Wetting

Wetting is the ability of a liquid to maintain contact with a solid surface by displacing another substance or material – either a gas, or other liquid not miscible with the wetting liquid – due to the differential strength of intermolecular interactions with the surface. The degree of wetting, or wettability, is dependent on the force balance between adhesive and cohesive forces, occurring when liquid and solid make contact in the presence of another physical phase. As such, wetting is of importance to bonding and adherence of substances in different phases. The wetting power of a liquid, and surface forces contributing to its wettability, are also responsible for capillary action. Surfactants can be used to increase the wetting power of liquids (i.e., water) by reducing surface forces. There are two types of wetting: non-reactive and reactive wetting. Wetting has gained increased attention in nanotechnology and nanoscience research following the development of nanomaterials, such as graphene, carbon nanotubes, and boron nitride nanomesh.

Wetting is the ability of a liquid to maintain contact with a solid surface by displacing another substance or material – either a gas, or other liquid not miscible with the wetting liquid – due to the differential strength of intermolecular interactions with the surface. The degree of wetting, or wettability, is dependent on the force balance between adhesive and cohesive forces, occurring when liquid and solid make contact in the presence of another physical phase. As such, wetting is of importance to bonding and adherence of substances in different phases. The wetting power of a liquid, and surface forces contributing to its wettability, are also responsible for capillary action. Surfactants can be used to increase the wetting power of liquids (i.e., water) by reducing surface forces. There are two types of wetting: non-reactive and reactive wetting. Wetting has gained increased attention in nanotechnology and nanoscience research following the development of nanomaterials, such as graphene, carbon nanotubes, and boron nitride nanomesh. \gamma_{\alpha\theta} + \gamma_{\theta\beta}\cos\left(\theta\right) + \gamma_{\alpha\beta}\cos\left(\alpha\right) &= 0 \\ \gamma_{\alpha\theta}\cos\left(\theta\right) + \gamma_{\theta\beta} + \gamma_{\alpha\beta}\cos\left(\beta\right) &= 0 \\ \gamma_{\alpha\theta}\cos\left(\alpha\right) + \gamma_{\theta\beta}\cos\left(\beta\right) + \gamma_{\alpha\beta} &= 0 \end{align} where α, β, and θ are the angles shown and γ ij is the surface energy between the two indicated phases. These relations can also be expressed by an analog to a triangle known as Neumann's triangle, shown in Figure 4. Neumann's triangle is consistent with the geometrical restriction that \alpha + \beta + \theta = 2\pi , and applying the law of sines and law of cosines to it produce relations that describe how the interfacial angles depend on the ratios of surface energies. Because these three surface energies form the sides of a triangle, they are constrained by the triangle inequalities, , meaning that not one of the surface tensions can exceed the sum of the other two. If three fluids with surface energies that do not follow these inequalities are brought into contact, no equilibrium configuration consistent with Figure 3 will exist. \gamma_\text{SG} = \gamma_\text{SL} + \gamma_\text{LG}\cos\left(\theta\right) which relates the surface tensions between the three phases: solid, liquid and gas. Subsequently, this predicts the contact angle of a liquid droplet on a solid surface from knowledge of the three surface energies involved. This equation also applies if the "gas" phase is another liquid, immiscible with the droplet of the first "liquid" phase. Consider the interface as a curve y(x) for x\in I=[0,L] where L is a free parameter. The free energy to be minimized is {\cal F}[y,L] = \int_0^L \left( \gamma_{LG}\sqrt{1+y'^2} + (\gamma_\text{SL}-\gamma_\text{SG}) \right) dx with the constraints y(0)=y(L)=0 , which we can write as \int_I y'dx=0 and fixed volume \int_I ydx= A . The modified Lagrangian, taking into account the constraints is therefore {\cal L} = \gamma_\text{LG}\sqrt{1+y'^2} + (\gamma_\text{SL}-\gamma_\text{SG}) -\lambda_1 y' - \lambda_2 y where \lambda_i are Lagrange multipliers. By definition, the momentum p=\partial_{y'}{\cal L} and the Hamiltonian {\cal H} = p y' - {\cal L} which is computed to be: {\cal H} = -\gamma_\text{LG}\frac{1}{\sqrt{1+y'^2}} - (\gamma_\text{SL}-\gamma_\text{SG}) + \lambda_2 y Now, we recall that the boundary is free in the x direction and L is a free parameter. Therefore, we must have: \frac{\partial{\cal F}}{\partial L} = -{\cal H} = 0 At the boundary y(L)=0 and (1+y'^2)^{-1/2} = \cos\theta , therefore we recover the Young equation. \theta_\mathrm{c} = \arccos\left( \frac{r_\mathrm{A}\cos\left(\theta_\mathrm{A}\right) + r_\mathrm{R}\cos\left(\theta_\mathrm{R}\right)} {r_\mathrm{A} + r_\mathrm{R}} \right) where r_\mathrm{A} = \left( \frac{\sin^3\left(\theta_\mathrm{A}\right)}{2 - 3\cos\left(\theta_\mathrm{A}\right) + \cos^3\left(\theta_\mathrm{A}\right)} \right)^\frac{1}{3} ~;~~ r_\mathrm{R} = \left( \frac{\sin^3\left(\theta_\mathrm{R}\right)}{2 - 3\cos\left(\theta_\mathrm{R}\right) + \cos^3\left(\theta_\mathrm{R}\right)} \right)^\frac{1}{3} Equations for the advancing and receding contact angles have been derived also based on a purely thermodynamic theory S = \gamma_\text{SG} - \left(\gamma_\text{SL} + \gamma_\text{LG}\right) When , the liquid wets the surface completely (complete wetting). When , partial wetting occurs. Combining the spreading parameter definition with the Young relation yields the Young–Dupré equation: S = \gamma_\text{LG}\left(\cos\left(\theta\right) - 1\right) which only has physical solutions for θ when . \delta w = \gamma_\text{LV}dA_\text{LV}+\gamma_\text{SL}dA_\text{SL}+\gamma_\text{SV}dA_\text{SV}-\kappa dL-PdV-VdP At constant volume in thermodynamic equilibrium, this reduces to: 0= \frac{dA_\text{LG}}{dA_\text{SL}} + \frac{\gamma_\text{SL}-\gamma_\text{SG}}{\gamma_\text{LG}} - \frac{\kappa}{\gamma_\text{LG}}\frac{dL}{dA_\text{SL}}-\frac{V}{\gamma_\text{LG}} \frac{dP}{dA_\text{SL}} Usually, the VdP term has been neglected for large droplets, however, VdP work becomes significant at small scales. The variation in pressure at constant volume at the free liquid-vapor boundary is due to the Laplace pressure, which is proportional to the mean curvature of the droplet, and is non zero. Solving the above equation for both convex and concave surfaces yields: \cos(\theta\mp\alpha)=A+B\frac{\cos(\alpha)}{a}\pm C\sin(\theta\mp\alpha)(\cos(\theta)+1)^2\biggl(\frac{\sin(\alpha)(\cos(\alpha)+2)}{(\cos(\alpha)+1)^2}\mp\frac{\sin(\theta)(\cos(\theta)+2)}{(\cos(\theta)+1)^2}\biggr), where the constant parameters A, B, and C are defined as: A = \frac{\gamma_\text{SG}-\gamma_\text{SL}}{\gamma_\text{LG}} , B = \frac{\kappa}{\gamma_\text{LG}} and C = \frac{\gamma}{3\gamma_\text{LG}} This equation relates the contact angle \theta , a geometric property of a sessile droplet to the bulk thermodynamics, the energy at the three phase contact boundary, and the curvature of the surface α. For the special case of a sessile droplet on a flat surface (α=0), \cos(\theta) = \frac{\gamma_\text{SG}-\gamma_\text{SL}}{\gamma_{LG}} + \frac{\kappa}{\gamma_\text{LG}} \frac{1}{a} -\frac{\gamma}{3\gamma_\text{LG}}(2+\cos(\theta)-2\cos^2(\theta)-\cos^3(\theta)) The first two terms are the modified Young's equation, while the third term is due to the Laplace pressure. This nonlinear equation correctly predicts the sign and magnitude of κ, the flattening of the contact angle at very small scales, and contact angle hysteresis. \text{H} = \,\theta_{a} - \,\theta_{r} When the contact angle is between the advancing and receding cases, the contact line is considered to be pinned and hysteretic behaviour can be observed, namely contact angle hysteresis. When these values are exceeded, the displacement of the contact line, such as the one in Figure 3, will take place by either expansion or retraction of the droplet. Figure 6 depicts the advancing and receding contact angles. The advancing contact angle is the maximum stable angle, whereas the receding contact angle is the minimum stable angle. Contact angle hysteresis occurs because many different thermodynamically stable contact angles are found on a nonideal solid. These varying thermodynamically stable contact angles are known as metastable states. Such motion of a phase boundary, involving advancing and receding contact angles, is known as dynamic wetting. The difference between dynamic and static wetting angles is proportional to the capillary number, Ca , When a contact line advances, covering more of the surface with liquid, the contact angle is increased and is generally related to the velocity of the contact line. If the velocity of a contact line is increased without bound, the contact angle increases, and as it approaches 180°, the gas phase will become entrained in a thin layer between the liquid and solid. This is a kinetic nonequilibrium effect which results from the contact line moving at such a high speed that complete wetting cannot occur. A well-known departure from ideal conditions is when the surface of interest has a rough texture. The rough texture of a surface can fall into one of two categories: homogeneous or heterogeneous. A homogeneous wetting regime is where the liquid fills in the grooves of a rough surface. A heterogeneous wetting regime, though, is where the surface is a composite of two types of patches. An important example of such a composite surface is one composed of patches of both air and solid. Such surfaces have varied effects on the contact angles of wetting liquids. Cassie–Baxter and Wenzel are the two main models that attempt to describe the wetting of textured surfaces. However, these equations only apply when the drop size is sufficiently large compared with the surface roughness scale. When the droplet size is comparable to that of the underlying pillars, the effect of line tension should be considered. \cos\,\left(\theta^*\right) = r \cos\,\left(\theta\right) where \theta^* is the apparent contact angle which corresponds to the stable equilibrium state (i.e. minimum free energy state for the system). The roughness ratio, r, is a measure of how surface roughness affects a homogeneous surface. The roughness ratio is defined as the ratio of true area of the solid surface to the apparent area. θ is the contact angle for a system in thermodynamic equilibrium, defined for a perfectly flat surface. Although Wenzel's equation demonstrates the contact angle of a rough surface is different from the intrinsic contact angle, it does not describe contact angle hysteresis. \cos\,\left(\theta^*\right) = r_f\,f \,\cos\,\left(\theta_\text{Y}\right) + f - 1 Here the r f is the roughness ratio of the wet surface area and f is the fraction of solid surface area wet by the liquid. When and , the Cassie–Baxter equations becomes the Wenzel equation. On the other hand, when there are many different fractions of surface roughness, each fraction of the total surface area is denoted by f_i . A summation of all f_i equals 1 or the total surface. Cassie–Baxter can also be recast in the following equation: \gamma\cos\,\left(\theta^*\right) = \sum_{n=1}^N f_i\left(\gamma_\text{i,sv} - \gamma_\text{i,sl}\right) Here \gamma is the Cassie–Baxter surface tension between liquid and vapor, \gamma_{i,sv} is the solid vapor surface tension of every component, and \gamma_{i,sl} is the solid liquid surface tension of every component. A case that is worth mentioning is when the liquid drop is placed on the substrate and creates small air pockets underneath it. This case for a two-component system is denoted by: \gamma\cos\,\left(\theta^*\right) = f_1\left(\gamma_\text{1,sv} - \gamma_\text{1,sl}\right) - \left(1 - f_1\right)\gamma Here the key difference to notice is that there is no surface tension between the solid and the vapor for the second surface tension component. This is because of the assumption that the surface of air that is exposed is under the droplet and is the only other substrate in the system. Subsequently, the equation is then expressed as . Therefore, the Cassie equation can be easily derived from the Cassie–Baxter equation. Experimental results regarding the surface properties of Wenzel versus Cassie–Baxter systems showed the effect of pinning for a Young angle of 180 to 90°, a region classified under the Cassie–Baxter model. This liquid/air composite system is largely hydrophobic. After that point, a sharp transition to the Wenzel regime was found where the drop wets the surface, but no further than the edges of the drop. Actually, the Young, Wenzel and Cassie-Baxter equations represent the transversality conditions of the variational problem of wetting. \cos\,\left(\theta_\text{C}\right)= \frac{\phi - 1}{r - \phi} where θ C is the critical contact angle Φ is the fraction of solid/liquid interface where drop is in contact with surface r is solid roughness (for flat surface, r = 1) The penetration front propagates to minimize the surface energy until it reaches the edges of the drop, thus arriving at the Wenzel state. Since the solid can be considered an absorptive material due to its surface roughness, this phenomenon of spreading and imbibition is called hemiwicking. The contact angles at which spreading/imbibition occurs are between 0 and π/2. The Wenzel model is valid between θ C and π/2. If the contact angle is less than Θ C , the penetration front spreads beyond the drop and a liquid film forms over the surface. Figure 11 depicts the transition from the Wenzel state to the surface film state. The film smooths the surface roughness and the Wenzel model no longer applies. In this state, the equilibrium condition and Young's relation yields: \cos\,\left(\theta^*\right) = \phi\cos\,\left(\theta_C\right) + \left(1 - \phi\right) By fine-tuning the surface roughness, it is possible to achieve a transition between both superhydrophobic and superhydrophilic regions. Generally, the rougher the surface, the more hydrophobic it is. r(t) = r_e\left[ 1 - \exp\left(-\left(\frac{2\gamma_{LG}}{r^{12}_e} + \frac{\rho g}{9r^{10}_e}\right)\frac{24\lambda V^4 \left(t + t_0\right)}{\pi^2\eta}\right) \right]^\frac{1}{6}. For the complete wetting situation, the drop radius at any time during the spreading process is given by r(t) = \left[ \left(\gamma_{LG} \frac{96\lambda V^4}{\pi^2 \eta} \left(t + t_0\right)\right)^\frac{1}{2} + \left(\frac{\lambda(t + t_0)}{\eta}\right)^\frac{2}{3} \frac{24\rho g V^\frac{8}{3}}{7 \cdot 96^\frac{1}{3} \pi^\frac{4}{3} \gamma_{LG}^\frac{1}{3}} \right]^\frac{1}{6}, where γ LG is surface tension of the fluid V is drop volume η is viscosity of the fluid ρ is density of the fluid g is gravitational constant λ is shape factor, 37.1 m¹ −1 t 0 is experimental delay time r e is drop radius in equilibrium \sigma = \gamma S + PV + \pi\,R^2 \left(\gamma_\text{SL} - \gamma_\text{SV}\right) γ is the liquid–vapor interfacial tension γ SL is the solid–liquid interfacial tension γ SV is the solid–vapor interfacial tension S is the area of liquid–vapor interface P is the excess pressure inside liquid R is the radius of droplet base Based on this equation, the excess free energy is minimized when γ decreases, γ SL decreases, or γ SV increases. Surfactants are absorbed onto the liquid–vapor, solid–liquid, and solid–vapor interfaces, which modify the wetting behavior of hydrophobic materials to reduce the free energy. When surfactants are absorbed onto a hydrophobic surface, the polar head groups face into the solution with the tail pointing outward. In more hydrophobic surfaces, surfactants may form a bilayer on the solid, causing it to become more hydrophilic. The dynamic drop radius can be characterized as the drop begins to spread. Thus, the contact angle changes based on the following equation: \cos\,\left(\theta(t)\right) = \cos\,\left(\theta_0\right) + \left(\cos\,\left(\theta_\infty\right) - \cos\,\left(\theta_0\right)\right)\left(1 - e^{-\frac{t}{\tau}}\right) θ 0 is initial contact angle θ ∞ is final contact angle τ is the surfactant transfer time scale As the surfactants are absorbed, the solid–vapor surface tension increases and the edges of the drop become hydrophilic. As a result, the drop spreads.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1362483628 under CC BY-SA 4.0; modifications have been made.