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Steric-Effect Induced Alterations in Streaming Potential and Energy Transfer Efficiency of Non-Newtonian Fluids in Narrow Confinements Aditya Bandopadhyay and Suman Chakraborty* Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur-721302, India ABSTRACT: In this work, we explore the possibilities of utilizing the combined consequences of interfacial electrokinetics and rheology toward augmenting the energy transfer efficiencies in narrow fluidic confinements. In particular, we consider the exploitation of steric effects (i.e., effect of finite size of the ionic species) in non-Newtonian fluids over small scales, to report dramatic augmentations in the streaming potential, for shear-thickening fluids. We first derive an expression for the streaming potential considering strong electrical double layer interactions in the confined flow passage and the consequences of the finite conductance of the Stern layer, going beyond the DebyeH€uckel limit. With a detailed accounting for the excluded volume effects of the ionic species and their interaction with pertinent interfacial phenomena of special type of rheological fluids such as the power law fluids in the above-mentioned formalism, we demonstrate that a confluence of the steric interactions with the non-Newtonian transport characteristics may result in giant augmentations in the energy transfer efficiency for shear-thickening fluids under appropriate conditions.
1. INTRODUCTION Interfacial phenomena over small scales may give rise to interesting physicochemical interactions in narrow fluidic confinements, several of which may be nontrivial in nature. Such interactions, in turn, are critical to the understanding of functionalities governing devices and systems characterized with microscopic and nanoscopic features. Interestingly, several devices of these kinds transport electrolytic solutions, which may give rise to the formation of a charged interfacial layer (also known as the electrical double layer; EDL),1 as a consequence of electrochemical interactions between the solution and the fluidic substrate. As a result, a surplus of counterions is likely to be present in the solution. Thus, a pressure-driven fluidic transport occurring through the conduit may result in the preferential net advective migration of counterions toward the downstream direction (the corresponding current being known as streaming current),2 thereby establishing a potential difference across the channel. This potential difference, also known as the streaming potential,2 is established in a manner such that the overall electroneutrality of the solution is preserved, that is, the forward advective current of ions is exactly balanced by the reverse electromigration current (or equivalently, conduction current). The establishment of streaming potentials across narrow fluidic confinements may have far-ranging scientific and technological consequences. As such, because of the development of streaming potential, it may be possible to drive a net current through an external load resistor, thereby representing a means of transferring hydraulic energy into electrical power.315 Such implications may be particularly useful and interesting for fluidic devices r 2011 American Chemical Society
characterized with nanoscopic spatial scales. This may be attributed to the fact that, for such narrow confinements, the Debye length may turn out to be of comparable scale as that of typical nanochannel dimensions, which may permit the exploitation of strong EDL interaction regimes (including EDL overlap1619), where the energy transfer efficiency is expected to be a maximum. It is also important to mention in this context that EDL overlap phenomenon in nanochannels is often accompanied with the scenario of high surface charge density conditions, so that the ionic species may no more be specified as point charges and their size effects become significantly more consequential. This effect, also known as the steric effect,20 stems fundamentally from the fact that a critical limit of the interfacial potential may be reached beyond a threshold value of the ionic charge densities, signifying perceptible excluded volume effects.2133 These artifacts may be accounted for in a mean field formalism by including the entropy contributions of the finite sized ions in the free energy calculation, the minimization of which results in a modified Poisson Boltzmann type formalism. Such effects, however, have been ignored by far in the literature on streaming current and streaming potential estimations in narrow fluidic confinements. Interestingly, the above deficit in literature holds true for nonNewtonian fluids as well. The non-Newtonian fluids,34,35 nevertheless, are getting progressively more relevant in microfluidic and nanofluidic applications, mostly as a consequence of their Received: June 16, 2011 Revised: August 23, 2011 Published: August 24, 2011 12243
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Langmuir emerging applications in chemistry and life sciences. Many complex fluids, such as biofluids, protein chains in solvents, colloids, gels, and cell suspensions, fall in this category. An essential feature of such systems is the presence of discrete entities over small scales. Under an imposed velocity field, the equilibrium distribution of these particles may get significantly disturbed. At sufficiently high shear rates, this perturbation may be considerably prominent to overcome the restoring Brownian effects, giving rise to interesting rheological characteristics. With strengthened magnitudes of interparticle interactions, such deviations from Newtonian behavior may tend to get prominently manifested over reduced values of the shear rates. In addition to this, such systems may also exhibit wall depletion36 in a sense that a depleted layer of pure solvent establishes at the solidliquid interface, despite the existence of discrete particulate entities in the bulk. The wall depletion layer has a thickness typically of the order of the radius of gyration of the particles. Interestingly, their length scales may turn out to be of comparable order of dimensions of typical microfluidic and nanofluidic confinements, giving rise to important implications on the consequent transport characteristics, particularly in reference to the interfacial electrokinetic effects. Recognizing the scientific importance and technological relevance of the transport processes outlined as above, a vast body of literature has been reported over the past few years on various aspects of electrokinetic transport of non-Newtonian fluids,3753 including the issues of streaming current and streaming potential.5456 However, until now, no study has been addressed to highlight the consequences of EDL overlap and steric interactions on streaming potential and energy conversion efficiency of non-Newtonian fluids in narrow fluidic confinements. The aim of the present study is to address the streaming current and streaming potential development as well as the resultant energy transfer characteristics in non-Newtonian fluids with arbitrary EDL thicknesses and in the presence of steric interactions in narrow fluidic confinements. As a demonstrative example, we consider the power law model, which is a common rheological model of a fluid with shear dependent viscosity, for a wide range of shear rates. Despite the fact that this model does not asymptotically exhibit Newtonian behavior for limitingly low and high shear rates, it has advantage with regard to both applicability and simplicity to justify its application in steady shear dependent flow behavior through narrow conduits, even in the presence of other influencing parameters such as electrokinetic effects.3643 To capture electrochemical-hydrodynamic effects in such fluids, we employ a chemical equilibrium based interfacial transport consideration (instead of a given zeta potential, which is a poor description under overlapped EDL conditions) and take the Stern layer conductance also into account. We effectively consider nontrivial interplay between the fluid rheology and the interfacial electrokinetic transport in the presence of steric interactions and delineate the effect of the corresponding confluence on energy transfer characteristics over small scales. Proceeding further, we demonstrate that the energy conversion efficiencies may be substantially elevated from those that have hitherto been obtained may be potentially realized over appropriate regimes of the power law index (conforming to shear-thickening fluids) as well as the steric factor.
2. MATHEMATICAL MODELING 2.1. Fluid Flow Equations. We consider the pressure-driven flow of an ionic solution through a slit-type fluidic confinement
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(with half-height H), for which the Cauchy (Navier) momentum equation may be expressed as F
Dv ¼ ∇p þ ∇ 3 τ þ F Dt
ð1Þ
where v is the flow velocity, F is the fluid density,τ is the stress tensor, and F is the body force per unit volume (originating from electrokinetic effects, as a consequence of establishment of the streaming potential). For low Reynolds number flows typical to microfluidic and nanofluidic channels, the inertia terms may be neglected, so that one may write 0 ¼ ∇p þ ∇ 3 τ þ F
ð1aÞ
For non-Newtonian fluids, τ = μapp[2Γ], where Γ = (1/2)[rv + rvT] is the rate of strain tensor. In case of power law type of nonNewtonian fluids, μapp = β(2Γ)α1, where Γ = [(1/2)(Γ:Γ)]1/2. Here β is the flow consistency index, and α is the flow behavior index. Shear thinning (pseudoplastic) behavior is observed for α < 1 (i.e., the apparent viscosity decreases with increasing shear rate), whereas shear thickening (dilatant) behavior is observed for α > 1 (i.e., the apparent viscosity increases with increasing shear rate). For describing the velocity field in the channel, we consider the case of depletion layer formation (see Figure 1) such that there are two regions in the channel.36 The region close to the wall is characterized by the lack of discrete particles and presence of pure solvent (Newtonian behavior). The zone away from this region is characterized by non-Newtonian behavior which is dictated by the rheological aspects the solution. Let u1 be the axial (x-component) velocity in this outer layer and u2 be the axial velocity in the wall-depletion layer. Accordingly, one may write (following eq 1a)
dp d½βðdu1 =dyÞα þ þ Fe E ¼ 0 dx dy
ð2aÞ
dp d2 u2 þ μ 2 þ Fe E ¼ 0 dx dy
ð2bÞ
where Fe is the charge density, E is the induced electric field due to streaming effects, and y is the transverse direction. Boundary conditions, consistent with eqs 2a and 2b, may be described as follows: u2 ¼ u1 at u2 ¼ 0 at α du1 β ¼ dy du1 ¼ 0 at dy
y¼δ y¼0 du2 at μ dy
y¼δ
ð2c f Þ
y¼H
It is important to mention here that, in the solution methodology employed in this work, we do not consider an equivalent effect of the EDL via the HelmholtzSmoluchowski slip velocity, primarily because of the fact that such a slip velocity consideration is restrictive to thin EDL limits only. Instead, we completely resolve the hydrodynamics within the EDL (with the aid of an appropriate forcing term in the momentum equation), and as a result, using a no-slip boundary condition at the channel wall is justified. It is also important to mention here that the problems of determination of u1, u2, and E are nonlinearly coupled, as constrained 12244
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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u zeψ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiu 1 u 1 þ 2ν cosh dψ 2kB Tn0 u k T B uln ¼ ( zeψc dy εν t 1 þ 2ν cosh 1 kB T ð6Þ Nondimensionalizing the above, considering ψ ̅ = (zeψ)/(4kBT) and y = (y/H), it follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 1 þ 2ν½cosh 4ψ ̅ 1 dψ ̅ ¼ ( pffiffiffi ln ν 1 þ 2ν½cosh 4ψ ̅ c 1 dy̅
Figure 1. A schematic depiction of the physical domain.
by the requirement of electroneutrality of each cross section of the channel. Nevertheless, for a consistent determination of these parameters, the ionic charge density field needs to be first evaluated, taking the steric effects appropriately into account. The same is outlined in the subsequent subsection. 2.2. Ionic Charge Density Calculation and EDL Potential Estimation. The ionic charge density field may be evaluated by noting that the electrochemical potential (taking steric effects appropriately into account) is uniform for equilibrium, so that one may write for the ith ionic species:57 [kBT ln ai] (chemical potential) + [zieψ] (electrical potential) = constant, where ai = (ni/n0)/(1 ν∑k(nk/n0)); ν = 2n0a3 being the steric factor (a is a representative ionic length scale), n0 being the nominal ionic concentration at the reservoir, kB being the Boltzmann constant, T being the absolute temperature, e being the protonic charge, zi being the valency of the ith ionic species, and ψ being the potential field within the EDL. Considering a z:z symmetric electrolyte solution and channel-to-reservoir equilibrium conditions, it is trivial to write from the above (noting that the electrochemical potential at anywhere in the channel is same as that in the reservoir) n0 zeψ exp n( ¼ zeψ kB T 1 þ 2ν cosh 1 kB T
ð3Þ
where n+ and n are the number densities of positive and negative ions, respectively. The EDL potential distribution is governed by the Poisson equation, which may be described as ∇2 ψ ¼
Fe ε
ð4Þ
where ε is the permittivity of the medium. For the z:z symmetric electrolyte, the charge density, Fe, may be expressed as Fe ¼ ezðnþ n Þ
ð4aÞ
Substituting eq 3 in eq 4a and subsequently in eq 4, one gets zeψ 2 sinh d2 ψ n0 ze k T B ¼ zeψ dy2 ε 1 þ 2ν cosh 1 kB T
ð5Þ
Equation 5 may be integrated across the channel section with (dψ/dy) = 0 and ψ = ψc at the channel centerline, to yield
ð6aÞ
where V = [(zeH)/(4kBT)][(2kBTn0)/ε]1/2. It is evident that the final solution of the EDL potential ψ can only be obtained with the correct specification of the condition at the channel walls, as well as the specification of the potential at the channel centerline. To achieve this purpose, we consider here an illustrative example that the channel walls are made of bare silica so that the zeta potential is determined from the equilibrium of the chemical reaction between the bare silica, the hydrogen ions, and the added cations. Thus, the zeta potential depends on the bulk ionic concentration, n0, and the buffer pH. Accordingly, one can write, after accounting for the appropriate chemical reaction at the channel wall (for details of the reaction description one may refer to Behrens and Grier58), the interrelationship between the zeta potential (ζ) and the bare wall charge density (σ) as: ζ¼
kB T σ kB T σ ln ðpH0 pKa Þln 10 e eΓ þ σ e CSt
ð7Þ
where Γ is the fraction of dissociated chargeable sites, pH0 is the value of the pH of the solution in the reservoirs, Ka is the dissociation constant of the silicawater interface, and CSt is the capacitance of the Stern layer. Also we can apply eq 6 at the channel walls [where σ = ( ε(dψ/dy)|wall = (4kBTε)/(zeH)[(dψ̅ )/(d y)]y=1; or, in a dimensionless form: σ̅ = [(dψ̅ )/(d y)]y=1 , where σ̅ = (zeHσ)/(4kBTε)], to close the system, coupled with a self-consistent estimation of ψc. Toward that, we first assume a dimensionless zeta potential (ζ̅ ). We use this to obtain ψ̅ c (for a given set of parameters) by numerically integrating eq 6 under the condition that at the centerline ψ̅ = ψ̅ c. With this ζ̅ and ψ̅ c, we first obtain σ̅ . We use this value in eq 7 to obtain a new value of ζ̅ . This iteration is continued until all of the variables ζ̅ , ψ̅ c, and σ̅ cease to change. It may be pointed out here that the centerline potential solution follows naturally from the methodology described above. A nonzero centerline potential would imply a more nonuniform velocity profile rather than the traditional plug flow profile which is seen for thin and nonoverlapping EDL cases. 2.3. Velocity Field, Streaming Potential, and Energy Transfer Efficiency. We introduce the following dimensionless parameters: ui = (ui/Uref) where Uref = (dp/dx)(H2/2μ) and i = 1 or 2, depending on the region (outer and inner layer, respectively), δ1 = (δ/H), E = (E/Eref), where Eref = (dp/dx)(H2/2εζ) and ϕ = (dp/dx)(H/β). With the determination of the EDL potential field and the charge density field in a manner as outlined in 12245
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the amount of power required to create the flow.
Section 2.1, eqs 2a and 2b can be integrated twice to yield: ! ψ ̅ δ1 2 u̅ 1 ¼ ð2δ1 δ1 Þ þ E̅ 1 ̅ ζ Z
E̅ dψ ̅ ð1 yÞ ̅ 2ζ ̅ dy̅ δ1 ! ψ̅ 2 u̅ 2 ¼ 2y̅ y̅ þ E̅ 1 ζ̅ Hϕ1=α þ Uref
̅y
ηconv ¼
!!1=α dy̅
ð8aÞ
ð8bÞ
Importantly, the velocity field described by eqs 8a and 8b can be closed only with a correct estimation of the streaming electric field, E. For that purpose, one may use the following constraint of electroneutrality through each channel section, so that15 Z H Iion ¼ 2ze ðnþ n Þu dy fstreaming currentg þ2
0 2 2
Z zeE H ðnþ þ n Þdy fconduction currentg f 0
þ 2σstern E fStern layer currentg ¼ 0
ð9Þ
where σb = (2n0z e )/f is the ionic conductivity, f is the ionic friction factor, and σstern is the stern layer conductivity.15 To further express eq 9 in terms of dimensionless parameters, we introduce the following normalization variables: Ic,ref = (2n0z2e2ErefH)/f is the reference conduction current, Is,ref = 2zeUrefHn0 is the reference streaming current. On the basis of these variables, upon nondimensionalizing and dividing by Is,ref, the constraint given by eq 9 reads EðI ̅ 2 þ I4 þ JðI6 þ DuÞÞ þ I5 ðEÞ ̅ þ I1 þ I3 ¼ 0 ð10Þ 2 2
where J = (Ic,ref/Is,ref) = (zeμ)/(fεζ), Du = σstern/(Hσb) is the Dukhin number, which is the ratio of Stern layer conductivity to the bulk ionic conductivity. The integrals appearing in eq 10 are defined as Z δ1 ð2y̅ y̅ 2 Þsinhð4ψ̅ Þ dy; I1 ¼ ̅ ̅ Þ 1Þ 0 1 þ 2νðcoshð4ψ Z δ1 ð1 ψ̅ =ζ ̅ Þsinhð4ψ̅ Þ I2 ¼ dy ̅ Þ 1Þ ̅ 0 1 þ 2νðcoshð4ψ Z 1 ð2δ1 δ1 2 Þsinhð4ψ̅ Þ dy; I3 ¼ ̅ Þ 1Þ ̅ δ1 1 þ 2νðcoshð4ψ Z 1 ð1 ψ̅ δ1 =ζ̅ Þsinhð4ψ ̅ Þ I4 ¼ dy 1 þ 2νðcoshð4ψ ̅ Þ 1Þ ̅ δ1 8 !1=α 9 Z 1 2), it can be observed that the dimensionless potential increases with the steric factor and then plateaus off. For shear thinning fluids (A < 2), on the other hand, we find that the nondimensional streaming potential decreases initially with an enhancement in the steric factor and subsequently remains constant. Figure 5b exhibits the steric effect dependent alteration in the streaming potential, as a function of the steric factor, for different values of the parameter A. Interestingly, it is revealed that, for shear thickening fluids (A > 2), there is a monotonic enhancement in the dimensionless streaming potential with a corresponding increment in the steric factor, beyond the Newtonian limit, as compared to the cases in which steric effects are not considered. Such trends are observable from Figure 5c as well, which depicts simultaneous implications of non-Newtonian behavior and steric interactions. Again, it may be noted from Figure 5ac that the cases represented 12248
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Figure 5. (a) E/ESteric+Newtonian, (b) E/EnoSteric+non-Newtonian, and (c)E/ EnoSteric+Newtonian as a function of ν, for different values of the parameter A. Other dimensionless parameters are taken as follows: δ1 = 0.2, J0 = 4.72, V = 3.535, Du = 0.
Figure 6. (a) η/ηSteric+Newtonian, (b) η/ηnoSteric+non-Newtonian, and (c) η/ηnoSteric+Newtonian, as a function of ν, for different values of the parameter Du considering the parametric variations to be the same as those considered in Figure 2.
by Figure 5b,c, which are typically reminiscent of analyses delineating the implications of steric effects, exhibit remarkably distinctive qualitative behavior as compared to Figure 5a in which steric effects are already considered in the reference case. Figure 6a shows the variations in the relative conversion efficiency, normalized with respect to a case with Newtonian behavior and steric effects are taken into consideration, as a function of steric factor for different values of Du. These plots essentially correspond to the cases presented in Figure 2a. As seen from the figure, for low values of steric factor, the ratio is greater than unity, which drops off rapidly for low values of the steric factor and then tends to become more or less constant. Figure 6b shows the variations in the relative conversion efficiency, normalized with respect to a case with non-Newtonian behavior, and no steric effects are taken into consideration, as a function of steric factor for different values of Du. These plots essentially correspond to the cases presented in Figure 2b. It is seen that the ratio is always less than unity and decreases with the increase in steric factor. Figure 6c shows the variations in the relative conversion efficiency,
normalized with respect to a case with Newtonian behavior, and no steric effects are taken into consideration, as a function of steric factor for different values of Du. These plots essentially correspond to the cases presented in Figure 2c. Again the trend is similar to Figure 6a where the ratio is greater than unity for low values of the steric factor but decreases with initial increases in steric factor and becomes constant for higher values of the steric factor. It may be seen from all three figures that, for a high Du, the efficiency is low which is expected because a larger Du provides alternate path for the conduction current and hence causing a drop in the ratio of the conversion efficiency. Figure 7a depicts variations in a relative energy conversion efficiency as a function of the steric factor for different nondimensionless depletion layer thickness (δ1), with a Newtonian fluid exhibiting steric effects as a reference. These plots essentially corroborate the cases depicted in Figure 3b. For low values of the steric factor, the ratio is greater than unity, which falls off with an increase in the steric factor. Figure 7b shows the variations in the 12249
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Figure 7. (a) η/ηSteric+Newtonian, (b) η/ηnoSteric+non-Newtonian, and (c) η/ηnoSteric+Newtonian, as a function of ν, for different values of the parameter δ1 considering the parametric variations to be the same as those considered in Figure 3.
Figure 8. (a) η/ηSteric+Newtonian, (b) η/ηnoSteric+non-Newtonian, and (c) η/ηnoSteric+Newtonian, as a function of ν, for different values of the parameter V considering the parametric variations to be the same as those considered in Figure 4.
relative conversion efficiency, normalized with respect to a case with non-Newtonian behavior and no steric effects taken into consideration, as a function of steric factor for different values of δ1. There is a decreasing trend with increasing steric factor, for all values of δ1. Figure 7c shows the variations in the relative conversion efficiency, normalized with respect to a case with Newtonian behavior and no steric effects taken into consideration, as a function of steric factor for different values of δ1. These plots essentially correspond to the cases presented in Figure 3c. Analogous to Figure 7b, it is revealed from Figure 7c that there is a decreasing trend in the relative efficiency with increasing steric factor, for all values of δ1. Figure 8a shows variations in a relative energy conversion efficiency as a function of steric factor for different values of V, with a Newtonian fluid exhibiting steric effects as a reference. These plots essentially correspond to the cases depicted in Figure 4a. A monotonically decreasing trend of the relative efficiency with increments in the steric factor is seen for low values of the
parameter V (i.e., for large EDL thickness compared to the channel width). However, the ratio still remains greater than unity for large values of the parameter V, over the entire regime of the steric factor investigated. On the other hand, for smaller values of the parameter V, steeper decrements do occur in the normalized energy conversion efficiency, so that the ratio falls rapidly below unity as the steric factor is progressively lowered. Figure 8b shows the variations in the relative conversion efficiency, normalized with respect to a case with non-Newtonian behavior and no steric effects taken into consideration, as a function of steric factor for different values of the parameter V. These plots essentially correspond to the cases presented in Figure 4b. It is seen that, for increasing steric factor, the relative energy conversion efficiency decreases progressively, irrespective of the value of V. Figure 8c depicts the variations in the relative conversion efficiency, normalized with respect to a case with Newtonian behavior and no steric effects taken into consideration, as a function of steric factor for different values of the parameter V. 12250
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efficiency, normalized with respect to a case with non-Newtonian behavior and no steric effects taken into consideration, as a function of steric factor for different values of the parameter A. These curves essentially correspond to the cases depicted in Figure 5b. It is seen that, for increasing steric factor, the relative efficiency decreases, irrespective of the rheological nature of the fluid. However, the decrement appears to be more rapid for shear thinning fluids, as compared to shear thickening ones. Overall, it can be inferred that for the same non-Newtonian fluid, an increase in the steric factor leads to a decrease in the dimensionless conversion efficiency. Figure 9c shows the variations in the relative conversion efficiency, normalized with respect to a case with Newtonian behavior and no steric effects taken into consideration, as a function of steric factor for different values of the parameter A. Analogous to Figure 9b, all of the curves in this figure show a decreasing trend, with qualitatively similar characteristics. Again, the contrasting features of Figure 9a, as compared to the features of either Figure 9b or c, clearly implicate the distinctive implications of non-Newtonian behavior and steric effects and their suitable combinations on the energy transfer characteristics in narrow fluidic confinements.
Figure 9. (a) η/ηSteric+Newtonian, (b) η/ηnoSteric+non-Newtonian, and (c) η/ηnoSteric+Newtonian, as a function of ν, for different values of the parameter A considering the parametric variations same as those considered in Figure 5.
These plots essentially correspond to the cases presented in Figure 4c. The normalized efficiency turns out to be greater than unity for low values of the steric factor and decreases as the steric factor increases. The decrement is steeper as the value of the parameter V is progressively increased. Figure 9a shows variations in a relative energy conversion efficiency as a function of the steric factor for different values of the parameter A, with a Newtonian fluid exhibiting steric effects as a reference. The plots shown in this figure correspond to the cases exhibited in Figure 5a. For a shear thinning fluid (α < 1), it is seen that the normalized efficiency is greater than unity for low values of the steric factor and falls off as the steric effects is enhanced. For a shear thickening fluid (α > 1), on the other hand, the ratio is less than unity for small values of the steric factor and increases as the steric factor increases. It is seen that for A = 2, which corresponds to α = 1, the ratio, as expected, is equal to unity. Figure 9b shows the variations in the relative conversion
4. CONCLUSIONS We have explored in detail the implications of employing pressure-driven flows of non-Newtonian fluids in the presence of steric effects toward altering hydraulic to electrical energy transfer in narrow fluidic confinements. By employing a semianalytical formalism, we have demonstrated that the exploitation of nonNewtonian effects, under certain conditions (typically for shear thickening fluids), may be effectively considered in conjunction with steric interactions for augmenting energy conversion efficiencies in narrow fluidic confinements to a considerable extent, as corroborated by Figure 9a. This proposition of augmenting energy transfer capabilities may turn out to be particularly appealing, especially in perspective of the rapid advancements in miniaturization technologies over the past few years. Utilizing intricate interactions between electrokinetic and rheological interactions over small scales, our study has revealed that the consequent advective-electromigrative transport of the ionic species may be effectively utilized to drive an external load in an energy efficient manner, particularly with the employment of non-Newtonian fluids of certain kinds. With rigorous theoretical considerations, we have derived a semianalytical formalism to depict relationships among the concerned energy transfer performance, the flow behavior index, and the steric factor, considering electrohydrodynamic interactions over small scales aptly into account. The most interesting finding from our work is that the energy transfer efficiency can improve dramatically by employing higher flow behavior index fluids in the presence of steric interactions, as compared to the case of a reference Newtonian fluid. By utilizing optimal combinations of the parameters in the Discussion, the conversion efficiency can be maximized. Despite the fact that low energy transfer efficiencies have been realized by most experimental researchers until date, the present theoretical considerations exemplify the possibility of improving this scenario considerably, by exploiting combinations of nonNewtonian flow behavior instead of a traditional Newtonian fluid, steric effects, and large interfacial potentials. The only possible limiting constraint may be the Stern layer conductance, which acts so as to reduce the efficiency by virtue of providing an additional mechanism for power dissipation. 12251
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Langmuir Interestingly, the present considerations may further be extended to futuristic fluidic devices approaching a few molecular dimensions, consistent with the fact that, in strongly interacting EDL limits, co-ions are effectively expelled, so their power dissipating effects are avoided. Such possibilities may favorably act with the complex rheological behavior of the fluids being employed for the transport (in particular, shear-thickening fluids), as well as the steric effects, so as to dramatically augment the energy transfer efficiencies to unprecedented limits. Despite such promising features, the exact mechanism of interaction of all pertinent interfacial interaction phenomena in devices having extremely small characteristic scales still remains unclear, in the quest of producing the best efficiency and simultaneously generating the maximum power density. Further activities on nanofluidic research, particularly on the experimental front, therefore need to be directed to demonstrate the real promise in giant augmentation of energy transfer transfer efficiency through intricate interactions between steric effects and non-Newtonian flow rheology in extreme narrow confinements.
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*E-mail:
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