Xianliang Sheng*a,
Hongming Baic and
Jihua Zhang*b
aCollege of Science, Inner Mongolia Agricultural University, Hohhot, 010018, P. R. China. E-mail: shengxl@iccas.ac.cn; Fax: +86 04714300726; Tel: +86 04714300726
bAerospace Research Institute of Material and Processing Technology, Beijing, 100076, P. R. China. E-mail: zjhicca@iccas.ac.cn; Fax: +86 01068382974; Tel: +86 01068383313
cMeinders School of Business, Oklahoma City University, 73106-1493, USA. E-mail: brianceo@gmail.com
First published on 17th March 2015
To understand sweat transportation along a single hair fiber, its capillary rise was observed. But the effects of hair cuticular cells on the process were difficult to identify due to their submicron scales. In order to deal with the problem, an analogical model of a conical frustum with ratchet-like sides was developed. Side microstructures of the model block the rise of water; moreover, the contact line and contact angle (θ) are more sensitive to such microstructures than wetting height, and there holds the relation between θ and time (t): θ ∼ t−n (n > 0.5). Similarly, the dynamic θ is verified to fit the wetting of the hair fiber. In contrast with a smooth artificial fiber, the introduction of cuticular cells positively speeds up the transportation of liquid (e.g. sweat) along the hair fiber. In addition, hair cleaning is emphasized because oily liquid easily covers the hairs, preventing the transportation of sweat. We believe that our findings are of important significance for cosmetic applications.
Lately, we noticed an interesting wetting phenomenon from a sports TV show. In the program, the player shook off sweat along his hairs to keep cool. Sweat beads even broke away from the hair fibers and were thrown into the air (see Fig. 1a). We wonder how sweat can be excreted along the hair fibers. Although previous literature has revealed that the liquid transportation along the hair fiber arrays mainly comes from their elastocapillary response,28,29 the key transferring step from sweat gland to hair is still unclear. High resolution SEM images show that scale-like cells (their ridge thicknesses are ∼500 nm, see Fig. 1b) grow on the surface of a single hair fiber. To our best knowledge, rough microstructures are always important for the wettability of natural fibers.30 Herein, the initial sweat transportation may be associated with hair roughness induced capillary rise. However, due to the submicron scales, the role of cuticular cells for the dynamic wetting of hair fibers is difficult to identify.31–33 Therefore, it becomes a challenge to reveal the effects of cuticular cells on the capillary rise of hair fibers.
Further observations show that cuticular cells give the hair surface ratchet-like sides (from the viewpoint of a planar projection, see the inset of Fig. 1b). Such microstructures catch our eyes because they always give rise to some possible applications, such as self-propelled Leidenfrost droplets on them.23,24 What’s more, this periodic up-and-down structure on the solid surface is easy to be modelled and then analyzed.24 Therefore, it generates an analogy to dig into the underlying principles of sweat transportation along the rough hair fiber by a model method.
In this article, the capillary rise of a conical frustum with a ratchet-like microstructure was modelled and theoretically analyzed. In order to verify the analysis validity, some model samples were machined. Subsequently, their dynamic wetting in water or hexane was experimentally recorded through a high-speed CCD camera. Finally, deep discussions on the capillary transportation of a liquid, e.g. sweat or an oily liquid, along the single hair fiber were conducted by making an analogy with the models. We believe that these studies may provide some insight into the wetting processes of the human body or other organisms, thereby giving inspiration to the cosmetic industry.
The capillary wetting process on a conical frustum’s sides is schematically shown in Fig. 2. Ratchet-like microstructures cover the sides of the conical frustum (see Fig. 2a). Once the conical frustum descends to contact the liquid (see Fig. 2b), it will wet these micron triangle units at an instinct CA of θ0 (i.e. the CA on the smooth and flat surface, a boundary condition) and then form a macroscopic meniscus as shown in Fig. 2c, or be pinned to the top edge of the model. The critical condition for capillary rise to occur is given by Gibbs’ equation.34 But owing to the ratchet-like microstructures, it is modified:35
θc = θ0 + ω + δ | (1) |
After the liquid wets the rough side of the conical frustum, the meniscus becomes a stable one (see Fig. 2d). The static meniscus height he can be calculated by balancing the hydrostatic pressure with the Laplace pressure (see Fig. S1†):
![]() | (2) |
cos![]() ![]() ![]() | (3) |
cos![]() ![]() ![]() | (4) |
A dynamic meniscus is more important in contrast with a static one, because it develops the macroscopic θ (see Fig. 2d) and represents the liquid transfer rate along the conical frustum. Such a process is analyzed by viscous dissipation theory. On one hand, the viscosity (η) of the liquid generates an energy dissipation to block the capillary flow, which is proportional to ηv2/θ (v is the edge speed of the meniscus and θ is in radians).10,11 On the other hand, the dynamics of the meniscus are driven by the capillary force, fc, which creates a flow work of fcv to move the contact line.12 Owing to surface roughness, fc is modified as fc = γ(cosθe − cos
θ) = γ(rf
cos
θ0 + f − 1 − cos
θ) (see eqn (3)) where γ is the liquid–vapour surface tension. In the case of small θ (θ is close to zero in radians), it holds as: cos
θ ≈ 1 − (θ2/2). Then the capillary force reduces to fc = γ(r
cos
θ0 − cos
θ) ≈ γ[(r − 1) + (θ2 − rθ02)/2]. For a low-viscosity liquid, a quick balance of the flow work with the viscous dissipation leads to the equation:39 ηv2/θ ∝ γ[(r − 1) + (θ2 − rθ02)/2]v. So the relation between v and θ is predicted by40
v ∝ θ(r − 1) + θ(θ2 − rθ02)/2 | (5) |
Another expression of v (v ≡ d(h(t))/dt) is also given where dynamic h(t) can be eliminated by eqn (2):
![]() | (6) |
![]() | (7) |
Subsequently, PTFE and Al conical frusta were ultrasonically cleaned with benzene, acetone and alcohol. PMMA cylinders were ultrasonically washed by petroleum ether and methanol, respectively. PA6 cylinders were ultrasonically washed by cyclohexane and methanol. Finally, all samples were ultrasonically washed by Milli-Q water for 15 min. The intrinsic CA of θ0 was measured on the top of each asperity by a contact angle system (DataPhysics OCA 20, Germany).
Two low-viscosity liquids, water and hexane, were used. Table 1 lists the basic physical properties of the two liquids. Their boiling points in the experiments are about 40–80 °C higher than room temperature. The density of the liquids comes from the literature,13 and the surface tension was measured by the Wilhelmy plate technique at a room temperature of ∼25 °C.
The capillary force of the hair fiber was checked by dynamic contact angle measurement (DCAT11, DataPhysics Instruments GmbH, Germany). A single hair fiber was simply washed with a surfactant solution (20% sodium lauryl sulfonate) and then rinsed with distilled water prior to use. Then it was vertically hung on a microelectronic balance. A vessel filled with water or hexane was placed on the balance table. The vessel was moved upward until the hair fiber contacted the water or hexane. Instantaneous data of the capillary force were recorded automatically by a computer.
Material | θ0,w/θ0,ha/° | d0/mm | ωb/° | βb/° | δb/° | θ*e,w/θ*e,hc/° | θc,w/θc,hd/° | Capillary rise of water | Capillary rise of hexane | |
---|---|---|---|---|---|---|---|---|---|---|
a Intrinsic CA values, θ0,x of liquid x (w and h represents water and hexane, respectively) on the smooth top of the conical frustum.b The angle accuracy of ω, β and δ is ±1°.c Apparent CA, θ*e,x of liquid x meniscus calculated by the average value of θ0,x according to eqn (4).d Critical CA values, θc,x of liquid x meniscus calculated by eqn (1).e The sign “—” indicates capillary rise was not observed.f The θc value of beyond 180° means that the contact line is limited around the edge and capillary rise cannot occur. | ||||||||||
Al | 67.8 ± 1.5°/<5° | 3 | 30 | 7 | 12 | 1.013 | 67/0 | 109.8/<47 | Occurred | Occurred |
45 | 16 | 13 | 1.033 | 67/0 | 125.8/<63 | Occurred | Occurred | |||
60 | 7 | 40 | 1.045 | 67/0 | 167.8/<105 | Occurred | Occurred | |||
90 | 53 | 12 | 1.110 | 65/0 | 169.8/<107 | Occurred | Occurred | |||
4 | 30 | 4 | 7 | 1.004 | 68/0 | 104.8/<42 | Occurred | Occurred | ||
45 | 16 | 62 | 1.184 | 63/0 | 174.8/<112 | Occurred | Occurred | |||
60 | 40 | 24 | 1.168 | 64/0 | 151.8/<89 | Occurred | Occurred | |||
90 | 46 | 11 | 1.085 | 66/0 | 168.8/<106 | Occurred | Occurred | |||
5 | 30 | 7 | 5 | 1.005 | 68/0 | 102.8/<40 | Occurred | Occurred | ||
45 | 15 | 33 | 1.081 | 66/0 | 145.8/<83 | Occurred | Occurred | |||
60 | 17 | 22 | 1.060 | 66/<0 | 149.8/<87 | Occurred | Occurred | |||
90 | 48 | 10 | 1.081 | 66/0 | 167.8/<105 | Occurred | Occurred | |||
20 | 30 | 48 | 20 | 1.170 | 64/0 | 117.8/<55 | Occurred | Occurred | ||
45 | 47 | 41 | 1.388 | 58/0 | 153.8/<91 | Occurred | Occurred | |||
60 | 28 | 23 | 1.107 | 65/0 | 150.8/<88 | Occurred | Occurred | |||
90 | 17 | 35 | 1.099 | 65/0 | >180f/<130 | —e | Occurred | |||
30 | 90 | 17 | 31 | 1.086 | 66/0 | >180f/<126 | — | Occurred | ||
PMMA | 72.1 ± 0.9/<5° | 3 | 90 | 6 | 17 | 1.016 | 72/0 | 179.1/<112 | Occurred | Occurred |
5 | 90 | 11 | 14 | 1.024 | 72/0 | 176.1/<109 | Occurred | Occurred | ||
10 | 90 | 10 | 6 | 1.009 | 72/0 | 168.1/<101 | Occurred | Occurred | ||
20 | 90 | 17 | 17 | 1.046 | 71/0 | 179.1/<112 | Occurred | Occurred | ||
PA6 | 68.2 ± 2.1/<5° | 3 | 90 | 36 | 30 | 1.191 | 64/0 | >180f/<125 | — | Occurred |
4 | 90 | 35 | 26 | 1.157 | 65/0 | >180f/<121 | — | Occurred | ||
10 | 90 | 23 | 17 | 1.063 | 67/0 | 175.2/<112 | Occurred | Occurred | ||
20 | 90 | 19 | 24 | 1.074 | 67/0 | >180f/<119 | — | Occurred | ||
PTFE | 113.7 ± 2.3/33.2 ± 1.8 | 3 | 90 | 19 | 55 | 1.191 | 119/1 | >180f/178.2 | — | Occurred |
4 | 30 | 23 | 17 | 1.063 | 115/27 | 160.7/80.2 | Occurred | Occurred | ||
20 | 90 | 41 | 26 | 1.189 | 116/3 | >180f/149.2 | — | Occurred |
Fig. 5a shows a series of CCD images of the capillary rise along the Al conical frustum with d0 = 3 mm and ω = 60°. The meniscus rapidly climbs along the rough sides after it contacts the hexane. It takes about 40 ms to reach the quasi-balanced state of the meniscus. To more accurately describe this process, the relations between the meniscus height h, and radius rx versus time t are recorded in detail (see Fig. 5b). Apparently, various meniscus sizes (h and rx) increase with prolonged contact time until the meniscus becomes static. Moreover, at early the stage of the rise, the meniscus even climbs in a self-similar manner, i.e. a constant ratio of its radius to its height13 (see Fig. S4†). In addition, the experimental animation in Video S1† interestingly exhibits an obvious oscillation around the meniscus as the liquid wets the rough sides. As described in the theory part, the wetting of hexane along the rough sides gives rise to an inconsistent movement between the front of the contact line and the whole meniscus, and thus induces the oscillations.
Subsequently, we vary the geometric structures of the conical frusta’s rough sides to discuss their effect on the meniscus. Two characteristic parameters are tested: the wetting height h of the meniscus and its CA (θ). Fig. 6a shows the dynamic h of the meniscus for Al cylinders with diameters of 3, 4, 5, 20 and 30 mm (the side microstructures are displayed in Table 2). It is worthwhile to note that the initial climbing speed (dh/dt) of hexane grows with the increase of cylinder diameter. But it hardly affects the balance time (te) to reach a stable meniscus. For example, at t = 5 ms, the meniscus height of h reaches 0.8 mm for the cylinder with d0 = 5 mm; however, when the size of the cylinder (d0 = 30 mm) becomes larger, the meniscus rises to a height of 1.2 mm. Anyhow, they develop a stable one at almost the same time (te = ∼43 ms). The relation between he of the meniscus and ω of the conical frustum is also plotted (see Fig. 6b). The experimental he decreases with the growth of ω as predicted by eqn (2). But there are some slight deviations between them. What’s more, their relative errors seem to be associated with top diameters of the conical frusta. For example, the relative error of the Al cylinder with d0 = 30 mm is 5.6%, less than that with d0 = 20 mm (8.9%). In contrast, the deviations of the Al cylinders with small sizes (d0 = 3, 4 and 5 mm) are larger (below 50%). Thus eqn (2) fits better to describe the conical frusta with large sizes.
![]() | ||
Fig. 6 (a) Effects of Al cylinder sizes on the dynamic wetting height h of the meniscus in hexane; (b) the dependence of steady height he on the angle of ω for Al cylinders with various diameters, where the solid line is calculated by eqn (2). |
In fact, the variation of meniscus height cannot reflect too many effects of the rough sides on the capillary rise of the conical frustum due to its complete wetting. But Video S1† suggests that the contact line may be more sensitive to such microstructures, so the dynamic CA (θ) was checked. Fig. 7a illustrates the time dependence of the CAs for the cylinders with different surface tensions. All cylinders exhibit the similar trend that the CAs decrease from an initial 180° − ω to the static θe with longer contact time. But all the θ–t curves are not smooth, together with obvious fluctuations, reflecting the surface microstructures, i.e. the triangle units of the ratchet-like sides. When the CA values approach θe, their fluctuations are weakened, and finally become steady. On the other hand, size effects of the cylinders on the dynamic CAs are barely seen during the rise (see Fig. 7b). Thus the differences in the θ–t plots for various cylinders mainly come from the surface topography of the solid sides. The relations of θ versus t are fitted as the power functions of θ ∝ t−0.513 and θ ∝ t−0.536 (see Fig. S5†). Lots of literature has reported that there is a relation of θ ∝ t−0.5 on a smooth surface.13,41 So it suggests that ratchet-like structures give a faster meniscus rise than a smooth surface.
![]() | ||
Fig. 7 Plots of dynamic θ versus time t: (a) solid cylinders with various surface tensions; (b) Al cylinders with d0 = 3, 4, 5, 20 and 30 mm. |
Model studies tell us that the ratchet-like microstructures of conical frustra have significant impacts on its capillary wetting. Consider the similar microstructures, the analogy is made for hair fibers that cuticular cells may play the same role during sweat transportation. So the capillary rise along a a clean hair fiber was examined (see Fig. 8, its average diameter is ∼72 μm). Sweat is replaced by water due to its high water content (beyond 98%).
![]() | ||
Fig. 8 Time-sequence CCD images of capillary wetting along the hair fiber when it contacts (a) water and (b) hexane. The total recorded time is 15 ms. |
In the experiments, the hair fiber constantly keeps its axis vertical to the water surface. Moreover, it is arranged along the direction against the scales of the cuticular cells (AS direction, see the insert of Fig. 1b) to contact the water, which is consistent with the growth direction of cuticular structures on the head. The total recorded time is 15 ms. At t = 0.60 ms, a stable meniscus forms (see Fig. 8a), suggesting that a clean hair fiber has the capacity of rapidly transporting sweat. The CA of the meniscus and its height are measured as ∼84° and ∼28 μm, respectively. In fact, there are always the problems of experimental determination of the water contact angles or meniscus heights because the hairs are rough (not an equilibrium state). Pinning prevents the meniscus from reaching its equilibrium profile. So contact angles or meniscus heights are complicated by pinning of the contact line. The measured contact angles or heights on the hair are higher or lower than the equilibrium ones in the experiments (the equilibrium contact angle can be theoretically obtained as in the ESI according to ref. 42, see Fig. S6†). It is therefore addressed that our contact angle or meniscus height measured by the static meniscus is only a steady one, not necessarily the equilibrium one.43
Unfortunately, in this case, there are insufficient data to describe the dynamic wetting process of the hair fiber due to the limit of CCD recording rates (see Fig. S7†). Instead, we observe the complete wetting case (in hexane) because the ultra-low θe (<10°) of hexane on the hair prolongs the rise time. Fig. 8b shows the dynamic meniscus of hexane along the hair fiber. As expected, it takes more time (∼3.5 ms) to reach the stable state. Moreover, the steady height of he reaches ∼170 μm. Careful examination reveals that, in either water or hexane, the rise time of the meniscus along the hair fiber is much shorter than those of the model conical frusta (see Fig. 5), showing the robust micro-scale effects. Plots of h and θ of hexane versus t are drawn in Fig. 9. The height h quickly increases with the increase of t. It is well fitted by the power function h ∼ t0.47, which is very close to the exponent of 0.5 proposed by Liu’s work.22 On the contrary, θ along the hair fiber decreases rapidly until it becomes steady. Due to the scale of the hair fiber, gravity can be neglected during its meniscus rise. Thus assumption (2) in the above theory part becomes invalid, and then eqn (2) is replaced by James’ equation:7
![]() | (8) |
![]() | (9) |
Fig. 10 shows the plots of capillary force (F) versus distance when the hair fiber contacts the water and hexane, where the capillary force F acting on the hair fiber is deduced as a function of dynamic θ: F = 2πr0γcos
θ (r0 is the radius of the hair fiber). The steady capillary forces of water and hexane are measured as 1.42 ± 0.10 μN and 4.35 ± 0.22 μN, respectively. Clearly, the capillary force of hexane is larger than that of water. This situation suggests that if a hair fiber is wetted by oil (e.g. hexane), water would not be able to detach it and then wet. As we known, oil and grease can be excreted from the scalp glands to hair fibers. Thus, these oily attachments on the scalp may block the wetting or transport of sweat. Naturally, when hairs are not cleaned frequently, more sweat would stay on our heads (not transported), which does not refresh us. To prove it, simple tests for hair fibers not washed for seven days were conducted. As expected, water rise along the unclean hair fiber is prohibited (see Fig. S8†). So timely hair-care (e.g. cleaning) is important as it can really promote the excreting of sweat to maintain a good metabolism. If you want to win a sport game, why not clean your hair before the match?
![]() | ||
Fig. 10 Plots of dynamic capillary force (F) versus distance for a hair fiber after it contacts water and hexane, respectively. |
Footnote |
† Electronic supplementary information (ESI) available: Graphs show additional information concerning the experimental setups, detailed information about the static meniscus shapes and optical images of clean and unclean hair fibers contacting water, and the videos show the oscillation of the whole meniscus along the solid surface with ratchet-like microstructures and dynamic wetting of hexane and water along the hair. See DOI: 10.1039/c4ra15078c |
This journal is © The Royal Society of Chemistry 2015 |