Capillary Systems (Part 4 of 4): Capillary Force
In part three of this series, we introduced capillary systems and discussed how a capillary might encourage or discourage movement of fluid within or along its length. In this final video, we’ll introduce the Capillary Force formula to present you with the full technical description of the forces at work in a capillary system.
Hi, my name is Ken Milam. I’m an application engineer here at ThermoPore. Welcome back to Thermo.TV. Armed with an understanding of the capillary force formula, you’ll be able to predict the response of a capillary system for a host of liquid-solid combinations with materials of varying pore sizes.
Let’s start by discussing a capillary system made from a high surface energy material. As you might recall, in the third video of this series, we used a glass capillary as our high surface energy material and we discussed the fluid’s movement up the glass capillary tube. But you might ask, how long will this wicking process continue? Is there no end? It’s a good question. So, let’s explore some of the forces at work in this arrangement.
When a fluid wets a capillary, we know that the wall of the capillary is trying to pull the fluid over its own surface. We also know that the fluid will respond like a three dimensional magnetic chain by pulling its neighboring molecules along for the ride. However, if we assume the capillary is oriented in a vertical fashion, then the weight of the fluid located inside the capillary gets heavier and heavier as the liquid moves up the capillary tube. Let me say this again, provided we’re analyzing a capillary oriented in a vertical fashion, the weight of the fluid in the capillary gets heavier and heavier as the high surface energy capillary pulls the liquid up the capillary tube.
What we’ll eventually find in a capillary wetted by a liquid is capillary equilibrium. At equilibrium, the upward pulling force of the capillary is equal to the downward force of the liquids weight. The height of the capillary is referred to as the capillary height or the capillary head and it quantifies the capillary force that is acting on of the system.
So let’s look at these two forces, the upward capillary force and the downward weight force in more detail. Let’s start with the downward weight force. We’ll start by calculating the capillary’s volume which is equal to the capillary height times the area of the capillary, π r2 h. To calculate the mass, we’ll multiply this volume times the fluid’s density, π r2 h ρ. Lastly, to arrive at the column’s weight, we’ll multiply the mass by gravity h π r2 ρ g. Okay, so that gives us the downward force component. Now, remember, at equilibrium, this downward force component is balanced by an upward capillary force component which we’ll define next.
At equilibrium, the upward capillary force acting on the liquid can be represented by a line drawn through the top of the liquid’s surface. This line effectively represents the attractive force of the capillary acting on the liquid. The contact angle that is made between the liquid and the solid can be measured at the solid, liquid, gas interface. Using the contact angle, we can show the capillary’s attractive forces in “x” and “y” components. We’re primarily interested in the vertical or the “y” component which can be describe as the cosine of the contact angle multiplied by the liquids surface tension. So that calculation gives us a unit upward force. Now, because the attractive force exists around the perimeter of the capillary, we’ll need to multiply the Unit Upward force by the capillary’s circumference to obtain the capillary’s total force acting in the upward direction. Upward force = Fy = λ cos θ 2 π r.
Setting the Weight and the Capillary forces equal to each other and solving for the capillary height, we get the traditional formula for capillary height.
So, this formula predicts that the capillary height of a system will increase when 1) the surface tension of the liquid increases, 2) the contact angle approaches zero, 3) the density of the fluid decrease, and/or 4) the radii of the capillary decreases.
Now, what about the other scenario – whereby a fluid comes into contact with a low surface energy capillary, i.e., the fluid does not wet the capillary. In this second scenario, we know that the wall of the capillary is trying to resist the fluid entry. But we know that an external force might enable the fluid to gain entry into the capillary. So, let’s see what our capillary force formula predicts for this arrangement.
In this scenario, we don’t have the fluids weight as a force that counter balances the capillary force. Instead, we might have a pressure that is acting on the fluid across the capillaries diameter. The pressure that a fluid exerts on a capillary is a function of the capillary depth under the water’s surface. You might recall this effect in a swimming pool. As you swim deeper and deeper, the water pressure increases. By multiplying a fluid’s pressure times a capillary’s area, we’ll derive a force that should be equal to and opposite the resistive capillary force at equilibrium.
So the external upward force will be F = Pressure X Capillary’s Area. One formula for pressure is P = ρ g h. The capillary’s area is easily calculated by π r2. So, if we were to submerge the tip of a capillary underwater, the inward Force exerted by the fluid onto the capillary’s opening will be F = π r2 ρ g h.
As we did in our previous example at equilibrium, the downward capillary force acting on the liquid can be represented by a line drawn through the top of the liquid’s surface. This line effectively represents the repulsive force of the capillary acting on the liquid. The contact angle that is made between the liquid and the solid can be measured at the solid, liquid, gas interface. Using the contact angle, we can show the capillary’s resistive forces in “x” and “y” components. As was the case earlier, we’re primarily interested in the vertical or the “y” component which can be describe as the cosine of the contact angle multiplied by the liquids surface tension. Unit downward force Fy = λ cos θ . So that calculation gives us a unit downward force. Now, because the repulsive force exists around the perimeter of the capillary, we’ll need to multiply the Unit Downward force by the capillary’s circumference to obtain the capillary’s total force acting in the upward direction. Capillary downward force Fy = λ cos θ 2 π r.
Again, if we solve for the h to determine the point of water entry, we’ll derive the following formula.
Did you notice that this is the capillary force equation is the same one that we derive earlier. It conveniently addresses both wetting and non-wetting capillary systems! The height will be a positive value for systems with small contact angles, or wetting systems. This means that the capillary will draw fluids vertically up a capillary a distance “h”. The height will be a negative value for systems with large contact angles, or non-wetting systems. This means that the capillary will resist a fluids entry until submerge downward to a capillary distance “h”.
ThermoPore maintains a diversified portfolio of porous materials that can help you achieve any number of hydrophilic wicking or hydrophobic venting applications and I hope that this four part tutorial has provided you with some insight into the type of variables that we’ll be able to tweak to satisfy the needs of your next development project.
Stay on the look out for additional videos by signing up for our RSS feed and as always, if you have any additional questions or if there are some topics that you’d like to see added to the Thermo.TV channel – give us a call or drop us a line. For now, I’m Ken Milam saying thanks for watching this installment of Themro.TV – we’ll see you next time.