We develop a one-dimensional model for transient diffusion of gas between ridges into a quiescent liquid suspended in the Cassie state above them. In the first case study, we assume that the liquid and gas are initially at the same pressure and that the liquid column is sealed at the top. In the second one, we assume that the gas initially undergoes isothermal compression and that the liquid column is exposed to gas at the top. Our model provides a framework to compute the transient gas concentration field in the liquid, the time when the triple contact line begins to move down the ridges, and the time when menisci reach the bottom of the substrate compromising the Cassie state. At illustrative conditions, we show the effects of geometry, hydrostatic pressure, and initial gas concentration on the Cassie to Wenzel state transition.

Introduction

Superhydrophobic surfaces (SHPos), i.e., hydrophobic surfaces that are topographically rough, may support a droplet or a liquid column in an unwetted (Cassie) state [1]. Then, in the case of a droplet, the apparent contact angle exceeds the Young angle and can approach 180 deg. It is also well known that the use of SHPos in pressure-driven flows in microchannels reduces frictional drag [2]. However, it is challenging to design and fabricate SHPo surfaces that can permanently maintain a stable Cassie state. Indeed, due to phenomenon such as evaporation [3], condensation [4], pressurization [1], impact [5], and gas diffusion [6], the Cassie state may be gradually compromised resulting in a transition to a wetted (Wenzel) state over a time period known as longevity [6,7].

Limited longevity presents a key challenge in the industrial application of SHPo surfaces. In the context of microgap cooling, structured surfaces that have high longevity at high operating pressure are desirable because they enable high channel pressure drops which result in high volumetric flow rate and, thus, low caloric resistance in comparison to planar Poiseuille flow [8]. Similarly, in underwater applications, structured surfaces that have high longevity under high hydrostatic pressure may enable vehicles to operate at high speeds with low energy consumption [9,10]. Among the methods of enhancing longevity suggested in the literature are gas injection and electrolysis of water to produce inert gas [7,10,11]. To implement such techniques, there is a need to model the physics of Cassie to Wenzel state transition as a result of gas diffusion.

Theoretical models and experimental studies show that the shape of the meniscus changes in two primary stages. Initially, the meniscus curvature increases and, thus, the contact angle (θ) increases. A stable Cassie state may be reached in which the contact angle reaches a terminal value. Otherwise, it becomes equal to the advancing contact angle (θA) at the critical time (tcr). Then, the triple contact line depins and moves down into the cavity with the contact angle fixed at θA [12,13]. The meniscus touches the bottom of the cavity at the final time (tf) in a symmetric or asymmetric fashion, where the latter is thought to be caused by the presence of impurities in the cavity walls [14]. A broad literature review on gas diffusion-induced Cassie to Wenzel state transition was done by Xue et al. for structures whose gas cavities are closed to the atmosphere [15]. To validate the aforementioned theory, researchers have experimentally tracked the menisci using techniques such as photography [11], light reflection [6], confocal microscopy [14], and acoustic tracking [16].

Emami et al. [13,17] were the first to analytically consider wetting transition in a submerged ridge-type SHPo surface where, on the menisci, the gas pressure in the cavity and the surface tension counter-balance the atmospheric pressure (p) exerted on top of a liquid (water) domain and hydrostatic pressure. At t = 0, they assumed that the gas in the cavity instantaneously undergoes either isothermal or adiabatic compression and that the water is saturated with air at atmospheric pressure in accordance with Henry's law. For t > 0, gas diffuses into the water at a volume flow rate per unit depth $dVg′/dt$ that equals the interfacial area (A) times an empirically determined average invasion coefficient for air $(ξ¯)$ times the difference between the gas pressure in the cavity (pg) and the total pressure of dissolved gas in the bulk liquid (p) as per
$dVg′dt=ξ¯A[p∞−pg(t)]$
(1)

Neglecting inertial and viscous effects in the liquid and applying the ideal gas law in the gas cavity, Emami et al. predict the longevity of a submerged ridge-type SHPo surface using Eq. (1), an equilibrium force balance on the menisci and the ideal gas law [17]. Their results predict that, for a given geometry, longevity decreases with increasing hydrostatic pressure because the solubility of gases increases with the gas pressure in the cavity [17]. Prior experimental studies by Samaha et al. [6] and Lv et al. [18] show that longevity decreases exponentially with hydrostatic pressure in superhydrophobic fibrous coatings and micropore arrays, respectively. The use of Eq. (1) by Emami et al. [13,17] and others [11,18] in experimental studies implies a convective as opposed to purely diffusive mass transfer process in the liquid phase. Rahn and Paganelli show that the value of $ξ¯$ is equivalent to the ratio of the diffusion coefficient and an assigned effective boundary layer thickness [19]. Therefore, the accuracy of the empirical model depends on the careful choice of $ξ¯$, which can be arbitrary [11].

For applications dominated by diffusion, longevity can be predicted by solving the transient diffusion equation, with no need for computing $ξ¯$, as per the Epstein–Plesset model for gas diffusion in bubbles surrounded by liquid [20]. Sogaard et al. successfully correlated experimental results of diffusion-induced Cassie-to-Wenzel state transition on a microcavity array with a purely numerical solution of the 1D, transient gas-diffusion equation [21]. Results in their study show that one can predict, with reasonable accuracy, the longevity of a SHPo surface based on geometry, thermophysical properties, and initial and boundary conditions [21]. In this paper, we analytically solve the 1D transient diffusion equation for the dissolved gas concentration field in the liquid domain during a Cassie-to-Wenzel state transition to predict the longevity of a parallel ridge-type SHPo surface. We, and not Sogaard et al., also consider the cases where (a) gas diffusion into the liquid is driven by hydrostatic pressure rather than an external pressure due to a piston or high-pressure gas pressurizing the liquid phase and (b) the initial dissolved gas concentration in the liquid is arbitrary.

Mathematical Modeling

When measuring and modeling the lubricating properties of flows in the Cassie state over structured surfaces, the liquid pressure (p) is typically higher than what could be supported if the gaps between the menisci and underlying substrate are under a vacuum. This is due to the presence of inert gas and/or vapor. However, inert gas support can be compromised by diffusion of it into the liquid phase. Assuming that the vapor pressure of the liquid is small compared to the partial pressure of the gas, the force balance along the meniscus is governed by the Young–Laplace equation as per
$pg(t)=p∞+ρℓg h(t)−σR(t)$
(2)

where pg(t) is the pressure of the gas in the cavity, ρ is the density of the liquid, g is the acceleration due to gravity, h(t) is the height of the liquid domain, σ is the surface tension along the meniscus, and R(t) is the radius of curvature of the meniscus.

We consider gas diffusion in quiescent liquid initially in the Cassie state above a ridge-type structured surface as depicted in Fig. 1. The liquid is supported above one period of the ridges (2d), the gas cavity height is s0, and the width of the meniscus is 2a. The meniscus has a contact angle of θ(t) measured from the vertical and the corresponding radius of curvature is R(t). The thin dashed lines represent the shape of the meniscus at critical time tcr, intermediate time t where tcr < t < tf, and final time tf. The triple contact line is initially pinned at y = 0, but it is displaced by s(t) for tcr < t < tf. We analyze two representative studies where the boundary condition at y = h and the role of hydrostatic pressure are the main differences. In case study I, a massless, impermeable piston applies a constant pressure at y = h (for t > 0) which far exceeds the hydrostatic pressure of the liquid, i.e., p ≫ ρgh as per Fig. 1(a); therefore, the gas pressure in the cavity is initially equal to the liquid pressure. In case study II, the top of the liquid is exposed to the atmosphere and hydrostatic pressure of the liquid is important as per Fig. 1(b); therefore, there is an initial pressure difference across the meniscus. It follows from the Young–Laplace equation that the meniscus is initially flat for case study I and has a finite radius of curvature R0 for case study II as shown in the figure.

Fig. 1
Fig. 1
Close modal
The dissolved gas concentration field in the liquid phase is governed by
$∂ρg∂t=Dgl(∂2ρg∂x2+∂2ρg∂y2)$
(3)
where ρg is the partial density of gas in the liquid phase and Dgl is the molecular diffusivity of the gas in it. By symmetry, only the liquid domain on 0 < x < d needs to be considered. At x = 0 and x = d, the symmetry boundary conditions are as per Eqs. (4), where ym (x, t) is the deflection of the meniscus relative to y = 0, and Eq. (5) below. The top of the domain (y = h) for case study I and the solid–liquid interface (a < x < d and y = 0) are assumed to be impermeable to gas diffusion as per Eqs. (6) and (7)
$∂∂xρg(0,ym
(4)
$∂∂xρg(d,0
(5)
$∂∂yρg(0
(6)
$∂∂yρg(a
(7)
For case study II, boundary condition (6) is replaced by
$ρg(0
(8)
where H is Henry's constant. At time t = 0, the partial density of gas in the liquid domain (ρg,0) is uniform and the initial gas pressure in the cavity is equal to atmospheric pressure as per
$ρg(x,y,0)=ρg,0$
(9)
$pg(0)=p∞$
(10)

The gas concentration field in the gas phase is always assumed to be uniform because the molecular diffusivity in the gas phase is much larger than that in the liquid phase.

For t > 0, gas diffusion through the meniscus decreases the gas pressure in the cavity. Surface concentration on the meniscus is computed from Henry's Law, i.e., ρsat = pg(t)H. As the gas pressure decreases, the contact angle increases until it equals the advancing contact angle at the critical time. The boundary condition on the meniscus is
$ρg(0
(11)
The shape of the meniscus is a circular arc of radius R(t) centered at xc = 0 and $yc=∓R(t)2−a2−s(t)$ where s(t), which is always ≥0, is the vertical distance between the ridge tip (y = 0) and the triple contact line and the − and + signs are for menisci protruding upward and downward, respectively. It follows that
$ym(x,t)=±[R(t)2−x2−R(t)2−a2]−s(t)$
(12)

where the − sign corresponds to a downward protruding meniscus and vice versa. Only the downward protruding case is considered herein, so only the − sign is needed in subsequent equations.

After the critical time, the meniscus radius of curvature and gas pressure are constant and denoted by Rcr and pcr, respectively. From geometry
$Rcr=a cos(π−θA)$
(13)
Upon substituting Eq. (13) into Eq. (2), pcr is given by
$pcr=p∞+ρℓg h(tcr)− cos(π−θA)σa$
(14)
We substitute Eq. (14) into Eq. (11) to determine the constant surface concentration boundary condition on the meniscus after critical time, but before complete wetting, as per
$ρg(0
(15)
The aforementioned problem is likely analytically insoluble because (a) the meniscus is deforming before tcr and moving after it, (b) the boundary condition along the meniscus is time-dependent before tcr, and (c) there is a mixed boundary condition along the bottom of the domain, i.e., solid–liquid interface plus meniscus. However, the problem can be reduced to a transient, 1D diffusion problem by assuming that the solid fraction approaches zero and approximating the meniscus as flat so that the liquid domain of height h moves as a plug. The first assumption is valid in fluid flow applications where low frictional drag is important because the lubrication effect of SHPo surface increases as its solid fraction approaches zero [22]. In the case of water on fluoropolymer surface, the assumption of a flat meniscus is realistic because the advancing contact angle is 110 deg [23]. However, the assumption breaks down for, e.g., water on surfaces with re-entrant structures where the contact angle can be as high as 150 deg [24,25]. We change to a reference frame that moves with the piston such that y = 0 corresponds to the moving, flat meniscus. The governing equation becomes
$∂ρg∂t=Dgl∂2ρg∂y2$
(16)
Although the dissolved gas concentration field is assumed to be one-dimensional, meniscus curvature is captured as per the Young–Laplace equation, Eq. (2), in the calculation of gas pressure in the cavity and, subsequently, the volume of the cavity. Application of the ideal gas law yields
$pg(t)Vg′(t)=mg,c′(t)MgRuT$
(17)
where $mg,c′$ is the residual mass of gas in the cavity per unit depth, $Ru=8.314 J mol−1 K−1$ is the universal gas constant, Mg is the molar mass of the gas, $Vg′$ is the volume of gas in the cavity per unit depth, and T is the absolute temperature. In subsequent sections, mass and volume quantities are normalized to the depth into the page. It follows that
$Vg′(t)=a s0+∫0aym(x,t)dx$
(18)
The integral evaluates to
$Vg′(t)=a s0−a s(t)−12[R(t)2 arcsin[aR(t)]−a R(t)2−a2]$
(19)

where R(t) can be expressed in terms of pg (t) as per Eq. (2).

Given that the meniscus boundary condition is time-dependent until critical time, after which it remains constant, we split the problem into two parts: (a) before critical time and (b) after critical time. Boundary condition (11) is replaced by Eq. (15) at tcr and, too, initial condition (9) is replaced by the computed dissolved gas concentration field. In part (a) of both case studies, Duhamel's theorem may be used to determine the partial density of the gas in the liquid in the presence of the time-dependent boundary condition given by Eq. (11) [26]. The standard form of Duhamel's theorem requires a homogeneous Dirichlet initial condition, so the dependent variable is changed to
$ρg*=ρg−ρg,0$
(20)
The governing equation becomes
$∂ρg*∂t=Dgl∂2ρg*∂y2$
(21)
where
$ρg*(y,0)=0$
(22)

Case Study I: Piston Exerts Pressure on Liquid

Assuming that the atmospheric pressure is much greater than the hydrostatic pressure, the pressure in the gas cavity becomes
$pg(t)=p∞−σR(t)$
(23)

where pg is initially the same as the liquid pressure, i.e., pg (0) = p. It follows that the meniscus is initially flat, i.e., R(0) =  and θ(0) = π/2.

Before Critical Time.

The one-dimensional forms of Eqs. (6) and (8) are prescribed to account for the dissolved gas concentration at the meniscus and the impermeability of the massless piston at the top of the liquid domain, respectively, as per
$ρg*(0,t>0)=pg(t)H−ρg,0$
(24)
$∂∂yρg*(h,t>0)=0$
(25)
Duhamel's superposition integral consists of a summation term, which captures the effect of discontinuities in the boundary condition, and an integral term, which captures those of transient, but continuous, portions [26]. The gas phase pressure varies continuously with time; hence, only the latter is relevant here, except at t = 0. It follows that
$ρg,bc*(y,t)=ρg*(0,0+)Φ(y,t)+∫0tΦ(y,t−τ)dρg*(0,τ)dτdτ$
(26)
where the subscript bc denotes the period of the solution before critical time, 0+ denotes a time > 0, but infinitesimally close to zero, τ is a dummy variable and Φ(y, t) is a solution to an auxiliary problem, where the time-dependent boundary condition is replaced by a step input as per
$∂Φ∂t=Dgl∂2Φ∂y2$
(27)
$Φ(0,t>0)=1$
(28)
$∂∂yΦ(h,t>0)=0$
(29)
$Φ(y,0)=0$
(30)
The solution to this auxiliary problem is
$Φ(y,t)=1−2h∑n=0∞1λn sin(λny)exp[−λn2Dglt]$
(31)
where
$λn=2n+12hπ, for n=0,1,2,…$
(32)
Substituting Eq. (26) into Eq. (20), the solution to the original problem is
$ρg,bc(y,t)=ρg,0+ρg*(0,0+)Φ(y,t)+H∫0tΦ(y,t−τ)dpg(τ)dτdτ$
(33)
We discretize Eq. (33) explicitly because the boundary condition along the meniscus (at y = 0) is not prescribed a priori. Instead, it must be computed from an expression for the mass of gas in the cavity at discrete time-steps, $mg,c′(tj−1)$. In discrete form, Eq. (33) becomes
$ρg,bc(y,tj)=∑i=0j[ρg*(0,τi)−ρg*(0,τi−1)]Φ(y,tj−τi)+ρg,0$
(34)

where $ρg*(0,τ−1)=0$ to account for the (discontinuous) change in the partial density of the gas along the meniscus from ρg,0 to pg (0)H at τ0 = 0+. Overall, the time-varying dissolved gas concentration on the liquid side of the meniscus is a continuous function, ρg (0, t), as depicted by the thick solid line in Fig. 2. This function has been approximated by discrete values evaluated at times τ1, τ2,…, τj−1, τj where j denotes the time-step.

Fig. 2
Fig. 2
Close modal
After the first time-step, i.e., when j = 1
$ρg,bc(y,t1)=ρg,0+[pg(0+) H−ρg,0]Φ(y,t1)+H[pg(τ1)−pg(0+)] Φ(y,t1−τ1)$
(35)

where pg (0+) = p and the last term on the right-hand side, i.e., corresponding to i = j = 1 or t1 − τ1 = 0, is zero as per the initial condition.

The mass of gas that was initially in the cavity and is now dissolved in the liquid can be computed by integrating Eq. (35) from y = 0 to y = h and multiplying by a. It follows that
$mg,c′(t1)=a[ρg,0−p∞H][h−∑n=0∞2 exp(−λn2 Dgl t1)λn2 h]+m0′$
(36)
where $mg,c′$ and $m0′$ are the current and initial mass of gas in the cavity, respectively. It also follows from the ideal gas law that
$m0′=p∞a s0MgRuT$
(37)
The gas pressure in the cavity at time t1, for use in the second time-step, is determined by substituting Eqs. (19) and (36) into the ideal gas law. This yields
$mg,c′(t1)=pg(t1)Mg aRuT[s0+R(t1)2−a2]−pg(t1)Mg2 RuT{R(t1)2 arcsin[aR(t1)]}$
(38)
where R(t1) is related to pg (t1) as per Eq. (23) and pg (t1) is computed numerically. Proceeding to the second time-step, we find that
$ρg,bc(y,t2)=ρg,0+[pg(0+) H−ρg,0]Φ(y,t2)+H [pg(τ1)−pg(0+)]Φ(y,t2−τ1)$
(39)

where pg (τ1) = pg (t1).

The foregoing approach is followed in order to compute the dissolved gas concentration field in the liquid at the end of subsequent time-steps, i.e., (a) the ideal gas law is used to compute the new gas pressure in the cavity, thus accounting for the total mass of gas that is dissolved into the liquid during the previous time-step and (b) then, the new gas pressure is used in Duhamel's superposition integral to compute pg. At the beginning of the jth time-step, i.e., at tj−1, the total mass of gas in the cavity is
$mg,c′(tj−1)=m0′−h a ∑i=0j−2[ρg*(τi)−ρg*(τi−1)]+2 a ∑i=0j−2[ρg*(τi)−ρg*(τi−1)]∑n=0∞ exp[−λn2Dgl(tj−1−τi)]λn2h$
(40)
where the summation on the right-hand side is up to i = j − 2 because the (j − 1)th term evaluates to zero. Equation (40) is solved numerically. A sufficient number of terms in the infinite series are retained in order to satisfy the convergence criterion
$|mg,ℓ′(tj,N)−mg,ℓ′(tj,N−1)mg,ℓ′(tj,N)|≤10−5$
(41)

where $mg,ℓ′=m0′−mg,c′+a h ρg,0$ is the mass of gas in the liquid domain and N denotes one plus the number of terms in the sum over n.

Substituting Eq. (19) into the ideal gas equation, Eq. (17), the equation governing pg takes the form
$m0′−h a ∑i=0j−2[ρg*(τi)−ρg*(τi−1)]+2 a ∑i=0j−2[ρg*(τi)−ρg*(τi−1)]∑n=0∞ exp[−λn2Dgl(tj−1−τi)]λn2h=−pg(tj−1)Mg2 RuT{R(tj−1)2 arcsin[aR(tj−1)]}+pg(tj−1)Mg a RuT[s0+R(tj−1)2−a2]$
(42)

where pg (tj−1) is the gas pressure in the cavity for the interval tj−1 < t < tj and R(tj−1) can be expressed in terms of pg (tj−1) as per the Young–Laplace equation. The variable pg (tj−1) is then the only unknown and it is computed numerically and then substituted into Eq. (34) to compute the dissolved gas concentration field, ρg,bc (y, tj). If, at the end of time-step j, the total mass of gas left in the cavity is such that pg (tj) > pcr, then ρg,bc (y, tj+1) is computed in a similar manner. Conversely, if pg (tj) < pcr, then the critical time is located in the interval tj < t < tj+1 in which case the dissolved gas concentration field is computed using the “after critical time” model.

Critical time depends on the geometry of the structures, height of the liquid domain, and initial dissolved gas concentration in the liquid. If the liquid solution is saturated with gas before critical time, then there is no diffusion-induced loss of gas in the cavity. It follows that the gas pressure in the cavity and meniscus deflection are constant as per the ideal gas law and Young–Laplace equations, respectively.

After Critical Time.

Once the advancing contact angle is reached, the pressure difference between the liquid and gas phases remains constant as per Eq. (14). The triple contact line moves down the ridges to account for the mass of gas leaving the cavity maintained at a constant pressure [17]. The governing equation and boundary conditions are
$∂ρg,ac∂t =Dgl∂2ρg,ac∂y2$
(43)
$ρg,ac(0,t−tcr)=pcrH$
(44)
$∂∂yρg,ac(h,t−tcr)=0$
(45)
where the subscript ac denotes the dissolved gas concentration field after critical time. The initial condition, Eq. (46), is the Fourier series representation of the previous problem, evaluated at critical time, in the form
$ρg,ac(y,t−tcr)=pcr H−∑n=0∞Bn sin (λny), for t=tcr$
(46)
where Bn are the Fourier coefficients of Eq. (34) at the time instance j = jcr, i.e.,
$Bn=2Hλn h∑i=0jcr[pg(τi)−pg(τi−1)] exp[−λn2Dgl(tcr−τi)]$
(47)
The solution after critical time is then given by
$ρg,ac(y,t−tcr)=−∑n=0∞Bn sin (λny)exp [−λn2Dgl(t−tcr)]+pcrH$
(48)
The residual mass of gas in the cavity is given by the difference between the initial mass of gas in the cavity, $m0′$, and the mass of gas in the liquid domain as per
$mg,c′(t−tcr)=m0′−pcrH a h+a∑n=0∞Bnλn exp[−λn2Dgl(t−tcr)]$
(49)

Longevity.

Cassie to Wenzel state transition occurs when the meniscus reaches the bottom of the gas cavity as depicted in Fig. 1(a) at the time tf. The volume of the gas cavity at t = tf in terms of the advancing contact angle and width of the cavity is found from geometry to be
$Vf′=14a2 sec2θA[sin 2θA−4 cos θA−2θA+π]$
(50)
The residual mass of gas in the cavity at tf is determined by substituting $Vf′$ into the ideal gas law as per
$mf′=pcrVf′MgRuT$
(51)
Longevity is then computed by equating $mf′$ with $mg,c′$, which is given by Eq. (49), as per
$mf′=m0′−pcrH a h+a∑n=0∞Bnλn exp[−λn2Dgl(tf−tcr)]$
(52)

and solving for tf.

The longevity described in Eq. (52) requires a numerical computation of tcr as per the solution of Eq. (42). When that critical time tcr is much smaller than the longevity tf, we can assume that the gas pressure in the cavity is constant from t = 0 to remove the time dependence of boundary condition (11). Doing so leads to an analytical series solution of the form Eq. (31), and
$mf′=m0′−a hp∞ H[1−∑n=0∞2λn2 h2 exp[−λn2Dgl tf]]$
(53)
In the model described herein, transition to Wenzel state occurs only when the liquid height is greater than $hmin$, which is defined as the minimum liquid column height that has the capacity to hold $m0′−mf′$ mass of gas in a saturated solution. For the above-mentioned further simplified model with tcrtf, this is given by
$hmin=m0′−mf′ a (p∞H−ρg,0)$
(54)

Case Study II: Liquid Exposed to Atmospheric Pressure

We adapt our mathematical model to capture the effects of hydrostatic pressure in the absence of the massless, impermeable piston as shown in Fig. 1(b). The gas in the cavity is initially at atmospheric pressure, but it instantaneously undergoes compression under the weight of the liquid to a pressure of pg,0. Assuming that it is a quasi-equilibrium process
$p∞(a s0)γ=pg,0Vg,0′ γ$
(55)
where γ = 1 for isothermal compression, γ is the ratio of the gas' specific heat capacity at constant pressure to that at constant volume for adiabatic compression and $Vg,0′$ is the volume of gas per unit depth in the cavity at t = 0. The latter can be expressed in terms of initial radius of curvature (R0) as
$Vg,0′=a s0−12[R02 arcsin(aR0)−a R02−a2]$
(56)

The pressure of the compressed gas can be computed numerically by substituting Eq. (2) into Eq. (56) and substituting the resulting expression into Eq. (55). The temperature in the gas cavity may rise due to the moving boundary work done on the gas, therefore, we considered both adiabatic and isothermal compression.

Before Critical Time.

There are two menisci where gas diffusion can occur, namely, at y = 0 and at y = h, as per
$ρg*(0,t)=pg(t)H−ρg,0$
(57)
$ρg*(h,t)=p∞H−ρg,0$
(58)
The solution to $ρg*(y,t)$ is a superposition of two transient problems, $ρI*(y,t)$ and $ρII*(y,t)$, where each problem contains only one inhomogeneous boundary condition as per
$ρg*(y,t)=ρI*(y,t)+ρII*(y,t)$
(59)
where
$ρI*(0,t)=pg(t)H−ρg,0$
(60)
$ρI*(h,t)=0$
(61)
and
$ρII*(0,t)=0$
(62)
$ρII*(h,t)=p∞H−ρg,0$
(63)
The solution to $ρII*$ is obtained using separation of variables while that to $ρI*$ is determined using Duhamel's theorem where Φ(y, t) is the solution to the corresponding auxiliary problem as per
$ρI*(y,t)=[pg(0+)H−ρg,0]Φ(y,t)+H∫0tΦ(y,t−τ)dpg(τ)dτdτ$
(64)
The governing equation of Φ(y, t) is
$∂Φ∂t=Dgl∂2Φ∂y2$
(65)
subjected to
$Φ(0,t)=1$
(66)
$Φ(h,t)=0$
(67)
$Φ(y,0)=0$
(68)
The solution to the auxiliary problem is
$Φ(y,t)=(1−yh)−∑n=1∞2βn h sin(βn y)exp(−βn2 Dglt)$
(69)
where
$βn=πnh, for n=1,2,…$
(70)
For the interval tj−1 < t < tj (time-step j), $ρI*$ is given by
$ρI*(y,t)=∑i=0j−1[ρg*(τi)−ρg*(τi−1)]Φ(y,t−τi)$
(71)
where $ρg*(t)=pg(t)H−ρg,0$ is unknown a priori. The solution for $ρII*$ is
$ρII*(y,t)=(p∞H−ρg,0)yh+∑n=1∞Cn sin(βn y)exp(−βn2 Dglt)$
(72)
where
$Cn=2βnh[p∞H−ρg,0]cos(πn)$
(73)
Therefore, the dissolved gas concentration in the liquid is given by
$ρg,bc(y,t)=ρg,0+ρI*(y,t)+ρII*(y,t)$
(74)
In case study II, the mass of gas in the cavity cannot be computed from the approach that was employed in case study I, i.e., integrating over the liquid domain, because gas molecules can enter the liquid domain through two menisci. Instead, the mass of gas in the cavity for case study II is computed by subtracting the total mass of gas that has crossed the meniscus from the initial mass of gas in the cavity as per
$mg,c′(tj)=m0′+a Dgl∫0tj(∂ρI*∂y|y=0+∂ρII*∂y|y=0)dt$
(75)
where the convergence criterion is given by Eq. (41). Substituting Eqs. (71) and (72) into Eq. (75) yields
$mg,c′(tj)=m0′−a Dgl∑i=0j−1[ρg*(τi)−ρg*(τi−1)][∑n=1∞2βn2 Dglh{ exp[−βn2 Dgl (tj−τi)]−1}−tj−τih]−a Dgl{(p∞H−ρg,0)tjh−∑n=1∞Cnβn Dgl [exp(−βn2 Dgl tj)−1]}$
(76)
Substituting Eq. (76) into the ideal gas law, the governing equation for the pressure in the gas cavity for time-step j is
$mg,c′(tj−1)=pg(tj−1)Mg RuT{a s0+a R(tj−1)2−a2}−pg(tj−1)Mg2RuTR(tj−1)2 arcsin[aR(tj−1)]$
(77)

with $mg,c′(tj−1)$ given by Eq. (76), but evaluated at tj−1. The pressure of gas, pg (tj−1), is computed numerically, then ρg (y, tj) is computed from Eqs. (71), (72), and (74).

After Critical Time.

The initial condition of the solution after critical time is the computed dissolved gas concentration field before critical time. Equation (74) is rearranged and evaluated at t = tcr to give
$ρg,bc(y,tcr)=pcrH+H(p∞−pcr)yh+∑n=1∞En sin(βn y)$
(78)
where
$En=−2∑i=0jcr[ρg*(τi)−ρg*(τi−1)] exp[−βn2 Dgl(tcr−τi)]βn h+Cn exp(−βn2 Dgl tcr)$
(79)
The governing equation and boundary conditions after critical time are
$∂ρg,ac∂t =Dgl∂2ρg,ac∂y2$
(80)
$ρg,ac(0,t−tcr)=pcrH$
(81)
$ρg,ac(h,t−tcr)=p∞H$
(82)
Using separation of variables method, the solution after critical time is
$ρg,ac(y,t−tcr)=∑n=1∞En sin(βny)exp[−βn2Dgl(t−tcr)]+pcrH+H(p∞−pcr)yh$
(83)

Longevity.

Transition to Wenzel state occurs when the residual mass of gas in the cavity becomes equal to $mf′$ (given by Eq. (51))
$mf′=mg,c′(tcr)+a Dgl∫tcrtf ∂ρg,ac(y,t−tcr)∂y|y=0 dt$
(84)
which evaluates to
$mf′=mg,c′(tcr)−a∑n=1∞Enβn{ exp[−βn2Dgl(tf−tcr)]−1}+a Dgl(p∞−pcr)(tf−tcr)Hh$
(85)

Results and Discussion

The before and after critical time solutions described herein were implemented using MATLAB® for both case studies. The gas in the cavity and above the liquid was assumed to be 100% nitrogen, Mg = 0.0280 kg/mol [27], and the liquid to be water. Thermophysical properties, evaluated at an ambient temperature of 25 °C and at pressure p = 1 atm, were set to the following values: $H=1.82×10−7 kg/(m3Pa)$ [27], $Dgl=2.01×10−9 m2/s$ [28], and σ = 0.073 N/m [29].

In order to demonstrate the functionality of the model presented in case study I, a sample set of dimensions and initial condition of $s0=10 μm, a=5 μm, h=800 μm$ and ρg,0 = 0 was considered. The step sizes before and after critical time were set to 0.001 s and 1 s, respectively. Figure 3 shows that the gas pressure in the cavity decreases exponentially from a maximum pressure of p to pcr at tcr = 1.82 s. The longevity is 176.7 s; therefore, tcr is 1% of tf and thus insignificant. Nevertheless, it is important to use the “before critical time” model for applications where the meniscus is always stable, for example, when ρg,0 = 0.018 kg/m3 as shown in Fig. 3.

Fig. 3
Fig. 3
Close modal

The computed dissolved gas concentration fields at time tcr and tf are presented in Fig. 4 where two cases are compared: (a) a simplified case where the gas pressure is held constant at pg (t) = p and (b) a case where the gas pressure is allowed to decrease from p to pcr during the interval 0 < t < tcr as depicted in the plot of pg (t) in Fig. 3. It follows that, in the latter case, within the interval 0 < t < tcr, the surface boundary condition decreases from pH to pcrH as shown in the inset of Fig. 5. In the vicinity of the meniscus, the dissolved gas concentration gradient is initially steep and surface concentration is at its maximum (pH). Gradually, the dissolved gas concentration gradient flattens as the gas diffuses into the bulk of the liquid domain and as the surface concentration decreases. It is for that reason that the plot of ρg (y, 0.001 s) crosses that of ρg (y, 1.82 s) in Fig. 5.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

The mass of gas in the cavity, $mg,c′$, is presented in Fig. 6. It decreases with time until complete wetting is achieved. Figure 7 presents the longevity for the case at hand as a function of h. The effect of h on tf is shown to be more important when the height is close to the minimum height $hmin$; longevity approaches infinity as h approaches $hmin$ from the right. Consequently, it is possible to achieve infinite longevity, i.e., stability, when $h.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

Case study II accounts for hydrostatic pressure and gas diffusion at the exposed top surface. The dimensions s0 = 85 μm, a = 73.5 μm, and h = 16.5 cm were chosen based on an experimental study by Xu et al. [11]. The temperature rise in the gas cavity as a result of adiabatic compression evaluates to only 1.3 K and the difference in initial gas pressure in the cavity is 0.01% as per Eq. (55) where γ = 1.4; therefore, the process was assumed to be isothermal. For samples of the same geometry, longevity varies with the initial dissolved gas concentration as shown in Fig. 8. As the initial gas concentration approaches 0.8 pH, critical time and longevity approach 25 min and 48 h, respectively. For initial gas concentration greater than 0.8pH, the rate of gas diffusion across the meniscus is so slow that the meniscus can be maintained in the Cassie state for indefinite periods. Xu et al. measured critical time and longevity, on a single trench submerged under a water column of the same dimensions and ρg,0 = pH, to be about 2 h and 10 h, respectively [11]. The reason for the differences is that the model in case study II is applicable only to SHPo surfaces with low solid fraction, whereas the experimental setup in Xu et al. implies large solid fraction.

Fig. 8
Fig. 8
Close modal

Figure 9 shows that tcr decreases with the increasing cavity depth while tf increases. The cavity depth is limited to a minimum of 15 μm to prevent the sagging meniscus from contacting the substrate before critical time. For s0 = 15 μm, the triple contact line depins at tcr = 110 s and transition to Wenzel state starts almost immediately, at tf = 170 s. This highlights another extreme case where it is necessary to consider the time-dependent boundary condition for 0 < t < tcr. However, tcrtf for a deep cavity, i.e., s0 = 85 μm, where tcr = 89 s and tf = 2.7 h. In a deeper cavity, the initial mass of gas in the cavity is larger, hence it takes less time to reach critical time, but more time to deplete the gas in the cavity through gas diffusion.

Fig. 9
Fig. 9
Close modal

Validation

The computed results before and after critical time were validated using the partial differential equation Toolbox in MATLAB®. In this case, Eq. (21) subjected to the boundary conditions (24) and (25) was solved iteratively using a finite element method and the solution process was as follows. At the time instance tn (t0 = 0 s), the algorithm computes the pressure of the gas pg (tn) from Eq. (38) using the current mass of the gas in the cavity $mg,c′(tn)$, where $mg,c′(t0)=m0′$. If the computed pg (tn) < pcr, pg (tn) is set equal to pcr for all subsequent time-steps. Next, the algorithm updates the boundary condition (24) and Eq. (43) is solved over one time interval Δt, i.e., until tn+1 = tn + Δt, to compute the new dissolved gas concentration field in the liquid ρg (y, tn+1). Consequently, the algorithm integrates ρg (y, tn+1) over the domain to compute the total mass of the gas that has diffused into the liquid from t0 until tn+1 denoted as $mg,ℓ′(tn+1)$. Thence, the net change of this mass of the diffused gas over the current time-step $dmg,ℓ′(tn)$ is computed by subtracting $mg,ℓ′(tn)$ from $mg,ℓ′(tn+1)$. It is noted that $mg,ℓ′(t0)=0$. Next, $mg,c′(tn+1)$ is computed by subtracting $dmg,ℓ′(tn)$ from $mg,c′(tn)$. Then, $mg,c′(tn+1)$ is set equal to $mg,c′(tn)$ and the solution process is repeated until the maximum specified time instance. The domain was discretized with 1920 elements. The time-step was set equal to 5 × 10−4 s and 5 × 10−2 s for tn < 2 s and tn ≥ 2 s, respectively, i.e., equal to the half of the corresponding time-steps that were used in the semi-analytical analysis. For case study I, the results are in excellent agreement, with the discrepancies for tcr and tf to be less than 0.09% and 0.14%, respectively, and the maximum discrepancy for $mg,ℓ′$ found to be less than 0.49%. For case study II, as represented in Fig. 9, the discrepancies for tcr and tf were less than 1.15% and 0.9%, respectively.

Conclusion

We present a semi-analytical, 1D, transient model for gas diffusion-induced Cassie to Wenzel state transition on ridge-type structured surfaces with low solid fractions. Our model accounts for meniscus curvature insofar as surface tension forces and gas cavity volume. For water on fluoropolymer-coated surfaces, it is sufficient to assume 1D gas diffusion because the meniscus is almost flat given that the advancing contact angle of water on a fluoropolymer-coated silicon surface is close to 110 deg. In addition, our model captures the effects of time-dependent surface concentration on the meniscus, hydrostatic pressure, and initial dissolved gas concentration on the longevity of SHPo surfaces. In a future publication, we will extend the analysis to 2D liquid domains in order to fully capture the effects of meniscus curvature and solid fraction.

Based on results from selected case studies, we demonstrate that, with appropriate dimensions and initial conditions, it is possible to stabilize the meniscus in a pinned or depinned Cassie state (metastable) for long periods if the liquid domain reaches saturation. We then conclude that precharged liquid domains combined with SHPo surfaces that have deep cavities tend to last longer in the Cassie state. We see potential applications of this 1D solution in the design of water-based microgap cooling systems where the primary concern is to optimize channel pressure drop, longevity, geometry, and materials for superior cooling performance.

Acknowledgment

The work of Toby Kirk was supported by an EPSRC doctoral scholarship. The computations in this paper were executed on the Tufts High-Performance Computing Research Cluster at Tufts University.

Funding Data

• National Science Foundation (1402783)

Nomenclature

a =

width of the gas cavity (m)

A =

surface area (m2)

Bn =

nth coefficient of the series for ρg,ac (kg/m3), case I

Cn =

nth coefficient of the series for ρg,ac (kg/m3), case II

d =

pitch of parallel ridges (m)

Dgl =

molecular diffusivity of gas in liquid (m2/s)

g =

acceleration due to gravity (m/s2)

h =

height of liquid domain above triple contact line (m)

H =

Henry's constant (kg m−3 Pa−1)

M =

molar mass (kg/mol)

$m′$ =

mass per unit depth (kg/m)

n =

positive integer

N =

positive integer

p =

pressure (Pa)

p =

atmospheric pressure (Pa)

R =

Ru =

universal gas constant (J mol−1 K−1)

s =

distance moved by the triple contact line (m)

t =

time (s)

T =

absolute temperature (K)

$V′$ =

volume per unit depth (m3/m)

Greek Symbols

Greek Symbols
β =

eigenvalue

γ =

exponent for adiabatic or isothermal compression

θ =

λ =

eigenvalue

$ξ¯$ =

average invasion coefficient (m2 s/kg)

ρ =

density of liquid phase (kg/m3)

ρg =

partial density of gas in liquid phase (kg/m3)

ρsat =

equilibrium concentration of gas (kg/m3)

$ρg*$ =

instantaneous partial density of gas minus the initial partial density of gas in the liquid phase (kg/m3)

σ =

surface tension (N/m)

τ =

time variable (s)

Φ =

auxiliary solution

Subscripts

Subscripts
A =

ac =

after critical time

bc =

before critical time

c =

center of circular arc

cr =

critical time

f =

final time

g =

gas phase

g, c =

gas in cavity

i =

ith term

j =

jth term

=

liquid phase

m =

meniscus

min =

minimum value of quantity

n =

nth term

0 =

initial quantity

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