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.
Explanation
Wetting of a solid material with a liquid substance occurs when adhesive forces allow the liquid (as a droplet) to spread across the surface of the solid at the solid-liquid interface. However, cohesive forces acting on the liquid – at the liquid–vapor interface – counteract the adhesive forces to prevent the droplet from making full contact with the surface.
The contact angle (θ), as seen in Figure 1, is the angle at which the liquid–vapor interface meets the solid–liquid interface, and is determined by the balance between adhesive and cohesive forces. As the tendency of a drop to spread out over a flat, solid surface increases, the contact angle decreases. Thus, the contact angle is used as an inverse measure of wettability. A contact angle less than 90° (low contact angle) usually indicates that wetting of the surface is very favorable, and the fluid will spread over a large area of the surface. Contact angles greater than 90° (high contact angle) generally mean that wetting of the surface is unfavorable, so the fluid will minimize contact with the surface and form a compact liquid droplet. For water, a wettable surface may also be termed hydrophilic and a nonwettable surface hydrophobic. Superhydrophobic surfaces have contact angles greater than 150°, showing almost no contact between the liquid drop and the surface. This is sometimes referred to as the "Lotus effect". The table describes varying contact angles and their corresponding solid/liquid and liquid/liquid interactions. For nonwater liquids, the term lyophilic is used for low contact angle conditions and lyophobic is used when higher contact angles result. Similarly, the terms omniphobic and omniphilic apply to both polar and apolar liquids.
Molecular energetic perspective of hydrophobicity Recent studies have introduced a quantitative molecular definition of hydrophobicity and wetting transitions, based on the energetic cost required to stabilize hydrogen-bond defects in the hydration shell of a surface or solute. According to this approach, a system is considered hydrophobic if it fails to compensate the energetic penalty of missing hydrogen bonds with an energy at least as favorable as the defect cost in bulk water, estimated around −6 kJ/mol (approximately 30% of a typical hydrogen bond energy). This criterion, known as the Defect Interaction Threshold (DIT), aligns remarkably with the classical 90° contact angle threshold, thus providing a non-arbitrary molecular basis for the onset of hydrophobic behavior. Moreover, by comparing local hydration interactions (such as the weakest and second weakest hydrogen-bond sites of the water molecules, often denoted V4S and V3S) against the DIT, it becomes possible to delineate regimes of full wetting, partial wetting (or partial drying), and complete drying (dewetting).
High-energy vs. low-energy surfaces Liquids can interact with two main types of solid surfaces. Traditionally, solid surfaces have been divided into high-energy and low-energy solids. The relative energy of a solid has to do with the bulk nature of the solid itself. Solids such as metals, glasses, and ceramics are known as 'hard solids' because the chemical bonds that hold them together (e.g., covalent, ionic, or metallic) are very strong. Thus, it takes a large amount of energy to break these solids (alternatively, a large amount of energy is required to cut the bulk and make two separate surfaces), so they are termed "high-energy". Most molecular liquids achieve complete wetting with high-energy surfaces. The other type of solid is weak molecular crystals (e.g., fluorocarbons, hydrocarbons, etc.) where the molecules are held together essentially by physical forces (e.g., van der Waals forces and hydrogen bonds). Since these solids are held together by weak forces, a very low amount of energy is required to break them, thus they are termed "low-energy". Depending on the type of liquid chosen, low-energy surfaces can permit either complete or partial wetting. Dynamic surfaces have been reported that undergo changes in surface energy upon the application of an appropriate stimuli. For example, a surface presenting photon-driven molecular motors was shown to undergo changes in water contact angle when switched between bistable conformations of differing surface energies.
Wetting of low-energy surfaces Low-energy surfaces primarily interact with liquids through dispersive (van der Waals) forces. William Zisman produced several key findings: Zisman observed that cos θ increases linearly as the surface tension (γLV) of the liquid decreased. Thus, he was able to establish a linear function between cos θ and the surface tension (γLV) for various organic liquids. A surface is more wettable when γLV and θ is low. Zisman termed the intercept of these lines when cos θ = 1 as the critical surface tension (γc) of that surface. This critical surface tension is an important parameter because it is a characteristic of only the solid. Knowing the critical surface tension of a solid, it is possible to predict the wettability of the surface. The wettability of a surface is determined by the outermost chemical groups of the solid. Differences in wettability between surfaces that are similar in structure are due to differences in the packing of the atoms. For instance, if a surface has branched chains, it will have poorer packing than a surface with straight chains. Lower critical surface tension means a less wettable material surface.
Ideal solid surfaces An ideal surface is flat, rigid, perfectly smooth, chemically homogeneous, and has zero contact angle hysteresis. Zero hysteresis implies the advancing and receding contact angles are equal. In other words, only one thermodynamically stable contact angle exists. When a drop of liquid is placed on such a surface, the characteristic contact angle is formed as depicted in Figure 1. Furthermore, on an ideal surface, the drop will return to its original shape if it is disturbed. The following derivations apply only to ideal solid surfaces; they are only valid for the state in which the interfaces are not moving and the phase boundary line exists in equilibrium.
Minimization of energy, three phases
Figure 3 shows the line of contact where three phases meet. In equilibrium, the net force per unit length acting along the boundary line between the three phases must be zero. The components of net force in the direction along each of the interfaces are given by:
γ α θ + γ θ β cos ( θ ) + γ α β cos ( α ) = 0 γ α θ cos ( θ ) + γ θ β + γ α β cos ( β ) = 0 γ α θ cos ( α ) + γ θ β cos ( β ) + γ α β = 0 {\displaystyle {\begin{aligned}\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{aligned}}}
where α, β, and θ are the angles shown and γij 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 α + β + θ = 2 π {\displaystyle \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, γij < γjk + γik, 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.
Simplification to planar geometry, Young's relation
If the β phase is replaced by a flat rigid surface, as shown in Figure 5, then β = π, and the second net force equation simplifies to the Young equation,
γ SG = γ SL + γ LG cos ( θ ) {\displaystyle \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.
Simplification to planar geometry, Young's relation derived from variational computation Consider the interface as a curve y ( x ) {\displaystyle y(x)} for x ∈ I = [ 0 , L ] {\displaystyle x\in I=[0,L]} where L {\displaystyle L} is a free parameter. The free energy to be minimized is
F [ y , L ] = ∫ 0 L ( γ L G 1 + y ′ 2 + ( γ SL − γ SG ) ) d x {\displaystyle {\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 {\displaystyle y(0)=y(L)=0} , which we can write as ∫ I y ′ d x = 0 {\displaystyle \int _{I}y'dx=0} and fixed volume ∫ I y d x = A {\displaystyle \int _{I}ydx=A} . The modified Lagrangian, taking into account the constraints is therefore
L = γ LG 1 + y ′ 2 + ( γ SL − γ SG ) − λ 1 y ′ − λ 2 y {\displaystyle {\cal {L}}=\gamma _{\text{LG}}{\sqrt {1+y'^{2}}}+(\gamma _{\text{SL}}-\gamma _{\text{SG}})-\lambda _{1}y'-\lambda _{2}y}
where λ i {\displaystyle \lambda _{i}} are Lagrange multipliers. By definition, the momentum p = ∂ y ′ L {\displaystyle p=\partial _{y'}{\cal {L}}} and the Hamiltonian H = p y ′ − L {\displaystyle {\cal {H}}=py'-{\cal {L}}} which is computed to be:
H = − γ LG 1 1 + y ′ 2 − ( γ SL − γ SG ) + λ 2 y {\displaystyle {\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 {\displaystyle x} direction and L {\displaystyle L} is a free parameter. Therefore, we must have:
∂ F ∂ L = − H = 0 {\displaystyle {\frac {\partial {\cal {F}}}{\partial L}}=-{\cal {H}}=0}
At the boundary y ( L ) = 0 {\displaystyle y(L)=0} and ( 1 + y ′ 2 ) − 1 / 2 = cos θ {\displaystyle (1+y'^{2})^{-1/2}=\cos \theta } , therefore we recover the Young equation.
Non-ideal smooth surfaces and the Young contact angle The Young equation assumes a perfectly flat and rigid surface often referred to as an ideal surface. In many cases, surfaces are far from this ideal situation, and two are considered here: the case of rough surfaces and the case of smooth surfaces that are still real (finitely rigid). Even in a perfectly smooth surface, a drop will assume a wide spectrum of contact angles ranging from the so-called advancing contact angle, θ A {\displaystyle \theta _{\mathrm {A} }} , to the so-called receding contact angle, θ R {\displaystyle \theta _{\mathrm {R} }} . The equilibrium contact angle ( θ c {\displaystyle \theta _{\mathrm {c} }} ) can be calculated from θ A {\displaystyle \theta _{\mathrm {A} }} and θ R {\displaystyle \theta _{\mathrm {R} }} as was shown by Tadmor as,
θ c = arccos ( r A cos ( θ A ) + r R cos ( θ R ) r A + r R ) {\displaystyle \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 A = ( sin 3 ( θ A ) 2 − 3 cos ( θ A ) + cos 3 ( θ A ) ) 1 3 ; r R = ( sin 3 ( θ R ) 2 − 3 cos ( θ R ) + cos 3 ( θ R ) ) 1 3 {\displaystyle 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
Young–Dupré equation and spreading coefficient The Young–Dupré equation (Thomas Young 1805; Anthanase Dupré and Paul Dupré 1869) dictates that neither γSG nor γSL can be larger than the sum of the other two surface energies. The consequence of this restriction is the prediction of complete wetting when γSG > γSL + γLG and zero wetting when γSL > γSG + γLG. The lack of a solution to the Young–Dupré equation is an indicator that there is no equilibrium configuration with a contact angle between 0 and 180° for those situations. A useful parameter for gauging wetting is the spreading parameter S,
S = γ SG − ( γ SL + γ LG ) {\displaystyle S=\gamma _{\text{SG}}-\left(\gamma _{\text{SL}}+\gamma _{\text{LG}}\right)}
When S > 0, the liquid wets the surface completely (complete wetting). When S < 0, partial wetting occurs. Combining the spreading parameter definition with the Young relation yields the Young–Dupré equation:
S = γ LG ( cos ( θ ) − 1 ) {\displaystyle S=\gamma _{\text{LG}}\left(\cos \left(\theta \right)-1\right)}
which only has physical solutions for θ when S < 0.
Generalized model for the contact angle of droplets on flat and curved surfaces With improvements in measuring techniques such as AFM, confocal microscopy and SEM, researchers were able to produce and image droplets at ever smaller scales. With the reduction in droplet size came new experimental observations of wetting. These observations confirm that the modified Young's equation does not hold at the micro-nano scales. In addition the sign of the line tension is not maintained through the modified Young's equation. For a sessile droplet, the free energy of the three phase system can be expressed as:
δ w = γ LV d A LV + γ SL d A SL + γ SV d A SV − κ d L − P d V − V d P {\displaystyle \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 = d A LG d A SL + γ SL − γ SG γ LG − κ γ LG d L d A SL − V γ LG d P d A SL {\displaystyle 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 ( θ ∓ α ) = A + B cos ( α ) a ± C sin ( θ ∓ α ) ( cos ( θ ) + 1 ) 2 ( sin ( α ) ( cos ( α ) + 2 ) ( cos ( α ) + 1 ) 2 ∓ sin ( θ ) ( cos ( θ ) + 2 ) ( cos ( θ ) + 1 ) 2 ) , {\displaystyle \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 = γ SG − γ SL γ LG {\displaystyle A={\frac {\gamma _{\text{SG}}-\gamma _{\text{SL}}}{\gamma _{\text{LG}}}}} , B = κ γ LG {\displaystyle B={\frac {\kappa }{\gamma _{\text{LG}}}}} and C = γ 3 γ LG {\displaystyle C={\frac {\gamma }{3\gamma _{\text{LG}}}}}
This equation relates the contact angle θ {\displaystyle \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 ( θ ) = γ SG − γ SL γ L G + κ γ LG 1 a − γ 3 γ LG ( 2 + cos ( θ ) − 2 cos 2 ( θ ) − cos 3 ( θ ) ) {\displaystyle \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.
Computational prediction of wetting For many surface/adsorbate configurations, surface energy data and experimental observations are unavailable. As wetting interactions are of great importance in various applications, it is often desired to predict and compare the wetting behavior of various material surfaces with particular crystallographic orientations, with relation to water or other adsorbates. This can be done from an atomistic perspective with tools including molecular dynamics and density functional theory. In the theoretical prediction of wetting by ab initio approaches such as DFT, ice is commonly substituted for water. This is because DFT calculations are generally conducted assuming conditions of zero thermal movement of atoms, essentially meaning the simulation is conducted at absolute zero. This simplification nevertheless yields results that are relevant for the adsorption of water under realistic conditions and the use of ice for the theoretical simulation of wetting is commonplace.
Non-ideal rough solid surfaces
Unlike ideal surfaces, real surfaces do not have perfect smoothness, rigidity, or chemical homogeneity. Such deviations from ideality result in phenomenon called contact angle hysteresis, which is defined as the difference between the advancing (θa) and receding (θr) contact angles
H = θ a − θ r {\displaystyle {\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, C a {\displaystyle 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