Normalized order parameter ψ(z)/|ψ∞| vs z/ξ∞ at T = T c in the semi-infinite case at ψ0 = 20|ψ∞| for positive and negative small ψ∞. The decay is algebraic for small z as in Eq. (2.15) and is eventually exponential as in Eq. (2.18). For ψ∞ < 0, the approach to ψ∞ is slow and the adsorption is large as in Eq. (2.20).
Normalized order parameter ψ(z)/ψ D vs z/D at T = T c in a film in the half region 0 < z < D/2 for s = ψ∞/ψ D = 2, 1, 0, −1.11, and −1.5 from above with ψ0 = 20ψ D in the symmetric boundary conditions, where ψ D is defined by Eq. (2.9). The decay is algebraic for small z as in Eq. (2.15).
Left: m = ψ m /ψ D vs s = ψ∞/ψ D for a film at T = T c in the limit ψ0 → ∞, where ψ m is the midpoint value, ψ∞ is the reservoir value, and ψ D ∝D −β/ν. Here, m stays at 0.8 in the weak response region (−0.9 ≲ s ≲ 0.7), changes steeply in the catastrophic region (−1.2 ≲ s ≲ −0.9), and tends to s in the strong reservoir region (|s| ≳ 1.1). Right: Normalized adsorption Γ/Dψ D vs s at T = T c . It approaches 2B + s −k for s ≳ 1 and 2B −|s|−k for s ≲ −1 (dotted lines) with k = ν/β − 1 from Eq. (2.20). The lines of Γ/Dψ D and 2B −|s|−k crosses at s = −0.90, where the Casimir amplitude Δ(s) is maximized from Eq. (2.52).
Casimir amplitude ratios for the force density and Δ(s)/Δcri for the grand potential vs s = ψ∞/ψ D at T = T c in the limit ψ0 → ∞. These quantities are maximized at s ∼ −1 upon the changeover between the weak response and catastrophic regions (see the text).
Normalized singular free energy density f (ψ)/f cri (left) and normalized distance from the criticality w/|τ| = 1/|S| (right) vs ψ/ψcx, where and ψcx = b cx|τ|β (both for τ > 0 and τ < 0). The upper (lower) curves correspond to those for τ > 0 (τ < 0). For τ < 0, w is independent of ψ within the coexistence curve |ψ|/ψcx < 1 in our model.
Normalized order parameter ψ(z)/ψ D vs z/D for μ∞ = 0 in a film in the half region 0 < z < D/2 for t = 30, 15, 5.51, −4, and −15 from below with ψ0 = 20ψ D in the symmetric boundary conditions. Here, ψ∞ = 0 for t > 0 and ψ∞ = ψcx for t < 0. The decay is algebraic for small z as in Eq. (2.15).
Left: Normalized midpoint order parameter m(t) = ψ m (t)/ψ D vs t = τ(D/ξ0)1/ν for a film with thickness D, where s = 0 for τ > 0 and s = ψcx/ψcx for τ < 0 (μ∞ = 0). The coexistence curve ψ cx (t)/ψ D vs t in Eq. (3.3) is also shown in the region t < 0. Right: Normalized excess adsorption Γ/Dψ D vs t for μ∞ = 0, whose maximum is 1.51 at t = 0. It tends to 2g ±|t|β − ν for t ≳ 10 and for t ≲ −1 with g + = 1.245 and g − = 0.471, respectively (see Eq. (3.45)).
vs t = τ(D/ξ0)1/ν along the critical path μ∞ = 0 from our theory (red bold line) and from Borjan and Upton's theory28 (blue broken line), which are in good agreement with the Monte Carlo data (+).25 Also plotted is Δ(t)/Δcri vs t from our theory on the critical path.
Normalized midpoint order parameter m(s, t) = ψ m /ψ D (top left), normalized excess adsorption Γ/ψ D D (top right), (bottom left), and Δ(s, t)/Δcri (bottom right) for t > 0 in the s-t plane. The latter three quantities are peaked for s ∼ −1. The amplitudes and Δ(s, t) are very small for t ≫ 1 and |s| ≫ 1.
Phase diagram of a near-critical fluid in a film for large adsorption in the s-t plane, where s = ψ∞/ψ D (∝ψ∞ D β/ν) and t = τ(D/ξ0)1/ν. Two-phase region is written in the right (in gray). There appears a first-order phase transition line (red) of capillary condensation with a critical point at (s, t) = (−1.27, −3.14). Plotted also are a line of maximum of Δ(s, t) with (∂Δ/∂s) t = 0 and a line of maximum of (broken lines). The former approaches the coexistence curve for t ≲ −6 and the latter is very close to the capillary condensation line. On paths (a)–(g), m, Γ, , and Δ are shown in Figs. 11 and 12.
Normalized midpoint order parameter m = ψ(D/2)/ψ D (top) and normalized excess adsorption Γ/ψ D D (bottom) vs s in a film with thickness D for t = −0.3, −2, −3.1, −4, −6, −8, and −10 from the right. See Eqs. (2.34) and (2.35) for Γ. For t < −3.1, these quantities are discontinuous across the capillary condensation line in Fig. 10.
Casimir amplitude ratios (top) and Δ(s, t)/Δcri (bottom) vs s for t = −0.3, −2, −3.1, −4, −6, −8, and −10 from the right in the range s < −ψcx/ψ D = −0.66|t|β. Curve (c) corresponds to the capillary-condensation critical point. Those (d)–(g) exhibit a first-order phase transition, where is discontinuous but Δ is continuous.
Left: Susceptibility (∂m/∂s) t = (∂ψ m /∂ψ∞)τ at t = −3.1 slightly above the capillary-condensation critical point (). Right: Relations between deviations and near the capillary-condensation critical point at t = −3.1 and away from it at t = −2 on a logarithmic scale. The former can be fitted to the mean-field form (3.48).
Normalized order parameter ψ(z)/ψ D vs z/D in a film in the half region 0 < z < D/2 at t = −4, with ψ0 = 20ψ D in the symmetric boundary conditions. From above, (A) s = −1.10 (at maximum of Δ), (B) −1.29 (near maximum of ), (C) −1.33, and (D) −1.50. A first-order phase transition occurs between (B) and (C). Lines of ψ∞ = ±ψcx are written, where ψcx/ψ D = 1.04.
(top), Δ(s, t)/Δcri (middle), and Γ*(s, t) (bottom) for t = −4 (left) and t = −10 (right). For t = −4, points (A), (B), (C), and (D) correspond to the curves in Fig. 14. There are equilibrium and metastable branches near the transition. Amplitude Δ is maximized at (A) (left) and at (a) (right), where Γ* = 0 from Eq. (3.42). The transition occurs between (b) and (b’), where Δ is continuous.
Scaled inverse susceptibility for a film R ∞ in Eqs. (2.52) and (3.42) in the s-t plane, which is of order 40|s|δ − 1 for |s| ≳ 1.
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