First, we derive the surface formation energy of droplets attached to cone shaped protrusions or cavities. Afterwards, we discuss the properties of specific surface patterns composed of cones with varying apex angles.
Properties of droplets attached to cone shaped protrusions or cavities
Consider a sphere-like droplet of radius R attached to an axially symmetric, cone shaped protrusion or cavity with apex half-angle ε = 0…180° forming a contact angle θ = 0…180° (see ). We assume that the droplet is attached in a symmetry preserving way (i.e., the symmetry axes of cone and droplet coincide). We further assume that the droplet consists of a fluid (“fluid #1” in what follows) and is surrounded by a second fluid (“fluid #2” in what follows). One of the fluids must be a liquid, if both fluids are liquids they should be immiscible.
Figure 1 Sphere like droplets of fluid #1 (grey) attached to an axially symmetric solid cone (the broken line represents the symmetry axis). θ: contact angle, R: droplet radius, s: radius of contact circle (= line where solid, fluid #1 and fluid #2 are (more ...)
In order to form a droplet in contact with a solid, a surface formation energy W
has to be provided. If we consider only droplets with constant volume this energy is given by the expression [9
where S denotes the attachment area between droplet and solid, M is that part of the droplet surface which is in contact with fluid #2 and S
tot – S is the area where plane and fluid #2 are in contact.
denote the surface tensions (or surface energies) with respect to fluid #1/fluid #2, solid/fluid #1 and fluid #2/solid interfaces, respectively. The product σ2s
yields a constant value. Since W
is defined only up to an arbitrary constant, we can ascribe to it the value –σ2s
which yields the second version of Eq. 1
. This choice is equivalent to ascribing a vanishing droplet (i.e., M
→ 0, S
→ 0) zero surface energy.
Droplets of constant volume in equilibrium with respect to the surface tensions pulling at them obey the Young Law
Inserting this relation into Eq. 1
we obtain the surface energy W
of a droplet of fixed volume V
in equilibrium with respect to surface tensions:
Expressing the surface segments M and S as well as the contact circle radius s and the droplet volume V in terms of the quantities R, ε and θ, we find from
Since the formulas in Eq. 4
through Eq. 7
encompass cone shaped protrusions and cavities, we designate them both in what follows by the common term “cone”.
Inserting Eq. 4
and Eq. 5
into Eq. 3
, we obtain for the surface energy W
of an equilibrated droplet
where we have employed equation Eq. 7
to replace the droplet radius R
in favour of the (constant) droplet volume V
in the second expression.
denotes the surface formation energy of a spherical droplet of volume V which consists of fluid #1 and floats (i.e., without contact to the solid) within fluid #2.
A closer look at the terms in the braces of expression Eq. 7
reveals that certain (ε, θ)-combinations have to be excluded because they represent (non-physical) negative droplet volumes. The admissible (ε, θ)-ranges (equivalent to droplets with V
≥ 0) are given by (see also ):
Figure 2 Thin lines: Curves of constant surface formation energy W (ε, θ) of an equilibrated droplet of volume V, according to Eq. 8. Values of W are given as multiples of the surface energy W
float of an unattached spherical droplet (see Eq. 9 (more ...)
The function Θ0
(ε) is calculated by setting V
= 0 in Eq. 7
. Comparison of Eq. 7
and Eq. 8
shows the equivalence of the conditions V
≥ 0 and W
(ε, θ) ≥ 0.
Below, the question will arise whether a freely floating droplet of surface energy W
gains energy if it attaches to a cone defined by a given (ε, θ)-pair or whether this process consumes energy. The answer is found by equating the expressions in Eq. 8
and Eq. 9
: Solving for θ, one obtains a curve
The curve Θ1(ε) = θ () divides the (ε, θ)-plane into two regions: Cones that are generated by (ε, θ)-pairs below it imply W(ε, θ) < Wfloat, that is, a freely floating droplet of surface energy W
float gains surface energy if it chooses to attach to such a cone. Cones characterised by (ε, θ)-pairs above Θ1(ε) > θ require for attachment the energy W (ε, θ) > W
float, i.e., attachment of a freely floating droplet would consume energy.
If ε is held constant, W
(ε, θ) is a continuous and increasing function of θ, i.e., θ2
) ≥ W
). This can be seen by calculating the slope of Eq. 8
with respect to θ which is positive for ε, θ = 0…180°.
illustrates the behaviour of W (ε, θ) if θ is kept constant: W (ε, θ) exhibits extrema with respect to ε whose positions depend on the value of θ. For 0° ≤ θ ≤ 90°, the curves W (ε, θ = const.) show maxima within the range ε = 0°…90°, more precisely
Figure 3 Surface formation energy W (ε, θ) of an equilibrated droplet of volume V. Values of W are given as multiples of the surface energy W
float = of a floating spherical droplet of volume V. The curves are distinguished by the value of the (more ...)
whereas the curves with 90° ≤ θ ≤ 180° have minima within a range ε = 90°…180°, i.e.,
Curves with 90° ≤ θ ≤ 180° have also saddle points at εsaddle = θ – 90°. For ε → 90° both protrusions and cavities degenerate to flat surfaces.
The main results of the previous section are as follows (see also and ):
1. The surface energy W (ε, θ) of a droplet attached to a cone of apex half-angle ε in a symmetry-preserving way is smaller than the surface energy W
float of a freely floating droplet, provided that the (ε, θ)-pair lies between the curves Θ0(ε) = θ and Θ1(ε) = θ ().
2. If the value of the contact angle is fixed and lies within the interval 0° < θ < 90°, the surface energies of droplets sitting on cones whose apex half-angle ε are close to the value εmax
given in Eq. 14
are higher than the surface energies of droplets attached to cones with greater or smaller apex half-angles.
3. If the value of the contact angle is fixed and lies within the interval 90° < θ < 180°, W
(ε, θ) exhibits a minimum at ε = εmin
). Thus, the surface energies of droplets sitting on cones which are very differently shaped (in terms of apex half-angle ε) are higher than the surface energies of droplets attached to cones whose shape is more similar to εmin
The features just discussed permit speculation about constructing “band-conveyors” for droplets. Such a “band-conveyor”, capable of “passing down” droplets from cone to cone, might be generated by arranging cones with increasing values of ε (but fixed θ) in one- or two-dimensional patterns. illustrates the basic idea: Upper and lower part of the figure show the same line-up of cones. The apex half-angle ε increases from left to right. If the fixed contact angle lies within 0° < θ < 90° (upper part of ), the function W (ε, θ) has a maximum at point B (i.e., εmax(θ)). Thus, the energy difference ΔW := W
float – W (ε, θ) which is required to detach a droplet from a cone has its minimum at point B. With increasing distance from B, droplets are increasingly stronger bound to their substrate, that is, ΔW increases towards A and D. If the fixed contact angle lies between 90° < θ < 180° (lower part of ), the minimum of W (ε, θ) is located at point C (i.e., εmin(θ)). Hence, ΔW increases if point C is approached from A or D.
Figure 4 Line-up of cones. The apex half-angle ε increases from left to right. For θ = const. and 0° < θ < 90° (upper part), the surface energy W (ε, θ) has a maximum at point B and for 90° (more ...)
If both lyophilic (i.e., 0° < θ < 90°) and lyophobic droplets (i.e., 90° < θ < 180°) reside on the landscape of cones of , it appears that both droplet species experience an increase of ΔW from point B towards C. In sections A–B and C–D, however, the variation of ΔW points into opposite directions for the two droplet species.
Perhaps, these findings can be utilised to construct a “band-conveyor” for droplets. We present two ideas how this might be achieved:
1. Consider . The double line-up of cones similar to a zip fastener is constructed from the left part of . The apex half-angle ε of the cones increases from point A (ε ≈ 0°) to point B (ε = εmax
). According to Eq. 14
, a lyophilic droplet attached to cone #2 is in a lower state of surface energy than a droplet at cone #1, but in a higher energetic state than the droplet at cone #3. If the dimensions of droplet and cones, the contact angle between them and the temperature of the arrangement are suitably chosen, thermal oscillations of the droplet around its position of symmetry at cone #2 may bring it in contact with cone #1 or cone #3. Due to the gradient of W
(ε, θ) with respect to ε, for a lyophilic droplet it is energetically attractive to move to cone #3 (towards lower values of W
(ε, θ)), but not to cone #1. Thus, lyophilic droplets should finally get to point A. For lyophobic droplets, a similar reasoning applies, which starts, however, from Eq. 15
instead of Eq. 14
. Hence, lyophobic droplets should migrate towards point B.
Figure 5 Double line-up of cones similar to a zip fastener constructed from the left part of . The apex half-angle ε of the cones increases from point A (ε ≈ 0°) to point B (ε = εmax). Apart from one or two (more ...)
2. If the cones are farther apart than in (more like in ), the droplets have to detach completely from a cone before they come in contact with the next one. Similarly as above, if droplet and cone dimensions, contact angle and temperature are favourable, the interaction of the droplet with the thermally agitated particles constituting fluid #2 may outweigh the binding energy ΔW = W
float – W (ε, θ) between droplet and solid substrate. Since the thermal agitation fluctuates randomly, the transfer of low amounts of energy onto the droplet occurs more frequently than the transfer of high amounts of energy, on an average. Consequently, the mean residence time of a droplet sitting on a cone increases with increasing binding energy ΔW. Therefore, lyophilic droplets attached to cones close to point B (see upper part of ) leave the cones more often than droplets farther away from B. Doing so, a droplet may jump – with equal probability – either to the left or to the right: In case its next attachment is closer to B, its residence time is shorter than before, if it moves to a cone farther away from B, it will remain there longer, on an average. The overall effect of this is a (net) movement of lyophilic droplets towards B.
Lyophobic droplets behave similarly, they move also in the direction of decreasing binding energy ΔW. This means, in terms of the lower part of , that they move away from point C, towards points A and D.
Up to now, we have simply assumed that detached droplets re-attach to the tips of the cones. Of course, this cannot be taken for granted. One may exclude this eventuality by assembling the landscape of cones of from two materials with different contact angles: (a) bulk material with a very high contact angle (dotted areas in ), implying a very small ΔW
, and (b) employing material with smaller contact angle for the cones (white areas in ). Alternatively, one might apply hair- or pillar-like structures which are smaller than the cones by an order of magnitude or so to the dotted areas in . Droplets coming in contact with these structures should experience a Cassie state, leading also to very high effective contact angles [8
Notice that the mechanisms depicted in and predict droplet migration in opposite directions. Concentrating on , a possible application of the suggested mechanism arises, for example, from exploiting only parts of : Arranging the landscape of cones in a twodimensional, radial pattern such that point A is close to the centre and B represents the outer fringe of a circular disc, lyophilic droplets would migrate towards the centre whereas lyophobic droplets would migrate away from it. A reverse migration order should result if the roles of A and B are interchanged or if point C is chosen as the centre and point D as the periphery.