Capillary wave fluctuations and intrinsic widths of coupled fluid-fluid interfaces : An x-ray scattering study of a wetting film on bulk liquid

Masafumi Fukuto,* Oleg Gang, Kyle J. Alvine, and Peter S. Pershan Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA Received 17 April 2006; revised manuscript received 25 July 2006; published 25 September 2006


I. INTRODUCTION
X-ray and neutron reflectivity is the principal method for nondestructive characterization of the structure buried within nanometers of surfaces ͓1-4͔.In addition to individual layer densities, thicknesses, and interfacial roughness, the reflectivity techniques can be used to extract more subtle structural features that may be technologically relevant.In particular, as demonstrated by extensive studies of solid multilayer films, the simultaneous measurements of x-ray specular reflectivity ͑XR͒ and off-specular diffuse scattering ͑XDS͒ provide a powerful tool for probing the layer-to-layer correlations in the roughness of neighboring interfaces ͓5-9͔.The combination of XR and XDS has also been used to study the nature of roughness and/or the correlated roughness of multiple interfaces in substrate-supported thin films of liquids ͓10-12͔, polymers ͓13-16͔, and organic Langmuir-Blodgett multilayers ͓17-20͔.
For thin liquid films without solid substrates, the elucidation of correlated roughness, or equivalently film thickness fluctuations, is of fundamental interest since these effects reflect the form of the interfacial potential ͓21-23͔.For example, in the first study of its kind for liquid films, Daillant et al. ͓21,24͔ measured and theoretically analyzed the XDS from free-standing soap films and estimated the strength of the coupling between the capillary fluctuations of the two ͑equivalent͒ liquid-vapor interfaces.More recently, Schlossman et al. ͓22,25,26͔ carried out a combined XR/XDS study on an incomplete wetting film on bulk liquid and demonstrated the presence of partially correlated ͑i.e., neither independent nor perfectly conformal͒ capillary fluctuations at the liquid-vapor and liquid-liquid interfaces; furthermore, they used the extracted coupling constant to estimate the parameters for the short-and long-range interaction terms in the interfacial excess free energy ͓22͔.
Despite these capabilities, however, the application of XR/XDS to purely fluid ͑or fluidlike͒ systems with multiple interfaces has been very limited ͓21-23,27͔.This can be attributed partly to the experimental difficulties associated with liquid surfaces and films ͑flatness, stability, etc.͒.Another likely reason is that due to the subtleties of liquid surface scattering, proper interpretation of the data is nontrivial and tends to be technically involved.Thermal diffuse scattering from capillary fluctuations exhibits an algebraic singularity at the specular direction ͓28-30͔, and this causes the measured XR to depend on the experimental resolution ͓29-34͔.For layered surfaces, not only is it more difficult to distinguish the capillary roughness at individual interfaces from non-step-like intrinsic interfacial profiles, but also the scattering is further complicated by its dependence on the partial correlations between the fluctuations of different interfaces.While the few published XR/XDS studies ͓21,22,27͔ have adequately dealt with some of these complications, not all of these technical issues have been explicitly addressed.For example, the data analysis in these previous studies follows the same general approach as used for solid surfaces that is based on the separation of scattering into the specular and diffuse contributions ͓28͔.This approach is questionable given that the two contributions are experimentally indistinguishable for liquid surfaces ͓29,30͔.Moreover, it appears that the effect of partially correlated roughness is usually neglected in the analysis of XR data and that the intrinsic widths of multiple interfaces are either too small to be extracted reliably or assumed to be zero.
We present here an XR/XDS study of the coupled thermal capillary fluctuations and the intrinsic profiles in complete wetting films adsorbed on the surface of bulk liquid.The purpose of this paper is twofold.The first objective is to present approximate methods for describing the surface x-ray scattering from two fluctuating fluid-fluid interfaces and to demonstrate their accuracy and utility for data analysis.Our approach follows the well-known ͑and well-practiced͒ hybrid model of fluid surfaces ͓35,36͔ in which each interface is assumed to have an intrinsic width and the thermally excited capillary waves are described in terms of macroscopic ͑i.e., lateral wave-vector-independent͒ surface tension and an upper wave-vector cutoff ͓29,31,33,34,37-39͔.On the basis of these assumptions and the same Hamiltonian that Schlossman et al. used for the coupled capillary fluctuations ͓22͔, approximate behaviors of the lateral correlation functions for interfacial height and film thickness fluctuations are obtained.These approximate correlation functions are used to calculate, in the kinematic ͑Born͒ limit, normalized scattering intensities that can be compared with the data.In particular, rather than making an artificial distinction between the specular and diffuse contributions, the theoretical XR and XDS intensities are obtained by integrating one common expression for the differential cross section over realistic resolution functions.In addition, an approximation of the above integrals by Debye-Waller factors is used to derive a simple analytic form of specular reflectivity that takes into account the partial correlations between capillary fluctuations at the two interfaces.
The second objective is to present the results of XR/XDS measurements on wetting films of perfluoromethylcyclohexane ͑PFMC, C 6 F 11 -CH 3 ͒ on eicosane ͓C20; CH 3 -͑CH 2 ͒ 18 -CH 3 ͔.Our initial XR study, described in Ref. ͓40͔, demonstrated the complete wetting of the bulk C20 surface by PFMC vapor.The present paper focuses on characterization of the coupled capillary fluctuations and the intrinsic interfacial widths at the liquid-liquid and liquid-vapor interfaces, over a range of film thickness D ͑=30-160 Å͒.PFMC and C20 are nonpolar, immiscible liquids and their intermolecular interactions are dominated by the long-range van der Waals ͑vdW͒ forces; therefore, PFMC wetting films on C20 should provide a simple model system for testing the expected coupling of capillary fluctuations at the two interfaces.The analysis of XR/XDS measurements indicate that ͑i͒ the capillary fluctuations at the two interfaces are consistent with the presence of interfacial coupling that is dominated by ͑nonretarded͒ vdW interactions, ͑ii͒ the intrinsic liquid-liquid interfacial width is in the range of 4 -6 Å for D տ 50 Å, comparable to both the radius of gyration R g Ϸ 5.3 Å for the C20 molecule ͓34͔ and an estimated bulk correlation length b Ϸ 4.8 Å for molten long-chain alkanes ͓38͔, and ͑iii͒ the intrinsic liquid-vapor interfacial width is sharper ͑ϳ2 Å͒ and remains nearly constant over the entire probed range of D.
This paper is organized as follows.Section II provides a brief background on x-ray scattering in order to establish the notations and describe the general formalism used to estimate XR/XDS intensities.In Sec.III, the capillary wave fluctuations at the two fluid-fluid interfaces of the wetting film and their effects on x-ray scattering are considered in some detail.Section IV provides a brief description of experimental details.In Sec.V, the experimental results and the analysis of XR and XDS measurements are discussed.In Sec.VI, the main conclusions are summarized.Appendix A briefly describes the main characteristics of liquid surface scattering in the single-interface limit, which form the basis for the approximations developed in Sec.III.

A. X-ray kinematics
The kinematics for surface x-ray scattering is illustrated in Fig. 1͑a͒.The average surface lies in the x-y plane, and the plane of incidence is taken to be the y-z plane.The incident and scattered wave vectors are denoted as k in and k out , respectively, and the wave-vector transfer is given by q = k out − k in .In terms of the incident angle ␣ and the output angles ␤ and relative to the surface and the incidence plane, respectively, the components of q are given by q z = k͓sin͑␣͒ + sin͑␤͔͒, ͑1a͒ where k =2 / and is the x-ray wavelength.The specular x-ray reflectivity ͑XR͒ is the fraction R͑q z ͒ = I͑q z ͒ / I 0 of the incident intensity I 0 that is reflected in the specular direction, i.e., at ␤ = ␣ and =0 ͑or equivalently, q xy =0͒, and it is measured as a function of ␣ or q z =2k sin͑␣͒.The diffuse scattering ͑XDS͒ from a surface is measured at nonspecular conditions ␤ ␣, typically within the incidence plane ͑ =0͒.For Ϸ 0, the full width at half-maximum ͑FWHM͒ detector resolutions in terms of wave vectors and scattering angles are related to each other as ⌬q x Ϸ k⌬, ⌬q y Ϸ k sin͑␤͒⌬␤, and ⌬q z Ϸ k⌬␤.In most XR and XDS experiments, the lateral wave-vector resolution is much wider transverse to the plane of incidence than along the incidence plane, i.e., ⌬q x ӷ⌬q y .Therefore, in typical measurements the smallest lateral wavevector that can be resolved is roughly of the order of q xy,min ϳ ⌬q y .

B. X-ray scattering from a rough surface with two interfaces
Figure 1͑b͒ shows the convention used in this study to describe a two-interface surface, consisting of a film of average thickness D on a subphase.The quantities referring to the film-subphase and film-vapor interfaces are denoted with subscript "1" and "2," respectively.Deviations in the local interfacial heights from the average heights are represented by ͕h 1 ͑r xy ͒ , h 2 ͑r xy ͖͒, where ͗h i ͑0͒͘ ϵ 0.
X-ray scattering from rough surfaces has been reviewed extensively ͓4,24,28,30,41-45͔.The measured intensity I͑q xy , q z ͒ is proportional to the integration of the differential cross section d / d⍀ over the solid angle ⌬⍀ subtended by the detector slits.Equivalently, using the relation ␦⍀ = cos͑␤͒␦␦␤ Ϸ ␦q x ␦q y / ͓k 2 sin͑␤͔͒, the scattering intensity can be obtained by integration over the experimental resolutions in reciprocal space.For a film with two separate fluctuating interfaces, the normalized intensity can be expressed as where q c =4 ͱ r 0 ϱ = ͑4 / ͒sin͑␣ c ͒ is the critical wave vec- tor, r 0 is the classical radius of the electron, and ϱ is the electron density in the bulk subphase far below the surface ͓1͔.Apart from the inclusion of the Fresnel transmission factors T F ͑͒ to account for the surface enhancement effect near ϳ ␣ c ͓28,29͔, Eq. ͑2͒ is based on the kinematic ͑Born͒ approximation.For a simple case in which the interfacial profiles vary monotonically across each interface, the local ͑in-trinsic͒ form factors for the two interfaces are commonly taken to be where w i represents the intrinsic interfacial width and 1 = 1 / ϱ is the electron density in the film relative to the bulk subphase.The dependence of observed intensity on both the interfacial roughnesses and the detector resolutions is contained in the factors where A is the surface area.The height-height correlation functions are defined as The resolution function ⌶͑q xy − q xy Ј ͒ in Eq. ͑4͒ satisfies ⌶͑0͒ = 1 at the detector center ͑q xy Ј = q xy ͒ and ⌶ → 0 well out- side the resolution widths.Note that for the most relevant case of scattering within the plane of incidence ͑ =0͒, q z implicitly depends on q y Ј and varies as ␦q z Ϸ k␦␤ Ϸ −␦q y Ј/ sin͑␤͒.As is appropriate for typical experiments, Eq. ͑2͒ assumes that either the facor ⌰ i ͑q z ͒ varies slowly enough with q z or the resolution is sufficiently fine that the minute variation in ⌰ i ͑q z ͒ over the resolution volume can be neglected.
A kinematic expression for the specular reflectivity R͑q z ͒ = I͑0,q z ͒ / I 0 is given by the evaluation of Eq. ͑2͒ at The Fresnel reflectivity R F ͑q z ͒ corresponds to an ideally flat and sharp surface for which the electron density changes discontinuously from the bulk value ϱ to zero ͓1͔.R F ͑q z ͒ is equal to unity for q z ഛ q c ͑total external reflection͒ and has the form for q z ӷ q c .The main objectives of XR/XDS experiments are to obtain measures of interfacial height fluctuations from the resolution-dependent roughness factors R ij ͑q xy , q z ͒ and to ex-tract the intrinsic profiles from the form factors ⌰ i ͑q z ͒.These two types of effects can be separated if the nature of height fluctuations is known and the factors R ij ͑q xy , q z ͒ can be theoretically accounted for.This is done in the following way.By taking the Fourier transform where the thickness-thickness correlation function is given by In Eqs.͑9͒-͑11͒, the surface is assumed to be laterally isotropic and r xy = ͉r xy ͉.The upper cutoff wave vector q max corresponds to the smallest wavelength of the allowed height fluctuation modes, which is usually limited by the molecular size a, i.e., q max ϳ / a ͓29,34,36͔.
Once g ij ͑r xy ͒ are known, the factors R ij ͑q xy , q z ͒ in Eq. ͑4͒ can be obtained, for example, by numerical integration using the slit-determined resolutions ͑⌬q x , ⌬q y ͒ ͓46-51͔.On the other hand, as demonstrated previously ͑see, e.g., Refs.͓15,21,24,29,52,53͔͒, the integration can be simplified considerably for scans at = 0 and small ␤ ͑i.e., for ⌬q y Ӷ⌬q x ͒ if the q xy resolution function is taken to be infinitely wide in the q x direction.Substituting ⌶͑q xy − q xy Ј ͒ = ⌶͑q y − q y Ј͒ and taking the Fourier transform of the resolution function

͑12͒
the integration of Eq. ͑4͒ over q xy Ј yields R ij ͑q y ,q z ͒ = ͵ dye iq y y e −͑1/2͒g ij ͑y͒q z 2 ⌶ ˜͑y͒.͑13͒ Since our measurements satisfy the condition ⌬q y Ӷ⌬q x , this simplified form is used to evaluate R ij in Sec.III.

III. LIQUID SURFACE WITH TWO INTERFACES
Appendix A briefly describes ͑i͒ the peculiarities of thermal diffuse scattering arising from the capillary waves on liquid surfaces and ͑ii͒ approximations that have been used successfully to account for measured intensities for simple surfaces, i.e., those with either a single interface or conformal roughness.The aim of the present section is to develop similar approximations for surfaces in which there are two interacting fluid-fluid interfaces and their fluctuations are partially correlated with each other.

A. Wetting and interfacial fluctuations
The experimental surface to be discussed consists of a complete wetting film of PFMC on the surface of bulk liquid C20 ͓40͔.The two fluctuating interfaces are illustrated in Fig. 1͑b͒.The equilibrium thickness D of a complete wetting film depends on the balance between ͑i͒ the energy cost of forming a liquid layer where the vapor is the stable bulk phase and ͑ii͒ the van der Waals ͑vdW͒ attractions between the wetting and subphase molecules.Taking ⌬ to be the intrinsic chemical potential of the wetting molecules relative to the bulk liquid-vapor coexistence, n to be the number density of the wetting liquid, and A eff to be the effective Hamaker constant for the planar bulk-film-vapor system, the thickness-dependent part of the surface excess free energy ͑per area͒ is given by ͓54,55͔ The average film thickness D is determined by minimization of Eq. ͑14͒, . ͑15͒ The 1 / 3 power-law dependence of D on ⌬ signifies complete wetting ͓56,57͔.Following Li et al. ͓22͔, the energy cost for the interfacial height fluctuations ͕h 1 ͑r xy ͒ , h 2 ͑r xy ͖͒ can be taken to be

͑16͒
where ͕␥ 1 , ␥ 2 ͖ are the surface tensions of the two interfaces, B͑D͒ is a thickness-dependent coupling constant between the capillary waves at the two interfaces, and ⌬ m,i refers to the mass density difference across each interface.In addition to the usual ␥ i -dependent gradient terms and the gravitational terms for the individual interfaces, the vdW coupling term ͓ϰB͑D͔͒ is included to account for the work required to change the local thickness from the average equilibrium value D. If the amplitudes of the fluctuations are small compared to D, this term can be taken to be the leading term in the power series expansion of F͑D + ␦D͒ − F͑D͒ ͓via Eq.
A eff 2D 4 .͑17͒ The fact that the coupling term is minimized if h 2 ͑r xy ͒ = h 1 ͑r xy ͒ implies that in the limit of small thickness or strong vdW attractions between the film and the subphase ͓B͑D͒ → ϱ͔, the fluctuations of the two interfaces are primarily conformal.In the opposite limit of large D or small A eff ͓B͑D͒ → 0͔, Eq. ͑16͒ reduces to the sum of Hamiltonians at the two individual interfaces, and therefore the two interfaces should fluctuate independently, as expected for macroscopically thick films.
The above treatment of the coupling term is implicitly based on the Derjaguin approximation ͑DA͒ ͓58-60͔.In the present context, the DA implies that ͑i͒ the local energy density ͑per unit area͒ at a given point r xy for the interactions between the two fluctuating interfaces is assumed to depend only on the local thickness d͑r xy ͒ = D + h 2 ͑r xy ͒ − h 1 ͑r xy ͒, and ͑ii͒ this local energy density is approximated with the value that would be obtained for a pair of flat and parallel interfaces that are separated by d͑r xy ͒.As shown by Andelman et al. ͓59,60͔ for wetting films on rough solid surfaces, the DA should be valid for q xy Ӷ 1/D, i.e., for describing the coupling between interfacial fluctuations of long wavelengths, but is expected to break down for q xy ӷ 1/D.This limitation of the DA is not a serious problem here since the x-ray scattering measurements in the present study were taken at relatively small magnitudes of the lateral wave vectors ͓͉q y ͉ ഛ max͕͉q y ͉͖ ϳ 2 ϫ 10 −3 Å −1 Ͻ 1 / max͕D͖ϳ6 ϫ 10 −3 Å −1 ͔ and should thus be dominated by the longwavelength behavior.
Using the Fourier transform of h i ͑r xy ͒ in Eq. ͑8͒, the equipartition of the Fourier modes in the Hamiltonian ͓Eq.͑16͔͒ yields the mean-square amplitudes of height and thickness fluctuations, where ␦ ij is the Kronecker delta.The presence of two interfaces leads to two lateral correlation lengths, where m,ϱ = ⌬ m,1 + ⌬ m,2 is equal to the mass density of the bulk liquid.The vdW-limited correlation length ϳ D 2 is a measure of how far the correlations in thickness fluctuations persist over the surface ͓see Eq. ͑19͔͒; it ranges from intermolecular length scales for very thin films to ϳ1 m for films that are a few hundred angstroms thick.As in the case of simple liquid surfaces ͑Appendix A͒, the gravity-limited length g is of the order of ϳ1 mm and sets the correlation length for the height fluctuations.It is clear from the above that the two types of height fluctuations contribute to ͉͗h ˜i͑q xy ͉͒ 2 ͘.For each interface ͑i = j͒, the first term in Eq. ͑18͒, which depends only on the surface tension ␥ i of that interface and has the same form as for the simple liquid-vapor interface ͓31,36͔, describes the component of independent fluctuations.The second term in Eq. ͑18͒, which is exactly the same for all ͕i , j͖ and depends on the effective surface tension ␥ eff = ␥ 1 + ␥ 2 , represents the component of conformal fluctuations.The mean-square amplitudes above will be used below to derive approximate expressions for the real-space correlation functions and to estimate the Debye-Waller-like effects on the reflectivity.

B. Approximate correlation functions
For simple liquid surfaces ͑Appendix A͒, it has proven practical to approximate the scattering cross section by extending the logarithmic behavior of g͑r xy ͒ for q max −1 Ӷ r xy Ӷ g ͓Eq.͑A3͔͒ to the entire range r xy ജ q max −1 .An equivalent approximation for two fluctuating interfaces can be developed by considering the limiting behavior of the correlation functions in Eqs.͑9͒-͑11͒.On the basis of Eqs.͑18͒ and ͑19͒, the correlation functions behave approximately as follows ͓61͔: For each correlation function, note that the limiting forms in adjacent ranges of r xy are continuous and extrapolate to identical values at r xy = and at r xy = g , and the substitution of r xy = q max −1 in the expression for r xy ӷ q max −1 results in the mean-square value at r xy = 0.These limiting forms are plotted in Figs.2͑a͒ and 2͑b͒, for a PFMC film on C20 at 42 °C ͑see Table I͒ and D  For the purpose of estimating the scattered intensity ͓Eqs.͑2͒, ͑4͒, and ͑6͔͒, the correlation functions can be simplified further.First, for the typical film thickness probed here ͑D = 30-160 Å͒, the inequalities q max −1 ͑ϳ2 Å͒ Ӷ ͑=150 -4400 Å͒ Ӷ g ͑ϳ1 mm͒ are generally satisfied.Since these lengths differ significantly in magnitude and are independent of the precise form of correlation functions, a reasonable approximation is to neglect the gradual crossover behaviors near r xy = ͕q max −1 , , g ͖ and apply the limiting forms in Eqs.͑22͒-͑25͒ to all values of r xy within each length scale region.Second, since it is impossible to resolve g in practical x-ray experiments ͑q xy,min ϳ ⌬q y =10 −5 -10 −4 Å −1 ӷ g −1 ͒, the limiting forms of g ij ͑r xy ͒ for Ӷ r xy Ӷ g , which are independent of g , will be extended to large r xy → ϱ.This same approximation has been used for g͑r xy ͒ for simple surfaces ͑Appendix A͒.Applying these approximations to Eqs. ͑23͒ and ͑24͒, obtains  I for a PFMC wetting film on bulk C20 at 42 °C and D = 100 Å, corresponding to = 1700 Å. ͑a͒ ͗h i ͑r xy ͒h j ͑0͒͘ ͑labeled ij͒ from Eq. ͑22͒; ͑b͒ g ij ͑r xy ͒ ͑labeled ij͒ and ͗␦D͑r xy ͒␦D͑0͒͘ ͑labeled "DD"͒, from Eqs. ͑23͒-͑25͒; ͑c͒ exp͓− 1 2 g ij ͑r xy ͒q z 2 ͔ at q z = 0.15 Å −1 , from Eqs. ͑26͒ and

͑27͒
where the q z -dependent exponents are defined as Figure 2͑c͒ shows the behavior of these approximate functions over the range r xy Ͻ g , calculated for q z = 0.15 Å −1 and the same parameters as in Figs.2͑a͒ and 2͑b͒.
Using the above approximate forms of exp͓− 1 2 g ij ͑r xy ͒q z 2 ͔, the evaluation of R ij ͑q y , q z ͒ is reduced to a single numerical integration of simple analytic functions in real space.This is illustrated below for XDS and XR separately, choosing a realistic resolution function that is appropriate for each technique.

C. Off-specular diffuse scattering
In order to maximize the dynamic range of the XDS data, it is often advantageous to have access to as small a lateral wave vector ͑q xy,min ϳ ⌬q y ͒ as possible.Ultimately, the resolution is limited by either the angular divergence of the incident beam ͑⌬␣ ϳ 10 −4 rad for the measurements below͒ or the finite curvature of the liquid surface.The detector angular resolution ⌬␤ for XDS should be chosen such that it is as close to this limit as possible and still wide enough to yield measurable intensities at higher q z .The XDS measurements in the present study were taken with ⌬␤ = 4.3ϫ 10 −4 rad ͑Sec.IV B͒; a typical rocking curve scan around the specular condition is shown in Fig. 3.It is clear that the FWHM of the central peak is close to the slit-based width ⌬q y = ͑q z /2͒⌬␤, and that the experimental resolution is well represented by a Gaussian ͓15,29,52͔ of the form

͑29͒
This expression satisfies ⌶͑0͒ = 1 at the detector center, and the integration of ⌶ over ͑−ϱ , ϱ͒ yields the FWHM resolution ⌬q y .Substituting the Fourier transform of Eq. ͑29͒ in Eq. ͑13͒,

͑30͒
This integration can be carried out numerically using the approximate factors exp͓− 1 2 g ij ͑r xy ͒q z 2 ͔ in Eqs.͑26͒ and ͑27͒.
Due to the relatively fast decay of the integrand with y, the numerical integration converges without difficulties.The normalized intensity I͑q y , q z ͒ / I 0 to be compared with XDS data can be obtained using Eq.͑2͒.A eff = A HF − A FF =0.9ϫ 10 −20 J Liquid-liquid surface tension b ␥ 1 = 6.5 dyn/ cm at T 1 ␥ 1 = 6.3 dyn/ cm at T 2 Liquid-vapor surface tension c ␥ 2 = 12.5 dyn/ cm Lateral correlation length ͑van der Waals͒ d = ͑0.17 Lateral correlation length ͑gravitational͒ e g = 1.6 mm Upper wave-vector cutoff f q max = / a = 0.5 Å −1 Relative layer electron density g PFMC / C20 = 1.9 a For fluorocarbon film on bulk hydrocarbon ͓66͔. b For bulk PFMC-C20 interface, via pendant drop method ͓65͔.
g Nominal value estimated from mass densities; treated as a fitting parameter for R / R F . FIG.
3. An XDS rocking scan in the incidence plane ͑q x =0, =0͒ at q z = 0.14 Å −1 , taken with an experimental resolution of ⌬q y Ϸ͑q z /2͒⌬␤ = 3.0ϫ 10 −5 Å −1 .The data ͑᭺͒ correspond to a PFMC wetting film on C20 at 42 °C and D = 155 Å and is identical to the data in Fig. 7͑c͒.The observed intensity is shown as the product sin͑␣͒I͑q y , q z ͒, which accounts for the change in the x-ray footprint on the surface with varying q y and should be symmetric with respect to the sign of q y for fixed q z .The solid curve corresponds to the Gaussian resolution function ⌶͑q y − q y Ј͒ in Eq. ͑29͒.For com- parison, the data sin͑␣͒I͑q y , q z ͒ are normalized to be unity at q y =0.
In the limit of very weak ͑ → g or ϱ͒ or strong ͑ → q max −1 ͒ interfacial coupling, the factors exp͓− 1 2 g ij ͑r xy ͒q z 2 ͔ are simplified such that the integral in Eq. ͑30͒ can be solved analytically, in the manner similar to the case of simple liquid-vapor interfaces ͓29͔.The limit of weak coupling or independent fluctuations yields lim where ⌫ is the Gamma function and 1 F 1 is the Kumer function.The asymptotic behavior of R ij is consistent with the fact that the diffuse scattering from independently fluctuating interfaces contains no interference term.In the opposite limit of strong coupling or conformal fluctuations, the three R ij factors become identical, lim for ͉q y ͉ ӷ ⌬q y .͑33͒ For 12 Ӷ 1, the asymptotic form reduces to the singlesurface expression Eq. ͑A4͒ with ⌶͑q xy − q xy Ј ͒ = ⌬q y ␦͑q y − qЈ y ͒ and = 12 .
In Sec.V the measured XDS intensities will be compared with the predicted form based on Eq. ͑30͒ for the partially correlated interfaces as well as the above limiting cases.

D. Specular reflectivity
In general, XR from liquid surfaces is measured with a resolution that is sufficiently coarse as to integrate over the near-singular cusplike peak in the differential cross section at q xy =0 ͑Appendix A͒.In slits-based XR measurements, the resolution is usually closely approximated by the rectangular form ⌬q y 0 otherwise.

· ͑34͒
One method to calculate R ij ͑0,q z ͒ for XR ͓Eq.͑6͔͒ is to follow the same approach as for XDS.Substituting the Fou-rier transform of Eq. ͑34͒ in Eq. ͑13͒ and setting q y =0,

͑35͒
This integration can be carried out numerically using the approximate forms of exp͓− 1 2 g ij ͑r xy ͒q z 2 ͔ in Eqs.͑26͒ and ͑27͒.While the numerical integration is straightforward to implement, this approach has two disadvantages.First, although the effects of various interfacial parameters on reflectivity can in principle be calculated, an intuitive understanding of these effects is difficult to gain using this approach.Second, both the fast oscillation and the slow decay of the integrand for some values of y cause the numerical integration process to be relatively slow and inconvenient for use in the XR data analysis that involves iterative fitting.
The alternative approach that works well for small q z is to approximate the factors R ij ͑0,q z ͒ by Gaussians of Debye-Waller ͑DW͒ form, where each capillary-wave roughness parameter c,ij depends on g ij ͑r xy ͒ and the resolution function.The simplest way to derive the expressions for c,ij is to expand exp͓− 1 2 g ij ͑r xy ͒q z 2 ͔ in Eq. ͑4͒ as a power series and identify c,ij 2 with the coefficient for the q z 2 term, leading to The mean-square roughness ͗h i 2 ͑0͒͘ is given by Eq. ͑22͒, and q xy Ј integration can be carried out analytically for simple resolution functions using the mean-square Fourier amplitudes in Eqs.͑18͒ and ͑19͒.For the rectangular resolution ͓Eq.͑34͔͒, the resulting expressions for c,ij 2 in the limit g → ϱ are given by where

͑40͒
Using these DW factors and the intrinsic form factors ⌰ i ͑q z ͒ in Eq. ͑3͒, Eq. ͑6͒ reduces to where the effective total roughness i at each interface is defined as ͓31,34,62͔

͑42͒
The first square term in Eq. ͑41͒ corresponds to the reflectivity that would be obtained if ⌬ c,12 2 = 0, or equivalently, if the film thickness did not fluctuate ( i.e., → q max −1 or ͓͗␦D͑0͔͒ 2 ͘ → 0).The second, oscillatory term arises from fluctuations in the thickness of the fluid layer.
Some features of the capillary roughness c,ij ͓Eqs.͑38͒-͑40͔͒ should be emphasized.First, c,ij depends weakly ͑logarithmically͒ on q z through the resolution ⌬q y Ϸ͑q z /2͒⌬␤, just as for c for the simple liquid surface ͑Appendix A͒.Second, c,ij depends on the film thickness D through the correlation length ϰ D 2 .Third, c,ij reduces to the expected single-interface forms in the extreme cases of interfacial coupling.In the limit of weak coupling or independently fluctuating interfaces, the limiting forms recover the corresponding single-interface expressions for c 2 and ͗h 2 ͑0͒͘ ͓Eqs.͑A6͒ and ͑A7͔͒, lim In the opposite limit of strong coupling or conformal tuations, lim for all ij, and ⌬ c,12 → 0 as → 0, consistent with ͗␦D͑r xy ͒␦D͑0͒͘ =0.
The two methods described above for calculating reflectivity are compared quantitatively in Appendix B. The use of the analytic approximation based on the DW factors ͓Eqs.͑38͒-͑42͔͒ is shown to be valid so long as q z is sufficiently small that the exponents in Eq. ͑28͒ satisfy i Ͻ 1, corresponding to q z Ͻ 0.3 Å −1 for the PFMC wetting films on C20.The calculations of R ij ͑0,q z ͒ for two nonrectangular resolution functions ͑Gaussian and error-function-based͒ are described in Appendix C.

IV. EXPERIMENTAL DETAILS
A. X-ray measurements X-ray experiments were carried out at Beamline X22B of National Synchrotron Light Source, using the Harvard/BNL liquid surface spectrometer operated at an x-ray wavelength of = 1.56 Å.The scattered intensities were measured using a NaI scintillation detector.Two sets of crossed Huber slits were placed between the sample and the detector, one set located at S 1 = 233 mm and the other ͑detector slits͒ at S 2 = 700 mm from the sample center.The height and width of the slit opening at S i are denoted as ͑H i , W i ͒ below.
XDS was measured using a rocking scan method, in which intensities scattered in the incidence plane ͑ =0͒ are measured as a function of the output angle ␤ while keeping the surface-normal component q z of the wave vector ͑or the sum ␣ + ␤͒ fixed.The background intensities were measured at offsets of b = ± 0.3°, and the background-subtracted intensity was normalized to the incident beam to obtain the normalized intensity I͑␣ , ␤͒ / I 0 .The detector slits were set to ͑H 2 , W 2 ͒ = ͑0.3mm, 3.0 mm͒, corresponding to angular resolutions of ⌬␤ = 0.025°and ⌬ = 0.25°.The equivalent q-space resolutions are given by ⌬q x = 0.017 Å −1 and ⌬q z = 0.0017 Å −1 , while the q y resolution varied with ␤ as ⌬q y Ϸ k sin͑␤͒⌬␤ = ͑0.0017Å −1 ͒sin͑␤͒.The slits at S 1 were set to ͑H 1 , W 1 ͒ = ͑2.5 mm, 4.0 mm͒.For rocking scans with fixed angular resolutions ⌬␤ and ⌬, sin͑␣͒I͑q y , q z ͒ / I 0 is symmetric with respect to the sign of q y ͓64͔.

B. Materials, wetting chamber, and sample preparation
PFMC ͑Lancaster, 95%͒ and C20 ͑Aldrich, 99%͒ were used as received.The relevant physical parameters for the PFMC film on bulk C20 are listed in Table I ͓65͔.
The wetting chamber, sample preparation, and the method for forming wetting films are the same as in Ref. ͓40͔.The thickness of a wetting film was controlled through the temperature difference ⌬T = T − T res Ͼ 0 between the film and a bulk reservoir of the same liquid ͓10,40,66,67͔, where the chemical potential offset ⌬ ͑Sec.III A͒ is proportional to ⌬T for small ⌬T.The chamber consisted of two concentric temperature-controlled cells, where the hermetically sealed inner copper cell ͑internal diameter 110 mm, height 100 mm, temperature stability ±0.001 K͒ contained the sample ͑PFMC film on bulk C20͒ in the middle and a reservoir of wetting liquid ͑PFMC͒ at the bottom.A circular copper disk ͑2 in.diam, 3 mm thick͒ for holding the liquid subphase ͑C20͒ was placed flat on a copper platform such that the liquid surface was approximately at 50 mm above the cell bottom.The platform rested on copper legs with Teflon spac-ers inserted in between.Two thermistors ͑YSI, 30 k⍀͒ that had been calibrated against each other were used to monitor the temperature of the liquid reservoir and the sample platform.The temperature offset ⌬T was controlled via an adhesive-free Kapton heater sandwiched between two plates of the sample platform assembly.
Prior to each sample preparation, the inner cell and the inside parts were wiped and/or rinsed thoroughly with chloroform and methanol, soaked in warm methanol, and dried on a heating plate.The circular copper disk for holding the sample was cleaned by wiping with methanol and soaking for 3 min in an acidic solution of H 2 SO 4 ͑20% by volume͒ and HNO 3 ͑10%͒, followed by a thorough rinse with Millipore water and drying on a heating plate.After the assembly of the inner cell, the sample platform was heated above the bulk melting point ͑ϳ35.7 °C ͓63͔͒ of C20 and a few milligrams of C20 were spread uniformly over the copper disk using a spatula.The cell was then sealed without PFMC and the entire chamber was mounted on the liquid surface spectrometer.The cleanliness of the pure C20 surface was ensured by taking XR scans around the surface freezing temperature ͑ϳ38.7 °C͒ and checking that the surface freezing property of C20 ͓63͔ was reproducible.
For the formation of a wetting film, a reservoir of PFMC was introduced by opening a small screw hole ͑diameter 1.5 mm͒ on the cell lid, injecting 3 mL of PFMC to the cell bottom via a Hamilton syringe, and immediately resealing the hole.After setting the temperatures for given film temperature and offset ⌬T, the sample was left undisturbed until the temperatures stabilized, typically for 1 -2 h, prior to starting x-ray measurements.At each temperature setting, XR measurements were repeated multiple times over a few hours to ensure that the wetting film was stable.With this setup, it was possible to obtain stable PFMC films up to D ϳ 160 Å ͓40͔.

A. Specular reflectivity
Representative XR data from PFMC wetting films on C20 at 37 and 42 °C are shown as R͑q z ͒ / R F ͑q z ͒ in Figs.4͑a͒ and  4͑b͒, respectively.The shown data sets represent films of different thicknesses ͑D = 37 to 142 Å͒, obtained at different temperature offsets ⌬T between the C20 subphase and the PFMC reservoir.The period ͑in q z ͒ of the interference ͑Kies-sig͒ fringe is a direct measure of the inverse thickness D −1 , and the topmost ͑bottommost͒ data set in each figure corresponds to the thinnest ͑thickest͒ film, respectively.The 1 / 3 power-law behavior D ϳ͑⌬T͒ −1/3 of PFMC wetting films on C20, which signifies complete wetting, has already been discussed in Ref. ͓40͔.The primary goal of the present XR study is to extract the interfacial structures and their dependence on the interfacial separation D.
Since the kinematic Born approximation ͑BA͒ is not accurate for q z / q c Շ 5 ͓31͔ ͑q c = 0.0195 Å −1 for bulk C20͒, the analysis of the XR data has been carried out using different formalisms for q z ജ 0.1 Å −1 ͑ϳ5q c ͒ and for q z Ͻ 0.1 Å −1 .For q z ജ 0.1 Å −1 , the fitting curve was calculated by using the BA expression based on the DW factors ͓Eq.͑41͔͒.As shown in Appendix B, this BA form should be accurate for the PFMC-C20 system up to q z ϳ 0.3 Å −1 , i.e., the highest q z probed in the present measurement.In the fitting, the four parameters D, 1 , 1 , and 2 were varied.The factor ⌬ c,12 ͑→0 as q z → 0͒ in the correlation correction term in Eq. ͑41͒ was calculated from Eq. ͑40͒, using the known resolution ⌬q y = ͑q z /2͒⌬␤, the appropriate correlation length for given D ͓Eq.͑20͔͒, and the parameters in Table I.
For q z Ͻ 0.1 Å −1 , the reflectivity was estimated by a combination of the Parratt formalism ͓1,68,69͔ and the BA.For the lowest values q z Ͻ 0.06 Å −1 , the fitting curve was calculated by replacing the first ͑square͒ term in Eq. ͑41͒ with the R / R F based on the matrix method of the Parratt formalism ͓69͔.For this calculation, a single-layer electron density profile ͑"box model;" see, e.g., Refs.͓48,50,66͔͒ was first constructed from the same four parameters ͑D, 1 , 1 , and 2 ͒ as used in the BA; the profile was then divided into a stack of constant-density, 0.5-Å-thick slabs, and the exact reflectivity was calculated for this set of slabs.Unfortunately, there is a systematic error in the BA analysis that is due to neglecting refraction effects and is responsible for a small difference between the Parratt and BA forms ͓70͔.This difference is not FIG.4. Measured Fresnel-normalized specular reflectivity R / R F vs q z ͑symbols͒, obtained from PFMC wetting films on bulk C20: ͑a͒ at 37.2 °C and thickness D = 37, 51, 87, and 142 Å ͑top to bottom͒; and ͑b͒ at 42.2 °C and D = 38, 66, 97, and 142 Å ͑top to bottom͒.Each data set is accompanied by the best-fit R / R F curves based on fit 1 ͑the gray line͒ and fit 2 ͑the black line͒ of the XR analysis; the two fits are almost identical except for the slight differences at high q z .physically important, and in order to provide a smooth transition in the region 0.06Ͻ q z Ͻ 0.1 Å −1 , the square term in Eq. ͑41͒ was replaced by a weighted sum of the Parratt and BA forms, where the relative weight of the BA contribution was increased linearly from 0 to 1 over this q z range.
The theoretical R / R F curve based on these high-q z , lowq z , and intermediate forms was fitted simultaneously to the observed data in the range q z Ͼ 0.03 Å −1 .For all q z Ͼ 0, the calculated R / R F contained the BA-based correlation correction term ͓Eq.͑41͔͒, which vanishes at q z = 0.The actual fitting was carried out using two different types of parametrizations.The two methods, denoted hereafter as fit 1 and fit 2, differ in whether the total roughnesses i ͓Eq.͑42͔͒ or the intrinsic widths w i are treated as fitting parameters.Both approaches produce best fits of similarly good quality.

Fit 1: Effective total roughness i
In fit 1, the weak q z dependence of i ͑through c,ii ͒ is completely neglected, and i are simply treated as fitting parameters.Thus, the four directly varied parameters are D, 1 , 1 , and 2 .The best-fit R / R F curves obtained by fit 1 are indicated by the gray lines in Fig. 4, and the best-fit parameters for the shown curves are listed in Table II.The presence of a nearly immiscible PFMC wetting film on C20 is supported by the fact that the extracted values for the relative electron density 1 of the layer agree, within 7% or less, with the expected nominal value of 1 = PFMC / C20 ϳ 1.9.The best-fit effective total roughness i are plotted as a function of the film thickness D in Fig. 5, where the upper data set corresponds to the liquid-liquid interface ͑ 1 ͒ and the lower set to the liquid-vapor interface ͑ 2 ͒.The observed relation 1 Ͼ 2 is consistent with the lower surface tension ͑␥ 1 Ͻ ␥ 2 ͒ of the liquid-liquid interface as compared to that of the liquid-vapor interface.
For comparison, Fig. 5 also shows three different sets of theoretical curves for the capillary wave roughness c,ii .The black solid curves correspond to the expected case of coupled interfacial fluctuations mediated by vdW interactions ͓Eq.͑38͔͒.The two horizontal solid lines in gray correspond to the limit of weak interfacial coupling, i.e., two independently fluctuating interfaces ͓Eq.͑43͔͒.The gray dashed line on the bottom describes the limit of strong coupling, in which two interfaces fluctuate together and obey perfect conformality so that c,11 ͑conf.͒= c,22 ͑conf.͓͒Eq.͑45͔͒.All these curves were calculated by using the parameters in Table I and assuming the resolution of ⌬q y = 3.2 ϫ 10 −4 Å −1 at q z = 0.15 Å −1 .The upper wave-vector cutoff, on which c,ii 2 depends logarithmically ͑Sec.III D͒, was fixed at q max = / a = 0.5 Å −1 , on the basis of a ϳ 6 Å for the diameter of the PFMC molecule ͓66͔.This value is also close to the published value of q max = 0.44± 0.06 Å −1 for the free surface of bulk C20 ͓34͔.In Fig. 5, the capillary roughness c,ii ͑vdW͒ for the vdW-coupled fluctuations displays the expected D-dependent behavior ͓12,15,16,37,66͔.At small D, c,ii ͑vdW͒ is partially quenched below the value c,ii ͑indep.͒for the independent fluctuations due to the close proximity of the two interfaces; however, as D increases, c,ii ͑vdW͒ increases and saturates to c,ii ͑indep.͒at large D.
On the basis of Fig. 5, the following inferences can be made.͑i͒ The observed total roughness i at each interface is dominated by the capillary roughness c,ii .͑ii͒ While the possibility of perfectly conformal fluctuations is clearly ruled out, the measured roughnesses i are of the same order of magnitude as both c,ii ͑vdW͒ for the vdW-coupled fluctuations and c,ii ͑indep.͒for the independent fluctuations.In view of this, the XR result here is not sufficient justification for discriminating unambiguously between the coupled and 2 as a function of PFMC film thickness D at 37.2 °C ͑᭺͒ and 42.2 °C ͑b͒, representing the best-fit values extracted from fit 1 of the XR analysis.The upper data set corresponds to 1 for the C20-PFMC ͑liquid-liquid͒ interface and the lower set to 2 for the PFMC-vapor ͑liquid-vapor͒ interface.The lines describe the theoretical effective capillary roughness c,ii for three different scenarios of interfacial coupling, calculated using the physical parameters in Table I and a fixed resolution of ⌬q y = ͑q z /2͒⌬␤ = 3.2ϫ 10 −4 Å −1 at a midpoint q z = 0.15 Å −1 of the XR measurement.The black solid lines that vary with D are based on Eq. ͑38͒ and correspond to the expected coupling of interfacial fluctuations that is mediated by ͑nonretarded͒ vdW interactions.The two horizontal solid lines in gray describe the limit of weak coupling or independent fluctuations ͓Eq.͑43͔͒.The bottommost gray dashed line represents the limit of strong coupling or perfectly conformal fluctuations ͓Eq.͑45͔͒.independent fluctuations.We will return to this point below when we discuss XDS results.On the other hand, the fact that the measured i remain close to or above c,ii ͑vdW͒ does point toward the plausibility of the vdW-coupled capillary fluctuations and small but non-negligible intrinsic widths w i .͑iii͒ The observed 1 is consistently higher than c,11 ͑vdW͒, although only by small amounts for the smallest D probed.This suggests that the intrinsic width w 1 at the liquid-liquid ͑C20-PFMC͒ interface is nonzero.This point, as well as the following, is better addressed by the second fitting method to be discussed below.͑iv͒ The fact that the observed 2 is fairly close to c,22 ͑vdW͒Ϸ c,22 ͑indep.͒at all D, implies that the intrinsic width w 2 of the liquid-vapor ͑PFMC-vapor͒ interface is smaller than w 1 and remains so for all D.

Fit 2: Intrinsic widths w i
In fit 2, the capillary roughnesses c,ii are assumed to be given by c,ii ͑vdW͒ for the coupled interfacial fluctuations and are calculated exactly using Eq.͑38͒.Therefore, their dependence on q z through the resolution ⌬q y is completely accounted for, and the only unknown in the calculation of c,ii ͑vdW͒ is the film thickness D, which is a fitting parameter.Thus, for fit 2, the total roughnesses i are varied only indirectly, and the four directly varied parameters are D, 1 , and the intrinsic widths w 1 and w 2 .The best-fit R / R F curves are shown as black lines in Fig. 4, and the corresponding best-fit parameters are listed in Table III.The quality of the fits is similar to that for fit 1, except that fit 2 tends to reproduce the data slightly better than fit 1 at high q z .The two fitting methods yield the values for D and 1 that agree within their uncertainties.The best-fit intrinsic widths w i are plotted as a function of D in Fig. 6.
While the uncertainties are relatively large, the behavior of the extracted intrinsic widths w i is consistent with the inferences made earlier in Sec.V A 1. Figure 6͑a͒ indicates that w 1 at the liquid-liquid ͑C20-PFMC͒ interface is nonzero, with the best-fit values in the range 2 -6 Å.The previous XR study of water-alkane interfaces by Mitrinovic et al. ͓38͔ has provided evidence that the intrinsic width of an immiscible, bulk liquid-liquid interface ͑i.e., D → ϱ͒ is determined by either the molecular size or the bulk correlation length associated with the larger of the two molecular components.In particular, the intrinsic width that they extracted for the water-alkane interface was found to agree with the radius of gyration R g of the alkane molecule for short-chain alkanes ͑Cn with n Շ 16͒ and with the bulk correlation length b of liquid alkane for long-chain alkanes ͑n տ 16͒.For similar comparison, Fig. 6͑a͒ shows three horizontal lines representing the gyration radius R g Ϸ 5.3 Å for C20 ͑solid line͒ ͓34͔, an estimated bulk correlation length b Ϸ 4.8 Å for molten long-chain alkanes ͑long dashed line͒ ͓38͔, and the molecular radius a /2Ϸ 3 Å for PFMC ͑short dashed line͒ ͓66͔.Due to the large uncertainties, it is not possible to discern whether the measured width w 1 is in better agreement with R g or b for C20.However, the fact that w 1 observed for large D ͑տ50 Å͒ agrees, within error, with these two lengths associated with C20 is consistent with the expected behavior of the bulk liquid-liquid interface.Another interesting observation is that w 1 for D Շ 50 Å is consistently smaller than R g or b for C20, possibly indicating a reduction in the intrinsic liquid-liquid interfacial width for very thin films.Although the uncertainties associated with this observation are relatively large, it is interesting to note that a similar thicknessdependent or confinement-induced reduction of the intrinsic width at a buried liquid-liquid interface has been observed in thin binary wetting films on a solid substrate ͓66͔.
By contrast, Fig. 6͑b͒ shows that the intrinsic width w 2 of the liquid-vapor interface is always small ͑w 2 =2±1 Å͒ and remains nearly constant as the interfacial separation D is varied.The narrow width of the PFMC-vapor interface is  consistent with the general observation that the intrinsic width of a liquid-vapor interface is usually no more than the size of atoms, as demonstrated previously for the surface of bulk water ͓32,33,46͔, ethanol ͓29͔, and normal alkanes ͓34͔, as well as for the free surface of organic and polymer liquid films on silicon substrates ͓12,16,66͔.

B. Off-specular diffuse scattering
Due to its q y dependence, XDS is better suited than XR for probing the effect of interfacial coupling, which should manifest itself through the lateral correlation length .In order to effectively discriminate between the vdW-coupled and independent fluctuations, the dynamic range ͑in q y ͒ of the XDS measurement must be sufficiently wide and the correlation length ϰ D 2 ͓Eq.͑20͔͒ must be long enough to meet the condition ⌬q y Ӷ −1 Ӷ max͕͉q y ͉͖.This condition is satisfied in the present experiments for the thickest stable films of D ϳ 150 Å ͑ ϳ 4000 Å͒, provided that the fixed q z for a rocking scan is roughly in the range 0.1-0.15Å −1 .The restriction in q z arises for two reasons.First, the range of rocking scans is limited to a maximum value of max͕͉q y ͉͖ ϳ q z 2 /2k, and if q z Ͻ 0.1 Å −1 , the dynamic range of the scans is too small to gain a true measure of the algebraic cusp.Second, the upper limit on q z arises because the XDS intensities are significantly reduced for higher q z .Within this range, the exact value of the fixed q z should be chosen such that the difference in the expected I / I 0 between the two types of fluctuations is maximized.This is best done by comparing rocking scans that are measured at q z for minima and maxima of the Kiessig fringes in R͑q z ͒ / R F ͑q z ͒ ͓i.e., cos͑q z D͒ = ±1͔ ͓22͔.As discussed in Sec.III C, for the independent fluctuations, the R 12 ͑q y , q z ͒ term makes no contribution to I / I 0 at ͉q y ͉ ӷ⌬q y ͓Eq.͑32͔͒.Therefore, observed differences between these scans would provide a direct indication of nonindependent fluctuations.
The results of rocking scans from films at D = 140, 148, and 155 Å are shown as open symbols in the lower panels of Figs.7͑a͒-7͑c͒, respectively, in terms of the normalized intensity sin͑␣͒I͑q y , q z ͒ / I 0 .The q z position of each scan is close to either a maximum or a minimum in R / R F , as indicated in the upper panels that show the XR data taken immediately prior to the XDS scans.The XDS data for q y Ͻ 0 and q y Ͼ 0 are plotted on top of each other and show good overlap between the two sides as expected for fixed-q z scans ͓64͔.Each rocking scan is characterized roughly by three regions: the specular part at small ͉q y ͉ Ͻ⌬q y , the off-specular part at larger ͉q y ͉ Ͼ⌬q y , and a surface enhancement ͑"Yoneda"͒ peak near max͕͉q y ͉͖, which occurs at ␣ or ␤ ϳ ␣ c ͓through T F ͑͒ in Eq. ͑2͔͒ ͓28,29͔.The off-specular scattering at reflectivity maxima ͓᭝ in Figs.7͑a͒ and 7͑c͔͒ falls off monotonically with increasing ͉q y ͉, with a constant slope in the log-log plot.By contrast, the falloff of the offspecular scattering at reflectivity minima ͓᭺ in Figs.7͑a͒ and 7͑b͔͒ appears to change its slope in the log-log plot near ͉q y ͉ = −1 .A similar kink in the XDS rocking scan has been observed for thin organic and polymer liquid films on silicon substrates ͓12,14-16͔.This is a direct demonstration that the correlation length affects the scattering behavior and pro-vides qualitative evidence for the expected presence of vdWcoupled capillary fluctuations at the two interfaces.
For quantitative comparison, the predicted normalized intensities sin͑␣͒I / I 0 are shown as thick, gray solid lines in Fig. 7.The factors R ij ͑q y , q z ͒ for the vdW-coupled fluctuations were calculated by performing the numerical integration in Eq. ͑30͒ for the Gaussian resolution function, using the approximate forms of exp͓− 1 2 g ij ͑r xy ͒q z 2 ͔ in Eqs.͑26͒ and ͑27͒, the physical parameters in Table I, and the experimental resolution ⌬q y = k sin͑␤͒⌬␤ that corresponds to the nominal XDS slit setting ͑Sec.IV A͒.The remaining parameters ͑D, 1 , w 1 , w 2 ͒ were obtained from the fit 2 analysis of the associated R / R F data.As a result, the calculation of I / I 0 involves no adjustable parameters.Figures 7͑b͒ and 7͑c͒ also show the two limiting cases of theoretical off-specular scattering for ͉q y ͉ ӷ⌬q y , where the black solid and dashed lines correspond to the independent and perfectly conformal fluctuations, respectively.These I / I 0 curves are based on the asymptotic forms of R ij ͑q y , q z ͒ in Eqs.͑31͒-͑33͒ and, again, calculated with no adjustable parameters.
It is clear from Fig. 7 that the measured XDS data are consistent with the vdW-coupled capillary fluctuations of the two interfaces.The predicted I / I 0 curves reproduce not only the uniform falloff in the data for q z at the R / R F maxima but also the observed crossover behavior around ͉q y ͉ϳ −1 for q z at the R / R F minima.By contrast, regardless of the q z position, the independent and perfectly conformal fluctuations would produce only a uniform falloff in I / I 0 .Moreover, for q z at the R / R F minima, the independent fluctuations would overestimate I / I 0 for ͉q y ͉ Ͻ −1 , while the conformal fluctuations would underestimate I / I 0 especially at large ͉q y ͉.
Finally, it should be noted that the XDS results presented here support the reliability of the fit 2 analysis of the XR data, from which the intrinsic interfacial widths w i have been extracted by assuming the vdW-coupled capillary fluctuations.

VI. SUMMARY
The thermal capillary fluctuations and the intrinsic widths of two interacting fluid-fluid interfaces have been studied by a combination of x-ray specular reflectivity and off-specular diffuse scattering measurements.The primary objective of this study has been to probe for the expected vdW-dominated coupling of capillary fluctuations at the liquid-liquid and liquid-vapor interfaces of PFMC wetting films on the surface of bulk C20.The analysis of the XR and XDS data indicates that the observed magnitudes and wave-vector dependence of specular and off-specular intensities are consistent with the presence of the vdW-coupled capillary fluctuations, provided that each interface possesses a small but finite intrinsic width w i 0. The presence of the coupled capillary fluctuations is evidenced most convincingly by the fact that the XDS rocking scans taken at q z near R / R F minima display a crossover behavior near ͉q y ͉ϳ −1 , i.e., the expected inverse correlation length for the vdW-mediated film thickness fluctuations.Moreover, we have demonstrated that both this crossover behavior and its absence at R / R F maxima are well reproduced by the theoretically calculated intensity that takes the effects of into account.The intrinsic widths of the two interfaces have been extracted from the XR data analysis by theoretically accounting for the effects of the vdW-coupled capillary fluctuations on R / R F .The intrinsic width of the liquid-liquid ͑C20-PFMC͒ interface for D տ 50 Å is found to be comparable to the value expected for the bulk liquidliquid interface ͑D → ϱ͒, determined by either the gyration radius of the C20 molecule ͑5.3 Å ͓34͔͒ or the bulk correlation length in molten long-chain alkanes ͑4.8 Å ͓38͔͒; for D Շ 50 Å, the width appears to be reduced below these expected bulk values, possibly indicating the effect of confinement.By contrast, the liquid-vapor ͑PFMC-vapor͒ interface has a sharper intrinsic width of w 2 =2±1 Å that is essentially independent of D.
Another major objective of the present study has been to develop a relatively simple and practical approach for estimating the XR and XDS intensities that fully account for both the subtlety of liquid surface scattering and the complexity associated with the partially correlated capillary roughness of the two fluid-fluid interfaces.For this purpose, we presented two different methods for approximating the theoretical intensities and illustrated their practical utility and accuracy by applying them to the data analysis.The first method, which is more useful for XDS than XR, is to make use of approximate behavior of the height-height and thickness-thickness correlation functions.Despite their simplicity, the approximate correlation functions that we used ͓Eqs.͑22͒-͑27͒, Fig. 2͔ capture the basic effects of the coupled capillary fluctuations of the two interfaces.The combination of these correlation functions and the realistic resolution functions that closely represent actual experiments can be used to calculate the normalized intensities I / I 0 for XDS ͑Sec.III C͒ and XR ͑Sec.III D͒.The second approximation method, discussed in Sec.III D, results in a simple analytic form of R / R F ͓Eq. ͑41͔͒ that is based on Debye-Waller factors and effective capillary roughnesses ͓Eqs.͑38͒-͑40͔͒.This approximate form of R / R F properly accounts for the effects of both the resolution and the coupled FIG. 7. Demonstration of the vdW-dominated coupling of capillary fluctuations at the liquid-liquid and liquid-vapor interfaces.The lower panels show measured off-specular diffuse rocking scans in the incidence plane ͑q x =0, =0͒, from PFMC wetting films on bulk C20 at 42.2 °C: ͑a͒ D = 140 Å ͑ = 3300 Å͒ at q z = 0.11 Å −1 ͑᭝͒ and 0.14 Å −1 ͑᭺͒; ͑b͒ D = 148 Å ͑ = 3700 Å͒ at q z = 0.13 Å −1 ͑᭺͒; and ͑c͒ D = 155 Å ͑ = 4100 Å͒ at q z = 0.10 Å −1 ͑᭝͒.The data for both q y Ͼ 0 and q y Ͻ 0 are plotted on top of each other in terms of the normalized intensity sin͑␣͒I / I 0 vs ͉q y ͉.The vertical dotted line indicates the position of the expected inverse correlation length −1 .For each D, the upper panel shows the associated R / R F data ͑b͒ obtained immediately before the diffuse scan, together with the corresponding best-fit curve based on fit 2 ͑solid curve͒.The fixed q z position of each rocking scan is close to either a maximum or a minimum in R / R F , as indicated in the upper panels.In the lower panels, the thick solid lines in gray represent the theoretical sin͑␣͒I / I 0 for the expected vdW-mediated coupling of interfacial fluctuations.The calculation, which involves no adjustable parameters, is based on the expressions for I / I 0 and ⌰ i ͑q z ͒ in Eqs.͑2͒ and ͑3͒; the factors R ij ͑q y , q z ͒ in Eq. ͑30͒ for the Gaussian resolution function with the experimental ⌬q y = k sin͑␤͒⌬␤; the approximate correlation functions in Eqs.͑26͒ and ͑27͒; the physical parameters in Table I; and the best-fit parameters ͑D, 1 , w 1 , and w 2 ͒ from the fit 2 analysis of the XR data.In ͑b͒ and ͑c͒, the black solid lines represent the limit of weak coupling or independent interfacial fluctuations ͓via the asymptotic forms of R ij ͑q y , q z ͒ in Eqs.͑31͒ and ͑32͔͒; the black dashed lines are for the limit of strong coupling or perfectly conformal fluctuations ͓via the asymptotic form of R ij ͑q y , q z ͒ in Eq. ͑33͔͒.
capillary fluctuations of the two interfaces.Its use in the XR analysis should be valid for q z / q c տ 5 so long as both of the two exponents satisfy i = ͑k B T /2␥ i ͒q z 2 Ͻ 1 ͑Appendix B, Fig. 8͒.The various expressions that we presented for the correlation functions, the scattering intensities and the effective capillary roughness all reduce to the proper forms in the limit of weak and strong interfacial coupling, corresponding to the limits of independent and perfectly conformal fluctuations.

͑A1͒
This decomposition is experimentally possible provided that the height-height correlations are short-ranged and the inverse correlation length −1 is significantly greater than the typical resolution q xy,min ͑ϳ⌬q y =10 −5 -10 −4 Å −1 ͒.For −1 ӷ q xy,min , the surface diffuse scattering intensity, represented by the second ͑integral͒ term in Eq. ͑A1͒, is usually both small and smoothly varying at q xy = 0 and can easily be separated empirically from the specular reflection ͓the first term in Eq. ͑A1͔͒ ͓71͔.Thus, with ⌶͑q xy =0͒ = 1, the principal effect of surface roughness is to reduce the magnitude of specular reflectivity R͑q z ͒ from the flat surface values through Debye-Waller ͑DW͒ factors,

͑A2͒
By contrast, the decomposition into true specular and diffuse terms in Eq. ͑A1͒ cannot be directly applied to liquid surfaces because the presence of thermal capillary waves causes the height-height correlations to be quasi-long-ranged ͓29,35,72͔.For simple surfaces with either a single interface or perfectly conformal roughness, the characteristic features of liquid surface scattering can be summarized as follows ͓30͔.For given surface tension ␥ and liquid mass density m , the height-height correlations are limited by the gravitational capillary length g = ͱ ␥ / m g, which is of the order of mm for typical liquids, i.e., g −1 Ӷ q xy,min ͓73͔.Moreover, because of the logarithmic behavior of g͑r xy ͒ϵ͓͗h͑r xy ͒ − h͑0͔͒ 2 ͘ ϳ 2͑k B T /2␥͒ln͑r xy ͒ for q max −1 ͑ϳÅ͒ Ӷ r xy Ӷ g ͑ϳmm͒ ͓28,30͔, thermal diffuse scattering ͑TDS͒ from liquid surfaces displays an algebraic, cusplike singularity d / d⍀ ϳ 1/q xy 2− with ϵ͑k B T /2␥͒q z 2 ͓28,30͔ that is only truncated at q xy ϳ g −1 .Since the resolution widths ͑⌬q x , ⌬q y ͒ of virtually all practical x-ray reflectometers are large compared to g −1 , distances of order g cannot be resolved ͓31,71,73͔.Consequently, for liquid surfaces there is no practical way to make a simple distinction between the true specular signal and the small-angle diffuse scattering that is sharply peaked at q xy =0.
The above problem can be circumvented by recognizing that it is not necessary to consider the detailed behavior of g͑r xy ͒ for the large values of r xy ͑ӷ1/q xy,min ϳ 1/⌬q y ͒ that cannot be resolved experimentally ͓30,46,47͔.One approach that has proven to be successful is ͑i͒ to assume the singular form d / d⍀ ϳ N / q xy 2− over the entire range 0 Ͻ q xy Ͻ q max by taking the limit g → ϱ; and ͑ii͒ to obtain the proper normalization factor N by applying an appropriate sum rule.
Step ͑ii͒ is equivalent to setting The resulting single-interface analogue of R ij is approximately given by ͓30,46,47͔ This single expression applies to both XR ͑q xy =0͒ and XDS ͑q xy 0͒.The underlying assumption is that for all practical purposes, the measured intensity, whether specular or offspecular, corresponds to the integration of TDS over the resolution.For q xy that are small compared to the reciprocal of the molecular lengths, this assumption follows from general hydrodynamic considerations ͓74͔.Empirically, the quantitative accuracy of Eq. ͑A4͒ is supported by the XR/ XDS measurements on various simple, laterally uniform liquid surfaces ͓46-51͔.
One important implication of Eq. ͑A4͒ is that even for XR, R 1 ͑0,q z ͒ is resolution-dependent.Nevertheless, as shown previously, it is often possible to make a DW-like approximation with an effective capillary-wave roughness c ͓30,31,33,47͔,

͑A5͒
In contrast to the solid case ͓Eq.͑A2͔͒, however, the measured roughness c for liquids depends on the resolution ͓31,33,34,38͔ through ⌶͑q xy ͒ in Eq. ͑A4͒.For the resolution function that is rectangular in q y and infinitely wide along q x ͓Eq.͑34͔͒, Eq. ͑A5͒ can be shown to be correct to order q z 2 with ͓31,34,38͔

͑A6͒
This apparent roughness c for liquid surfaces is smaller than the root-mean-square amplitude ͗h 2 ͑0͒͘ 1/2 of height fluctuations, where for capillary waves on bulk liquid surfaces ͑⌬q y ӷ g −1 ͒ ͓31,36͔.Moreover, the relation ⌬q y Ϸ k sin͑␤͒⌬␤ = ͑q z /2͒⌬␤ at the specular condition ͑␣ = ␤͒ implies that for typical XR measurements with fixed angular resolution ⌬␤, c should in principle display a weak logarithmic dependence on q z through ⌬q y ͓31,34͔.
In order to extract the intrinsic profile of liquid surfaces from XR, these subtle effects of small-angle TDS on the apparent measured reflectivity need to be properly accounted for ͓30͔.This requires prior knowledge of parameters such as the experimental resolutions, T and ␥.If one of these parameters is unknown or involves large uncertainties, simultaneous measurements of XR and XDS are necessary in order to quantitatively separate the intrinsic form factor ͉⌰͑q z ͉͒ 2 from the DW factor R 1 ͑0,q z ͒.In Sec.III approximations similar to those that give rise to Eqs. ͑A4͒-͑A6͒ are developed for liquid surfaces with two interacting fluid-fluid interfaces.

APPENDIX B: APPLICABILITY OF ANALYTIC APPROXIMATION FOR XR
The two methods described in Sec.III D for calculating reflectivity are compared quantitatively in Fig. 8.In the plot of R / R F in Fig. 8͑a͒, the open circles represent the numerical integration method ͓Eq.͑6͒ with Eq. ͑35͔͒, and the solid line corresponds to the analytic small-q z approximation with the DW factors ͓Eq.͑41͒ with Eqs.͑38͒-͑40͔͒.The calculations are based on the parameters for a PFMC film on C20 and the resolution used in the actual XR measurements.The period of the interference ͑Kiessig͒ fringes is a measure of the layer thickness ͑D = 100 Å͒, and the decay rates for both the overall magnitude and the oscillation amplitude depend on the various height fluctuations.The solid lines in Figs.8͑b͒-8͑e͒ represent the relative differences, ⌬R / R or ⌬R ij / R ij , between the numerical integration and the DW method, where each difference has been normalized to the numerically integrated value; the dashed curves in Figs.8͑b͒ and 8͑e͒ show the same except that we set ⌬ c,12 = 0 for the small-q z approximation.Also plotted in Figs.8͑c͒-8͑e͒ ͑dotted lines͒ is the q z dependence of the exponents i and 12 ͓Eq.͑28͔͒.The inequality 1 Ͼ 2 Ͼ 12 at any given q z originates from the fact that the surface tension of the liquid-liquid interface ͓i = 1, Fig. 1͑b͔͒ is lower than that of the liquid-vapor interface ͑i =2͒, i.e., ␥ 1 Ͻ ␥ 2 Ͻ ␥ 1 + ␥ 2 ͑Table I͒.
According to Fig. 8, the DW representation of R / R F ͓Eq. ͑41͔͒ should provide an accurate description of the reflectiv-ity for the PFMC film on C20 up to q z ϳ 0.3 Å −1 ͑⌬R / R ϳ 5%͒, provided that the ⌬ c,12 -dependent term is properly accounted for.If this term were neglected, the error in the oscillation amplitude of R / R F would become significant at even smaller q z ͓Figs.8͑b͒ and 8͑e͔͒.Another more general observation is that the DW representation of R ij ͑0,q z ͒ is ac-FIG.8. ͑a͒ Theoretical Fresnel-normalized specular reflectivity R / R F vs q z , calculated using the parameters in Table I for a PFMC wetting film on C20 at 37 °C and D = 100 Å, the rectangular FWHM resolution ⌬q y = ͑q z /2͒⌬␤ of the XR experiment, and zero intrinsic widths w i = 0 for simplicity.The symbols ͑᭺͒ represent the calculation in which the factors R ij ͑0,q z ͒ are evaluated through numerical integration ͓Eqs.͑6͒ and ͑35͔͒.The solid curve is based on the analytic form of R / R F ͓Eq. ͑41͔͒ for which R ij ͑0,q z ͒ are approximated by Debye-Waller ͑DW͒ factors with effective capillary roughness parameters c,ii ͓Eqs.͑36͒ and ͑38͒-͑40͔͒; the calculation accounts for the interfacial correlation-correction term ͑⌬ c,12 0͒ in Eq. ͑41͒.The relative difference between the numerical integration method and the DW factor method is shown as the solid curve for ͑b͒ R͑q z ͒, ͑c͒ R 11 ͑0,q z ͒, ͑d͒ R 22 ͑0,q z ͒, and ͑e͒ R 12 ͑0,q z ͒.The dashed curves in ͑b͒ and ͑e͒ represent, respectively, the relative difference in R and in R 12 that would result if the correlation correction for the DW approach were neglected, i.e., if ⌬ c,12 = 0.The dotted curves in ͑c͒, ͑d͒, and ͑e͒ show the q z dependence of exponents 1 , 2 , and 12 , respectively ͓Eq.͑28͔͒.
curate so long as the associated exponent ͑= i or 12 ͒ Ӷ1.In particular, the condition i ϳ 1 for the larger of the two exponents i can be used to set an upper limit on q z above which the DW approximation of R / R F should be considered unreliable ͓see Figs.8͑b͒ and 8͑c͔͒.In this regard, it is to be noted that for the TDS from the simple liquid surface ͑Appendix A͒, the integration of d / d⍀ ϳ 1/q xy 2− over the infinitely wide resolution in q x causes the scattered intensity to display a singularity of form ϳ1/q y 1− ͓29,71͔.As pointed out previously ͓30,49,51͔, this singularity vanishes as → 1, and the distinction between surface and bulk diffuse scattering becomes meaningless for Ͼ 1.

APPENDIX C: XR FOR GAUSSIAN AND ERROR-FUNCTION-BASED RESOLUTION FUNCTIONS
In Sec.III D specular reflectivity is calculated using the rectangular resolution function ͓see Eqs.͑34͒ and ͑35͔͒, due to the relatively wide detector resolution used in the present XR measurements.We summarize here the results of equivalent calculations for two other resolution functions that may be more appropriate for narrower resolutions than used in this study.As before, the conditions q max −1 Ӷ Ӷ g and ⌬q y ӷ g −1 are assumed to hold below.
For the Gaussian resolution function ͓Eq.͑29͔͒, the factors R ij ͑0,q z ͒ for XR can be obtained by setting q y = 0 in Eq. ͑30͒ and using the approximate correlation functions of form Eqs. ͑26͒ and ͑27͒.The results can be expressed in terms of analytic functions as follows: R ii ͑0,q z ͒ Ϸ erf ͩ ⌬q y /q max 2 ͱ ͪ + ⌬q y /q max 2 ͭ ⌫ͩ 1 − i where ⌫͑␣ , x͒ is the ͑upper͒ incomplete Gamma function.
Note that the q z dependence of R ij ͑0,q z ͒ is buried in the parameters ͕ 1 , 2 , 12 , ⌬q y ͖.

͑C4͒
In the limit of ␦ → 0, Eqs.͑C3͒ and ͑C4͒ reduce to the forms for the rectangular resolution, i.e., Eqs.͑34͒ and ͑35͒, respectively.The integration in Eq. ͑C4͒ can be carried out numerically and should converge faster than for the rectangular resolution due to the presence of the factor e −␦ 2 y 2 /2 in the integrand.
It should be noted that the above expressions for R ij ͑0,q z ͒ are not particularly suited for understanding the qualitative behavior of specular reflectivity and its dependence on various parameters.This disadvantage can be overcome for small q z by approximating Eqs.͑C1͒, ͑C2͒, and ͑C4͒ with the Debye-Waller factors, as discussed in Sec.III D.

FIG. 1 .
FIG. 1. ͑a͒ X-ray scattering geometry.͑b͒ Illustration of a wetting film of average thickness D on the surface of a bulk liquid.The liquid-liquid and liquid-vapor interfaces are denoted with indices 1 and 2, respectively.The local fluctuation in thickness is equal to the difference in the height fluctuations of the two interfaces at the same lateral position.
= 100 Å.The essential properties of the correlation functions are captured by the above limiting forms, including the basic features of ͗h i ͑r xy ͒h j ͑0͒͘ that have been discussed previously by Daillant et al. ͓24͔.In addition, it is shown explicitly that sets the lateral correlation length for thickness fluctuations ͓͗␦D͑r xy ͒␦D͑0͒͘, Fig. 2͑b͔͒ and g sets the correlation length for height fluctuations ͓͗h i ͑r xy ͒h j ͑0͒͘, Fig. 2͑a͔͒.The correlation functions always display logarithmic dependence on r xy over the ranges of r xy where they vary; however, the behavior of each correlation function is different above and below .For r xy Ӷ , in addition to the common constant term in all three functions ͗h i ͑r xy ͒h j ͑0͒͘ ͓Eq.͑22͒ and Fig. 2͑a͔͒, each self-correlation function ͑i = j͒ contains a logarithmic term that depends only on the surface tension ␥ i of the relevant interface.Therefore, both the independent and conformal fluctuations contribute to the self-correlations ͗h i ͑r xy ͒h i ͑0͒͘ 0, whereas the cross-correlations ͗h 1 ͑r xy ͒h 2 ͑0͒͘ 0 originate solely from the conformal part.The presence of independent height fluctuations is reflected in the different growth rates of g 11 ͑r xy ͒ and g 22 ͑r xy ͒ for r xy Ͻ and in the r xy dependence of the nonconformal thickness fluctuations ͗␦D͑r xy ͒␦D͑0͒͘ 0. For Ӷ r xy Ӷ g , the fluctuations are completely conformal ͓͗␦D͑r xy ͒␦D͑0͒͘ =0͔; all three functions ͗h i ͑r xy ͒h j ͑0͒͘ become identical and follow a common logarithmic falloff with r xy that depends on the effective surface tension ␥ eff = ␥ 1 + ␥ 2 .For r xy ӷ g , the height fluctuations become uncorrelated, which necessarily implies ͗␦D͑r xy ͒␦D͑0͒͘ =0.

FIG. 6 .
FIG. 6. Intrinsic or local width ͑a͒ w 1 of the C20-PFMC ͑liquid-liquid͒ interface and ͑b͒ w 2 of the PFMC-vapor ͑liquid-vapor͒ interface, as a function of PFMC film thickness D at 37.2 °C ͑᭺͒ and 42.2 °C ͑b͒.The symbols represent the best-fit values extracted from fit 2 of the XR analysis.In ͑a͒, the gray horizontal lines correspond to the radius of gyration R g Ϸ 5.3 Å for C20 ͑solid line͒ ͓34͔, the bulk correlation length b Ϸ 4.8 Å for long-chain alkanes ͑long dashed line͒ ͓38͔, and the molecular radius a /2ϳ 3 Å for PFMC ͑short dashed line͒ ͓66͔.
xy e iq xy •r xy h i ͑r xy ͒,

TABLE I .
Values of the parameters relevant to the present experiment.

TABLE II .
List of best-fit parameters, based on fit 1, for fitting to the XR data in Fig.4.

TABLE III .
List of best-fit parameters, based on fit 2, for fitting to the XR data in Fig.4.