Tricritical Casimir forces and order parameter profiles in wetting films of ${}^3\mathrm{He}-{}^4\mathrm{He}$ mixtures

Tricritical Casimir forces in ${}^3\mathrm{He}-{}^4\mathrm{He}$ wetting films are studied, within mean field theory, in terms of a suitable lattice gas model for binary liquid mixtures with short-ranged surface fields. The proposed model takes into account the continuous rotational symmetry O(2) of the superfluid degrees of freedom associated with ${}^4\mathrm{He}$ and it allows, inter alia, for the occurrence of a vapor phase. As a result, the model facilitates the formation of wetting films, which provides a strengthened theoretical framework to describe available experimental data for tricritical Casimir forces acting in ${}^3\mathrm{He}-{}^4\mathrm{He}$ wetting films.

Figure 1

Schematic bulk phase diagram of ${}^3\mathrm{He}-{}^4\mathrm{He}$ mixtures (black curves and surfaces) and two specific surfaces (blue and brown) in the $(T,Z,P)$ space, where $P$ is the pressure and $Z=exp({\mu }_3/T)$ is the fugacity of ${}^3\mathrm{He}$, with ${\mu }_3$ as the chemical potential of ${}^3\mathrm{He}$ atoms [1]. ${\mathrm{A}}_1$ shows the surface of first-order solid-liquid phase transitions, whereas ${\mathrm{A}}_2$ is the surface of first-order vapor-liquid phase transitions. The phase transitions between the normal fluid and the superfluid phase are either of second or of first order, which are shown by the surfaces ${\mathrm{A}}_3$ and ${\mathrm{A}}_4$, respectively. The surfaces ${\mathrm{A}}_3$ and ${\mathrm{A}}_1$ intersect along a line ${\mathrm{ce}}^{+}-{\mathrm{tce}}^{+}$ of critical end points. The surfaces ${\mathrm{A}}_3$ and ${\mathrm{A}}_4$ are separated by a line ${\mathrm{tce}}^{+}$-tce of tricritical points TC. This line meets ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ at the tricritical end points ${\mathrm{tce}}^{+}$ and tce, respectively. The surfaces ${\mathrm{A}}_3$ and ${\mathrm{A}}_2$ intersect along a line ce-tce of critical end points. The surface ${\mathrm{A}}_2$ terminates at a line of critical points, starting from c in the plane $Z=0$. The phase diagram in the plane $Z=0$ corresponds to that of pure ${}^4\mathrm{He}$. The dashed lines indicate that the corresponding surface continues. On the blue surface the total density is constant, which corresponds to the situation studied in Refs. [20, 21]. The brown surface ${\mathrm{A}}_{2,b}$ lies in the vapor phase slightly below the liquid-vapor coexistence surface ${\mathrm{A}}_2$. Although the thermodynamic fields along the thermodynamic paths taken in the experiment in Ref. [3] have been tuned to their values at the liquid-vapor coexistence surface, due to gravity the actual measurements have been carried out for thermodynamic states which lie on a surface resembling the brown one. At the thermodynamic states on the brown surface, in addition to the stable vapor phase, there are metastable liquid phases. These metastable liquid phases undergo transitions similar to the liquid-liquid phase transitions tied to ${\mathrm{A}}_2$. Therefore, for each point tce, ce, and c, there is a metastable counterpart ${\text{tc}}_{\text{m}}$, ${\text{ce}}_{\text{m}}$, and ${\text{c}}_{\text{m}}$, respectively, on the brown surface.

Figure 2

Liquid-liquid bulk phase transitions at coexistence with the vapor phase for ${}^3\mathrm{He}-{}^4\mathrm{He}$ mixtures and the thermodynamic paths taken in the experiments reported in Ref. [3]. The black curves denote the first-order phase transitions between the normal fluid phase and the superfluid phase, which terminate at the tricritical end point tce. The red curve shows the second-order $\lambda$ transitions between the normal fluid phase and the superfluid phase. The dashed dotted lines indicate three distinct thermodynamic paths corresponding to three fixed values of the concentration ${\mathcal{C}}_3=X_3/(X_3+X_4)$ [see, cf., Eq. (17)] of the ${}^3\mathrm{He}$ atoms as done experimentally. $X_3$ and $X_4$ are the bulk number densities of ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$, respectively. Upon decreasing the temperature, the bulk liquid undergoes a first-order phase separation at some demixing temperature $T_{\text{d}}\left({\mathcal{C}}_3\right)$. Upon further decrease of the temperature the thermodynamic paths follow that branch of the coexistence curve, which they hit (see the the brown and green arrows).

Figure 3

Projection of the brown surface ${\mathrm{A}}_{2,b}$ in Fig. 1 (which lies in the vapor phase) onto the $(P,T)$ plane. The solid and dashed brown lines are the projections of the corresponding ones for ${\mathrm{A}}_{2,b}$. The dashed-dotted lines are the projections of the thermodynamic paths shown in Fig. 2 and which lie on the brown surface. Upon lowering $T$ the brown dashed-dotted line first crosses the full line $T_{\lambda }\left({\mathcal{C}}_3\right)$ in Fig. 2, continues through the superfluid phase and then encounters the two-phase region. This sketch is based on our numerical results (see Sec. 3b) in the vicinity of ${\text{tc}}_{\text{m}}$.

Figure 4

Liquid-liquid demixing phase transitions in the bulk at coexistence with the vapor phase (the vapor phase is not shown here) in the $(X_3$, $T)$ plane, with $X_3=\langle N_3\rangle /\left(L\mathcal{N}\right)=D-X$ for (a) $(C/K,J/K,J_s/K)=(1,5.714,0)$, (b) $(C/K,J/K,J_s/K)=(1,5.714,3.674)$, and (c) $(C/K,J/K,J_s/K)=(1,9.107,3.701)$. The inset of panel (b) shows the same phase diagram in the $(T,{\mathcal{C}}_3)$ plane, where ${\mathcal{C}}_3=X_3/D$ denotes the concentration of ${}^3\mathrm{He}$. The phase diagrams in (a) and (b) have been discussed in detail in Ref. [33]. (Note that in Ref. [33] the coupling constants are rescaled by a factor of 6 and the total number of lattice sites are denoted as $\mathcal{N}$, whereas here the total number of lattice sites is given by $L\mathcal{N}$.) In (b) and (c) the black curves denote the binodals of the first-order phase transitions between the normal fluid (N) and the superfluid (S). The lines of first-order phase transitions in (a) terminate at the critical end point ce with $T_{\text{ce}}/K=6.286$, whereas in (b) and (c) the lines of first-order phase transitions terminate at a tricritical end point tce. In (b) and (c) the red curve denotes the $\lambda$ line of second-order phase transitions between the normal fluid and the superfluid. The temperature of the tricritical end point in panels (b) and (c) are $T_{\text{tce}}/K=8.782$ and $T_{\text{tce}}/K=8.47974$, respectively. In (b) the thin vertical line indicates $X_3=X_3^{\text{tce}}$, whereas in the inset of this figure the thin vertical line indicates ${\mathcal{C}}_3={\mathcal{C}}_3^{\text{tce}}$. The short dotted strokes indicate the character (S, N, rich in species 4, or rich in species 3) of the corresponding binodal. (See also Appendix pp2.)

Figure 5

Fluid parts of the phase diagram of a classical binary liquid mixture in the $(T,P,{\mu }_4-{\mu }_3)$ space (schematic diagram). ${\mathrm{B}}_1$ is the surface of first-order liquid-vapor phase transitions, whereas ${\mathrm{B}}_2$ is the surface of first-order liquid-liquid demixing transitions, between phases rich in either species 3 or 4. The surface ${\mathrm{B}}_1$ terminates at a line ${\mathrm{L}}_1$ of critical points. ${\mathrm{L}}_2$ denotes the line of critical points of the liquid-liquid demixing transitions, which ends at the surface ${\mathrm{B}}_1$ at the critical end point ce. The surfaces ${\mathrm{B}}_1$ and ${\mathrm{B}}_2$ intersect along a line ${\mathrm{L}}_3$ of triple points. The dashed curves indicate that the corresponding surfaces continue. The demixing two-phase region in terms of temperature and the number density $X_3$ of species 3 for the liquid phases coexisting along ${\mathrm{L}}_3$ are shown in Fig. 4; the vapor phase is not shown there.

Figure 6

Number density profiles for weak surface fields $(\widetilde{f_{+}},\widetilde{f_{-}})/K=(0.857,0)$ and for several values of temperature $T$ (same color code in all panels) above $T_{\text{ce}}/K=6.286$ and at $X_3=X_3^{\text{ce}}=0.431$ in the liquid phase, with (a) $\mathrm{\Delta }{\mu }_{+}/K=-8.57\times {10}^{-4}$ and (b) $\mathrm{\Delta }{\mu }_{+}/K=+8.57\times {10}^{-4}$. In (a) the bulk phase is the vapor phase, whereas in (b) it is the coexisting liquid phase with slight offsets. Panel (c) shows the liquid-vapor coexistence line (red curve) in the $(P,T)$ plane, emerging under the constraint $X_3=X_3^{\text{ce}}$ in the liquid phase. The colored dots in panel (c) indicate the thermodynamic states with the same temperature values as in panels (a) and (b), which, unlike in these two panels, lie at liquid-vapor coexistence. The star denotes the liquid-vapor critical point $(P_{\text{c}}/K,T_{\text{c}}/K)=(1.422,7.230)$. The thermodynamic paths in panels (a) and (b) follow the red curve in panel (c) but with the corresponding offset values $\mathrm{\Delta }{\mu }_{+}$. In panel (a) the corresponding thermodynamic states are: $(T/K,D=X_4+X_3,X=X_4-X_3,M)=\{(6.286,0.181,-0.121,0),(6.793,0.229,-0.137,0),(7.388,0.304,-0.155,0),(7.756,0.391,-0.174,0)\}$. The vapor bulk phase in (a) is preferred by the wall. Accordingly, there are no liquidlike wetting films. In panel (a) near $T_{\text{c}}$ critical adsorption (see Fig. 8 in Ref. [39]) of the preferred vapor phase occurs, which is indicated by the increased depth and range of the minimum in $D_l$. In panel (b), due to $\mathrm{\Delta }{\mu }_{+}>0$ the stable bulk phase is the liquid. Since the vapor phase is preferred by the wall drying films form there upon increasing the temperature. The corresponding thermodynamic states are $(T/K,D,X,M)=\left\{\right(6.286,0.913,0.050,0),(6.793,0.878,0.015,0),(7.388,0.815,-0.048,0),(7.756,0.722,-0.141,0\left)\right\}$. The value of ${\mu }_{-}$ can be obtained from Eq. (15) using the values of $(T,D,X,M)$ for the corresponding thermodynamic states as provided above. Note that the nonmonotonic behavior of the red curve in (c) is caused by the constraint $X_3=X_3^{\text{ce}}$. For (c), in order to identify the vapor and liquid phases, in addition to $T$ and $P$ also the value of the chemical potential ${\mu }_{-}$ is required, which is not shown.

Figure 7

Order parameter profiles $D_l$ and $X_l$ at $X_3=X_3^{\text{ce}}=0.431$ and for $\mathrm{\Delta }{\mu }_{+}/K=-8.57\times {10}^{-4}$ for several temperatures and for two sets of surfaces fields. In panels (a) and (b) the surface fields are $(\widetilde{f_{+}},\widetilde{f_{-}})/K=(5.143,0)$, whereas panels (c) and (d) correspond to $(\widetilde{f_{+}},\widetilde{f_{-}})/K=(5.143,0.857)$. The bulk phase is the vapor phase and the wall prefers the liquid phase, giving rise to wetting films. The positive value of $\widetilde{f_{-}}$ not only results in the increase of the number density $X_{4,l}$ of species 4 and hence also of $X_l$ in the first layer [see panel (d)], but also increases the total number density $D_l$ in the first layer [see panel (c)]. The stable thermodynamic states of the vapor phase in the bulk are $(T/K,D,X,M)=\left\{\right(6.286,0.180,-0.121,0),(7.388,0.304,-0.155,0),(7.649,0.355,-0.165,0),(7.756,0.391,-0.174,0\left)\right\}$. All three bulk states lie in the vapor phase close to liquid-vapor coexistence on the left side of the red line shown Fig. 6. The value of ${\mu }_{-}$ can be obtained from Eq. (15) using the values of $(T,D,X,M)$ for the corresponding thermodynamic states as provided above.

Figure 8

Film thickness $y/a$ versus $|\mathrm{\Delta }{\mu }_{+}|/K$ for $(\widetilde{f_{+}},\widetilde{f_{-}})/K=(8.571,0)$ at $X_3=X_3^{\text{ce}}$ and for several temperatures. Upon approaching the liquid-vapor coexistence surface at low temperatures, the film thickness increases smoothly and reaches a plateau. The height of this plateau increases gradually by increasing $T$, indicating $7.097<T_w/K<7.123$. The jumps are due to first-order layering transitions induced by the lattice model. Above the roughening transition they are an artifact of mean field theory [38]. For $T>T_w$ one has $y(\mathrm{\Delta }{\mu }_{+}\to 0^{-})\sim \kappa \text{ln}\frac{1}{|\mathrm{\Delta }{\mu }_{+}|/K}$ with a slight increase of $\kappa \left(T\right)$ as a function of $T$.

Figure 9

Layering transitions in the $({\mu }_{+},T)$ plane for $(\widetilde{f_{+}},\widetilde{f_{-}})/K=(8.571,0)$. Each line of first-order layering transition ends at a critical point. The color code does not carry a specific meaning. The lines are colored differently so that it is easier to distinguish them.

Figure 10

Equilibrium film thickness $y/a$ as a function of temperature for $\mathrm{\Delta }{\mu }_{+}/K=-8.57\times {10}^{-8}$ and $X_3=X_3^{\text{ce}}$. Since $\mathrm{\Delta }{\mu }_{+}$ is nonzero, $y\left(T\right)$ does not diverge but attains a maximum upon passing by $T_w$. This maximum diverges for $\mathrm{\Delta }{\mu }_{+}\to 0$. The jumps are bunched together around $T/K\simeq 7.125$ and spread-out otherwise.

Figure 11

Equilibrium film thickness $y/a$ versus $|\mathrm{\Delta }{\mu }_{+}|/K$ for $(\widetilde{f_{+}},\widetilde{f_{-}})/K=(10.714,16.071)$ and for four temperatures. Unlike the situation in Fig. 8 with $J_s=0$, the thickness of the wetting films as a function of $|\mathrm{\Delta }{\mu }_{+}|$ is a nonmonotonic function of $T$. The most rapid increase occurs at $(T-T_{\text{tce}})/T_{\text{tce}}\approx -0.016$, whereas for lower and higher temperatures the growth of the wetting film as a function of $\mathrm{\Delta }{\mu }_{+}$ is slower. Upon approaching the liquid-vapor coexistence surface, the ${}^4\mathrm{He}$-rich layers within the wetting films become superfluid. At each temperature, the continuous surface transition to superfluidity occurs for values of the offset $|\mathrm{\Delta }{\mu }_{+}|$ smaller than the one indicated by the corresponding tick on the abscissa with the same color. For $(T-T_{\text{tce}})/T_{\text{tce}}=-0.016$ and $(T-T_{\text{tce}})/T_{\text{tce}}=-0.042$, the number density of ${}^3\mathrm{He}$ on the superfluid branch of the binodal [Fig. 4] is $X_3=0.201$ and $X_3=0.188$, respectively, whereas for $T\ge T_{\text{tce}}$ the number density of ${}^3\mathrm{He}$ is fixed at $X_3=X_3^{\text{tce}}=0.20845$.

Figure 12

Order parameter profiles $D_l$, $X_l$, and $M_l$ for $(\widetilde{f_{+}},\widetilde{f_{-}})/K=(10.714,16.071)$ at $\mathrm{\Delta }{\mu }_{+}/K=-1.07\times {10}^{-4}$ for the bulk states (a) $T=T_{\text{tce}}$, $X_3=X_3^{\text{tce}}$, (b) $\mathrm{\Delta }T/K=(T-T_{\text{tce}})/K=0.18$, $X_3=X_3^{\text{tce}}$, and (c) $\mathrm{\Delta }T/K=(T-T_{\text{tce}})/K=-0.964$, $X_3=0.1461$ (which is on the superfluid branch of the binodal [Fig. 4]. For these bulk states, in panels (a)–(c) the stable vapor phase (i.e., $l\to \infty$) exhibits the order parameters $(D=X_4+X_3,X=X_4-X_3)=\{(0.0523,-0.0206),(0.0559,-0.0204),(0.0422,-0.0255)\}$, respectively ($M=0$ for the vapor phase). The bulk parameters of the system are those for Fig. 4. The keys for the OP profiles are the same for all panels. The value of ${\mu }_{-}$ can be obtained from Eq. (15) using the values of $(T,D,X,M)$ for the corresponding thermodynamic states as provided above.

Figure 13

The bulk liquid-liquid phase transitions at coexistence with the vapor phase as in Fig. 4 plotted in the $(T,{\mathcal{C}}_3)$ plane, with ${\mathcal{C}}_3=(D-X)/\left(2D\right)$ as the concentration of ${}^3\mathrm{He}$ (the vapor phase is not shown here). The blue line $T_{\text{s}}\left({\mathcal{C}}_3\right)$ represents the continuous surface transition. Upon crossing this transition line a thin film near the wall becomes superfluid although the bulk remains a normal fluid. This line merges with the $\lambda$ line [red line denoted as $T_{\lambda }\left({\mathcal{C}}_3\right)$] at the special point s*. The inset shows the vertical thermodynamic paths (at liquid-vapor coexistence) taken experimentally. The numerical paths in our calculations are located in the vapor phase parallel to the ones in the inset. $T_{\text{d}}^{\left(\text{s}\right)}\left({\mathcal{C}}_3\right)\left[T_{\text{d}}^{\left(\text{n}\right)}\left({\mathcal{C}}_3\right)\right]$ denotes the superfluid $\left[\text{normal}\text{fluid}\right]$ binodal of the two-phase region. The arrows indicate how the vertical thermodynamic paths continue after encountering the demixing curve. The path shown by the black dotted line can follow both binodals.

Figure 14

Thickness of ${}^3\mathrm{He}-{}^4\mathrm{He}$ wetting films extracted from capacity measurements (Fig. 4 in Ref. [3]). The values of $X$ refer to various concentrations of ${}^3\mathrm{He}$. The concentration of ${}^3\mathrm{He}$ at the tricritical point is $X_{\text{t}}=0.672$. Panels (a) and (b) correspond to $X\ge X_{\text{t}}$ and $X\le X_{\text{t}}$, respectively. Thin arrows show the points, where the bulk liquid phase separates. The large headed arrow indicates the tricritical point. In (b) the arrows with double lines indicate the onset temperature of superfluidity. For the thermodynamic path on the superfluid side [panel (b)], the growth of the film thickness exhibits a characteristic shoulder between the tricritical temperature and the superfluid transition temperature on the $\lambda$ line. The growth of the film thickness as a function of temperature is due to repulsive TCFs between the solid wall and the liquid-vapor interface, arising near the tricritical point. For small $X$ the wetting film resembles a film of pure ${}^4\mathrm{He}$, which corresponds to $(O,O)$ BCs for the CCFs arising near the temperature of the $\lambda$ transitions, which are attractive [2] [see the dip in panel (c)]. This figure is, with permission, reprinted.

Figure 15

Numerical results for the film thickness $y/a$ corresponding to the thermodynamic path at fixed ${\mathcal{C}}_3={\mathcal{C}}_3^{\text{tce}}$ and $\mathrm{\Delta }{\mu }_{+}/K=-1.07\times {10}^{-4}$ (i.e., slightly shifted thermodynamic path shown by the vertical black dashed line in Fig. 13). The arrows indicate the tricritical end point $T_{\text{tce}}$ and the onset temperature $T_{\text{s}}\simeq T_{\text{s}}\left({\mathcal{C}}_3\right)$ for superfluidity at the surface transition. [The deviation of $T_{\text{s}}$ from $T_{\text{s}}\left({\mathcal{C}}_3\right)$ (see Fig. 13) is due to the offset from liquid-vapor coexistence.] Below the tricritical temperature the thermodynamic path follows $T_{\text{d}}^{\left(\text{s}\right)}\left({\mathcal{C}}_3\right)$ indicated in Fig. 13 (infinitesimally on the superfluid side). For further discussions see the main text. The bulk parameters of the system are those belonging to Figs. 4 and 13. $T_{\text{peak}}/K=8.3346$ is the position of the peak.

Figure 16

Film thickness $y/a$ versus temperature $T$ at ${\mathcal{C}}_3={\mathcal{C}}_3^{\text{tce}}$ for four values of $\mathrm{\Delta }{\mu }_{+}/K$. By increasing the offset value $|\mathrm{\Delta }{\mu }_{+}|$ the tricritical Casimir effect and complete wetting become less pronounced.

Figure 17

Film thickness $y/a$ as function of temperature $T$ for three values of ${\mathcal{C}}_3\ge {\mathcal{C}}_3^{\text{tce}}$, i.e., $\mathrm{\Delta }{\mathcal{C}}_3={\mathcal{C}}_3-{\mathcal{C}}_3^{\text{tce}}\ge 0$, and at $\mathrm{\Delta }{\mu }_{+}/K=-1.07\times {10}^{-4}$. The sudden drop in the green and in the violet curve occurs at $T_{\text{d}}$ close to the demixing temperature $T_{\text{d}}^{\left(\text{n}\right)}\left({\mathcal{C}}_3\right)$ (see Fig. 13 with the same color code). Below $T_{\text{d}}^{\left(\text{n}\right)}\left({\mathcal{C}}_3\right)$ the violet and the green curve merge with the black curve and follow the binodal denoted by $T_{\text{d}}^{\left(\text{n}\right)}\left({\mathcal{C}}_3\right)$ in Fig. 13. The black curve is similar to the one in Fig. 15 except that below $T_{\text{tce}}$ it follows the normal branch of the binodal [see Fig. 4]. The jumps are due to first-order layering transitions. This figure corresponds to Fig. 14. Note that $X$ and $X_{\text{t}}$ in Fig. 14 here correspond to ${\mathcal{C}}_3$ and ${\mathcal{C}}_3^{\text{tce}}$, respectively. Due to the offset from liquid-vapor coexistence the values of $T_{\text{d}}$ and $T_{\text{s}}$ differ slightly from $T_{\text{d}}^{\left(\text{n}\right)}\left({\mathcal{C}}_3\right)$ and $T_{\text{s}}\left({\mathcal{C}}_3\right)$ as shown in Fig. 13. The bulk parameters of the system are the same as in Figs. 4 and 13. $T_{\text{peak}}/K=8.3346$ is the position of the peak.

Figure 18

Film thickness $y/a$ as function of temperature $T$ for three values of ${\mathcal{C}}_3\le {\mathcal{C}}_3^{\text{tce}}$ with $\mathrm{\Delta }{\mathcal{C}}_3={\mathcal{C}}_3-{\mathcal{C}}_3^{\text{tce}}\le 0$ and at $\mathrm{\Delta }{\mu }_{+}/K=-1.07\times {10}^{-4}$. The black curve is the same as the one in Fig. 15. The sudden drop in the blue and in the red curve occurs at $T_{\lambda }$ close to the temperature of the $\lambda$ transition $T_{\lambda }\left({\mathcal{C}}_3\right)$ (see Fig. 13). The red and the blue curve merge with the black curve at $T_{\text{d}}$ and follow the binodal denoted by $T_{\text{d}}^{\left(\text{s}\right)}\left({\mathcal{C}}_3\right)$ in Fig. 13. This figure corresponds to panel (b) in Fig. 14. Note that $X$ and $X_{\text{t}}$ in Fig. 14 here correspond to ${\mathcal{C}}_3$ and ${\mathcal{C}}_3^{\text{tce}}$, respectively. Due to the offset from liquid-vapor coexistence the value of $T_{\text{d}}$, $T_{\text{s}}$, and $T_{\lambda }$ differ slightly from $T_{\text{d}}^{\left(\text{s}\right)}\left({\mathcal{C}}_3\right)$, $T_{\text{s}}\left({\mathcal{C}}_3\right)$, and $T_{\lambda }\left({\mathcal{C}}_3\right)$ as introduced in Fig. 13. The bulk parameters of the system are the same as in Figs. 4 and 13. $T_{\text{peak}}/K=8.3346$ is the position of the peak.

Figure 19

Scaling functions of the TCF calculated from the data in Fig. 17 within (a) the wetting film geometry and (b) the slab approximation. Concerning the definition of the slab thickness $L_0$ see the main text. The violet curve and the green curve merge with the black curve at their corresponding demixing point indicated by $T_{\text{d}}$ in Fig. 17, using the same color code. The corresponding curves in the two panels agree qualitatively but differ in detail, e.g., in height (see the horizontal lines). In panel (a) the thermodynamic states are off the liquid-vapor coexistence surface, whereas in panel (b) the thermodynamic states lie on the liquid-vapor coexistence surface. The reduced temperature is $t=(T-T_{\text{tce}})/T_{\text{tce}}$, where $T_{\text{tce}}$ is the temperature of the tricritical end point. Due to the smoothing procedure and within the presently available numerical accuracy, the small difference between the positions of the maxima in (a) and (b) cannot be resolved reliably.

Figure 20

Scaling functions of the TCF calculated from the data in Fig. 18 within (a) the wetting film geometry and (b) the slab approximation. Concerning the definition of the slab thickness $L_0$ see the main text. The blue curve and the red curve merge with the black curve at their corresponding demixing point, indicated by $T_{\text{d}}$ in Fig. 18. The corresponding curves in the two panels agree qualitatively but differ in detail, e.g., in height (see the horizontal lines). The dashed blue curve shows the region, where the blue curve is multivalued and scaling does not hold anymore. The same holds also for the right parts of the red curves, because the drops of the curves exhibit a slightly positive slope. In panel (a) the thermodynamic states are off the liquid-vapor coexistence surface, whereas in panel (b) the thermodynamic states lie on the liquid-vapor coexistence surface. The reduced temperature is $t=(T-T_{\text{tce}})/T_{\text{tce}}$, where $T_{\text{tce}}$ is the temperature of the tricritical end point. Due to the smoothing procedure and within the presently available numerical accuracy, the small difference between the positions of the maxima in (a) and (b) cannot be resolved reliably.

Figure 21

Scaling functions of the TCF calculated from the data in Fig. 16 within (a) the wetting film geometry and (b) the slab approximation. Concerning the definition of the slab thickness $L_0$ see the main text. The maxima of the scaling functions in panel (a) differ from each other, whereas the ones in panel (b) are almost equal. The reduced temperature is $t=(T-T_{\text{tce}})/T_{\text{tce}}$, where $T_{\text{tce}}$ is the temperature of the tricritical end point.

Figure 22

Adjusted scaling functions (see the main text) obtained for the slab geometry as in Refs. [20, 21] and inferred from the wetting films compared with the corresponding experimental curve [3]. All data correspond to the tricritical concentration of ${}^3\mathrm{He}$. $\overline{L}$ is the film thickness of the wetting films, whereas in Refs. [20, 21] it denotes the width of the slab. The reduced temperature $t=(T-T_{\text{tc}})/T_{\text{tc}}$ is relative to tricritical point in Refs. [20, 21] and relative to the tricritical end point for the wetting film. The experimental data has been obtained via private communication with the corresponding authors.

Figure 23

Schematic representation of the transformation of the bulk phase diagram of a classical binary liquid mixture for fixed values of $(C_0/K_0,J_0/K_0)$ and $J_s=0$ (the dotted curve) into that of ${}^3\mathrm{He}-{}^4\mathrm{He}$ mixtures with $J_s=J_s^0\ne 0$ (solid curves). In a first step, for suitable, fixed values of $(C_0/K_0,J_0/K_0)$ and $J_s=0$ one has the phase diagram of a classical binary mixture with ce as in Fig 4. In a second step, one has to find a nonzero value of $J_s=J_s^0$ (which produces the superfluid phase) such, that the critical end point ce of the phase diagram for $(C_0/K_0,J_0/K_0,J_s=0)$, is in thermodynamic coexistence with a superfluid phase. For this new set of coupling constants $(C_0/K_0,J_0/K_0,J_s>J_s^0)$, the phase diagram with $(C_0/K_0,J_0/K_0,J_s=0$) lies in the two-phase region of the phase diagram with $(C_0/K_0,J_0/K_0,J_s>J_s^0)$.