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Nanofluid Surface Wettability Through Asymptotic Contact Angle Saeid Vafaei,†,‡ Dongsheng Wen,† and Theodorian Borca-Tasciuc*,‡ † ‡
School of Engineering and Materials Science, Queen Mary University of London, London, United Kingdom Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, New York, United States ABSTRACT: This investigation introduces the asymptotic contact angle as a criterion to quantify the surface wettability of nanofluids and determines the variation of solid surface tensions with nanofluid concentration and nanoparticle size. The asymptotic contact angle, which is only a function of gas-liquid-solid physical properties, is independent of droplet size for ideal surfaces and can be obtained by equating the normal component of interfacial force on an axisymmetric droplet to that of a spherical droplet. The technique is illustrated for a series of bismuth telluride nanofluids where the variation of surface wettability is measured and evaluated by asymptotic contact angles as a function of nanoparticle size, concentration, and substrate material. It is found that the variation of nanofluid concentration, nanoparticle size, and substrate modifies both the gas-liquid and solid surface tensions, which consequently affects the force balance at the triple line, the contact angle, and surface wettability.
1. INTRODUCTION The spreading of liquids on solid substrates is usually quantified through the contact angle and is of interest to many practical applications and industrial processes (such as coating, boiling, condensation, etc.). Extensive experimental investigations and theoretical work have been performed to understand, predict, and model the interactions of phases on the triple line, where solid, gas, and liquid meet each other. The surface wettability is believed to depend typically on several factors: (a) physical properties of solid, liquid, and gas; (b) homogeneity of solid substrate and liquid; (c) solid surface roughness. The physical properties of solid, gas, and liquid such as solid surface tensions and gas-liquid surface tension have a significant role on the force balance at the triple line and consequently the triple line behavior. The solid surface roughness and homogeneity are the main effective elements in contact angle hysteresis. Finally, the surface roughness could change the wetted area and contact angle.1-5 The mechanisms of wetting behavior of pure liquids on solids are still not completely revealed, in spite of many investigations.1 Surface wettability becomes even more complicated for mixture of base liquid and nanoparticles. Adding nanoparticles into a base liquid to formulate nanofluids not only changes the physical properties of the working fluid, but also modifies the surface wettability. Indeed, nanofluids have shown great potential to control surface wettability6-9 through the effects of nanoparticle material, shape, size, and concentration.10,11 Sefiane et al.6 investigated the spreading of aluminum-ethanol nanofluid on a hydrophobic Teflon-AF coated substrate where the variation of contact angle and velocity of the triple line with nanofluid concentration was obtained. They observed that the nanoparticles in the vicinity of the triple line could enhance the wettability for particle concentrations up to 1% by weight. Deegan et al.12,13 studied the cause of ring stains from dried coffee drops and realized that the solids dispersed in a drying drop could migrate to the edge of the drop and form a solid ring. The liquid evaporating from the edge can be refilled by liquid from the interior; therefore, the resulting outward flow could carry dispersed particles to the edge. Their r 2011 American Chemical Society
measurement confirmed the capillary flow mechanism for contact line deposition and also predicted that the shape and thickness of the deposit could be controlled by the speed of evaporation. Shen et al.14 studied the evaporation of nanofluid droplets and the pattern of deposited polystyrene latex beads and carboxyl latex beads. The size of nanofluid droplets was in the range of 3 μm to 1 mm contact diameter. The dried patterns indicate that the suspended nanoparticles are transported to the droplet perimeter and can form a ring at the edge of the contact line when the droplet wetting diameter is larger than the threshold size. For droplets with diameters smaller than the threshold size, the suspended nanoparticles are dispersed homogeneously on the hydrophilic surfaces. The threshold size decreases with increasing particle concentration. The ring structure was found to be clearly formed for 100 nm nanoparticles; however, when droplets containing 20 nm nanoparticles were evaporated the nanoparticles distributed uniformly over the surface. Vafaei et al.7 investigated the effect of bismuth telluride nanoparticle size and concentration on surface wettability and discovered the great potential of nanoparticles to engineer wettability on the solid substrate. By solving the Young-Laplace equation, the effects of bismuth telluride nanoparticle concentration and size on the liquid-gas surface tension was determined.10 However, to fully understand the mechanisms of nanoparticles on surface wettability, it is essential to also determine the effects of nanoparticles on the solid surface tension. Surface wettability has a significant role in an array of phenomena such as multiphase fluid flow,15-17 boiling heat transfer, and critical heat flux.18-23 The effects of nanoparticles on surface wettability enhancement were examined19 and measured by equilibrium droplet contact angle on flat,11,18 round,21 and wire24 surfaces. It was realized that the wettability of the heating surface has an important role on critical heat flux enhancement of nanofluids, and as Received: July 14, 2010 Revised: December 22, 2010 Published: February 21, 2011 2211
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Figure 1. Schematic of acting forces at the triple line due to the solid surface tensions and liquid-gas surface tension.
a result, the mechanisms of surface wettability enhancement and surface wettability measurement become more and more the focus of researchers in the field. Experimental investigations could help conclude whether the main reasons for the variation of surface wettability of nanofluids are a combination of the effects of nanoparticles on gas/liquid/solid interaction at the triple line, physical properties of nanofluids, and modification of substrate. The effects of nanoparticles can be controlled by the chemical compositions of nanoparticles, gas, liquid, and solid, as well as characteristics and specifications of nanoparticles (particle size and coating properties). To recognize and distinguish the effects of nanoparticles on wettability, a precise criterion to measure the surface wettability is essential. However, to date the surface wettability is measured predominantly by the droplet contact angle method. The droplet contact angle is size dependent,25 and the inferred surface wettability will change by droplet size, so it cannot be a unique and precise criterion to measure surface wettability. The objectives of the current paper are as follows: (1) to establish a criterion to measure the surface wettability and the effects of nanoparticles on surface wettability; and (2) to demonstrate the method by investigating the effects of functionalized bismuth telluride nanoparticle based fluids on solid surface tensions and consequently the force balance at the triple line. Sections 2 and 3 provide the necessary background on triple force balance and surface wettability of nanofluids, while sections 4-8 present the asymptotic contact angle, analysis of the droplet shape, experimental setup, model validation, and experimental results and discussion.
2. TRIPLE LINE FORCE BALANCE Interfacial behavior at the triple line such as motion and contact angle depends on resultant forces on the triple line. The acting forces on the triple line are forces due to liquid-gas surface tension,σlg, solid-liquid surface tension, σsl, and solid-gas surface tension, σsg. Figure 1 shows the forces at the triple line due to solid surface tensions and liquid-gas surface tension. The gravity force is another parameter affecting the droplet contact angle. The droplet contact angle changes with contact line curvature and volume;25 consequently, it is impossible to have a unique parameter to express and evaluate the surface wettability by only measuring droplet contact angle. For an equilibrium system, a force balance method is usually proposed at the triple line to relate the macroscopic contact angle with gas-liquid-solid physical properties. A classical example is the Young’s equation, which is typically written as ð1Þ σ lg cos θo ¼ σsg -σsl where θo is the equilibrium contact angle. The Young’s equation is associated with several restrictive conditions, and its application is limited to the situations where the substrate is ideal and contact angle is size-independent.1,4 The Young’s equation cannot be applied directly except for long droplets27 or for no-gravity situations. In
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general, the left side of eq 1 is droplet size dependent through the contact angle, while the right side of the equation contains the physical properties, which make it inconsistent in many situations. The resultant force due to the solid-gas, solid-liquid, and liquid-gas surface tensions acting on the triple line has a significant role in defining the shapes of bubbles and droplets in general. The modeling of acting forces at the triple line and the dynamics of moving contact lines are still challenging tasks. The liquid-gas surface tension is tabulated for different materials; however, the solid-liquid and solid-gas surface tensions are not readily available. Several independent approaches have been employed to calculate the solid surface tensions, such as direct force measurement, contact angles, capillary penetration into columns of particle powder, sedimentation of particles, solidification front interaction with particles, film flotation, gradient theory, Lifshitz theory and van der Waals forces, and the theory of molecular interactions.4,27 Several correlations have been developed to calculate 4 the solid-liquid such as Berthelot’s combining √ surface √ tension, 28 2 rule, σsl = ( σlg - σsg) , modified Berthelot’s rule,29 σsl = σlg þ σsg - 2(σlgσsg)1/2 exp-β(σlg - σsg)2, the alternative formulation,30,31 σsl = σlg þ σsg - 2(σsgσlg)1/2(1 - β√ o(σlg 2 32,33 σ ) ), and equation of state formulation, σ = ( σlg sg sl √ 2 σsg) /(1 - 0.015(σlgσsg)1/2), where β = 0.000 115(m2/mJ)2 and βo = 0.000 105 7(m2/mJ)2. The solid-liquid surface tension calculated by these correlations has been compared against experiments for some materials and a relatively good agreement has been reached.27 The solid surface tensions were also obtained after considering the effects of adsorption of gas and liquid on solid substrate at the interface. The expressions for the solid-gas and solid-liquid surface tensions were determined by adding an equilibrium adsorption isotherm at the solid-gas interface to Gibbsian thermodynamics.34 The effect of adsorption at the solid-liquid interface was determined35 for an axisymmetric liquid-gas interface by measuring the dependency of contact angle with the liquid-phase pressure at the three-phase line. The relation between the contact angle and the radius of the contact line was investigated for small droplets, and it was found that the contact angle also depends on adsorption.36,37 Theoretical prediction of the surface tensions remains a challenging task with relatively limited results reported in the literature. Experiments are still the main avenue to quantify the surface tensions.
3. NANOFLUID SURFACE WETTABILITY The affinity of liquids for solid surfaces is referred to as the wettability of the fluid. A fixed volume of liquid will spread more over the surface as the contact angle decreases.1 Liquids with weak affinities for a solid substrate will collect themselves into beads, while those with high affinities for the solid surface will form films to maximize the liquid-solid contact area. Therefore, there is a relationship between wetting area and contact angle. As surface wettability increases, the liquid spreads more over the substrate, and for a given droplet volume, the contact angle decreases. It is also important to realize that the gravity is in favor of spreading droplet liquid over solid substrate. The liquid would spread over the substrate until the acting forces would be balanced at the triple line (see Figure 1). There have been many attempts to formulate the force balance at the triple line, including the Young’s equation; however, the effect of gravity and droplet size25 is still not modeled properly. In spite of that, the Young’s equation can be useful to interpret the force balance at the triple line, regardless of the gravity effect. The solid-gas, solid2212
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Langmuir liquid, and liquid-gas surface tensions will have a significant role in balancing forces at the triple line. In fact, variation of solid, gas, and liquid materials would change the solid-gas, solid-liquid, and liquid-gas surface tensions, and consequently the contact angle for a given droplet volume. In addition, the surface homogeneity and roughness of the substrate could be important in variation of the contact angle.5,38-41 For a smooth substrate having a contact angle over 90o, the presence of surface roughness would increase this angle, and vice versa. The surface roughness will increase the actual substrate area and increases the total liquid-solid interaction.42 Similar behavior for nanofluid droplets is expected. The nanoparticles change the gas-liquid surface tension,10 as well as the solid surface tensions. Consequently, the droplet contact angle and surface wettability will be changed. The presence of nanoparticles at the triple line can cause many complexities, where issues such as nanoparticle distribution at the interface and inside the nanofluid bulk and nanoparticle interactions with gas, liquid, and solid molecules at the triple line need to be addressed. The effects of nanoparticles on the spreading of the bubble triple line were examined15-17 by observing the formation of gas bubbles inside water and gold nanofluids on top of a stainless steel needle. It was observed that the bubble triple line can spread freely to the outer edge of the needle inserted in the water. Interestingly, the bubble triple line did not spread freely inside nanofluids but was pinned somewhere in the middle of the needle wall thickness. It was clearly observed that the radius of the triple line inside gold nanofluids was smaller than that inside water. In other words, the nanoparticles promoted the pinning behavior of the bubble triple line inside gold nanofluids. Therefore, the wetting area for a bubble inside gold nanofluids is bigger. Unlike droplets, for bubbles the gravity force is not in favor of the spreading of the triple line. Practically, the buoyancy force pushes the bubble upward and helps the bubble grow more in a vertical direction rather than laterally on the triple line. That might help to clarify the pinning behavior of the bubble triple line. The effects of nanoparticles on physical properties of nanofluids and surface wettability depend on their material, concentration, size, and shape, as well as base liquid. The nanoparticles are an effective parameter to engineer the wettability by changing the liquid-gas and solid surface tensions and consequently the variation of force balance at the triple line. To determine suitable and specific nanofluid wettability, proper combinations of environment, substrate, base liquid, and nanoparticles are required. An effective way to measure the effects of nanoparticles on the solid surface tension and consequently justify the effects of nanoparticles on surface wettability is presented below.
4. ASYMPTOTIC CONTACT ANGLE It is customary nowadays to evaluate the surface wettability by measuring the droplet contact angle. The droplet shape depends on gravity, adsorption, and physical properties of the solid, gas, and liquid. The issue is that the droplet contact angle varies with droplet size,25 for the same solid, gas, and liquid materials. A sizedependent contact angle cannot be the unique and accurate criterion to measure the surface wettability. It is necessary to introduce an accurate criterion to measure surface wettability. As the effect of gravity decreases, the droplet shape gradually changes and eventually becomes a spherical cap under the condition of zero gravity. The droplet contact angle then becomes only a function of the physical properties of solid, gas,
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and liquid, which is defined as the asymptotic contact angle θS.26 The asymptotic contact angle can be considered a physical property that explains the interactions between gas, liquid, and solid at the triple line. Indeed, in the case of no gravity, the force balance at triple line can be expressed precisely as σ lg cos θS ¼ σ sg -σ sl ð2Þ In fact, eq 2 is a precise force balance equation at the triple line and can be called the modified form of Young’s equation, assuming that the substrate is smooth and homogeneous. The right- and left-hand sides of eq 2 are size-independent, and all elements of the equation are physical properties. The asymptotic contact angle expresses a unique criterion to measure surface wettability as a function of solid, gas, and liquid physical properties, regardless of droplet size or gravity effect. For instance, the difference between asymptotic contact angle of fluid with and without nanoparticles in the same gas environment and on the same substrate is an accurate criterion to measure the effects of nanoparticles (shape, size, concentration, etc.) on surface wettability, without interference of gravity effect. The asymptotic contact angle can be measured experimentally or calculated theoretically. In this paper, the asymptotic contact angle is theoretically determined, based on experimental parameters of a droplet in normal condition (g = 9.8 m/s2). As indicated by hypothesis 2 from ref 26, “For given liquid, gas and solid materials, the interfacial force on a droplet, normal to the solid surface, is uniquely defined; in particular, this force is independent of the acceleration of gravity”. That means that the normal component of the droplet interfacial force is independent of gravity and can be written as Fni , a ¼ Fni , s ð3Þ where Fin,a is a normal component of interfacial force of an axisymmetric droplet and Fin,s is a normal component of interfacial force of a spherical droplet. Since only the liquid-gas surface tension presents a normal component, it can be written as Fni , a ¼ σgl 2πrd sin θo ð4Þ Fni , s ¼ σgl 2πrd, S sin θS
ð5Þ
where the radius of contact line at zero-gravity condition is " #1 = 3 3V ð6Þ rd, s ¼ sin θS πð2 þ cos θS Þð1 - cos θS Þ2 Knowing eqs 4-6, eq 3 can be written in following form ð7Þ rd sin θo ¼ rd, s sin θS " 3V rd sin θo ¼ πð2 þ cos θS Þð1 - cos θS Þ2
#1 = 3 sin2 θS
ð8Þ
In addition, the combination of eqs 2 and 7 expresses a new formulation at the triple line as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u rd sin θo t ð9Þ ¼ σsg - σsl σlg 1rd, s Equations 8 and 9 can be used for both nanofluids and base liquids. In general, the Young-Laplace equation can be solved to 2213
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interfaces, to obtain the bubble shape.
Figure 2. Schematic of a droplet shape and coordinates system for solving the Young-Laplace equation.
predict the droplet characteristics (e.g., droplet volume, V, contact angle, θo), knowing physical properties, and two droplet characteristics such as the radius of the contact line and the location of apex. The asymptotic contact angle, θS, can then be calculated using eq 8, with known droplet volume, V, contact angle, θo, and radius of contact line, rd.
5. THEORETICAL ANALYSIS OF DROPLET SHAPE Mathematically, the Young-Laplace equation represents a mechanical equilibrium condition between two fluids separated by an interface. The Young-Laplace equation shows that the pressure difference across the interface is equal to the product of the curvature multiplied by the gas-liquid surface tension. The Young-Laplace equation has been solved to predict the shape of axisymmetric liquid pendants and sessile drops on an ideal solid surface.4,7,10,26 In addition, the prediction of axisymmetric droplet shapes by the Young-Laplace equation has been examined by experiments,4,7,10 and a good agreement was reached. In this paper, the Young-Laplace equation was applied to predict the droplet shape. The solid surface is assumed to be perfectly smooth and homogeneous without any contact angle hysteresis. Under such conditions, the equilibrium of the interfacial pressure on the bubble surface can be described by the Young-Laplace equation. Assuming that the droplet is deposited in a quasi-steady state, the Young-Laplace equation on the interfacial surface can be written as 1 1 þ ð10Þ σ gl Δp ¼ R1 R2 where R1 and R2 are the radii of curvatures, i.e., R1 is the radius of curvature describing the latitude as it rotates and R2 is the radius of curvature in a vertical section of the droplet describing the longitude as it rotates. The center of R1 and R2 are on the same line but different location. Δp is the pressure difference between the gas, pg, and liquid phase, pl (see Figure 2), which can be written as 2σ g ln þ Po þ Fl gz ð11Þ pl ðzÞ ¼ Ro pg ðzÞ ¼ Po þ Fg gz R1 ¼ ds=dθ
and
R2 ¼ r=sin θ
ð12Þ ð13Þ
where Po is the ambient pressure, h is the hydrostatic head, and θ is the running angle. Substituting eqs 11-1213 into eq 10, the Young-Laplace equation is obtained as dθ 2 gz sin θ ¼ þ ðFl -Fg Þ ds Ro σg ln r
ð14Þ
The Young-Laplace equation can be solved, with the following system of ordinary differential equations for axisymmetric
dr ¼ cos θ ds
ð15Þ
dz ¼ sin θ ds
ð16Þ
dV ¼ πr 2 sin θ ds
ð17Þ
This system of ordinary differential equations avoids the singularity problem at the bubble apex, since sin θ 1 ð18Þ ¼ r s¼0 Ro where Ro is radius of curvature at apex. Knowing two parameters of a droplet shape (such as contact angle, radius of contact line, droplet volume, or location of the apex), the system of ordinary differential eqs 14-17 can be solved to obtain the axisymmetric droplet shape, using the following boundary conditions.4 In this study, four first-order differential equations are employed [see eqs 14-17)], therefore four boundary conditions are needed [see eq 19]. Knowing nanofluid density, Fl, gas density, Fg, and gas-nanofluid surface tension,10 σg ln, two parameters are needed to solve eqs 14-17, since the radius of curvature at apex, Ro, and where the integration has to stop are unknown. These two parameters are the radius of contact line and the droplet height. The gas-liquid surface tension of nanofluids was obtained10 by solving the Young-Laplace equation using a similar method by Rotenberg et al.43 Numerous examples of solving the Young-Laplace equation to predict the droplet4,7,10,26 and bubble15,17,44 shapes are given in the literature. rð0Þ ¼ zð0Þ ¼ θð0Þ ¼ V ð0Þ ¼ 0
ð19Þ
6. EXPERIMENTAL SETUP Contact angle measurements were performed on sessile droplets of nanofluids formulated by dispersing 2.5 and 10.4 nm bismuth telluride nanoparticles in deionized water. The experimental details were presented elsewhere on the synthesis and characterization of the 2.5 and 10.4 nm bismuth telluride nanoparticles,7,45 as well as the nanofluid preparation and contact angle measurements.7,10 In brief, the nanofluid is made by dispersing dried and functionalized bismuth telluride nanoparticle powders into pure water. The nanoparticles are functionalized with thiol groups that evaporate unless they are attached to nanoparticles, this ensuring that only ligated thiol is present in the nanofluid. The thioglycolic acid molecules remain attached to the nanoparticles unless the temperature is increased above 300 °C.7 The segregation time of the nanofluids was a couple of days (relatively very long compared with the timeline of the experiments). The nanofluids were homogenized by ultrasonic agitation for 10 min before each individual experiment. No packing, accumulation, or depletion was observed with the current apparatus. A goniometer was used to measure the droplet parameters including contact angle, θo, radius of contact circle, rd, and height of the droplet, δ. The nanofluid was injected slowly on the solid surface of the substrate by a syringe. Surface characterization of the substrates employed in this work was performed using a tappingmode atomic force microscopy (AFM). The mean roughness was found to be
