Finite-size effects in presence of gravity: The behavior of the susceptibility in ${}^3\text{H}\text{e}$ and ${}^4\text{H}\text{e}$ films near the liquid-vapor critical point

We study critical-point finite-size effects on the behavior of susceptibility of a film placed in the Earth’s gravitational field. The fluid-fluid and substrate-fluid interactions are characterized by van der Waals type power-law tails, and the boundary conditions are consistent with bounding surfaces that strongly prefer the liquid phase of the system. Specific predictions are made with respect to the behavior of ${}^3\text{H}\text{e}$ and ${}^4\text{H}\text{e}$ films in the vicinity of their respective liquid-gas critical points. We find that for all film thicknesses of current experimental interest the combination of van der Waals interactions and gravity leads to substantial deviations from the behavior predicted by models in which all interatomic forces are very short ranged and gravity is absent. In the case of a completely short-ranged system exact mean-field analytical expressions are derived, within the continuum approach, for the behavior of both the local and the total susceptibilities.

Figure 1

(Color online) The schematic phase diagram of a $d$-dimensional film system for various thicknesses $L$ in the presence of a gravity subject to boundary conditions that strongly favor the liquid phase at the plates bounding the fluid.

Figure 2

(Color online) The behavior of the finite-size susceptibility $\chi$ in thin films with thickness $L=1000$, 2000, 4000, 6000, and 8000 layers at the bulk critical temperature $T=T_c$ of the corresponding infinite system as a function of the scaling variable $x_{\mu }=\left(\beta \Delta \mu \right)L^{\Delta /\nu }$. The corresponding scaling variable that governs the dependence on the gravity is proportional to $x_g\sim \left(\beta g\right)L^{\Delta /\nu +1}$ and, thus, the gravitational effects gradually set in with increase in $L$ in the behavior of the finite-size susceptibility.

Figure 3

(Color online) The behavior of the finite-size susceptibility $\chi$ at $T=T_c$ in thin films with short-ranged interaction and with $L=1000$, 2000, 4000, and 8000 layers. Note that the maximum of the susceptibility is at negative values of $x_{\mu }=\left(\beta \Delta \mu \right)L^{\Delta /\nu }$. Note the excellent scaling in the behavior of the curves for the different $L$’s, contrary to the behavior of the same system but with gravity and van der Waals interactions with the bounding the system plates taken into account—see the previous figure.

Figure 4

(Color online) The scaling function of the total susceptibility $X\left(x_t\right)$.

Figure 5

(Color online) Plot of the function $X_{\chi }\left(\zeta \mid x_t=0\right)$.

Figure 6

(Color online) The scaling function of the total susceptibility $\chi L^{-\gamma /\nu }$, calculated via the lattice model, compared versus the scaling function $X\left(x_t\right)$ of the same quantity as derived analytically within the continuum approach.

Figure 7

(Color online) The behavior of the finite-size susceptibility $\chi$ at $T=T_c$ as a function of $x_{\rho }=\Delta \rho L^{\beta /\nu }-3.2\text{ln}L$ for films with thickness $L=1000$, 2000, 4000, and 8000 layers. Both van der Waals forces and gravity effects are taken into account.

Figure 8

(Color online) The behavior of the finite-size susceptibility $\chi$ at $T=T_c$ as a function of $x_{\rho }=\Delta \rho L^{\beta /\nu }-2.182\text{ln}L$ for films with thickness $L=1000$, 2000, 4000, and 8000 layers for systems with only short-ranged interaction present.

Figure 9

(Color online) The behavior of the finite-size susceptibility $\chi$ at $T=T_c$ as a function of $\Delta \rho$ as obtained within the mean-field-like treatment of the model. All the curves are for $L=1000$ or $L=8000$ layers thick film but with gravity and/or van der Waals interaction neglected or taken into account. One observes that both van der Waals interactions, as well as gravity, suppress the divergence of the susceptibility.

Figure 10

(Color online) The same as in Fig. 9 but now the behavior of the finite-size susceptibility $\chi$ is considered as a function of the scaling variable $\left(\beta \Delta \mu \right)L^{\Delta /\nu }$.

Figure 11

(Color online) (a) The density profile for a system with $L=1000$ layers at $T=T_c$ and $\mu ={\mu }_c$. (b) The density profile for a system with $L=1000$ layers at $T=T_c$ and $\Delta \mu =\Delta {\mu }_{\text{max}}$ for which the susceptibility reaches its maximum. (c) The density profile for a system with $L=8000$ layers at $T=T_c$ and $\mu ={\mu }_c$. (d) The density profile for a system with $L=8000$ layers at $T=T_c$ and $\Delta \mu =\Delta {\mu }_{\text{max}}$ for which the susceptibility reaches its maximum. In all the figures the bold (red) curve shows the profile of the local susceptibility normalized by $L^2$ for the system with size $L$. In order to have a better visibility of the order-parameter profile simultaneously with the profile of the local susceptibility, the magnitude of the local susceptibility has been further reduced eight times for $L=1000$ and 16 times for $L=8000$.

Figure 12

(Color online) The behavior of the finite-size susceptibility as a function of the scaling variable $x_{\mu }=\Delta \mu L^{\Delta /\nu }$ for a film with thicknesses $L=1000$ and $L=8000$ for $t=0,t=\pm {10}^{-6}$ and $t=\pm {10}^{-5}$.

Figure 13

(Color online) The behavior of the finite-size susceptibility as a function of $\Delta \rho$ for a film with thickness $L=1000$ and for $t=0,t=\pm {10}^{-6}$ and $t=\pm {10}^{-5}$.

Figure 14

(Color online) The behavior of $\Delta \rho$ as a function of the scaling variable $x_{\mu }=\Delta \mu L^{\Delta /\nu }$ for a film with thicknesses $L=1000$ and $L=8000$ for $t=0,t=\pm {10}^{-6}$ and $t=\pm {10}^{-5}$.

Figure 15

(Color online) Finite-size behavior of the susceptibility for a film with $L=1000$, 2000, 4000, and 8000 layers as a function of the scaling variable $x_{\mu }=\Delta \mu L^{\Delta /\nu }$ in the case in which both the intrinsic van der Waals interactions as well as the presence of gravity are taken into account. For a comparison the behavior of a system with completely short-ranged interaction is also presented.

Figure 16

(Color online) Plot of the profile function $X_m\left(\zeta \mid x_t\right)$—see Eq. (B17).

Figure 17

(Color online) Plot of the function ${\psi }_1\left(\zeta \mid x_t\right)$—see Eq. (B23).

Figure 18

(Color online) Plot of the function ${\psi }_2\left(\zeta \mid x_t\right)$—see Eq. (B29).

Figure 19

(Color online) Plot of the function ${\psi }_i\left(\zeta \mid x_t\right)$—see Eq. (B31).

Figure 20

(Color online) Plot of the function ${\chi }_l\left(\zeta \right)$.