Article pubs.acs.org/Langmuir
Liquid−Infused Surfaces with Trapped Air (LISTA) for Drag Force Reduction A. A. Hemeda and H. Vahedi Tafreshi* Department of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond, Virginia 23284-3015, United States
ABSTRACT: Superhydrophobic (SHP) surfaces are known for their drag-reducing attributes thanks to their ability to trap air in their surface pores and thereby reduce the contact between water and the frictional solid area. SHP surfaces are prone to failure under elevated pressures or because of air-layer dissolution into the surrounding water. Slippery liquid-infused porous surfaces (SLIPS) or liquid-infused surfaces (LIS) in which the trapped air is replaced with a lubricant have been proposed in the literature as a way of eliminating the air dissolution problem as well as improving the surface stability under pressure. While an LIS surface has been shown to reduce drag for flow of water−glycerol mixture (ref 18), no significant drag reduction has yet been reported for the flow of water (a lower viscosity fluid) over LIS. In this concern, we have designed a new surface in which a layer of air is trapped underneath the infused lubricant to reduce the frictional forces preventing the LIS to provide drag reduction for water or any fluid with a viscosity less than that of the lubricant. Drag reduction performance of such surfaces, referred to here as liquidinfused surfaces with trapped air (LISTA), is predicted by solving the biharmonic equation for the water−oil−air three-phase system in transverse grooves with enhanced meniscus stability thanks to double-reentry designs. For the arbitrary dimensions considered in our proof-of-concept study, LISTA designs showed 20−37% advantage over their LIS counterparts. contacts with viscous fluids like crude oil, such surfaces are not likely to show a measurable drag reduction when the working fluid is a low-viscosity fluid like water.18 LIS may also suffer from additional problems such as lubricant drainage due to shear or gravitational forces as examined in the recent LIS literature.23−26 In the current paper, we present a new design for LIS in which an air layer is placed underneath the infused lubricant to improve the drag-reduction benefits of the surface when used with low-viscosity fluids like water. The air layer is expected to reduce the frictional forces acting against the formation of a vortical flow in the lubricant layer, and therefore allow the surface to provide a slip velocity at the lubricant− water (working fluid) interface (LWI). Inspired by a recent study reported in ref 27, a double-reentry geometry is considered in our design to enhance the mechanical stability of the lubricant layer (Figure 1) (see also ref 28). For the sake of brevity, a lubricant-infused surface with trapped air is referred
1. INTRODUCTION A surface promoting an apparent water contact angle of 150° or higher is often referred to as a superhydrophobic (SHP) surface.1 This is in part thanks to the peculiar ability of such a surface to trap air in its surface pores. SHP surfaces can potentially be used for drag reduction and/or self-cleaning applications among many others.2−5 The SHP surfaces designed for underwater drag reduction applications are often affected adversely by the water hydrostatic pressure or the time in service.6−13 Excessive hydrostatic pressures can imbalance the mechanical forces acting on the air−water interface (AWI) that forms over a SHP surface. In addition, the dissolution of the trapped air into the surrounding water can lead to the collapse of an AWI over time. 11−17 In this concern, investigators in ref 18 used a lubricant to impregnate the pores of their drag-reducing surface and thereby they extended the surface lifespan almost indefinitely. Such surfaces have been referred to as slippery liquid-infused porous surfaces (SLIPS) or lubricant-infused surfaces (LIS) and were first used in applications such as antifouling and anticoagulation.19−22 Despite the success of LIS surfaces in providing slippery © XXXX American Chemical Society
Received: December 30, 2015 Revised: February 23, 2016
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DOI: 10.1021/acs.langmuir.5b04754 Langmuir XXXX, XXX, XXX−XXX
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counterpart are discussed in detail. Our numerical calculations are conducted for a Couette flow formed between a moving upper plate and a stationary bottom plate. The bottom plate is composed of transverse parallel grooves impregnated with lubricant (LIS) or lubricant and air layers (LISTA). The remainder of the paper is organized as follows. Design and stability of LISTA are discussed in Section 2. Problem formulations, boundary conditions, and comparison between the present and previous studies conducted for SHP transverse grooves are presented in Section 3. Detailed comparison between the slip effects of the LIS and LISTA surfaces are discussed in Section 4 for a Couette flow with arbitrary dimensions. This is followed by our conclusions in Section 5.
2. LISTA DESIGN AND ITS MECHANICAL STABILITY In a recent study reported in ref 18, a drag reduction benefit of 16% was reported for glycerol−water mixture flowing over an LIS surface with Silicon oil as the lubricant (a viscosity ratio of 261 between the working fluid and the lubricant). The LISTA design is aimed at decreasing the skin-friction drag of the LIS surfaces and so to potentially expand their applications to working fluids with viscosities less than that of the infused lubricant (see Figure 1a−e). In addition to the low dragreduction benefits, LIS surfaces may suffer from problems involving lubricant stability. A stable LIS surface is generally designed by targeting the configuration that minimizes the total interfacial energy per unit area of the four phases involved, i.e., air, working fluid, lubricant, and solid, in comparison to all other possible configurations (see ref 26). We conjecture that the entrapped air in LISTA helps to stabilize the lubricant in the grooves. The compressibility of the air layer can potentially dampen the pressure fluctuations in the flow field and relax some of the shear forces acting on the lubricant layer (see refs 23−25 for problems with lubricant stability in LIS, and ref 29 for the effects of pressure fluctuations on the stability of submerged SHP grooves). In fact, thinner air layers better resist the elevated pressures.12,15 Generally speaking, SHP surfaces are made of hydrophobic materials. However, as shown in ref 28, superomniphobicity can be achieved with a so-called single-reentry design even with “philic” solid materials. The study reported in ref 27 shows that a surface with doubly reentry features can exhibit superomniphobic (or superrepellent) behavior regardless of the YLCA between the solid material and the working fluid. This concept can be used here to better enhance the mechanical stability of the lubricant layer in our LISTA design. For instance, a single or double reentry design can help to stabilize the air−lubricant interface (ALI) in an LISTA made preferably of an oleophobic material, except for the interior walls of the lubricant groove as shown with red color (and marked with an “s”) in Figure 1, which should be oleophilic. Depending on the wettability of the groove’s walls, the LWI and ALI may become pinned to different points at different walls. For instance, the LWI and ALI will probably anchor themselves to points A and B, respectively, if all walls are “phobic” except the s-labeled walls (see Figure 1b). However, the ALI may move along the BC wall and pin itself to point C if the BC wall is not oleophobic, as shown in Figure 1c. To better stabilize the lubricant layer inside a LISTA design exposed to negative pressures (or fluctuating pressures), one can use another double reentry design for the surface in contact with water (or working fluid) as shown in Figure 1e. This new design can promote the LWI to pin itself to points D or A
Figure 1. Schematic illustration of a unit cell of a liquid-infused surface with trapped air (LISTA) composed of transverse grooves with a double reentry inlet is given in (a). All walls are oleophobic except for the AB wall (shown in red), which must be oleophilic. The air− lubricant interface (ALI) and lubricant−water interface (LWI) are shown in (b) and (c) for oleophobic and oleophilic BC walls, respectively. Schematic illustration of a liquid-infused surface (LIS) is shown in (d). A LISTA with an additional double reentry design is shown in (e) as a means of further enhancing lubricant stability in the groove under a negative pressure.
to here as LISTA. The drag reduction performance of LISTA is examined in the current paper by solving the biharmonic equation, and the advantages of this new design over its LIS B
DOI: 10.1021/acs.langmuir.5b04754 Langmuir XXXX, XXX, XXX−XXX
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Langmuir (depending on the wettability of DA surface) when the surface is exposed a negative pressures.30−32 Note that the menisci curvatures (ALI and LWI) shown in Figure 1e are calculated from the force balance equations discussed in our previous work.30 For the sake of simplicity in the numerical calculations presented in the next sections, we assume the ALI and LWI to be flat (Figure 1a). Note that the Weber number We = and the Capillary number Ca =
τ1 σ / w1
ρV12w1 σ
(3b)
∂ 2v ∂ 2v 1 ∂P + 2 = 2 μ ∂y ∂x ∂y
(3c)
where u and v are the velocities in the x and y-directions, respectively, P is pressure, and μ is the dynamic viscosity. Using ∂ψ the stream function definition for the velocity field u = ∂y and
should be smaller than 1
(the case here) for a fluid−fluid interface to be stable in the absence of a pressure difference across the interface (V1 and τ1 are the average velocity and shear stress along the interface, respectively, ρ is the density, w1 is groove’s width, and σ is the surface tension).33,34 For the fluid−fluid interfaces discussed in this paper, We and Ca numbers are found to be O(10−8) and O(10−4), respectively. It should also be note that, the effects of surfactants or impurities on LWI or ALI are not included in the study presented here. Such contaminations are known to rigidify a fluid−fluid interface and prevent momentum transfer between different phases.35−37 This can obviously affect the performance of LISTA surfaces. The mole fraction of dissolved air in the lubricant layer adjacent to the ALI in an LISTA can be found according to Henry’s law,
Xaal =
∂ 2u ∂ 2u 1 ∂P + 2 = 2 μ ∂x ∂x ∂y
∂ψ
v = − ∂x , and by multiplying eqs 3b and 3c by their complementary partial derivatives and adding them together, we obtain
∇4 ψ = 0
where ψ is the stream function, and ∇ is the gradient operator. The flow is driven by a given shear stress at the upper wall τ∞ at y = H, where H is the gap between the plates. For simplicity, a flat profile is considered for the interface between water and the trapped fluid as mentioned earlier (see ref 41 for more information about the importance of interface curvature in slip length calculation). The no-slip boundary condition (i.e., u = v = 0) is considered for flow along the solid boundaries, i.e.,
Pa
uf =
H al
(1)
where the superscript “l” refers to lubricant, the letter “a” (either subscript or superscript) refers to air, Hal is Henry’s constant for air transport across an ALI, and Pa is the air pressure in the entrapped bubble.38,39 Neglecting capillary forces for simplicity, one can assume that Pl = Pa, and hence the Pa2
Likewise, the mole fraction of the dissolved air in the working fluid next to LWI can be written in terms of the pressure of the entrapped air bubble as,
∂x
Xalf
Pa2 al
1 = H H af
∂ 3ψf ∂x 3
∂y
∂ψf
∂ψl ∂y
= vl = −
∂x
=0
∂ψl ∂x
=0
(5a)
(5b)
=0
(6a)
=0
(6b)
Constant shear stress τ∞ and no-pressure-gradient boundary conditions are used for the upper boundary at y = H, i.e.,42
(2)
where the superscript “f” refers to the working fluid (water), and Haf is Henry’s constant for air transport across an LWI. Note that in the absence of the lubricant layer (i.e., an SHP surface simple with air-filled grooves), the mole fraction of dissolved air in the working fluid can be estimated as Paaf . In
∂ 2ψf ∂y
2
∂ 3ψf
H
∂y 3
other words, the lubricant layer in LISTA can reduce the rate of air dissolution into the working fluid (e.g., water) by a factor of Pa (about 2 to 4 orders of magnitude since Haf is about 102 to af
=
τ∞ μf
(7a)
=0 (7b)
At y = 0 along the AWI, or the LWI, the velocity and shear stress are assumed to be identical for the fluids in contact, i.e.,43−45
H
104 bar for most fluids).40
uf =
3. MATHEMATICAL FORMULATIONS AND VALIDATION In this section, we present the governing equations for the flow of water (working fluid) between a moving upper plate and a stationary hydrophobic bottom surface comprised of transverse grooves (Figure 1). The Naiver−Stokes equations for this problem can be written as ∂u ∂v + =0 ∂x ∂y
= ul =
Periodic boundary conditions (PBCs) are considered for the inlet and outlet boundaries at x = −L/2 and x = L/2, i.e., ∂ψf
H al
∂ψf
vf = −
.
air partial pressure in the lubricant becomes equal to
(4)
∂ψf ∂y
= ul =
∂ψl ∂y
2 ∂ 2ψf ∂u f lf ∂ul lf ∂ ψl = = N = N ∂y ∂y ∂y 2 ∂y 2
(8a)
(8b)
where Nlf = μl/μf is the viscosity ratio between the lubricant and the working fluid. A constant stream function is considered along the fluids’ interface line (e.g., ψ = 0). It is worth mentioning that the shear stress along the fluid−fluid interface can often be neglected if the trapped fluid is air, and this can
(3a) C
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2
∂ ψl
=
∂y 2
=
1 ∂ua N la ∂y
2
=
1 ∂ ψa 2 N la ∂y
∂ψl ∂y
= ua =
∂ψa ∂y
and
where Nla = μl/μa is the viscosity
ratio between the lubricant (subscript or superscript “l”) and air (subscript or superscript “a”). Eq 4 is discretized using a central finite difference (five-point) scheme and is solved via a successive over-relaxation (SOR) Gauss-Seidel (iterative) algorithm,48 i.e., ψin, j+ 1 = ωψin, j + (1 − ω)ψi*, j
(9a)
where ψi*, j =
1 {8(ψin+ 1, j + ψin− 1, j + ψin, j + 1 + ψin, j − 1) 20
− 2(ψin+ 1, j + 1 + ψin− 1, j − 1 + ψin− 1, j + 1 + ψin+ 1, j − 1) − (ψin+ 2, j + ψin− 2, j + ψin, j + 2 + ψin, j − 2)}
(9b)
where the indices i and j represent the spatial steps in the x and y directions, respectively, and n denotes the iteration level. The over-relaxation factor is taken to be ω = 0.8 for convenience. In order to quantify the effective slip length, the Navier slip condition is used at y = 0, i.e., ∂ψf
b̅ =
∂y ∂ 2ψf ∂y 2
(10)
where ⟨ ⟩ represents average value.46 The governing equations and boundary conditions used for LIS are also considered for LISTA. We benchmarked our work using the results reported in the fluid dynamics literature. We start by solving eq 4 for the classical lid-driven cavity problem (Figure 2a) and then move on to the case of a Couette flow with LIS with transverse grooves as the bottom surface. Figure 2b,c shows the solution of the classical lid-driven cavity problem for cavities with two different aspect ratios ar = h1/w1, where w1 and h1 are the width and height of the rectangular groove, respectively. The position of the center of the main vortex from the upper plate de is obtained from our numerical simulations and is compared to the predictions obtained from the equation given in ref 49., i.e., de = c0 erf(c1ar) w1
Figure 2. Classical lid-driven cavity problem is considered for benchmarking our numerical calculations (a). Stream function contour plots are shown for cavities with aspect ratios of ar = 1 and 2 in (b) and (c), respectively. Comparison between the results of present calculations (dotted line) and those from the analytical solution of ref 44 (solid line) are given in (d).
(11)
where c0 = 0.251 and c1 = 1.319. In the work of ref 49, the de/w1 ratios were chosen to be 0.088, 0.160, 0.240, and 0.250 for cavities with ar of 0.25, 0.5, 1.0, and 2.0, respectively. With excellent agreement with the aforementioned results (or eq 11), the de/w1 ratios are obtained from our calculations to be 0.084, 0.163, 0.240, and 0.244 for ar values of 0.25, 0.5, 1.0 and 2.0, respectively. To further benchmark our numerical calculations, we considered a Couette flow geometry similar to the one shown in Figure 1d but with air as the trapped fluid, and compared our resulting flow field with that from the analytical solution given in ref 44, i.e.,
ψex =
τ∞w12 2y/w1 [2y/w1 8μf 1 + 4Dt N fa + 4Dt N fa Im( (2x /w1 + 2iy/w1)2 − 1 )]
where τ ∞ is the applied shear at y = H, i = Dt = d0,t erf D
( ) with d π a 8d0,t r
0,t
(12)
−1 ,
= 0.124, and Nfa = μf/μa is the DOI: 10.1021/acs.langmuir.5b04754 Langmuir XXXX, XXX, XXX−XXX
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Figure 3. Stream function contour plots and velocity vector fields are shown in (a) and (b) for an LIS and its LISTA counterpart, respectively. Normalized slip velocity u/umax and velocity gradient (∂u/∂y)/(∂u/∂y)max along lubricant−water interface (LWI) are shown in (c) and (d), respectively. Here umax and (∂u/∂y)max are the maximum slip velocity and the maximum velocity gradient obtained for LIS at x = 0 and w1/2, respectively. All dimensions are in micrometers.
the bottom surface. Figure 3a,b show the velocity field for an LIS and its LISTA counterpart using streamline contour plots and velocity vectors, respectively. Note the large dark blue region in the lubricant layer of the LISTA in comparison to that in the LIS in Figure 3a, indicating a stronger lubricant circulation in LISTA. Similarly, larger velocity-vectors can be seen in the lubricant layer in the LISTA relative to those its LIS counterpart. The resulting slip-velocity along the LWI on the bottom plate is shown in Figure 3c and is normalized with the maximum slip velocity (umax = 78 μm/s) predicted for the LIS at x = 0. It can be seen that the LISTA design allows water to achieve a greater slip velocity near the bottom plate leading to a larger overall slip length. Figure 3d shows the velocity gradient ∂u/∂y along y = 0 for LIS and LISTA normalized with the maximum velocity gradient (∂u/∂y)max = 118.5 s−1 obtained for the LIS at x = −w/2 and w/2. It can be seen that overall velocity gradient (i.e., overall shear stress) across the LWI is smaller in the case of LISTA surface. This is thanks to the air layer underneath the infused lubricant, allowing it to slip along the ALI at y = −h1. Effective slip length values of b ̅ = 0.51 and 0.66 μm are obtained for water over the LIS and LISTA surfaces shown in Figure 3, respectively (29% increased slip length for LISTA). Figure 4 presents a comparison between the slip length values obtained from the above-mentioned simulations in the form of slip length gain defined as
viscosity ratio between water and air. For this comparison, we considered a computational domain with H = 500 μm, L = 3000 μm, w1 = 50 μm, and h1 = 50 μm, and used a constant shear stress of 0.1 Pa for the moving wall at y = H. Figure 2d shows excellent agreement between the streamlines obtained from our calculations (dotted lines) and those produced using the exact mathematical solution given in eq 12 (shown with solid lines). Note that the exact solution is obtained only for the water above the grooves. Therefore, these results are superimposed to our flow field calculation results, which were for both the water and air phases (see Figure 2d).
4. SLIP EFFECT FROM LISTA In this section, we compare the performance of an LIS comprised of transverse grooves with its LISTA counterpart. A Couette flow with arbitrary specifications, a height of H = 50 μm a pitch of L = 100 μm, and an upper plate shear stress of τ∞ = 0.1 Pa, is considered for this study. We consider a solid area w fraction (SAF) of φ = 1 − L1 for both LIS and LISTA surfaces. The rectangular groove encloses the lubricant film with a width of w1 = 76 μm (leading to SAF of φ = 0.24) and height of h1 = 10 μm and it is placed on the stationary bottom plate. For the double reentry design in this work, we assumed l1 = l2 = l3 = 4 μm (note that w2 = w1 + 2l1 + 2l3) and h2 = 15 μm. Dynamic viscosities of 10−3, 1.789 × 10−5, and 3.2 × 10−3 Pa·s are considered for water, air and lubricant (e.g., hexadecane C16H34), respectively. Unless otherwise stated, these parameters will be used in the remainder of this section. To predict the slip effect generated by LISTA, eq 4 is solved for water in the gap between the plates as well as for the lubricant and air in the grooves. Similar calculations are also conducted for the same Couette geometry but with the LIS as
⎛ b̅ − bLIS ̅ ⎞ E = ⎜ LISTA ⎟100 bLIS ̅ ⎝ ⎠
(13)
In this figure, slip length gain E is predicted for different groove aspect ratios ar = h1/L and SAFs φ. It can be seen that E decreases with increasing ar. It is interesting to note that, all E
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can be seen that increasing Nlf further improves the slip length of an LISTA over its LIS counterpart (note that the effective slip length of a LIS decreases with Nlf). Similarly, E increases with Nlg. This is because friction in LIS is higher for a more viscous lubricant in comparison to its LISTA counterpart. As can be seen in Figure 4b, E varies more with Nlf than it does with Nla especially at low Nlf values. Note that for air Nlg = 55 (see Figure 5).
Figure 4. Slip length gain for LISTA over LIS for transverse grooves with different aspect ratios and solid area fractions are shown in (a). μ μ Effects of the fluids viscosity ratios N lf = μ l and N la = μl (where μg is f
a
the viscosity of the entrapped gas) on the slip length gain are shown in (b). Figure 5. Slip velocity along LWI is shown in (a) for LIS and LISTA having longitudinal grooves. The velocity values are normalized using the maximum slip velocity umax obtained for LIS at x = 0. Slip length gain for LISTA over LIS with longitudinal grooves having different aspect ratios and SAFs are shown in (b). Note that these results are only accurate for longitudinal grooves with no ends.
parameters held constant, effective slip length increase with h1 for an LIS but decreases for an LISTA. This is because increasing h1 increases the area of the frictional (oleophilic) side walls of the groove relative to that of its shear-free bottom boundary. Therefore, increasing h1 results in a decrease in the slip advantage of an LISTA. Nevertheless, slip length obtained from an LISTA design is always greater than that from its LIS counterpart (E is always positive). It can also be seen that E decreases with increasing SAF, when other parameters are held constant. This is because the shear-free surface along the ALI decreases with increasing SAF, while the surface area of the frictional side walls of the lubricant cavity remain the same. Figure 4b shows slip length gain E for lubricants with different viscosities (i.e., different Nla = μl/μg where μg is the viscosity of trapped gas) versus Nlf which is the ratio of the lubricant viscosity to that of the working fluid (e.g., water). It
For completeness of the study, we also compare the performance of an LISTA comprised of longitudinal grooves with its LIS counterpart by solving a simplified form of eq 4 for both cases. Note that using 2-D formulations to describe longitudinal grooves can only be accurate when the grooves are infinitely long, or when the grooves have no ends (e.g., concentric circular grooves such as those used in the refs 5, 18, and 50). The 2-D approach may also be relevant because the lubricant is continuously pumped into the grooves in the same direction as the working fluid flows (like the LIS in refs F
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23−25). For such special cases, the Navier−Stokes equations can be simplified to ∇2u = 0, where u is the velocity in the z∂u direction (flow direction) with ∂xf = 0 at x = −L/2 and x = L/ 2. The boundary conditions along the LWI (y = 0) are uf = ul and and
∂uf ∂y ∂ul ∂y
= =
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS H.V.T. acknowledges partial support of this research from the National Science Foundation CBET (1402655) and CMMI (1029924) programs.
∂u N lf ∂yl . Similarly, the boundary conditions are ul = ua 1 ∂ua along ALI (y = −h1). We solved this differential N la ∂y
■
equation using a three-point finite different scheme and benchmarked our solution by comparing its results with the work in refs 44 and 46 (the comparison is not shown for the sake of brevity). Figure 5a shows the u-velocity near the bottom plate with longitudinal grooves in the aforementioned Couette geometry for both the LIS and LISTA designs. Similar to the case of transverse grooves, these velocities are normalized with the maximum slip velocity predicted for the LIS at x = y = 0. It can be seen that LISTA provides a greater slip velocity than its LIS counterpart. Once again, note that these results are only relevant to surfaces comprised of grooves with no ends. For longitudinal grooves with finite length, we can only conjecture that the performance will be similar or perhaps somewhat better than that of transverse grooves (more accurate assessment of this argument requires complicated 3-D calculations and will also depend on the dimensions of such longitudinal grooves). The inset in Figure 5a shows the velocity profile across the gap between the upper and bottom plates at x = 0 for a bottom plates with LIS and LISTA grooves. This figure shows how the trapped air in LISTA allows the lubricant layer to acquire a velocity comparable to that of water, in contrast to the case with LIS. Figure 5b shows a parameter study on the effects of h1 and φ on the relative slip length gain of LISTA similar to the study presented in Figure 4a. Similar to the case of transverse grooves, it can be seen that E decreases with h1 and φ.
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5. CONCLUSIONS A new design in which a layer of air is entrapped underneath the lubricant layer in a liquid-infused surface (LIS) is introduced in this paper (i.e., LISTA), and is examined using in-house numerical simulations. To further enhance the stability of the lubricant in the LISTA design, the groove’s inlet was enhanced with double-reentry geometry. The slip effect generated with our design is computed by solving the biharmonic equation for the water−lubricant−air three-phase system in transverse grooves placed in a Couette flow geometry, and is compared to that of its LIS counterpart. This paper is a proof-of-concept study to introduce the LISTA as a means of providing long-lasting measurable drag reduction effects for water (or other low-viscosity fluids) as the working fluid over an LIS surface. For the arbitrary dimensions considered in our study, LISTA designs showed 20−37% advantage over their LIS counterparts. It was also found that the drag reduction benefits of an LISTA over its LIS counterpart increases with increasing the viscosity of the lubricant. Effects of lubricant layer thickness and solid area fraction of LISTA are studied in relation to their LIS counterpart and discussed in detail.
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DOI: 10.1021/acs.langmuir.5b04754 Langmuir XXXX, XXX, XXX−XXX
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DOI: 10.1021/acs.langmuir.5b04754 Langmuir XXXX, XXX, XXX−XXX
