Probing the Universal Critical-Adsorption Profile by Neutron Reflectometry

Generalized skew-symmetric interfacial probability distribution in reflectivity and small-angle scattering analysis

Generalized skew-symmetric probability density functions are proposed to model asymmetric interfacial density distributions for the parameterization of any arbitrary density profiles in the `effective-density model'. The penetration of the densities into adjacent layers can be selectively controlled and parameterized. A continuous density profile is generated and discretized into many independent slices of very thin thickness with constant density values and sharp interfaces. The discretized profile can be used to calculate reflectivitiesviaParratt's recursive formula, or small-angle scatteringviathe concentric onion model that is also developed in this work.

Critical adsorption profiles around a sphere and a cylinder in a fluid at criticality: Local functional theory

We study universal critical adsorption on a solid sphere and a solid cylinder in a fluid at bulk criticality, where preferential adsorption occurs. We use a local functional theory proposed by Fisher et al. [M. E. Fisher and P. G. de Gennes, C. R. Acad. Sci. Paris Ser. B 287, 207 (1978); M. E. Fisher and H. Au-Yang, Physica A 101, 255 (1980)PHYADX0378-437110.1016/0378-4371(80)90112-0]. We calculate the mean order parameter profile ψ(r), where r is the distance from the sphere center and the cylinder axis, respectively. The resultant differential equation for ψ(r) is solved exactly around a sphere and numerically around a cylinder. A strong adsorption regime is realized except for very small surface field h_{1}, where the surface order parameter ψ(a) is determined by h_{1} and is independent of the radius a. If r considerably exceeds a, ψ(r) decays as r^{-(1+η)} for a sphere and r^{-(1+η)/2} for a cylinder in three dimensions, where η is the critical exponent in the order parameter correlation at bulk criticality.

Time-of-flight Extreme Environment Diffractometer at the Helmholtz-Zentrum Berlin

The Extreme Environment Diffractometer (EXED) is a new neutron time-of-flight instrument at the BER II research reactor at the Helmholtz-Zentrum Berlin, Germany. Although EXED is a special-purpose instrument, its early construction made it available for users as a general-purpose diffractometer. In this respect, EXED became one of the rare examples, where the performance of a time-of-flight diffractometer at a continuous source can be characterized. In this paper, we report on the design and performance of EXED with an emphasis on the unique instrument capabilities. The latter comprise variable wavelength resolution and wavelength band, control of the incoming beam divergence, the possibility to change the angular positions of detectors and their distance to the sample, and use of event recording and offline histogramming. These features combined make EXED easily tunable to the requirements of a particular problem, from conventional diffraction to small angle neutron scattering. The instrument performance is demonstrated by several reference measurements and user experiments.

The relationship between charge density and polyelectrolyte brush profile using simultaneous neutron reflectivity and in situ attenuated total internal reflection FTIR

We report on a novel experimental study of a pH-responsive polyelectrolyte brush at the silicon/D2O interface. A poly[2-(diethylamino)ethyl methacrylate] brush was grown on a large silicon crystal which acted as both a substrate for a neutron reflectivity solid/liquid experiment but also as an FTIR-ATR spectroscopy crystal. This arrangement has allowed for both neutron reflectivities and FTIR spectroscopic information to be measured in parallel. The chosen polybase brush shows strong IR bands which can be assigned to the N-D(+) stretch, D2O, and a carbonyl group. From such FTIR data, we are able to closely monitor the degree of protonation along the polymer chain as well as revealing information concerning the D2O concentration at the interface. The neutron reflectivity data allows us to determine the physical brush profile normal to the solid/liquid interface along with the corresponding degree of hydration. This combined approach makes it possible to quantify the charge on a polymer brush alongside the morphology adopted by the polymer chains.

Effect of surface curvature on critical adsorption

We studied critical adsorption on curved surfaces by utilizing spherical nanoparticles immersed in a critical binary liquid mixture of 2,6 lutidine+water. The temperature dependence of the adsorbed film thickness and excess adsorption was determined from fluorescence correlation spectroscopy measurements of the enlarged effective hydrodynamic radius of the particles. Our results indicated that the adsorbed film thickness is of the order of correlation length associated with concentration fluctuations. The excess adsorption per unit area increases following a power law in reduced temperature with an exponent of -1, which is the mean-field value for the bulk susceptibility exponent. This has been confirmed with silica particles of two different radii, 10 and 25 nm. The results were also compared with the theoretical mean-field scaling function.

Comparison of critical adsorption scaling functions obtained from neutron reflectometry and ellipsometry

Carpenter et al. [Phys. Rev. E 59, 5655 (1999); 61, 532 (2000)] managed to explain ellipsometric critical adsorption data collected from the liquid-vapor interface of four different critical binary liquid mixtures near their demixing critical temperature using a single model. This was the first time a single universal function had been found which could quantitatively describe the surface critical behavior of many different mixtures. There have also been various attempts to investigate this surface critical behavior using neutron and x-ray reflectometries. Results have been mixed and have often been at variance with Carpenter et al. In this paper, the authors show that neutron reflectometry data collected from a crystalline quartz-critical mixture interface, specifically deuterated water plus 3-methylpyridine, can be quantitatively explained using the model of Carpenter et al. derived from ellipsometric data.

Adsorption at the liquid-vapor surface of a binary liquid mixture

In a binary liquid mixture, the component possessing the lowest surface tension preferentially adsorbs at the liquid-vapor surface. In the past this adsorption behavior has been extensively investigated for critical binary liquid mixtures near the mixture's critical temperature T(c). In this fluctuation-dominated regime the adsorption is described by a universal function of the dimensionless depth zxi where xi is the bulk correlation length. Fewer studies have quantitatively examined adsorption for off-critical mixtures because, in this case, one must carefully account for both the bulk and surface crossover from the fluctuation-dominated regime (close to T(c)) to the mean-field dominated regime (far from T(c)). In this paper we compare extensive liquid-vapor ellipsometric adsorption measurements for the mixture aniline+cyclohexane at a variety of critical and noncritical compositions with the crossover theory of Kiselev and co-workers [J. Chem. Phys. 112, 3370 (2000)].

Neutron reflectivity studies of critical adsorption: behavior of the surface scaling function

Neutron reflectometry has been employed to examine the nature of the critical adsorption surface scaling function for a near-critical mixture of hexane-d14+perfluorohexane adsorbing to a solid substrate from the liquid one-phase region. The analysis method of Dietrich and Schack has been applied to examine the nature of the power-law part of the critical adsorption surface scaling function, which has been found to behave as m(z) approximately P0z(-mu) as the critical point is approached. Values of mu = 0.514+/-0.018 and P0 = 0.90+/-0.04 have been obtained. These values are consistent with theoretical expectations (mu(th) = 0.516+/-0.004; P0(th) = 0.94+/-0.05), the value determined from Monte Carlo simulations (P(MC)0 = 0.866, and other experimental determinations (P(ex)0 = 0.955=/-0.08).

X-ray specular reflectivity study of a critical binary fluid mixture

We have used direct inversion of x-ray reflectivity data to extract the liquid-vapor interface composition profile and the related critical scaling function of a binary mixture of dodecane and tetrabromoethane. The mixture was in the one-phase region above its critical point. The results indicate the formation of a monolayer of the lower surface tension component followed by an abrupt change to a mixed composition which gradually relaxes to the bulk composition deep within the fluid.

Adsorption from alkane+perfluoroalkane mixtures at fluorophobic and fluorophilic surfaces. II. Crossover from critical adsorption to complete wetting

Using neutron reflectometry, adsorption from an equimolar mixture of hexane + perfluorohexane to a fluorophobic, octadecyl-coated, silicon substrate has been investigated as a function of temperature in the one-phase region upon approach to liquid-liquid coexistence. The composition of the investigated mixture, x(F) = 0.50, is well removed from the critical composition of x(F) = 0.36, where x(F) is the perfluorohexane mole fraction. To aid the modeling, mixtures with three different neutron refractive index contrasts have been used: namely, mixtures of C(6)H(14) + C(6)F(14) (H-F), C(6)D(14) + C(6)F(14) (D-F), and a mixture of C(6)H(14) + C(6)D(14) + C(6)F(14) which has been adjusted to have the same refractive index as silicon (CMSi). For all three contrasts, the principal features of the composition profile normal to the interface follow similar trends as the temperature T is reduced towards T(0), the coexistence temperature. These features consist of: (i) a hexane-rich primary adsorption layer appended to the octadecyl coupled layer. This primary layer is 22 +/- 5 A thick and becomes increasingly enriched in hexane as T(0) is approached. (ii) A tail that decays exponentially towards the bulk composition with a characteristic decay length zeta. As T(0) is approached, zeta increases. The scattering length density profiles have been converted to volume fraction profiles and the surface excess of hexane Gamma has been determined as a function of temperature for all three contrasts. As T(0) is approached Gamma increases, and its behavior can be represented using the scaling law Gamma approximately |T - T(0)|(-m). The resulting values of m are 0.71 +/- 0.09, 0.68 +/- 0.04, and 0.68 +/- 0.06 for the D-F, H-F, and CMSi contrasts, respectively. The behavior of Gamma with temperature does not adhere to the Gamma approximately |T - T(0)|(-1/3) law expected for complete wetting in systems with van der Waals interactions nor does it correspond to Gamma approximately |T - T(c)|(-0.305) expected for critical adsorption. The magnitude of the exponent m indicates that the adsorption resides in the crossover region between critical adsorption and complete wetting.

Phase behaviour of n-hexane/perfluoro-n-hexane binary thin wetting films

We present X-ray reflectivity investigations of the concentration distribution in binary liquid thin films on silicon substrates. The liquid-vapor coexistence of the binary mixture investigated, hexane and perfluorohexane, is far from criticality. Therefore, a sharp interface separates the liquid film from the vapor. The data reveal a separation of the film in layers parallel to the substrate. A phase diagram is constructed as a projection to the (composition difference, temperature) space, covering a temperature range corresponding to the one-phase and the two-phase regime of the bulk liquid. Although the composition data indicate a mixing gap similar to that of the bulk system, there are two major differences: i) only the near-surface phase changes its composition significantly, and ii) a composition gradient in the film exists also at higher temperatures where in the bulk system the one-phase regime exists.

Ellipsometric study of undersaturated critical adsorption

At the liquid-vapor surface of a critical AB binary liquid mixture, if the surface energies sigma(A)<<sigma(B), then component A completely saturates the surface and one is in the regime of strong critical adsorption where the local volume fraction at the surface is solely a function of z/xi where xi is the bulk correlation length and z is the depth into the liquid. If, however, sigma(A) approximately sigma(B) then the surface composition is a sensitive function of the surface energy difference Deltasigma=sigma(A)-sigma(B) and the surface is "undersaturated." We study this undersaturated critical adsorption regime using a homologous series of critical binary liquid mixtures. Component B (methyl formate) is fixed, while component A is varied from n-undecane to n-tetradecane. With increasing carbon chain length Deltasigma changes systematically from a negative to a positive value. We find that the experimental results, in both the one- and two-phase regions, are well described by a universal surface scaling function G(z/xi,h(1)t(-Delta(1))), where h(1) approximately Deltasigma, t=[T(c)-T]/T(c) is the reduced temperature relative to the critical temperature (T(c)), and Delta(1) is a universal surface critical exponent. These results are in conformity with theoretical predictions.

Ellipsometric determination of universal critical adsorption scaling functions

In this paper we determine and compare a number of theoretical models which describe the universal scaling functions for critical adsorption in the strong surface field limit. The P1 and P3 models, which are continuous up to and including the first and third derivatives, respectively, provide excellent descriptions of the ellipsometric data for four different critical binary liquid mixtures. The exponential-Pade model, initially proposed by Liu and Fisher [Phys. Rev. A 40, 7202 (1989)], provides a reasonable but less accurate description of the one-phase experimental data. This later model has the advantage, however, that it is continuous in all derivatives.

Neutron reflectivity studies of critical adsorption: the correspondence between a critical adsorption profile and specular neutron reflection

For interfacial order parameter profiles which decay as Az(-mu), such as the composition profile of the noncritical interface of a binary liquid mixture at a critical end point, there is a reported one-to-one correspondence between the profile and the reflectivity which can be described by an analytical theory [S. Dietrich and R. Schack, Phys. Rev. Lett. 58, 140 (1987)]. Evidence for mu=0.53+/-0.02 for adsorption at the hydrophilic silicon/liquid surface of the mixture (2-butoxyethanol+deuterium oxide) near its lower critical end point, determined from neutron reflectivity measurements, is presented. This value is in good agreement with the theoretical prediction of mu=0.516+/-0.004. Further examination of the data permits the determination of the asymptotic surface enrichment scaling factor amplitude P0 approximately 0.11, which is not in agreement with the theoretical value P(0) = 0.94+/-0.05 and values determined by other experimental methods.

Universal surface scaling function for critical adsorption

We present an accurate determination of the one- and two-phase universal surface scaling function which describes critical adsorption at noncritical interfaces of critical binary liquid mixtures in the strong surface field limit. The 95% confidence levels for this function are also determined. This function quantitatively describes ellipsometric critical adsorption data for three upper critical and one lower critical binary liquid mixtures.