Phase diagram of fluid phases in ${}^3\mathrm{He}-{}^4\mathrm{He}$ mixtures

Fluid parts of the phase diagram of ${}^3\mathrm{He}-{}^4\mathrm{He}$ mixtures are obtained from a mean-field analysis of a suitable lattice gas model for binary liquid mixtures. The proposed model takes into account the continuous rotational symmetry O(2) of the superfluid degrees of freedom associated with ${}^4\mathrm{He}$ and includes the occurrence of vacancies. This latter degree of freedom allows the model to exhibit a vapor phase and hence can provide the theoretical framework to describe the experimental conditions for measurements of tricritical Casimir forces in ${}^3\mathrm{He}-{}^4\mathrm{He}$ wetting films.

Figure 1

Schematic bulk phase diagram of ${}^4\mathrm{He}$ exhibiting the vapor, superfluid, normal fluid, and solid phases. The liquid-vapor critical point is denoted by c, whereas ${\text{ce}}^{+}$ and ce are critical end points. The $\lambda$-line is the line of second-order phase transitions between the superfluid and the normal fluid.

Figure 2

Schematic bulk phase diagram of ${}^3\mathrm{He}-{}^4\mathrm{He}$ mixtures at fixed pressure. $X_3$ is the concentration of ${}^3\mathrm{He}$ and $T_{\lambda }\left(X_3\right)$ is the line of continuous superfluid transitions, which turn into first-order superfluid transitions at the tricritical point tc. Note that $T_{\lambda }\left(X_3\right)$ meets the two-phase region at its top. If $T_{\lambda }\left(X_3\right)$ met the two-phase region below $T_{\text{tc}}$, this would imply that there is a discontinuous phase transition either between two normal fluid phases or between two superfluid phases, which is not the case.

Figure 3

Schematic phase diagram of ${}^3\mathrm{He}-{}^4\mathrm{He}$ mixtures in the $(T,Z,P)$ space, where $Z=exp({\mu }_3/T)$ is the fugacity of ${}^3\mathrm{He}$ and $P$ is the pressure. ${\text{A}}_1$ and ${\text{A}}_2$ are the surfaces of the first-order solid-liquid and vapor-liquid phase transitions, respectively, whereas ${\text{A}}_3$ and ${\text{A}}_4$ are the surfaces of second- and first-order phase transitions between the normal fluid and the superfluid, respectively. ${\text{A}}_3$ intersects ${\text{A}}_1$ and ${\text{A}}_2$ along a line of critical end points connecting ${\text{ce}}^{+}$ with ${\text{tce}}^{+}$ and ce with tce, respectively. The surfaces ${\text{A}}_3$ and ${\text{A}}_4$ are separated by a line of tricritical points TC which meets ${\text{A}}_1$ and ${\text{A}}_2$ at the tricritical end points ${\text{tce}}^{+}$ and tce, respectively. ${\text{A}}_2$ terminates at a line of critical points, starting from c in the plane $Z=0$. The phase diagram in the plane $Z=0$ is the same as the one in Fig. 1. The dashed lines have no physical meaning; they indicate that the corresponding surface continues. The arrow ${\text{l}}_w$ indicates the thermodynamic path along which tricritical end point wetting occurs.

Figure 4

The schematic phase diagram for $H=0$ and $j_s=0$ (i.e., without coupling between the superfluid degrees of freedom). The phase diagram in the plane $Z=0$ is that of a one-component system consisting of particles “4.” Upon increasing the fugacity $Z$ of particles “3,” the transition lines in the plane $Z=0$ extend to form three distinct surfaces. The surface SL is a surface of first-order phase transitions between the solid and the liquid phases. The transition surfaces between the vapor and the liquid phases, and between the solid and the liquid phases, are denoted by VL and SV, respectively. The surface VL terminates at a line of critical points (green line). The critical point of the pure system of “4” particles is denoted by e. The liquid can be either mixed or demixed. Concerning the demixed phases, LL denotes the surface of first-order phase transitions between the phase rich in component 3 (large $Z$) and the phase rich in the component 4 (small $Z$). This surface terminates at a line of critical points (brown line), which meets the surfaces SL and VL at the critical end points b and c, respectively. The point a is a quadruple point, whereas d is a triple point. The lines a-b, a-c, and a-d are triple lines. The dashed lines have no physical meaning; they indicate that the corresponding surfaces continue.

Figure 5

Phase diagrams for the coupling constants $(c/k,j/k,j_s/k)=(1,5.714,0)$ and $H=0$. Along the triple line of three-phase coexistence (see line a-c in Fig. 4) the figures show the first-order demixing transitions of the liquid at coexistence with the vapor phase (a) in the $(X_3^l$, $T)$ plane, with $X_3^l={\langle N_3\rangle }_l/\mathcal{N}$ corresponding to the 3-particles in the liquid phase; (b) in the $(X_v$, $T)$ plane at coexistence with the two liquid phases, where $X_v=({\langle N_4\rangle }_v-{\langle N_3\rangle }_v)/\mathcal{N}$ in the vapor phase; (c) in the $(D_l$, $T)$ plane, where $D_l=({\langle N_4\rangle }_l+{\langle N_3\rangle }_l)/\mathcal{N}$ in the liquid phase; and (d) in the $(D_v$, $T)$ plane, where $D_v=({\langle N_4\rangle }_v+{\langle N_3\rangle }_v)/\mathcal{N}$ in the vapor phase. The indices $l$ and $v$ refer to the values of the order parameters in the liquid and in the vapor phase, respectively. The critical end point ce here corresponds to the point c in Fig. 4. Panels (e) and (f) show the dependencies of the pressure $P$ and of the fugacity $Z=exp({\mu }_3/T)$ on the temperature along the triple line a-c in Fig. 4. The two liquid states become identical at the critical end point ce at $T_{\text{ce}}/k=1.947$, above which the liquid is mixed. The coexisting liquid and vapor phases at ce are $(X_l^{\text{ce}},D_l^{\text{ce}},M_l^{\text{ce}})=(0.050,0.913,0)$ and $(X_v^{\text{ce}},D_v^{\text{ce}},M_v^{\text{ce}})=(-0.120,0.180,0)$, respectively. The transitions between the vapor phase and the liquid phases are always first order. According to (e) and (f), along a-c both $P$ and $Z$ vary as function of $T$, with the requirement of staying in coexistence with the vapor phase. This implies that the white domains in (a) and (c) are not projections of a three-dimensional surface, given by the equation of state, onto the $(T,X_3)$ and $(T,D)$ plane, respectively. The black lines provide only the $T$ dependence of $X_3$ and $D_l$ along the line a-c, which contains two branches. Similar remarks hold for Figs. 9–12.

Figure 6

Schematic phase diagram for small values of $j_s$. SL, VL, and SV are surfaces of first-order phase transitions with the same meanings as in Fig. 4. The points denoted as g and h are critical end points of the continuous superfluid transition of the 4-pure fluid (i. e., $Z=0$); g-h is the line of critical points for the continuous superfluid transition of the 4-pure fluid. ${\text{LL}}_3$ is a surface of continuous superfluid transitions bounded by the red lines g-h, h-f, i-g, and the violet line f-i. The triple point of the solid, vapor, and superfluid phases of the 4-pure fluid is denoted as d. In the plane $Z=0$ the line to the left of d is the sublimation curve of the 4-pure fluid; d-h-e is the liquid-vapor coexistence line of the 4-pure fluid, which ends at its critical point e; the extension of the latter to $Z>0$ forms the green line. The line d-g is the melting curve of the 4-pure solid into the superfluid and above g into the normal fluid. The line d-a is the triple line along which solid, vapor, and superfluid coexist; beyond a this line extends into a triple line along which solid, vapor, and normal fluid coexist. At the quadruple point a solid, vapor, normal fluid, and superfluid coexist. The surface ${\text{LL}}_1$ $\cup$ ${\text{LL}}_2$, which corresponds to the surface LL in Fig. 4, is the surface of first-order transitions between the 4-rich liquid at the back and the 3-rich liquid in the front; it is bounded by the brown line b-c of critical points which connects the critical end points b and c. The surface ${\text{LL}}_1$ $\cup$ ${\text{LL}}_2$ of first-order liquid-liquid demixing transitions terminates the surface ${\text{LL}}_3$ of continuous superfluid transitions. At this intersection this gives rise to the violet line f-i of critical end points, which themselves end at the end points i and f of this line of critical end points. At the surface ${\text{LL}}_2$ there are first-order phase transitions between two normal fluids, whereas at the surface ${\text{LL}}_1$ there are first-order phase transitions between a normal fluid with high concentration of 3-particles and a superfluid with high concentration of 4-particles. Accordingly, the superfluid phase forms a dome formed by the plane $Z=0$, SL, VL, ${\text{LL}}_3$, and ${\text{LL}}_1$ with the vertices d, a, f, h, g, and i.

Figure 7

Schematic phase diagram for a value of $j_s$, for which both critical (b-c and ${\text{LL}}_3$) and tricritical (j-k) phase transitions between liquids occur. SL, VL, and SV are the surfaces of first-order phase transitions similar to those in Fig. 4. The superfluid dome is characterized the vertices a, f, k, h, g, d, j, and i. Outside this region the liquid is a normal fluid. ${\text{LL}}_1$ is the surface of first-order phase transitions between the 3-rich normal fluid and the 4-rich superfluid, ${\text{LL}}_2$ (enclosed by the lines i-b, b-c, c-f, and f-i) is the surface of first-order demixing phase transitions between 3-rich and 4-rich normal fluids, whereas ${\text{LL}}_3$ is the surface of second-order phase transitions between the normal fluid and the superfluid. The blue line (j-k) is the line of tricritical points where the surface ${\text{LL}}_3$ connects to the new surface ${\text{LL}}_4$ of first-order phase transitions between the normal fluid and the superfluid liquid phases both being 4-rich. The surfaces ${\text{LL}}_2$ and ${\text{LL}}_3$ $\cup$ ${\text{LL}}_4$ meet at the line of triple points (light blue line i-f). The brown and the green lines are lines of critical points; a, i, and f are quadruple points, d is a triple point, whereas b, c, g, and h are critical end points. The line of triple points (i-f) ends on the surfaces SL and VL at the points i and f, respectively. The points j and k are tricritical end points. Note that in Fig. 6 the line i-f is a line of critical end points, whereas here it is a triple line. This different character motivates their different color code (violet versus light blue). This different character also implies that the lines a-i-b and a-f-c have a break in slope at i and f, respectively, here, but not in Fig. 6.

Figure 8

Schematic phase diagram for a value of $j_s$, for which only a tricritical line occurs. In this case, first-order phase transitions between liquid phases occur only between the superfluid and the normal fluid so the model exhibits only a (blue) line of tricritical points (j-k). SL, VL, and SV are the surfaces of first-order phase transitions as described in Fig. 4. The superfluid dome is characterized by the vertices a, k, h, g, d, and j. Outside this region the liquid is a normal fluid. The surface ${\text{LL}}_2$ from Fig. 7 does not exist anymore and the transitions between liquid phases are either second-order phase transitions between the normal fluid and the superfluid mixed liquid (${\text{LL}}_3$) or first-order phase transitions between the normal fluid and the superfluid liquid (${\text{LL}}_1\cup {\text{LL}}_4$). The surfaces ${\text{LL}}_3$ and ${\text{LL}}_1\cup {\text{LL}}_4$ meet at the line j-k of tricritical points (blue line). The points g and h are critical end points, whereas j and k are tricritical end points. The point a is a quadruple point and d is a triple point.

Figure 9

Phase diagram for the coupling constants $(c/k,j/k,j_s/k)=(1,5.714,1.717)$ and $H=0$ corresponding to Fig. 6. Along the triple line a-f in Fig. 6 the figures show the coexistence between vapor, normal fluid, and superfluid (a) in the $(X_3^l$, $T)$ plane, with $X_3^l=\langle N_3\rangle /\mathcal{N}$ corresponding to the number density of 3-particles in the liquid phase; (b) in the $(X_v$, $T)$ plane at coexistence with the two liquid phases, where $X_v=({\langle N_4\rangle }_v-{\langle N_3\rangle }_v)/\mathcal{N}=\langle X_4^v\rangle -\langle X_3^v\rangle$ in the vapor phase; (c) in the $(D_l$, $T)$ plane, where $D_l=({\langle N_4\rangle }_l+{\langle N_3\rangle }_l)/\mathcal{N}$ is the total number density in the liquid phase; and (d) in the $(D_v$, $T)$ plane, where $D_v=({\langle N_4\rangle }_v+{\langle N_3\rangle }_v)/\mathcal{N}$ is the total number density in the vapor phase. The indices $l$ and $v$ refer to the values of the order parameters in the liquid and the vapor phases, respectively. Panels (e) and (f) show the temperature dependence of $P$ and $Z$ along the triple line a-c Fig. 6 (black) and along the red line f-h near f in Fig. 6. The red line is the line of second-order transitions between the normal fluid and superfluid at coexistence with vapor, which ends at the demixing curve at the green critical end point e (i. e., f in Fig. 6). At e the liquid state $(X_l^e,D_l^e,M_l^e)=(0.773,0.994,0)$ coexists with the vapor state $(X_v^e,D_v^e,M_v^e)=(-0.051,0.059,0)$ at $T_e/k=0.758$. N and S denote normal fluid and superfluid, respectively. The transitions between the vapor and the liquid phases are always first order. The points ce and e here correspond to the points c and f in Fig. 6.

Figure 10

Phase diagram for the coupling constants $(c/k,j/k,j_s/k)=(1,5.714,2.231)$ and $H=0$ which corresponds to Fig. 7. Along the triple lines a-f and f-c in Fig. 7 the figures show the first-order demixing transitions of the liquid phase at coexistence with the vapor phase (a) in the $(X_3^l$, $T)$ plane, with $X_3^l={\langle N_3\rangle }_l/\mathcal{N}$ as the number density of 3-particles in the liquid phase; (b) in the $(X_v$, $T)$ plane at coexistence of the vapor with the two liquid phases, where $X_v=({\langle N_4\rangle }_v-{\langle N_3\rangle }_v)/\mathcal{N}$ in the vapor phase; (c) in the $(D_l$, $T)$ plane, where $D_l=({\langle N_4\rangle }_l+{\langle N_3\rangle }_l)/\mathcal{N}$ in the liquid phase; and (d) in the $(D_v$, $T)$ plane, where $D_v=({\langle N_4\rangle }_v+{\langle N_3\rangle }_v)/\mathcal{N}$ in the vapor phase. The indices $l$ and $v$ refer to the values of the order parameters in the liquid and the vapor phases, respectively. Panels (e) and (f) show the temperature dependence of $P$ and $Z$ along the triple lines a-f, f-c, and f-k in Fig. 7. The red line corresponds to second-order phase transitions between normal fluids and superfluids (line k-h in Fig. 7). At tce, the liquid state $(X_l^{\text{tce}},D_l^{\text{tce}},M_l^{\text{tce}})=(0.662,0.982,0)$ coexists with the vapor state $(X_v^{\text{tce}},D_v^{\text{tce}},M_v^{\text{tce}})=(-0.074,0.100,0)$ at $T_{\text{tce}}/k=0.917$. The point ce remains as in the case $j_s=0$. N and S denote normal liquid and superfluid, respectively. At the quadruple point qp the four coexisting states at $T_{\text{qp}}/k=0.887$ are two normal liquids $(X_l^{\text{qp}},D_l^{\text{qp}},M_l^{\text{qp}})=\{(-0.493,0.904,0),(0.594,0.982,0)\}$, a superfluid $(X_l^{\text{qp}},D_l^{\text{qp}},M_l^{\text{qp}})=(0.701,0.987,0.281)$, and the vapor state $(X_v^{\text{qp}},D_v^{\text{qp}},M_v^{\text{qp}})=(-0.082,0.105,0)$. The points ce, tce, and qp here correspond to the points c, k, and f, respectively in Fig. 7. In (b), (d)–(f), the long black coexistence curves are expected to exhibit a break in slope at qp; on the present scales this is not visible.

Figure 11

The same as in Fig. 10 but for the coupling constants $(c/k,j/k,j_s/k)=(1,5.714,2.747)$ and $H=0$. At tce the order parameters of the liquid and the vapor phases are $(X_l^{\text{tce}},D_l^{\text{tce}},M_l^{\text{tce}})=(0.644,0.961,0)$ and $(X_v^{\text{tce}},D_v^{\text{tce}},M_v^{\text{tce}})=(-0.071,0.135,0)$, respectively, and $T_{\text{tce}}/k=1.101$. Again, ce remains as in the case of $j_s=0$. N and S denote normal liquid and superfluid, respectively. The four coexisting states at the quadruple point qp are given by $(X^{\text{qp}},D^{\text{qp}},M^{\text{qp}})=(-0.222,0.888,0),(0.334,0.954,0),(0.742,0.984,0.456),\text{and}(0.110,0.157,0)$ at the temperature $T_{\text{qp}}/k=1.001$. The long black lines in (b) and (d) and the ones in (e) and (f) ending at ce are expected to exhibit a break in slope at qp; on the present scales this is not visible.

Figure 12

Phase diagrams for the coupling constants $(c/k,j/k,j_s/k)=(1,5.714,3.674)$ and $H=0$, corresponding to Fig. 8. The panels show the same as in Figs. 10 and 11; however, the critical end point ce has disappeared. At tce the liquid state $(X_l^{\text{tce}},D_l^{\text{tce}},M_l^{\text{tce}})=(0.693,0.900,0)$ coexists with the vapor state $(X_v^{\text{tce}},D_v^{\text{tce}},M_v^{\text{tce}})=(0.010,0.176,0)$ at $T_{\text{tce}}/k=1.462$. For $T>T_{\text{tce}}$ there is only a second-order phase transition from a normal mixed liquid to a superfluid. For $T<T_{\text{tce}}$ the phase transitions between the normal fluid and the superfluid are first order. N and S denote normal liquids and superfluids, respectively.

Figure 13

The $(T,P)$ phase diagram for the coupling constants $(c/k,j/k,j_s/k)=(1,5.714,3.674)$ and $H=0$ for pure ${}^4\mathrm{He}$, i.e., $Z=0$. The dashed green line shows the $\lambda$-line of second-order phase transitions between normal liquids and superfluids. N, S, and V denote the normal liquid, the superfluid, and the vapor phases, respectively. The blue line of first-order liquid-vapor transitions terminates the $\lambda$-line at the critical end point ce and ends at the critical point c of the liquid-vapor coexistence line. (We have been unable to find a set of coupling constants for which the dashed $\lambda$-line of second-order phase transitions exhibits a negative slope as it is the case for actual ${}^4\mathrm{He}$.)

Figure 14

Numerical results for the fluid parts of the phase diagram for the choice of the coupling constants $(c/k,j/k,j_s/k)=(1,5.714,3.674)$ and $H=0$ in $(P$, $Z$, $T)$ space. The points o, ce, c, and $\text{c}{}^{\prime \prime }$ lie in the zero fugacity plane ($Z=0$) and the points $\text{c}{}^{\prime \prime }$, t, and $\text{b}{}^{\prime }$ lie in the constant pressure plane $P/k=0.560$. The points o and $\text{b}{}^{\prime \prime }$ have the same temperature $T/k=1.071$, while b, $\text{b}{}^{\prime }$, and $\text{c}{}^{\prime }$ share the same value of the fugacity $Z=0.139$. The surface ($\text{o-ce-tce-b-b}{}^{\prime \prime }$-o) corresponds to first-order phase transitions between the vapor phase (V) and the superfluid phase (S), whereas ($\text{ce-c-c}{}^{\prime }$-b-tce-ce) is the surface of first-order phase transitions between the vapor phase and the normal liquid phase (N); their union corresponds to ${\text{A}}_2$ in Fig. 3. The surface ($\text{b-tce-t-b}{}^{\prime }$-b) is the surface of first-order phase transitions between the superfluid and the normal liquid phase corresponding to ${\text{A}}_4$ in Fig. 3 and ($\text{tce-ce-c}{}^{\prime \prime }$-t-tce) is the surface of second-order phase transitions between the superfluid and the normal liquid phase corresponding to ${\text{A}}_3$ in Fig. 3. The black line b-tce and the light red line tce-ce are the lines of first- and second-order liquid-liquid transitions at coexistence with the vapor phase, respectively, which meet at the tricritical end point tce. The solid blue line $\text{c-c}{}^{\prime }$ is the line of critical points of the liquid-vapor phase transitions and the dark red curve (tce-t) is the line of tricritical points. The lowest pressure is $p/k=0$, whereas the highest temperature is $T/k=2.179$. The line $\text{o-b}{}^{\prime \prime }$ is the intersection of ${\text{A}}_2$ and the plane $T/k=1.071$, the line $\text{c}{}^{\prime }\text{-b-b}{}^{\prime \prime }$ is the intersection of ${\text{A}}_2$ and the plane $Z=0.139$; the line $\text{b-b}{}^{\prime }$ is the intersection of ${\text{A}}_4$ with the plane $Z=0.139$; the line $\text{b}{}^{\prime }$-t and $\text{t-c}{}^{\prime \prime }$ are the intersection of ${\text{A}}_4$ and ${\text{A}}_3$, respectively, with the plane $P/k=0.560$. We note that at ce the line o-ce-c does not exhibit a break in slope (see Fig. 13).

Figure 15

The liquid-liquid phase transitions at fixed pressure $P/k=0.560$ in the $(X_3$, $T)$ plane for the choice of the coupling constants $(c/k,j/k,j_s/k)=(1,5.714,3.674)$ and $H=0$. The figure provides the temperature dependence of $X_3$ along the line $\text{b}{}^{\prime }\text{-t-c}{}^{\prime \prime }$ in Fig. 14. For $T_t<T<T_{c{}^{\prime \prime }}$ the light blue dashed line represents continuous phase transitions, whereas for $T_{b{}^{\prime }}<T<T_t$ the lines indicate the coexisting superfluid (S) and normal liquid (N) states at first-order phase transitions. The two-phase region is shaded in gray. The point t corresponds to a tricritical point.