• Chinese Physics B
  • Vol. 29, Issue 9, (2020)
Zechao Jiang1,2, Xiuyuan Yang1,2, Mengmeng Wu1,2, and Xingkun Man1,2,†
Author Affiliations
  • 1Center of Soft Matter Physics and Its Applications, Beihang University, Beijing 009, China
  • 2School of Physics, Beihang University, Beijing 100191, China
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    DOI: 10.1088/1674-1056/ab8ac7 Cite this Article
    Zechao Jiang, Xiuyuan Yang, Mengmeng Wu, Xingkun Man. The drying of liquid droplets[J]. Chinese Physics B, 2020, 29(9): Copy Citation Text show less

    Abstract

    The drying of liquid droplets is a common phenomenon in daily life, and has long attracted special interest in scientific research. We propose a simple model to quantify the shape evolution of drying droplets. The model takes into account the friction constant between the contact line (CL) and the substrate, the capillary forces, and the evaporation rate. Two typical evaporation processes observed in experiments, i.e., the constant contact radius (CCR) and the constant contact angle (CCA), are demonstrated by the model. Moreover, the simple model shows complicated evaporation dynamics, for example, the CL first spreads and then recedes during evaporation. Analytical models of no evaporation, CCR, and CCA cases are given, respectively. The scaling law of the CL or the contact angle as a function of time obtained by analytical model is consistent with the full numerical model, and they are all subjected to experimental tests. The general model facilitates a quantitative understanding of the physical mechanism underlying the drying of liquid droplets.
    x=Rsinα,r=Rsinθ,H=R(1cosθ),h=R(cosαcosθ),V=13πR3(1cosθ)2(2+cosθ).(1)

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    V˙(t)=V0.r(t)r0=V0.R(t)sinθR0sinθ0,(2)

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    J(t)=V˙(t)πr2(t)=V0.πr0r(t).(3)

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    =Φ+F˙.(4)

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    F=(γLSγSV)πr2+γLV0θ2πxRdα=πR2γLV[sin2θcosθe+2(1cosθ)],(5)

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    cosθe=γSVγLSγLV.(6)

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    F˙=2γLVπR[R˙(22cosθsin2θcosθe)+Rsinθθ˙(1cosθcosθe)].(7)

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    Φ=120r2πx3ηhv2dx+πξclrr˙2.(8)

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    h˙=1xx(xvh)J.(9)

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    v(x,t)=R˙(cosα1)(cosαcosθ+cosα+cosθ)sinα(1+cosθ).(10)

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    Φ=3πηRR˙2C1(θ)+πξclRsin3θR˙2+πξclR3sinθcos2θθ2.+2πξclR2sin2θcosθR˙θ˙,(11)

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    C1(θ)=116cos3θcos2θ92cosθ+23371+cosθ2(1+cosθ)2+cos3θ(1cosθ)(2+cosθ)2(1+cosθ)3ln1cosθε+2(1+cosθ)3ln21+cosθ,(12)

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    (Φ+F˙)R˙=6πηRR˙C1(θ)+2πξclRR˙C2(θ)+2ξclV˙RC3(θ)+2πγLVRC4(θ)=0,(13)

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    C2(θ)=(1cosθ)2sinθ(1+cosθ)2,C3(θ)=cosθsinθ(1+cosθ)2,C4(θ)=1cosθ1+cosθ(cosθcosθe).()

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    τev=V0V0.,τre=ηr0γLV,(14)

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    kev=τreτev,(15)

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    {(Φ+F˙)R˙=0,V˙=0.(16)

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    {τrer˙=(1cosθ)23C1(θ)sinθ(1+cosθ)2(cosθcosθe),τreθ˙=(1cosθ)2(2+cosθ)3C1(θ)(1+cosθ)2r(cosθcosθe),(17)

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    τrer˙k1θ3=k2r9,(18)

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    R˙=γLVC4(θ)3ηC1(θ).(19)

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    τevr˙=(1cosθ0)2(2+cosθ0)cotθ3sin3θ0r+(1cosθ)2(cosθecosθ)3C1(θ)kevsinθ(1+cosθ)2,(20)

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    τevθ˙=(1cosθ0)2(2+cosθ0)3sin3θ0r2(1cosθ)2(2+cosθ)(cosθecosθ)3kevC1(θ)(1+cosθ)2r.(21)

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    r=123t.(22)

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    τevr˙=(1cosθ0)2(2+cosθ0)cosθ3sin3θ0sinθr+1cosθ1+cosθsinθ3sinθ(1+cosθ)2C1(θ)+kcl(1cosθ)2×[kcl(1cosθ0)2(2+cosθ0)cosθ3sin3θ0r+sinθ(cosθecosθ)kev],(23)

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    τevθ˙=(1cosθ0)2(2+cosθ0)3sin3θ0r2(1cosθ)2(2+cosθ)3sinθ(1+cosθ)2C1(θ)+kcl(1cosθ)2×[kcl(1cosθ0)2(2+cosθ0)cosθ3sin3θ0r2+sinθ(cosθecosθ)kevr].(24)

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    θ˙=(1cosθ0)(2+cosθ0)3τevsinθ0(1+cosθ0)(1+cosθ)2.(25)

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    Zechao Jiang, Xiuyuan Yang, Mengmeng Wu, Xingkun Man. The drying of liquid droplets[J]. Chinese Physics B, 2020, 29(9):
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