Article pubs.acs.org/Langmuir
Condensation Enhancement by Surface Porosity: Three-Stage Mechanism Michal Yarom and Abraham Marmur* Department of Chemical Engineering, Technion − Israel Institute of Technology, Haifa, Israel S Supporting Information *
ABSTRACT: Surface defects, such as pores, cracks, and scratches, are naturally occurring and commonly found on solid surfaces. However, the mechanism by which such imperfections promote condensation has not been fully explored. In the current paper we thermodynamically analyze the ability of surface porosity to enhance condensation on a hydrophilic solid. We show that the presence of a surface-embedded pore brings about three distinct stages of condensation. The first is capillary condensation inside the pore until it is full. This provides an ideal hydrophilic surface for continuing the condensation. As a result, spontaneous condensation and wetting can be achieved at lower vapor pressure than on a smooth surface.
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INTRODUCTION The condensation of vapor is a fundamental, ubiquitous process. In principle, condensation can occur in a pure vapor−liquid system (homogeneous nucleation);1 however, this is highly unlikely, since it requires very high supersaturation. Also, it is rare to find a system that is completely devoid of any impurities or irregularities that may serve as nucleation sites. Most natural condensation phenomena, such as the formation of dew and raindrops,2 and most industrial condensation applications, such as heat transfer,3 dehumidification, desalination,4 and distillation,5 involve heterogeneous nucleation. Thus, understanding and enhancing the nucleation process is of great importance. The presence of a solid surface, whether flat6 or curved,7 is known to enhance condensation by heterogeneous nucleation. This process, as is also well known, is usually associated with an energy barrier, which allows the growth of the nucleus and the formation of a new phase only if surpassed. The nucleus radius of curvature that is associated with the top of this barrier (maximum in the energy curve) is defined as the critical radius. If the radius of curvature of the nucleus is smaller than its critical value, then the new phase will disappear. If it gets to be larger than the critical value, then the new phase will continue to grow. The presence of the solid surface does not change the critical radius; it enhances condensation by reducing the number of vapor molecules (volume) needed to reach the critical radius.8 While condensation on smooth, flat, and curved surfaces has been well researched, the effect of other topographies of solid surfaces has not been fully explored. Since many surfaces are rough or porous, it is the objective of the present paper to determine how surface porosity affects the condensation process. As shown below, surface porosity may indeed lead to © XXXX American Chemical Society
an appreciable enhancement of condensation and does so by employing an interesting combination of mechanisms.
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MODEL SYSTEM The proposed model consists of a flat, hydrophilic solid surface with an embedded pore, as shown in Figure 1. The surface outside the pore is assumed to be ideal; namely, the contact angle equals the Young contact angle, and there is no contact angle hysteresis. The pore is supposed to simulate the local behavior of a rough surface. For simplicity, the condensing vapor is assumed to consist of a single component. The saturation ratio, SR (= PV/PS = vapor pressure/saturation pressure), is determined by setting the external pressure (which is the vapor spressure) and temperature (which determines the saturation pressure). The pore is assumed, for simplicity, to be conical. If the geometrical condition of θ + α < π/2 is fulfilled (where θ is the contact angle and α is the cone half angle), then capillary condensation takes place at a vapor pressure below the saturation pressure9 (SR < 1, Figure 1a). The meniscus of the condensed liquid in the pore has a negative (concave) and stable curvature.10,11 As the external pressure increases and as long as SR < 1, the liquid condenses inside the cone, reaching equilibrium at a height that corresponds to the critical radius, rc, associated with the set pressure. This capillary condensation process, which occurs at external (vapor) pressures below the saturation pressure, is considered to be the first stage in the overall condensation process discussed here. As the external pressure is further increased toward the saturation pressure (SR → 1), the interface reaches zero Received: June 1, 2015 Revised: July 22, 2015
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DOI: 10.1021/acs.langmuir.5b02003 Langmuir XXXX, XXX, XXX−XXX
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Langmuir
the inside wall of the pore to the outside horizontal surface. At each point along this transition, the actual contact angle between the liquid and the solid must remain the same, equal to the Young contact angle. Consequently, the apparent (macroscopic) contact angle (between the liquid−air interface and the outside solid surface) appears to change by up to π/2 − α. The beginning of the second stage is shown by line 4 in Figure 2, and the end of it is shown by line 7 in the figure, which corresponds to the end of the pore, namely, where the top, flat solid surface begins. It is worth noticing that the radius of curvature decreases during the second stage (e.g., lines 4−6 in Figure 2). The third stage involves the spreading of the drop as condensation proceeds, with the contact angle remaining the Young contact angle. This is demonstrated in Figure 2 by lines 8 and 9. The radius of curvature increases during the third stage (lines 7−9 in Figure 2). In order to thermodynamically analyze the system, the formation energy12 should be used. The formation energy describes the energy change due to the creation of a new phase and is used to determine the equilibrium, stability, and spontaneity of the phase transition. Since condensation in this case takes place at constant environmental pressure and temperature, the function to be used is the Gibbs formation energy,13,10 ΔG. It is given, for a single component, by (see Appendix 1 in the Supporting Information for the complete derivation)
Figure 1. Schematic representation of condensation on a porous surface following an increase in the saturation ratio. (a, b) First stage: capillary condensation. (b, c) Second stage: condensation on top of the pore. (d) Third stage: condensation over the solid surface.
curvature (from the negative side). Reaching SR = 1 and consequently a liquid interface with zero curvature (Figure 1b) marks the beginning of the second stage. In this state, condensation is, in principle, possible without further increases in the external pressure. However, unless the contact angle is zero, further condensation on the liquid surface requires increasing the SR to beyond 1 in order to support the increased pressure required to form a spherical cap. The SR is increased until the interface reaches the edge of the pore (Figure 1c). This marks the end of the second stage. The third and final stage describes the formation of a liquid spherical cap outside the pore perimeter, that is, on the flat surface as well as on the liquid-filled pore (Figure 1d). Figure 2 schematically demonstrates the gradual change in meniscus curvature during the second stage, as the contact line of the drop moves up along the edge of the pore corner. The underlying assumption is that the macroscopically sharp corner is actually, on a microscopic scale, a continuous transition from
ΔG = (PV − P L)V L + (μL − μ V )n L +
∑ (σ jAj − σ joAjo) j
(1)
where V, P, A, n, σ, and μ denote the volume, pressure, surface area, number of moles, liquid−vapor surface tension, and chemical potential, respectively. Superscripts L, V, and o indicate the liquid, vapor, and initial states (defined as the system under the same external conditions, before liquid condensation), respectively. Superscript j indicates the identity of the surface. The first term on the right-hand-side of eq 1 is the work term which stems from the inequality of pressures inside and outside the droplet; the next is the contribution of molecular transfer; and the last term represents the contribution of the surface energy. The formation energy describes the behavior of the new phase that is formed. As is well known, the extremum points represent the equilibrium state. The height of the energy barrier at equilibrium defines the likelihood of formation, and the extremum type determines its stability: a maximum point indicates an unstable equilibrium, while a minimum point indicates a stable one. It is important to note that the critical curvature is a function of external conditions alone (PV, T),14 independent of the solid geometry. This critical curvature can be formed only if and where the solid structure enables it. In other words, the external, intensive variables determine the critical curvature, while the solid geometry and chemistry determine the shape, size, and formation energy of the new phase nucleus. A thermodynamic analysis of the proposed model, as presented in the next section, enables an understanding of the effect of pre-existing, liquid-filled pores on the surface condensation process.
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RESULTS AND DISCUSSION The formation energy, eq 1, can be expressed as a function of the radius of curvature only,8 using the predetermined, relevant intensive variables (PV, T) and geometrical parameters. The
Figure 2. Schematic representation of the variation of contact line location and radius of curvature due to an increase in the saturation ratio. Lines 1−3: PV < PS. Line 4: PV = PS. Lines 5−9: PV > PS. B
DOI: 10.1021/acs.langmuir.5b02003 Langmuir XXXX, XXX, XXX−XXX
Article
Langmuir
Figure 3. Formation energy of water condensation on a surface with a pore for increasing saturation ratios at SR > 1. (a) As a function of liquid volume and (b) as a function of the meniscus radius. Lines 1−3: SR < SRmax (SR = 1.003, 1.004, and 1.005)). Line 4: SR = SRmax (SR = 1.007). Line 5: SR > SRmax (SR = 1.008). T = 300 K, a = 10−7 m, and α = 26.5°.
first stage that consists of capillary condensation is well known. Therefore, we focus on the formation energy above the saturation pressure, in the second and third stages, where condensation forms a convex liquid−vapor interface. This energy is given by (see Appendix 2 in the Supporting Information for the complete derivation) ⎡ VL PV RT ln S + πσ ⎢2r 2(1 − cos θ) ⎣ υL P ⎤ ⎞ 1 2 2 + 1 + r sin θ − a2⎟cos θ ⎥ 2 ⎦ ⎠ tan α
(2)
πa 3 r3 + π (1 − cos θ)2 (2 + cos θ) 3 tan α 3
(3)
To further understand the effect of the pore, Figure 4 shows a comparison of the formation energy for condensation at a
ΔG(r ) = − ⎛ − ⎜a2 ⎝
where VL =
In this equation, υL is the liquid molar volume, r is the radius of curvature of the drop, R is the universal gas constant, and T is the temperature. The initial state is defined as an empty pore of width 2a and half angle α (Figure 1). Figure 3 shows the formation energy of water condensation for a given pore and SR (>1). The formation energy is expressed as a function of the new liquid phase volume (Figure 3a) as well as of the radius of curvature of the liquid−vapor interface (Figure 3b). As the saturation ratio is set slightly above 1 (e.g., Figure 3, line 1), the formation energy plot shows two extremum points. One is a maximum and therefore an unstable equilibrium, and the other is stable (a minimum). The two states have identical critical curvature, as seen in Figure 3b that shows the energy as a function of the radius of curvature. It is possible to have two states with the same radius of curvature but different volumes because of the geometry of the pore corner, as shown in Figure 2: one extremum occurs along the corner, and the other occurs outside the pore (e.g., lines 6 and 8 in Figure 2). The critical nucleus outside the pore is unstable, and overcoming an energy barrier is required in order to get condensation. In contrast, the critical nucleus along the pore edge is stable. Further increasing the SR (Figure 3, lines 2 and 3) lowers the energy barrier until eventually an inflection point is reached (Figure 3, line 4). This state, which is formed on the upper edge of the pore, is defined here as rmin c (Figure 2, line 7). Reaching SRmax, the saturation ratio that corresponds to rmin c , is a key point in the condensation process: it represents the onset of surface wetting. At SR > SRmax (Figure 3, line 5) there is no extremum point in the energy plot; therefore, condensation (surface wetting) is spontaneous.
Figure 4. Formation energy of water condensation on smooth (dashed line) and porous (continuous lines) surfaces. T = 300 K, θ = 40°, a = 5 × 10−8m, SR = SRmax = 1.013, and α = 47.7°.
given SR on a flat, nonporous surface and on a porous surface. The classical, well-known energy plot for a smooth surface is given by the dashed line, showing a single, nonspontaneous, unstable equilibrium. Condensation requires overcoming an energy barrier, namely, having a nucleus with a radius of curvature larger than the critical one. On a porous surface, condensation is spontaneous since the same critical radius is at the limit of metastability (inflection point) and is energetically favorable to that of flat surface condensation. As mentioned above, the key stage for achieving wetting over the surface is getting to rmin by reaching SRmax. rmin is c c determined geometrically by pore width, a, and the surface hydrophilicity. Then, SRmax is calculated using the Kelvin equation. Figure 5 displays SRmax as a function of the pore width and contact angle. As the figure shows, wider pores and more hydrophilic surfaces enable the initialization of surface wetting at lower levels of supersaturation. In the case of complete wetting, namely, a Young contact angle of zero, an infinite radius will be reached only at the pore edge (on contact with the flat surface), essentially eliminating the second stage discussed above. In this case, further condensation on the liquid surface should proceed without the need to increase the SR beyond 1. This is somewhat indirectly shown in Figure 5 for a contact angle of 1° since for 0° the curve would coincide with the coordinates. C
DOI: 10.1021/acs.langmuir.5b02003 Langmuir XXXX, XXX, XXX−XXX
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Article
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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Figure 5. SRmax as a function of pore width and contact angle for water at T = 300 K.
The value of the contact angle that characterizes a surface is usually one out of a range of contact angles because of the wellknown phenomenon of contact angle hysteresis. The lowest possible contact angle is the receding contact angle, and the highest is the advancing one. Roughly speaking, the advancing contact angle is the one that determines the wetting behavior when the wetting contact line advances and vice versa. Thus, the “tendency” to wet is decreased by the contact angle being the advancing one. Of course the opposite is true for dewetting processes.
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SUMMARY AND CONCLUSIONS Heterogeneous nucleation in naturally found scratches, pits, and cracks on the surface is extremely common. Bubble nucleation in conical cavities has been previously explored, focusing on liquid−gas solutions due to its importance in biological systems.15 However, condensation that starts off in such pores has not been researched. The current paper analyzes a model system in order to understand the mechanisms of condensation on porous surfaces. The overall condensation process is divided into three distinct stages: the first is capillary condensation (occurring at pressures below saturation pressure), the second is condensation at the pore edge (occurring slightly above saturation), and the third is liquid condensation outside the pore perimeter. Thermodynamic analysis of the system shows that each stage differs in its formation energy and stability. Condensation on a porous surface is shown to be energetically favorable to that on a smooth surface. This is due to the first stage occurring before saturation and providing an ideal, hydrophilic contribution to the solid surface that is to be wet. This liquid part of the surface reduces the formation energy and therefore promotes spontaneous condensation and wetting of the surface. When comparing pore geometries and surface wettabilities, it is evident that a wider pore and a more hydrophilic surface initiate surface wetting at a lower saturation ratio.
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REFERENCES
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b02003. General formation energy. Formation energy for the proposed model. (PDF) D
DOI: 10.1021/acs.langmuir.5b02003 Langmuir XXXX, XXX, XXX−XXX
