An ideal solid apparent is one that is flat, rigid, altogether smooth, chemically homogeneous, and has aught acquaintance bend hysteresis. Aught hysteresis implies that the advancing and abbreviating acquaintance angles are equal. In added words, there is alone one thermodynamically abiding acquaintance angle. If a bead of aqueous is placed on such a surface, the appropriate acquaintance bend is formed as depicted in Fig. 1. Furthermore, on an ideal apparent the bead will acknowledgment to its aboriginal appearance if it is disturbed.25 The afterward derivations administer alone to ideal solid surfaces. In added words, they are alone accurate for the accompaniment in which the interfaces are not affective and the appearance abuttals band exists in equilibrium.
edit Minimization of energy, three phases
Figure 3: Coexistence of 3 aqueous phases in alternate contact. α, β, and θ represent both the labels of the phases and the acquaintance angles.
Figure 4: Neumann's triangle apropos the apparent energies and acquaintance angles of 3 aqueous phases circumstantial in changeless equilibrium, as depicted in Figure 3.
Figure 3 shows the band of acquaintance area three phases meet. In equilibrium, the net force per assemblage breadth acting forth the abuttals band amid the three phases have to be zero. The apparatus of net force in the administration forth anniversary of the interfaces are accustomed by:
\gamma_{\alpha\theta}+\gamma_{\theta\beta}\cos{\theta}+\gamma_{\alpha\beta}\cos{\alpha}\ = 0
\gamma_{\alpha\theta}\cos{\theta}+\gamma_{\theta\beta}+\gamma_{\alpha\beta}\cos{\beta}\ = 0
\gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{\alpha\beta}\ = 0
where α, β, and θ are the angles apparent and γij is the apparent activity amid the two adumbrated phases. These relations can aswell be bidding by an analog to a triangle accepted as Neumann’s triangle, apparent in Figure 4. Neumann’s triangle is constant with the geometrical brake that α + β + θ = 2π, and applying the law of sines and law of cosines to it aftermath relations that call how the interfacial angles depend on the ratios of apparent energies.6
Because these three apparent energies anatomy the abandon of a triangle, they are accountable by the triangle inequalities, γij < γjk + γik acceptation that no one of the apparent tensions can beat the sum of the added two. If three fluids with apparent energies that do not chase these inequalities are brought into contact, no calm agreement constant with Figure 3 will exist.
edit Simplification to collapsed geometry, Young's relation
If the β appearance is replaced by a collapsed adamant surface, as apparent in Figure 5, again β = π, and the additional net force blueprint simplifies to the Young equation,7
Figure 5: Acquaintance bend of a aqueous atom wetted to a adamant solid surface.
\gamma_{SG}\ =\gamma_{SL}+\gamma_{LG}\cos{\theta}8
which relates the apparent tensions amid the three phases: solid, aqueous and gas. Subsequently this predicts the acquaintance bend of a aqueous atom on a solid apparent from ability of the three apparent energies involved. This blueprint aswell applies if the "gas" appearance is addition liquid, immiscible with the atom of the aboriginal "liquid" phase.
edit Absolute bland surfaces and the Young acquaintance angle
The Young blueprint assumes a altogether collapsed and adamant surface. In abounding cases surfaces are far from this ideal situation, and two are advised here: the case of asperous surfaces (see Non-ideal asperous solid surfaces) and the case of bland surfaces that are still absolute (finitely rigid). Even in a altogether bland apparent a bead will accept a advanced spectrum of acquaintance angles alignment from the so alleged advancing acquaintance angle, θA, to the so alleged abbreviating acquaintance angle, θR. The calm acquaintance bend (θc) can be affected from θA and θR as was apparent by Tadmor9 as,
\theta_\mathrm{c}=\arccos\left(\frac{r_\mathrm{A}\cos{\theta_\mathrm{A}}+r_\mathrm{R}\cos{\theta_\mathrm{R}}}{r_\mathrm{A}+r_\mathrm{R}}\right)
where
r_\mathrm{A}=\left(\frac{\sin^3{\theta_\mathrm{A}}}{2-3\cos{\theta_\mathrm{A}}+\cos^3{\theta_\mathrm{A}}}\right)^{1/3} ~;~~ r_\mathrm{R}=\left(\frac{\sin^3{\theta_\mathrm{R}}}{2-3\cos{\theta_\mathrm{R}}+\cos^3{\theta_\mathrm{R}}}\right)^{1/3}
edit The Young–Dupré blueprint and overextension coefficient
The Young–Dupré blueprint (Thomas Young 1805, Lewis Dupré 1855) dictates that neither γSG nor γSL can be beyond than the sum of the added two apparent energies. The aftereffect of this brake is the anticipation of complete wetting if γSG > γSL + γLG and aught wetting if γSL > γSG + γLG. The abridgement of a band-aid to the Young–Dupré blueprint is an indicator that there is no calm agreement with a acquaintance bend amid 0 and 180° for those situations.
A advantageous constant for appraisal wetting is the overextension constant S,
S\ = \gamma_{SG}-(\gamma_{SL}+\gamma_{LG})
When S > 0, the aqueous wets the apparent absolutely (complete wetting). If S < 0, there is fractional wetting.
Combining the overextension constant analogue with the Young affiliation yields the Young–Dupré equation:
S\ = \gamma_{LG}(\cos\theta-1)
which alone has concrete solutions for θ if S <
edit Minimization of energy, three phases
Figure 3: Coexistence of 3 aqueous phases in alternate contact. α, β, and θ represent both the labels of the phases and the acquaintance angles.
Figure 4: Neumann's triangle apropos the apparent energies and acquaintance angles of 3 aqueous phases circumstantial in changeless equilibrium, as depicted in Figure 3.
Figure 3 shows the band of acquaintance area three phases meet. In equilibrium, the net force per assemblage breadth acting forth the abuttals band amid the three phases have to be zero. The apparatus of net force in the administration forth anniversary of the interfaces are accustomed by:
\gamma_{\alpha\theta}+\gamma_{\theta\beta}\cos{\theta}+\gamma_{\alpha\beta}\cos{\alpha}\ = 0
\gamma_{\alpha\theta}\cos{\theta}+\gamma_{\theta\beta}+\gamma_{\alpha\beta}\cos{\beta}\ = 0
\gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{\alpha\beta}\ = 0
where α, β, and θ are the angles apparent and γij is the apparent activity amid the two adumbrated phases. These relations can aswell be bidding by an analog to a triangle accepted as Neumann’s triangle, apparent in Figure 4. Neumann’s triangle is constant with the geometrical brake that α + β + θ = 2π, and applying the law of sines and law of cosines to it aftermath relations that call how the interfacial angles depend on the ratios of apparent energies.6
Because these three apparent energies anatomy the abandon of a triangle, they are accountable by the triangle inequalities, γij < γjk + γik acceptation that no one of the apparent tensions can beat the sum of the added two. If three fluids with apparent energies that do not chase these inequalities are brought into contact, no calm agreement constant with Figure 3 will exist.
edit Simplification to collapsed geometry, Young's relation
If the β appearance is replaced by a collapsed adamant surface, as apparent in Figure 5, again β = π, and the additional net force blueprint simplifies to the Young equation,7
Figure 5: Acquaintance bend of a aqueous atom wetted to a adamant solid surface.
\gamma_{SG}\ =\gamma_{SL}+\gamma_{LG}\cos{\theta}8
which relates the apparent tensions amid the three phases: solid, aqueous and gas. Subsequently this predicts the acquaintance bend of a aqueous atom on a solid apparent from ability of the three apparent energies involved. This blueprint aswell applies if the "gas" appearance is addition liquid, immiscible with the atom of the aboriginal "liquid" phase.
edit Absolute bland surfaces and the Young acquaintance angle
The Young blueprint assumes a altogether collapsed and adamant surface. In abounding cases surfaces are far from this ideal situation, and two are advised here: the case of asperous surfaces (see Non-ideal asperous solid surfaces) and the case of bland surfaces that are still absolute (finitely rigid). Even in a altogether bland apparent a bead will accept a advanced spectrum of acquaintance angles alignment from the so alleged advancing acquaintance angle, θA, to the so alleged abbreviating acquaintance angle, θR. The calm acquaintance bend (θc) can be affected from θA and θR as was apparent by Tadmor9 as,
\theta_\mathrm{c}=\arccos\left(\frac{r_\mathrm{A}\cos{\theta_\mathrm{A}}+r_\mathrm{R}\cos{\theta_\mathrm{R}}}{r_\mathrm{A}+r_\mathrm{R}}\right)
where
r_\mathrm{A}=\left(\frac{\sin^3{\theta_\mathrm{A}}}{2-3\cos{\theta_\mathrm{A}}+\cos^3{\theta_\mathrm{A}}}\right)^{1/3} ~;~~ r_\mathrm{R}=\left(\frac{\sin^3{\theta_\mathrm{R}}}{2-3\cos{\theta_\mathrm{R}}+\cos^3{\theta_\mathrm{R}}}\right)^{1/3}
edit The Young–Dupré blueprint and overextension coefficient
The Young–Dupré blueprint (Thomas Young 1805, Lewis Dupré 1855) dictates that neither γSG nor γSL can be beyond than the sum of the added two apparent energies. The aftereffect of this brake is the anticipation of complete wetting if γSG > γSL + γLG and aught wetting if γSL > γSG + γLG. The abridgement of a band-aid to the Young–Dupré blueprint is an indicator that there is no calm agreement with a acquaintance bend amid 0 and 180° for those situations.
A advantageous constant for appraisal wetting is the overextension constant S,
S\ = \gamma_{SG}-(\gamma_{SL}+\gamma_{LG})
When S > 0, the aqueous wets the apparent absolutely (complete wetting). If S < 0, there is fractional wetting.
Combining the overextension constant analogue with the Young affiliation yields the Young–Dupré equation:
S\ = \gamma_{LG}(\cos\theta-1)
which alone has concrete solutions for θ if S <
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