Simulation of Evaporating Droplets on AFM-Cantilevers Thomas Haschke 1, Daniel Lautenschlager 1, Wolfgang Wiechert 1, Elmar Bonaccurso 2, Hans- Jürgen Butt 2 1 University of Siegen, Faculty 11, Department of Simulation, 57076 Siegen, Germany, {haschke, lautenschlager, wiechert}@simtec.mb.uni-siegen.de, 2 Max-Planck-Institute for Polymer Research, Polymer Physics Group, Ackermannweg 10, 55128 Mainz, Germany, {bonaccur, butt}@mpip-mainz.mpg.de 1 Introduction The design of micro and nanoscale systems is a great challenge for modelling and simulation. In particular Lab-on-a-Chip-technology offers a wide spectrum of possible applications. Because micro and nano scale systems behave very differently from macro scale systems, the behaviour of physico-chemical processes on this scale is still subject to research. Hence several research projects for Lab-on-a-Chip-applications are run to understand the behaviour of these systems and get familiar with the different effects on that scale (e. g. Cµ 2002, Nano 2002). At least since Deegan s publication about drying drops of coffee in Nature (Deegan et al. 1997), evaporating micro droplets are a frequently investigated topic. Other authors dealt with special effects like the behaviour of evaporating droplets on solid surfaces (Erbil et al. 2002), fluid flow inside the drop (Hu & Larson 2002), Marangoni-Benard-instability (Nguyen & Stebe 2002) or surface morphology (Shmuylovich et al. 2002), only to mention some of them. Stupperich et al. (2005) and Cordeiro & Pakula (2005) presented simulation models for the production of microwells which takes into account some of the phenomena mentioned before from other authors. A general problem for many systems of drying droplets is the relatively short evaporation time from milliseconds up to a few seconds. So it is very difficult to study the evaporation process and related effects. Recently Bonaccurso and Butt (Bonaccurso & Butt 2005) presented a new approach to investigate droplet evaporation: They placed water droplets on rectangular, silicon atomic force microscope (AFM) cantilevers so that the resonance frequency and the inclination of the cantilever versus time could be measured (Fig. 1). With the help of these values the evaporation process can be described and conclusions about the acting forces and interactions between solid, fluid and gas can be drawn. This contribution deals with a simulation model for these experiments, the resonance frequency analysis as well as the simulation of the evaporating droplet on the cantilever. In the following chapters the experimental setup and analytical basics, the simulation model and the simulation results are presented. 2 Experimental Setup and Analytical 2D Model As shown in Fig. 1 the water drops are placed on the cantilever with a dispenser. The liquid drop exerts forces on the cantilever, which finally result in an upward bending free end of the beam. The upward bending is due to the fact that on this scale mass and gravitation can be neglected and surface forces are dominating.
Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt Fig. 1: Evaporation sequence of a water drop on an AFM-cantilever: Images taken from an experiment (left) and experimental data contact radius and inclination, respectively, versus evaporation time (right) taken from cantilever deflection. Forces responsible for the deflection (Fig. 2) are, according to Bonaccurso and Butt (2005), 2J / R - Laplace pressure of the fluid: P - Normal component of the liquid surface tension: - Change in surface stress of the cantilever: Lateral component of the liquid surface tension: Line tension at the rim of the drop: J normal J sin 4 'V J lateral J cos 4 N 2J sin 4 / a Here, R is the radius of curvature, a the contact radius and 4 the contact angle between solid and liquid. With the Young s modulus E of the cantilever and the moment of inertia I of a rectangular cantilever (I = WH³/12; W and H are the cantilever s width and height, respectively), the differential equation for the cantilever inclination in a simplified 2D approximation is given by (Bonaccurso & Butt 2005) dz dx S a3 ª 2d N º J sin 4 J cos 4 'V» «4 EI a a ¹¼ (1) The first term in (1) is caused by the normal component of the liquid surface tension and the Laplace pressure and leads to an upward bending of the cantilever towards the drop. J cos4, 'V and N act on the three-phase contact line. They hamper the bending in the length direction of the cantilever and simultaneously lead to a bending in the cantilever s width direction. If Young s equation is valid, the term J cos4 - 'V disappears because the difference in the solid surface tensions J is equal to the change in surface stress 'V. The last term (line tension N) and the gravity are negligible so that (1), in a first approximation, is reduced to the first term. The analytical model (1) predicts inclinations that are about 20% higher than the inclinations measured in the experiments. The reason for this discrepancy may be the fact that the analytical 2D model does not take the deflection of the cantilever in its width direction into account. This deflection in width direction leads to a stiffer cantilever and so to a smaller inclination. Hence a 3D simulation model with FEMLAB which considers this effect is set up. The resonance frequency of the cantilever with (fi) and without (f0) a drop is calculated from the analytical model by (Bonaccurso & Butt 2005) as
Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt f0 1 2S K 0.243M 0.1614 H L2 E U fi 1 2S K mi 0.243M (2) Here, K is the spring constant calculated with K = EWH³/4L², L the cantilever s length, U its density and M its mass. The two equations are basically alike, but for the mass mi of the drop, which is added to the mass of the cantilever. Again this has to be considered in the 3D simulation in FEMLAB. 3 Simulation Models The experiments are divided into three parts and therewith three different simulation models. The main focus is of course on the simulation of the cantilever bending. On the other hand the determination of the resonance frequency with and without a drop on the cantilever, respectively, plays a role, too. For the simulation, 3D-FEMLAB models (structural mechanics module) were set up which take into account the cantilever geometry, the drop properties and the above mentioned forces. Fig. 2: Forces causing the deflection (left) and simulation for a cantilever with fixed left side (right). The cantilever, with given dimensions (L, W, H) and material properties (E, U and Poisson s ratio Q) is modelled as a rectangular solid in Cartesian coordinates in all three models. The left side of the cantilever is fixed (Rx, Ry, Rz = 0) and the contact area of the drop is placed as a circle on an embedded workplane at the solid (Fig. 2). Because of the critical ratio of length and width to height, the mesh in all models is scaled in z-direction with the factor 7. 3.1 Simulation of the Cantilever Bending The bending of the cantilever is simulated at discrete times, at which also the contact radius, the contact angle and the inclination were measured (compare Fig. 1). The model takes only the Laplace pressure and the normal component of the liquid surface tension into account (see (1)). The contact area is defined by the contact radius a and the middle of the drop in xdirection. In y-direction the drop s center is placed on the middle line. The Laplace pressure is defined as force per area on the contact area in negative z-direction. The three-phase contact line is represented by the border of the contact area. Here the normal component of the liquid surface tension is set as force per length in positive z-direction. For the solution process two steps of grid adaptation with a maximum of 106 cells are allowed. To evaluate the simulation, the simulated inclination of the cantilever has to be compared with the measured one. For this purpose a MATLAB routine is used which calculates the inclination between adjacent points in cantilever s length direction. Therefore a fine meshgrid of points is laid on the cantilever s top and the displacements in z-direction at these points are computed with the postinterp-command. In a last step the difference between the calculated inclinations from the free to the fixed end of the beam are summarized. If the sum of differences exceeds a defined tolerance H, the summation is stopped and the mean of the
inclinations from the fixed end up to this point is calculated. This way, several mean inclinations in cantilever s length direction are calculated. The simulated inclination is finally set to the mean of all determined inclinations in x-direction. 3.2 Simulation of the Resonance Frequencies The simulation of the resonance frequency of the unstressed cantilever gives above all the proof that the experiment can be simulated with satisfying results in FEMLAB. In particular the scaling of the mesh with factor 7 due to the dimension s ratio is critical and has to be revised. For this purpose the cantilever s resonance frequency with the described setup is computed without grid adaptation in the resonance frequency mode. For the validation of the model also the stressed cantilever is simulated. In this experiment the impact of the drop and the resulting initial resonance frequency is measured. Hence only the drop s mass just before the impact and the location of the impact, i.e. the x-location, are important. Instead of a spherical cap we used a cylinder with the same volume as the falling drop before the impact on the cantilever and the contact radius measured in the experiments. The Poisson s Ratio of the water drop was set to 0.25, the density to 1000 kg/m³ and the Youngs s Modulus to 0.015 GPa, i.e. the Young s Modulus of rubber, so that the cantilever s stiffness is not increased through the additional mass. Again no grid adaptation is allowed. 3.3 Parameter Studies and Sensitivity Analysis For the purpose of parameter studies and sensitivity analysis the FEMLAB-models are parameterized so that the simulations can be controlled with MATLAB. Hence the cantilever s dimensions and properties, the experimental data and the forces are set as variables. In the first place the material properties of the cantilever (E,, ), but also the cantilever s height H are varied to investigate their influence on the cantilever s inclination. The aim of these studies is a verification of the underlying models for the cantilever bending due to droplet evaporation on micro and nano scales. 4 Results The simulations described in sections 3.1 and 3.2 were made for three different experiments each, i.e. there are three different evaporation sequences and three resonance frequencies, each with and without drop. The results of the parameter studies (section 3.3) are presented for one cantilever. 4.1 Simulation of the Cantilever Bending The results of the simulation are in good agreement with the experiment (see section 3). Assuming that the deflection of the cantilever can be described in a first approach by (1) we see a nearly constant error between the measured and simulated inclination for all three cantilevers. The mean error variation from +4% (Cantilever E1, cf. Fig. 3) to -12% (Cantilever E3) can be due to the measurement errors in the experiments, or to wrong material parameters (see section 4.3).
Fig. 3: Error between measured and simulated inclination for 2 different cantilevers 4.2 Simulation of the Resonance Frequencies The resonance frequencies for the unstressed cantilevers show very satisfying results (Tab. 1). The error between measured and simulated frequencies is only up to 1.4%. The resonance frequencies for the stressed cantilevers are not as satisfying as for the unstressed ones, but they are in the range of the experimental errors of about 10% (cf. 4.1). Tab. 1: Resonance frequencies for the stressed and unstressed cantilevers Cantilever Properties Results Drop (Cylinder) Properties Results Cantilever F_I Cantilever F_II Cantilever F_III Length [µm] 500 770 470 Width [µm] 100 100 50 Height [µm] 0.81 0.90 2.56 Young s Modulus [GPa] 178.43 173.19 94.33 Poisson s Ratio [ - ] 0.26 0.26 0.26 Density [kg / m3] 2330 2330 2330 Unstressed Cantilevers f0_exp [Hz] 4579 2121 11890 f0_sim [Hz] 4641 2141 12048 Error [%] 1.35 0.94 1.33 Stressed Cantilevers Contact Radius [µm] 42 21.5 25 Height [µm] 24.72 19 15.08 Center in x-direction [µm] 458 450 435 Mass [kg] 1.37e-10 2.76e-11 2.96e-11 f i_exp [Hz] 2272 1975 8785 f i_sim [Hz] 2241 2014 9323 Error [%] -1.36 1.97 6.12 4.3 Parameter Studies and Sensitivity Analysis With the help of the parameter studies for one cantilever, the effect of the Young s Modulus, the Poisson s Ratio, the density and the height of the cantilever are investigated. Each parameter was varied around a reference value with one higher and one lower value, respectively. Again, the change of the mean error between the measured and simulated inclination is chosen as degree of the influence of each parameter. Tab. 2 shows quantitative results, where ++ indicates a very significant increase of the mean error, + a significant increase, 0 not noticeable, - a significant decrease and --" a very significant decrease with respect to the reference.
Tab. 2: Results of the parameter studies Young s Modulus [GPa] Influence 200 - - (reference 180) 160 + + Poisson s Ratio [ - ] 0.33 - (reference 0.26) 0.17 + Density [kg / m³] 2500 0 (reference 2330) 2000 0 Cantilever Height [µm] 1.00 - - (reference 0.89) 0.80 + + 5 Conclusions and Outlook The results presented in section 4 show first of all that the simulation of the bending of a cantilever due to drop evaporation on AFM-cantilevers with FEMLAB leads to better results than the analytical 2D model. The error between experiment and simulation is in the range of the experimental errors so that the simulations can serve as a basis for further studies. This is an important step towards a better understanding of physico-chemical processes, and in particular with regard to Lab-on-a-Chip-applications. With the help of the parameter studies and sensitivity analysis (see 4.3, Tab. 2) we are not only able to verify the underlying models for the cantilever bending, but we are also able to study the influence of the cantilever parameters. For example, it is very important to know the exact height and Young s Modulus of the cantilevers because they influence the bending very distinctly. Only if all parameters are determined correctly, the simulations can describe the experiments correctly. The next step will be to study the influence of other parameters and of the forces neglected so far, especially the change in surface stress. It is already known that Young s equation is not valid for small drop diameters, but up to now there are no secure statements about its behaviour. We are confident to be able to make statements about the Young s equation with the help of parameter studies combined with optimization. Perhaps we are also able to explain a negative bending of the cantilever at the end of the evaporation process noticed in the experiments with a better understanding of the Young equation. Literature Bonaccurso, E.; Butt, H.-J. (2005) Microdrops on Atomic Force Microscope Cantilevers: Evaporation of Water and Spring Constant Calibration, J. Phys. Chem. B, 2005, 109, 253-263 Cµ, Center for Micro- and Nanochemistry and Engineering (2002) Integrated Multidisciplinary Research Institute, University of Siegen, http://www.cu.uni-siegen.de Cordeiro, M.; Pakula, T. (2005) Behavior of Evaporating Droplets at Nonsoluble and Soluble Surfaces: Modelling with Molecular Research, J. Phys. Chem. B, 2005, 109, 4152-4161 Deegan, R.D.; Bakajin, O.; Dupont, T.F.; Huber, G.; Nagel, S.R.; Witten, T.A. (1997) Capillary Flow as the Cause of Ring Stains from Dried Liquid Drops, Nature 389, 6653 Erbil, H.Y.; McHale, G.; Newton, M.I. (2002) Drop Evaporation on Solid Surfaces: Constant Contact Angle Mode, Langmuir 2002, 18, 2636-2641 Hu, H.; Larson, R.G. (2002) Evaporation of a Sessile Drop on a Substrate, J. Phys. Chem. B, 2002, 106, 1334-1344 Nano, Nano- and Microfluidics (2002), DFG Priority Programm 1164, http://www.merkur.rz.unisaarland.de/softmatter/index.php Nguyen, V.X.; Stebe, K.J. (2002) Patterning of Small Particles by a Surfactant-Enhanced Marangoni- Bénard Instability, Physical Review Letters, Vol. 88, No. 16, April 2002 Shmuylovich, L.; Shen, A.Q.; Stone, H.A. (2002) Surface Morphology of Drying Latex Films: Multiple Ring Formation, Langmuir 2002, 18, 3441-3445 Stupperich-Sequeira, C.; Butt, H.-J.; Graf, K.; Wiechert, W. (2005) Modelling of Microwell Formation, MCMDS, in press