Slow evaporation of a small sessile liquid drop: new solutions based on the electrostatic analogy

Evaporating sessile drop of liquid deposited on horizontal surface is an important object of applications and theoretical investigations. The evaporating flow density dramatically depends on contact angle of the droplet. Previously, the dependence of the flux density on the polar angle was established for arbitrary contact angles. The expression has the form of an integral and is rather complicated for use in modelling algorithms. In this paper, we propose new exact solutions for the set of some specific contact angles (135-180 degrees). The analogy of the drop evaporation problem with a similar problem from the field of electrostatics of equipotential surfaces, which has been solved in general terms before us, was used. The results of British mathematician H.M. Macdonald and Soviet scientist G.A. Grinberg were used in the derivation. Thus, relatively simple expressions for the evaporation flux density were found.


Introduction
Evaporating sessile drop of liquid solution deposited on horizontal surface is an important object of applications (printing technologies, electronics, sensorics, medical diagnostics, hydrophobic coatings, etc) and theoretical investigations (evaporation processes, microfluidics inside the drop, self-assembly of nanoparticle, crystallization of the solute molecules, etc.) [1][2][3][4].J.C. Maxwell was the founder of the theory of evaporation of droplets in a gaseous medium.In 1877, in an article for the Encyclopedia Britannica under the subtitle "Theory of the Wet Bulb Thermometer", he considered the simplest case of stationary evaporation of a spherical drop with respect to an infinitely extended homogeneous medium.Maxwell calculated the diffusive drift of vapor from the surface of the evaporating droplet into the air, assuming that the vapor concentration at the surface of the droplet determines by saturated vapor density at the given temperature, ns.This is valid for a drop radius much larger than the mean free path length of vapor molecules in the air, i.e., for example, this is not true for droplets smaller than 100 nm in the air.In the following, we consider the droplets for which the Maxwell model is valid [5].
Consider the evaporation of a stationary isolated spherical drop suspended in an infinitely extended air medium under normal conditions with vapor concentration n0..It is necessary to solve the equation of solvent diffusion from the drop surface: where D is the vapor diffusion coefficient in the surrounding air; n(x,y,z,t) is the solvent concentration in the gas phase, with the following initial boundary conditions: on the drop surface (S) ( , ) const s n S t n   , t > 0. At the initial moment (t = 0) everywhere outside the drop the vapor concentration is determined by the asymptotic value of the vapor concentration in the atmosphere (for aqueous solution, it is the relative humidity of the air 0 ( , 0) const n n    ).In spherical coordinates, if there is no dependence on angles, we obtain the equation whose solution is given by the formula where R is the droplet radius, r > R is radial coordinate.From these formulas we obtain an expression for the integral flux where corresponds to the quasi-stationary (very slow) evaporation process.The smallness of the second term in brackets in formula (4) compared to unity is a condition of fairness of the quasi-stationarity approximation.For water droplets evaporating at 20°C in dry air, as shown by substitution of characteristic values, this condition is fulfilled with high accuracy [6].Thus, in the case of quasi-stationary evaporation, the diffusion equation ( 1 ).A deviation from quasi-stationarity can occur for droplets with high vapor pressure (very rapidly evaporating liquids), but in this case, one should not neglect the drop in temperature of the droplet surface, i.e., one should consider both concentration and temperature relaxation of the system.In this case, the heat conduction equation should be included in the system of initial equations.Problem ( 5) is mathematically identical to the electrostatics problem of finding the electrostatic potential φ near an equipotential surface.Such equipotential surfaces can be surfaces with high electrical conductivity, for example, metallic surfaces.The equation for the electrostatic potential outside charged surfaces is known to be of the form: (6) Solving equation (6) for electrostatics has a long history, so that previously developed electrostatics methods can be applied to the droplet evaporation problem (5).

Small evaporating droplet deposited on the flat horizontal substrate
The problem of evaporation of a sessile droplet is solved similarly to the case of a spherical pendant droplet considered above, only with boundary conditions corresponding to the contact of the droplet with the substrate (figure 1).If the characteristic size of a drop, which is its height h, is much smaller than the capillary constant, i.e., satisfies the inequality , where  is the surface tension at the boundary "solution-air",  is the density of the liquid, g is the acceleration of gravity, then the equilibrium shape of the drop is determined mainly by the surface tension.For aqueous solutions, this condition is fulfilled for droplets whose size is much smaller than 3 mm.The equilibrium form of such a drop on a flat horizontal substrate is a spherical segment with a given edge angle (wetting angle).
Although under such assumptions the shape of the drop on the substrate is quite simple, there is a significant complication that distinguishes this case from the spherical drop.It is necessary to take into account the interaction between the solution and the substrate.This interaction is multifactorial and, in general, nonlinear in the contribution of individual components.
Wetting forces, substrate roughness, the presence of impurities of dissolved substances, each of which has its own specificity in drying and interaction with the substrate, as well as the ionic strength of the solvent, external conditions, temperature regime of the substrate and other factors are important.A universal way to take into account these factors for any type of droplet systems in a single model is hardly possible from a practical point of view.But consideration of some limiting model systems, where only the most universal factors are taken into account, allows us to describe the main features of the processes.
(a) (b) In the last 50 years, at the new stage of development of fundamental sciences and technologies, a large number of papers on the microfluidics of an evaporating drop on a flat substrate have appeared, for example, [7][8][9][10][11][12][13]. First of all, the works [7-9] laid the foundation for understanding the processes in an evaporating drop of capillary size.The importance of taking into account the behavior of the three-phase boundary (between the solution, substrate, and air medium), i.e., the contact line, for describing the hydrodynamics of the processes inside the drop was realized.Depending on the properties of the substrate and the solution, as well as the size of the droplet, which determines the relative magnitude of capillary forces, there are two main scenarios for the displacement of the contact line.In the first scenario, the contact line may stand still or move very slowly during the drying of the drop, so that the contact angle between the liquid and the substrate decreases and becomes nonequilibrium.In the second scenario, the movement of the contact line keeps the contact angle constant, ensuring an equilibrium shape of the droplet.The first scenario is realized if the capillary forces pulling up the contact line are weak compared to the forces holding it down.This occurs when the substrate is well wetted by the solution, such as a drop of water on glass, when the height of the drop is much smaller than its radius, and the capillary forces are relatively weak.The line retention is also facilitated by substrate irregularities, and slow motion is facilitated by the rapidity of evaporation.The second scenario is characteristic of the case of weak wetting of the substrate by the solution (a drop on a hydrophobic surface).In that case the capillary forces are relatively large and the substrate does not hold the contact line [7].When the contact line is pinned by the substrate, the so-called effect of pinning the contact line takes place.It leads to the establishment of radial flow of compensatory nature, which moves the dissolved particles in the drop from the center to the periphery.It is forming a characteristic annular thickening, an excess of solid phase at the periphery of the drop (the so-called «coffee ring effect» [8])

Exact solutions for arbitrary contact angles
Previously, exact analytical solutions have been obtained to describe the evaporation rate and evaporation flux density of a small sessile liquid droplet having the shape of an axisymmetric spherical segment deposited on a horizontal substrate (figure 1(b), figure 2) [14][15][16][17].The analogy of the drop evaporation problem with a similar problem from the field of electrostatics of equipotential surfaces, which has been solved in general terms before, was used.For the total evaporation rate, the following expression was obtained by O. Yu.Popov [14]: Two alternative expressions are obtained for the evaporation flux density.First, in 2000-2005, the following solution was proposed and tested [14][15]   where is the Legendre function of the first kind.Here, toroidal coordinate  ranges in the interval from 0 (top of the drop) to ∞ (contact line).So, this coordinate is related to the cylindrical coordinate r by where  is the contact angle of the droplet (figure 2).Eq. ( 8) is a double integral, being an implicit function of cylindrical coordinate r, which makes Eq. ( 8) extremely difficult to use in calculations.An alternative solution in polar coordinates was published in 2021-2022 [16,17], which allows to calculate the flux density as a function of the polar angle φ explicitly (figure 2): Expression ( 10) is mathematically equivalent to formula (8), but it much more convenient for calculations; since formula (10) defines an explicit dependence of the evaporation flux density on the polar angle and represents a single integral.Therefore, we could recommend using the new expression (10) instead of equation ( 8).

Exact solutions for some values of contact angles
Previously [16,17], an expression was obtained that describes the vapor concentration near an evaporating drop: This expression can also be represented as: The results of British mathematician H.M. Macdonald [18] and Soviet scientist G.A. Grinberg [19] were used in obtaining these expressions.Using these equations, relatively simple expressions for the evaporation flux density can be found for some specific contact angles [20].
The integral (13) can be represented as the sum of a finite number of terms for some specific contact angles [19].It was shown that expression (13) under the condition , where j=1,2,3, … can be rewritten as Then, evaporation flux density is given by [16,17] ( where By differentiating in the formula ( 16), we get ( ) where It can be shown that the first sum in formula (16) gives identically zero (can be verified by direct calculation), and the terms in brackets in the second sum are equal in absolute value.With this in mind, the expression is greatly simplified Taking into account ( 14), we get sin 0.5 2 ( ) (cosh cos0.5 ) cosh cos 0.5 2 Thus, for any j, the evaporation flux density is given by   If k=j, the term under summation in (22) has the form sin 0.5 2 sin 0.5 cosh cos 0.5 2 cosh cos 0.5 j j j j j j j j If k=0, the term under summation in (22) has the same form sin 0.5 sin 0.5 cosh cos 0.5 cosh cos 0.5 Taking into account ( 23) and (24), equation ( 22) can be transformed as Expression (25) is the equivalent to Eq. (22).To apply expressions (22) or (25) for calculations, it is necessary to take into account the formula ( 14) and following geometric relationship [16][17]: To establish a relationship between the parameter j and the corresponding contact angle  , one has to use the geometric relation , where j=1,2,3… First three solution of the expression (25) (j=1,2,3) placed into Table 1.
Table 1.First three solutions for evaporation flux density J (contact angles  of 90 o , 135 o and 150 o )

Examples
Using Table 1, one can compare the solutions for contact angles 135 o and 150 o obtained from Eq. ( 22) or Eq. ( 25) with the general solution ( ) f  that works for arbitrary contact angles given by equation (10):      It is easy to verify by direct calculation that formula (10) gives the same curves, which confirms the correctness of the mathematical transformation that led to the formula (22) or the expression (25).

Conclusion
New complex solution (25) was derived for the evaporation flux density of a small liquid droplet having the shape of an axisymmetric spherical segment deposited on a horizontal substrate for the set of discrete that do not contain integral dependencies.They can also be used as approximate expressions for a narrow range of contact angles around the specified values.

(
with the following initial boundary conditions: on the drop surface ( ) const s n S n   .Outside the drop the vapor concentration is determined by the asymptotic value of the vapor concentration in the atmosphere (for aqueous solution, it is the relative humidity of the air 0

Figure 1 .
Figure 1.(a) aqueous droplet deposited on a polystyrene substrate (drop diameter about 3 mm), (b) geometry of the droplet on the substrate: wetting angle θ; radius of a contact line of the droplet R, evaporation flux density J(r).

Figure 2 .
Figure 2. The geometry of the sessile droplet: θ is a contact angle, φ is a polar angle, ρ is a spherical segment radius, R is the radius of contact line, (z, r) are the cylindrical coordinates.

Figure 3
Figure 3 represents the dependences of the evaporation flux density (dimensionless) on the polar angle, which is given by formulas (30) and (31).

,
where j=1,2,3… The analogy of the drop evaporation problem with a similar problem from the field of electrostatics of equipotential surfaces, which has been solved in general terms before, was used.As an example, very simple exact expressions (30) and (31) were obtained explicitly for the evaporation flux density for droplets with contact angles