,%(K) 2
+
bh
+jil
bjP(KZj)
h(K) >
.
(12)
I
Together with parametrized forms for the distributions n,(z), nh(z), and P({u}), Eq. (12) constitutes a quite general model, which in principle may be applied to neutron reflectivity data for any air-surfactant-solvent interface. In practice, it may in some cases be a reasonable approximation to assume linear chains (see Fig. 1b ) . For instance, in Langrnuir monolayers, where the chains are relatively densely packed, this is usually quite a valid assumption. In the more loosely packed adsorbed monolayers the chain conformations are more likely to deviate from linearity due to the presence of gauche defects. Nevertheless, for a sufficiently low defect frequency the conformations may be considered essentially linear. The orientational distribution function then reduces to P((U))=P(U,)@U,--%) x&u,-U,)...&U,-u,)
)
(13)
wherep( U) is a simple tilt angle distribution function that by itself completely characterizes the conformation, and from Eqs. (9) and ( 11 c),
+
bh+ g b,exp(-iKjxz.&) j=l
A,,(K) >
.
(16)
I
Compared with the model of Thomas et al., our model differs in essentially two respects. First, by including a more detailed representation of the chain layer scattering length density, the model is sufflciently general that it may be applied to systems of chains with arbitrary conformations and isotopic compositions. Second, since the static or geometrical thickness of the chain layer is now decoupled from the overall chain layer thickness, which includes a significant dynamic contribution from capillary wave roughness, orientational ordering of the chains may be studied more directly. In section 4 we proceed to apply the model to neutron reflectivity data for a surfactant monolayer at the air-water interface and attempt to extract chain tilt angles.
4. Application to a surfactant monolayer at the airwater interface To illustrate the model described in section 3 we now apply it to a reanalysis of data from a recent neutron reflectometry study [ 71 of a monolayer of the ionic surfactant tetradecyltrimethylammonium bromide ( Ci4TAB) adsorbed at the air-water interface. (The full chemical formula of the protonated species is C14H29N(CH3)3Br.) In the original experiment, reflectivity profiles were measured for two bulk surfactant concentrations and for a variety of isotopic compositions. The concentrations were c= 3 mM and
315
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
c=4.5 mM, corresponding to areas/molecule of roughly 48 and 43 A*, respectively. (For comparison, the critical micelle concentration for this system is c= 3.7 mM. ) The four isotopic compositions that we consider are C&TAB-d (fully deuterated chain and head group) and C&-TAB-h (fully deuterated chain, protonated head group ) , each adsorbed either from null reflecting water [ 3 ] (nrw ) or DzO. Reflectivities were also measured for protonated-chain species, but since the signals are considerably weaker in these cases, and the statistical errors correspondingly greater, we have chosen not to include these data in our analysis. Following Simister et al. [ 7 1, we model the head group and solvent number densities as normalized Gaussian and hyperbolic tangent profiles, respectively, their centres separated by a distance &,. Taking the centres of the head group and solvent profiles to be at z= -&, and z= 0, respectively, we thus assume nh(Z)
=
where A is the area/molecule and ch the full width at (l/e) x (maximum), and n,(z) =InsO
11+taWz/C) 1 ,
(18)
where nso (~0.0332 A-’ for water) is the bulk solvent density and [the characteristic width of the profile. The corresponding Fourier transforms, required for the kinematic approximation, are Ah(7c)= f exp( - hrc’at) exp( -iK&)
head group layer width ah, the head group layer-solvent (water) separation &, and the chain layer width. The numerical value and physical interpretation of the latter parameter, however, are somewhat imprecise due to the approximate treatment of the chain layer. The purpose of the present reanalysis, then, is simply to refine the characterization of the chain layer structure by means of our more detailed model. In applying the model, we assume for simplicity a bond length X= 1.265 A, corresponding to fully extended chains [ 2 1 ] (no defects). The validity of this assumption is discussed below. The required scattering lengths may be obtained from standard tables (e.g., ref. [ 17 ] ) and are given in Table 1. Although a complete implementation of the model would involve independently varying the four parameters c, 4, &,, and uo, we have chosen here to simply f= the first three, which characterize the air-solvent interface and the head group layer, at those values previously determined [ 22,231 (see Table 2) and to vary only the tilt angle parameter uo. For each of the two concentrations, we fit the reflectivity profiles of all four isotopic species by a single tilt angle parameter. Plots of the resulting least-squares error x2 versus uo, shown in Fig. 2, are seen to exhibit well-defined minima corresponding to optimal values of uo, or equiv-
,
Table 1 Average scattering lengths used in fitting the structural model to the neutron reflectivity data for C1,TAB
(19)
and fi,(rc) = tilr: m, csch( hrt C;K).
(20)
It may be shown that if the head group number density is any even function of z, then for KZ<<1 its Fourier transform is well approximated by Eq. ( 19 ) . The initial analysis performed by Simister et al. [ 71 made use of a model very similar to ours with regard to the solvent and head group layer densities, but less detailed in its treatment of the chain layer. A leastsquares tit to the reflectivity data produced optimal values for various structural parameters of the interface, including the air-water interfacial width c, the
Unit
Scattering length (lo-4A)
N(CD&Br N(CW3Br CD2 CD3 Dz0 nrw
9.618 0.249 1.999 2.666 1.914 0
Table 2 Structural parameters for (&TAB. The values of C, 4, and S, were obtained from refs. [ 22,231 C
c
4
s,
(mW
(A)
(A)
(A)
3.0 4.5
5.2kO.S 5.9kO.5
lOf2 12+2
If1 2+1
a3
& (de&
0.71 0.67
45 48
316
A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
Fig. 2. Least-squares error x2 versus tilt angle parameter u,, for a fit of the structural model to neutron reflectivity data for a C14TAB monolayer adsorbed at the air-water interface. The solid and dashed curves correspond, respectively, to bulk surfactant concentrations of c= 3 mM and ~~4.5 mM.
alently the tilt angle $. The latter are calculated using 0, = cos- ’ u. and are included in Table 2. The experimental reflectivity profiles themselves, and our best numerical fits to them, are shown together in Fig. 3. Although we have considered scaling the reflectivity data according to the kinematic correction formula of Crowley (241, given the statistical errors in the data and the approximations underlying our model, we find that the effects of scaling are in this case quite insignificant.
5. Discussion The accuracy with which we are able to tit the reflectivity data would seem to lend support to our assumption of linear conformations with a fixed tilt angle. As a check on that assumption though, we have also explored other tilt angle distributions (e.g., a twoparameter Gaussian). In general, the accuracy of the fit is quite insensitive to the functional form of the distribution function, and the width of the distribution tends to be rather narrow (e.g., 2-4 A for a truncated Gaussian). It should be noted, however, that in a recent molecular dynamics simulation, Backer et al. [ 25 ] computed a relatively broad tilt angle distribution for a similar system, namely a Cr6TAC monolayer at the air-water interface. The basic form of the tilt angle distribution - or, more generally, the ori-
entational distribution function - and its dependence on such factors as surface density and chain length are interesting issues for future study. Although a model of surfactant chains with strictly linear conformations and fixed tilt angles must certainly be an idealization, nevertheless, the fact that the C&TAB data may be fit at all with such a model - let alone with such accuracy-strongly suggests the possibility of some orientational ordering of the chains in the monolayer. Quantitative results for the tilt angle are subject to some uncertainty due to an unknown frequency of gauche defects in the chains, each defect acting to shorten a chain by roughly 1 A. In the simulation of Biicker et al. the defect frequency was found to be about 200/bfor Ci6TAC. If the same frequency were present in the case of C,,TAB, then the average bond length would be reduced to xx 1.065 and the average tilt angle range to roughly 33”-37”, illustrating the magnitude of uncertainty in our tilt angle determinations. In any case, it is clear that the chains are significantly tilted with respect to both the vertical and the horizontal. It is interesting to compare this conclusion with a similar conclusion reached in recent studies of the &E,,, series of nonionic surfactants. From analyses of neutron reflectivity data for Cr2E3 [ 121 and Cr2E6 [ 131, Lu et al. conclude that the alkyl chains are significantly tilted away from the normal over a range of concentrations, and that the ethylene glycol chain orientation varies strongly with concentration, with the chains oriented close to vertical at high concentration (Ax 36 A’), but close to horizontal at substantially lower concentrations (A~89 A’). Although the molecular structures of these nonionic surfactants are rather different from those of the ionic Ci4TAB surfactant, nevertheless the occurrence of sizeable chain tilt angles away from the normal appears to be a feature common to both systems. Finally, we note that the small difference in coverage of only about 10% between the two bulk concentrations is unfortunately too small to permit any conclusions regarding dependence of tilt angle on coverage. This dependence, however, is an interesting issue to which the model may be applied in future.
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A.R. Denton et al. /Chemical Physics Letters 219 (1994) 310-318
-6 0
I 0.05
01K ',&';i
'0'2''
0.25 '
&TAB
&TAB on m-w c=4.5 mM
on D,O
-6 -6
0
,,,"'~,,',,,"""""' 0.05
0.2
0.25
Fig. 3. Reflectivity R(K) versus wavevector transfer K normal to the interface for the surfactant C14TABadsorbed at the air-water interface, for two different bulk concentrations and four different isotopic compositions. The symbols are the data from ref. [ 71, and the curves are fits to the data of our structural model (Fig. lb). The solid curves and solid symbols correspond to the isotopic composition C&TAB-h, the &shed curves and open symbols to Cr.,-d-TAB-h. 6. Summary and conclusions
In summary, we have proposed a new model for the structure of surfactant monolayers at interfaces which, in conjunction with the kinematic (or first Born) approximation, may be used to calculate neutron reflectivities. Compared with previous models, the new model is distinguished by a more realistic and detailed representation of the chain scattering length density. As a practical demonstration, we have applied the model to a reanalysis of data from a recent neutron reflectometry study of a &TAB monolayer adsorbed at the air-water interface. Assuming, for simplicity, fully extended linear chains with fned tilt angles, a least-squares fit of the model to reflectivity data for two different bulk concentrations and four differ-
ent isotopic species yields chain tilt angles in the range 45”-48”. Although considerable uncertainty must be attached to the tilt angle and its distribution due to an unknown frequency of gauche defects in the chains, it is at least qualitatively clear that the chains are significantly tilted away from the normal to the interface and quite possibly orientationally ordered. It is hoped that future refinements and applications of the model to neutron reflectivity data for a range of adsorbed surfactant monolayers, with various coverages and isotopic compositions, will aid in illuminating fundamental issues concerning the nature of orientational ordering of monolayers at interfaces.
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A.R. Denton et al. / ChemicalPhysics Letters 219 (1994) 310-318
Acknowledgement We are grateful to Professor P.A. Egelstaff, Dr. J. Penfold, Dr. J.R. Lu, and Professor R.K. Thomas for helpful discussions, and especially to Dr. Lu and Professor Thomas for kindly providing us with the neutron reflectivity data from ref. [ 7 1. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). One of us (ARD) gratefully acknowledges support through an NSERC Postdoctoral Fellowship during the early stages of the work.
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[6] J.E. Bradley, E.M. Lee, R.K. Thomas, A.J. Willatt, J. Penfold, R.C. Ward, D.P. Gregory and W. Waschkowski, Langmuir 4 (1988) 821. [7] E.A. Simister, E.M. Lee, R.K. Thomas and J. Penfold, J. Phys. Chem. 96 (1992) 1373. [ 81 J.R. Lu, E.A. Simister, E.M. Lee, R.K. Thomas, A.R. Rennie and J. Penfold, Langmuir 8 (1992) 1837. [9] J.R. Lu, E.M. Lee, R.K. Thomas, J. Penfold and S.L. Flitsch, Langmuir 9 (1993) 1352. [ lo] E. Staples, L. Thompson, I. Tucker, J. Penfold, R.K. Thomas and J.R. Lu, Langmuir 9 (1993) 1651. [ 111 J.R. Lu, 2.X. Li, T.J. Su, R.K. Thomas and J. Penfold, Langmuir 9 (1993) 2408. [ 121 J.R. Lu, M. Hromadova, R.K. Thomas and J. Penfold, Langmuir 9 (1993) 2417. [ 131 J.R. Lu, Z.X. Li, R.K. Thomas, E.J. Staples, I. Tucker and J. Penfold, J. Phys. Chem. 97 (1993) 8012. [ 141 M.L. Goldberger and F. Seitz, Phys. Rev. 71 (1947) 294. [ 151 J. Lekner, Theory of reflection (Martinus Nijhoff, Dordrecht, 1987). [ 161 V.F. Sears, Neutron optics (Oxford Univ. Press, Oxford, 1989). [ 171 G.E. Bacon, Neutron diffraction, 3rd Ed. (Clarendon Press, Oxford, 1975). [ 181 M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford, 1970). [ 191 T.L. Crowley, E.M. Lee, E.A. Simister, R.K. Thomas, J. Penfold and A.R. Rennie, Colloids Surfaces 52 ( 1990) 85. [20] J. Ah-Nielsen, Z. Physik B 61 (1985) 411. [ 2 1 ] C. Tanford, J. Phys. Chem. 76 ( 1972) 3020. [ 22 ] J.R. Lu, private communication. [23] J.R. Lu, E.A. Simister, R.K. Thomas and J. Penfold, J. Phys. Chem. 97 (1993) 6024. [24] T.L. Crowley, Physica A 195 (1993) 354. [ 25 ] J. Biscker, M. Schlenkrich, P. Bopp and J. Brickman, J. Phys. Chem. 96 (1992) 9915.