Bounded amplification of diffusioosmosis utilizing hydrophobicity

Abstract

This study is to investigate if and how much surface hydrophobicity can augment diffusioosmotic velocity. Molecular dynamics simulations are utilized to estimate the slip length of diffusioosmosis through slit microducts of different surface conditions. Having the slip length in hand, hydrodynamic analyses based on the Navier slip conditions are performed to derive semi-analytical solutions of the locally developed velocity field. It is shown that, unlike electroosmosis, there is an upper limit for diffusioosmotic flow enhancement by means of hydrophobic surfaces: only about 1 order of magnitude amplification of the flow rate is achievable. This is attributed to the decreasing dependency of the velocity gradient at the wall and the electric field on the slip length for diffusioosmosis, even though the latter is very weak. Moreover, we demonstrate that hydrophobicity is more effective at low zeta potentials; based on this fact, full analytical solutions are presented for the velocity field invoking the Debye–Hückel linearization. The slip effects are intensified when the diffusivity of the counterions increases. Furthermore, slippage increases the chance of flow toward higher concentrations, revealing that hydrophobicity not only is a tool to increase the flow rate, but also may be utilized as a mechanism for diffusioosmotic flow control.

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1. Introduction

In the recent years, microfluidics has become one of the most interesting areas in the scientific community since it is capable of providing a wide range of applications, from cell biology to fuel cells.^1–3 Lab-on-a-chip (LOC) devices are fully eligible candidates to take on the microfluidics capabilities. Nowadays, they are prepared for advanced diverse implementation such as in the detection of toxicity of drugs or even medical diagnosis where LOCs diminish the costs which heretofore were very high. In LOC technology, the high hydrodynamic resistance that a liquid experiences as it flows in a network of microchannels invokes a suitable pumping system to be incorporated into the microfluidic device. Since the pump should be of micron size to retain the advantages of the whole miniaturized instrument, the conventional methods of fluid pumping requiring complex moving components are not appropriate to be used in LOCs.⁴ In this respect, electrokinetic (EK) phenomena are of paramount importance.

The electrokinetic phenomena may be defined as any motion which is a result of interaction between charged surfaces, ionic clouds, and electric fields. These phenomena provide an ingenious way of flow generation and particle manipulation in microfluidic devices.⁵ Depending on the type of the driving force and the object to be actuated, different electrokinetic phenomena like electroosmosis, electrophoresis, streaming potential, and sedimentation potential may be recognized. Unlike the classical EK, mentioned before, for which the driving force is usually either of the pressure gradient or external electric field, there are more recently explored phenomena that utilize the gradients of temperature⁶ or ionic concentration⁷ for flow generation. The first is thermoosmosis and the latter, which is to be dealt with in this work, is termed diffusioosmosis (DO). Indeed, this phenomenon is the superposition of two contributing components: chemiosmosis, a result of the pressure gradient arisen due to the ionic concentration gradient, and electroosmosis which is created upon the application of the induced electric field and the net electric charge within the electric double layer (EDL) near the wall.

Although, in comparison with the classical electrokinetic phenomena, DO has received much less attention, its unique characteristics are indicative of many potential applications. As one of the cases, the concentration-dependent velocity of DO can regulate the rate of drug release as a function of the physiological change in pH through self-regulated drug delivery systems.⁸ Another case worth mentioning is the importance of DO in LOC devices or biosensors wherein it is capable of intensifying the current recognition, which gives rise to enhancing the performance of moving charged particles⁹ or making a micropump.¹⁰ Besides, DO velocity is an important parameter in diffusiophoresis (moving charged particles under an ionic concentration gradient).^8,11

After the introduction of diffusioosmotic flow (DOF) very firstly by Derjaguin et al.⁷ in early 70's and then by Prieve^12,13 in 80's, Keh and his research group started publicizing their comprehensive works in a series of papers. Their studies were evolutionary in geometry^14–16 (from flat surface to fibrous porous media), in charged surface^17–19 (from bare to polyelectrolyte coated), in the induced electric field^14,20,21 (incorporating or not the convection term), and in the solving procedure^22–24 (from approximations to implicit relations). Huang and coworkers performed pioneering studies on DOF of non-Newtonian fluids.^25–28 Among the very small number of relevant experimental works are the recent studies of Kar et al.²⁴ and Lee et al.²⁹ Up to this point, all provided discussions illuminate that DOF is such a complicated phenomenon which demands a case-by-case investigation.

Apart from microfluidics, the drag reduction has traditionally been an interesting topic to achieve higher velocities and/or more efficient consumption of energy.³⁰ The classic hydrodynamics, usually used to formulize the electrokinetic phenomena in microfluidics, is founded on continuum point of view and is supposed to remain valid up to small scales. The no-slip boundary condition for fluid velocity at the wall, nonetheless, could not hold true if the solid surface comes with some specific properties of wettability (low-energy material coats such as organosilane or fluorine) or systematic roughness (micro-structured surface).³¹ Such properties lead to promising surface characteristics called hydrophobicity. Hydrophobicity is best defined quantitatively by contact angle and slip length. Hydrophobic surfaces are those for which the contact angle between the surface and the droplet is >90° whereas that of super-hydrophobic would be ≥150°. It should be mentioned that utilizing surface chemistry cannot improve the hydrophobicity higher than 120° unless the surface structure is modified physically. The fluid in hydrophobic ducts slips on the surface with a velocity proportional to the shear stress at the surface. The slip degree is characterized by the slip length which is an extrapolated distance from the wall within which the tangential velocity reduces to zero.³² Hydrophobic surfaces provide actually diverse applications including anti-ice/frost surfaces,³³ drag reduction,³⁴ reduction of bacterial adhesion (manufacture of antibacterial surfaces),³⁵ separation of oil–water,³⁶ water desalination, sensors, and microfluidics;³⁷ as for EK, the probable enhancement in flow velocity is favorable. There are effective reviews in this field that indicate massive interest in both fundamental and applied aspects.^38,39

One of the main deficiencies of DOF is its very low flow rate (mean velocities of the order 10⁻⁵ m s⁻¹) that has prevented this technique to be widely used in microfluidic instruments. Utilizing hydrophobic or superhydrophobic surfaces may remedy this defect to some extent. Using molecular dynamics (MD) simulations coupled with simple hydrodynamic analyses, Bocquet and coworkers^40,41 reported 2 and 3 orders of magnitude amplification in DO flow rate for hydrophobic and superhydrophobic surfaces, respectively. The method used by these researchers can be considered as a type of multiscale modeling. Since experimental observations for determination of the slip length is extremely difficult and expensive at mesoscale, utilizing such multiscale modelings is inevitable. Nevertheless, performing MD simulations, as a main part of such analyses, has its own complexities.⁴² Molecular dynamics requires high computational resources for few thousands of atoms, while the time-scale is about femtoseconds that is quite short for many dynamic problems. Considering these limitations, MD is only used for determination of surface characteristics like the slip length.

Although Bocquet and coworkers^40,41 conducted elegant multiscale analyses, the simplified assumptions they used such as considering identical diffusion constants for positive and negative ions, taking the slip length of DOF to be the same as that of electroosmotic flow, and neglecting the advection contribution to the electric field may significantly affect the results. The latter is especially important since considering the advection of ions in evaluating the electric field has a damping effect and prevents getting irrationally large velocities. In the present work, by performing a multiscale analysis within which the above-mentioned premises are relaxed, it is shown that the actual increase in DO flow rate by means of slip effects is only about 1 order of magnitude. The bounded amplification of DOF due to surface hydrophobicity is quite different from that of electroosmosis for which a linear increase in the flow rate has been reported.⁴³ The method we consider includes molecular dynamics simulations to estimate the slip lengths of DOF in hydrophobic ducts that are used in continuum analyses based on Navier slip conditions. This method provides insight into realistic physics and bridges the gap between atomistic and continuum scales by means of the slip length that is held in common.⁴² For convenience, a slit geometry is considered and the flow is assumed to be locally developed. In general, a semi-analytical solution is presented for the velocity field; nevertheless, full analytical solutions, valid for small potentials, are also obtained by performing the Debye–Hückel linearization of the electrical potential equation. This work is a step forward in presenting close-to-reality models for DOF in hydrophobic and super-hydrophobic microducts.

2. Mathematical formulation

Diffusioosmotic flow through a (super-) hydrophobic slit microchannel of height 2H is to be theoretically investigated. An illustration of the paradigm of the physical problem including the direction of decrease in the number concentration in the bulk n^∞(z) as well as the coordinate system is depicted in Fig. 1. It is presumed that the flow is steady and locally developed and the liquid contains a symmetric electrolyte of valence [Doublestruck Z]. It is also assumed that the zeta potential is constant and uniform. Finally, the ionic concentration gradient ∇n^∞ is supposed to be negligible when compared to the bulk number density, in order to allow the neglect of the axial variations in the electrostatic potential and ionic concentration.

Fig. 1 Schematic representation of the slit microchannel with super-hydrophobic walls including the coordinate system and direction of decrease in the number density in the bulk.

2.1. Electrical potential distribution

Evaluating the electrical potential distribution is usually the starting point when analyzing the electrokinetic phenomena. In this section, we pay attention to the EDL potential, ψ, which together with that corresponding to the induced electric field (IEF) constitute the total electrical potential within the flow domain. The electric potential within hydrodynamically developed flow is governed by the Poisson equation which here readswhere ε represents the fluid permittivity while ρ_e stands for the net electric charge density given as:

ρ_e = [Doublestruck Z]e(n₊ − n₋) (2)

with e denoting the proton charge. For determination of the ionic concentrations we make use of the Boltzmann distribution which is valid for a rectilinear flow and is written asin which k_B and T are Boltzmann constant and absolute temperature, respectively. Inserting eqn (2) into eqn (1) after substitution for the Boltzmann distribution of ions, the electrical potential equation, also known as the Poisson–Boltzmann equation, takes the following dimensionless form

Here, ψ* = [Doublestruck Z]eψ/k_BT, y* = y/H, and K = H/λ_D wherein λ_D = (2n^∞e²[Doublestruck Z]²/εk_BT)^−1/2 is the Debye length, a measure of the extent of EDL. Writing d²ψ*/dy*² as (dψ*/dy*)[d(dψ*/dy*)/dψ*] and multiplying both sides of eqn (4) by dψ*, it can be integrated across the channel height subject to the symmetry condition at centerline where ψ* = ψ^*_c to yield:

Since we consider the upper half of the channel, the positive sign is adopted in eqn (5). Further integration, with the consideration of ζ as the electrical potential at the wall, gives:

in which [scr A, script letter A] = [2/(cosh ψ^*_c + 1)]^1/2, ζ* = [Doublestruck Z]eζ/k_BT, and [scr F, script letter F] is incomplete elliptic integral of the first kind, that is

As is evident from eqn (6), what we have in hand is an implicit relationship for the electrical potential in the form y* = f(ψ*). The major problem in using eqn (6) is the evaluation of ψ^*_c which is not known a priori. It has been already reported that this parameter may be set to zero for K ≤ 5 without losing accuracy.⁴⁴ The shooting method might be used to obtain ψ^*_c for smaller values of the Debye–Hückel parameter, K, using the criterion that the corresponding value of y* be sufficiently close to zero.

2.2. Induced electric field (IEF) distribution

As mentioned before, DOF is described best as a superposition of two flow components namely chemiosmosis and electroosmosis. The former, to be dealt with in subsection 2.3, is created due to the so-called osmotic pressure gradient that is arisen because of axial variations in the ionic concentration. The establishment of an induced electric field (IEF) is the reason for generation of the second part of the flow, that is electroosmosis; regarding the fact that dissimilar species of different diffusivities tend to move against an applied concentration gradient under different rates, one can expect a larger number of the ions having a higher diffusion coefficient to be present at the region of lower concentration. Consequently, an electrical potential difference sets up between the channel ends. Due to the presence of EDL within which there is an excess of counterions over coions, such a potential difference exists even when the species are of the same diffusivity. The associated IEF creates an electroosmotic flow upon acting on the net electric charge inside EDL. For evaluation of IEF, we make use of the fact that the total ionic flux in the axial direction should vanish at the steady conditions. The ionic flux vector, J, is predicted by the Nernst–Planck equation given as:where D_± denotes the diffusion coefficient of the cat/an-ions, u represents the velocity vector, and E is the induced electric field which acts in the axial direction only. Equating J_+,z with J_−,z and solving for E_z, we come up with

The new dimensionless parameters in this equation include the dimensionless axial velocity u* = u_z/U, the dimensionless diffusivity difference β = (D₊ − D₋)/(D₊ + D₋), and the Péclet number Pe = 4n^∞U/(D₊ + D₋)|∇n^∞|. Moreover, sgn is the sign function and is given by sgn(∇n^∞) = |∇n^∞|/∇n^∞. Note that the velocity scale used for normalization is the characteristic velocity of DOF, given as:

wherein μ is the dynamic viscosity. The electric field away from the wall where the electric potential approaches zero is obtained from eqn (9) to be E^∞ = k_BTβ∇n^∞/[Doublestruck Z]en^∞.

2.3. Velocity distribution

For evaluation of the velocity field, we should start with the Navier–Stokes equations with a body force vector of F = ρ_e(E − ∇ψ). Under the problem assumptions, the following equations are derived for both y and z directions, respectively

By integrating eqn (11a) across the channel height with the consideration of the associated boundary conditions, the pressure field is found to be:

where p^∞ stands for the centerline pressure and is constant in the present study as a consequence of no applied pressure force. Differentiating eqn (12) with respect to z and substituting the resultant expression into eqn (11b), the z-momentum equation becomes

The dimensionless form of eqn (13) along with the symmetry condition at the centerline and Navier slip boundary condition at the wall may be written as

where E* = E_z/E^∞ and b* = b/H with b representing the slip length. Integrating eqn (14) with respect to y* from the centerline to an arbitrary point by considering the symmetry conditions and integrating the resultant equation once more from the wall to an arbitrary place by making use of the slip conditions, the dimensionless velocity is obtained as:

Note that b* = 0 recovers the velocity distribution obtained in our previous work for diffusioosmotic flow in slit microchannels subject to the no-slip boundary conditions.⁴⁵

2.4. Special solutions

Although the foregoing analysis is valid for any situation, nevertheless, it requires a try and error approach because of the coupling between the electric and velocity fields. For some special cases, the associated physics allow significant simplifications and we may therefore utilize much more efficient solutions. Here, we pay attention to a well-known special case that is the situation for which the zeta potential is low enough to allow the Debye–Hückel (DH) linearization. Assuming a small ψ*, the term βE* in eqn (14) may be approximated by expanding the two main terms of eqn (9) in Taylor series and discarding all the terms that are cubic or of higher orders in ψ* to get

βE* = β + (β² − 1)ψ* + β(β² − 1)ψ*² + sgn(∇n^∞)Pe(ψ* + βψ*²)u*/2 (17)

Performing the same for cosh ψ*, cosh ψ^*_c, and sinh ψ* with the consideration of the fact that the electrical potential under DH linearization is given by ψ* = ζ* cosh(Ky*)/cosh K, eqn (14) reduces to

It can be shown that the solution of eqn (18) subject to the boundary conditions (15) is as follows

where and C, S, C′, and S′ are Mathieu functions of cosine, sine, and their derivatives, respectively. In addition, η^±(y*) = (Peζ*²/4 cosh² K, −Peζ*²/8 cosh² K, ±Ky*i) and the coefficients a₁ and a₂ are given as

Further simplification is possible when the Peclet number is so small that the advection contribution to the electric field may be ignored. Setting Pe = 0 into eqn (18), it is easy to verify that the velocity field becomes

Since we have a closed form solution of the velocity in hand, it is convenient to evaluate the mean velocity given in dimensionless form as . Performing the integration provides

3. Results and discussion

First, and before proceeding further parametric sweep studies, one must make an estimate of the dimensionless slip length of DOF through hydrophobic and super-hydrophobic surfaces. This parameter is directly proportional to the surface energy and thus a molecular dynamic model based on the Lennard-Jones potential was used to accurately measure it. In this respect, the Lennard-Jones parameter of energy interaction of the channel wall ϵ_ww was varied between 0.165 and 2.065 kcal mol⁻¹ to gradually change the wall from a hydrophobic to a hydrophilic surface (contact angle changing from ∼140° to ∼55°). All the details regarding the MD simulations are provided by the ESI† and we only show the dimensionless slip lengths predicted in Fig. 2 which are presented as a function of the dimensionless surface energy, ϵ^*_ww = ϵ/k_BT. As it is shown in Fig. 2, the slip length is higher at lower surface energies. This is because the less the surface energy is, the less the fluid tends to dissipate energy due to the friction adjacent to the wall, leading to easier slip of the fluid. The dimensionless slip length is also found to be an increasing function of the dimensionless driving force mainly because of lowering the fluid-surface engagement. The results presented in Fig. 2 reveal that b* may reach values of the order of 1 for hydrophobic surfaces. We may expect higher values of b* for a super-hydrophobic surface as it could be proven by the indentation of the surface in the MD simulations; we will soon figure out why there is no need to invoke such a time-consuming process.

Fig. 2 Dimensionless slip length of DOF as a function of dimensionless surface energy at different values of dimensionless driving forces f^*_x = f_x/(k_BTH) acting on the fluid molecules in the axial direction. The surface characteristics turn from hydrophilic into hydrophobic by decreasing ϵ^*_ww.

The advection contribution to the electric field, characterized by the Peclet number, plays as a self-regulating role, that is it tends to control the velocity to remain in a specific range. This is justified by Fig. 3 that illustrates the velocity distribution at different values of b* and Pe. It can be seen that there is a trade-off between Pe and the slip velocity, i.e. if Pe is infinitesimal the slip velocity is rather high, whereas a Peclet number of the order of 1 gives rise to quite small slip velocities unless very large slip lengths are invoked. The physical mechanism behind this behavior is clarified in the following for the data set considered in Fig. 3; when D₊ < D₋, that is in cases for which β < 0, a positive electric field is established for a positive zeta potential due to the accumulation of anions within EDL at the region of lower concentration. This electric field creates a negative velocity upon acting on the negative net charge in the channel. Larger Peclet numbers correspond to higher transfer of ions by advection and it means that IEF is more and more being depleted by the accumulation of mostly negative ions in EDL at the positive pole, thereby slowing down the flow and lowering the slip velocity due to smaller shear rates close to the wall.

Fig. 3 Velocity distribution at different slip lengths for (a) Pe = 1, (b) Pe = 0.1, and (c) Pe = 0.01. The lines exhibit the results of the full solution eqn (16), whereas the symbols represent the predictions of the approximate solution given by eqn (19).

Besides showing the variations of the velocity profile with both b* and Pe, Fig. 3 may be used as a tool for the evaluation of the approximate solution, given by eqn (19). Despite the fact that ζ* = 2, corresponding to ζ ≅ 50 mV, is not sufficiently small to allow DH approximation, the results for no-slip conditions show a surprisingly high degree of agreement. The error increases for higher slip lengths and reaches about 25% for b* = 1, mainly because of the fact that the shear rates predicted based on the Debye–Hückel linearization are not sufficiently accurate; hence the slip velocities are not predicted accurately, leading to profiles that are similar to the exact ones in shape but follow them with an offset. Since the zeta potential of hydrophobic surfaces is usually small due to low solid–liquid interactions,⁴³ the inadequacy of eqn (19) in predicting the velocity field at large zeta potentials should not cause major problems for most practical applications.

Now that the accuracy of the approximate solution (19) has been evaluated it is the time to check the validity of eqn (21) which was derived assuming Pe = 0. The ratio of the flow rate predicted by eqn (21) to the exact flow rate calculated by means of eqn (16) is depicted in Fig. 4. The dimensionless zeta potential is fixed to 1, a reasonable value for hydrophobic surfaces. We see that, for the range of K considered, the error remains within 18% of the exact value and is mainly below 10%. Although these errors are generally not small enough to be satisfactorily neglected, the ease of use of closed form solutions makes eqn (21) suitable for design and optimization purposes which require a large number of simulations; after coarsely determining the optimum design, eqn (16) then may be used for further modifying of the results.

Fig. 4 The ratio of approximate and exact flow rates vs. dimensionless Debye–Hückel parameter for different b* and β.

In the next illustration, we are going to figure out the direct effect of the slip length on the dimensionless mean velocity. It is noteworthy that, here and in what follows, all the results are obtained by means of the general solution given by eqn (16). As seen from Fig. 5, at certain conditions, one may find drastic amplification of the flow rate by employing higher slip lengths. The unexpected point is that the mean velocity tends to asymptotic values at high values of b* which is quite different from EOF for which a linear increase occurs in the mean velocity by increasing b*, as reported by Sadeghi et al.⁴³ This difference may be attributed to the fact that, unlike EOF for which the electric field and the shear rate at the wall are independent of the slip length,⁴³ both of these parameters are decreasing functions of b* for DOF. Note that the cause of the electric field getting weaker in the presence of surface hydrophobicity is grounded in the higher accumulation of anions by the flow in the positive pole of the electric field. The sharp increase in u^*_m for high values of K occurs within a reasonable range of b*; nevertheless, to amplify the flow rate for small Debye–Hückel parameters like K = 5 extremely large slip lengths are required that are only possible when invoking super-hydrophobic channels of high surface indentation. Since indentation of the wall leads to smaller zeta potentials, the slip length above which there is no gain in the flow rate should be practically smaller than that predicted here.

Fig. 5 Dimensionless mean velocity vs. slip-length at different K.

As depicted above, DOF induces unforeseen consequences for slip effect such as a maximum attainable mean velocity. We now turn our attention to the zeta potential dependency of the dimensionless slip velocity, u^*_s, and see how the degree of surface hydrophobicity can alter the trends. Fig. 6 demonstrates that for a given slip length there is a specific zeta potential that provides the maximum slip velocity. This response can be attributed to the dissimilar variations of the charge density and IEF with ζ. Although the charge density is an increasing function of the surface potential, quite the opposite is true for the electric field; the existence of more anions in the channel paves the way for more accumulation of them by the flow at the region of higher concentration at which the positive pole of IEF locates, thereby depleting the electric field. The ultimate outcome of increasing the zeta potential depends on its relative influences on the electric charge and IEF; at smaller zeta potentials the enlargement of the electric charge wins whereas the opposite holds at higher values of ζ. The reason the zeta potential corresponding to the maximum u^*_s is lower for larger slip lengths is that, as noted previously, the slip effects tend to decrease the strength of the electric field. Another point taken from Fig. 6 is that utilizing hydrophobic surfaces for creating higher flow rates is much more effective at low and moderate zeta potentials.

Fig. 6 Slip velocity as a function of zeta potential at different dimensionless slip lengths.

The impact of β on the slip length dependency of u^*_s is the subject of our next investigation that is based on the results presented by Fig. 7. It is evident that the effects of β on u^*_s are dramatically pronounced at higher values of b*. Moreover, we notice from the graphs that for a positive zeta potential, the impacts of the slip length are stronger when β gets smaller, that is when either D₋ amplifies or D₊ becomes smaller. It is also deduced that one can get higher slip velocities and, consequently, higher flow rates by lowering the dimensionless diffusivity ratio. This trend is observed since by increasing the diffusion coefficient of anions while retaining that of cations one can amplify the accumulation of the former at the channel end where negative pole of IEF exists, leading to a stronger body force.

Fig. 7 Dimensionless slip velocity vs. dimensionless slip length at various β.

The influences of both ζ* and b* on the IEF distribution are shown in Fig. 8; the main curves are IEF distributions based on b* = 1 while insets are those of adjacent the wall for ζ* = 1 and 6 at different b*. The insets, even though implying that (super-) hydrophobicity has a weak impact on IEF, justify our previous deduction that wall slippage tends to decrease the electric field strength. The figure further indicates a noticeable influence of the zeta potential on IEF especially near the wall in which higher electric fields are achieved by increasing ζ*.

Fig. 8 IEF distribution at different ζ* and b*.

Traditionally, the discussions of DOF analyses are terminated by presenting the curves of the flow direction. Each curve divides the figure into 4 zones bounded by the curve and the line ζ* = 0. Such a curve is obtained by a highly time-consuming procedure that includes finding β of zero flow rate at each ζ* by setting a tolerance of the order of 10⁻⁴. The curves of zero flow rate depicted in Fig. 9 for different dimensionless slip lengths reveal that whereas βζ* < 0 ensures a flow toward higher concentrations, the opposite depends not only on ζ* and β but also on b*. The influence of the hydrophobicity on this diagram is to increase the chance of creating a flow toward higher concentrations but this increment is supposed to have an upper bound since the curves will not notably change for 10 < b*. A behavior that is not seen for conventional ducts is that by using a hydrophobic surface, for a given electrolyte, increasing ζ* may change the flow direction twice. Another interesting finding is that all the curves collapse into one at large |ζ*|, confirming our previous conclusion that making use of hydrophobicity is not efficient for surfaces of high zeta potential.

Fig. 9 Flow direction diagram at various dimensionless slip lengths while keeping K = 5.

4. Conclusions

We studied the feasibility of using hydrophobic surfaces to enhance the flow rate of diffusioosmosis. To this end, molecular dynamics simulations were used to obtain the slip length of diffusioosmotic flow in different hydrophobic microducts. This data was then used as input for continuum analyses considering Navier slip conditions to provide semi-analytical solutions of the locally developed velocity field in a slit microchannel. A complete parametric study revealed that hydrophobicity is efficient only if the wall imposes low or moderate electrical potentials. Therefore, adopting the Debye–Hückel linearization, an analytical solution of the velocity profile was presented for low zeta potentials. The results also showed that, unlike electroosmotic flow, for which the flow rate is a linearly increasing function of the slip length, there is an upper bound for amplifying the diffusioosmotic flow; only about 1 order of magnitude increase in the mean velocity is possible. This difference arises because of the decreasing dependency of the shear rate at the wall and the electric field on the slip length for diffusioosmosis. In addition, it was observed that surface hydrophobicity increases the chance of diffusioosmotic flow toward higher concentrations.

Acknowledgements

The research council at Iran University of Science and Technology (IUST) and Iran's National Elites Foundation (INEF) are highly acknowledged for their support during the course of this research. The authors gratefully acknowledge the Sheikh Bahaei National High Performance Computing Center (SBNHPCC) for providing computing facilities and time. We also appreciate Dr S. M. Rahimian for his masterful advices for construction and diagnosis of the MD model.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra05846a

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