Supplementary Information: Numerical investigation of micro- and nanochannel deformation due to discontinuous electroosmotic flow
Josephde Rutte1∙ Kjeld Janssen1∙ Niels Tas2∙Jan Eijkel2∙ Sumita Pennathur1
- Department of Mechanical Engineering, University of California, Santa Barbara, Engineering Science Building, Room 3231C, Santa Barbara, CA, USA
- MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500AE Enschede, The Netherlands
In the electronic supplementary information, we include the following: 1)COMSOL simulation results that show the geometric constraints for which the proportionality constant c is within 1% of our reported values; 2) Validation of 2D deformation profile using 3D structural COMSOL model; 3) error approximations showing that hydraulic resistance is accurately depicted by our model in the limit that deflection is small in comparison to the channel width ();4) data showing that -potential dependence on concentration has little effect on induced pressures in channels with thick EDLs; 5) comparison of pressure results using bulk conductivity approximation and local ion concentration dependent conductivity and 6) data showing that the final deflection ratio calculated using an iterative solver is not dependent on the conductivity ratio.
1 Validation of proportionality constant c
To determine the geometric constraints for which our scaling arguments for 2D deflection holds we calculated the proportionality constant c over a range of channel aspect ratios () and glass thickness to channel width ratios (t/w) using a 2D COMSOL model. The results indicate that c holds within 1% for channel aspect ratios greater or equal than 11.0 () and substrate thickness to width ratios greater or equal then 10.9 (). Inside of this range channel wall deflection is accurately defined by , where p is pressure and is elastic modulus.
Figure S1: (a) Proportionality constant c vs channel width to height ratio determined using 2D structural COMSOL model. (b) Proportionality constantcvs glass thickness to channel width ratios (t/w)determined using a 2D COMSOL model.
2. Validation of 2D deformation model using 3D structural model
For our coupled 2D flow and deformation model we assume that the channel length is much larger than the channel width such the deformation along the length of the channel is dependent only on the local pressure. In order to verify this assumption and determine the regime in which this assumption holds true we performed simulations of structural deformation in a full 3D model using COMSOL and compared the results those predicted with our coupled 2D flow and deformation model (Figure S2). Our results confirm that for large length to width ratios the local pressure assumption holds and the results of the two models closely agree (max deflection within 0.03% for L/w = 177). As this ratio is decreased the results begin to deviate. Based on our observations, the deformations predicted with our coupled 2D model are a good approximation for L/w ratios above 50.
Figure S2: Comparison of channel deformation predicted by our 3D structural model and coupled 2D model.
3Validity of rectangular cross-section modelfor deformation
In order to model flow in our system using only a 2D flow model we assume that the channel cross-section can be approximated as rectangular as in previous work (Gervais et al. 2005, van Honschoten et al. 2007). To show that this approximation is appropriate we examinedthe effect of shape on the hydraulic resistance in the channel in the limit of high aspect ratios. The hydraulic resistance of a channelR with an arbitrary cross-section can be approximated as
where P is the perimeter, and A is the cross-sectional area. Using this expression we compare the effect of shape on hydraulic resistance by determining the percent difference in R between therectangular cross-section (Rr)and a cross-section with parabolic deflected walls (Rp) while maintaining the cross-sectional area.
where h0 is the channel height prior to deflection, is the averaged cross-sectional deflection, and sis the arc length of the parabolic cross-section defined as
Figure S3: Percent difference in hydraulic resistance between (a)rectangular cross-section(b) parabolic deflection cross-section is shown in (c). For relatively small deflection () and/or high aspect ratio channels () the parallel plate model well approximates the actual hydraulic resistance in the channel.
4 Constant —potential vs concentration dependent —potential (Numerical)
In section 4 of the main text we showed that the dependence of —potential on concentration has minimal effects on induced pressure. To confirm that this is also true for systems with finite EDLs, we determined numerically for both constant —potential (Figure S4a) and concentration dependent —potential (Figure S4b). Our results indicate that in both cases the numerical simulations show the trends in comparison to the analytical solution.
Figure S4: Nondimensional maximum pressurevs conductivity ratio a for various relative EDL thicknesses (). Results show that for both (a) constant —potential and (b) concentration dependent —potential, the numerical model is comparable to the analytical solution (--) for relatively thin EDL (squares). Deviation from the thin EDL approximation shows the same behavior in both cases.
5 Conductivity method comparison for 2D flow model
In this section we compare pressure results using bulk conductivity and conductivity based on the ion distribution in the channel. Conductivity, K, can be directly related to local ion concentration and ion mobility by:
where is the ionic mobility of the i’th ion type in the solution. Ionic mobility may be related to ion diffusivity D using the Einstein-Smoluchowski relation:
which gives the following relationship for conductivity.
Lastly the local ion concentration niis related to the potential distribution in the channel () using the Boltzmann equation.
Figure S5 shows our COMSOL results using both bulk conductivity and conductivity based on the local ion distribution in the channel. KCL is used as the background electrolyte for all simulations, with diffusion coefficients of and for K+ and Cl-, respectively. Our results show that resulting pressure is in close agreement for both conductivity evaluations, especially for thin electric double layers, less then a 2.5 % difference for and less than a 0.25% difference for . For a relative EDL thickness of 0.5, however, there is a noticeable difference in pressure, a maximum just above 30% as conductivity ratio approaches 1 and a minimum of approximately 10% as the conductivity ratio approaches 0. Thus, in general assuming bulk conductivity is a reasonable approximation, especially since we are concernedwith cases where pressure is maximized, which in this case is where there is the least amount of deviation. Although it is not necessarily a good approximation for studying all electrokinetic phenomenon, for the purposes of calculating deformation, in the majority of cases of interest, this approximation will give accurate results.
Figure S5: (a) Comparison of calculated using bulk conductivity and local ion concentration dependent conductivity for a range of conductivity ratios and relative electric double layer thickness . In both cases the numerical model converges to the analytical solution for thin EDLs. (b) Relative difference in for the two conductivity methods. Relative difference is maximimzed as EDL thickness is increased and conductivity ratio approaches 0. Compared to the maximum possible value of the relative difference is quite small (1 vs 0.012). (c) % difference in between the bulk and local conductivity methods. Percent difference is maximized for thick EDLs and conductivity ratio approaching 1. In general, the percent difference is low in regimes of highest pressure, thin EDLs and conductivity ratio approaching 0.
6 Dependence of channel deflection on conductivity ratio
Using our iterative COMSOL model we compute the dependence of final deflection relative to initial deflection () on the initial relative deflection () for a range of conductivity ratios (Figure S6). The results show the same trends for all of the conductivity ratios indicating that the deformation behavior is independent of the conductivity ratio.
Figure S6: Relative final deflection () vs relative initial deflection () for various conductivity ratiosa and a relative EDL thickness of 0.01 calculated using our iterative COMSOL model.