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Kharkats–Ulstrup size effect at the air-water interface: (a) Single-ion electrostatic free energy of the alkali ions at the air-water interface as a function of ; (b) differences between free energies of monovalent alkali ions at the air-water interface. For the radius for , for , and for (Refs. 30 and 31).
The four-layer slab model of hydrosol’s surface-normal structure. Each layer has a thickness and an electron density . In addition, parameters determine the interfacial width between slabs of electron density (the standard deviation of their locations ). At only three interfaces (top two layers with adsorbed ions) contribute to the reflectivity since . The density is the only parameter of the surface-normal structure that depends strongly on the composition of the alkali metal.
The surface-structure factors of the 22 nm particle sols: The rhombi represent sol stabilized by NaOH, ; the filled and open triangles are for potassium-enriched sols with and ; the dots and circles are for rubidium-enriched sols with and ; the filled and open squares are for cesium-enriched sols with and . Here, filled and open symbols on each curves refer to samples with different equilibration histories. The crosses and stars are for mixtures of cesium- and potassium-enriched sols with , , and . The lines denote the first Born approximation that is discussed in the text. Insert: and are, respectively, wave vectors of the incident beam, and the beam scattered toward the point of observation, and is the wave-vector transfer, . At reflectivity conditions there is only one component of the wave-vector transfer, , where and are the angles of the incident and scattered beams in the plane normal to the surface. The reflectivity was measured with the detector’s vertical slit gap of ~0.8 mm at the distance of ~700 mm form the footprint or angular acceptance at (twice higher than in Ref. 34) and its horizontal acceptance at ~0.8° (~10 mm gap).
(a) Model distributions of electron density in layer 1, normalized to the density of bulk water. (b) Integral density of layer 1 vs the number of electrons in the alkali ion, where , , , and , respectively, are the numbers of electrons in , , , and . The circles and squares correspond to the reflectivity curves in Fig. 3 and the data obtained from Ref. 34, respectively. A solid line is the linear fit of these data.
Estimates of the model parameters in Eq. (4) (see also Fig. 2). is the bulk concentration of sodium in the hydrosols; is the bulk concentration of alkali ions in the enriched sols; are the thicknesses of the interfacial layers with electron densities , normalized to the density of bulk water under normal conditions ; . Parameters and in the rows shifted upward and downward correspond, respectively, to the data in Fig. 3 shown by the open and solid symbols. The bulk electron densities of the sols, , were established from their densities and known chemical compositions. The error bars were estimated utilizing the conventional -criteria at the confidence level of 0.95.
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