Towards an improved understanding of liquid transportation along a hair fiber: ratchet-like microstructure induced capillary rise

Abstract

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.

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1. Introduction

Partly or completely wetting a solid surface through the capillary rise of liquid has been widely applied in the coating of material surfaces.¹ It has become an important subject of many research work over the past decade.² Up to now, researchers have sufficient capacity to accurately calculate the static wetting profile of liquid during capillary rise.^3–9 Moreover, its dynamic wetting can be characterized by the contact angle (CA) or the evolution to reach a static one using analysis of the energy dissipation,^10,11 mechanical analysis,^12,13 the lattice-Boltzmann method⁴ and molecular kinetic theory.^14,15 Both theory and experiment have verified that the variation of CA (θ) with the contact time (t) always has the relationship of θ ∼ t^−0.5 for smooth plates, cylinders or fibers.^10–12 However, wetting studies on the capillary rise along rough surfaces are few, although newly developed knowledge of roughness-induced superhydrophobicity or superwetting has been skilfully adopted in scientific research.^16–27 The capillary phenomena of a rough Cu₆Sn₅/Cu intermetallic substrate with various liquids (different surface tensions and viscosities) were recently observed and modelled by mechanical analysis.²² The wetting behaviors of liquids along rough surfaces depends on the surface topography, the wetting properties of the solid and the physical characteristics of the liquid. This gives us some hints that the capillary processes on solid surfaces may be adjusted well by varying the surface microstructures, which is of value for the industrial field.

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.

Fig. 1 (a) Sweat bead transportation and separation from hair fibers when a human being shakes their head. (b) FE-SEM image of an Asian human hair; the whole hair fiber with cuticular cells and its magnified structure. The bottom left inset shows a two-dimensional ratchet-like schematic illustration where the AS and WS directions are against and with the scale direction of the cuticular cells, respectively.

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.

2. Theoretical basis

The objective of the model is to describe the process of capillary rise on rough surfaces. The assumptions are given as follows: (1) the impact of the evaporation of the wetting liquid is negligible, and liquid’s properties are constant during the meniscus rise; (2) the liquid source is large enough; (3) the top of the conical frustum keeps parallel to contact the liquid surface; (4) gravitational force cannot be neglected due to the large size of the conical frustum; (5) pressure equilibrates quickly as the rise occurs almost instantaneously; and (6) the top edge of the conical frustum is smooth and undamaged.

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

θ_c = θ₀ + ω + δ (1)

where the microstructures are composed by periodic arrays of triangle units with three side lengths (x₁, x₂ and y) and two included angles (β and δ), ω is the rise angle of the conical frustum, θ₀ is the instinct contact angle and θ_c is the critical CA value when the liquid just moves over the edge. Apparently, capillary rise will occur for the condition of θ_c < 180°. Otherwise, when the calculated θ_c is beyond 180°, the liquid surface must deform to a convex one to drive the rise, that is, the contact line is pinned at the edge and the rise cannot occur in a natural state.

Fig. 2 Schematic illustration of the wetting process of a liquid on a rough conical frustum through capillary rise. (a) The rough conical frustum is hung over the liquid. (b) Once the top of the conical frustum contacts the liquid, the liquid will be pinned or climb along its sides which depends on the Gibbs’ equation. (c) The liquid wets the rough side of the conical frustum with prolonged contact time. (d) Finally, the meniscus reaches a stable one with a height of h_e and a contact angle of θ_e. The dotted line indicates the dynamic rise process of the meniscus. The microstructures of the conical frustum are described by the internal angles (δ and β) and side lengths (x₁, x₂ and y).

After the liquid wets the rough side of the conical frustum, the meniscus becomes a stable one (see Fig. 2d). The static meniscus height h_e can be calculated by balancing the hydrostatic pressure with the Laplace pressure (see Fig. S1†):

where a is the capillary length, i.e. a = (γ/ρg)^1/2.^36,37 The static CA, θ_e for the composite interface, is typically computed using the Cassie–Baxter relation, which combines the contribution of the fractional area of the wet surface and the fractional area with air pockets:²⁵

cos θ_e = rf cos θ₀ + f − 1 (3)

where f is the fractional flat geometrical area of the solid–liquid interface, and r is a roughness factor defined as a ratio of the solid–liquid area to its projection on a flat plane. On the condition that liquid invades the ratchet-like microstructures of the conical frustum (especially θ₀ ≤ π − (δ + β), see Fig. S2†), eqn (3) directly reduces to the Wenzel equation,³⁸

cos θ_e = r cos θ₀ (4)

where r is approximately calculated:

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, f_c, which creates a flow work of f_cv to move the contact line.¹² Owing to surface roughness, f_c is modified as f_c = γ(cos θ_e − 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 f_c = 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):

In the case of complete wetting (θ_e ≈ 0) and surface roughness r ≫ 1, combining eqn (5) with eqn (6) produces a new expression:which yields the linear evolution θ = (π − ω)(1 − rt) at a short time, and a slow relaxation for the contact angle θ ∼ e^−rt at a long time. Alternatively, eqn (7) resumes a cubic law on the condition of r ≈ 1. For the matching ones (r > 1), the v–θ relation follows a power law, v ∼ θ^m (m is a certain value between 1 and 3), which causes a simple power law behavior of θ ∼ t^1/(1−m). As a result, the roughness of the conical frustum theoretically modifies its capillary process during the rise.

3. Materials and methods

3.1 Sample preparation

Single asperities of different shapes and sizes were machined from polymethyl methacrylate (PMMA), polyamide 6 (PA6), polytetrafluoroethylene (PTFE) and aluminium alloy (Al) cylinders (Yuhangtuo Factory, Beijing, China). All the cylinders were machined into small-size conical frusta (the accuracy of the finish is ±0.01 mm) with the rise angles ω = 30°, 45°, 60° and 90° (a right angle, cylinder). Their top diameters were between 3 mm and 30 mm. The ratchet-like microstructures on the sides of the conical frusta were achieved with cutter shape, cutter orientation, and cutter speed. All tops of the asperities were carefully ground with various sand papers and then polished with a 100-grit diamond rag wheel. The roughness of the edge on the bottom surface is below 1 μm (FE-SEM observations, JOEL JSM 6700F, Japan).

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.

Table 1 Physical properties of the two liquids

Liquid	Density, ρ/g cm^−3	Surface tension, γ/mN m^−1	Viscosity, η/× 10^−3 Pa s	Boiling point/°C	Capillary length^a, a/mm
a The capillary length is calculated by a = (γ/ρg)^1/2.
Water	0.998	72.75	1.002	99.9	2.727
Hexane	0.660	18.43	0.308	68.7	1.688

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3.2 Instruments and observation

A schematic of the experimental setup is shown in Fig. S3.† A conical frustum (or hair fiber) was vertically hung above the liquid surface. A fluid-drive device was used to lift these solids up or down at the rate of 0.05 mm s⁻¹. The whole setup was placed in a heavy air cushion table. The meniscus rise was measured by a backlight scattering method and restored in image format with a high speed CCD camera (Photron, Fastcam SA-1). The recording rates varied from 1800 frames per second (fps) to 10 000 fps. The software Image-Pro plus V6.0 was used to deal with all image information. The image resolution was 768 × 768 pixels. A vessel with a diameter of 60 cm (20 times larger than that of the conical frusta to avoid the effect of the glass wall) and height of 10 cm was slightly overfilled to reduce reflection and refraction problems. To avoid the impact of evaporation, they were placed into an experiment chamber where a nearly saturated liquid vapor atmosphere was provided. The same samples were used in water or hexane. Before measurement, the CA on the sides of each sample was tested to ensure no contamination.

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.

4. Results and discussion

Fig. 3 displays the representative (optical) image of side microstructures on the PTFE cylinder. Clearly, the ratchet arrays cover its whole side. The angles of β and δ (see Fig. 2d) in the ratchet units are measured as ∼19° and ∼55°, respectively. Fig. 4a shows its dynamic wetting while being descended to contact the water surface. Water cannot wet over the edge of the PTFE cylinder (θ_0,PTFE is ∼113°). Even if we continue to move the cylinder downward for some distance, the water surface is not pierced and a large dimple is formed (see Fig. 4b). It is well explained by eqn (1) that the critical θ_c (=∼113° + (90° + 55°) = ∼258°) is beyond 180°, implying the contact line is pinned at the edge and does not bypass the sides of the PTFE cylinder. Table 2 lists structure parameters of conical frusta and observations for their capillary rises. Interestingly, not all hydrophilic sides of conical frusta are wetted after they contact water. This situation is far different from the capillary phenomena on smooth surfaces. For example, intrinsic PA6 has a hydrophilic surface (θ_0(PA6) < 90°), but its rough cylinder (β = 36° and δ = 30°) fails to be wetted by water. In contrast, hexane wets all conical frusta because the relations constantly hold: θ_c < 180°. Observations are in good agreement with those calculated by eqn (1). Thus the introduction of the ratchet-like microstructures and the enhancement of surface hydrophobicity (instinct θ₀) can prevent the wetting process of water along the sides. In addition, static CA along the sample sides is always consistent with the one predicated by eqn (4) in our cases. Take the Al conical frustum with d₀ = 20 mm and ω = 45° as an example. Its roughness is estimated as 1.388 and thus the static CA is calculated as ∼58°, which is very close to our observation (60.2 ± 0.5°).

Fig. 3 Optical graph of the ratchet-like microstructures on the sides of a PTFE cylinder. The inset at the top right corner schematically illustrates the two dimensional microstructures of the rough cylinder in detail.

Fig. 4 Optical images of the PTFE cylinder (d₀ = 3 mm, ω = 90°, β = 19° and δ = 55°) contacting the water surface at various times: (a) just contacting the water surface (t = 0 s); (b) at a contact time of t = 1.647 s.

Table 2 Geometric parameters of the conical frusta’s sides and experimental observations of their capillary rise

Material	θ0,w/θ0,h^a/°	d0/mm	ω^b/°	β^b/°	δ^b/°		θ^*e,w/θ^*e,h^c/°	θc,w/θc,h^d/°	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	>180^f/<130	—^e	Occurred
		30	90	17	31	1.086	66/0	>180^f/<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	>180^f/<125	—	Occurred
		4	90	35	26	1.157	65/0	>180^f/<121	—	Occurred
		10	90	23	17	1.063	67/0	175.2/<112	Occurred	Occurred
		20	90	19	24	1.074	67/0	>180^f/<119	—	Occurred
PTFE	113.7 ± 2.3/33.2 ± 1.8	3	90	19	55	1.191	119/1	>180^f/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	>180^f/149.2	—	Occurred

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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 r_x versus time t are recorded in detail (see Fig. 5b). Apparently, various meniscus sizes (h and r_x) 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.

Fig. 5 (a) Time-sequence CCD images of capillary rise for the Al sample with d₀ = 3 mm and ω = 60°. Initial time (t = 0 ms) is determined when it just contacts with the hexane. (b) The recorded relation between the wetting height (h) and the radius (r_x) of the meniscus and the contact time.

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 (t_e) 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 (t_e = ∼43 ms). The relation between h_e of the meniscus and ω of the conical frustum is also plotted (see Fig. 6b). The experimental h_e 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 h_e 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 θ_e 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 θ_e, 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 θ_e (<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 h_e 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 ∼ t^0.47, 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

where C ≡ 0.58 is the Euler constant. The wetting speed (v) of the meniscus is correspondingly rewritten as v = −r₀dθ/dt for the complete wetting case (θ is approximately zero), and its differential equation is revised as:The solution of eqn (9) is discussed next: on a smooth surface (r = 1), the relation between θ and t holds: . For the surfaces with a roughness of r ≫ 1, it is given by θ ∼ e^−(r−1)t. In the case of r > 1, there follows a power law, v ∼ θ^m (m is a certain value between 1 and 3), which yields a similar expression to the conical frustum model, θ ∼ t^1/(1−m). Accordingly, the power function of θ ∼ t^−0.71 for the hair fiber was achieved by Origin software (version 8.0). To illustrate well the effects of rough cuticular cells on capillary transportation, a smooth nylon fiber with an average diameter of ∼62 μm is used to compare and then its capillary rise is measured. The relation of θ ∼ t^−0.46 exists for the nylon fiber which is in agreement with the solution of eqn (9) (i.e. θ ∼ t^−0.5 for a smooth surface). The conclusion is also drawn that rough cuticular structures give a quicker wetting process along a hair fiber than that without them. In other words, for a clean hair fiber, its rough cuticular cells speed up the capillary wetting of sweat transport. Besides, the waves of the meniscus cannot be observed in Video S2† during the capillary rise. It makes sense that the submicron-scale cuticular cells of the hair fiber could not provide enough surface topography to induce inconsistent movements between the contact line and the whole meniscus.

Fig. 9 The plots of wetting height h (the square symbol) and contact angle θ (the circular symbol) versus contact time t for the hair and nylon fibers during the capillary rise of hexane. Solid lines are fitted by Origin software.

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 = 2πr₀γ cos 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.

5. Conclusion

To investigate the effects of submicron-scale cuticular cells on the sweat transportation along a hair fiber, the analogical model of a conical frustum with ratchet-like sides was proposed and then its capillary rise was examined. The side microstructures of the model prevent the rise of water, and vary the dynamic contact angle (θ): θ ∼ t⁻ⁿ (n > 0.5), which allows liquid to rapidly wet the rough sides of the conical frustum. In the same way, the cuticular cells of clean hair fibers speed up excreting sweat from the head. Moreover, oily pollutants are easier to wet on the hair than water, stressing the importance of hair cleaning. Therefore, roughness induced capillary rise is the key to initially transport sweat along the hair fiber. This work will give some inspiration in cosmetic applications.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (51103033).

Notes and references

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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

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