One of the most common fluid mechanical effects exploited in microfluidics is capillarity, e.g., induced fluid motion in very small channels. As we have seen, curved surfaces introduce a pressure gradient that can be exploited in order to drive fluids. In Eq. 20.11, we can see that for small radii, this pressure drop can amount to significant values.
This is exploited by using channels with very small diameters; in the simplest case a circular capillary is used (see Fig. 21.1a). From Eq. 20.11, we can deduce that for a circular tube for which r1 = r2 = r, the pressure difference is given as
From geometry it follows that
Therefore, Eq. 21.1 can be rewritten as
Eq. 21.2 expresses the capillary pressure drop across the curved surface in a thin capillary. The pressure drop is positive, i.e., the pressure inside the liquid column is higher than in the ambiance. Therefore the capillary will pull the liquid up. This driving pressure is often referred to as the capillary pressure.
The rising liquid column will experience gravitational forces that eventually balance the capillary forces. At this equilibrium, the column will have reached a total height of hc, which can be derived as
The parameter hc is often referred to as capillary height or Jurin1height named after English physician James Jurin. However, there are several accounts of earlier studies on capillarity, including work by Leonardo da Vinci in the 15th century and even work by Greek mathematician and engineer Hero of Alexandria who used capillarity in a number of setups more than 2000 years ago [2].
Please note that in Eq. 21.4, we introduced a parameter that is referred to as the capillary length Lc, which we will briefly introduce.
The capillary length Lc is an important number that we will use in the following sections. As we will see, numerous fluid mechanical phenomena originating from the effects of surface tension are dependent on this parameter. For a given liquid, it is sometimes also referred to as capillary constant and defined as
This number is a constant for a given liquid. As we can see in Eq. 21.4, the introduction of this constant makes the equation independent of the fluid in question. This means that the capillary height can be given normalized to Lc, which makes it valid for every fluid. We will see that there are a number of occasions where this constant can be used for a given fluid mechanical phenomenon thus making the derived equations valid for all fluids by multiplying it with the capillary length.
Analytically, the capillary length puts the surface tension forces in relation with the inertia forces. As stated, it can be calculated for any given fluid. It can also be calculated for fluid immersed in a different fluid, in which case the difference in densities and the interfacial surface tension have to be used in Eq. 21.5. Using water in air as the two fluids and standard conditions, the capillary length is calculated to be Lc = 2.71 mm. For a droplet, the capillary length indicates that for all droplets with diameters in the range of 2.71 mm, the effects of gravity can be ignored. This is an interesting consideration, e.g., for rain drops (which rarely are more than a millimeter in diameter). Although often drawn in tear-shaped form, raindrops will always be near-perfect spheres.
Using the capillary length, the capillary heights of tubes with different diameters can be calculated in Eq. 21.4. The results are depicted in Fig. 21.2. Fig. 21.2a shows the capillary height hc normalized to . The values shown are calculated for different wall contact angles Θ starting with Θ = 0° (perfectly wetting) as the topmost curve all the way down to Θ = 180° (perfectly non-wetting) in increments of 20°. For Θ = 90°, the line is horizontal.
The calculated values are valid for any fluid and depend only on the wall contact angle Θ of this fluid. Fig. 21.2b shows the capillary height hc for water, again calculated for the different wetting cases.
One interesting aspect can be seen in Eq. 21.3 and Fig. 21.2: for contact angles of Θ > 90°, the capillary pressure and the capillary height are negative (see Fig. 21.1b). For these cases, the capillary will represent a pressure barrier, i.e., an overpressure is required to push the liquid into the capillary. This is the case, e.g., when pumping aqueous solutions through hydrophobic capillaries made from polytetrafluoroethylene, for example.
We will discuss the fluid mechanics of the meniscus in section 21.3. However, for small capillary diameters, this meniscus can be adequately approximated by a circle (see Fig. 21.1). The depth of the meniscus hr can be calculated by using the following geometrical relations
Inserting Eq. 21.7 into Eq. 21.6 results in
The relevant dimensionless quantity to assess the influence of the capillary forces compared to the viscous forces of the fluid is the capillary number Ca (see section 9.9.10). In strongly porous materials, the capillary number is usually in the range of Ca < 1 × 10−5, in which case the viscous forces of the fluid can be ignored, and the flow is purely driven by capillary forces. Driving microfluidic flows with capillarity is one of the oldest and still most commonly used ways of controlling fluid motion.
Capillarity is the reason why a liquid forms meniscus at the wall of a vessel (see Fig. 21.3). In the following section, we will derive the equations that describe the curvature of the surface in close proximity of the wall. As we will see, there is one (surprisingly) easy way of deriving the curvature, which, however, yields (surprisingly) incorrect results. The second approach we will take allows deriving the curvature implicitly. This approach yields correct results.
The first approach we will take is based on the curvature of the surface because of the Young-Laplace pressure drop across the curved interface. The shape of the meniscus is due to the balance of forces from gravitation and the Young-Laplace pressure drop at the curved surface. For a fluid segment dx, the force balance is
where we used Eq. 3.103 to approximate the curvature. Noted that this equation is valid only if the changes in curvature are small because the function is a linearization of Eq. 3.102. As we will see, this approximation is incorrect, especially in close proximity to the wall where the meniscus is strongly curved. However, using the full equation of the surface curvature results in a differential equation that can only be solved numerically.
Eq. 21.9 is an ordinary second order differential equation with constant coefficients (see section 8.2.3), which is solved by a function of type
The details of this solution can be found in section 8.2.3.6. We see that the surface smooths out the farther we move away from the vessel wall; therefore z (x) → 0 for x → ∞, in which case we find that c1 = 0. The second integration constant is found for the boundary condition at x = 0 where because at x = 0. Therefore we find
which leads to the final solution
The meniscus therefore decays exponentially. In Eq. 21.12 we used the capillary length (see Eq. 21.5), which is characteristic length scale for the decay.
As already stated, this result is a crude approximation because the meniscus near the wall is strongly curved, which is in contradiction to our initial assumption of a not-strongly curved meniscus. This becomes especially obvious for liquid that wets the vessel wall well. For (perfect wetting) Eq. 21.12 will converge to infinity, which is (obviously) incorrect.
In order to find the correct solution, we need to apply the surface curvature without linearizing it. As already stated, this results in a nonlinear ordinary differential equation that is extremely difficult to solve analytically.
However, there are a couple of ways of solving this problem. Looking at Fig. 21.3, we first note the following relations
Using Eq. 21.15, we can rewrite Eq. 21.9 to
this time applying the nonlinearized form to express the surface curvature. We now take the derivative with respect to s and apply Eq. 21.14
We now apply a small trick in order to convert the double integral to a single integral (see section 3.3.2) by exploiting the fact that
which we can employ after multiplying Eq. 21.16 by
Eq. 21.17 can be integrated to result in
The integration constant is determined by the fact that for ϕ → 0: , in which case c1 = Lc, which results in
Here we can exploit the fact that (see Eq. 3.35), which results in
Eq. 21.19z contains the implicit solution to the contour of the meniscus. It is expressed as a function of the angle φ and the path variable s. As we will see, it is analytically possible but very tedious to derive the function z (x) from Eq. 21.19. It is significantly easier to derive two functions from it, i.e., x (φ) and z (φ). Using these two equations, it is possible to calculate the contour implicitly by finding x and y as functions of the angle φ. The value range for φ is given by the initial contact angle and φ → 0 for x → ∞.
Deriving x (φ). We will start by deriving x (φ). For this we replace ds in Eq. 21.19 using Eq. 21.13, resulting in
which can be integrated to result in
The integral is not straightforward, and it is best to locate it in a suitable integral table or by using, e.g., Maple. The integration constant c2 can be found for x = 0: φ = φ0, in which case we find
This allows us to rewrite Eq. 21.19 as
Eq. 21.21 allows us to find the values for x for a given angle φ.
Deriving z (φ). We now turn to finding the function z (φ), which can be derived by replacing ds in Eq. 21.19 using Eq. 21.14, resulting in
which can be integrated as
Again, the integral is best located in an integration table or by using an algebra tool. Eq. 21.22 is the sought equation. We now have two equations to calculate the values x (φ) and z (φ) and thereby implicitly find the meniscus contour.
One important aspect that we can already see when looking at Eq. 21.21 and Eq. 21.22 is that they are both scaled to the capillary length Lc. Therefore the derived equations are correct for all liquids, irrespective of their physical properties because for each liquid, the capillary length is a constant. Therefore it is not necessary to calculate the meniscus shape for different liquids. If the wetting contact angle Θ0 at the wall is known, the profile can be derived from Eq. 21.21 and Eq. 21.22.
Fig. 21.4 shows the calculated profiles for the meniscus at different wetting angles at the wall, starting from Θ0 = 0° (perfectly wetting) as the topmost profile all the way to Θ0 = 180° (perfectly non-wetting) as the bottommost profile in increments of 20°. Please note that Eq. 21.21 is discontinuous for Θ0 = 90°, which is why this profile is not shown. However, for Θ0 = 90°, there is no meniscus formation, and the surface remains flat. Fig. 21.4a shows the calculated meniscus shapes for water at STP conditions. As stated, the profiles will look identical for all liquids as long as Θ0 is known. Fig. 21.4b shows this general meniscus shape normalized to the capillary length, the latter of course being different for different liquids.
In this section we discussed the concept of capillarity, which is a direct consequence of the surface tension of fluids. Capillarity is a major driving mechanism for microfluidic flows and is well known, e.g., from the fact that a capillary of a hydrophilic material (such as glass) as well as a substrate with inherent porosity (such as a natural fiber) will transport fluids due to the fact that water wets them. Capillarity is one of the easiest functional mechanism to use in order to drive a microfluidic system and numerous platforms, e.g., the lateral flow devices make ample use of this fact. We discussed the equations from which we can derive the capillary pressure as well as the capillary heights. We also detailed the shape of the liquid meniscus.