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Structure-Related Simultaneous X-ray and Neutron Reflectivity Analysis Method for Langmuir Films Erich Politsch* Medizinische Biophysik, Technische Universita¨ t Mu¨ nchen, Ismaninger Strasse 22, 81675 Mu¨ nchen, Germany Received August 15, 2002. In Final Form: November 19, 2002 A novel, structure-related, smeared-boxes based method suitable for simultaneous analysis of specular X-ray and neutron reflectograms is presented. The method properly considers molecular subunits and employs an improved weighting function, thereby enabling the deduction of a detailed volume fraction profile. In contrast to existing box-based methods, the new method simulates the subphase medium separately, thus allowing distinction between intrinsic (caused by molecular arrangement) and extrinsic (caused by capillary waves) roughness. Another benefit is a well defined subphase surface origin. In contrast to the recently introduced String Fit method, the new approach is not suited to model whole molecular ensembles but is much faster and easier to implement. The performance of such a doubly enhanced boxfitting approach is illustrated with simulated as well as measured data for lipopolymer monolayers at an air/water surface. It is also compared with the results of previous, more laborious String Fit analyses.
1. Introduction During the last two decades, X-ray and neutron reflectometry have proven to be excellent tools for studying Langmuir and other films.1-4 Considering certain constraints, the measured reflectograms can be converted into scattering length density (SLD) profiles in the analytical process. Apart from the fact that the underlying inversion is normally ambiguous, the relation of SLD profiles to corresponding molecular structures is nontrivial. The adequacy and molecular justification of the constraints used in the conversion process are therefore of paramount importance for the final success of data analysis. It is undisputed that simultaneous analysis of neutron and X-ray data can greatly enhance data analysis. However, a necessary prerequisite of such an approach is a suitable structural model that serves as a link between different data sets. The simplest method for the constraining approach is to divide a given molecule into n parts and then try to fit the measured data with a SLD profile consisting of n smeared, that is, convolved, boxes; the length and height of each box are taken to provide a rough estimate of the underlying associated molecular segment. Although widely used, this method has at least two very serious shortcomings: first, it is not a priori clear that a physically possible molecular conformation can yield the respective SLD profile; second, without additional constraints, no simultaneous analysis of neutron and X-ray data is possible. Wiener and White5 proposed a somewhat different method (so-called “composition space refinement”) for * Telephone: +49 89 15919539. Fax: +49 89 15919540. E-mail:
[email protected]. Current address: IMBP, Pegnitzstrasse 12, 80638 Mu¨nchen, Germany. (1) Als-Nielsen, J.; Mo¨hwald, H. In Handbook on Synchrotron Radiation; Elsevier Science Publishers: 1991; Chapter 1, pp 1-51. (2) Als-Nielsen, J.; Kjaer, K. In Phase Transitions in Soft Condensed Matter; Riste, T., Sherrington, D., Eds.; The Proceedings of the NATO Advanced Study Institute; Plenum Press: New York, 1989; pp 113138. (3) Russell, T. P. Mater. Sci. Rep. 1990, 5, 171-271. (4) Baltes, H.; Schwendler, M.; Helm, C. A.; Mo¨hwald, H. J. Colloid Interface Sci. 1996, 178, 135-143. (5) Wiener, M. C.; White, S. H. Biophys. J. 1991, 59, 174-185.
correlating molecular constituents to SLD fractions, which was originally designed for diffraction measurements but can also be applied to reflectivity analysis.6-10 Within the framework of Wiener and White’s5 method, volume fractions of molecular subunits with “associated” solvent are approximated by Gaussian functions; the succession order is assumed to be given and serves as a constraint. Another constraint is given by the necessity to get a volume fraction sum of one for positive depths z g 0, the positive z direction pointing into the solvent and having its origin z ) 0 at the solvent surface. Although difficult to match, this constraint must be fulfilled to a satisfying degree. String Fit11,12 is perfectly suited to solve said problems; however, it is a method which is computationally demanding and difficult to implement. It is very useful for detailed analyses but is not suited to serve as a “quick shot” tool for fast estimations. The aim of this work is to show how the shortcomings of previous methods for the analysis of X-ray and neutron reflectograms can be overcome in a simple manner. Effectively, the development of String Fit produced a gap between the simple, fast, and easy-to-handle but inaccurate box-fit methods and the sophisticated, highly accurate, but difficult-to-implement and time-consuming structure-related String Fit method. It is the objective of this work to close this gap with a fast method that is easily implemented in programs and nevertheless supersedes said existing unsatisfactory box-fit approaches. It is important to notice that the newly proposed method cannot substitute String Fit, as it is completely unsuitable for the simulation of molecular ensembles. However, due (6) Vaknin, D.; Kjaer, K.; Als-Nielsen, J.; Lo¨sche, M. Biophys. J. 1991, 59, 1325-1332. (7) Schalke, M.; Kru¨ger, P.; Weygand, M.; Lo¨sche, M. Biochim. Biophys. Acta 2000, 1464, 113-126. (8) Lu, J. R.; Hromadova, M.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys. Chem. 1994, 98, 11519-11526. (9) Lu, J. R.; Su, T. J.; Thomas, R. K.; Penfold, J.; Richards, R. W. Polymer 1996, 37 (1), 109-114. (10) Lee, E. M.; Milnes, J. E.; Wong, I. C. Physica B 1996, 221, 159167. (11) Politsch, E.; Cevc, G. Physica B 2000, 276-278, 390-391. (12) Politsch, E. J. Appl. Crystallogr. 2001, 34, 239-251.
10.1021/la020722z CCC: $25.00 © 2003 American Chemical Society Published on Web 01/15/2003
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to its simplistic approach, the novel method is perfectly suited to quickly estimate the molecular structure underlying reflectograms and is vastly superior to the classical “composition space refinement” technique. 2. Classical Approach to Simultaneous Analysis The conventional method includes the solvent in the single boxes that represent molecular subunits. This allows reflectivity to be calculated by superposition, according to the Fourier transform addition, convolution, and shift theorem (see e.g. ref 13):
R(q) 1 )| F RF(q) ∞
dF(z) -iq′z e dz|2 dz
∫-∞∞
1 )| F∞
∑i (FT of box i including solvent) + (FT of bulk solvent box)|2 (1)
(q ) (4π/λ) sin θ is the wavevector transfer. θ is the angle between the surface and the incident beam. q′ ) (q2 2 1/2 qc,∞ ) . RF(q) ) |(q - q′)/(q + q′)|2 is the Fresnel reflectivity of a sharp vacuum/substrate interface. See also ref 14.) The Fourier transform term “FT of box i including solvent” can be explicated further:
FT of box i including solvent ) Fi[exp(-iaiq′) -
(
)
q′2(σi2 + σext2) (2) exp(-ibiq′)] exp 2 ai and bi denote the starting and end points of box i, respectively, and bi - ai ): di is the thickness of box i. σi is the roughness of box i. The term “FT of bulk solvent box” is given by
FT of bulk solvent box )
(
F∞ exp(-izsolvq′) exp -
)
q′2(σsolv2 + σext2) (3) 2
zsolv is the starting point of the solvent box. σsolv is the required solvent roughness for a smooth transition to the bulk phase. Its value is adapted during the fit run to ensure a smooth, space-filling transition between the investigated molecular constituent region and the bulk solvent region. No physical meaning can be attributed to this value. Moreover, a defined separation between extrinsic (capillary wave) roughness σext and intrinsic box roughness values σi is therefore impossible. The solvent volume proportion of nearly the whole region to which the investigated molecules can extend is taken into account by exact multiples, or fractions, of the simulated molecular constituent volumes. It is easily conceivable, and will become apparent when considering an example, that the required space-filling is not guaranteed by such a procedure. Even worse, moderately satisfactory space-filling can be achieved only for rather simple molecular arrangements, to which the approach is naturally restricted therefore. The coupling of solvent and molecules simplifies mathematical handling. The space-filling constraint is normally (13) Bracewell, R. N. The Fourier Transform and Its Applications; McGraw-Hill Series in Electrical Engineering; McGraw-Hill Publishing Company: New York, 1986. (14) Hamley, I. W.; Pedersen, J. S. J. Appl. Crystallogr. 1994, 27, 29-35.
considered by some kind of regularization parameter, which is chosen arbitrarily. All parameters are then optimized using a χ2 minimization algorithm. To overcome the aforementioned problems, the novel method discussed in this work introduces two essential, following alterations. 3. Enhancement 1: Separation of Solvent and Surface-Covering Molecules The first analytical enhancement that I propose concerns the relation between molecular subunits and solvent. The underlying idea is to divide a molecule into linear subunits and then to relate each resulting part to a corresponding smeared box in the volume fraction profile. Figure 1 illustrates how the molecular topology then restricts positions and lengths of all boxes to physically meaningful values. Each box can be described by its length di, -li e di e li, and height, vi ) (li/di)(Ai/AM) (in terms of volume fractions). Here, li denotes the length of molecular part i, Ai is its cross-sectional area, and AM is the occupied surface area per molecule. While in String Fit a computationally demanding subdivision of single (sub)segments into slices is carried out, the instant “box-fit” method disregards such fine details in favor of ease and speed of treatment. As real surface-adsorbed layers consist of a plethora of molecules, different smearing parameters σi pertaining to individual boxes i ∈{1, n} must be introduced. They describe the intrinsic roughness, due to the molecular ensemble conformation, and are realized by convolution; a Gaussian function is a natural and good choice, as it enables easy mathematical handling of the kinematical approximation formula. It is clear that this approach is restricted to unimodal volume fraction distributions of the modeled molecular units.15 A height vi ) ∞ can occur for di ) 0. This is allowable, since the only required constraint is that the sum of effective volume fractions, that is, the volume fraction boxes (di, vi) convolved by their respective roughness values σi, is less than (for z e 0) or equal to (for z > 0) one, with a smooth transition between these two regions. A maximum immersion depth zsmax is assumed, corresponding to the maximum possible separation between the water surface and a given point in the molecules (which corresponds to a hydrophobic/hydrophilic boundary within an amphiphilic molecule). The solvent is simulated separately by a simple box, starting at z ) 0, smeared with roughness σext, the subscript ext emphasizing that this parameter describes the extrinsic roughness resulting from capillary waves. To get correct space-filling, all other individually smeared boxes are subtracted from the solvent box for z g 0. In contrast to existing box-based methods, this separation of solvent and of the investigated molecules gives the fit parameter σext a well-defined meaning. Volume fraction profiles can easily be translated into SLD profiles. Different SLDs for neutron and X-ray scattering results as well as different solvents for a series of neutron measurements (e.g. H2O and D2O) can thus easily be considered. Solvent separation from the surface-associated Langmuir layer causes a problem, however. The resulting solvent distribution cannot be mathematically handled with ease. The problem can be overcome with “brute force”, (15) For multimodal distributions the String Fit11,12 is an appropriate means.
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Figure 1. Molecular topology is taken to restrict the positions and lengths of all boxes to physically meaningful values. Each molecular segment corresponds to a box (here, volume fractions are depicted; they can, however, be easily translated into SLD fractions). The thin boxes are directly related to a certain conformation of a single molecule, as shown on the left-hand side. Real surface-adsorbed layers consist of a plethora of molecules. The simulating boxes must therefore be smeared (here by convolution with different Gaussian functions for distinct boxes), as is shown by the thick lines. The gray dashed-dotted line represents the sum of molecular parts; the dashed line stands for the solvent (which exactly fills the volume to give the overall volume fraction shown as a solid gray line).
Figure 2. Conventional (classical) vs novel approach for fast reflectogram analysis. The sum of volume fraction portions that correspond to molecular constituents should never exceed one and should remain constant for z > 0. The volume that is not occupied by the molecules of the Langmuir layer should be filled with water. To comply roughly with this condition, the classical approach must be constrained, for example, by a regularization parameter. The novel approach fulfills this condition intrinsically. In contrast to the old one, only the novel method allows reliable determination of surface roughness, as the solvent box is an exact step function. This fact also gives the origin of the z axis a well-defined meaning. (The symbolic operation “/ extrinsic roughness” stands for the convolution considering extrinsic (capillary) roughness.)
by dividing the SLD profile into arbitrarily resolved layers from which the reflectivity is then evaluated. Such an approach is rather time-consuming, however.16 Another possibility is to subtract the scattering length of the solvent (for the volume occupied by box i) from the scattering
length of box i (see Figure 2):
R(q)
1 )| F∞ RF(q)
∑i (FT of modified box i) +
(FT of solvent box) + (correction term)|2 (4) (16) It can, and should, nevertheless be applied to systems for which kinematical approximation seems to be insufficient.
“Solvent box” here represents the (smeared) steplike
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SLD profile of the pure solvent interface, as would be present in the absence of any surface-adsorbed additional molecules. In other words, the volume occupied by a box i is taken into account by adding a “negative water volume” of the same magnitude. Unfortunately, this method breaks down for z < 0, where no space-filling solvent exists. For that reason, one is forced to introduce a correction term that must be evaluated by dividing the parts of the boxes for which z < 0 into thin slices. For molecules with large hydrophilic parts, such as lipopolymers, the superposition with an included correction term is nevertheless much faster than complete discretization into thin layers. As already explained, the extrinsic roughness parameter σext is applied to all boxes in Gaussian form. Consequently, each of the three terms in eq 4 contains a factor exp(q′2σext2/2), which is preferably accounted for by first disregarding the extrinsic roughness on the right-hand side of (4) and subsequent multiplication by the factor exp(-q′2σext2). This procedure has the advantage of not widening the correction term region z < 0 by the extrinsic roughness and therefore being faster. 4. Enhancement 2: Modification of the χ2 Deviation Function The typically used, error-weighted quadratic deviation function
∆i2 2 2 2 χj ) δi ; δi ) ; ∆i ) R(qi) - Rexp(qi), (5) i)1 σi2 Mj
∑
depends on theoretical and experimental reflectivity, respectively (R(qi), Rexp(qi)); the number of measured data points Mj for the data set j; and the corresponding errors σi. I herewith advocate the use of a modified quantity, which combines features of the logarithmic deviation function and said classical error-weighted quadratic deviation. The improved deviation function, δ′′, has already been introduced in ref 12 and is given by ′′2
δi
{
(∆i/σi)2 exp[-(∆i/5σi)2] for |∆i| e 5σi ) 25e-1 + 104{ln[Rexp(qi) - 5σi] - ln R(qi)}2 for -∆i > 5σi χj′′2 )
(6)
Mj
δi′′2 ∑ i)1
(7)
The motivation for its use shall only be briefly highlighted here. The lower graph of Figure 3 illustrates the modified local deviation function, that is, the penalty function corresponding to a given deviation from a single measured data point. In Figure 3 the measured reflectivity of Rexp ) 1.0 and two different experimental errors, σ1 ) 0.001 and σ2 ) 0.005, have been assumed. The modified deviation function is then seen to weigh both data points similarly outside the (5σ region, whereas the classical deviation function weighs the two data points differently by a factor (0.005/0.001)2 ) 25 throughout the R region. This reveals that the conventionally used weighting function causes strong suboptima in the optimization landscape. An erroneously assigned low error bar can therefore pin the optimization algorithm to such a strong suboptimum. The upper graph of Figure 3 illustrates the effect of the modified deviation function for three randomly generated
Figure 3. Lower graph: modified local deviation function for an assumed experimental reflectivity Rexp(q) ) 1. Middle and upper graph: effect of the modified deviation function. Due to the strong barriers bounding the suboptima in the classical chi square function χ2 (see e.g. the region -1 < a < -0.5) and the practically indistinguishable fitness differences between the different suboptima and the global optimum, a once found suboptimum will hardly be discarded. The modified deviation functions δ′′ and the sum thereof, χ′′, avoid being trapped in such strong suboptima by leading all local deviations into the 5σ error region with logarithmic, error-independent absolute scaling.
functions, which represent linear deviations, shown in the middle graph in Figure 3. These functions have a distinct minimum at an arbitrarily chosen parameter a ) 0. During the fitting process, the variation of a parameter can cause a decrease of the deviation between a certain measured data point and the corresponding theoretically evaluated data point, whereas it causes an increase of the deviation at another point. The set of the three random deviation functions in Figure 3 simulates this possibly complicated relation between different points of one data set or different data sets fitted simultaneously. The corresponding conventional and modified sum of deviation functions is shown in the upper graph of Figure 3. For illustration purposes, the deviation function sums, according to the two different approaches, have been rescaled to a maximum value of 1.0. Otherwise, the conventional deviation function sum would exceed the modified function by a factor of 105! This highlights that a global minimum is much easier to find with the aid of the modified deviation function. The fitting task is then to minimize the quantity
χsum′′
)
1
N
∑
χj′′2
Nj)1 Mj
(8)
N being the number of data sets fitted simultaneously; Mj is the number of measured data points for data set j. The
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Figure 4. Results of a fit run based on simulated data. All dashed lines relate to the original simulated starting data set. Left-hand side: new method introduced in this work. Right-hand side: comparison with the semiclassical approach. The classical approach, which utilizes a gradient search or Levenberg-Marquardt method and the classical χ2 function, gets stuck much earlier in local minima and therefore delivers much worse results that are not shown. For details see the text.
rationale for using this minimization function is explained in ref 12. It has turned out that this modification, together with the application of an evolutionary algorithm,17-21 dramatically reduces the probability of getting stuck in a local minimum. Moreover, single outliers affect the fitting process much less due to the modified deviation function. 5. Examples 5.1. Simulated Ensemble of Lipopolymers at an Air/Water Interface Compared with a Semiclassical Approach. To test the usefulness of the model described in previous sections, I will analyze a Langmuir layer consisting of a randomly simulated ensemble of lipopolymers. “Experimental” data are derived as follows. The simulated ensemble is taken to consist of 40 molecules, which is sufficient to simulate conformational freedom of a real ensemble. For the randomly generated ensemble using a quasi-molecular model comparable with that shown in Figure 1, but equipped with flexible segments, the corresponding X-ray and neutron reflectograms are evaluated assuming H2O and D2O as two different subphases. The resulting reflectograms are smeared realistically: relative noise 1% for X-rays and 5% for neutrons, plus an (17) See refs 18-21. Another nondeterministic optimization method could be appropriate too. (18) Rechenberg, I. Evolutionsstrategie, problemata; FrommannHolzboog: Stuttgart-Bad Cannstatt, 1973. (19) Rechenberg, I. Evolutionsstrategie ′94, Werkstatt Bionik und Evolutionstechnik; Frommann-Holzboog: Stuttgart, 1994. (20) Schwefel, H.-P. Evolution and Optimum Seeking; Sixth-Generation Computer Technology Series; John Wiley & Sons: New York, 1995. (21) Ba¨ck, T. Evolutionary Algorithms in Theory and Practice; Oxford University Press: Oxford, 1996.
absolute noise proportional to q4 (0.02 at q ) 0.5 Å-1 for X-rays and 0.05 at q ) 0.2 Å-1 for neutrons); the cutoff value is assumed to be q ) 0.6 Å-1 for X-ray data and q ) 0.2 Å-1 for neutron data. The resulting reflectograms serve as the basis for following fit runs. In the fitting run, a molecule is divided into six segmental boxes (alkyl double chain, glycerol group, two equally long hydrogenated polymer segments, and two deuterated polymer segments of equal length). The fit parameters are then optimized as described in the previous section with an evolution strategy. This optimization method, together with the modified χ2 function, greatly reduces the probability of getting prematurely stuck in a local minimum. The left-hand side of Figure 4 shows the results of a typical fitting run of the method presented here and confirms good agreement between the original and the deduced volume fraction distributions. The simulated and the fitted reflectograms agree nicely as well. To obtain the corresponding volume fraction graph, the distributions of the boxes corresponding to the hydrogenated and deuterated segments are combined. The fit quality is about 1 order of magnitude worse than that obtained from the String Fit model.11,12 However, the advantage of the presented method is that it allows rapid system parametrization of a quality that suffices in many cases. Structural features beyond the volume fraction distribution are not derivable from a box fit. The right-hand side of Figure 4 shows the corresponding results of the semiclassical approach. Although the modified χ2 function and an evolution strategy have been employed for the fit, the result is still far inferior to the corresponding result of the new method. Even worse experience has been encountered by other authors without
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said enhancements. This is because of the coupling of water to the molecular segments, which does not allow sufficient freedom to obtain a realistic molecular distribution. Due to the inability of the conventional model to represent the underlying molecular structure adequately, the volume fractions of all segments deviate strongly from the original, “correct” distribution; here, the limits of conventional boxfitting methods become clear and obvious. 5.2. Measurements with Lipopolymers at an Air/ Water Interface. The results obtained with real experimental data pertaining to a lipopolymer layer (distearoylpolymethyloxazolin, DS-PMeO) adsorbed at an air/water interface reveal similar trends. As a reference, the results of a previous very detailed analysis are used.22,23 It has been previously concluded that it is impossible to fit the data given in Figure 5 satisfactorily with a boxlike method.23 The classical box methods give totally inadequate results. I succeeded in overcoming the limitation of conventional box analysis by using the method described in this work, provided that enhancements 1 and 2 described in previous sections are combined. Without such combination the desired result cannot be obtained. The method introduced in this work offers the first practicable solution for a quick analysis. Figure 5 shows the results of simultaneous analysis of neutron and X-ray reflectograms done within the framework of the instant method. For comparison, the results of String Fit are included. The quality of the new results obtained with the doubly enhanced box model, as described in this work, is about 1 order of magnitude worse than that of String Fit (which is obviously mainly due to the inability of the instant simple method to fit the details of the X-ray reflectogram). However, a quantitatively similar molecular picture is gained in either case. An eye-catching difference is the shift of the alkyl chain region volume fraction toward higher z values in comparison with the outcome of String Fit runs. This is due to the inability of the present, simpler model to mimic the complicated interfacial structure with a small number of boxes. (22) Politsch, E.; Cevc, G.; Wurlitzer, A.; Lo¨sche, M. Macromolecules 2001, 34, 1328-1333. (23) Wurlitzer, A.; Politsch, E.; Hu¨bner, S.; Kru¨ger, P.; Weygand, M.; Kjaer, K.; Hommes, P.; Nuyken, O.; Cevc, G.; Lo¨sche, M. Macromolecules 2001, 34, 1334-1342.
Politsch
Figure 5. Comparison between the doubly enhanced box-fit model and the detailed String Fit model in the case of real data analysis of X-ray and neutron reflectivities which stem from a lipopolymer layer at an air/water interface. Dashed lines relate to String Fit runs. For details see the text.
6. Conclusion The doubly enhanced box-based fitting method presented in this work is especially well-suited for investigating Langmuir layers. It has the advantage of being computationally nondemanding but nevertheless reasonably accurate even for rather complex underlying molecular arrangements (which dominate real systems). The main clou to success is the introduction of a modified χ2 deviation function, given by eq 6. Using this function prevents getting prematurely stuck in a local minimum, which often happens in conventional simultaneous reflectogram analysis. Acknowledgment. I thank Prof. G. Cevc for his invaluable support. This work was financially supported by the Deutsche Forschungsgemeinschaft (SFB 266, TP C8) and the EU (TMR contract ERBFMGECT950059). LA020722Z
