Liquid transport in non-uniform capillary fibrous media

Abstract

Liquid transport in porous materials is affected significantly by the geometry of the non-uniform capillaries. In this study, an N-section lotus-rhizome-node-like non-uniform capillary model was for the first time proposed based on the plane Poiseuille flow and capillary pressure equation to investigate the liquid transport in porous fibrous media. Normalized total flow time of the non-uniform capillary was obtained as a function of the height and width ratio between the converging and diverging nodes and their total number. The results indicated that the velocity of liquid transport greatly depended on the number of nodes in a certain liquid transport length. The non-uniform capillaries with frequent alterations between converging and diverging nodes have low liquid transport efficiency. The thick capillary exhibits fast liquid transport efficiency in those capillaries with the same self-similar geometry. The model was verified using polypropylene filament yarns and different liquids. The results agreed well with the theoretical prediction. This work not only provides a deeper understanding of liquid transport inside porous fibrous media with non-uniform capillaries, but can also guide the novel design and optimization of functional fibrous materials.

Keywords

liquid transport non-uniform capillary wicking porous media

Liquid transport in the porous materials is of considerable importance for many fields and applications, such as fabrics,^1–3 microfluidics,^4,5 separators for batteries,^6,7 water collection^8,9 and solar-steam generation.^10,11 Initially, capillary flow in a uniform cross-section was widely employed to describe the process of liquid wicking in a porous medium using the Lucas–Washburn equation.^12,13 Lately, more and more researchers have realized that the non-uniformity of the capillary cross-section along capillary flows in porous media affects the flow behavior significantly in actual applications.^14,15 Erickson et al.¹⁶ found theoretically that the non-uniform cross-sectional capillaries exhibited remarkably slower wetting behavior than straight capillaries; they also noticed that the deviation in the capillary diameter tends to slow the overall wetting speed. Multiple irregular regions were considered in their extended model, but they found that the wetting time in a given length is independent of the number of irregular regions (up to three converging–diverging regions). Young¹⁷ obtained the same result by constructing a simple formulation based on analysis of the Lucas–Washburn equation for simulation of the interface progression in a non-uniform capillary. However, Young also pointed out that gravity can affect the results. A periodically constricted tube with sinusoidal shape was modeled by Sharma and Ross¹⁸ to study the kinetics of liquid penetration without considering gravity. Later, Staples and Shaffer¹⁹ considered a similar sinusoidal model including gravity to study wicking flow in glass bead beds. Liou et al.²⁰ further developed the non-uniform cross-sectional capillary flow model based on the Navier–Stokes equations, incorporating inertial and viscous terms, which is applicable to different wall variations. All of these investigations on the capillary flow in non-uniform capillaries suggest that the configuration of the non-uniform capillary has a substantial effect on the capillary flow.²¹ However, the majority of these works are theoretical or analytical and mainly focus on a confined structure, namely a glass tube. Simple but more realistic porous materials, such as paper and fiber or yarn, are less studied.

Recently, Shou et al.²² established a model to investigate the geometry-induced capillary flow in multi-section porous paper layers with variation in width and height against the flow time. They found that the two-section structures with a negative gradient of radius against the absorption direction have faster absorption rates than those with uniform radius. Benltoufa et al.²³ investigated the capillary rise in a jersey knitted structure considering both macro and micro pores, but the flow channel between yarns was simplified as parallel plates and the flow path between fibers was regarded as smooth tortuous capillary tube composed of parallel fibers/filaments. In another study,²⁴ a twist coefficient was introduced to consider the influence of twist of the yarn on the wicking mechanism. However, the yarn was idealized as an assembly of cylinders and its cross-section was treated as a uniform capillary. In fact, the capillary tubes in fibrous media, such as yarns and fabrics, are rarely smooth and uniform because of the unparalleled arrangement of fibers and yarns due to twist. The difference of centripetal component pressure of the spinning force in the yarn cross-section caused by twist results in the fiber migration regularly from the outside to the center of yarn and then back to the outside.²⁵ Consequently, the non-uniform capillary is ubiquitous in fibrous material. It is worth noting that the non-uniform construction of the capillary tube can greatly affect the wicking behavior of fibrous media.

As known to all, moisture or liquid transport in porous fibrous materials not only determines the wearing comfortability,^23,24 but also affects post-processing, including dyeing, finishing and coating. Furthermore, fibrous materials have shown great potential in microfluidic devices^26–29 used for medical diagnostics and environmental monitoring. Therefore, it is critical to understand the liquid transport in porous fibrous materials, especially fiber and yarn, which can work as a model of general porous materials. Besides, a universal multi-section model for porous materials with a non-uniform capillary is essential and indispensable to understand the liquid transport in porous-materials-based microfluidic devices.

Unlike previous capillary flow investigations established on the uniform capillary composed of parallel fiber/filament, in this study a simple but effective one-dimensional (1D) N-section lotus-rhizome-node-like capillary model was firstly proposed based on the plane Poiseuille flow and capillary pressure equation to investigate the non-uniform capillary flow behavior in porous materials. The capillary flow time equation and the normalized capillary flow time equation were established based on the capillary construction parameters, including the width ratio and height ratio between the converging section and the diverging section, the total section number, the total length of the capillary and the equivalent diameter of the tube. Comparing with the existing non-uniform capillary model, the N-section lotus-rhizome-node-like capillary model provides a simple but effective method to illustrate how the structure of a non-uniform capillary affects the mass transport in the fibrous porous medium driven by capillary force. In order to verify this model, polypropylene filament (with a round cross-section) yarns of different twist factors and filament fineness were constructed to mimic non-uniform capillaries with different structural parameters. The capillary flow behaviors in a non-uniform capillary were compared with the yarn wicking experiments. We aim to deepen the fundamental understanding of liquid transport in porous materials, and pave the way from design of these materials toward microfluidic device fabrication.

Theoretical model

In the capillary flow, liquid will spontaneously find a continuous channel driven by the capillary force. For wicking in a fibrous material, the smooth capillary channel will be pinned at the position where filament migration occurs (assuming the yarn has an ideal structure).²⁵ To continue the capillary process, the original smooth capillary channel (converging section) will enter a wider capillary intersection (diverging section) to find a new capillary channel to move forward. This process will be continuously repeated along the capillary flow.

To describe the above capillary flow, a 1D N-section non-uniform lotus-rhizome-node-like capillary model constructed by heteromorphic parallel plates, as depicted in Figure 1(a), was proposed as a simplified three-dimensional (3D) or two-dimensional (2D) system. When capillary wicking happens, liquid flows along the capillary length (z-axis). The non-uniform capillary in Figure 1(a) is composed of a number of alternative converging and diverging sections, just like the lotus root with alternative converging and diverging sections, as was shown in Figure 1(c). In the case of yarn, the width, the height and the alternate frequency of the converging and diverging sections in Figure 1(a) are determined by the filament arrangement in the yarn. A uniform capillary constructed by plane parallel plates in Figure 1(b) serves as the control sample. The uniform control sample can be regarded as the capillary channel of the ideal untwisted yarns.²⁴
Figure 1.
Physics model of the capillary tube: (a) N-section lotus-rhizome-node-like capillary channel; (b) uniform capillary channel; (c) picture of a lotus root.

The assumptions of the model in Figure 1 are as follows.
The N-section lotus-rhizome-node-like non-uniform capillary (Figure 1(a)) and the uniform capillary (Figure 1(b)) have the same height H in the z-axis, the unit length in the y-axis and the same area in the x–z plane. Thus, for a certain wicking height H (the non-uniform capillary is composed of an even section number), the non-uniform capillary and the uniform capillary have the same volume. The distance between the parallel plates of the uniform capillary, A, is defined as the equivalent width of the N-section lotus-rhizome-node-like non-uniform capillary (see Equation (23)).

Gravity is ignored since it is three powers of 10 smaller than the capillary pressure for delicate capillary wicking in a relatively short wicking length (less than 0.1 meters).³⁰

The capillary wicking is set to start from the diverging section, as shown in Figure 1(a).

The total number of the converging and diverging sections is N (even number).

The width and the height of the diverging section are a₁ and h₁, and the width and the height of the converging section are a₂ and h₂, respectively. For twisted yarns, the diverging section is shorter than the converging section, that is, h₁ < h₂. Meanwhile, the width of the diverging section is larger than that of the converging section, that is, a₁ > a₂.

The contact angle is considered as constant during the wicking process.

Flow time in the uniform capillary

Based on the plane Poiseuille flow,³¹ the flow rate of the uniform capillary (with unit width in the y direction) in Figure 1(b) is
$Q = A \cdot u = - A \cdot \frac{A^{2}}{12 η} \frac{\partial p}{\partial z} with u = \frac{\partial z}{\partial t}$
(1)
where Q is the flow rate, u is the average flow velocity, A is the width of the uniform capillary, η is the liquid dynamic viscosity and $\frac{\partial p}{\partial z}$ represents the pressure change per unit length.

Thus, the hydrostatic pressure drop accounting for the viscous flow can be obtained as below
$p = - η Q \int_{0}^{z} \frac{12 d x}{A^{3}} = - η \frac{\partial z}{\partial t} \int_{0}^{z} \frac{12}{A^{2}} d s = - η \frac{\partial z}{\partial t} \frac{12}{A^{2}} z$
(2)

The capillary pressure p_c in the uniform capillary composed of parallel planes (with unit width in the y direction) is given by Benltoufa et al.³²
$p_{c} = - \frac{2 γ cos θ}{A}$
(3)
where γ is the surface tension of water and θ is the contact angle between water and the solid capillary surface.

According to Equations (2) and (3), we have
$\partial t = η \frac{6 z}{A γ cos θ} \partial z$
(4)

Thus, the flow time of the uniform capillary can be obtained by integrating Equation (4)
$T_{u} = \frac{6 η}{γ cos θ} \int_{0}^{H} \frac{z d z}{A} = \frac{6 η}{γ cos θ} \frac{H^{2}}{2 A} = \frac{3 η}{γ cos θ} \frac{H^{2}}{A}$
(5)

It can be simplified as
$T_{u} = 3 C \frac{H^{2}}{A}$
(6)
where C is a constant, $C = \frac{η}{γ cos θ}$ .

The flow time for the penetration distance of H in the uniform capillary is
$T_{u} = \frac{3 η}{γ cos θ} \frac{H^{2}}{A}$
(7)

Flow time in the non-uniform capillary

We define
$h_{2} = {nh}_{1}$
(8)

$a_{2} = {ma}_{1}$
(9)

Here, n is the length ratio and m is the width ratio between the converging section and diverging section, respectively; let n ≥ 1 and 0 < m < 1.

To make the model straightforward, we only consider the scenario when N is an even number in this study. Because the uniform capillary and the N-section lotus-rhizome-node-like capillary have the same height and area
$\frac{N}{2} (h_{1} + h_{2}) = H$
(10)

$\frac{N}{2} (a_{1} h_{1} + a_{2} h_{2}) = AH$
(11)

Flow time in each converging section and the diverging section of the N-section non-uniform capillary are
$t_{2 i} = 3 C \frac{h_{1}^{2}}{a_{1}} [2 {inm}^{2} + (2 i - 1) \frac{n^{2}}{m}] (i = 1, 2, \dots, ., \frac{N}{2})$
(12)

$t_{(2 i - 1)} = 3 C \frac{h_{1}^{2}}{a_{1}} [(2 i - 1) + \frac{2 (i - 1) n}{m^{3}}] (i = 1, 2, \dots, ., \frac{N}{2})$
(13)
where N is an even number.

Thus, the total flow time with penetration distance of H in the non-uniform capillary is
$\begin{matrix} T_{N} = \sum_{i = 1}^{\frac{N}{2}} (t_{N - 2 i} + t_{N - (2 i - 1)}) \\ = \sum_{i = 1}^{\frac{N}{2}} 3 C \frac{h_{1}^{2}}{a_{1}} [(2 i - 1) + 2 {inm}^{2} + (2 i - 1) \frac{n^{2}}{m} + \frac{2 (i - 1) n}{m^{3}}] \\ = 3 C \frac{h_{1}^{2}}{a_{1}} \sum_{i = 1}^{\frac{N}{2}} [(2 i - 1) + 2 {inm}^{2} + (2 i - 1) \frac{n^{2}}{m} + \frac{2 (i - 1) n}{m^{3}}] = 3 C \frac{1}{(\frac{N}{2})^{2}} \frac{H^{2}}{A} \frac{1 + mn}{(1 + n)^{3}} \\ \times \sum_{i = 1}^{\frac{N}{2}} [(2 i - 1) + 2 {inm}^{2} + (2 i - 1) \frac{n^{2}}{m} + \frac{2 (i - 1) n}{m^{3}}] \\ = 3 C \frac{1}{(\frac{N}{2})^{2}} \frac{H^{2}}{A} \frac{1 + mn}{(1 + n)^{3}} [(\frac{N}{2})^{2} + ((\frac{N}{2})^{2} + \frac{N}{2}) {nm}^{2} \\ + (\frac{N}{2})^{2} \frac{n^{2}}{m} + ((\frac{N}{2})^{2} - \frac{N}{2}) \frac{n}{m^{3}}] \\ = 3 C \frac{H^{2}}{A} \frac{1 + mn}{(1 + n)^{3}} \\ \times [1 + (1 + \frac{2}{N}) {nm}^{2} + \frac{n^{2}}{m} + (1 - \frac{2}{N}) \frac{n}{m^{3}}] \end{matrix}$
(14)

The normalized total flow time of the N-section lotus-rhizome-node-like capillary in Equation (15) is defined as the ratio between the total flow time in the non-uniform capillary and that in the corresponding uniform capillary with the same wicking length H, which illustrates the influence of the unevenness of the capillary tube on wicking performance. Thus, the normalized total flow time is expressed as
$\begin{matrix} R_{N} = \frac{T_{N}}{T_{u}} = \frac{(1 + mn)}{(1 + n)^{3}} + \frac{(1 + \frac{2}{N}) m^{2} n (1 + mn)}{(1 + n)^{3}} \\ + \frac{n^{2} (1 + mn)}{m (1 + n)^{3}} + \frac{(1 - \frac{2}{N}) n (1 + mn)}{m^{3} (1 + n)^{3}} \end{matrix}$
(15)

Experimental section

Materials

A non-uniform capillary tube with different structural parameters can be easily constructed using yarns by twisting. Here, polypropylene filament yarns (Dongqiao Chemical Fiber Co., Ltd, Guangdong, China) with round cross-sections were employed to investigate the capillary flow behavior in a non-uniform capillary tube, because the capillary shape in polypropylene filament yarn will not be affected in the hygroscopic state due to its low water absorption capability (0.3%) and the round cross-section fiber does not consist a cannelure, which will affect the wicking in the capillary between fibers. Two twist factors (73.4 and 204) were respectively applied to polypropylene filament yarns with different filament fineness (300D/64F and 600D/58F) to construct four kinds of non-uniform capillaries, which have different alternate frequencies between the converging and diverging sections and different capillary channels.

In the textile industry, the twist factor of yarn is an empirical parameter that has been established by experiments and practice and can be expressed as
$α = W \sqrt{D / 9}$
(16)

Here, W is the twist of yarn, α is the twist factor of yarn and D is the yarn denier.

The spinning parameters of the polypropylene filament yarns are listed in Table 1.
Table 1.
Spinning parameters of the polypropylene filament yarns

Sample Low twist factor (73.4)
High twist factor (204)

Twist of yarn (twist/10 cm)

300D/64F 12.7 35.4

600D/58F 9 25

Equipment and test procedure

The filament yarns were twisted with a digital doubler twister (DSTW-01, Tianjin Jiacheng Mechatronic Equipment Co., Ltd, China). Bead rings of 58.3 and 158 mg were adopted for 300D/64F and 600D/58F polypropylene filament yarns, respectively, to provide the appropriate spinning tension. The twisted yarns were set in pressure vessel under 115℃ and about 1.7 atmospheric pressure for 2 h. The filament yarns with different filament fineness and twist were characterized using a scanning electron microscope (SEM, TM1000, Hitachi, Japan). An image analysis software, ImageJ, was used for measurement. The twist of yarns was tested using a yarn twist counter (YG(B)331A, Wenzhou Darong Textile Instrument Co., Ltd, China).

For wicking that happened in the small scale of the capillary tube between fibers, the gravity effect can be ignored due to the capillary driven force being larger by more than three powers of 10 compared to the gravity in a short wicking length.³⁰ As a result, the vertical wicking experiment was carried out using the experimental apparatus, as shown in Figure 2. Yarns were mounted on the iron support with 10 g pre-tension added on the lower end of each filament yarn. A ruler was set parallel to the yarns to measure the wicking height (meters). A liquid reservoir containing colored liquid was hoisted on a lab jack. When the test started, the liquid reservoir was quickly elevated and the lower ends of the fixed yarns were immediately immersed into the liquid reservoir. The capillary rise at different times was continuously recorded with a video recorder during a certain wicking length.
Figure 2.
Schematic drawing of the experimental apparatus.

To verify the validity of the N-section non-uniform capillary model, two additional liquids, 0.5 wt% methylene blue solution and 0.5 wt% red colored soybean oil, were used as the colored liquid for the wicking test. The contact angle of the colored liquids on the polypropylene filament was measured using a contact angle tester (Shanghai Zhongchen Digital Technology Apparatus Co., Ltd., China) at 20℃. The spraying method was adopted to produce small droplets that can stay on the filament steadily. Its surface tension was measured using a surface tension tester (JYW-200A, Chengde Dingsheng Testing Equipment Co., Ltd., China) at 20℃. Its viscosity was measured using a digital viscometer (SNB-1, Shanghai FangRui Instrument Co., Ltd. China) at 20℃.

Results and discussion

Normalized total flow time of the N-section lotus-rhizome-node-like capillary

According to Equation (15), the normalized total flow time curves of N-section lotus-rhizome-node-like capillaries versus section number N with different length and width ratios are shown in Figures 3 and 4. The results reveal that in most cases, except for the section number N = 2 when the two-section capillary is a negative gradient radius capillary with the a larger diameter section below and a smaller diameter section above, the normalized total flow time of the N-section lotus-rhizome-node-like capillary is larger than 1, suggesting that the non-uniform capillary has a lower liquid transport efficiency compared with that of the uniform capillary, which is consistent with previous studies.^16,17 For capillaries with the same length ratio n and width ratio m, the normalized flow time increases with increasing section number N. This means that compared with the uniform capillary, the wicking velocity of the non-uniform capillary decreases due to the increased unevenness of the capillary caused by more frequent alternations between the converging section and the diverging section.
Figure 3.
Normalized total flow time curves of N-section lotus-rhizome-node-like capillaries versus different section numbers N with different length ratios n when m = 0.3.

Figure 4.
Normalized total flow time curves of N-section lotus-rhizome-node-like capillaries versus section number N with different width ratios m when n = 3.

In the scenario of the same section number N and identical width ratio m (shown in Figure 3), the normalized flow time decreases with increasing length ratio n: the greater the difference between converging and diverging length (namely, larger n), the faster the liquid transports. When the length ratio is large, such as n = 100, the liquid transport time in the non-uniform capillary approaches that in the uniform capillary.

This is because in the minimum cycling unit (one converging section and one diverging section) of the capillary, the converging section takes almost the whole length of the non-uniform capillary length, making the overall diameter more uniform and possibly decreasing the diameter deviation, thereby increasing the wetting speed.¹⁶ When the length ratio n approaches the convergence condition $n \to \infty$ , Equation (15) is
$\begin{matrix} {lim}_{n \to \infty} R_{N} = {lim}_{n \to \infty} \frac{T_{N}}{T_{u}} \\ = {lim}_{n \to \infty} [\frac{(1 + mn)}{(1 + n)^{3}} + \frac{(1 + \frac{2}{N}) m^{2} n (1 + mn)}{(1 + n)^{3}} \\ + \frac{n^{2} (1 + mn)}{m (1 + n)^{3}} + \frac{(1 - \frac{2}{N}) n (1 + mn)}{m^{3} (1 + n)^{3}}] = 1 \end{matrix}$
(17)

Equation (17) indicates that when the length ratio n approaches its limitation ∞, the normalized total flow time approaching its limitation is 1, suggesting that when the length ratio n is very large the non-uniform capillary tends to be the corresponding uniform capillary. The wicking performance of the non-uniform capillary is just the same as that in the uniform capillary. In addition, it is noticed that the change of normalized flow time becomes not obvious when the section number N is larger than 15 for a certain wicking length.

On the contrary, for the same section number N and identical length ratio n (as shown in Figure 4), the closer the width of the converging and diverging sections (namely, larger m), the higher the liquid transport efficiency is. With the increasing width ratio m, the width difference between the converging section and diverging section gradually decreases. Namely, the non-uniform capillary tends to change into the uniform capillary. When the width ratio m approaching the limitation is 1, Equation (15) is
$\begin{matrix} {lim}_{m \to 1} R_{N} = {lim}_{m \to 1} \frac{T_{N}}{T_{u}} \\ = {lim}_{m \to 1} [\frac{(1 + mn)}{(1 + n)^{3}} + \frac{(1 + \frac{2}{N}) m^{2} n (1 + mn)}{(1 + n)^{3}} \\ + \frac{n^{2} (1 + mn)}{m (1 + n)^{3}} + \frac{(1 - \frac{2}{N}) n (1 + mn)}{m^{3} (1 + n)^{3}}] = 1 \end{matrix}$
(18)

Equation (18) suggests that when the width ratio m approaching its limitation is 1, the normalized total flow time approaching its limitation is 1 as well. That means that with the disappearance of the width difference between the converging section and the diverging section, the wicking performance of the non-uniform capillary is just the same as that in the uniform capillary. Figure 4 also indicates that small a capillary diameter tends to have low liquid transport efficiency, agreeing with previous studies.^16,17

When N approaches infinity, the normalized total flow time in Equation (15) is
$\begin{matrix} {lim}_{N \to \infty} R_{N} = {lim}_{N \to \infty} \frac{T_{N}}{T_{u}} \\ = {lim}_{N \to \infty} [\frac{(1 + mn)}{(1 + n)^{3}} + \frac{(1 + \frac{2}{N}) m^{2} n (1 + mn)}{(1 + n)^{3}} \\ + \frac{n^{2} (1 + mn)}{m (1 + n)^{3}} + \frac{(1 - \frac{2}{N}) n (1 + mn)}{m^{3} (1 + n)^{3}}] \\ = \frac{(1 + mn)}{(1 + n)^{3}} + \frac{m^{2} n (1 + mn)}{(1 + n)^{3}} \\ + \frac{n^{2} (1 + mn)}{m (1 + n)^{3}} + \frac{n (1 + mn)}{m^{3} (1 + n)^{3}} \end{matrix}$
(19)

The corresponding contour plot according to Equation (19) is shown in Figure 5. It can be seen from Figure 5 that when N approaches infinity, theoretically, the normalized total flow time decreases with the increase of the width ratio m. Meanwhile, the normalized total flow time decreases with the increase of the length ratio n. These results are consistent with Figures 3 and 4. Moreover, the non-uniform capillary has the slowest capillary flow at about n = 1.
Figure 5.
Contour plot of the normalized total flow time when the section number N approaches infinity.

Total flow time of the N-section lotus-rhizome-node-like capillary

For theoretical analysis, suppose the surface tension and the viscosity of liquid are 7.28 × 10^–2 N/m and 1 × 10^–3Pa·s, respectively. The contact angle between the liquid and solid material is 66˚, the wicking length H = 70 mm, the length ratio n = 3 and the width ratio m = 0.3. According to the total flow time expression in Equation (14), it is easy to obtain the total flow time curve of the N-section lotus-rhizome-node-like capillaries versus the section number N under different equivalent width A, as shown in Figure 6. It is obvious that for a lotus-rhizome-node-like capillary with certain equivalent width A, the total flow time increases with the increase of the section number N. For a certain wicking height, a larger section number N means a decreasing length of the diverging section (h₁) and converging section (h₂), and larger alternate frequency between the diverging and converging sections. The result suggests that it will take a longer time for the N-section lotus-rhizome-node-like capillary with larger alternate frequency between the diverging section and converging section to reach a certain wicking height, which is in line with the study done by Liu et al.²⁴ This phenomenon can be attributed to the increased unevenness of the non-uniform capillary tube, which caused frequent alternation between the diverging section and converging section.
Figure 6.
The total flow time curve of N-section lotus-rhizome-node-like capillaries versus section number N with different equivalent widths A, while H = 70 mm, m = 0.3 and n = 3.

Figure 6 also suggests that for a certain section number N (i.e. the length of the diverging section h₁ and converging section h₂ are constant), the total flow time of the N-section lotus-rhizome-node-like capillaries increases with the decrease of the equivalent width A (i.e. smaller width of the diverging section a₁ and converging section a₂), suggesting that for a certain diameter range of capillary, liquid transports faster in wider capillary tubes at a certain wicking distance.^16,17

Yarn appearance and experimental wicking results

In order to understand the morphology of the employed yarns, microscopy images were captured and are displayed in Figure 7 for polypropylene filament yarns with different deniers (fineness or diameter) and twist factors. It can be found that the filaments of the low twisted yarns are more parallel to each other than those in high twisted yarns, and the migration between the inner layer filaments and the outer layer filaments is less intense under the smaller centripetal component pressure of spinning force. The less intense migration of the low twisted yarn results in the longer migration period of filaments (one migration period includes one converging section and one diverging section) along the yarn axis. In contrast, the high twisted filament yarns exhibit a high intense migration between the inner layer filaments and the outer layer filaments under the larger centripetal component pressure of spinning force. Thus, the high twisted filament yarns have a shorter migration period along the yarn axis. Filament migration frequency is determined by the tension difference between the inner layer and outer layer filament in the yarn cross-section,²⁴ and this phenomenon can be directly observed on the surface of the yarn. When filament migration happens, the original converging section (see Figure 7(d)) composed of parallel arranged filaments is broken due to the movement of the filament in the yarn cross-section, and instead a diverging section (see Figure 7(d)) constructed by unparalleled arranged filaments appears. It is noticed that yarns with the same twist factor (or twist level) almost have the same appearance (Figures 7(a)–(d)), even though they are of different filament fineness. The filament migration caused by the twisting of yarn determines the smoothness of the capillary channel between the yarn filaments; less migration gives smoother and uniform liquid transport channels that are also less compacted compared with those with a high twist level.^24,25
Figure 7.
Scanning electron microscopy images of 300D/64F and 600D/58F filament yarns with different twist factors: (a) 300D/64F (twist factor 73.4); (b) 300D/64F (twist factor 204); (c) 600D/58F (twist factor 73.4); (d) 600D/58F (twist factor 204). The converging/diverging sections and the transit region are denoted in Figure 7(d) as a representation of the other pictures.

The yarn twist value and the corresponding calculated twist factor are listed in Table 2. It can be seen from Table 2 that the twist of the yarn sample is consistent with the design value in Table 1.
Table 2.
Twist of yarn samples

Sample Low twist factor (73.4 ± 1.7)
High twist factor (204 ± 2)

Twist of yarn (twist/10 cm)

300D/64F 12.7 ± 0.3 35.4 ± 0.4

600D/58F 9 ± 0.2 25 ± 0.3

The corresponding wicking experiments for the above four different filament yarn specimens in different liquids are illustrated in Figures 8 and 9, where the liquid fronts of the polypropylene filament yarns were recorded at different wicking times. It is apparent that the wicking length is affected by the filament fineness and twist factor, that is, a higher twist factor results in a shorter wicking length (compare yarns a and b, yarns c and d in Figures 8 and 9), while a larger filament fineness has a longer wicking distance (compare yarns a and c, yarns b and d in Figures 8 and 9), which are in accordance with previous reports.^3,24,33 Generally, it is believed that high twist and small filament fineness decrease the size and continuity of the inter-filament capillaries,^3,34 increasing the resistive force to capillary action^34,35 and leading to the above phenomena. Moreover, it can been seen that the wicking velocity of the yarn sample in 0.5 wt% methylene blue solution (Figure 8) is much faster than those in 0.5 wt% red colored soybean oil (Figure 9), mainly due to the high viscosity of soybean oil.
Figure 8.
Photographs of the liquid front change in polypropylene filament yarns at different times in 0.5 wt% methylene blue solution: (a) 300D/64F (twist factor 73.4); (b) 300D/64F (twist factor 204); (c) 600D/58F (twist factor 73.4); (d) 600D/58F (twist factor 204). Scale bar = 1 cm.

Figure 9.
Photographs of the liquid front change in polypropylene filament yarns at different times in 0.5 wt% red colored soybean oil: (a) 300D/64F (twist factor 73.4); (b) 300D/64F (twist factor 204); (c) 600D/58F (twist factor 73.4); (d) 600D/58F (twist factor 204). Scale bar = 1 cm.

The contact angles of the 0.5 wt% methylene blue solution and 0.5 wt% red colored soybean oil on the polypropylene filament at 20℃ are shown in Figures 10(a) and (b), respectively, where the small droplet can cover and steadily stay on the surface of the polypropylene filament by the spraying method. The mean contact angle is measured to about 66° for 0.5 wt% methylene blue solution and 20° for 0.5 wt% red colored soybean oil.
Figure 10.
Contact angle of (a) 0.5 wt% methylene blue solution and (b) 0.5 wt% red soybean oil with on polypropylene filaments.

The average surface tension of the 0.5 wt% methylene blue solution at 20℃ is about 65.5 × 10^–3 N/m, which is lower than that of the water due to the surfactant effect of methylene blue. The mean surface tension of the 0.5 wt% red colored soybean oil is 27 × 10^–3 N/m, which is the same as that of the soybean oil without red oil. The addition of red oil did not affect the surface tension of soybean oil.

The viscosities of the 0.5 wt% methylene blue solution and the 0.5 wt% red colored soybean oil at 20℃ are measured as about 1 × 10^–3 and 51 × 10^–3Pa·s, respectively. It can be seen that the viscosity of red soybean oil is higher than that of the blue water. The addition of the small amount of dyes did not change the viscosity of either water or soybean oil.

Evaluation of the theoretical result

To verify the model, we compare the theoretical results with the experimental results. Structure parameters of the non-uniform capillary in each yarn sample were obtained by measuring the SEM images of the corresponding yarns.

Determination of n

By measuring the length of the diverging and converging section of each yarn sample in the SEM images, it could be found that the length ratio between the converging and diverging sections of the less twisted yarns (twist factor 73.4) is about 3, while that of the higher twisted yarns (twist factor 204) is about 2. The percentage of the diverging section tends to increase with the increase of the alternate frequency between the converging and diverging sections in high twisted yarns. Yarns with the same twist factor have the same length ratio between the converging and diverging sections.

Determination of $a_{1}, a_{2}, m$

Suppose the filaments of yarn are closely packed under the normal force vertical to the yarn axes by twisting. When there is no filament migration, the filaments are closely packed and parallel to each other, constructing the converging section of the capillary. The converging section is composed of three closely packed filaments, shown in Figure 11(a). The diameter of the converging section a₂ equals the equivalent diameter of the pore among the three filaments.
Figure 11.
Schematic drawings of the cross-section of the converging section (a), filament migration (b) and the diverging section (c).

The schematic drawing of filament migration is shown in Figure 11(b). When the filament migration occurs, the filament moves from the outer layer to the inner layer, or from the inner layer to the outer layer of the yarn (filament No. 5). The original closely packed pattern of the converging section (top surface of Figure 11(a)) changed into an incompactly packed pattern of the diverging section constructed by five filaments (bottom surface of Figure 11(c)). The diameter of the diverging section a₁ equals the equivalent diameter of the pore among the five filaments.³⁶

The diameter of filament d can be obtained from Equation (20)
$d = \sqrt{\frac{4 \times 10^{3}}{9 π} \cdot \frac{D / n_{f}}{ρ}}$
(20)

Here, d is the diameter of the filament (µm), D is the yarn denier, n_f is the number of filaments in the yarn and ρ is the density of the filament (g/cm³).

By simple geometric calculation, it is easy to obtain that the diameter of the converging section a₂ = 0.2257d, and the diameter of the diverging section a₁ = 0.8317d. Thus, the width ratio m = a₂/a₁ ≈ 0.3.

Determination of $h_{1}, h_{2}, N$

SEM images suggest that filament migration occurs at about every twist. The length of yarn in one twist can be calculated according to Equation (21)
$l = \frac{100}{W}$
(21)
where l is the length of yarn in one twist (mm) and W is the twist of the yarn (twist/10 cm).

The height of the diverging section h₁, the converging section h₂ and the corresponding total section number N in a certain yarn length H can be calculated according to Equations (8) and (21)–(23)
$l = h_{1} + h_{2}$
(22)

$N = \frac{2 H}{l}$
(23)

Determination of A

Based on the above analysis, the equivalent width A can be calculated according to Equation (24)
$A = \frac{a_{1} h_{1} + a_{2} h_{2}}{h_{1} + h_{2}}$
(24)

All structure parameters of the 300D/64F and 600D/58F filament yarns are calculated and listed in Table 3. Here we set H = 80 mm.
Table 3.
Structure parameters of the 300D/64F and 600D/58F filament yarns

Sample Twist factor a₁ (µm) a₂ (µm) m A (µm) Length of yarn in one twist (mm) h₁ (mm) h₂ (mm) n N

300D/64F 73.4 27.2 8.16 0.3 12.9 7.87 1.97 5.90 3 20.33

204 27.2 8.16 0.3 14.5 2.87 0.96 1.91 2 55.75

600D/58F 73.4 40.3 12.1 0.3 19.2 11.11 2.78 8.33 3 14.40

204 40.3 12.1 0.3 21.5 4.00 1.33 2.67 2 40.00

Based on the above data, we can plot the predicted flow time curves of the N-section lotus-rhizome-node-like capillaries and the corresponding experimental flow time of 300D/64F and 600D/58F yarns versus wicking length with different colored liquids. Figures 12 and 13 suggest that the model agrees well with the experimental results for yarn samples with different structures and different liquids. Both the theoretical and the experimental results follow the same trend, and the wicking height value of yarn samples was in agreement with the previous research report.³⁷ For yarns with a certain filament fineness, the higher twist factor gives a slower wicking velocity and a shorter wicking length than yarn with a low twist factor. It can be seen from Table 1 that yarns with a high twist factor have a larger total section number N in a certain yarn length. A larger total section number means more frequent alternating between the converging section and the diverging section and increased unevenness of the capillary tube in yarns, which results in the slower wicking velocity of yarn with a high twist factor.¹⁶ The influence of filament fineness on the flow time can also be found by comparing Figures 12(a) and (b), and 13(a) and (b). The result reveals that for yarns with the same twist factor, the thicker filaments tend to have faster wicking velocity.^3,29 SEM images suggest that yarns with the same twist factor but different filament fineness have the same appearance (Figure 7). Accordingly, we deduced that the non-uniform capillaries in yarns with the same twist factor should have the same morphology and structure as well, that is, the height and the width ratio between the converging and diverging sections of the non-uniform capillaries are the same. The difference is that the capillaries constructed with the thick filaments are larger than those constructed with the fine filaments (see Table 1). As a result, the yarns composed of fine filaments have a relatively low wicking velocity.³
Figure 12.
Flow time of (a) 300D/64F filament yarn and (b) 600D/58F filament yarn versus wicking length in 0.5 wt% methylene blue solution.

Apart from the yarn structure, it can be found from Figures 12 and 13 that the properties of liquid have a great influence on the wicking performance of yarns. The wicking velocity of yarn samples in the 0.5 wt% methylene blue solution (Figure 12) is much faster than that in the 0.5 wt% red colored soybean oil (Figure 13). It also can be found from Equation (6) that the three important liquid property parameters, namely viscosity, surface tension and contact angle, determine the C value. For a certain yarn in which the capillary pipe structure is fixed, Equation (14) can thus be expressed as
$T_{N} = {kCH}^{2}$
(25)
where k depends on the capillary pipe structure, and it is a constant when the capillary pipe structure is fixed. According to Equation (14)
$k = \frac{3}{A} \frac{1 + mn}{(1 + n)^{3}} [1 + (1 + \frac{2}{N}) {nm}^{2} + \frac{n^{2}}{m} + (1 - \frac{2}{N}) \frac{n}{m^{3}}]$
(26)

Figure 13.
Flow time of (a) 300D/64F filament yarn and (b) 600D/58F filament yarn versus wicking length in 0.5 wt% red colored soybean oil.

It can be found from Equation (25) that for certain wicking height/certain wicking time, variation of the C value due to different liquid properties will result in changing of the corresponding wicking time/wicking height, which means the wicking velocity of a certain yarn will be greatly influenced by liquid properties. Thus, the wicking velocities of yarn samples in 0.5 wt% methylene blue solution and in 0.5 wt% red colored soybean oil are different.

It should be noted that the calculated wicking times are inaccurate at low wicking height, especially for the case that when yarn samples have a low twist factor and the liquid has a high velocity (see Figure 13). However, in practice we usually consider the wicking of water or sweat in yarns for garment fabric. The theoretical results in Figure 13 suggest that the N-section lotus-rhizome-node-like capillary model is more appropriate for low-viscosity liquids, such as water and sweat.

Conclusions

A 1D N-section lotus-rhizome-node-like capillary model was established based on the plane Poiseuille flow and capillary pressure equation to investigate the movement of liquid in non-uniform capillaries. Normalized total flow time of the N-section non-uniform capillary was obtained as a function of the number of converging and diverging sections, the height ratio and the width ratio between the converging and diverging sections. The results indicated that the velocity of liquid transport greatly depends on the structural morphology of the non-uniform capillary. The capillaries with frequent alternating between converging and diverging sections (large number of converging and diverging sections) in a certain wicking length have low liquid transport efficiency. Meanwhile, the non-uniform capillaries with a large equivalent width are favorable for quick liquid transport. The experimental results from 300D/64F and 600D/58F polypropylene filament yarns with different twist factors in different liquids agreed well with the theoretical predictions from the N-section lotus-rhizome-node-like capillary model. The present model can be further extended to a real 3D N-section lotus-rhizome-node-like capillary model based on the Poiseuille law for pipes with circular cross-section by considering additional parameters, such as the total cross-sectional area as a function of the number of filaments, the precise geometry change of the transport channel and the liquid diffusion in the radial direction, which will facilitate a more precise description of the real morphology of the internal yarn spaces and the wicking process of the non-uniform capillary. This work provides an initial theoretical exploration of the liquid transport in porous fibrous material with complex geometry and will potentially cast light in the area of modeling and microfluid device design.

Sample	Low twist factor (73.4)	High twist factor (204)
300D/64F	12.7	35.4
600D/58F	9	25

Sample	Low twist factor (73.4 ± 1.7)	High twist factor (204 ± 2)
300D/64F	12.7 ± 0.3	35.4 ± 0.4
600D/58F	9 ± 0.2	25 ± 0.3

Sample	Twist factor	a₁ (µm)	a₂ (µm)	m	A (µm)	Length of yarn in one twist (mm)	h₁ (mm)	h₂ (mm)	n	N
300D/64F	73.4	27.2	8.16	0.3	12.9	7.87	1.97	5.90	3	20.33
204	27.2	8.16	0.3	14.5	2.87	0.96	1.91	2	55.75
600D/58F	73.4	40.3	12.1	0.3	19.2	11.11	2.78	8.33	3	14.40
204	40.3	12.1	0.3	21.5	4.00	1.33	2.67	2	40.00

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Key R&D Program of China (Grant Number 2017YFB0309100) and The Science and Technology Plans of Tianjin (Grant Number 15PTSYJC00230).

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Sample	Low twist factor (73.4)	High twist factor (204)
Sample	Twist of yarn (twist/10 cm)
300D/64F	12.7	35.4
600D/58F	9	25

Liquid transport in non-uniform capillary fibrous media

Abstract

Keywords

Theoretical model

Flow time in the uniform capillary

Flow time in the non-uniform capillary

Experimental section

Materials

Equipment and test procedure

Results and discussion

Normalized total flow time of the N-section lotus-rhizome-node-like capillary

Total flow time of the N-section lotus-rhizome-node-like capillary

Yarn appearance and experimental wicking results

Evaluation of the theoretical result

Determination of n

Determination of a 1 , a 2 , m

Determination of h 1 , h 2 , N

Determination of A

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References

Determination of $a_{1}, a_{2}, m$

Determination of $h_{1}, h_{2}, N$