DERIVATION OF (Subscript[dA, LV])/(Subscript[dA, SL])

areas and volumes

Asl=2 r δ Sin[θ]
Alv=2 r δ θ
Vr= δ r^2(2θ-Sin[2θ])/2

find the radius as a function of V and theta

rv= r/.Solve[Vr==Vol,r][[2]]

derivatives of the area

FullSimplify[(( 2δ(rv+ θ D[rv,θ]) )/(2δ(rv Cos[θ]+Sin[θ] D[rv,θ] )))]

we could also do things more cleanly, computing dLog[r]/dθ

lr=(Log[Vol/δ]-Log[θ-Sin[2θ]/2])/2

(1/2) (Log[(Vol/δ)]-Log[θ-(1/2) Sin[2 θ]])

dlr=FullSimplify[D[lr,θ]]

FullSimplify[(( 2δ(1+ θ dlr) )/(2δ( Cos[θ]+Sin[θ] dlr )))]

SOLID-LIQUID INTERFACE (SMOOTH)

DEFINITIONS

V[r_,th_]:=Pi/6 r^3 *(1-Cos[th])(3Sin[th]^2+(1-Cos[th])^2)

S12[r_,th_]:=Pi r^2 Sin[th]^2
S23[r_,th_]:=2Pi r^2 (1-Cos[th])

rv=r/.FullSimplify[Solve[v==V[r,th],r][[2]]]

rad[th_]:=(((2^(2/3)) ((((3/π)))^(1/3)) )/(((8-9 Cos[th]+Cos[3 th]))^(1/3)))

sc[th_]:=Show[Graphics[{
RGBColor[1.0,0.5,0.2],
Disk[{0,-rad[th]Cos[th]},rad[th]],
RGBColor[0.3,0.4,1.0],Rectangle[{-2,-1},{2,0}]
}],PlotRange->{{-2,2},{-1,2}}]

sc[Pi*2/3]

interactive

Manipulate[Show[sc[ArcCos[(Gsv-Gsl)/Glv]],
Graphics[
{Text[Dynamic["(Subscript[W, SVL])="<>ToString[Wsvl=Gsv+Glv-Gsl]],Offset[{5,-5},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, SV])="<>ToString[Gsv=Wsvl+Gsl-Glv]],Offset[{5,-15},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, LV])="<>ToString[Glv=Wsvl+Gsl-Gsv]],Offset[{5,-25},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, SL])="<>ToString[Gsl=Gsv+Glv-Wsvl]],Offset[{5,-35},{-2,2}],{-1,1}],
Text[Dynamic[Style["θ="<>ToString[SetAccuracy[ArcCos[(Gsv-Gsl)/Glv]*180/Pi,2]]],Bold],Offset[{5,-50},{-2,2}],{-1,1}]
}
]]
,
{{Wsvl,1,"(Subscript[W, SVL])"},0,10},{{Gsv,1,"(Subscript[γ, SV])"},0,10},{{Glv,1,"(Subscript[γ, LV])"},0,10},{{Gsl,1,"(Subscript[γ, SL])"},0,10},SaveDefinitions->True]

SOLID-LIQUID INTERFACE (Wenzel)

DEFINITIONS

V[r_,th_]:=Pi/6 r^3 *(1-Cos[th])(3Sin[th]^2+(1-Cos[th])^2)

S12[r_,th_]:=Pi r^2 Sin[th]^2
S23[r_,th_]:=2Pi r^2 (1-Cos[th])

rv=r/.FullSimplify[Solve[v==V[r,th],r][[2]]]

rad[th_]:=(((2^(2/3)) ((((3/π)))^(1/3)) )/(((8-9 Cos[th]+Cos[3 th]))^(1/3)))

surface area for the “cosine surface”

D[am*Cos[x*2Pi/lm],x]

(*w=Integrate[Sqrt[1+ (pre Sin[2Pi x/lm])^2],{x,0,lm},Assumptions->{y>0,lm>0,pre>0}]/(lm)/.{pre->am 2Pi/lm}*)

phi[r_]:=2 EllipticE[-4 (r^2) (π^2)]/π

Remove[sc]
Module[{th},
Off[FindRoot::cvmit];
Off[FindRoot::lstol];
scwenzel[th0_,r_,am_:0.1]:=
Show[Graphics[{
th=ArcCos[Cos[th0]*phi[r]];
RGBColor[1.0,0.5,0.2],
Disk[{0,-rad[th]Cos[th]},rad[th]],
Rectangle[Re[{-x,-1}],
{Re[x],am Cos[2Pi Re[ x] r/am]}]/.FindRoot[Sqrt[rad[th]^2-x^2]*Sign[Cos[th]]-rad[th]Cos[th]==am Cos[2Pi x r/am],{x,0.01}]
}],

Plot[am Cos[2Pi x r/am],{x,-2,2},Filling->-1, PlotStyle->RGBColor[0.3,0.4,1.0], FillingStyle->RGBColor[0.3,0.4,1.0]]
,
PlotRange->{{-2,2},{-1,2}}]
]

sc[Pi 0.77,0.2,0.02]

interactive

Manipulate[Show[scwenzel[ArcCos[(Gsv-Gsl)/Glv],r,h],
Graphics[
{Text[Dynamic["(Subscript[W, SVL])="<>ToString[Wsvl=Gsv+Glv-Gsl]],Offset[{5,-5},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, SV])="<>ToString[Gsv=Wsvl+Gsl-Glv]],Offset[{5,-15},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, LV])="<>ToString[Glv=Wsvl+Gsl-Gsv]],Offset[{5,-25},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, SL])="<>ToString[Gsl=Gsv+Glv-Wsvl]],Offset[{5,-35},{-2,2}],{-1,1}],
Text[Dynamic[Style["(Subscript[θ, 0])="<>ToString[SetAccuracy[ArcCos[(Gsv-Gsl)/Glv]*180/Pi,2]]],Bold],Offset[{5,-50},{-2,2}],{-1,1}],
Text[Dynamic[Style["(Subscript[θ, macro])="<>ToString[SetAccuracy[ArcCos[phi[r](Gsv-Gsl)/Glv]*180/Pi,2]]],Bold],Offset[{5,-60},{-2,2}],{-1,1}],
Text[Dynamic[Style["ϕ="<>ToString[SetAccuracy[phi[r],2]]],Bold],Offset[{5,-70},{-2,2}],{-1,1}]
}
]]
,
{{Wsvl,1,"(Subscript[W, SVL])"},0,10},{{Gsv,1,"(Subscript[γ, SV])"},0,10},{{Glv,1,"(Subscript[γ, LV])"},0,10},{{Gsl,1,"(Subscript[γ, SL])"},0,10},"Surface roughness",
{{r,0.01,"σ"},0,0.5},
{{h,0.02,"h"},0.01,0.2}, SaveDefinitions->True
]

SOLID-LIQUID INTERFACE (Cassie)

DEFINITIONS

Remove[sc]
Module[{th},
Off[FindRoot::cvmit];
Off[FindRoot::lstol];
sccassie[th0_,phi_,am_:0.1]:=
Show[Graphics[{
th=ArcCos[(1+Cos[th0])*phi-1];
RGBColor[1.0,0.5,0.2],
Disk[{0,-rad[th]Cos[th]},rad[th]],
White,
Rectangle[{-2,-1},{2,am}]
}],

Plot[am If[2Abs[5(x+100)-Round[5(x+100)]]<=phi,1,-1],{x,-2,2},PlotPoints->1000,Filling->-1, PlotStyle->RGBColor[0.3,0.4,1.0], FillingStyle->RGBColor[0.3,0.4,1.0]]
,
PlotRange->{{-2,2},{-1,2}}]
]

sc[Pi 0.2,0.8,0.1]

interactive

Manipulate[Show[sccassie[ArcCos[(Gsv-Gsl)/Glv],r,0.05],
Graphics[
{Text[Dynamic["(Subscript[W, SVL])="<>ToString[Wsvl=Gsv+Glv-Gsl]],Offset[{5,-5},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, SV])="<>ToString[Gsv=Wsvl+Gsl-Glv]],Offset[{5,-15},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, LV])="<>ToString[Glv=Wsvl+Gsl-Gsv]],Offset[{5,-25},{-2,2}],{-1,1}],
Text[Dynamic["(Subscript[γ, SL])="<>ToString[Gsl=Gsv+Glv-Wsvl]],Offset[{5,-35},{-2,2}],{-1,1}],
Text[Dynamic[Style["ϕ="<>ToString[SetAccuracy[r,2]]],Bold],Offset[{5,-70},{-2,2}],{-1,1}],
Text[Dynamic[Style["(Subscript[θ, 0])="<>ToString[SetAccuracy[ArcCos[(Gsv-Gsl)/Glv]*180/Pi,2]]],Bold],Offset[{5,-50},{-2,2}],{-1,1}],
Text[Dynamic[Style["(Subscript[θ, macro])="<>ToString[SetAccuracy[ArcCos[(1+(Gsv-Gsl)/Glv)*r-1]*180/Pi,2]]],Bold],Offset[{5,-60},{-2,2}],{-1,1}]
}
]]
,
{{Wsvl,1,"(Subscript[W, SVL])"},0,10},{{Gsv,1,"(Subscript[γ, SV])"},0,10},{{Glv,1,"(Subscript[γ, LV])"},0,10},{{Gsl,1,"(Subscript[γ, SL])"},0,10},"Surface roughness",
{{r,1,"ϕ"},0.1,1},SaveDefinitions->True
]