When the size of a droplet is smaller than the capillary length, the gravity acting on the droplet could be ignored.^[12] In fact, all the experimental results indicated that the smaller a droplet is, the easier its C– W transition, which means that gravity is not the primary cause. Additionally, there are no other obvious external forces, such as the hydraulic pressure, acting on a small droplet during its evaporation. Then we propose here that the IFEG possessed by a droplet is the driving force to complete its C– W transition, as shown in Fig. 2(a). This is quite similar to the case that a solid object on an inclined wall slides along the surface under the action of its potential energy gradient (gravity) as shown in Fig. 2(b). Just because the energy gradient is a kind of driving force, ^{[32, 33]} the object can move from a high energy position to a lower one. For the object in Fig. 2(b), its initial potential energy is E_p0, and its initial velocity is zero. If the friction along the surface is F_F, the energy of this object at any position x is (internal energy is not included)

where is the energy loss due to the friction. The energy E here includes both dynamic and potential energies, and the dynamic one is

where E_dx and E_px are the local dynamic and potential energies at position x, respectively. If differentiating each term in the above equation to x, we can find

where − dE_px/dx indicates the potential energy gradient of the object from a high position to a lower one, i.e., the gravity or the driving force making the object slide. It is thus clear that the potential energy gradient of an object is the driving force for its spontaneous movement from high energy position to lower one. Besides, dE_dx/dx in Eq. (3) is the dynamic energy gradient, namely, the inertial force of the object. Therefore, equation (3) is just the same as the one of Newtonian’ s second law.

Especially, at the initial position x = 0, the critical condition for the object starting to move is

For the inclined surface with slope decreasing continuously shown in Fig. 2(b), the potential energy gradient of the object may become equal to or less than the friction force when the object arrives at a certain position x₀, i.e., − dE_px/ dx| _{x≥ x0} ≤ F_F. But the object will not stop at this location because it already possesses a certain speed or dynamic energy. Only after the object continuously moves forward for a certain distance with its speed declining sustainedly until zero, it will stop. Therefore, the criterion of the object stopping moving forward is not the driving force equal to or less than the resistance, but is its dynamic energy or speed equal to zero. Namely, the criterion is

The process of an initial Cassie droplet transiting to the Wenzel state on a textured surface is very similar to that of the above case. The initial Cassie droplet has a certain interface free energy (IFE) which corresponds to the potential energy of the above object. If the TPCL below an initial Cassie droplet is able to move downwards, its IFEG, − dE_x/dx, must be greater than the resistance preventing the TPCL to depin. Quite similar to Eq. (4), the critical condition of the TPCL below an initial Cassie droplet starting to depin is

where x is the distance of the TPCL moving downward, and E_x is the IFE of the droplet, which is derived as follows.

Fig. 2.

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	Fig. 2. Schematics of (a) bending and depinning of droplet base TPCL acted by its IFEG and (b) a solid object on an inclined surface acted by its potential energy gradient.

For a composite droplet, when its base TPCL shifts to position x along posts (Fig. 2(a)), its IFE can be expressed as^[22]

where A_cap and A_b are the sphere cap and the base area of the droplet, respectively; σ _LG, σ _SG, and σ _SL are the liquid– gas, solid– gas, and solid– liquid interface tensions, respectively; f and r are the solid area fraction and the Wenzel roughness factor of the structural surface, respectively; H is the height of the pillars; and A_total is a fixed surface area to calculate the IFE of the droplet.^[22] The IFE of the composite droplet in Eq. (7) covers all interfacial energy in the selected area including the drop.

After simplification, equation (7) becomes

where θ _I is the intrinsic contact angle determined by the Young equation

And the expressions of A_cap and A_b are

where θ _d is the apparent contact angle, which is determined by the following formula when the volume of the droplet is

Here, the whole drop volume V is divided into two parts, the segment volume above posts V_up and the volume into structures V_in.

Then the final equation of the droplet IFE is

For cylindrical posts, the Wenzel roughness factor r and the solid area fraction f in the equation are given as

where d, H, and p are the diameter, the height, and the pitch of the pillars, respectively.

Furthermore, the IFEG, − dE_x/dx, in Eq. (6) is just the driving force of the droplet spontaneous C– W transition, which can be found by the relevant partial differentiations with Eqs. (12) and (13),

The validity and the detailed derivation process of this equation are shown in the supplementary information.

It should be pointed out that there may be three types of modes for the TPCL below a droplet to shift downward. The first one is by decreasing the apparent contact angle of the droplet while the base radius unchanging; the second mode is by reducing the base radius while the contact angle kept constant; the third one is by changing both the contact angle and base radius. It can be proved that the TPCL moves downward only in the first mode, as described in the supplementary information and Eq. (16).

Finally, F_W in Eq. (6) is the resistance preventing the TPCL moving down to wet the sides of the posts, which is explained as follows.

The C– W transition pressure P_t acting on a Cassie state droplet can be divided into the following three circumstances.

a) P_t ≤ 0, in this case there is no transition pressure acting on the droplet, and the droplet stays at the Cassie state without its bottom liquid– air interface bending.

b) 0 < (1 – f)P_t ≤ R_t, in this case the droplet is affected by the pressure, and its bottom liquid– air interface is bending while the contact angle on the side wall of the posts is less than the advancing contact angle, so the TPCL is still pinned. The pressure acting on the droplet is balanced by the resistance shown in Eq. (18)

c) (1 – f)P_t > R_t, the TPCL depins in this situation.

For a certain P_t, the contact angle of the curved liquid– air interface can be firstly found according to Eq. (21), and then the other parameters of the meniscus can be calculated with the following relations:

In the above formulas, R₁ and R₂ are the two arc radii on the curved liquid– air interface beneath the droplet respectively locating between the two adjacent and diagonal pillars with a square array. And h_x1 and h_x2 are the bending depths of the above two arcs, respectively, as shown in Fig. 1.

For short posts, the meniscus below a droplet will sag and touch the structural substrate before the TPCL depins. While for high pillars, the TPCL begins to move after the liquid– air interface bends to maximum until the meniscus contacts the pillar base. Touching the texture substrate of the meniscus is used as the criterion for droplets completing the C– W transition for both short and high posts in this model,

For short posts, x = 0; while for high pillars, x > 0.

For high posts, it is also necessary to judge whether the shifted TPCL beneath the composite droplet is able to continue to move forward until it touches the substrate of the textures. The judgment method is quite similar to that for the solid object to move forward sustainedly, as shown in Fig. 2(b) and Eq. (5). Now whether the “ dynamic” or the “ moving forward” energy of the composite droplet, when its base TPCL shifts to x, is greater than zero should be used as the criterion, namely,

where E_C represents the IFE of the Cassie droplet, which can be found when x in Eq. (13) is zero,

in which θ _EC is the equilibrium contact angle of the Cassie drop

The calculation diagram and the corresponding calculating program of the droplet C– W transition are shown in the supplementary information.

Figure 3 shows the C– W transition pressure P_t, the C– W transition resistance R_t, and the parameters of the curved liquid– air interface of a 2 μ L sessile droplet on textured surfaces with different structural parameters. Firstly, it can be seen from Fig. 3(a) that P_t is always 0 while R_t is very large if p/d of posts with a certain diameter is relatively small. Only when p/d is greater than a critical value, P_t starts to increase rapidly while R_t decreases sharply with the increase of p/d. Meanwhile, the smaller the micro/nano pillar diameter, the larger the critical value of p/d, and the greater the R_t, which means that large pitch is needed for small diameter posts to make P_t acting on the droplet exceed R_t so that the droplet can complete the C– W transition. Consequently, the thinner the pillars, the more favorable the prevention of the C– W transition. Namely, nano rough structures are able to maintain droplets in the superhydrophobic state more effectively. The fundamental of this result is explained as follows. The r will increase as d declines according to Eq. (14) if p/d keeps unchanging. Then R_t will increase while P_t decreases under this condition.

The effect of the textured parameters on the C– W transition can be more clearly seen by comparing Fig. 3(a) with Fig. 3(b). When P_t acting on the droplet is larger than 0 (Fig. 3(a)), the two bending depths, h_x1 and h_x2, of the meniscus begin to correspondingly increase quickly (Fig. 3(b)). Moreover, when P_t ≥ R_t (Fig. 3(a)), the TPCL beneath the droplet starts to move downward suddenly, that is x/H in Fig. 3(b) changing abruptly from 0 to 1. This fully shows that the liquid– air interface below the droplet becomes curved gradually and then obviously when P_t is greater than 0, and the TPCL begins to shift when P_t ≥ R_t. Besides, it can also be found from Fig. 3(b) that the larger the diameter of the posts, the more obvious the curvature of the meniscus, which is caused by the fact that the distance between the pillar side walls, p − d, is substantial when the diameter is large even if p/d is not great. For instance, if the diameter of one type of posts is 1 μ m and p/d is 30, the p − d will be 29 μ m; while the p − d will become 80 μ m if the diameter of another type of pillars is 10 μ m, even the p/d is 9 in this case.

Fig. 3.

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	Fig. 3. Effects of structural parameters on (a) C– W transition pressure and resistance and (b) liquid– air interface parameters below droplets. Here V = 2 μ L, H = 10 μ m, θ I = 110° , and θ A = 115° .

The droplet size affects the wetting state evidently. Generally, the smaller the droplet, the easier its C– W transition when it sits on a surface with fixed structural parameters. The calculation results of the model for this phenomenon are shown in Fig. 4. It is clear that R_t acting on the droplet has no relation with its volume, but is relevant to the structural parameters and the physicochemical properties of the material. However, P_t acting on the droplet has an obvious relation with its volume. There exists a critical volume for the droplet on a textured surface. When the volume of the droplet is less than the critical value, P_t increases obviously, but when it is larger than the critical value, P_t is 0, which means that a small droplet will suffer great P_t so that it is easy to transit to the Wenzel state while a large droplet may not be acted by P_t so that it will remain in the Cassie state. At the same time, the critical volume is clearly relative to the structural parameters. The smaller the pillar diameter, the smaller the critical volume, which also indicates that nano-structure is able to prevent droplets from the C– W transition. For instance, when the pillar diameter is 0.1 μ m, a droplet as small as a nano litre could keep the Cassie state, but when the pillar diameter becomes 1 μ m, a droplet of 0.1 mL will be in the Wenzel state.

Fig. 4.

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	Fig. 4. Effects of drop volume on (a) C– W transition pressure and resistance and (b) liquid– air interface parameters below droplets. Here θ I = 110° , θ A = 115° , p/d = 20, and H = 10 μ m.

The above results indicate that the essential reason for the droplet spontaneous transforming from Cassie to Wenzel state while its size reduces is that the transition pressure P_t will increase constantly with its size decreasing whereas the transition resistance R_t remains unchanging. When the driving force is greater than the resistance, the spontaneous C– W transition will be completed.

The wetting state of a droplet obviously depends on its volume and the superhydrophobicity of a tiny droplet is usually expected to be sustained, it is important to study the critical volume of droplets transiting from Cassie to Wenzel state on different structural surfaces. The calculation result is shown in Fig. 5. First of all is the effect of the pillar height H. When H is 1 μ m or lower, even a little bending of the liquid– air interface beneath the droplet will lead to the C– W transition through the sag mechanism. Consequently, it is almost impossible even for a large droplet, such as 10 μ L in this calculation, to maintain the Cassie state when H is low enough. When H is high enough, however, the critical volume decreases obviously. In this condition, the C– W transition takes place definitely through the TPCL depinning impalement. The second is the effect of p/d, which is similar to the result above. The smaller the value of p/d, the more difficult the C– W transition, and thus the smaller the critical volume. Besides, the effect of the pillar diameter can be found if comparing Fig. 5(a) with Fig. 5(b). Namely, the smaller the pillar diameter, the more difficult the C– W transition, and the smaller the critical volume. For instance, when d = 1 μ m, the critical volume is samller than one nano litre. Therefore, nano architectures can avoid the C– W transition of tiny droplets effectively.

Fig. 5.

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	Fig. 5. Effect of structural parameters on critical volume of droplet C– W transition. Here θ I = 110° , θ A = 115° ; (a) d = 10 μ m; (b) d = 1 μ m.

The above mainly investigates the effects of the structural parameters and the droplet volume on the C– W transition. There are two other parameters, the intrinsic contact angle and the advancing contact angle, which also influence the transition, as shown in Fig. 6. It is clear that the C– W transition pressure P_t decreases obviously while the transition resistance R_t rises with the increase of the intrinsic contact angle (Fig. 6(a)). As a result, the critical volume declines (Fig. 6(b)). Additionally, the increase of the advancing contact angle can also make R_t enlarge and P_t reduce in a certain extent (Fig. 6(a)). The critical volume is also decreased in this case, but not as obviously as that influenced by the intrinsic contact angle (Fig. 6(b)).

Fig. 6.

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	Fig. 6. Effects of intrinsic contact angle and advancing angle on (a) C– W transition pressure and resistance and (b) critical volume VC. Here p = 75 μ m, d = 10 μ m, H = 20 μ m, and V = 2 μ L.

Although increasing the intrinsic contact angle can effectively hinder the C– W transition, the intrinsic contact angle of current hydrophobic materials is generally under 120° . Consequently, it seems difficult to further restrain the C– W transition only relying on the change of the chemical properties of the materials. On the other hand, nano structures or micro-nano hierarchical textures have been proved to be able to effectively suppress the C– W transition.^{[16, 39, 40]} Thereby, the superhydrophobicity of droplets on these surfaces can be maintained. Furthermore, the sag mechanism may be no longer valid for the droplet C– W transition on a two-tier surface, i.e., a composite droplet will not transform to a Wenzel one even though its meniscus touches the micro structural substrate, ^{[41– 43]} the fundamental of which needs to study further.

There are two types of experiments to study the droplet C– W transition on superhydrophobic surfaces. One is to observe the contact angles of the droplets with fixed volume sitting on textured surfaces with different structural parameters. And the sudden change of the contact angles reflects the C– W transition. The other is to detect the abrupt variation of the shape parameters of the droplet before and after the C– W transition during its evaporation on a textured surface. The measurement results with these two types of experiments and the corresponding calculation outcomes with the model are shown in Figs. 7– 9.

Fig. 7.

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Fig. 7. Changes of contact angle and IFE barrier Eb of sessile droplets before and after C– W transition on different textured surfaces. (a) Experimental data originate from literature, [13]H = 10 μ m when d = 5 μ m, H = 30 μ m when d = 14 μ m; V = 5 μ L, θ I = 109° , θ A = 109° . (b)– (d) Experimental data originate from literature, [22]V = 3 μ L, θ I = 110° , θ A = 116° , (b) d = 10 μ m, H = 40 μ m; (c) p = 50 μ m, H = 20 μ m; (d) H = 2 × d. (e) Experimental data originate from literature, [21]d = 105 μ m, H = 150 μ m, V = 4.3 μ L, θ I = 107.5° , and θ A = 107.5° . Here the IFE barrier of the C– W transition is defined as the difference between the IFE of a composite droplet when its bottom TPCL moves to x = H and the IFE of Cassie state droplet.

Fig. 8.

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Fig. 8. Changes of base area As and IFE barrier Eb of a droplet before and after its C– W transition during evaporation. Experimental data originate from literature.[11] Here d = 3 μ m, p = 20 μ m, H = 4.8 μ m, θ I = 110° , θ A = 116° . Note: 1) the droplet volume required in the model calculation is solved by using the measured droplet base area and receding contact angle; 2) the IFE barrier of the C– W transition is defined as the difference between the IFE of a composite droplet when its bottom TPCL moves to x = H and the IFE of the Cassie state droplet.

Fig. 9.

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	Fig. 9. Changes of contact angle and drop radius Rd before and after the C– W transition of droplets during evaporation. Experimental data originate from literature, [13]θ I = 109° , θ A = 109° ; (a) d = 5 μ m, H = 10 μ m; (b) d = 14 μ m, H = 30 μ m. Here the droplet volume required in the model calculation is solved by using the measured droplet radius and the contact angle.

As shown in Fig. 7, the contact angles of the sessile droplets on different textured surfaces all decrease abruptly after they transit from Cassie to Wenzel state. The model calculation results are generally in good agreement with the measured ones.^{[13, 21, 22]} It can be seen from Figs. 7(a), 7(b), 7(d), and 7(e) that the increase of the pillar pitch eventually results in the droplet C– W transition. And from Fig. 7(c), it can be found that the droplets become easily appearing in the Cassie state with the increase of the pillar diameter while Wenzel droplets emerge on the surfaces with small diameters of posts.

It is generally considered that the C– W transition is restricted by the IFE energy barrier.^{[22, 26, 40, 44– 46]} Once this energy barrier decreases to 0, the C– W transition should be accomplished.^[11] However, the comparison between the calculation of this study and the measured results on different textured surfaces indicates that the droplet C– W transition usually does not happen when the energy barrier is zero, but often takes place when the energy barrier is obviously smaller than zero^[22] (Figs. 7(a)– 7(c)). The main reason for this is that the energy barrier is only a state function, which depends on the difference of the droplet IFE in two states.^[22] However, the C– W transition is also related to the process route, i.e., associated with the process of the TPCL moving along the structural side walls.^[46] Therefore, it is necessary to consider not only the energy barrier but also the process resistance, then the C– W transition can be accurately determined.

The Laplace pressure inside a droplet was considered as the driving force of the C– W transition in several reports.^{[1, 5, 15]} By the balance of capillary and Laplace pressure, a formula of drop radius at the moment of transition was found.^[25] Here the contact angles before and after drop the C– W transition in Fig. 7 are also calculated with this formula and compared with the experiments and our theoretical results. It can be seen that in most cases, the transition points calculated by the Laplace pressure method usually appear earlier or easier for the sessile drops on these surfaces as compared with the experiments.

When a Cassie state droplet evaporates on a structural surface, the C– W transition pressure acting on the droplet becomes larger as the drop size reduces. Consequently, the droplet will spontaneously transit into the Wenzel state at a certain moment. It can be seen from Fig. 8 that the base area of the droplet abruptly increases obviously when the C– W transition takes place. The C– W transition point calculated by this model fits the measured value^[11] very well. Meanwhile it is visible that the IFE energy barrier of the droplet is always negative during the evaporation process. Thus the droplet C– W transition during evaporation cannot be determined with the energy barrier being 0. Furthermore, the C– W transition point determined by the balance of capillary and Laplace pressure^[25] is obviously delayed.

As shown in Fig. 9, the size of an evaporating droplet when its C– W transition is taking place is obviously relative to the pillar pitch p. When p is small, there is no C– W transition happening for all the droplet sizes during evaporation since the contact angles do not decrease obviously. However, if p is large enough, the C– W transition occurs when the droplet size reduces to a critical value, at which the contact angle of the droplet decreases suddenly and the radius of the droplet increases obviously. Overall, the calculation results of the model agree well with the measured values^[13] except for the detailed transition points where there are deviations between the calculations and the measurements. For instance, the calculated critical radius of the droplet C– W transition is less than the measured value for the case of d = 5 μ m (Fig. 9(a)), while it becomes larger than the value for the case of d = 14 μ m (Fig. 9(b)). Considering the complexity of the dynamic process of droplet evaporation, the deviation between calculation and measurement of the C– W transition point is acceptable.

The model can be reliable to analyze the mechanism of the droplet C– W transition since the calculation results on the whole fit experiments well.