The capillary effect [1–3] appears everywhere in our everyday life and engineering, and the capillary fluid [4] can be used for enhancing heat conduction in micro/nanodevices or in a porous hierarchy [5, 6]. The capillary effect can also be effectively applied in the microelectromechanical system (MEMS) [7, 8] and energy-harvesting devices [9, 10] and can greatly affect the mechanical and thermal properties of porous materials [11–14]. The application of capillary oscillation to fabrics’ sweat permeability has significant implications in textile engineering. Sportswear garments that are designed with high sweat permeability are preferred to enhance breathability and prevent excessive moisture buildup. This is crucial as excessive moisture buildup can lead to discomfort and distraction during physical activity, ultimately hindering the performance of the wearer. Similarly, garments with excellent sweat permeability are necessary to allow for rapid absorption of sweat and minimize skin irritation in undergarments. The fabrics’ sweat permeability from the inner side to the environment has triggered rocketing interest in sportswear and military apparel in an extreme cold environment. If the sweat cannot be transferred through the cloth after an active motion, it will greatly affect the comfort property and even be life-threatening due to the icy fabric.

The capillary oscillation and its relationship with sweat permeability are crucial for developing new materials with improved performance characteristics in textile. However, the capillary fluid’s oscillation property was hardly analyzed. Jin et al. revealed the frequency property mathematically [1]. Han and He applied the capillary oscillation to fabric’s self-cleanliness [15]. Xiao, et al. studied the capillary oscillation in a short small tube [16]. Saxena studied moisture permeability through nylon and cotton fabrics [17]. Midha et al. researched the laundering times on moisture permeability [18]. As the capillary thread moves up and down, it displaces a small amount of liquid from the surface of the fabric into the surrounding air. The process reduces the local surface tension at the contact point between the fabric and the skin, making the fabric more hydrophilic and thus more susceptible to sweat absorption. Ha et al. showed that air and moisture permeability plays an important role in the clothing microclimate [19]. Raja et al. conducted an experiment on the sweat transfer of multi-weave structure fabrics [20]. Guan et al. studied the clothing–human body system [21]. Liu et al. researched the thermal property of a microstructure [22]. The oscillatory motion of the capillary wall can create micro-cracks and pores in the fabric, which can facilitate the passage of sweat molecules and enhance sweat permeability. Furthermore, these micro-cracks can also provide additional channels for moisture transfer from the skin to the surrounding environment, further improving sweat evaporation and cooling performance.

All the aforementioned theoretical and experimental studies revealed that the capillary effect on fabrics’ sweat permeability is of extreme importance for both everyday life and advanced applications. When sweat interacts with a fabric surface, capillary oscillation enhances the transfer of sweat from the skin to the fabric’s surface. This process results in increased sweat permeability, which is essential for regulating body temperature. In this paper, we will show the mechanism of the capillary oscillation and its great effect on the fabric’s air/moisture permeability.

When a small tube is gradually immersed into a fluid, the fluid rises along the tube. This phenomenon, commonly referred to as the capillary effect [2, 3], is a familiar observation to anyone who has witnessed the wetting process of a napkin when it comes into contact with water. It can be explained by the geometrical potential theory [23]. The capillary rise is vulnerable to an environmental perturbation, and the capillary fluid will vibrate periodically [1]. The periodicity of the capillary oscillator is determined by the combined action of the gravity and surface tension of the liquid, and this periodicity is more obvious in the pull-in solution. Pull-in instability and periodic behavior are two key phenomena in microelectromechanical system (MEMS) dynamics, and differential equations can well describe these nonlinear aspects [24–26]. The capillary rise without any perturbation can be expressed as [27]. h ∝ 1 r n , (1) where h is the capillary rise, r is the equivalent capillary radius, and n is a positive parameter depending on the tube’s geometry.

Equation 1 shows that a smaller porosity leads to a higher capillary rise, which can explain why a nanofiber membrane has high permeability [28]. The nanofiber membrane can be produced via the electrospinning technology [29] and has the potential application in optimizing design of the sportswear and military apparel in an extreme cold environment.

A mathematical model was established in Ref. [1], and Bin et al. conducted a numerical simulation of the capillary oscillation [2].

The dynamical motion of the capillary fluid through a zig-zag porous structure of a fabric can be modelled using the following equation [1]: x ″ + ε ⁡ sin ω x + ω 0 2 x = 0 , x 0 = A , x ′ 0 = 0 , (2) where x is the center of the capillary fluid when it is still, ω is the frequency of zig-zag porosity, ω 0 is a parameter relative to the capillary effect, ε is a geometric parameter relative to fabric’s geometry, and A is the amplitude.

When ω 0 = 0, Eq. 2 becomes a famous pendulum oscillator [30]. When x<<1, Eq. 2 can be approximately expressed as x ″ + ε ω + ω 0 2 x − 1 6 ε ω 3 x 3 = 0 , x 0 = A , x ′ 0 = 0 . (3)

This is the duffing oscillator [31]. To provide insights into the periodic property of Eq. 3, we quote the frequency formulation of Ji-Huan He [32, 33]. Consider a general nonlinear oscillator in the form x ″ + p x = 0 , x 0 = A , x ′ 0 = 0 , (4) where p is a nonlinear function of x and p/x > 0. Ji-Huan He’s frequency formulation reads [32, 33]. Ω 2 = p x x x = 3 2 A , (5) where Ω is the frequency of the nonlinear oscillator.

The square of the frequency of Eq. 3 is Ω 2 = ε ω + ω 0 2 x − 1 6 ε ω 3 x 3 x x = 3 2 A = ε ω + ω 0 2 − 1 6 ε ω 3 x 2 x = 3 2 A = ε ω + ω 0 2 − 1 8 ε ω 3 A 2 , (6) where Ω is the capillary fluid’s vibrating frequency and A is the amplitude. This frequency formulation is a simple yet effective tool for the fast and accurate identification of the periodic property of a nonlinear oscillator [34–36]. Eq. 4 can also be derived using various numerical methods, such as the homotopy perturbation method [37, 38], the variational iteration method [24, 39], Wang’s variational approach [40], or asymptotic methods [25, 26].

A higher vibrating amplitude of the capillary motion implies that sweat can be transferred to a farther distant. When the capillary rise is less than the thickness of the fabric, the capillary oscillation is the main factor for the sweat permeability. According to Eq. 4, a higher amplitude requires a low frequency, and this low-frequency property is the mechanism underlying the sweat permeability. The period of the capillary oscillation is T = 2 π Ω = 2 π ε ω + ω 0 2 − 1 8 ε ω 3 A 2 , (7) where A is the amplitude of the capillary fluid’s periodic motion. According to Eq. 7, a large amplitude implies a large period, which infers an extremely slowly motion. This can explain why the capillary rise seems to be stable at the initial stage when the small tube is immersed into water and keeps convincingly unchanged for few hours; however, it might change after 24 h or longer. As the permeability thickness reaches h+A, where h is the capillary height, the capillary fluid’s periodic motion will continue to oscillate within the capillary. This oscillation will cause the fluid to move back and forth along the length of the capillary, creating a wave-like motion. A will increase as the thickness of the permeability increases, leading to an increase in the overall flow rate through the capillary. When the fabric’s thickness is less than h+A, a good permeability is predicted.

The maximal amplitude of the capillary fluid reaches when the frequency becomes zero: Ω = ε ω + ω 0 2 − 1 8 ε ω 3 A 2 = 0 . (8)

The maximal amplitude reads A max = 8 ε ω + ω 0 2 ε ω 3 . (9)

Equation 9 shows that the frequency of zig-zag porosity is the main factor affecting the moisture/air permeability.

To ensure good sweat permeability, the fabric should have a large zig-zag period, as shown in Figure 1. L = 2 π ω , (10) where L is the zig-zag period. A larger L leads to a thicker fabric, and the multiple-layer structure is always adopted in practical applications.

In order to decrease the thickness of the fabric with good air/moisture permeability, we can adopt a hierarchical structure from a nano/micro-inner layer to a macro-outside layer, and the thickness of each layer should satisfy the following inequality 11: H n < r n + 8 ε n ω n + ω n 0 2 ε ω n 3 , (11) where H n is the thickness of the nth layer of the hierarchy, r n is the nth layer’s capillary rise, and ω n is the zig-zag frequency of the nth layer [1].

The inner wall of blood vessels is unsmooth, leading to a zig-zag inner surface, so that the blood can be transferred to a farther distance than that of the smooth inner surface.

If the porosity is smooth enough in each layer, inequality 11 should be modified as H n < r n , (12)

which means the thickness of each layer should be less than its capillary rise. Many natural hierarchical systems can transport water for a long distance [41].

This paper proposes the capillary oscillation model to design hierarchical fabrics with good air/moisture permeability, which is of critical importance for the clothing design, especially for sportswear and military apparel in an extreme cold environment, where the sweat permeability plays an important role in human’s comfort and safety. This theoretical analysis enables scientists to understand the capillary oscillation and its role in air/water transportation. Capillary oscillation is the periodic oscillation of liquid in a capillary tube due to the interaction between surface tension and viscous forces. In the context of fabric moisture absorption and perspiration, capillary oscillation plays an important role in determining the wicking and transport of moisture in the fabric. When a fabric comes into contact with moisture, capillary forces cause the liquid to be drawn into the fabric’s capillary channels. The capillary oscillation then helps distribute the moisture throughout the fabric, allowing it to be absorbed and transported more efficiently. This is because the oscillation creates a pumping effect that helps move the liquid through the capillary channels. In addition, capillary oscillation can also help enhance the fabric’s moisture absorption and perspiration properties. By promoting the movement of moisture through the fabric, capillary oscillation can help increase the rate of moisture absorption and perspiration, thereby improving the fabric’s overall comfort and performance. Overall, the relationship between capillary oscillation and moisture absorption and perspiration properties of fabrics is complex, with many factors influencing the process. However, it is clear that capillary oscillation plays an important role in determining the wicking and transport of moisture in the fabrics and can help enhance the fabric’s moisture absorption and perspiration properties. Understanding this relationship and applying advanced techniques like nanotechnology and optimized yarn structure can lead to the development of innovative textile materials with superior sweat permeability and performance. Because this paper gives a self-contained theoretical model for the sweat permeability for the first time, the future holds exciting possibilities for improving garment design and enhancing human wellbeing as technology advances and researchers continue to explore this topic.