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Measuring the dynamic surface tension by studying emanating liquid-air interfaces Cees vanRijn Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02413 • Publication Date (Web): 08 Oct 2018 Downloaded from http://pubs.acs.org on October 10, 2018
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Measuring the dynamic surface tension by studying emanating liquid-air interfaces Cees J.M. van Rijn, Microfluidics and Nanotechnology, Laboratory of Organic Chemistry, Wageningen University, Stippeneng 4, NL-6708 WE Wageningen, The Netherlands, Corresponding author,
[email protected] ABSTRACT: We have measured the dynamic surface tension of emanating surfaces with a new method by studying the dynamics of an emanating liquid-air interface and compared obtained data with literature at small time scales between 1 and 100 ms. The emanating liquid-air interface is being described with an equation related to the wellknown hydrostatic Young-Laplace equation for a pending liquid-air interface in which the only relevant forces are gravity g and surface tension γ. We explain that under specific constraints all mass parts of an emanating interface with density ρ will experience a same acceleration a(t). An asymptotic concave solution is subsequently numerically derived by making the corresponding Young-Laplace equation dimensionless and by dividing all lengths by a generalized capillary length λc; for a static pending interface it holds that λc =
𝛾/𝜌𝑔 and for the emanating interface we find λc(t) =
𝛾(𝑡)/𝜌(𝑎(𝑡) ― 𝑔). For an
emanating interface γ(t) can be derived by measuring both the time dependent acceleration a(t)-g and time dependent capillary length λc(t). Dynamic surface tension measurements of coffee show that the surface tension reduction according to the emanating jet
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method and with the maximum bubble pressure method are the same, herewith verifying the proposed model.
KEYWORDS Dynamic Surface Tension, Young-Laplace Equation, Emanating Jet, Capillary Length, Coffee, Asymptotic Shape Function.
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INTRODUCTION Various methods exist for measuring the dynamic surface tension γ(t) to study surfactant adsorption and surface tension dynamics at small and large timescales1,2. The dynamic surface tension is a function of surface age and usually decreases in time when surfactants are transported, adsorbed, and reoriented at the interface before the equilibrium surface tension value is reached. Known methods for measuring surface tension include force methods such as the Du Noüy ring1 and Wilhelmy plate1 technique, or shape methods such as the sessile and pendant drop technique, or pressure methods such as the small bubble surfactometer technique1. Normally the non-equilibrium or dynamic surface tension (DST) is a time dependent function of the age of an initially prepared surfactant-depleted surface and decreases in time when surfactants are being adsorbed and reoriented at the air-liquid surface. When a fresh new surface is created, mass transport of surfactants towards this surface will always occur by diffusion and convection. The contribution of convection depends on the method used. Examples of methods, in which convection is imposed deliberately but in a controlled and usually known manner, are the maximum bubble pressure method and the oscillating jet method.3-5
Here we present a new method to determine the dynamic surface tension at small time scales between 1 and 50 ms by studying the dynamics of an emanating liquid-air interface,
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for example when a coffee drop falls into a coffee bath and a short-lived emanating coffee jet is formed (see Figure 6). In nature one can encounter many different transient shapes of pendant drops, dripping threads and emanating jets, see Figure 1.
Figure 1. The Young-Laplace equation giving solutions that apply for the shape of various liquid bodies. (a) Sessile and pending drops, courtesy Feikje Breimer (b) Pending honey thread, courtesy Feikje Breimer. (c) Emanating jet from a water surface.
Using calculus of variations, Gauss6 unified earlier mathematical results of Young and Laplace7 to obtain equations and boundary conditions describing surface tension determined shapes8,9. It is known that solving the Young-Laplace equation is equivalent to minimizing the thermodynamic potential of a fluid-fluid system that may be in contact with a solid substrate10. This might imply that unstable liquid-air interfaces will always evolve towards a more stable solution of the Young-Laplace equation, such as described by the shape of sessile and pending drops11-18.
In Figure 1b a pending honey thread is depicted that slowly retracts to the spoon. The shape of the slowly moving honey-air interface can be described by a quasi-static solution
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of the Young-Laplace equation19,20. We show here that the Young-Laplace equation may be extended to describe the relatively fast varying shape of an emanating jet interface and to derive the dynamic surface tension (Figure 1c). Note a geometric similarity between Figures 1b and c, the pending honey thread and emanating jet seem (mirrored) images around a horizontal plane.
The well-known hydrostatic Young-Laplace equation10,12,19,20 expresses that the hydrostatic pressure ρgz with ρ mass density and g gravity constant at position z is compensated by the surface tension (γ) based Laplace pressure 2γ/Rc(z): 2𝛾
2𝛾
(1)
𝜌𝑔𝑧 = 𝑅𝑐(𝑧) ― 𝑅𝑐(𝑧 = 0)
with Rc(z) ≡1/2R1+1/2R2 denoting the mean radius of curvature and is determined by two principal radii of curvature (R1, R2) corresponding to tangent circles in two perpendicular planes. Please note that the z=0 level will be determined by physical considerations.
The asymptotic concave shape of a pending thread is a solution of the Young-Laplace equation provided that the surface tension-based forces are in equilibrium with the gravitational force (Figure2a).
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Figure 2. (a) Force balance between gravitation and surface tension forces of a pending thread. When the formed liquid thread including the drop remains stable for some time this implies that for every height value z the gravitation and surface tension forces are in equilibrium. (b) For an emanating jet the gravitation and surface tension forces are not in equilibrium, the jet will always accelerate a(z,t) in the downward direction with a value larger than g.
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For a pending thread (Figure2a) the net surface tension force Fsurface
tension
is directed
upward along the axial direction of the liquid thread having a circumference 2πR(zo) at height zo with magnitude F(zo)=2πR(zo).γ. At a height zo the hydrostatic pressure will counter-exert a downward directed force Fhydrostatic on the cross section πR2(zo) of the liquid with magnitude πR2(zo).ρgzo.
A downward gravitational force Fgravity on all the
thread mass between z = zo and z = ∞ will create an additional downward pulling force with magnitude g.mthread (zo). When the formed liquid thread including the drop remains stable for some time this implies that for every height value z the gravitation and surface tension forces are in equilibrium. For an emanating jet (Figure2b) the net surface tension force Fsurface
tension
is directed
downward along the axial direction of the liquid thread having a circumference 2πR(zo) at height zo with a magnitude F(zo)=2πR(zo).γ. The time dependent surface tension based downward acceleration d(t) (≡a(t)-g) will create (Newton’s third law) a positive pressure profile in the jet with magnitude ρd(t)z, herewith forming the boundary conditions for shaping the jet. At height zo an upward directed force Fpress.gradient on the cross section πR2(zo) of the liquid will therefore be present with magnitude πR2(zo).ρdzo due to the surface tension driven deceleration d of the jet. A downward gravitational force Fgravity on all the thread mass between z = zo and z = ∞ will create another downward directed force with magnitude g.mjet (zo). The sum of all these forces induce an acceleration with a magnitude a=g+d being larger than g towards the liquid bath. In Supporting Information
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a section Self Consistent Solution is included, showing that the downward acceleration based on the acting physical forces is/remains uniform throughout the whole jet, i.e. d(z,t)= d(t). Also here in the section Coaxial Cylinder Model an alternative derivation for the motion and shape of the jet is presented based on above considerations. A more elaborate theoretical model, including viscosity and other inertial contributions, will be presented next. THEORETICAL APPROACH Consider an emanating jet liquid-air interface, such as depicted in Figure 1c, 5a, 6a or 10a-c. Clearly, we cannot neglect here the dynamic movement of the jet fluid towards the bath, neither through the action of gravity, and neither through the surface tension-based attraction forces between the bath liquid surface and the jet surface. For an axisymmetric fluid element, the following momentum21 equation applies in the upward z direction: 𝜌∂𝑡𝑣 +𝜌𝑣∂𝑧𝑣 = ―𝛾∂𝑧
(
1 𝑅1
1
)
+ 𝑅2 +3
∂𝑧(𝑅2(∂𝑧𝑣)) 𝑅2
―𝜌𝑔
(2)
Here the velocity v=v(z,t) describes the flow field in the axial z direction and has been averaged in the radial direction. R(z,t) describes the radius of the emanating jet at time t and axial position z and R1, R2 = R1,2(z,t) the corresponding principal radii of curvature. If we assume that the inertial axial velocity gradient term is small with respect to the gravitational acceleration (v∂zv
