Chapter 1: Spreading of Complex Fluids Drops

When a droplet is positioned on a surface, it spreads outwards until it reaches its equilibrium shape. This spreading is hence inherently connected to wetting, as it involves the question of whether and how a liquid front advances on a surface. The kinematics of spreading may depend on the properties of the droplet, such as its composition of one or more fluids, surfactants, solid particles, etc., on the properties of the surface, such as its chemical composition and geometrical structure, and on other physical influence factors such as temperature distribution or initial speed of the droplet. Correspondingly, the droplet may also spread on a fluid phase.

Let us first consider a basic case of an ideal fluid and an ideal solid surface. The surface should be perfectly flat and chemically homogeneous, and the fluid should be a simple liquid without any particles, surfactants or the like, and without any complicating processes like evaporation taking place. After the droplet is deposited on the surface, it adjusts its shape until an energy minimum is reached. This typically corresponds to spreading outwards. The final shape usually corresponds to a lens- or circular-shaped form of the droplet, which forms a certain contact angle Θ with the surface (Figure 1.1). This is determined by Young's relation, whose work on the cohesion of fluid was published in 1805.¹  Notably, in his original work, Young did not use any equations, but instead discussed the cohesive and adhesive forces between a droplet and a surface. Nowadays, we write^2–4 

where γ is the liquid–vapor interfacial tension (or surface energy), γ_SV the interfacial tension at the solid–gas interface, and γ_SL at the solid–liquid interface, respectively. The interfacial tensions have dimensions measured in N m⁻¹ = J m⁻² and, for a simple surface, can be interpreted either as a force or a surface energy. Hence, Young's relation can also be understood from different perspectives. The simplest explanation is a force balance of the interfacial stresses as depicted in Figure 1.1. For solid surfaces, only the force balance parallel to the surface is considered. The vertical component of γ is taken up by the solid surface, which is however too stiff to deform notably. Only for a droplet sitting on a soft surface or liquid, this deformation may be significant, so that depending on the strength of the deformation the force balance perpendicular to the surface also may have to be evaluated.⁵ 

For the simple case considered here, the interfacial tensions are identical to the specific surface energies. Young's relation can then also directly be obtained by energy considerations. The most prominent non-simple case, where interfacial tension and specific surface energy are not identical, is elastic surfaces, which is known as the Shuttleworth effect.⁵  Here, expansion of the surface area due to stretching leads to a change in the surface energy.

In the simple case, where no such complications occur, Young's equation follows from an infinitesimal movement of the contact line along the surface (Figure 1.2). In this case, the work needed to perform a small movement of the contact line is

Here, the first part corresponds to the change in surface energy of the solid, and the second part is the additional fluid–gas interface that is newly created. In equilibrium, when the drop has come to a rest, this work is zero, so that we again obtain Young's relation (1.1).

Another method to obtain Young's relation is to minimize the free energy of a droplet.^2–4  For simplicity, this sometimes involves the assumption of a spherical drop, although, as has already been seen, Young's relation can be obtained by local considerations only. This is in line with the fact that Young's contact angle Θ is remarkably independent of various influences like gravitation or electrical fields. Even in electrowetting, where the macroscopic drop shape is manipulated significantly by electric fields, the local contact angle is still Young's.

The lens-shaped drop just described is not the only equilibrium shape. To what extent a droplet spreads on a surface is traditionally characterized by evaluation of the spreading coefficient S. This coefficient was introduced in 1922 by Harkins and Feldman⁶  as the difference between the work of adhesion of the interface of the droplet and the surface W_A and the work of cohesion of the liquid W_C:

They wrote: “Thus a liquid will not spread if its work of cohesion, which indicates its attraction for itself, is greater than the work of adhesion, which indicates its attraction for the body upon which the spreading will not occur”. The two works are

so that

Nowadays, the spreading coefficient is also often explained as the specific surface energy of a dry solid surface to that of a wetted one.⁷ 

In general, a liquid can either sit on the surface in a lens- or circular-shaped droplet as already shown in Figure 1.1, which is called partial wetting, or it can spread completely along the surface, which is only limited by molecular forces that in the end will lead to a film of nanoscopic thickness (Figure 1.3). This case is called complete wetting and, for example, corresponds to organic liquids spreading on water. If the spreading parameter S > 0, complete spreading will occur, while S < 0 leads to the common situation of partial wetting. A limiting case of partial wetting is complete dewetting, when S = −2γ. In practice, situations close to complete dewetting will however only be achieved upon appropriate geometric structuring of the surfaces, such as for superhydrophobic surfaces.

Consistently, combining Young's relation with the spreading coefficient

only gives valid solutions if S < 0.

In order for a droplet to spread on a surface, there are two important driving forces: capillarity and gravity. These forces can also be attributed to different regimes that are dominated by either of the two. Specifically, the Bond number

relates gravity and surface tension forces. Here, ρ is the density of the fluid, g the gravitational acceleration and r the radius of the droplet. If Bo ≪ 1 the shape of the droplet is dictated by the surface tension, and gravity can be neglected. The droplet takes a spherical shape and can also be modelled as such. In other words, the Bond number also describes whether a droplet is smaller or larger than the capillary length , the length scale up to which surface tension plays a role. It is therefore also said that, for small droplets (i.e. those that are smaller than the capillary length), spreading is driven by capillary forces, while large drops are dominated by gravity. A transition from the capillary to the gravitational regime can also occur during spreading. The formerly spherical drop then takes the form of a puddle, whose surface is only curved at the rim.

A simple standard case is a drop with small Bond and capillary numbers , with μ being the dynamic viscosity and U the contact line speed. The small capillary number ensures that viscous forces will not have any influence on the droplet shape, so it can be considered as a spherical cap. The drop is assumed to be sufficiently flat, i.e. the contact angle is small. In this case, lubrication theory can be applied to the flow towards the edge of the droplet. During spreading, the evolution of the droplet radius with time is described by what became known as Tanner's law:⁸ 

Here, V is the volume of the droplet and B is a constant that is estimated¹⁰  as . The same relationship had already been derived by Voinov three years earlier.⁹  The key point of Tanner's law is that the droplet radius evolves with time to a power n. In his early work, Voinov pointed out the relation of the contact line dynamics to dissipation, i.e. dissipation determines n. In droplet spreading, dissipation mainly takes place at the contact line itself and due to viscous forces. For viscous spreading n = 1/10, so

In terms of the contact angle for complete wetting, eqn (1.9) is equivalent to

It should be noted that under dynamic circumstances, the contact angle is generally not the same as in equilibrium. The exponents in the previous equations have also been found in experiments. For other mechanisms of dissipation, different exponents emerge. Also, for driving methods other than capillary forces, similar r(t) ∼ tⁿ relationships have been found. For example, gravity-driven spreading with viscous dissipation seems to follow an exponential law with n = 1/8, while a dominating contact line dissipation, both for viscous and gravitational dissipation, leads to n = 1/7 (for a summary, see Bonn et al.¹⁰ ).

In the end, dissipation is a very complex mechanism and strongly depends on both the surface and the fluid. Not only is it the case that most surfaces are not ideal, but have certain defects, roughness or impurities at which the contact line may stay trapped or pinned when moving across them, but the surfaces can also be engineered to have certain geometric or chemical nano- or micropatterns. We also know very well that there is no single static contact angle of a surface, but that most surfaces show a contact angle hysteresis with an advancing and receding contact angle at the front and back of a droplet that slides down the surface. This hysteresis also originates from the specific geometrical and chemical features of the surface. Real surfaces are not atomically smooth, and so every time the contact line passes a surface defect, it has to overcome the energy barrier associated with this structure, so that a single, simple spreading law is difficult to provide.

Theoretically, spreading dynamics is also very hard to access, since it touches the limits of standard continuum mechanics models. This is a general issue in contact line motion and not only applies to drops but also to any contact line moving on a surface. As Huh and Scriven pointed out in 1971,¹¹  viscous shear stress diverges in the vicinity of the contact line, if the standard continuum model assumptions, i.e. a no-slip condition for the fluid at the surface, are made. They give an example of a lubrication model of a moving contact line (that means under the condition that the contact angle is small). Then, the shear stress at the surface (y = 0) is

with u being the velocity parallel to the surface, h the height of the droplet and V the contact line velocity. As the height of the droplet moves towards zero (h → 0), the shear stress goes towards infinity. Together with the stress, dissipation also diverges, which determines contact line motion. Consequently, the contact line would not be able to move at all.

Hence, the classic macroscopic continuum mechanics description, as we typically employ it to describe the motion of fluids, does not work for a moving contact line, nor in the case of spreading. Consequently, specialized models are needed in order to be able to calculate the motion of a contact line. Numerous approaches have been proposed for how to resolve this issue. They may be motivated from a mathematical perspective, from a more detailed physical picture of the contact line region or from certain applications like a droplet rolling down an inclined surface.

A very common approach is the relaxation of the no-slip boundary condition at the solid–fluid interface, as proposed by Huh and Scriven.¹¹  In this case, a Navier slip condition¹²  is introduced as a boundary condition, which in its simplest form reads

The slip length b allows for a finite velocity at the wall (or a velocity difference to a neighbouring fluid) which is proportional to the shear stress. For flat, homogeneous surfaces, the intrinsic slip length can be considered a material property. In measurements, values ranging from zero to a few tens of nm have been found, depending on the forces between the two adjacent substances.¹³  Generally, for water, more hydrophobic surfaces tend to have a larger slip length. In fact, the intrinsic slip length is a physical property that is present in reality, but which is normally neglected because of its very small value and its minor influence on bulk flows. However, it can change the character of the solution significantly directly at the contact line. In a way, considering the slip length at the contact line can thus be regarded not as a specialized model, but as reconsidering a previously neglected quantity. However, this does not apply to all surfaces, as on many hydrophilic surfaces, it has not yet been possible to measure any slip length using current measurement techniques.

Furthermore, the lens-shaped picture of a droplet is often quite simplistic. While this is visible to the naked eye, significant physical processes can take place on a much smaller length scale which remain hidden from simple observation. An example is the precursor layer that forms in front of a spreading droplet. In this case, the contact angle is only a macroscopic quantity related to the spherical drop profile, while the actual interface gradually evolves into a very thin layer of fluid, in which the disjoining pressure plays a dominant role²  (Figure 1.4). With a thickness of the order of 10 Å,¹⁴  the layer itself is so thin that it is not visible to the naked eye. In 1919, Hardy¹⁵  noticed the presence of a thin layer on glass surfaces, which was able to provoke interactions between droplets of acetic acid that were sitting 1–2 cm apart from each other. Another mechanism that can influence the shape of droplet interfaces at very small scales is the presence of an electrical double layer.¹⁶ 

Any numerical calculation of spreading drops hence faces a number of challenges. Either multiple length scales have to be covered with different phenomena taking place on each scale, or the underlying phenomena occurring at small scales have to be modelled, e.g. as appropriate boundary conditions for a macroscopic flow simulation. The underlying phenomena may however be of varying significance depending on the considered materials. In this field, approaches are, for example, molecular dynamics simulations of very small droplets,¹⁷  the combination of continuum dynamics simulations with molecular dynamics simulations¹⁸  or diffuse interface methods.¹⁹  For macroscopic continuum models, a common approach is the mentioned introduction of a slip length in combination with a condition for the contact angle. A variety of slip models exist,²⁰  which differ, for example, in the values of slip, the region where it is applied and when it is applied, e.g. above a certain threshold in shear.

For simple liquids, spreading dynamics in accordance with an exponent of n = 1/10 in Tanner's law has been observed.^4,10  Anyhow, the difference to similar exponents may not be very large in practice. There are, however, a large number of influence factors that can make this simple relation much more complex.

From a theoretical perspective, non-Newtonian fluids are interesting, because a shear dependence in viscosity can compensate for the increasing stresses near the three-phase contact line and thereby theoretically eliminate the contact line singularity.^21,22  In practice, a shear-dependent viscosity and the presence of normal stresses have, however, been found to lead to a spreading behaviour that differs only slightly from the n = 1/10 Tanner's law.²² 

The presence of surfactants can either slow down or speed up spreading compared to a system with the same surface tension and viscosity. While most surfactants slow down spreading, trisiloxanes have become popular as so-called superspreaders, as they increase the spreading speed by orders of magnitude.²³  Firstly, the presence of a surfactant can influence all interfacial tensions between a fluid, solid and gas. This, in turn, modifies the spreading parameter S. The slowed-down velocity is commonly explained by the fact that, when a new section of interface is created during spreading, the transport of the surfactants to the interface is slower than the generation of a new interface. There is hence a region with less surfactant close to the edge of the droplet. This reduces the spreading parameter compared to a system with the same overall surface tension. Secondly, such gradients in surface tension may also lead to Marangoni flows. Superspreading is generally related to the existence of strong Marangoni stresses^24,25  that create a flow towards the three-phase contact line, where the concentration of surfactants at the newly created air–water interface is lower than that at a certain distance from the contact line, where transport of surfactants to the interface has already taken place (Figure 1.5). Correspondingly, in the opposite case, i.e. in dewetting, where the contact line retracts instead of spreads, Marangoni flows towards the contact line have been found to slow down dewetting.²⁶ 

Evaporation can compete with the extension of a droplet on a surface. Both the temperature and humidity of the air have been shown to modify the spreading velocity.^27,28  Using water–glycerol mixtures, a modified Tanner's law with an exponent of n = 0.0865 + 0.0290 RH, where RH is the relative humidity of the surrounding air, has been found.²⁸  In general, heated substrates can lead to Marangoni flows in the droplet that are driven by temperature gradients. Both for complete and partial wetting, the evolution of the droplet radius goes through several stages, increasing only during a very short stage at the very beginning, and then decreasing due to evaporation.²⁹  The evaporation itself is generally considered to be either limited by the diffusion of the fluid molecules through the gas or by the phase-change between liquid and gas.³⁰ 

For complex fluids, often several of the previously mentioned effects take place simultaneously. For example, mixtures of fluids may evaporate inhomogeneously, thereby creating concentration-gradient-driven Marangoni flows during spreading. Often, non-Newtonian fluids are also mixtures. Furthermore, colloids may be present in the fluid. Extensive research has been done on how to suppress the coffee-ring effect that is created by the remaining particles after evaporation of the liquid. Also in biological applications, fluids are typically complex, containing cells for example. General phenomena in such fluids may be concentration- or temperature-driven Marangoni flows, changes in interfacial tension by adsorption of substances or pinning of particles to the solid.

Also, the substrate significantly influences how a droplet spreads on a surface. After all, spreading is an interfacial process, and hence the surface (which is solid in most cases) plays an important role as a neighbour for both the fluid and the surrounding gas. Again, it is important to consider the different scales that occur in spreading. For the static contact angle, it is very well known that a geometric structure or a chemical pattern leads to a macroscopic contact angle on the scale of the droplet that differs from the local contact angle at the surface.⁷  This macroscopic contact angle may be smaller or larger than the local angle and depends both on the geometry and chemistry of the surface.

For example, on rough, wetted surfaces, the Wenzel law describes that the macroscopic contact angle becomes smaller than the local one, if the local contact angle is <90°, and it becomes larger if the local contact angle is >90°. Correspondingly, the spreading is affected also. On the one hand, the contact line has to overcome the local roughness features, so spreading on rough surfaces in some experiments has been found to take longer to reach a similar spreading state as on flat surfaces.³¹  On the other hand, the surface roughness may also guide the liquid and even pull it along the surface by capillary forces.³²  In this case, the roughness may act similarly to a porous medium: while a spherical cap remains in the centre, the liquid is gradually soaked up by the capillaries in between the surface roughness.³²  Also for rough surfaces, a Tanner-like spreading law has been found with r(t) ∼ ktⁿ. Increasing roughness also increases n, with exemplary values^31,32  ranging from 0.15 to 0.4. Also, the prefactor k was affected by roughness. The change in the apparent advancing and receding contact angles on rough, grooved surfaces was first modelled by Shuttleworth and Bailey³³  based on the slopes of the grooved walls.

Similarly to rough surfaces, spreading on porous surfaces is a competition between the advancement of the contact line and the imbibition of the fluid into the porous structure (Figure 1.6), which reduces the size of the droplet.^34,35 

On superhydrophobic or slippery surfaces, the fluid drop does not completely touch the surface everywhere, but air or another secondary fluid remains entrapped in the roughness features of the surface. This fact again leads to a different macroscopic contact angle for the droplet than the local contact angles at the surface, which vary according to the local surface or fluid material the contact line meets. Generally, the contact angle is very high on such surfaces, so there is only very little spreading if it is not forced by drop impingement. Under drop impact, spreading is often characterized by the maximum spreading radius as a function of parameters such as the Reynolds and Weber numbers.³⁶  After spreading, the drop will generally contract again and may also lift off.³⁷  It has been shown that when the drop advances on a superhydrophobic pillar array, the contact line first stays pinned to a pillar and the fluid–gas interface bends down until it touches the following pillar.³⁸ 

Overall, we have seen that spreading is not easily captured by a macroscopic, continuum viewpoint. Rather, it depends very much on the local phenomena taking place at various small (down to molecular) scales. It may, therefore, also be very specific to the considered situation. Influence factors are the fluid and its components, such as surfactants, colloids, polymers, etc., environmental conditions such as the temperature of the surface or the drop, and the substrate with its geometrical and chemical properties, such as porosity, roughness, hydrophilicity or hydrophobicity, or the presence of secondary fluids (Figure 1.7). Last but not least, there may be various dynamic processes taking place, induced by the creation of a new solid–liquid interface area during spreading, such as the formation of charge layers,³⁹  chemical reactions or swelling of surface-bound polymers. These processes currently are just at the start of being explored.