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Capillary Length and Mechanical Equilibrium in Capillary Actions

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Introduction - Capillary Length's Role in Mechanical Equilibrium and Menisci Behavior

Question.1 How does capillary length affect mechanical equilibrium and menisci behavior in capillary actions


Mechanical equilibrium in capillary actions can be discussed by introducing the terms capillary length or capillary constant. It is denoted by the symbol of q and the calculation of the capillary rise can be calculated by using the lateral force equation. Lateral force equation can be calculated by using the equation F (L) = 2πγ (tan ψ2)e^-qL. Here, F is the function of L, and L represents the center to center distance between two nuclei. Mechanical equilibrium is directly linked with the capillary length and it can be evaluated by introducing menisci behavior.


Lateral capillary forces can be calculated by using the following equation. Center to center distance between two particles can be determined by relating the equation of capillary length. Capillary forces can be determined by using the equation which can correlate with the laplace equation of pressure. Ψ represents the menisci constant which can be evaluated by using the equation of lateral capillary forces. The equation of the lateral capillary forces has been depicted below. This equation would help to determine the lateral capillary forces where center to center distance is 50 µm.

F (L) = 2πγ (tan ψ2) e^-qL [Here, γ denotes surface tension, ψ2 represents menisci constant, q denotes capillary length and L stands for center to center distance = 50 µm = 50×10^-6]

= 2× 3.14× 0.072× (tan 3) e^ (8.56 × 10^-3× 50× 10^-6)

= 0.45216 × 0.0524 × 1.0001

= 0.237 N

The force applied here is related to the attraction force because the force can integrate the particles in a capillary function. Other than, the calculated force also contains the positive integer (+ 0.237) and it indicates the attraction force (Xiao et al. 2021). Any capillary action is related to the attraction of the particles where anti-gravitational actions can be observed. Internal distances in capillary functions decrease with time and therefore the attraction force helps to integrate the particles more and more.


Capillary length is a constant term which can be discussed by considering two factors; gravity force and Laplace pressure. It is related to the floating of the substances in a water surface or other liquid surface. The capillary length is not related to the parameter of length rather it is related to a constant term. Capillary length is known as capillary constant and it can be measured by a length scaling factor. The length scaling factor is linked with gravity and surface tension. Menisci are related to this topic and it is a core physical property (Zhang et al. 2017). The fundamental of the physical property which is related to the capillary length governs the physical behavior of menisci. When the force of gravity equals the outward pressure of the liquid, then this condition can be achieved. Gravity here relates to the body forces and Laplace pressure is connected to surface forces. Menisci is a piece of rubbery cartilages or C-shaped tough that holds the knee joint concerning shinbone and thighbone. Our knee is a junction of two types of skeleton and menisci can hold those two types of bones and thus we can move our knee easily. Function of menisci is related to the functions of a joint that means to protect and cushion the knee. The upper bone which is known as femur and the lower bone are related to tibia. These two types of bones are connected to the joint and the function of the joint is to hold and move those two types of bones properly. It acts like a cushion of the knee. Capillary length is also related to the constant term of the force equilibrium.

Capillary length can be measured by taking the considerations which are related to the surface forces and gravity forces. Laplace pressure can be defined by using the equation of Laplace pressure and thus we can find the capillary length of the equilibrium. Menisci can be introduced by understanding the functions of a knee joint. Knee joint helps to connect the femur and tibia and it is a C-shaped tough that free to move the bones as per its functional criteria (O'Farrell et al. 2017). Particle motion in a fluid surface can be incorporated here because in case of particle motion on a fluid surface, floating property of the particle can be recorded in the liquid film. The motions of the particles are recorded here and then it would help to calculate the particle motion throughout the film. Subsequent bands can also be evaluated by practical experiment.

Capillary length can be calculated as q = √ (γ/?ρg)

Here, γ = 0.072 N m-1,

?ρ = (1-0.9) gm. cm^-3 = 0.1 gm. cm^-3 = 100 kg. m^-3 ,

g = 9.81 N.

After putting it in the equation, we can find capillary length of the set up and it can be donated by q.

q = √ (γ/ρg)

= √ {0.072/ (100 × 9.81)

= 0.00856

= 8.56 × 10^-3

Hence, the value of capillary length is 8.56 × 10^-3 (m^2. kg ^-1).

Question.2 Explore the basics of capillary properties: surface tension, energy, contact angle, and capillary rise. How do these terms help understand fluid behavior?

Although still not required, some basic understanding of capillaries properties will indeed be useful in understanding the subject. “Surface tension” (interfacial), “surface energy”, contact angle, “capillary length”, and “capillary rise'' are the important terms related to this study that have been calculated and discussed in this section. These fundamental principles are outlined shortly here. Because a fluid layer does have a significant power proportionate to its width, it continually desires to decrease (Alderete et al. 2019). As a result, every line on the top is pushed in both edges in the perpendicular direction to the line, as shown in figure (“the force vectors are on the surface; since the two forces are equal in magnitude and directed in the opposite directions with each other, the total force on the line is zero”). Interfacial tension, indicated as, is indeed the force per unit surface area. Similarly, every surface or solid–liquid interface (typically) has positive emotion; as a result of the power generation, any vector on the ground is dragged in perpendicular directions through both edges even by surface. S and SL represent the equivalent forces each unit distance for such a solid object as well as the solid–liquid contact, correspondingly.

Question.3 How can colloidal crystal formation be utilized to synthesize highly porous silica nanoparticles for enhanced efficacy in chemistry and medical nanotechnology?"

Silicon nanoparticles can be used in various fields of chemistry and medical nanotechnology. By using the method of colloidal crystal formation, micro porous silica can be synthesized. The pores of that silica cannot be seen in naked eye and this is why electron microscopes can be used to understand the morphology of the nanoparticles (Bae et al. 2017). Proper synthesis of the nanoparticles would help to enhance the efficacy of the components and therefore, the process of preparation should be monitored correctly. Calibration of the machine should be done properly otherwise; the outcomes would fail to meet the satisfactory result. For the manufacture of permeable silica nanoparticles, monodisperse macroporous poly (glycidyl methacrylate) (PGMA) microspheres have been employed as nothing more than a pattern (Berthiaume et al. 2018). The initial polymer nanoparticles measured 9.3 μ m in diameter and then were made using only a refined Ugelstad process inside a multistep foaming polycondensation. Following that, silica (SiO2) was absorbed on the surface and within the PGMA droplets to create poly (glycidyl methacrylate)-silica blended nanoparticles (PGMA-SiO2). Nanostructured SiO2 nanoparticles were generated during a calcination of the PGMA-SiO2 microparticles. Testing and scanning electron microscopy have been used to examine the morphological characteristics, droplet size, crystallinity, and interior architecture of silica nanoparticles. The quantity of silica produced as well as its large surface area was assessed using thermal decomposition analysis and reactive nitrogen adsorption. The thickness of the permeable silica nanoparticles reduced up to approximately 30% as contrasted to the beginning PGMA nanoparticles (Ma et al. 2018). Such highly porous droplets have great potential for chromatographic and biotechnological applications.

Question.4 How can you calculate the ideal number of microcapsules needed to encapsulate a 10 ml ibuprofen solution, considering the weight of microcapsules and the thickness of the polymer shell?


No of microcapsules used for the encapsulation of 10 ml ibuprofen can be calculated by using the volume analysis of the encapsulation bar. Encapsulation of ibuprofen can be done by using the polymerisation of microcapsules (Fatouh et al. 2018). Weight of microcapsules is measured as 250 gm. Hence weight of one microcapsule can be defined to calculate the number of micro capsules.

Encapsulation of 10 ml ibuprofen weighs 250 gm and it is the summative weight of the microcapsules and 10 ml ibuprofen.

Let's take x number of microcapsules used. Hence, volume of x number of microcapsules is equal to 4/3xπr^3 = x.4/3× 3.14 × (1× 10^-6) ^3 = 4.18x × 10^-18

According to the assumption 4.18x × 10^-18 = 10 ml = 0.01 liter = (0.01× 10) ^3 = 0.001 cm^3.

Therefore, x = 23.9 × 10^13

Hence, no of microcapsules used to cover up the ibuprofen solution is equal to 23.9 × 10^13.


Thickness of the microcapsules can be calculated by using the equation which is given below.

Da = (1 − √ 1 − F) R

Where, Da is the average thickness of the film, F is the force applied and R stands for radius.

If F and R are provided then Da can be calculated. Da denotes the thickness of the polymer shell (Li et al. 2017). By using the equation, it can be calculated and the value of the thickness of the wall is equal to 10^-6. This is nearly the thickness of animal membrane.

Reference list


Alderete, N., Zaccardi, Y.V., Snoeck, D., Van Belleghem, B., Van den Heede, P., Van Tittelboom, K. and De Belie, N., 2019. Capillary imbibition in mortars with natural pozzolan, limestone powder and slag evaluated through neutron radiography, electrical conductivity, and gravimetric analysis. Cement and Concrete Research118, pp.57-68.

Bae, J., Bende, N.P., Evans, A.A., Na, J.H., Santangelo, C.D. and Hayward, R.C., 2017. Programmable and reversible assembly of soft capillary multipoles. Materials Horizons4(2), pp.228-235.

Berthiaume, A.A., Grant, R.I., McDowell, K.P., Underly, R.G., Hartmann, D.A., Levy, M., Bhat, N.R. and Shih, A.Y., 2018. Dynamic remodeling of pericytes in vivo maintains capillary coverage in the adult mouse brain. Cell reports22(1), pp.8-16.

Fatouh, M. and Abou-Ziyan, H.J.A.T.E., 2018. Energy and exergy analysis of a household refrigerator using a ternary hydrocarbon mixture in tropical environment–Effects of refrigerant charge and capillary length. Applied Thermal Engineering145, pp.14-26.

Li, Y., Alibakhshi, M.A., Zhao, Y. and Duan, C., 2017. Exploring ultimate water capillary evaporation in nanoscale conduits. Nano letters17(8), pp.4813-4819.

Ma, L., Zhang, W., Wang, L., Hu, Y., Zhu, G., Wang, Y., Chen, R., Chen, T., Tie, Z., Liu, J. and Jin, Z., 2018. Strong capillarity, chemisorption, and electrocatalytic capability of crisscrossed nanostraws enabled flexible, high-rate, and long-cycling lithium–sulfur batteries. ACS nano12(5), pp.4868-4876.

O'Farrell, F.M., Mastitskaya, S., Hammond-Haley, M., Freitas, F., Wah, W.R. and Attwell, D., 2017. Capillary pericytes mediate coronary no-reflow after myocardial ischaemia. Elife6, p.e29280.

Saberian, M., Jahandari, S., Li, J. and Zivari, F., 2017. Effect of curing, capillary action, and groundwater level increment on geotechnical properties of lime concrete: Experimental and prediction studies. Journal of Rock Mechanics and Geotechnical Engineering9(4), pp.638-647.

Shou, D. and Fan, J., 2018. Design of nanofibrous and microfibrous channels for fast capillary flow. Langmuir34(4), pp.1235-1241.

Verbeke, K., Formenti, S., Vangosa, F.B., Mitrias, C., Reddy, N.K., Anderson, P.D. and Clasen, C., 2020. Liquid bridge length scale based nondimensional groups for mapping transitions between regimes in capillary break-up experiments. Physical Review Fluids5(5), p.051901.

Xiao, B., Huang, Q., Chen, H., Chen, X. and Long, G., 2021. A fractal model for capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media. Fractals29(1), pp.2150017-1867.

Zhang, P., Wittmann, F.H., Vogel, M., Müller, H.S. and Zhao, T., 2017. Influence of freeze-thaw cycles on capillary absorption and chloride penetration into concrete. Cement and Concrete Research100, pp.60-67.

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