## Abstract

We study the effect of small-amplitude fast vibrations and small-amplitude spatial patterns on various systems involving wetting and liquid flow, such as superhydrophobic surfaces, membranes and flow pipes. First, we introduce a mathematical method of averaging the effect of small spatial and temporal patterns and substituting them with an effective force. Such an effective force can change the equilibrium state of a system as well as a phase state, leading to surface texture-induced and vibration-induced phase control. Vibration and patterns can effectively jam holes in vessels with liquid, separate multi-phase flow, change membrane properties, result in propulsion and locomotion and lead to many other multi-scale, nonlinear effects including the shark-skin effect. We discuss the application of such effects to blood flow for novel biomedical ‘haemophobic’ applications which can prevent blood clotting and thrombosis by controlling the surface pattern at a wall of a vessel (e.g. a catheter or stent).

This article is part of the themed issue ‘Bioinspired hierarchically structured surfaces for green science’.

## 1. Introduction

Biomimetics (also called bionics or biomimicry) is the application of methods found in living Nature to the study and design of engineering systems and modern technology. Biomimetic materials constitute an important area of research. This is because many materials in living Nature have outstanding properties, which cannot be achieved by traditional engineering methods. A typical example is the spider's silk fibre, which is stronger than steel. The spider synthesizes fibre on site, at ambient temperatures and pressures. The human-made materials, such as steel, typically require high temperatures and pressures to produce them. Instead, materials in living Nature typically have a composite structure and hierarchical organization [1]. Such organization provides the flexibility needed to adapt to a changing environment. Furthermore, the hierarchical organization provides a natural mechanism for the repair or healing of minor damage in the material, thus leading to the new field of self-healing materials [2].

Some biomimetic surfaces are designed to enhance the fluid flow due to boundary slip, the suppression of turbulence (the shark-skin effect) and anti-biofouling (the fish-scale effect). Biofouling is the undesirable accumulation of microorganisms, plants and algae on structures which are immersed in water. Conventional antifouling coatings for ship hulls are often toxic and environmentally hazardous. On the other hand, in living Nature there are ecological coatings (e.g. fish scale), so a biomimetic approach is sought for liquid flow applications [3–6] and, in particular, for medical technology applications.

Owing to recent advances in nano/microtechnology, it became possible to design functional biomimetic surfaces with micro/nanotopography leading to various properties suitable for such applications as non-adhesive surfaces. In order to understand the structure–property relationships in these novel materials and surfaces, it is important to study the fundamental aspects of how micro/nanotopography affects surface properties and, in particular, fluid flow properties. There is a similarity between the effect of small-scale patterns and small-amplitude fast vibrations [7]. Micro/nanotopography usually constitutes a number of periodic spatial patterns, whereas small fast vibrations constitute periodic temporal patterns in the time domain. In certain situations, small fast vibrations can be substituted by an effective force perceived at the larger scale [7,8].

The stabilization of the inverted pendulum is a best-known example of vibration-induced stabilization. The normally unstable inverted equilibrium position of the pendulum becomes a stable equilibrium under certain small-amplitude and high-frequency vibrations. The effect of the vibrations is equivalent to a stabilizing ‘spring’ force with the magnitude proportional to the distance from the equilibrium. Vibration-induced stabilization is also observed in a dual pendulum and a system of multiple pendulums [9]. A natural generalization of the latter for the case of an infinite number of pendulums is a flexible stiff rope (i.e. a flexible rope which still possesses some stiffness akin to a stiff beam). Hardening of such a flexible stiff rope has been both predicted theoretically and observed experimentally [10]. Other related phenomena include shear thickening of non-Newtonian fluids, liquid droplet bouncing above a vibrating bath of bulk liquid [7,8], vibration-induced effective phase transitions, vibrojets, vibrational stabilization of beams, the transport and separation of granular material, soft matter, bubbles and droplets, and synchronization of rotating machinery [11,12]. In all of these cases, the small fast temporal patterns can be substituted by effective forces or torques that have a stabilizing effect on the system under certain conditions or lead to an effective change of properties.

Similar to the fast small vibrations, surface micro/nanopatterns can change effective material or surface properties. For example, properly controlled micro/nanotopography can affect the wetting properties of surfaces in the case of superhydrophobic [13] surfaces [14], as well as texture-induced phase transition [15,16]. In the case of both spatial and temporal patterns, the small-scale phenomenon, such as the micro/nanotopography or small-amplitude vibrations, can be effectively substituted by an effect perceived at the larger scale, such as a stabilizing or ‘vibro-levitation’ force or an effective energy term [8].

Another interesting manifestation of small patterns in the case of the fluid flow is the so-called shark-skin effect: the reduction of drag by small riblets parallel to the liquid flow simulating shark-skin scales, which suppress turbulence. In this paper, we discuss how small fast vibrations and micro/nanotopography affect the wetting and liquid flow properties, with potential application in the biomedical area of blood flow. We will discuss the analogy between vibrations and spatial patterns and the effect of topography and vibrations on membrane permeability and liquid flow. This analysis reveals the similarity between the molecular-scale effects governed by thermodynamics, such as the osmosis, micelle formation and thermal vibrations, and mesoscale phenomena, such as the filtration through a superhydrophobic mesh, liquid marble formation and hysteretic pinning of droplets, usually governed by phenomenological kinetics.

## 2. Effect of small fast vibrations and surface micro-patterns on the stability and phase transitions

In this section, we review the mathematical method of separation of motions. After that we discuss the mathematical analogies which correlate vibrations and spatial patterns.

### (a) Separation of motion and effective forces

The method of separation of motions was suggested by Kapitza [17] to study the stability of a pendulum on a vibrating foundation and then generalized for the case of an arbitrary motion in a rapidly oscillating field [18]. Let us consider a particle of mass *m* in a potential energy field *Π*(*x*). The force acting on the mass is given by −d*Π*(*x*)/d*x*, where *Π* is the potential energy of the system and *x* is the spatial coordinate. In addition, the mass is acted upon by an external periodic force with a small-amplitude *f* and high-frequency . Thus, is ‘fast’ compared with the ‘slow’ force −d*Π*(*x*)/d*x*. The equation of motion of the particle is . The particle will travel a smooth path due to the slow force, and at the same time execute small oscillations *ξ*(*t*) about the smooth path due to the fast force. The location of the mass can be expressed as
2.1where *X*(*t*) describes the smooth path of the particle averaged over the fast oscillations.

The mean values of the fast oscillations over its period 2*π*/*Ω* is zero, whereas *X*(*t*) changes only slightly during the same period . After separating the slow and fast terms, the equation of the slow averaged motion [8] is , where *Π*_{eff} is an effective potential energy given by
2.2The effect of the small-amplitude fast vibrations *ξ* when averaged over the time period 2*π*/*Ω* is equivalent to the additional term in the potential energy, , on the right-hand side in equation (2.2). Thus, small fast vibrations can be substituted by the additional term in the potential energy, resulting in the same averaged effect that the vibrations have on the system. An interesting case is when this term affects the state of the equilibrium of a system. Let us say, in the absence of the fast vibrations, a system has a local maximum of the effective potential energy *Π*_{eff}=*Π* (figure 1*a*). Small-amplitude fast vibration can turn the unstable equilibrium into a stable one due to the additional term in the effective potential energy discussed above (figure 1*b*). In such cases, the small fast vibrations have a stabilizing effect on the state of equilibrium.

The effect of the fast small-amplitude vibrations can also be expressed in terms of the force acting on the system. Besides the slow force −d*Π*(*x*)/d*x*, an additional slow stabilizing force (or torque for rotational systems) *V* acts on the system
2.3

For an inverted pendulum of length *L* on a harmonically vibrating () foundation, where *A* is the amplitude and *Ω* is the frequency, the inverted position can become stable if the condition *A*^{2}*Ω*^{2}>2*gL* is satisfied. By writing the equation of motion and substituting the nonlinear external force into equation (2.3), the effective stabilizing torque can be obtained as [8].

The method of separation of motions can be used to derive the effective stabilizing torques on inverted multiple pendulums on a vibrating foundation, and the increase in stiffness of a vibrating flexible rope (as a limiting case of an infinite number of pendulums connected to each other end to end). The fast vibrations of the foundation also increase the critical buckling load for an axially loaded flexible elastic beam [8]. Thus, small fast vibrations (temporal patterns) affect the equilibrium and can act as an effective stabilizing force. In the following sections, we discuss the analogies between spatial and temporal patterns.

### (b) Kirchhoff's dynamical analogy

We have seen how small temporal patterns can be substituted by an effective force. Small-amplitude spatial patterns can have the same effect. The so-called Kirchhoff's dynamical analogy establishes similarity between the static bending shape of an elastic flexible beam and the dynamics of motion of a rigid body [19]. The differential equation for the static bending of an elastic beam of area moment of inertia *I*, and modulus of elasticity *E*, whose endpoints are loaded by an axial force *F*, is given by
2.4where *ψ* is the slope of the beam at any point, *s* is the spatial coordinate, +*F* describes compressive loading and −*F* describes tensile loading. Equation (2.4) is similar to the differential equation of oscillation of a simple pendulum of length *L*,
2.5where the +*g* describes a regular pendulum and −*g* describes an inverted pendulum. Despite the difference in the independent variables, and the nature of the problem (boundary value versus initial value), an analogy exists between the motion of a pendulum and the shape of a buckled elastic rod.

On the basis of Kirchhoff's analogy, the stable regular pendulum corresponds to the beam under compressive loading (figure 2*a*), whereas the unstable inverted pendulum corresponds to the buckled beam under tensile loading with the bending moment (*M*) proportional to the displacement of the end of the beam (Δ*y*) (figure 2*b*). Since both the inverted pendulum and the buckled beam under tensile loading are governed by the same differential equation, the buckled beam can be stabilized by a spatial periodicity in the geometry of the beam (figure 2*c*), just as an inverted pendulum can be stabilized by external periodic vibrations [8]. The properties of the elastic rod can be changed in a periodic manner with small amplitude *h*≪1 and frequency *Ω* about the stationary value *EI*_{0} such that .

The effect of the distributed bending moments is equivalent to a stabilizing shear force *V* as shown in figure 2*c*. Therefore, we conclude that a pattern on the surface profile of a rod affects destabilization of the rod just as small fast vibrations affect the stability of an inverted pendulum.

### (c) Barenblatt's analogy between scaling of spatial solutions and nonlinear waves

There is another important analogy between the spatial and temporal phenomena in mechanics: the analogy between the classification of the self-similar solutions of partial differential equations and phenomena and the classification of nonlinear waves, which was discovered by Barenblatt [20]. The importance of this analogy is due to the fact that nonlinear vibration and waves play a significant role in many self-organized emergent phenomena such as Turing waves (or reaction–diffusion waves) and structures. In many nonlinear systems, it is impossible to obtain a closed-form solution of the equations of motion; however, it is possible to make judgements about the asymptotic behaviour of these solutions. On the other hand, the scaling behaviour and self-similar solutions provide an important tool to analyse spatial behaviour, especially near various critical points.

According to the classification of Barenblatt [20], there are two types of problems where self-similar solutions *u*(*x*,*t*) of a partial differential equation (involving temporal and spatial derivatives) can be sought. For the problems of the first kind, a transformation of variables *ξ*=*x*/*At*^{λ}, where *A* and λ are certain constants. This transformation reduces the partial differential equation for *u*(*x*, *t*) to an ordinary differential equation for *u*(*ξ*), which is easier to solve. The solution of these problems remains valid in the limit of *x*→0 (or *ξ*→0). For the problems of the second kind, the solution is not valid anymore for *ξ*→0. Therefore, a characteristic scale *η* should be introduced, so that the solution *u*(*ξ*,*η*) does not exist for *u*(*ξ*,0). However, in the limit of *η*→0 the solution can be presented asymptotically as *u*(*ξ*,*η*)=*η*^{α}*ϕ*(*ξ*/*η*^{α/2}), and the problem is again reduced to an ordinary differential equation for *ϕ*(*ξ*/*η*^{α/2}), while the parameter *α* can be found as an eigenvalue of a certain nonlinear equation.

These two types of scaling behaviour correspond to two types of nonlinear waves. Suppose there is a travelling wave
2.6where *ξ* and *τ* are the spatial and temporal variables, respectively, λ is the wave speed and *c* is the phase constant. It is known from the theory of nonlinear vibrations and waves that there are two types of travelling waves. For the waves of the first type, the wave speed λ is independent of the internal structure of the wave and, therefore, it can be found from the local laws such as the laws of conservation of mass, energy and momentum. An example is the shock waves in gas dynamics (as well as regular sound waves).

For the nonlinear waves of the second type, the wave speed depends on the structure of the global solution and of the wave. An example would be reaction–diffusion waves; thus, the Fisher equation has a solution for any value λ>2 with a unique wave shape for each value of λ. These are emergent structures associated with various reaction–diffusion waves [21,22]. The transformation of variables , , and yields
2.7where *x* and *t* are the space coordinate and time, respectively, *A* is a constant and λ is the critical exponent. The transformation equation (2.7) establishes the analogy between the two types of self-similar solutions and the two types of waves. Note that while Kirchhoff's analogy has direct relevance to vibration- and pattern-induced stabilization, Barenblatt's analogy can be used for the study of self-organized emergent phenomena such as nonlinear waves and patterns.

## 3. Vibration and patterns for propulsion in water flow

In the preceding section, we have discussed how small fast vibrations or small-amplitude spatial structures can be substituted by an effective energy term, which can either lead to an effective force or affect mechanical or phase equilibrium. In this section, we will concentrate on the effect of small vibrations and structures on wetting and liquid flow.

### (a) Surface patterns leading to the superhydrophobicity, phase transition and propulsion

Surface roughness and chemical heterogeneity can affect the wettability of a surface. The wettability of a surface is usually characterized by the contact angle (CA) of a liquid droplet with the surface. The equilibrium CA (*θ*_{0}) for ideally smooth surfaces are given by Young's equation [23]
3.1where *γ*_{SA}, *γ*_{SW} and *γ*_{WA} are the surface free energies of the solid–air, solid–water and water–air interfaces. The effects of surface roughness and chemical heterogeneity on surface free energies, and thus the CA, are incorporated in the Cassie–Baxter [24] and Wenzel [25] models.

Consider a solid rough surface of length *L* along the *x*-axis, with unit width, and the roughness profile described by the function *F*(*x*) and the local surface free energy *γ*(*x*). Similar to the temporal averaging in equation (2.3), the effective surface free energy can be written as an integral over the spatial coordinate *x*
3.2For a chemically homogeneous rough surface, equation (3.2) yields to the Wenzel equation in which the surface free energy is augmented by the roughness factor . For a chemically heterogeneous smooth surface, equation (3.2) yields to the Cassie–Baxter equation. The Wenzel CA (*θ*_{W}) and the Cassie–Baxter CA (*θ*_{CB}) for a rough composite interface are
3.3and
3.4respectively, where *R*_{f}≥1, *r*_{f} is the roughness factor of the wet area, and 0≤*f*_{SL}≤1 is the fractional solid–liquid interfacial area. Thus, we can substitute the micro/nanotopography with an effective surface free energy term. The micro/nanotopography manifests at the larger scale as a change in the macroscopic CA.

Closely related to the superhydrophobicity is the phenomenon of surface texture-induced phase transition. Marmur [15] first suggested that micro/nano topography could be used to stabilize air films on underwater surfaces. Micro-textured superhydrophobic surfaces can stabilize vapour film to sustain the Leidenfrost effect [16]. The surface texturing can potentially be applied to control other phase transitions, such as ice or frost formation, and to the design of low-drag surfaces at which the vapour phase is stabilized in the grooves of textures without heating.

Jones *et al.* [26] have demonstrated that rough textured surfaces may be used to manipulate the phase of water since the nanoscale roughness pattern stabilizes the vapour phase of water, even when the liquid is the thermodynamically favourable phase. Furthermore, the reverse phenomenon exists when patterned hydrophilic surfaces keep a liquid-phase layer of water under conditions for boiling. They used molecular dynamics simulations to demonstrate the stability of the vapour and liquid phases of water adjacent to textured surfaces. Patankar [27] also identified the critical roughness scale below which it is possible to sustain the vapour phase of water and/or trapped gases in roughness valleys, thus keeping the immersed surface dry [8].

Small asymmetric surface textures (saw-tooth profile) can induce self-propulsion in Leidenfrost droplets, even over steep inclines [28]. The temperature of the textured surface has to be much greater than the boiling point of the liquid. A thin film of the vapour phase thus formed at the textured surface supports the Leidenfrost droplets. The vapour phase is expelled from under the droplet due to the pressure gradient in the film between the peaks and the valleys of the surface profile. The vapour leaks out asymmetrically from under the droplet due to the inherent asymmetry of the surface texture (figure 3). The resulting viscous forces entrain the droplet in the same direction, thus driving the droplet over the steps in the saw-tooth profile against gravity [29–31].

Dupeux *et al.* [29] gives the viscous force generated per tooth of the saw-tooth profile as
3.5where *η* is the viscosity of the vapour, *U* is the velocity of the vapour flow, *h*_{f} is the average thickness of the vapour film, *r*_{c} is the contact radius of the droplet and λ is the tooth length. If there are *N* teeth below the droplet, then the net propulsion force can be obtained as
3.6In the above equation, the temporal patterns are substituted by an effective stabilizing force. Similarly, in equation (3.6) the surface topography manifests as a propulsion force.

We saw how surface topography can be substituted by an effective surface free energy term, or an effective force term that spontaneously propels a Leidenfrost droplet over steep inclines. In general, the phenomenon of surface texture-based phase transition can be described as suppressing the boiling point and thus it is similar to superheating or subcooling of water. Similar to the vibration-induced phase transitions, the effect of the small spatial pattern is in changing the phase state of the material.

### (b) Water penetration through a hole in a vibrating vessel

In this section, we briefly discuss the effect of vibrations on fluid flow through a hole. First, consider a pipe carrying a fluid at a mean flow velocity *v*_{0}, and the pressure loss Δ*P* given by the nonlinear relation Δ*P*=*av*^{2} where *a* is a constant (figure 4*a*). The quadratic term represents a nonlinearity, which may be a consequence of various factors, such as the turbulence, nonlinear viscosity or asymmetric variations in the pipe profile. The nonlinearity is essential since it results in hysteresis.

If the pipe is subjected to periodic fast external vibrations (figure 4*b*) of the form , then the resultant additional component of the velocity is , and the pressure loss is . The term changes negligibly over the period of the fast vibrations 2*π*/*Ω*. Similar to equation (2.2), where the effect of vibrations is averaged over the time period, the pressure loss Δ*P* over the period of the vibrations can be written as
3.7Thus, the effect of the fast vibrations manifests as an additional pressure difference *a*(*hΩ*)^{2}/2, which can affect the fluid flow through the pipe. A representative plot of equation (3.7) is shown in figure 4*c*. For any small change in velocity ±*δ*v due to the external vibrations, the corresponding change in Δ*P* is different, as shown. The pressure difference Δ*P*_{2} for a small increase in velocity is greater than the pressure difference Δ*P*_{1} for a small decrease in velocity. This hysteresis can affect the flow in the pipe and under certain conditions even stop the flow. The fluid in the vibrating pipe ceases to flow when
3.8where Δ*P*_{in} is the hydrostatic pressure head driving the fluid flow.

Thus, in fluid flow, vibrations can produce the same effect as a valve. If we consider a vibrating fluid container with a hole at the bottom, the fluid drainage through the hole can be arrested by controlling the amplitude and frequency of the vibrations. In the following section, we discuss the effect of vibrations on the porosity of membranes.

### (c) Membranes

Semipermeable membranes allow solvent molecules (usually water) to diffuse through, but prevent diffusion of larger molecules such as solutes and ions. Osmosis is the transport of solvent molecules through a semipermeable membrane from a region of higher to a region of lower solvent chemical potential, until the chemical potentials equilibrate. Osmosis is omnipresent in Nature at the cellular level. Transport across the semipermeable membrane of living cells is by osmosis. The driving force behind osmosis is the concentration gradient of solute across the membrane—in other words, the chemical potential difference of the solvent across the membrane.

Consider two columns of liquid separated by a semipermeable membrane as shown in figure 5. If both columns carry pure solvent at pressure *P*_{1}, osmotic equilibrium exists, i.e. the chemical potentials equate. In such a case, the heights of the pistons equilibrate and there is no net flow of solvent molecules across the membrane (figure 5*a*). Let the chemical potential of pure solvent at *P*_{1} and temperature *T* be given by *μ**_{solvent}. A small amount of solute is added to the left column to make it a dilute solution (at *P*_{1}) (figure 5*b*). This dilution results in lowering of the chemical potential of the solvent in the solution below that of the pure solvent. There is a net flow of solvent molecules from the pure solvent, across the membrane into the solution. The chemical potential of the solvent *μ*_{solvent} in the dilute solution at *P*_{1} is given by the relation
3.9where *R* is the gas constant, *T* is the absolute temperature and *x*_{solvent} is the mole fraction of the solvent in the solution. The equilibrium can be restored by increasing the pressure on the solution to a value *P*_{2}. This raises the chemical potential of the solvent in the solution to that of the pure solvent. The relation for change in chemical potential with pressure can be obtained from the Gibbs free energy relation,
3.10where is the partial molar volume of the solvent. Substituting equation (3.9) into equation (3.10) and integrating,
3.11The excess external pressure (*P*_{2}−*P*_{1}) that must be applied to prevent osmotic flow and restore equilibrium is called osmotic pressure, *π*. Thermodynamically, it is the excess external pressure that must be applied to the solution to raise the vapour pressure of the solvent to that of pure solvent. For a dilute solution, , *x*_{solute}≈*n*_{solute}/*n*_{solvent} and the volume of the solution . Using these relations, equation (3.11) can be written as
3.12which is the van’t Hoff equation, where *c*_{solute} is the molar concentration of the solute in the solution. Note its similarity to the ideal gas law. When an external pressure *P* greater than the osmotic pressure *π* is applied to reverse the flux of solvent molecules then the process is called reverse osmosis (RO) (figure 5*c*).

In the previous section, we saw that vibrations could affect macroscopic fluid flows. Equation (3.8), which describes the conditions for vibration to stop the fluid flow, can also be applied to the case of a fluid flow through a vibrating membrane when hysteresis is present. Vibrations can change membrane permeability if the effective pressure head due to the vibrations can overcome the osmotic pressure (figure 6).

### (d) Pattern-induced ‘pseudo-osmosis’

In the previous section, we briefly discussed the thermodynamics of osmosis, and the effect of vibrations on the permeability of membranes. One commonly employed method of desalination of water is the RO process. When a solution is passed through a porous RO membrane, the solvents take a tortuous path, thereby separating from the solute [32]. An analogous method using patterned superhydrophobic surfaces can be used to separate liquid mixtures [33]. Porous media/meshes which are selectively wetted by either water or organic solvents are used for separating the constituents of oil–water mixtures. The porous medium used is analogous to the RO membranes. Surfaces that are superhydrophobic and oleophilic, or hydrophilic and underwater oleophobic, can be used to separate oil from water. Underwater oleophobic surfaces [34] exhibit an oil CA greater than 90° in the solid–oil–water system.

Natural and artificial porous media have been used for oil–water separation [35]. Kapok plant fibre can separate diesel oil from water [36]. Artificial membranes are made by using porous/meshed structures with specific pore sizes, which may be roughened and coated with a surface agent to tailor the wetting property. The wetting properties depend on the pore size, surface roughness and surface agent used. Stainless steel [33,37–40] and copper meshes [41,42] and filter paper [43] were commonly used to separate mixtures in which oil is layered over water. If one of the phases in an oil–water mixture is dispersed in the other as small droplets (smaller than the pore size) the meshes become ineffective. Hydrophobic porous media have been developed for the separation of oil–water emulsions with and without surfactants [44–47].

The physics behind the oil–water separation using differentially wettable porous media is different from that of the RO process. The porous media used for oil–water separation employ surface micro/nanotopography. The pores are wetted partially by oil, water and air. The effective surface free energy of a rough chemically heterogeneous surface is given by equation (3.2). Consider a mesh with pores of diameter *w*, partially wetted by a liquid *L* and partially by air. The solid–liquid interface area in any single pore is augmented by the factor *f*_{SL}(*r*_{f})_{L}. The capillary pressure *P*_{cap} across the interface of the liquid *L* can be obtained from the force balance as
3.13Note that the effect of the surface micro/nanotopography is incorporated into equation (3.13) via the roughness factor. The roughness factor is the averaged surface profile over an area. This is similar to the effect of vibrations substituted by the temporal integral in equation (2.3). Equation (3.13) determines the spontaneous permeability of a liquid through the differentially wettable porous medium. For example, in the case of a hydrophobic, oleophilic mesh (*P*_{cap})_{water} is negative, whereas (*P*_{cap})_{oil} is positive, as a result of which oil selectively permeates though the pores; water permeates only if an external pressure is applied to negate (*P*_{cap})_{water}.

Although the oil–water separation is different from classical osmosis, we note the similarity between equation (3.12) for the osmotic pressure and equation (3.13) for the capillary pressure. The osmosis is a molecular-scale effect and the expression for the osmotic pressure (equation (3.12)) is derived thermodynamically. The pattern-induced liquid separation, which we suggest should be called ‘pseudo-osmosis’, is a mesoscale effect with a characteristic length scale, i.e. the superhydro/oleophobic/philic surface pattern, of nanometres.

In the following sections, we will discuss several mesoscale effects which are similar to the molecular-scale phenomena. These effects are induced either by smaller scale vibrations or by patterns involving hysteretic properties.

### (e) Vibrating clusters of particles and emergent effective surface tension

Small-amplitude fast vibrations have an important effect on the properties of granular materials. Thus, vibrations can overcome jamming of granular material due to friction. This is because vibrational acceleration causes an inertia force which can overcome dry Coulomb friction between the grains of a granular medium. As a result, the granular medium can flow into a narrow channel (figure 7), demonstrating an effective liquid-like behaviour [11]. Note that, from the viewpoint of rheological models, dry friction represents the key mechanisms of plasticity. Therefore, vibration-induced effective ‘melting’ of the granular flow can be interpreted as an elastic–plastic transition rather than as a true melting (which is a phase transition of the first kind).

Another effect of vibration on granular media is the emergence of the apparent surface tension. Clewett *et al.* [48] studied the vertical vibration of a layer of bronze spheres with diameter between 150 and 180 μm placed between flat glass substrates. The vibrated particles formed two-dimensional clusters that demonstrated behaviour similar to three-dimensional liquid droplets, thus suggesting the presence of an effective surface tension consistent with Laplace's equation and demonstrating the existence of an actual surface tension. The spheres inside the cluster had on average more collisions with neighbouring spheres than those at the border of the clusters. Since the collisions are not pure elastic and some energy is dissipated during the collisions, the average energy at the border of the clusters is larger than that inside the clusters, and the trend to minimize energy results in the clusters attaining the circular shape.

### (f) ‘Pseudo-Marangoni’ effect: vibro- and pattern-induced propulsion

Another vibrational effect which is worth mentioning is propulsion in a viscous medium due to small-amplitude fast vibrations, which is believed to be a principle of locomotion of many aquatic microorganisms [49]. Owing to the small size of these microorganisms, viscosity prevails over inertia on them, and a regular way of swimming as practised by large organisms would result in a back-and-forth motion rather than in successful locomotion. The so-called ‘scallop theorem’ states that to achieve propulsion at low Reynolds number in Newtonian fluids a swimmer must deform in a way that is not invariant under time reversal (motion is non-reciprocal).

Purcell [49] studied a microscopic swimmer with two hinged paddles in a Newtonian fluid. It swam in a loop by rotating its paddles. Using an approach similar to equation (2.3), the average propulsion force during the entire cycle is zero.

However, in non-Newtonian fluids reciprocal motions can propel microscopic swimmers [50]. The viscosity depends on the shear rate, and thus the angular velocities. Therefore, the net propulsion force over a cycle is non-zero, and the hysteresis in the viscosity of a non-Newtonian fluid, coupled with non-reciprocal motion, manifests as a net propulsion force.

The effect of vibrational locomotion is not limited to microorganisms and is widespread among aquatic animals, including whales [12]. Another kind of vibration-induced propulsion is the so-called ‘recoil locomotion’ in which oscillations of internal mass are used to supplement locomotion [51]. In general, a system should involve asymmetry to realize this effect. In the next section, we discuss the similarity between reaction kinetics and the multiple CAs observed on a surface.

### (g) Mesoscale versus molecular-scale effects: marbles versus micelles and contact angle hysteresis versus Arrhenius kinetics

In every phenomenon discussed previously, there has been some form of hysteresis or asymmetry involved. According to the classification by Blekhman [12], there are six main types of such asymmetry—force, kinematic, structural, gradient, wave and initial conditions—that can lead to an effective propulsion force. The effects of small fast vibrations and patterns are summarized in table 1 and they can be viewed as multi-scale phenomena where separation of motion or surface patterns at different scales can be performed.

Another important multi-scale aspect of the wetting and liquid flow phenomena is in the similarity of some macroscale effects to those at the molecular scale. Figure 8 summarizes some of these similarities. Thus, the so-called ‘liquid marbles’ are similar to spherical micelles. The energy barriers governing the range of the CAs on a surface are similar to the metastable states in a chemical reaction. Vibrations cause granular media to exhibit liquid-like behaviour including flow and apparent ‘surface tension’, while the surface topography results in the modification of the interfacial surface energy. Separation of immiscible and dispersed phases using selectively wettable meshes is similar to purification of solvents using osmotic membranes.

Liquid marbles are made by rolling a liquid droplet in a fine powder so that the droplet is spontaneously coated with a monolayer or multi-layer of grains of the powder. It is energetically favourable for hydrophilic as well as hydrophobic particles to attach to the liquid–vapour interface [52]. Also a variety of liquids of different surface tensions can be used to create liquid marbles. At small volumes, liquid marbles maintain an almost-spherical shape (figure 8*a*). Liquid marbles can be deformed, changed in size, manipulated and transferred to different surfaces. The liquid no longer wets any supporting surface and leaves no trace. The monolayer also slows down the liquid from evaporating.

The surface tension of the liquid marbles, which can be measured by different methods, is different from the liquid itself. It is generally seen that the chemistry as well as the micro/nanotopography of the powder coating affects the surface tension of liquid marbles. The surface tension of a liquid marble can be written as *γ*=*γ*_{LV}+*γ*_{int}, where *γ*_{int} is due to the capillary and electrostatic interactions between the covering particles, which can be positive or negative [53,54]. Liquid marbles are mesoscale entities.

Micelles are formed in a colloidal system when surfactant molecules arrange themselves into a spherical structure. Surfactants usually have a hydrophilic or ionic head and a hydrophobic–aliphatic tail. As surfactants are added to a colloid the surface energy of the interface between the phases continues to decrease. Once surfactants reach a critical concentration (called the critical micelle concentration) they can spontaneously agglomerate such that their heads are all arranged towards one phase of the colloid and their tails towards the other phase. When the sizes of the surfactant molecules are within a favourable limit, the agglomeration results in a spherical structure such as the one shown in figure 8*b*. Any further addition of surfactants beyond the critical micelle concentration does not alter the surface energy. Micelles are usually of nanometre scale [55].

Most obvious is the geometric similarity between liquid marbles and micelles. Powder coating modifies the surface energy of liquid marbles, while surfactants do the same for micelles. The formation of liquid marbles as well as micelles is driven by the minimization of the surface energy.

Contact angle hysteresis (CAH) is the difference between the maximum CA (advancing CA, e.g. in front of a moving droplet) and the minimum CA (receding CA, e.g. at the rear of a moving droplet). The three-phase (solid–liquid–vapour) contact line is pinned at the chemical or physical heterogeneities on the surface, resulting in CAH (figure 8*c*). As a result of this, any real surface exhibits a range of CAs. These CAs correspond to local minima of the Gibbs free energy for the surface. These minima are separated by energy barriers which are overcome with the help of the energy associated with different modes of vibration of the droplets [56]. Thus, the surface micro/nanotopography and vibrations affect the CAH of droplets on the mesoscale.

The energy barriers associated with CAH are similar to the activation energy observed between metastable states in a chemical reaction (figure 8*d*). The reaction kinetics are governed by the Arrhenius equation, in which the probability of attaining a metastable state is related exponentially to the inverse of the average kinetic energy of the molecules. Thus, temperature on the molecular scale is analogous to vibrations on the mesoscale. In §4, we discuss propulsion due to vibrations and patterns.

## 4. Shark-skin effect and blood flow: from hydrophobicity to ‘haemophobicity’

In the preceding sections, we discussed how the surface micropattern affects liquid flow. One area of application of patterns in the liquid flow is the so-called ‘shark-skin’ effect. Owing to the presence of micro-riblets (similar to those in the shark-skin scale) positioned parallel to the flow, the drag is reduced. Thus, the effect of the surface pattern can be substituted with an effective force, similar to the case of propulsion or water flow in a vibrating pipe, both of which have been considered in the preceding sections. The shark-skin effect is well known for water flow; however, here we discuss a similar effect for blood flow in biomedical applications and introduce the concept of surface ‘haemophobicity’ (similar to hydrophobicity or oleophobicity, from the Greek root *haemo* meaning ‘blood’). Owing to a properly tailored surface microstructure in a device such as a stent or artificial blood vessel, blood adhesion and stagnation can be controlled and, therefore, the risk of thrombosis can be reduced significantly.

### (a) Blood properties relevant to adhesion and drag resistance

Blood is a mixture of several different corpuscles: red blood cells (RBCs), white blood cells (WBCs), platelets and other proteins floating in the plasma. The accurate modelling of blood has to account for the interaction of all the particles and their corresponding deformation as they flow in the primary phase (plasma). Blood is a shear-thinning fluid with a yield stress. Depending on a particular problem or vascular territory, blood flow can be modelled as a Newtonian, non-Newtonian or a multi-phase flow. Blood is often modelled as a Newtonian fluid (i.e. the viscosity is a constant and is independent of the shear rate) when it is flowing through a larger diameter vessel, such as arteries (about 500 μm and larger), at high shear rates of over 100 s^{−1} [57]. However, blood is generally considered to be a non-Newtonian fluid at low shear rates, where its viscosity begins to differ with the rate of shear. Cardiovascular disease can cause blood to exhibit non-Newtonian behaviour in larger vessels since the flow in atherosclerotic or aneurysmal arteries is characterized by flow separation and recirculation regions with abnormally low shear rates.

Fahraeus & Lindqvist [58] showed that, in blood vessels with diameter less than 300 μm, the apparent blood viscosity decreases with decreasing diameter. However, with a further decrease in diameter in the range of 20–100 μm, viscosity starts to increase due to the presence of the cells in the flow [59]. Liepsch *et al.* [60] showed that, for shear rates less than 100 s^{−1}, blood flow exhibits non-Newtonian behaviour. Cokelet [61] conducted experiments on blood flow through narrow vessels and observed that a peripheral layer of plasma is free of cells while the suspension of RBCs is concentrated at the core.

The non-Newtonian behaviour of blood in smaller blood vessels (20–100 μm) with even lower shear rate (less than 10 s^{−1}) is due to the size of the suspended particles (e.g. RBCs) in the blood compared with the size of the blood vessels [62]. Since blood is a suspension containing particles, one can define its yield stress property which is due to RBC aggregation in the absence of shear. Tu & Deville [57] and Scott Blair & Spanner [63] showed that the Herschel–Bulkley (H-B) model, accounting for both shear-thinning and yield stress properties, can be used for the flow at very low shear rates where the yield stress is high and the diameter is less than 0.065 mm [64–66].

Biomimetic approaches, including the lotus and shark-skin effects, can be applied to the field of haemodynamics in order to enhance blood compatibility and to reduce the risk of thrombosis in biomedical applications. Kim *et al.* [67] studied the relation between the water CA and a thrombus, applying the multi-scale lotus effect on the surface and observing a decrease in platelet adhesion with an increase in hydrophobicity. Chen *et al.* [68] mimicked the inner cell layer topography of real blood vessel tissue with submicrometre ridges and nanoscale proturbances to reduce the platelet adhesion to the surface and concluded that the best results are obtained using a multi-scale surface. Liu *et al.* [69] designed a surface with microscale longitudinal aligned grafts and observed a reduced thrombus formation on the textured surface compared with that on the smooth one. In a study of the shark-skin effect on biofilm formation leading to blood thrombus, Chung *et al.* [70] designed riblets with a microtopography of 2 μm width, 3 μm spacing and various heights for their *in vitro* experiment on *Staphylococcus aureus*, a bacterial pathogen. The biofilm colonies started forming on the modified surface after day 21, while on the smooth surface the biofilm was formed at day 7 and matured at day 14.

The shark-skin effect leads to a reduction in drag resistance: Bechert *et al.* [71] found a reduction of up to 9.9%. Shear drag is the measure of the energy required to transfer momentum between the interacting fluid and the surface as well as the layers of the fluid flowing relative to each other due to the velocity gradients. The bursting and translation of vortices out of the viscous sublayer in the near-wall region, their collision and ejection and the chaotic flow in the outer layers are various types of momentum transfer that contribute to the flow drag. However, having riblets on the surface causes vortices to form and remain above the riblets, thus introducing zones of low-velocity fluctuation in the valleys between the riblets. Therefore, high-velocity vortices only interact with the small area at the tips of the riblets, counteracting the drag due to the addition of the riblets, which results in an overall reduction in drag [72–74]. Maani *et al.* [6] proposed a study of the shark-skin effect by computational fluid dynamics (CFD) modelling of the flow through a channel with micro-size riblets on the wall. In this study, the results are demonstrated.

### (b) Computational fluid dynamics of blood interaction with microstructured surfaces

In this study, blood is modelled as a non-Newtonian fluid using the Herschel–Bulkley model, which accounts for the shear-thinning and yield stress properties of the blood. The blood flows in a microchannel with one of its walls covered by riblets of different configurations and heights to study their effect on the near-wall flow, as shown in figure 9. Symmetry boundary conditions are assigned to the side walls of the channel in order to eliminate their effect on the flow over the textured surface on the bottom. In the first step, the height of the riblets is optimized to reduce the amount of overall drag on the surface. In the next step, the flow of blood over the surface with continuous, parallel and staggered configurations of riblets is simulated and the shear stress and drag in the channel are computed for each case in order to find the most effective configuration.

The channel cross-sectional area is (0.5 mm width × 0.8 mm height), and the length is 1.7 mm; the trapezoidal riblets are located in the region of fully developed flow. Figure 9*a* shows the dimensions of a riblet, while the parallel, staggered and continuous riblet configurations are shown in figure 9*b*, 9*c* and 9*d*, respectively. In all configurations, each row of riblets is placed *s*=50 μm apart from the next one in the continuous and aligned configurations and *s*=100 μm in the staggered configuration. These values correspond to the non-dimensional spacing values of *s*^{+}=2.77 and *s*^{+}=5.54 (the method of non-dimensionalizing is described in [73]).

The flow domain is discretized with a fully structured mesh generated with ANSYS ICEM CFD software; the finest mesh cell volume around the riblets is 1.5×10^{−17} m^{3}, and the whole channel contains 500 164 cells. The mesh is refined in the lower region of the channel to adequately resolve the flow around the riblets and in the boundary layer.

A constant flow velocity of 0.5 m s^{−1} (corresponding to *Re*=227) is prescribed at the model inlet, and the outflow boundary condition is assigned to the outlet. A coupled system of the momentum and pressure-based continuity equations is solved with the ANSYS Fluent pressure-based coupled algorithm. The second-order scheme is used for time discretization and the third-order Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) scheme is used for discretization in space. The shear stress, drag force and flow residence time are computed in the boundary layer near the surface and around the riblets and their values are compared for simulations with different geometries, including a smooth surface and the alternative riblet configurations. The riblets design is optimized in order to decrease the flow residence time in the near-wall region and thus prevent blood clot attachment to the surface.

In order to optimize the riblet height, several simulations were performed with continuous riblets of varying heights (*h*=10–60 μm, or non-dimensional *h*^{+}=0.554–3.325) and also with a smooth surface (*h*=0 μm) and the drag force on the patterned wall was calculated. The results are depicted in figure 10, where the *x*- and *y*-axes show the riblet height and the drag force on the wall, respectively. We observe that the minimum drag force is achieved when the height is 20 μm (*h*^{+}=1.113).

Note that, even though for riblet heights greater than *h*>20 μm the overall drag increases with height, the minimum shear stress continues to decrease. The reason is that the increased riblet height provides more volume for the flow in the valleys between the riblets, which results in lower momentum transfer and in a decrease in the shear stress. However, the riblet edge facing the flow acts like an obstacle opposing the flow which is associated with higher friction and specifically the pressure drag. Thus, the pressure drag increased by 260% while the viscous drag only decreased by 14% when the riblet height increased from *h*=20 to 60 μm. Therefore, the optimum height provides the minimum overall shear stress between the riblets (low-shear zones) and on their edges (high-shear zones).

In the next step, the overall shear stress along a line passing through the zone between the riblets (along the red line shown in figure 11) is calculated for the optimum height of *h*=20 μm. In figure 11, the *x*-axis is in the direction of the flow showing the section of the channel which contains the riblets and the *y*-axis shows the shear stress. Note that the section of the channel corresponding to the onset of the boundary layer is smooth. To study the effectiveness of the riblets, the shear stress and drag forces presented here exclude the entrance region. The line shear stress on the smooth surface is uniform and is also greater than that for the patterned walls.

On the wall with continuous riblets the shear stress is uniform and smallest. The shear stress fluctuates on the wall with aligned riblets since there are repeating patterns of riblet and smooth zones along the flow and the minimum shear stress is observed in the valleys between the riblets. The staggered riblet pattern produces a higher line shear stress than the parallel riblets. Calculating the overall average shear stress on the riblets (excluding the flat surface), we observe the same trend. Average shear stresses on the riblets are *τ*_{continuous}=30.29 Pa<*τ*_{staggered}=38.88 Pa<*τ*_{aligned}=39.64 Pa. In this study, we observe that, by patterning the wall with riblets, up to 6% drag force reduction is achieved while the surface area is increased by 25.4%.

These shear stresses are the highest stresses on the patterned surfaces while on the smooth surface the average shear stress is *τ*_{smooth}=44.035 Pa, still higher than the peak stress on the riblets, leading to the conclusion that the shark-skin pattern is effective for drag reduction in blood flow and the continuous riblet configuration is the most effective one.

The shark-skin effect is usually observed on the surface of high-speed objects in turbulent flow, as it is originally inspired by the skin patterns of fast-swimming sharks to reduce the drag force. However, in this study, it is applied to a laminar flow of blood in a microchannel to investigate the effect of riblets on haemodynamics. The presence of riblets is causing the drag reduction on the flat plate by introducing a zone with lower shear stress. Therefore, by manipulating the geometry of the surface we transport the high-shear stress to the riblet top and front edges.

Figure 12*a* depicts the flow distribution around the aligned riblets where we see lower velocities in the valleys and higher velocities on their top surfaces. The vectors are more uniform on the plane going through the valley (figure 12*b*). The contours of the shear stress are shown in figure 13, where the shear stress is at a minimum in the valleys around the riblets, while the concentration of shear stress is at the edges of the riblets.

To study the interaction of blood with the textured surface, we simulated an injection of neutrally buoyant particles in the flow and followed their trajectories. The transient blood flow with time step Δ*t*=0.0001 s was modelled with the particles continuously injected into the flow with the same time step. The particles were injected uniformly at the inlet surface with a total flow rate of 10^{−7} kg s^{−1}, and with each time step Δ*t* they were carried by the flow through the channel until they escaped from the outlet. The particle mass concentration contours for the smooth surface and the surface with the aligned riblet configuration at Δ*t*=0.008 s after the resumption of injection are shown in figure 14. We observe that, in the channels with the riblets, the particles flow over the top of the riblets, above the viscous boundary layer, and do not enter the valleys, while on the smooth surface they are flowing closer to the wall, which increases the chance of adhesion. Similar particle mass concentration is observed in all the riblet configurations.

## 5. Conclusion

We discussed how small fast vibrations (temporal patterns) and micro/nanotopography (spatial patterns) can affect wetting properties and liquid flow. The method of separation of motions at different scales was used to find an effective force that can be substituted for small fast vibrations and patterns. We applied this method to several examples including the flow of liquid though a vibrating pipe/membrane, vibrating granular matter and pattern-induced propulsion in Leidenfrost droplets and aquatic swimmers. In all these cases, we derived an expression for an effective force that can be substituted for vibrations or patterns.

We also discussed the similarities observed across multi-scale phenomena related to wetting. Vibrations can cause granular matter to behave like a flowing liquid with an apparent surface tension, whereas surface patterns can augment surface energy. Thus, the behaviour of large-scale grains is similar to the molecular-scale effect (surface tension). Furthermore, the separation of an oil–water mixture using selectively wettable membranes/meshes is similar to molecular-scale osmotic transport across a semipermeable membrane. Large-scale liquid marbles are similar to microscopic spherical micelles. In addition, CAH and the multiplicity of observed CAs on a surface are similar to the molecular-scale metastable states during the course of a chemical reaction. These observations provide additional insights on how different scale levels are related to each other and how the separation of effects at various scales can be performed.

In all the cases discussed here, vibrations or surface patterns lead to some nonlinearity or hysteresis, which results in peculiar behaviour including stabilization, phase control and propulsion. Thus, small-scale spatial and temporal patterns can affect macroscale material and surface properties. Potential applications include smart materials with tunable properties due to controlled microstructure and liquid flow, including the shark-skin effect.

Blood flow is a particularly novel area of application of the shark-skin effect. Using a properly tailored surface microstructure of a device such as a stent or artificial blood vessel, the risk of blood stagnation can be reduced significantly, thus preventing thrombosis.

## Authors' contributions

R.R. studied the effect of small vibrations and patterns and wrote §§2 and 3. N.M. performed CFD simulations of blood flow and wrote §4. V.L.R. and M.N. supervised the research and wrote parts of the paper.

## Competing interests

We have no competing interests.

## Funding

We received no funding for this study.

## Footnotes

One contribution of 12 to a theme issue ‘Bioinspired hierarchically structured surfaces for green science’.

- Accepted April 19, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.