Addition/Correction pubs.acs.org/Langmuir
Cite This: Langmuir XXXX, XXX, XXX−XXX
Correction to “Equilibrium Contact Angle and Adsorption Layer Properties with Surfactants” Uwe Thiele,* Jacco H. Snoeijer, Sarah Trinschek, and Karin John Langmuir 2018, 34 (24), 7210−7221. DOI: 10.1021/acs.langmuir.8b00513 The last line in eq 27 on page 7213 should read ÄÅ ÉÑ R ÅÅ ∂xh ÑÑ Å = dx [−P − κ ϒ]δh(x) + ÅÅÅ ϒ + λhÑÑÑÑδh(R ) ÅÅÇ ξ ÑÑÖ 0
∫
cos θe = (27)
Γd =
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In the caption for Figure 2a on page 7213 it should read (a) In the macroscopic approach, the equilibrium contact angle is determined by the solid−liquid interfacial tension ϒsl and the liquid−gas and solid−gas interfacial tensions ϒ and ϒsg that depend on the respective surfactant concentrations Γd and Γa on the droplet and the adsorption layer.
2
g (Γ) = ϒ + kBT Γ[ln(Γa ) − 1]
(45)
gsg (Γ) = ϒ 0sg + kBT Γ[ln(Γasg2 ) − 1]
(46)
(47)
ϒsg(Γ) = gsg − Γ∂ Γgsg = ϒ 0sg − kBT Γ
(48)
(49)
Γa δ
(50)
kBT ϒ0 kBT
cos θe ≈ cos θe0 +
ϒ0 kBT
cos θe ≈ cos θe0 −
χ (Γa) = 1 −
ϒ0
(cos θe0 − 1)Γ Γdcos θe0
(51)
(52)
Γa
(53)
kBT Γa ln δ f ̂ (h )
ÄÅ ÉÑ ÅÅ ÑÑ kT f (h ,Γ) = f ̂ (h)ÅÅÅÅ1 − B Γ lnδ ÑÑÑÑ ÅÅ ÑÑ f ̂ (ha) ÅÇ ÑÖ
(56)
a
This results in the following modifications in the subsequent formulas (pages 7215 and 7216) and Figures 3 and 4 (page 7217): ϒ(Γ) = g − Γ∂ Γg = ϒ 0 − kBT Γ
ϒ 0 − kBT Γd
cos θe = cos θe0 +
The free energies g(Γ) and gsg in section “Application for Simple Energy” on page 7215 written consistently in the surface number density Γ should read 0
ϒ 0 cos θe0 − kBT Γa
(57)
yz h2 Γ jij h3 zz jj zz Q hΓ yz jjj 3η 2 η zz zz = jj zz zz jj 2 zz 2 Q ΓΓ z jj h Γ h Γ zz { jj zz + Γ D jj zz 2 η η { k
The mobility matrix in eq 60 on page 7216 should read ijQ hh Q = jjjj jQ k Γh
(60)
Figure 3. Profiles of film height h (top) and surfactant concentration Γ (bottom) evolving in the numerical simulations for large times. The simulations are performed for three different ratios δ =
a2 asg 2
of the effective molecular length scales of the surfactant at a2Γ̅ = 0.04 in (a) and three
different mean surfactant concentrations Γ̅ at δ = 2 in (b) while keeping the remaining parameters fixed at ϵ1 = 0.2 and ϵ2 = 0.4. The insets show a zoom of the contact line region. Note that the surfactant concentration Γw which occurs on the wedge in the mesoscopic description corresponds to the concentration Γd on the droplet.
© XXXX American Chemical Society
A
DOI: 10.1021/acs.langmuir.9b00616 Langmuir XXXX, XXX, XXX−XXX
Langmuir
Addition/Correction
where η and D denote a viscosity and diffusivity, respectively. Formula 62 in the section “Generalization to Arbitrary Interface Energies” on page 7218 should read χ=
1 [g (Γ) − g (Γ) − ϒsl] fâ sg
(62)
resulting in the modifications -[h ,Γ] =
-a =
l o
∫ oomoooϒsl + n
f ̂ (h) [gsg (Γ) − g (Γ) − ϒsl] f̂
| o o + ξ[g (Γ) − λ Γ] − Pho dx } o o o ~
∫ {gsg − λ Γ} dx
a
(63)
(65)
Figure 4. Surfactant concentration on the droplet (top) and equilibrium contact angle (bottom) depending on the surfactant concentration in the adsorption layer. The analytically obtained equilibrium conditions (solid lines) are compared to values extracted from time simulations (diamonds) for three values of δ. The dashed (dotted lines) show the values obtained by parameter continuation for the domain size Lx/l = 400 (Lx/l = 6000). The discrepancy in the equilibrium contact angle between the numerical and the analytical result can be attributed to finite size effects.
■
ACKNOWLEDGMENTS We thank R. T. van Gaalen for carefully checking the manuscript and Mathis Plapp for his constructive criticism.
B
DOI: 10.1021/acs.langmuir.9b00616 Langmuir XXXX, XXX, XXX−XXX
