20.1 Fluid Effects at Interfaces

20.1.1 Origin of Surface Tension

In all materials a certain amount of intermolecular forces keep the individual molecules together. If this force is not strong enough, the material will not exist as a solid bulk decompose but would into individual molecules. In the volume of the material, these intermolecular forces are balanced in all directions. Depending on the nature of the primary bonds, these forces may be directed and more strongly expressed in one specific direction, as is the case for covalent bonds.

However, on the surface of the material, those forces are not balanced (see Fig. 20.1). The last layer of molecules has no interacting neighbors on top of it, and therefore the forces pointing toward the inside of the bulk are not balanced. This is why surfaces always have a net force F_net pointing inward, i.e., in the inward direction of the surface’s normal vector. In liquids in which the molecules can move more freely as compared to a solid, this net force causes the formation of a meniscus. The smaller the diameter of the surface is, the more expressed this meniscus is, i.e., the more strongly the surface is curved.

Now suppose the surface area of a system were to be increased. More molecules would have to be moved from the bulk to the surface. Consequently, more molecules would be left with unbalanced interaction forces. Obviously this is a thermodynamically less favorable state, and therefore increasing the surface requires energy. This state is the same for liquids and solids. A liquid will always try to minimize its surface area, which is why a water droplet will form a sphere. Similarly, solids will also try to minimize their surface, which is why energy is required to introduce, e.g., cracks in a solid, which effectively increase, the surface. However, in order to reach a stable state, a system will always try to strive toward the state of lowest energy. This is why it is thermodynamically more favorable to reduce the surface area as much as possible.

20.1.2 Free Surface Energy

The energy required to increase the surface area is referred to as the free surface energy or surface free energy γ. Not surprisingly its unit is energy per area, i.e., J m^{− 2}, as it defines the amount of energy required in order to create a certain surface area. It can be thought of as the degree of imbalance of the intermolecular forces of a surface. Intuitively, this imbalance is dependent on the material that is on top of the surface. If a second layer that has a similar free surface energy covers the surface, the two imbalances are balanced, because the surfaces find respective neighbors with which to interact. This is why the surface free energy of a material is always given in reference to the surrounding material or fluid. Naturally, these values are often given for surfaces exposed to air. In Fig. 20.3 we will see an example of surface free energy balance if a solid is deposited on a liquid’s surface.

20.1.3 Low and High Free Surface Energy Surfaces

From the preceding discussion, we can also roughly classify surfaces according to their free surface energy. Usually, there is a distinction between high free surface energy surfaces and low free surface energy surfaces. As discussed, the free surface energy results from an imbalance of forces at the surface. As a result, this free surface energy must be higher for surfaces that are held together by strong bonds because strong intermolecular bonds in the bulk will lead to strong imbalanced forces at the surface.

This approximation is surprisingly correct. Metals and crystals, which are held together by primary bonds, have relatively high free surface energies. Tab. 20.1 lists the free surface energies of the elements. As can be seen, metals have very high values.

In polymers the individual chains of the polymer are held together by secondary forces, i.e., van der Waals and hydrogen bonds. These bonds are significantly weaker, and consequently, these surfaces have very low free surface energies (see Tab. 22.2). Surfaces of materials with very low secondary interaction, e.g., pure hydrocarbon and fluorinated polymers have the lowest free surface energies. Tab. 22.2 lists a selection of commonly encountered polymers. As shown the values are significantly lower than those in metals.

20.1.4 Estimating the Free Surface Energy

Several methods can be used to measure the surface free energy. Here we will use a simple model that allows us to derive values that actually are not too far away from the values measured. In the following we will consider a solid to consist of individual atoms that are interconnected (see Fig. 20.2a). In the plane, each atom has four neighbors. Additionally, it is connected to one neighbor in the layer on top of it and to one neighbor in the layer below. Therefore each atom in the volume has a total of six neighbors.

At the surface, each atom is bound to only five neighbors because the last layer misses the neighbors above (see Fig. 20.2b). We can therefore estimate that the free surface energy will be one-sixth of the bond strength divided by the surface area of an atom:

$\gamma \approx \frac{1}{6}\frac{E_d}{d_{\text{atom}}}$

The bond energy can often be precisely predicted by the enthalpy of vaporization ∆H_vap per atom, i.e., by dividing ∆H_vap by the Avogadro constant N_A. This makes sense because transferring a liquid or a solid to the gaseous state is approximately equivalent to breaking all the intermolecular bonds. Therefore the energy required for this transfer should, with good precision, indicate the total bond energy. Therefore we find

$E_d\approx \frac{\Delta H_{\text{vap}}}{N_A}$

which yields

$\gamma \approx \frac{1}{6}\frac{\Delta H_{\text{vap}}}{N_A{d}_{\text{atom}}^2}$

For water we find the following values: ∆H_vapH2O = 43.990 kJ mol⁻¹ (see Tab. 6.5) and d_atom = 0.31 nm (see section 9.3.3). Using Eq. 20.1, we obtain an estimated surface free energy of γH₂O ≈ 126 mN m⁻¹. This value is definitely not correct (see Tab. 20.2); it should be around 72 mN m⁻¹, but it is surprisingly close.

20.1.5 Surface Tension

The term free surface energy is usually used only when talking about solid surfaces. When referring to liquids, the term surface tension γ is more commonly used. This term dates back to the work of English physicist Thomas Young¹. Please note that, despite the terminology, free surface energy and surface tension refer to the same physical quantity and share the same symbol. The unit of the free surface energy is J m⁻², and the unit of surface tension is N m⁻¹, which is, once multiplied by m m⁻¹ the same unit.

Tab. 20.2 lists surface tension values for commonly encountered liquids. As shown strongly polar liquids (such as alcohols and water) have very high surface tensions of around 50 mN m⁻¹ to 80 mN m⁻¹. This is due to the strong cohesive forces originating from secondary bonds. Most organic solvents have surface tensions in the range of around 20 mN m⁻¹ to 30 mN m⁻¹. Fluorinated substances have very low surface tensions approaching 15 mN m⁻¹. Fluorination leads to a significant reduction in surface tension as it reduces the amount of secondary interaction and therefore the cohesive forces. Tab. 20.2 lists examples of solvents in non-fluorinated and fluorinated form. As can be seen, the surface tension decreases significantly.

20.1.6 Temperature Dependency

In general heating a surface will cause the free surface energy to decrease. Following our rationale (see section 20.1.1), the free surface energy is due to the imbalance in the forces at the topmost atom layer of a surface. Heating increases the Brownian motion of the atoms thereby decreasing the interaction between the atoms. Eventually, heating will increase the motion such that the atoms can move freely in gaseous form (vaporization). This is due to the reduction of the interaction forces between the atoms. However, as surface tension is a consequence of the imbalance of the interaction force, it will decrease if the interaction forces decrease.

For example, heating a fluid will cause its surface tension to drop. Water has a surface tension of 74.23 mN m⁻¹ at 10 °C, 71.99 mN m⁻¹ at 25 °C, 67.94 mN m⁻¹ at 50 °C, 63.57 mN m⁻¹ at 75 °C, and 58.91 mN m⁻¹ at 100 °C [2].

20.1.7 Surface Tension Forces

Imagine that we want to immerse an object of a given size in a liquid (see Fig. 20.3). If the solid is placed on top of the liquid’s surface, the liquid will align to the surface. There will be no curvature, as we can see, on the free liquid surface because the imbalance of forces is no longer there. The surface molecules now have neighbors, and interaction forces will occur. Therefore, there is no further contribution to the surface tension. The surface tension thus acts only in the outer contact line of the object. This is the point where the intermolecular forces are (again) imbalanced.

Fig. 20.3 shows that the total force resulting from the surface tension is

$\begin{array}{ll}{F}_{\text{surface}\text{tensions}}\hfill & ={\gamma }_{\text{liquid}}{l}_c\hfill \\ \hfill & ={\gamma }_{\text{liquid}}2\left(w+l\right)\hfill \end{array}$

This is the maximum force that the surface tension will create. If the gravitational force of the object is higher than this value, it will immerse in the liquid. It is important to keep in mind that we need to calculate to circumference. For example, the legs of gerridae, small insects that can walk on water and are thus sometimes referred to as “water walkers” (see Fig. 20.4) have a contact area that can be approximated with a rectangle, similar to the one shown in Fig. 20.3. Usually, the width of the legs can be ignored, as w l. Eq. 20.2 can be simplified to

$\begin{array}{ll}{F}_{\text{surface}\text{tensions}}\hfill & ={\gamma }_{\text{liquid}}2\left(w+l\right)\hfill \\ \hfill & ={\gamma }_{\text{liquid}}2l\hfill \end{array}$

The multiplier 2 is often forgotten when indicating the formula.

20.1.8 Contact Angle

Wetting. After discussing the origins and the effect of surface tension and free surface energies we will now turn to the study of its effects. If a fluid is brought into contact with a solid, the molecules in the surfaces of the two substances interact. Suddenly they have neighboring molecules with which they can interact thus potentially reducing the force imbalance in the surface layer. This reduces the free surface energy of both the surface and the fluid. Depending on whether or not this reduction in energy is significant, the fluid will tend to increase its surface area.

The behavior of the liquid on a surface is usually referred to as the liquid’s wetting behavior on this specific surface. In general, the term wetting refers to the capability of a fluid, e.g., a liquid, to keep contact with a surface. Wetting is governed by the balance of adhesive (in the boundary) and cohesive (within the volume of the fluid) forces. We will discuss the individual forces contributing to the wetting behavior in a moment.

Contact Angle. Once the liquid drop settles on top of the surface, it will form a characteristic contact angle Θ (see Fig. 20.5). This contact angle is the visible boundary between the fluid and the solid. There is, in fact, a third partner involved, which is the fluid surrounding the system. Usually, the fluid deposited will be a liquid, and the fluid surrounding this drop of liquid will be a gas. However, this does not necessarily have to be so, as the surrounding fluid could also be a liquid, e.g., a silicone oil protecting the drop from evaporating. Please note that the contact angle is always measured “through the liquid”.

For contact angles Θ < 90° the liquid is said to be wetting or able to wet the surface. On the other hand, the solid is considered to be wettable or wetting. For water, the term hydrophilic is commonly used. A hydrophilic surface is a surface on which water will wet. For oils, the term oleophilic is used. Typical examples for hydrophilic surfaces are freshly cleaned glass surfaces. Examples for oleophilic surfaces include most polymer surfaces, e.g., polyethylene.

For contact angles Θ > 90°, the liquid is said to be non-wetting or not able to wet the surface. On the other hand, the solid is considered to be non-wettable or non-wetting. For water, the term hydrophobic is commonly used. A hydrophobic surface is a surface on which water will not wet. For oils, the term oleophobic is used. Typical examples for hydrophobic surfaces include most polymers, e.g., polyethylene. Examples for oleophobic surfaces include glass surfaces. As can be seen, hydrophilic surfaces are oleophobic and v.v. We will now turn to the intermolecular forces and their contribution to the formation of the contact angle.

Intermolecular Forces Between Wetting Fluid and Surface. Depending on their chemical composition, there will be a certain degree of adhesive forces between the wetting fluid and the wetted surface. These adhesive forces will force the wetting fluid to spread out as far as possible. On the other hand, the cohesive forces within the wetting fluid will try to counteract this spreading movement. So the degree of spreading is governed by the balance of adhesive forces between the wetted surface and the wetting fluid and the cohesive forces within the wetting fluid.

Intermolecular Forces Between Surrounding Fluid and Surface. The interaction between the surrounding liquid and the surface are likewise. If there are strong adhesive forces between the surrounding fluid and the surface, the fluid will spread as far as possible. On the other hand, the surrounding fluid will be retained by cohesive forces acting within its volume. Again, the wetting behavior is governed by the balance of these two forces.

Intermolecular Forces Between the Wetting and the Surrounding Fluid. The last of the three phase boundaries we need to discuss is the boundary between the wetting and the surrounding fluid. Likewise, there will be a tendency to increase the interaction area between the two fluids if there are adhesive forces between them. If this is the case, both fluids will form the largest possible contact area. On the other hand, cohesive forces within both fluids will try to counteract this spreading.

Rationale of the Balance of Forces. Returning to the concept of the free surface energy, we can now empirically balance the forces acting on the droplet. If the free surface energy of the solid is high, it will pull the droplet’s boundaries trying to increase the size of the droplet. By doing so, the imbalance in intermolecular forces at the surface is reduced, similarly to the example of the solid being placed on top of the liquid’s boundary (see Fig. 20.3). On the one hand, the droplet will try not to increase its surface area as this is energetically less favorable.

On the other hand, the droplet will form an interface with the surrounding fluid (usually air). As there is little interaction among gas molecules, the deficit in the balance of intermolecular forces is low in gases, which is why they tend to increase their surface area if the space is available. Therefore there is little gain for the free surface energy of the water droplet by forming a large contact area with air, which is why a drop of water will always try to decrease its size as a consequence of the cohesive forces.

As stated, if the surrounding fluid is a gas, it will have only a minor tendency to decrease or increase its surface as neither cohesive nor imbalance in intermolecular forces are critical in gases. However, if the surrounding fluid is a liquid, there will be a similar balance of cohesive forces versus reduction in free surface energy, as discussed for the wetting liquid.

Young’s Equation. We will now introduce the analytical equation that balances the individual force distribution. This equation is the so-called Young’s equation. It is given by

${\text{γ}}_{\text{gas/solid}}={\text{γ}}_{\text{liquid/solid}}+\cos \Theta {\text{γ}}_{\text{gas/liquid}}$

$\cos \Theta =\frac{{\text{γ}}_{\text{gas/solid}}-{\text{γ}}_{\text{liquid/solid}}}{{\text{γ}}_{\text{gas/liquid}}}$

From Eq. 20.4 we can derive a couple of important concepts. For 0 < cos Θ < 1, the contact angle is Θ < 90° and the liquid will wet the surface. This happens if the difference in the surface’s free energy between the “gas-wetted” and the “liquid-wetted” state is high, i.e., if there is a significant reduction in surface free energy gained by spreading of the liquid. In this case γ_gas/solid − γ_liquid/solid ≥ 0. This will be facilitated if the liquid’s surface tension is low, i.e., it is easy to pull the liquid apart.

For −1 < cos Θ < 0, the contact angle is Θ > 90°, and the liquid will not wet the surface. This is the case if γ_gas/solid − γ_liquid/solid < 0, i.e., there is no reduction in surface energy by spreading of the liquid.

We can also note that the surface tension of the liquid influences the spreading behavior. As this value is always positive and located in the denominator, it will not change the sign of the equation. However, it can be thought of as an amplification factor. If γ_gas/liquid > |γ_gas/solid − γ_liquid/solid|, it will merely “dampen” the reduction of surface energy by spreading, i.e., reduce the value. However, if γ_gas/liquid ≈ |γ_gas/solidγ_liquid/solid|, the surface tension of the liquid may be able to “amplify” the factor, i.e., the contact angle. A slightly negative nominator value would be amplified to −1 (perfectly non-wetting surface), a slightly positive nominator value would be amplified to almost 1 (perfectly wetting surface).

Spreading Parameter. As stated, the wetting of a liquid on a solid will decrease the solid’s surface free energy. The surface free energy of the liquid will decrease when spread as the contact with both the surface and the wetting liquid will decrease its surface free energy. At the same time, the contact area between the surrounding fluid (usually air) and the surface will decrease as the surface area wetted with the liquid will “dewet” from the surrounding liquid. The latter term will increase the free surface. Wetting will take place only if the increase in the wetted area of the liquid will dominate the decrease of surface free energy compared to the increase brought by the dewetting from the surrounding liquid. Analytically this can be described as

$\begin{array}{ll}d{\text{γ}}_{\text{total}}\hfill & ={\text{γ}}_{\text{gas/liquid}}dA+{\text{γ}}_{\text{liquid/solid}}dA-{\text{γ}}_{\text{gas/solid}}dA\hfill \\ \hfill & =\left({\text{γ}}_{\text{gas/liquid}}+{\text{γ}}_{\text{liquid/solid}}-{\text{γ}}_{\text{gas/solid}}\right)dA=SdA\hfill \end{array}$

where the change of surface area is denoted dA. Here we introduced the spreading parameter S

$S={\text{γ}}_{\text{gas/liquid}}+{\text{γ}}_{\text{gas/solid}}-{\text{γ}}_{\text{liquid/solid}}$

Thermodynamically, this increase of surface will happen only if d_γtotal < 1. Assuming dA > 0, it follows that the spreading parameter must be negative. Therefore for S < 0 the liquid will spontaneously wet the surface entirely. This wetting is driven entirely by surface tension forces. For S > 0, there will be partial wetting driven, e.g., by gravitation. Eventually, these forces will be balanced by the increase in the free surface energy as the wetted area increases. Eq. 20.5 is sometimes referred to as the Dupré¹equation. Eq. 20.5 can be combined with Eq. 20.3 to

$\begin{array}{l}S={\text{γ}}_{\text{gas/liquid}}+{\text{γ}}_{\text{liquid/solid}}+\cos \Theta {\text{γ}}_{\text{gas/liquid}}-{\text{γ}}_{\text{liquid/solid}}\hfill \\ ={\text{γ}}_{\text{gas/liquid}}\left(1+\cos \Theta \right)\hfill \end{array}$

Eq. 20.6 is often referred to as the Young-Dupré equation. Please note that there is another notation of this equation that assumes spreading of S > 0. In this case, the signs in Eq. 20.6 must be reversed.

20.1.9 Young-Laplace Pressure at Curved Interfaces

One of the many applications where we see surface tension at work is the formation of curved interfaces, e.g., in the case on a droplet deposited on the surface. Depending on the way the droplet was deposited, it will usually show an elongated shape that has an elliptical contact area with the underlying surface. If the two axes of the ellipse are identical, the surface area is a circle. We will consider the more general case of an ellipse in the following sections.

If the pressure inside of the droplet is greater than the pressure outside of the droplet, the droplet will tend to expand in order to decrease the pressure differences. This will increase the wetted surface area against the surface tension of the droplet. Expansion will proceed until the retaining forces of the surface tension are high enough to prevent further expansion. At this point, a static equilibrium is reached. In the following, we will derive the Young-Laplace equation, which is used to describe the two angles of the curved interfaces that result in equilibrium (see Fig. 20.6a).

The two forces acting are the forces resulting from the pressure difference and the surface tension. They act along the increase in diameters of the droplet and thus contribute energy. This contribution dE_p resulting from pressure differences and the contribution dE_w resulting from wetting are described by

$d{E}_p=\left(p_{\text{inside}}-p_{\text{outside}}\right)dAd\zeta$

$d{E}_w=-\text{γ}\left(d{A}_{\zeta }-dA\right)$

where dA is the surface area of the non-expanded droplet and dA_ζ the surface area of the expanded droplet. The linear increase in the respective diameters is denoted dζ. First, dA and dA_ζ have to be found. As we are considering a doubly curved surface, the surface area dA is given by the arc lengths ds₁· ds₂, which can be calculated as (see Fig. 20.6b)

$ds=\alpha r$

using the respective diameter of the circle. Eq. 20.9 is correct for small angles α, which, as we denote the infinitesimally small change in arc length ds, is a valid assumption. Using Eq. 20.9, dA can be calculated. dA_ζ can be derived from Fig. 20.6c. First we calculate the increase in length due to increase in diameter. As the angle α is not changed by the expansion of diameter, the increase in arc length is calculated using Eq. 20.9 as

$\begin{array}{l}d{s}_{\zeta }=\alpha \left(r+d\zeta \right)\hfill \\ =\frac{ds}{r}\left(r+d\zeta \right)\hfill \\ =ds\left(1+\frac{d\zeta }{r}\right)\hfill \end{array}$

from which dA_ζ can be calculated as

$\begin{array}{l}d{A}_{\zeta }=d{s}_{\zeta ,1}d{s}_{\zeta ,1}\hfill \\ =d{s}_1\left(1+\frac{d\zeta }{r_1}\right)d{s}_2\left(1+\frac{d\zeta }{r_2}\right)\hfill \\ =dA\left(1+\frac{d\zeta }{r_1}\right)\left(1+\frac{d\zeta }{r_2}\right)\approx dA\left(1+\frac{d\zeta }{r_1}+\frac{d\zeta }{r_2}\right)\hfill \end{array}$

where we used dA = ds₁ · ds₂ and ignored the term containing (dζ)².

In order to reach equilibrium, the change of energy brought forth by Eq. 20.7 and Eq. 20.8 must be zero; therefore

$\begin{array}{ll}d{E}_{\text{total}}\hfill & =d{E}_p+\gamma dA\hfill \\ \hfill & =\left(\left(p_{\text{inside}}-p_{\text{outside}}\right)d\zeta -\gamma \right)\left(d{A}_{\zeta }-dA\right)\hfill \end{array}$

Using Eq. 20.10 results in

$\begin{array}{ll}d{E}_{\text{total}}\hfill & =\left(\left(p_{\text{inside}}-p_{\text{outside}}\right)d\zeta +\gamma \right)\left(dA\left(1+\frac{d\zeta }{r_1}+\frac{d\zeta }{r_2}\right)-dA\right)\hfill \\ \hfill & =\left(\left(p_{\text{inside}}-p_{\text{outside}}\right)d\zeta +\gamma \right)dA\left(\frac{d\zeta }{r_1}+\frac{d\zeta }{r_2}\right)\hfill \\ \hfill & =\left(p_{\text{inside}}-p_{\text{outside}}-\gamma \left(\frac{1}{r_1}+\frac{1}{r_2}\right)\right)dAd\zeta \hfill \end{array}$

from which follows

$\begin{array}{l}{p}_{\text{inside}}-p_{\text{outside}}-\gamma \left(\frac{1}{r_1}+\frac{1}{r_2}\right)=0\hfill \\ p_{\text{inside}}-p_{\text{outside}}=\gamma \left(\frac{1}{r_1}+\frac{1}{r_2}\right)\hfill \end{array}$

Eq. 20.11 is the Young-Laplace equation. It describes pressure drops at cured interfaces and has a number of interesting applications.

Immiscible Flow. The Young-Laplace equation is valid for boundaries between two (largely) immiscible fluids in general. It is applicable for a liquid on a free surface in a gas atmosphere (e.g., ambient air), but it can also be used for two immiscible fluids such as an oil and a water layer. This is because we have, during the derivation of the Young-Laplace equation, considered the energy associated with the displacement of the boundary in general. If the boundary is displaced against a second fluid, the interphase surface tension γliquid A/liquid B has to be used instead of the surface tension of only one liquid.

Minimal Surfaces. A minimal surface is defined as a surface for which the right-hand side of the Young-Laplace equation, Eq. 20.11, is zero. As we can see from the equation, this needs to be the case for all surfaces that do not have a pressure drop across the surfaces, i.e., which are open. A good example is a soap films that is suspended at open profiles, e.g., rings. Fig. 20.7 shows an example of a soap film that is suspended between two open rings. There is no pressure difference between the inside and the outside of the film, thus $\frac{1}{r_1}+\frac{1}{r_2}=0$. As can be seen, r₁ decreases with increasing x and (by the same degree) r₂ increases in size. Interestingly, it may be very difficult to calculate these surfaces analytically. However, they will form readily when transferring the surface to models in a soap film experiment.

Surfaces Driven by Two Curved Interfaces. If a fluid plug is created inside a microfluidic channel that is slightly tapered, the plug will form two surfaces with different curvatures (see Fig. 20.8). This is because the contact angle is identical but the space restriction will squeeze a more strongly curved interface at the position with the smaller wall spacing. We know from Eq. 20.11 that the pressure drop across a surface increases if the radius decreases. In Fig. 20.8 the radius at the right side of the plug has a smaller radius; therefore the pressure drop on this side will be bigger. As a result, the plug will move toward the left. This mechanism can be used to drive fluid droplets or to extract air from microfluidic channels.

Pressure in Bubbles. The pressure inside of a bubble of radius r₀ is higher than the pressure outside of the bubble because of the curvature of the bubble. It is given by Eq. 20.11 as

$p_{\text{inside}}-p_{\text{outside}}=\gamma \frac{2}{r_0}$

The smaller the bubble, the higher the pressure difference. This is why smaller bubbles make significantly more noise when collapsing. This is the reason why champagne is “louder” than, e.g., mineral water where the bubbles are significantly bigger.

In soap bubbles, the pressure difference is even higher as a soap bubble consists of a bilayer (see section 20.3.1). This bilayer consists of a layer of surfactant molecules arranged such that the unpolar tails are oriented inward and the polar head groups toward the outside (see Fig. 20.11g). This is why we have to account for a set of two doubly curved surfaces with approximately the same radius (if we assume the bubble to be spherical in shape). This is because the thickness of a surfactant bilayer is usually in the range of a few nm and can therefore be ignored, which results in

$\begin{array}{ll}{p}_{\text{inside}}-p_{\text{outside}}\hfill & =\text{γ}\left(\frac{2}{r_0}+\frac{2}{r_0+d_{\text{bilayer}}}\right)\hfill \\ \hfill & \approx \text{γ}\frac{4}{r_0}\hfill \end{array}$

20.2 Contact Angle Measurement

20.2.1 Static Contact Angle

In general, there are two common methods of measuring the contact angle: static and dynamic. In static contact angle measurement, a droplet of a liquid is placed on top of a substrate, and the resulting contact angle is measured using a contact angle microscope. These instruments typically consist of a magnifying optical setup with attached camera that allows recording the contact angle, which is usually measured via software. Static contact angle measurement is important, e.g., for the determination of the free surface energy of a substrate. Please refer to section 22.3 for further details on these methods. Fig. 24.10a shows an example of a contact angle measured for water on poly(methyl methacrylate), which was recorded using a contact angle microscope of type DSA30 Drop Shape Analyzer by Krüss (www.kruss.de).

20.2.2 Dynamic Contact Angle Measurement

The term dynamic contact angle measurements refers to techniques that measure the contact angle during movement. This is usually accomplished by adding liquid to a static droplet on a surface and thus pushing the front of the liquid across the unwetted surface. The contact angle measured during droplet expansion is referred to as advancing contact angle Θ_a (see Fig. 20.9a). It is a measure of the dry-wetting behavior of a liquid or a surface, i.e., of the fluid mechanics on a freshly wetted surface. On the other hand, continuously removing liquid from a droplet will decrease its size and thus cause dewetting of the surface. The contact angle that is formed during shrinking of the droplet is referred to as receding contact angle Θ_r (see Fig. 20.9b). The receding contact angle is a measure of the remaining interaction forces between the liquid and the solid. The difference between the advancing and receding contact angle is referred to as contact angle hysteresis. The contact angle hysteresis is a means of assessing the mean roughness of the sample. If the surface is chemically sufficiently homogeneous, advancing and receding contact angles are constant and do not depend on the size of the droplet.

20.3.3 Applications of Surfactants

There are numerous applications for surfactants, the most important ones will be discussed in this section.

Emulsification. Surfactants are used for numerous applications in everyday life. From shampoos to soaps, cleaning agents, fat remover, and laundry detergents, all of these products use surfactants. They are also used in order to form emulsions, i.e., in order to increase the solubility of two substances that would normally not mix. Soap is a very good example. During hand washing, the water will dissolve all water-soluble substances on our hands leaving behind all substances that are not water-soluble. Using soap as surfactant, these substances can be encapsulated in small micelles and thus rendered water-soluble, which allows them to be washed off our hands. Usually surfactants with cleaning properties or surfactants that are used for cleaning are referred to as detergents.

Stabilization of Suspensions. Many suspensions use surfactants in order to increase the liquid’s capability to keep solids suspended and to avoid aggregation by forming repulsion layers around the objects (see Fig. 20.19). These layers rely on the repulsive action of equally charged surfaces that can be introduced using surfactants. Especially if an unpolar particle (such as a polymer bead) is suspended in a polar liquid (such as water), the particles tend to aggregate quickly. This is because the polarity of the solvent and the particles is different, and therefore the particles tend to cluster more easily. After adding a surfactant, the surface of the particles will be masked, thus exposing the same polarity as the solvent. Furthermore, if ionic surfactants are used, the particles will be equally charged and thus repel each other. Many synthesis protocols for nanoparticles rely on surfactants in order to keep the particles stable in solution.

Micelle Reaction Compartments. In many synthesis applications, micelles are formed inside of a solvent and act as small reaction compartments. These emulsion reactions use the local confinement of reactions partners, e.g., to synthesize components of only a given size. Again, the synthesis of nanoparticles is a good example. Here small micelles act as the boundary of the reaction and limit the growth of the nanoparticles. If the micelle size is varied, the size of the nanoparticles can be optimized.

Changing the Surface Tension. One of the main applications of surfactants is changing the surface tension of liquids. In many applications, liquids that need to be handled by a given technical system may not have sufficiently high surface tension. This may be critical, e.g., if a liquid is to be transported from a reservoir onto a given substrate via a stamping method or by pin transfer. If the liquid’s surface tension is too low, it may not be possible to transport a sufficient amount of liquid. In this case, a surfactant can be added that will migrate to the liquid’s interface and increase the surface tension. For example, a low surface tension oil (e.g., a silicon oil) can be blended with an ionic surfactant that has a polysiloxane tail. After migrating to the surface, the charged head groups of the surfactant will lead to a stronger interaction in the interface layer and lead to stronger unbalance forces and thus increasing the surface tension. This allows transporting larger quantities of the liquid via a liquid transfer method as the liquid/gas interface can sustain higher forces and thus retain larger quantities of liquid.

Another common example of optimization of the surface tension are inks used for inkjet printing. These are usually mixed with surfactants in order to optimize their ability to be spotted without the formation of satellite drops. Also, the wetting behavior on the paper onto which the ink is spotted can be altered by adding suitable surfactants.

Interface Stabilization. Many applications in microfluidics would not be possible without the use of surfactants in order to stabilize liquid/liquid interfaces. Whenever discrete droplets of liquids are to be handled, surfactants are used in order to stabilize the liquid/gas or liquid/liquid interface.

Digital microfluidics is a good example for this. Here the droplets to be moved are usually immersed in an inert oil phase. Because the electrowetting induces substantial shear stress on the boundaries, it is usually advisable to stabilize the boundary using a suitable surfactant. Also because the contact area between the droplet and the top and bottom substrates carrying the electrodes is usually large (the reason being that the droplet is squeezed in-between them), it is advisable to use a surfactant, which will reduce the unspecific interaction between the droplet and the substrates. This is especially true if the setup is being used for an analytical assay where the loss of analyte (e.g., a protein) must be avoided.

A second common example is droplet microfluidics. Here the droplets are dispersed in an oil phase exposing a significant surface area. Stabilization of the droplets is not only critical to maintain the droplet intact during formation but also to avoid droplets sticking to the channel walls and to reduce the risk of (undesired) droplet merging. By introducing a surfactant (either into the droplet itself or into the oil phase), the interface of the droplet is stabilized, inducing a certain repulsion between the individual droplets. This significantly reduces the risk of droplet coagulation and merging.

Passivation. Numerous applications in bioanalytics require biochemical passivation of channel walls and substrates in order to reduce the unspecific adsorption of, e.g., proteins. This can be accomplished using suitable surfactants which render the surfaces of the microfluidic system more hydrophilic, thus inhibiting hydrophobic interaction of the proteins and the (mostly) hydrophobic substrate. In many applications, the Triton surfactants are the first substance of choice here. Depending on the native polarity of the substrate, surfactants with a higher unpolar character may have to be used especially when working with polymers. In general, non-ionic surfactants are preferred in order to decrease the risk of altering and/or degenerating the proteins of interest.

Surface Coatings. Just as surfactants can change the surface tension of liquids, they can also alter the surface free energy of a substrate, thereby allowing or preventing the wetting of a given liquid. A typical example includes anti-icing coatings where, e.g., a glass window is coated with a layer of a surfactant that lowers the surface free energy. As a result, water will not wet the surface, and therefore the surface will not show icing.

Detergents. The term detergents in biochemistry usually refers to surfactants used for cell lysis. Detergents usually have to be very mild surfactant and biocompatible substances in order to ensure that they only disrupt the cell membranes and do not significantly alter any other compound of interest. Detergents are also used to break interactions between proteins or to destroy their quaternary and tertiary structure.

In general, ionic surfactants are not a good choice as detergents with a few exceptions. As stated, sodium dodecyl sulfate is commonly used for protein denaturing during gel electrophoresis. It adds a negative charge to all proteins irrespective of their pI, which allows a subsequent separation by size and weight only. Also, cholic acid and deoxycholic acid (both being very weak anionic surfactants) can be used to dissolve membranes. Sometimes sodium lauroyl sarcosinate can also be used.

Other detergents of choice are usually glycoside surfactants, e.g., n-dodecyl-β-D-maltoside or n-octyl-β-D-thioglucopyranoside, both of which are very mild reagents that maintain protein activity during cell lysis. Tween surfactants are often used as surfactants in washing buffers.

Environmental and Health Risks. Depending on their chemical nature, surfactants may have a significant half-life, and their effects on the environment must be considered. As surfactants allow such a wide variety of processes, they also act after being used if not properly disposed. Surfactants may significantly alter an ecosystem if probed in sufficient quantity. Another important aspect is their bioaccumulating capability, i.e., their potential to accumulate in animals or humans. This is especially critical for halogenated surfactants, with fluorinated and chlorinated surfactants being considered the major risk. For most surfactants, biotoxicity assays must be performed prior to use in order to assess these risks. Some surfactants (most prominently the halogenated ones) must be disposed of properly, whereas others (such as soap) may be disposed simply as waste water.

20.4.1 Temperature

Local heating or cooling causes changes in the free surface energy and can thus be used to drive fluids in a suitable technical system. As stated in section 20.1.6, heating a surface (or a liquid) will cause its surface tension to drop. Therefore applying a temperature gradient to a surface can be used to drive liquid flow. This is sometimes referred to as thermal creep. The dimensionless number often used when characterizing flows because of temperature gradient is the Marangoni number (see section 9.9.14).

20.4.2 Surfactants

A very good demonstration of the Marangoni effect is obtained by dropping a small amount of a surfactant (usually a drop of soap) to a static water reservoir (see Fig. 20.21). The surfactant will locally decrease the surface tension, thus inducing a fluid flow away from the region of lower surface tension. This flow can be observed by adding a small number of particles (dust or carbon black) to the water, in which case one can see that the particles completely vacate the region into which the surfactant was dropped.

20.4.3 Tears of Wine

Similar effects can also be observed for other liquids that have a lower surface tension than water. Dropping ethanol into water will result in a similar effect because of the lower surface tension (see Tab. 20.2). This results in the “tears in wine” phenomenon. It can be observed in wine glasses. Wine (which is roughly speaking a mixture of water and ethanol) wets the surface of the glass and forms a meniscus because of capillary action (see section 21). Because of the increased surface area, the evaporation is increased, and, as ethanol has a lower boiling point and a higher vapor pressure compared to water, ethanol will evaporate quicker leaving the liquid in the meniscus with a higher overall content of water. Because water has a higher surface tension, the surface tension gradient in the glass will increase toward to outside, i.e., toward the meniscus. This effectively pulls more and more liquid into the meniscus, which will eventually collapse into small droplets of water that flow back from the side of the glass forming the characteristic “tears of wine”.