Two-dimensional freezing in the liquid–vapor interface of a dilute Pb:Ga alloy

Abstract

We report the results of x-ray reflectivity and grazing incidence x-ray diffraction studies of the liquid–vapor interface of a dilute alloy of Pb in Ga over the temperature range of 23–76°C. Our data show that the liquid–vapor interface of this alloy is stratified for several atomic diameters into the bulk liquid and that a monolayer of Pb forms the outermost stratum of the interface. Over the temperature range of 23–56°C, the monolayer of Pb is in an ordered hexagonal phase. At about 58°C, this monolayer undergoes a first-order transition to a hexatic phase, which remains stable to 76°C. An analogy between the observed transition and the first-order melting transition in a one-component classical plasma is suggested.

The existence of a composition difference between the liquid–vapor interface and the bulk liquid phase of a binary mixture has been known since the work of Gibbs (see, e.g., ref. 1). However, it is only in the last few years that information about the atomic structure of that interface has become available. Theoretical studies of the liquid–vapor interfaces of pure metals and binary alloys (2–13) lead to the following predictions: (i) the (longitudinal) distribution of density along the normal to the interface is stratified for several atomic diameters into the bulk liquid, and (ii) the species in excess in the liquid–vapor interface of a dilute binary alloy forms a monolayer that is the outermost layer of that interface. Recent x-ray reflectivity (XR) and grazing incidence x-ray diffraction (GIXD) studies of the liquid–vapor interfaces of several pure metals and binary alloys have confirmed these predictions (14–27). In this paper, we report the results of XR and GIXD studies of the liquid–vapor interface of a dilute alloy of Pb in Ga over the temperature range of 23–76°C. These experiments explore the possibility of using the monolayer of the segregated species to study phase transitions in a quasi two-dimensional system. Our data suggest that the Pb monolayer that forms the outermost stratum of the liquid–vapor interface of the alloy is in a hexatic phase from 60°C to 76°C and that it undergoes a first-order two-dimensional hexatic-to-hexagonal solid transition in the vicinity of 58°C. Over the temperature range of 23–56°C, the range of the translational order in the hexagonal Pb monolayer is greater than our instrumental resolution.

Experimental Methods and Results

The high-vacuum sample chamber used for our experiments has been described elsewhere (24–27), as have the x-ray surface scattering spectrometer and the detection electronics used. We merely note that the sample chamber contains a residual gas analyzer to monitor the vacuum, an Auger spectrometer to monitor the liquid sample surface composition, and an ion gun and a sweep arm to clean the liquid sample surface. Two liquid alloy samples with slightly different compositions were studied. The results obtained from x-ray diffraction and XR studies of these samples were the same; thus, we focus attention on one of them: 0.054 atomic percent Pb in Ga (the composition of the alloy was determined by atomic absorption spectroscopy at the Schwarzkopf Microanalytical Laboratory, Woodside, NY). This sample was prepared at about 200°C in a separate vacuum chamber and was loaded into the evacuated experimental chamber via a capillary feed. Auger spectroscopy of the alloy liquid–vapor interface revealed the expected presence of excess Pb and also clearly indicated the absence of any oxide. Analysis of the composition of the residual gas over the sample showed that the partial pressure of oxygen was less than 10⁻¹² torr (1 torr = 133 Pa). All of our experiments were carried out with a background pressure of 3 × 10⁻¹⁰ torr.

XR measurements were carried out at many temperatures between 28°C and 76°C. The longitudinal density distributions inferred from the data obtained show, in every case, that the outermost stratum of the liquid–vapor interface is a monolayer of pure Pb. To within our experimental precision, for the alloy with 0.054 atomic percent Pb, the segregated layer of Pb forms a complete monolayer at all temperatures below 76°C. A sample of these longitudinal density distributions is shown in Fig. 1. Attempts to fit the reflectivity data with a longitudinal density distribution that has Pb in both the outermost layer and the next layer failed, thereby yielding an estimate of less than 5% Pb in the second stratum of the alloy liquid–vapor interface.

Figure 1.

XR as a function of momentum transfer along the normal from the liquid–vapor interface of a dilute Pb:Ga alloy. The data have been normalized to the Fresnel reflectivity from the interface. (a) Normalized reflectivity at 28°C (circles) and the fit of the model longitudinal density profile (solid line) to those data. (b–d) Inferred longitudinal density profile in the liquid–vapor interface of the Pb:Ga alloy at 28°C (b), 66°C (c), and 76°C (d).

GIXD measurements were carried out over the temperature range of 23–76°C. At all of the temperatures in this range, the diffraction patterns have a broad peak (centered at 0.247 nm) arising from scattering by liquid Ga. Subtraction of the Ga scattering and the diffuse background scattering arising from the chamber from the total GIXD scattering yields the results shown in Fig. 2. These diffraction patterns are reproducible under cycling of the temperature of the sample and from sample to sample. Studies of the position of the first sharp peak in the GIXD pattern as a function of the out-of-plane angle of detection clearly show that the diffraction comes from a monolayer of Pb atoms, supporting the conclusion drawn from the XR studies.

Figure 2.

GIXD from the liquid–vapor interface of a dilute Pb:Ga alloy. (a) Observed diffraction pattern at °C and, separately, the background scattering from Ga and the stray scattering. (a, Inset) A schematic diagram of the hexagonal lattice. (b–d) Observed diffraction pattern at 28°C (b), 66°C (c), and 76°C (d) after subtraction of the Ga and stray scattering.

When the temperature is below 56°C (see Fig. 2b), the GIXD pattern has five sharp peaks corresponding to lattice plane spacings (d) of 0.296, 0.224, 0.171, 0.148, and 0.112 nm. These diffraction peaks are attributed to scattering by Pb atoms and indicate that the Pb atoms in the liquid–vapor interface form a two-dimensional hexagonal lattice. For convenience, we chose to index the peaks observed in the basis of a degenerate two-dimensional face-centered orthorhombic lattice with a = 0.342 nm and b = 0.592 nm (see Fig. 2a). On this basis, the peaks are indexed as (1 1), (1 2), (2 0), (2 2), and (3 1) reflections, respectively. Our measurements also show that the peak intensities of the (1 1) and (2 0) reflections can be several times higher than the maximum intensity of the broad Ga peak. We note that, in a perfect two-dimensional hexagonal lattice, the (1 2) reflection is forbidden; thus, its intensity is expected to be very low, as we observed. The full width at half-maximal intensity of the (1 1) reflection is approximately equal to the angular resolution of the spectrometer as we used it. This observation implies that the translational correlation length (domain size) of the two-dimensional hexagonal lattice can be much larger than some tens of nanometers. However, we are not able to obtain a precise value of the correlation length from the data available. In qualitative support of this inference, we note that we found it difficult to record all the diffraction peaks in one measurement, which we attribute to the large size and the inhomogeneous distribution of orientations of the two-dimensional Pb crystal domains.

At the intermediate temperature of 66°C (see Fig. 2c), the (1 1) and (2 0) reflections are also observed. However, at this temperature, the peak width was much broader, corresponding to a much less well ordered phase of Pb.

At higher temperatures, for example 76°C (see Fig. 2d), the (1 1) reflection becomes very broad, but the (2 0) reflection is still observable. We infer that, at this temperature, and down to 60°C, the two-dimensional Pb monolayer forms a hexatic phase. The d spacings in the hexatic and solid phases are the same, although there is a very large change in translational correlation length across the transition (Fig. 3); the full width at half-maximal intensity of the (1 1) reflection in the hexatic phase corresponds to a translational correlation length of about 4.0 nm. Our data are insufficient to determine whether there is a density change across the hexatic-to-solid transition, but the coincidence of the d spacings corresponding to the (1 1) and (2 0) reflections suggests that it is very small or possibly zero. The apparently discontinuous change in the range of translational order across the transition (Fig. 3) suggests that the transition is first order.

Figure 3.

Translational correlation length as a function of temperature, obtained from deconvolution of the observed (1 1) reflection peak with the instrumental resolution function. The instrumental resolution function corresponds to L = 28 nm.

Discussion

From the observations described in Experimental Methods and Results, we infer the following: (i) at temperatures below 56°C, the Pb monolayer segregated in the liquid–vapor interface of the Pb:Ga alloy forms a two-dimensional hexagonal solid; (ii) a first-order solid-to-hexatic transition occurs at about 58°C; and (iii) up to the maximum temperature of our observations, the excess Pb in the liquid–vapor interface of the liquid alloy forms a hexatic phase.

The character of the phase transition in the Pb monolayer is reminiscent of the freezing transition in a two-dimensional one-component plasma. That transition is first order, although it occurs with zero density change (there is a nonzero entropy change). The zero density change across the transition is a consequence of the assumption that the neutralizing jellium background is rigid, whereupon the constraint of electroneutrality requires superposition of the charge densities of the jellium and the mobile component. In our case, the chemical potential of the electrons in the Pb monolayer must be the same as the chemical potential of the electrons in the bulk liquid, which is almost pure Ga. Because the Fermi temperature of Ga is very high, the chemical potential of the electrons will depend only on the electron density. Also, because the variation in the volume of Ga over the temperature range we have studied is vanishingly small, we infer that the chemical potential of the electrons in the monolayer corresponds to a system with fixed density. Thus, it is reasonable to expect that the freezing of the Pb ions into a hexagonal lattice will imitate the behavior of a one-component plasma, i.e., occur with zero density change. It is of interest, in this context, that the results of the most recent computer simulations of the crystallization of a classical two-dimensional one-component electron plasma hint, but do not prove, that the crystal melts to a hexatic phase (28).

Celestini et al. (29) have suggested, on the basis of results obtained from simulations that used the “glue model” Hamiltonian for liquid metals, that the outermost layer of the liquid–vapor interface of a metal will be ordered hexatically. These investigators do not report any results concerning the character of the transitions between liquid and hexatic packing or between solid and hexatic packing in the liquid–vapor interface. Although it is commonly expected that these transitions will be continuous, we note that first-order liquid-to-hexatic and hexatic-to-solid transitions have been observed in a quasi two-dimensional colloid system (30).

Abbreviations

XR

x-ray reflectivity

GIXD

grazing incidence x-ray diffraction