Liquid-vapor rectilinear diameter revisited

In the modern theory of critical phenomena, the liquid-vapor density diameter in simple fluids is generally expected to deviate from a rectilinear law approaching the critical point. However, by performing precise scannerlike optical measurements of the position of the ${\mathrm{SF}}_6$ liquid-vapor meniscus, in an approach much closer to criticality in temperature and density than earlier measurements, no deviation from a rectilinear diameter can be detected. The observed meniscus position from far ($10\mathrm{K}$) to extremely close ($1\mathrm{mK}$) to the critical temperature is analyzed using recent theoretical models to predict the complete scaling consequences of a fluid asymmetry. The temperature dependence of the meniscus position appears consistent with the law of rectilinear diameter. The apparent absence of the critical hook in ${\mathrm{SF}}_6$ therefore seemingly rules out the need for the pressure scaling field contribution in the complete scaling theoretical framework in this ${\mathrm{SF}}_6$ analysis. More generally, this work suggests a way to clarify the experimental ambiguities in the simple fluids for the near-critical singularities in the density diameter.

Figure 1

Top: Schematic diagram of coexisting liquid (upper branch) -gas (lower branch) density curve near the critical point (cp) of a simple fluid with critical temperature $T_c$ and critical density ${\rho }_c$. Inside the two-phase domain, the dashed line and the full curve (with a “viewed hook” close to cp) correspond to the expected rectilinear and singular density diameters, respectively. The horizontal line at ${\rho }_{\mathrm{cell}}$ evidences three characteristic points (a,b,c) along the thermodynamic path followed by cooling a cylindrical sample cell filled at a mean density $〈\rho 〉\equiv {\rho }_{\mathrm{cell}}$ slightly above the critical density ($〈\delta \tilde{\rho }〉=\frac{〈\rho 〉}{{\rho }_c}-1>0$). Bottom: Expected meniscus positions of a two-phase cell during cooling: a: above the volumetric median plane (VMP) of the cell at the coexistence temperature (very close to $T_c$); b: matching the VMP of the cell at the temperature $T_{\mathrm{cross}}$ crossing the density diameter curve; c: below the VMP of the cell far from $T_c$.

Figure 2

Top: Schematic cross-sectional orientations of the fluid volume and horizontal lines corresponding to the meniscus position close to its VMP. Bottom: Temperature dependence of both ($i=1$ and $i=2$) symmetrical pixel shifts of the meniscus position (with respect to each corresponding VMP), for the eight $\left(i,X\right)$ configurations. 1 pixel = 12 $\mu \mathrm{m}$.

Figure 3

Full circles: Ratio $\frac{h}{R}$ for the $\left(i,V\right)$ configuration, as a function of $1-\frac{T}{T_c}$. Full curve: Eq. (1) with ${〈\delta \tilde{\rho }〉}_{T_c}=0.002$, using a $\mathrm{\Delta }{\tilde{\rho }}_{LV}$ theoretical estimation without adjustable parameters (see text and Ref. [22] for details), ${\tilde{\rho }}_d=1+a_d\mathrm{\Delta }{\tau }^{*}$ with $a_d=0.84$, and capillary correction as defined in the text and the Supplemental Material [27]. Dashed curve: Normalized meniscus position $\frac{z}{R}$ obtained from CPM calculations for ${〈\delta \tilde{\rho }〉}_{T_{\mathrm{coex}}}=0.002$, with compressible effects due to gravity (see text). Dash-dotted curve: Eq. (1) for ${〈\delta \tilde{\rho }〉}_{T_{\mathrm{coex}}}=0.002$ and $\mathrm{\Delta }{\tilde{\rho }}_d$ given by Eq. (3), using Kim and Fisher's [12] parameters.