pubs.acs.org/Langmuir © 2010 American Chemical Society
Electrofluidic Gating of a Chemically Reactive Surface Zhijun Jiang and Derek Stein* Department of Physics, Brown University, Providence, Rhode Island 02912 Received November 25, 2009. Revised Manuscript Received January 22, 2010 We consider the influence of an electric field applied normal to the electric double layer at a chemically reactive surface. Our goal is to elucidate how surface chemistry affects the potential for field-effect control over micro- and nanofluidic systems, which we call electrofluidic gating. The charging of a metal-oxide-electrolyte (MOE) capacitor is first modeled analytically. We apply the Poisson-Boltzmann description of the double layer and impose chemical equilibrium between the ionizable surface groups and the solution at the solid-liquid interface. The chemically reactive surface is predicted to behave as a buffer, regulating the charge in the double layer by either protonating or deprotonating in response to the applied field. We present the dependence of the charge density and the electrochemical potential of the double layer on the applied field, the density, and the dissociation constants of ionizable surface groups and the ionic strength and the pH of the electrolyte. We simulate the responses of SiO2 and Al2O3, two widely used oxide insulators with different surface chemistries. We also consider the limits to electrofluidic gating imposed by the nonlinear behavior of the double layer and the dielectric strength of oxide materials, which were measured for SiO2 and Al2O3 films in MOE configurations. Our results clarify the response of chemically reactive surfaces to applied fields, which is crucial to understanding electrofluidic effects in real devices.
Introduction Lab-on-a-chip fluidic technology shares important similarities with the integrated circuit (IC) technology that inspired it. Chipbased micromachining techniques are employed to shrink the size of fluid-handling systems and thereby apply the “smaller, cheaper, faster” paradigm to chemical and biological analysis.1-3 Functional similarities have also emerged, as several ways of using electrostatic fields to manipulate fluidic systems have been demonstrated.4-13 A fluidic version of the field effect is possible because Coulomb forces are screened by counterions in solution over a characteristic distance called the Debye length, which is typically 1-100 nm.14-16 The electrical charge on a gate electrode located beneath an insulating dielectric can therefore influence the electrochemical potential within a thin layer of the fluid above. We use the term “electrofluidic gating” to describe such *To whom correspondence should be addressed. E-mail: derek_stein@ brown.edu. (1) Whitesides, G. M. Nature 2006, 442, 368–373. (2) Janasek, D.; Franzke, J.; Manz, A. Nature 2006, 442, 374–380. (3) Yager, P.; Edwards, T.; Fu, E.; Helton, K.; Nelson, K.; Tam, M. R.; Weigl, B. H. Nature 2006, 442, 412–418. (4) Hillier, A. C.; Kim, S.; Bard, A. J. J. Phys. Chem. 1996, 100, 18808–18817. (5) Schasfoort, R.; Schlautmann, S.; Hendrikse, J.; van den Berg, A. Science 1999, 286, 942–945. (6) Karnik, R.; Fan, R.; Yue, M.; Li, D.; Yang, P.; Majumdar, A. Nano Lett. 2005, 5, 943–948. (7) Fan, R.; Huh, S.; Yan, R.; Arnold, J.; Yang, P. Nat. Mater. 2008, 7, 303–307. (8) Vermesh, U.; Choi, J. W.; Vermesh, O.; Fan, R.; Nagarah, J.; Heath, R. J. Nano Lett. 2009, 9, 685–693. (9) Kalman, E. B.; Sudre, O.; Vlassiouk, I.; Siwy, Z. S. Anal. Bioanal. Chem. 2009, 394, 413–419. (10) Karnik, R.; Castelino, K.; Majumdar, A. Appl. Phys. Lett. 2006, 88, 123114. (11) Fraikin, J. L.; Requa, M. V.; Cleland, A. N. Phys. Rev. Lett. 2009, 102, 155601. (12) Schoch, R. B.; Renaud, P. Appl. Phys. Lett. 2005, 86, 253111. (13) Nam, S. W.; Rooks, M. J.; Kim, K. B.; Rossnagel, S. M. Nano Lett. 2009, 9, 2044–2048. (14) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: San Diego, CA, 1991. (15) Evans, D. F.;Wennerstr€om, H. The Colloidal Domain Where Physics, Chemistry, Biology and Technology Meet, 2nd ed.; Wiley-VCH: New York, 1999. (16) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: New York, 2001.
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voltage-actuated control over the ionic screening layer, which is also called the electric double layer. Recent experimental and theoretical work suggests intriguing technological possibilities for electrofluidic gating in micro- and nanofluidic devices. In this article, we focus on the crucial role played by surface chemistry at the solid-liquid interface, which has yet to be evaluated. We also evaluate how the nonlinear charging of the double layer and the dielectric strength of the insulator both impose limits on electrofluidic gating. The pioneering work of Hillier and Bard4 was an early and illustrative example of electrofluidic gating. They measured the interaction force between a gold surface and a negatively charged silica bead attached to the tip of an atomic force microscope cantilever in solution. They demonstrated that the electroosmotic force could be modulated by varying the electrochemical potential of the gold surface with respect to a reference electrode in the solution. Schasfoort et al. first employed electrofluidic gating in a microfluidic device called a flow field-effect transistor, or flowFET, which comprised a silicon gate electrode, separated from the inside of a microfluidic channel by an insulating layer of silicon nitride. The direction of electroosmotic flow in the channel, which depended on the potential at the surface, could be reversed by applying a sufficiently large positive potential to the gate.5 The flowFET concept was originally proposed by Ghowsi and Gale, who outlined the electrokinetic theory behind its operation.17 Electrofluidic gating has also been employed by several groups to modulate the conductance of nanoscale fluidic channels and pores, demonstrating behavior akin to an “ionic transistor”.6-9,12,13 Such control over ionic currents in nanofluidic structures is possible because the counterions in the double layer dominate transport at sufficiently low salt concentrations, and their density is governed by the charge at the channel surface.18 Similarly, the electrofluidic field effect was used to manipulate the electrical transport of charged proteins.10 Recently, Fraikin et al. (17) Ghowsi, K.; Gale, R. J. J. Chromatogr. 1991, 559, 95–101. (18) Stein, D.; Kruithof, M.; Dekker, C. Phys. Rev. Lett. 2004, 93, 035901.
Published on Web 03/01/2010
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demonstrated a radio frequency “Debye-layer transistor” in which the impedance between two electrodes immersed in solution was modulated by a third gate electrode by making use of the nonlinear behavior of the electric double-layer capacitance.11 Recent theoretical studies have focused on nanofluidic structures explicitly designed to behave as ionic analogs to fundamental semiconductor devices. The microscopic origin of electrofluidic phenomena is the response of mobile ions to electric fields and thermal fluctuations, for which the Poisson-Nernst-Planck (PNP) equations are the standard description. Because the PNP equations are nearly identical to the driftdiffusion equations used to describe charge carriers in a semiconductor, the parallels between electrofluidic and IC devices extend all the way to the underlying physics.19,20 Daiguji et al. applied the PNP description of ion transport and coupled it to the Navier-Stokes description of fluid flow in order to model ion transport inside various nanofluidic devices. Bipolar nanochannels, in which the surface charge density has the opposite polarity on either side of the midpoint, were predicted to show rectifying current-voltage characteristics that mimic semiconductor diodes.21,22 The same authors also predicted that tripolar nanochannels, in which the charge density of a central section has the opposite polarity of the two ends, behave like a bipolar junction transistor: the ends of the channel can be identified as the emitter and the collector, and the ionic current between them is controlled by the magnitude of the surface charge density in the central region, which acted as a gate. In this version of the ionic transistor, the gate exerts control over the transconductance by regulating the degree of ion accumulation and depletion that occur at the discontinuities in the surface charge density. Gracheva predicted the same behavior for ion transport across a nanopore formed through a stack of doped semiconductor films, whose surface potential could be electrically controlled.23 The notion of exerting field-effect control over charged objects in solution is appealing. Local forces can be applied to ions, charged colloidal particles, and biomolecules such as DNA, RNA, and proteins to control their motion in nanofluidic devices. In doing so, lab-on-a-chip technology can leverage the considerable know-how developed by the semiconductor industry. The same silicon-based fabrication methods are available to create nanofluidic networks articulated with gate electrodes, which can in principle be combined with electronic sensing and logic circuitry. Electrofluidic technology could potentially enjoy the same advantages in manipulating colloidal, chemical, and biological systems that integrated circuits enjoy in manipulating information. Despite experimental and theoretical attention to electrofluidic effects, however, the response of the electric double layer to applied electric fields is not properly understood. Importantly, the influence of surface chemistry has not been explored. Most materials spontaneously obtain a surface charge when they come into contact with water, which leads to the formation of an electric double layer, even if no external field is applied.14 The intrinsic surface charge density is determined by a chemical equilibrium, and this can shift in response to an applied field. This possibility was noted by Fan et al.,24 but its consequences were not evaluated.
The theory outlined by Ghowsi and Gale to describe the flowFET concept sought to account for the chemistry of amphoteric silanol groups and ion pairing at the surface of silica.17 However, they did not discuss the implications of the surface chemistry. Their model also ignored the nonlinearities that are inherent in the electrostatics of the double-layer, and it improperly applied the Boltzmann relation in specifying the activity of charged species near the surface (eqs 7-9 in ref 17), compromising the reliability of their predictions. The case of silicon dioxide (SiO2) illustrates the importance of considering surface chemistry. SiO2 is the most important dielectric material for integrated circuits and arguably the most common material in micro- and nanofluidic technology. The surface of SiO2 is highly negatively charged at neutral pH because of the dissociation of Hþ from silanol surface groups.25,26 Measured values of the surface charge density typically fall in the range of 10-100 mC/m2.6,7,12,18,25 When an electric field is applied normal to the surface, it will either draw in or repel Hþ, promoting either protonation or deprotonation, respectively. Any change in the double layer will therefore reflect the combined effects of the applied field and changes in the chemical surface charge density. The net outcome is not known. Furthermore, interesting applications of electrofluidics will likely require either significant swings in the double-layer potential;on the order of ∼kBT/e, where kBT is the thermal energy and e is the charge quantum;or the double layer to actually change polarity. However, consider a SiO2 dielectric layer between the parallel plates of a capacitor. To induce a surface charge density on SiO2 comparable to its 30 mC/m2 intrinsic chemical charge, an applied field of σ/εε0 ≈ 1 V/nm would be required. This value is close to the maximum dielectric strength of even the highest-quality silicon dioxide. After the gate dielectric breaks down, the local field effect is lost because the resulting leakage currents require longrange changes in the electrochemical potential. The surface chemistry and dielectric strength therefore interact to restrict the electrofluidic effects that can be achieved in a real device. It is possible to develop a predictive model of electrofluidic gating by building on extensive studies of the chemistry of oxide surfaces. For instance, the equilibrium charge density of the SiO2 surface depends on the pH and the ionic strength of the aqueous solution in a manner that is well described by the 1-pK basic Stern model.27 This model was extended by Behrens and Grier to account for the structure of the double layer better, especially at low ionic strength.26 Within these models, the native surface charge arises from the dissociation of Hþ from hydroxyl surface groups. Equilibrium is established between the surface charge density and the surface activity of Hþ and is characterized by a dissociation constant, pK. The surface activity of Hþ is in turn related to the bulk pH via the standard model of the double layer, comprising a phenomenological Stern layer capacitance and the Poisson-Boltzmann description of the diffuse layer. It is also possible to model the influence of the chemical surface charge on the potential in the adjacent dielectric material. Van Hal et al. applied such a model of oxide layers in their studies of ion-sensitive fieldeffect transistors (ISFETs), which use a buried semiconductor gate to sense changes in the equilibrium surface charge density induced by pH and the ionic content of a solution.28,29
(19) Eisenberg, R. S. J. Membr. Biol. 1996, 150, 1. (20) Chen, D.; Lear, J.; Eisenberg, B. Biophys. J. 1997, 72, 97. (21) Daiguji, H.; Yang, P.; Majumdar, A. Nano Lett. 2004, 4, 137–142. (22) Daiguji, H.; Oka, Y.; Shirono, K. Nano Lett. 2005, 5, 2274–2280. (23) Gracheva, M. E.; Vidal, J.; Leburton, J. P. Nano Lett. 2007, 7, 1717–1722. (24) Fan, R.; Yue, M.; Karnik, R.; Majumdar, A.; Yang, P. Phys. Rev. Lett. 2005, 95, 086607.
(25) Iler, R. K.; Aler, R. K. The Chemistry of Silica; Wiley: New York, 1978. (26) Behrens, S. H.; Grier, D. G. J. Chem. Phys. 2001, 115, 6716–6721. (27) Hiemstra, T.; Riemsdijk, W. H. W.; Bolt, G. H. J. Colloid Interface Sci. 1989, 133, 91–104. (28) van Hal, R. E. G.; Eijkel, J. C. T.; Bergveld, P. Adv. Colloid Interface Sci. 1996, 69, 31–62. (29) van Hal, R. E. G.; Eijkel, J. C. T.; Bergveld, P. Sens. Actuators, B 1995, 24, 201–205.
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Figure 1. Modeling an electrofluidic gate. (a) Schematic of a metaloxide-electrolyte (MOE) capacitor comprising the gate electrode, oxide insulator, electrical double layer, and bulk electrolyte. The variation of the electrochemical potential across the structure is sketched by the green curve, and its value is defined at the gate (Vg), the oxide surface (ψ0), the double-layer interface (ψDL), and the bulk electrolyte (Vb). The chemical reactivity of the oxide surface is represented by the presence of hydroxyl groups that are either protonated (OH) or deprotonated (O-). (b) Equivalent circuit model of the MOE capacitor. The oxide insulator and the Stern layer are modeled as capacitors of specific capacitance Cins and CStern, respectively, and the electric double layer is a nonlinear element modeled using the Poisson-Boltzmann equation.
In this article, we present a theoretical model of electrofluidic gating at a realistic, chemically reactive surface in contact with ionic solution. We account for the chemical equilibrium of ionizable surface groups that contribute significantly to the surface charge density. Our model can be applied to a surface with any number of ionizable groups, either positive or negative, that can be characterized by standard dissociation constants. We compare the expected gating of a chemically reactive surface with that of an “ideal”, chemically inert surface and generate predictions of two commonly used dielectric gate materials, SiO2 and Al2O3. We also consider the practical range over which applied fields can tune the surface charge density or the double-layer potential. Electrofluidic gating is limited by the nonlinear behavior of the double layer and by the strength of the oxide. We discuss these limitations and perform measurements of dielectric breakdown in thin films of SiO2 and Al2O3.
Electrochemical Model of Electrofluidic Gating Metal-oxide-Electrolyte (MOE) Capacitor. The metaloxide-electrolyte (MOE) structure depicted in Figure 1a can be naturally incorporated into chip-based fluidic devices to perform electrofluidic gating. Its geometry is that of a parallel-plate capacitor whose metallic gate electrode is separated from the conductive electrolyte by a thin insulating oxide. The electric double layer is modulated by applying a voltage across the capacitor, which generates an electric field normal to the solidliquid interface. Here we present a model for the charging Langmuir 2010, 26(11), 8161–8173
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behavior of the MOE capacitor. Our electrochemical model accounts for the chemistry of oxide surfaces and can be treated analytically. The planar gate electrode of the MOE capacitor is biased at voltage Vg. It is protected by an insulating oxide of thickness d and dielectric constant ε. The surface of the insulator is carpeted with ionizable groups that give rise to the chemical surface charge density, σchem, when in contact with solution. The sketch depicts hydroxyl groups possessed by the oxide materials that are commonly used in both integrated circuit and lab-on-a-chip devices. We denote the electrochemical potential at the insulator surface ψ0. The insulator is assumed to be in contact with a solution whose pH, bulk electrochemical potential, Vb, and concentration of monovalent salt, n, are all assumed to be constant far away from the surface. Close to the surface, the electric fields from the gate electrode and σchem induce the formation of an electric double layer, (i.e., an accumulation of counterions and repulsion of co-ions that screen electric fields over a length scale λD, known as the Debye length). Figure 1a illustrates the “standard model” of the double-layer structure. The innermost layer of counterions is known as the Stern layer, which is tightly bound to the surface. The Stern layer is separated from the surface by an effective distance δ comparable to the ionic radius. The electrochemical potential at the Stern layer is denoted ψDL. The diffuse layer is the portion of the electric double layer that extends farther away from the surface, where the electrochemical potential decays to the bulk value, Vb. The electric double layer screens electric fields, whether they originate from the chemical charge at the oxide surface or from the applied voltage across the capacitor. To treat these two contributions separately, we consider an equivalent circuit model of the MOE capacitor, shown in Figure 1b. This simple model allows us to determine the potential and the charge density at every location. It consists of three elements arranged in series: two linear capacitors representing the insulator and the Stern layer and a nonlinear element representing the electric double layer. The potential difference across the insulating layer is Vg - ψ0. The insulator is assumed to have a constant capacitance per unit area, Cins, that accurately describes the dielectric properties of common materials used in micro- and nanofluidic devices such as silicon dioxide (SiO2), aluminum dioxide (Al2O3), and poly(dimethylsiloxane) (PDMS).30-32 The capacitive charge density induced at the surface of the insulator, σins, is given by σ ins ¼ Cins ðVg - ψ0 Þ
ð1Þ
Within the basic Stern model, the potential drops linearly across the solid-liquid interface by an amount ψ0 - ψDL. The relationship between this potential drop and the charge density screened by the double layer, σDL, is given by σDL ¼ CStern ðψ0 - ψDL Þ
ð2Þ
where CStern is the phenomenological Stern capacitance per unit area. CStern reflects the structure and dielectric properties of the solid-liquid interface. The basic Stern model has been widely applied to model the charging of the double layer and is well supported by experimental evidence.26,27,33-35 (30) Sze, S. M. Semiconductor Devices, Physics and Technology, 2nd ed.; Wiley: New York, 2002. (31) McNutt, M. J.; Sah, C. T. J. Appl. Phys. 1975, 46, 3909–3913. (32) Klammer, I.; Hofmann, M. C.; Buchenauer, A.; Mokwa, W.; Schnakenberg, U. J. Micromech. Microeng. 2006, 16, 2425–2428. (33) Bolt, G. H. J. Phys. Chem. 1957, 61, 1166–1169.
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The net charge density on the surface of the insulator must equal the chemical charge density from the ionized surface groups, σchem; therefore, σchem ¼ σ DL - σins
ð3Þ
Chemistry of Oxide Surfaces. The extent to which a surface is positively or negatively charged in solution is determined by the chemical equilibrium between the ionizable surface groups and the solution. The solid surface has a number of reactive groups that can be either neutral or singly ionized. For example, the hydroxyl groups that are common to oxides are amphoteric and can become either negatively or positively charged according to the following dissociation reactions:27,36,37 -
þ
R 3 OH h R 3 O þ H
ð4Þ
R 3 OH2þ h R 3 OH þ Hþ
ð5Þ
Here, R is the element that specifies the type of oxide (e.g., Si, Al, Ti, Hf, etc.). The density of ionizable groups at the oxide surface is Γi, where the subscript i denotes the ith type of chemical group if there are more than one. For clarity, we will begin by modeling a surface with a single, negatively ionizable type of group at which only a single dissociation reaction can occur. We will later describe how the model can be generalized to accommodate multiple types of groups, including positively ionizable or amphoteric groups, and to account for chemical interactions between the surface and ions in solution. The density of negatively ionized groups is denoted by ΓO-. The chemical surface charge density associated with these groups, σchem, is given by O-
σ chem ¼ - eΓ
ð6Þ
where -e is the electron charge. Each surface group must either be neutral or negatively ionized, and the total number is constant; therefore, Γ ¼Γ
O-
þΓ
OH
ð7Þ
The degree to which the surface groups are ionized is determined by the chemical equilibrium between the reactive groups and the Hþ ions in solution. The law of mass action that expresses the chemical equilibrium for the dissociation reaction (eq 4) can be written as ½Hþ 0 ΓO ΓOH
-
¼ 10 - pK
ð8Þ
where [Hþ]0 is the proton activity at the solid surface and the pK value is the standard parametrization of the dissociation constant. Electric Double Layer. The electrochemical potential in ionic solution is determined by a competition between electrostatic and entropic forces acting on mobile ions in solution, which is commonly described by the mean-field Poisson-Boltzmann (PB) model. The Grahame equation, which is a consequence of the PB model, relates the potential drop across the electric double (34) van der Heyden, F. H. J.; Stein, D.; Dekker, C. Phys. Rev. Lett. 2005, 95, 116104. (35) Smeets, R. M. M.; Keyser, U. F.; Krapf, D.; Wu, M.; Dekker, N. H.; Dekker, C. Nano Lett. 2006, 6, 89–95. (36) Schindler, W. P.; Stumm, W. Aquatic Surface Chemistry; Stumm, W., Ed.; Academic Press: New York, 1987. (37) Westall, J. Aquatic Surface Chemistry; Stumm, W., Ed.; Academic Press: New York, 1987.
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layer to the total charge density contained within it for an isolated planar surface σDL
2εε0 K βeðψDL - Vb Þ ¼ sinh βe 2
ð9Þ
where β-1 = kBT, ε0 is the permittivity of free space, ε is the dielectric constant of the electrolyte, and κ-1 = [(εε0kBT)/(e2n)]1/2 defines the Debye length. The activity of protons in the double layer is related to the local electrochemical potential by the Boltzmann factor at thermal equilibrium. The proton activity at the solid surface is given by ½Hþ 0 ¼ ½Hþ b e - βeðψ0 - Vb Þ
ð10Þ
where [Hþ]b is the bulk activity of protons, also equal to 10-pH. Calculating the Response of an MOE Capacitor. The MOE capacitor model can be used to determine ψDL and σDL in terms of the applied voltages (Vg and Vb), the chemistry of the oxide surface, and the properties of the electrolyte. We begin by combining eqs 6-8 to obtain an equation for the chemical surface charge density attributable to the negatively ionized surface groups at equilibrium: σ chem ¼
- eΓ ½Hþ 0 1 þ - pK 10
ð11Þ
Equations 1-3 are combined with eq 10 to yield an expression for the potential at the insulator surface, ψ0, which is inserted into eq 11 above. An expression for σchem is then obtained: σchem ¼ - eΓ ! σ chem þ Cins Vg þ CStern ψDL ðpK - pHÞ 1 þ 10 exp - βe - Vb Cins þ CStern
ð12Þ The chemical reactivity of the oxide surface is reflected in the expression for σchem given by eq 12. Note that the applied gate voltage and the pH of the solution stand on equal footing in that expression. Both appear in the same quantity in the denominator of the right-hand side of eq 12, which suggests that a change in Vg, δVg, is expected to influence σchem in the same manner as a change in the pH, δpH. This equivalence can be expressed as δVg ¼ lnð10ÞðCins þ CStern ÞkB T Cins e
δpH
It is also possible to obtain an independent expression for σchem that is based on the properties of the double layer rather than the surface chemistry. Using eqs 1-3, we first determine σDL in terms of ψDL and σchem. This expression is then combined with the Grahame equation (eq 9) to give σ chem ¼
Cins 2εε0 K βeðψDL - Vb Þ þ Cins ðψDL - Vg Þ sinh 2 C βe ð13Þ
where C = (Cins-1 þ CStern-1)-1 is the capacitance of the insulator and the Stern layer added in series. Equations 12 and 13 provide two independent equations for σchem in terms of ψDL that can be solved self-consistently. Although this system of transcendental equations does not yield Langmuir 2010, 26(11), 8161–8173
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analytic expressions for σchem and ψDL, solutions for a particular choice of Vg, Vb, pH, and pK can be easily evaluated numerically. Solutions correspond to the intersection of the two σchem versus ψDL curves. We have applied this procedure using Matlab software. Solutions for σDL were determined by inserting the numerical solutions for ψDL and σchem into eq 9. Generalizing the Model to Describe Complex Surfaces. It is straightforward to extend our model to account for surfaces with different surface chemistries. First, the charge density from positively ionizable surface groups can be obtained by repeating the procedure described above σ chem ¼ eΓ
σchem þ Cins Vg þ CStern ψDL 1 þ 10ðpH - pKÞ exp βe - Vb Cins þ CStern
!
ð14Þ where the equilibrium of the dissociation reaction (eq 5) is given by ½Hþ 0 ΓOH ΓOH2
þ
¼ 10 - pK
ð15Þ
When a surface comprises multiple types of ionizable groups, each contributes an amount σi to the total chemical surface charge density, which is given by σ chem ¼
X
σi
ð16Þ
i
From eqs 12, 14, and 16, we obtain an expression for σchem at a
σchem ¼
X
σ chem ¼
X
zi eΓi
i
1 þ 10zi ðpH - pKi Þ expðzi βeðψ0 - Vb ÞÞ
-
þ
Γ ¼ ΓO þ ΓOH þ ΓOH2
ð18Þ
By following the same procedure that led to eq 17, we find an exact expression for the chemical charge density at a surface decorated with a single type of amphoteric surface group
expðzi βeðψ0 - Vb ÞÞ þ 10zi ð2pH - pK1 - pK2 Þ expð2zi βeðψ0 - Vb ÞÞ
where z1 = -1 and z2 = 1 are the two possible charge states of the ionizable groups. We point out that eq 19 differs from eq 17 in that the right-hand side of the former equation has an extra term in the denominator. The extra term is a consequence of the second reaction that was allowed to occur at the surface groups. In fact, for each additional reaction that is modeled at a particular surface group, the exact expression for the surface charge density acquires an additional term in the denominator. This fact complicates the modeling of complex surfaces, where multiple types of surface groups, multiple protonation states, ion pairing, and ion adsorption are all possible. Alternatively, we can account for groups with an arbitrary number of ionization states within our model by assigning the appropriate surface density and pK to each dissociation reaction. Treating each reaction as a distinct surface group allows us to apply our model for multiple types of groups (eq 17) to any surface. This simplified approach is accurate as long as there is sufficient separation between the pK values corresponding to the different reactions at a particular surface group. In the Supporting Information, we show that in the typical case where pK1 - pK2 > 2 the surface charge density predicted by the simplified method differs from the exact value by less than 1%. This simplification
ð17Þ
where zi is the valence of the ith group (i.e. þ1 for positive and -1 for negative) and where ψ0 = [(σchem þ CinsVg þ CSternψDL)/ (Cins þ CStern)]. Equation 17 is a generalized version of eq 12 and can be used, as previously described, to compute the electrofluidic gating response of a surface where distinct chemical groups each give rise to different dissociation reactions. The chemistry of oxide surfaces is generally more complex than the 1-pK model commonly used to describe SiO2. The hydroxyl groups of real oxide surfaces are typically amphoteric and are able to become either negatively or positively charged at the same surface site according to the chemical reactions described by eqs 4 and 5. A 2-pK model has been developed to account for these two ionization states.37 The protonation of the neutral state is typically characterized by a much lower pK value than for its deprotonation. We have applied a 2-pK model to obtain an exact expression for the surface charge density arising from amphoteric surface groups, equivalent to eqs 12 and 17. The full derivation appears in the Supporting Information. Here we present the main result. Describing amphoteric sites requires eq 7 to be modified as follows:
zi eΓ zi ðpH - pKi Þ
i ¼1, 2 1 þ 10
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surface where the number and polarity of different types of ionizable groups are arbitrary
ð19Þ
for multiple pK works because a single dissociation reaction typically dominates at any given time. In the unusual case that the multiple pK values are closely spaced, allowing significant fractions of that surface group to exist simultaneously in more than two charge states, significant errors can arise because our simplification does not constrain the total density of that multiply ionizable group. Although we have focused on hydroxyl groups of common oxides, it is straightforward to apply the same model to a wide range of other materials. There are several singly ionizable surface groups common to many organic materials, such as carboxylate,38 sulfide,39 and amine groups40,41 that can be modeled in the same way. In each case, the surface group interacts with protons in the electrolyte, either protonating or deprotonating depending on the local Hþ activity. Our model can even account for the adsorption and pairing of electrolyte ions (e.g., Naþ, Kþ, and Cl-) with trivial substitutions of the form pH f pNa provided the (38) Behrens, S. H.; Christl, D. I.; Emmerzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566–2575. (39) Bebie, J.; Schoonen, M. A. A.; Fuhrmann, M.; Strongin, D. R. Geochim. Cosmochim. Acta 1998, 62, 633–642. (40) Kalman, E. B.; Vlassiouk, I.; Siwy, Z. S. Adv. Mater. 2008, 20, 293–297. (41) Latini, G.; Wykes, M. Appl. Phys. Lett. 2008, 92, 013511.
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Figure 2. Response of the electric double layer to an applied electric field. The Eg dependences of (a) the double-layer potential, ψDL, (b) the charge density screened by the double layer, σDL, and (c) the percentage of surface groups that are ionized are plotted for an oxide surface with different densities of negatively ionizable surface groups. The electrolyte has n = 1 mM and pH = 7. The oxide layer has a dielectric constant of ε = 4.5 and negatively ionizable surface groups whose pK is 7.5, and densities of Γ = 0, 0.5, 1, and 5 ( 1017 m-2) are plotted as indicated. The ionizable surface groups act as a buffer, resisting rapid changes in ψDL and σDL with Eg via compensating shifts in their ionization state. (d) The Γ = 0.5 MOE capacitor is sketched in the four distinct regimes of behavior indicated in a-c. (i) The unbiased surface possesses a native charge density that forms a double layer. (ii) In the buffering regime, a small positive bias causes a greater fraction of surface groups to dissociate, resulting in little change in the double layer. (iii) In the saturated regime, a large positive bias ionizes all of the surface groups and flips the polarity of the double layer. (iv) In the neutralized regime, all of the surface groups are protonated by a large negative bias and the double layer becomes more negative.
adsorption or ion-pairing process can be well characterized by dissociation constants.42 Limitations of the MOE Capacitor Model. The electrochemical model of electrofluidic gating presented above has its limitations. First, it applies to isolated surfaces. In extremely narrow channels, where the double layers of opposing surfaces overlap significantly, the phenomenon of charge regulation is known to occur.43-46 The chemical equilibrium at a surface shifts because of the influence of the counterion cloud of the other. This effect can be taken into account by employing the exact solution of the Poisson-Boltzmann equation in a thin slit rather than the Grahame equation approximation for isolated surfaces.47 Our model also assumes that the oxide surface and the double layer are at thermal equilibrium. This assumption could be violated with the application of strong electric fields parallel to the surface or high fluid shear rates near the surface. Furthermore, it is not clear how quickly equilibrium is established after the (42) de Lint, W. B. S.; Benes, N. E.; Lyklema, J.; Bouwmeester, H. J. M.; van der Linde, A. J.; Wessling, M. Langmuir 2003, 19, 5861–5868. (43) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405–428. (44) Chan, D.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1975, 71, 1046–1057. (45) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 208–228. (46) Biesheuvel, P. M. Langmuir 2001, 17, 3553–3556. (47) Behrens, S. H.; Borkovec, M. Phys. Rev. E 1999, 60, 7040.
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voltage across the MOE capacitor is changed. Whereas the counterion distribution in the electric double layer is expected to relax quickly, on the order of nanoseconds or less,48 the timescale for establishing chemical equilibrium between the fluid and the reactive surface groups can range from a tenth of a second to a few minutes.49,50 The response of a real electrofluidic device may therefore exhibit a frequency dependence that is not captured here. Our equilibrium model may be completely inappropriate for describing high-frequency devices such as the rf Debye-layer transistor reported by Fraikin et al.11 Finally, our model ignores the microscopic structure of the solid-liquid interface. For instance, the widely used triple-layer model distinguishes between strongly bound, partially desolvated ions in the inner Helmholtz plane and more weakly bound, fully solvated ions in the outer Helmholtz plane, which are both on the inside of the diffuse layer. A four-layer model has even been proposed to account for the slight difference in separation from the surface between the innermost layers of cations and anions.51 (48) Gr€un, F.; Jardat, M.; Turq, P.; Amatore, C. J. Chem. Phys. 2004, 120, 9648– 9655. (49) Hachiya, K.; Yamamoto, K.; Inoue, T.; Takeda, K. J. Colloid Interface Sci. 1992, 150, 270–276. (50) Duc, M.; Adekola, F.; Lefevre, G.; Fedoroff, M. J. Colloid Interface Sci. 2006, 303, 49–55. (51) Charmas, R.; Piasecki, W.; Rudzinski, W. Langmuir 1995, 11, 3199–3210.
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It is possible to refine our model to account for such details, but we do not attempt this here because it would complicate the analysis and obscure the understanding that we seek. Furthermore, the nonlinear Poisson-Boltzmann description of the ion distribution already provides an excellent approximation of the solid-liquid interface, as evidenced by measurements of Coulomb forces between colloids and ion transport in nanochannels.52
Modeling Results for Double-Layer Gating The objective of electrofluidic gating is to control ψDL and σDL. The difference between ψDL and Vb determines the strength of the electrostatic forces on charged objects near the oxide surface. Controlling ψDL can also influence the magnitude and direction of electroosmotic flow in the presence of a tangential electric field or the magnitude and direction of the streaming current when a fluid is driven past the electrofluidic gate by an applied pressure. σDL, however, is the net ionic charge density screened by the double layer (or equivalently, the negative of the net charge contained within it). Because counterions can dominate the ionic conductance of nanofluidic systems that are sufficiently small or whose salt solution is sufficiently dilute,18 controlling σDL can translate into controlling the electrical transport properties of nanofluidic systems. In this section, we apply our model to explore the response of a chemically reactive surface to electrofluidic gating. We examine the influence of ionizable surface groups, including different types and densities. We also study how the properties of the electrolyte, namely, the pH and the salt concentration, affect gating behavior. In our calculations, the potential of the electrolyte was held at ground (Vb = 0) for convenience and the oxide thickness was assumed to be 10 nm. Our results are reported in terms of the nominal electrical field applied across the oxide, Eg Vg/d, so that they can be readily compared to devices with different oxide thicknesses. The nominal field is a good approximation of the actual field across the oxide ((Vg - ψ0)/d) when the voltage drop across the insulator exceeds that across the double layer. Chemically Reactive Surface Acting as a Buffer. Ionizable surface groups significantly influence the response of a MOE capacitor to electrofluidic gating. Figure 2a shows the dependence of ψDL on Eg for an oxide surface with various densities (Γ) of a negatively ionizable group (pK = 7.5) in contact with an n = 1 mM, pH = 7 eletrolyte. The polarity and the pK of the reactive surface groups were chosen to simulate the silanol groups at the surface of SiO2. Their densities were chosen to increase from Γ = 0, representing a chemically inert surface, to Γ = 5 1017 m-2. In the absence of ionizable groups (Γ = 0), ψDL increases monotonically with Eg and symmetrically about Eg = 0 V/nm. The nonlinear increase in ψDL with Eg reflects the inherent nonlinearity of the double-layer capacitance and the description of it based on the Grahame equation. Increasing Γ has two important effects on the behavior of ψDL. First, it results in an increasingly negative double-layer potential at Eg = 0 V/nm because of the increasingly negative chemical charge at the surface. For the highest density of surface groups, the potential is found to be -45 mV, a large value that is in good agreement with the -50 mV measured for SiO2.53 Second, increasing Γ weakens the influence of Eg on ψDL, especially for positive values of the applied field. For negative values of the applied field, ψDL collapses onto a curve that does not depend on the density of surface groups. (52) Ducker, W. A.; Senden, T. J. Langmuir 1992, 8, 1831–1836. (53) Sieger, H.; Winterer, M.; M€uhlenweg, H.; Michael, G.; Hahn, H. Chem. Vap. Deposition 2004, 10, 71–76.
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The dependence of σDL on Eg, shown in Figure 2b, offers an illuminating perspective on the same phenomenon. σDL increases linearly with Eg at the chemically inert Γ = 0 surface. Linear behavior is observed in this case because the counterions in the double layer completely screen electric fields far away from the surface, and Gauss’s law dictates that the charge be proportional to the applied field. The introduction of negatively ionizable chemical groups results in nonlinear charging behavior. Increasing Γ weakens the influence of Eg on σDL, resulting in a flattened region of the charging curve to the positive side of Eg = 0 V/nm. The slope of the curve in this region is inversely related to Γ. With increasing Eg, however, the slope eventually recovers and runs parallel to the Γ = 0 curve. The width of the flattened region is proportional to the density of surface groups. The applied field at which the slope recovers is 0 V/nm for Γ = 0, ∼0.4 V/nm for Γ = 0.5 1017 m-2, and ∼0.8 V/nm for Γ = 1 1017 m-2. The slope does not recover within the wide 1 V/nm range of applied fields for the highest surface group density of Γ = 5 1017 m-2. For negative values of Eg, the σDL curves quickly collapse onto the linear behavior of the Γ = 0 surface. Another effect of introducing ionizable surface groups is to increase σDL at Eg = 0 V/nm, which is a response to the increasing chemical surface charge density. An intuitive explanation of the observed behavior can be found in the fraction of surface groups that ionize as a function of the applied gate field, shown in Figure 2c. The chemically reactive surface resists rapid changes in ψDL and σDL with increasing Eg via compensating changes in the chemical surface charge density. In the absence of an applied field (Eg = 0 V/nm), a minority of the surface groups are ionized for the values of Γ considered. As Eg is increased (toward more positive values), neutral surface groups become ionized until no neutral groups remain, and the chemical surface charge density becomes saturated. The saturation field increases with the density of ionizable groups, from ∼0.4 V/nm for Γ = 0.5 1017 m-2 to ∼0.8 V/nm for Γ = 1 1017 m-2. The surface charge does not saturate within the 1 V/nm range of applied fields for the Γ = 5 1017 m-2 surface. For negative values of Eg, surface groups all become neutralized by ∼-0.2 V/nm regardless of the density of surface groups. The response of a chemically reactive surface to electrofluidic gating reveals how ionizable surface groups act as a buffer. The buffering range corresponds to the values of Eg for which the surface groups are neither completely ionized nor completely neutralized, and the buffering capacity is set by Γ. To illustrate the distinct electrofluidic gating regimes observed, the state of the MOE capacitor is sketched at representative values of Eg in Figure 2d (i-iv), taking the Γ = 0.5 1017 m-2 surface as an example. In the absence of an applied electric field, a small fraction of the surface groups ionize spontaneously, imparting a negative native charge density on the surface (Figure 2d, i). The limited ionization of the surface is the result of a self-regulation effect whereby the ionized groups have a strong influence on the surface potential, which in turn controls the local Hþ activity. In the buffering regime, positive applied fields are mostly screened by an increased number of dissociated surface groups (Figure 2d, ii). The double layer is consequently left relatively unaffected because only a minor shift in ψDL is required to induce a significant change in σchem via changes in the activity of Hþ at the surface. The surface groups continue to dissociate with increasing Eg until essentially all are charged (Figure 2d, iii)). In this saturated regime, further increases in the applied field can be screened only by further charging of the double layer, just as at an inert surface. Negative applied fields, however, quickly neutralize surface groups. In the regime where the surface charge has been completely neutralized, the surface again behaves as if it were inert and DOI: 10.1021/la9044682
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Figure 4. pK dependence of electrofludic gating. The dependence
of σDL on Eg is plotted for three surfaces with negatively ionizable groups whose pK values are 5.5, 7.5, and 9.5, respectively. The surface density of reactive groups is held constant at 1017 m-2. The electrolyte has n = 1 mM and pH = 7.
Figure 3. Electrofluidic gating of a surface with multiple ionizable groups. The dependence of σDL on Eg is plotted for surfaces with various densities of both a negatively ionizable chemical group (pK1 = 8) and a positively ionizable group (pK2 = 6). The ionic strength of the electrolyte is n = 1 mM, and its pH is 7. (a) Three surfaces for which the densities of positively and negatively ionizable groups are the same (Γ- = Γþ = 0.5, 1, and 2 in units of 1017 m-2). The response curves are symmetrical about Eg. (b) Three surfaces for which the density of negatively ionizable groups is held constant (Γ- = 2) but the density of positively ionizable groups is varied (Γþ = 0.1, 0.5, and 1). The three response curves show differences in behavior only for negative values of Eg. Varying the density of one type of group does not significantly affect the buffering effect of the other.
the polarity of the double layer can be flipped to negative (Figure 2d, iv). Our modeling results demonstrate that a chemically reactive surface influences electrofluidic gating in important and nontrivial ways. The main effect, buffering the double layer from the influence of applied fields, presents a technological challenge. Very large applied fields may be required to induce even modest changes in the double -layer potential or charge density. Multiple Ionizable Groups. Real oxide surfaces can play host to multiple dissociation reactions. Certain types of oxide, or particular chemical preparations, can leave the surface with multiple types of chemical groups, each with a different pK. For example, silane coupling chemistry is commonly used to tailor oxide surfaces by attaching a specific terminal group.54 A similar treatment of SiO2 with amine-terminated APTES molecules was used to obtain a more neutral surface for electrofluidic gating in an ionic transistor device.24 These facts highlight the importance of considering multiple chemical groups in order to understand electrofluidic gating properly. (54) Plueddemann, E. P. Silane Coupling Agents; Plenum Press: New York, 1982.
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Figure 3 presents the response of σDL to an applied gate field for surfaces with one type of negatively ionizable group (pK1 = 8) and one type of positively ionizable group (pK2 = 6). For these calculations, n = 1 mM and pH = 7. Comparing Figure 3 with Figure 2b, it is clear that the addition of the positively charged surface groups imparts a buffering capacity to negative applied fields. The slope of σDL is reduced on either side of Eg = 0 V/nm. When the densities of both types of surface groups are equal (Figure 3a), the response of σDL to Eg is symmetrical about Eg = 0 V/nm, as expected. As the surface group densities are increased in tandem, the response flattens out and the extent of the buffering regime extends to higher applied fields for both polarities. If the density of only the positively ionizable groups is increased while the negatively charged group density is held constant, then the response curve flattens out only for negative values of Eg, with no significant changes in the response to positive applied fields (Figure 3b). These results reinforce the notion that chemically reactive surfaces buffer the double layer against changes induced by applied fields. The presence of multiple surface groups extends the buffering range. Interestingly, there is little influence on the buffering behavior of one type of group by the other. This can be understood by realizing that for a given range of Eg, changes in σchem are dominated by one type of group but changes in the other are suppressed. The activity of Hþ at the surface is what determines which group prevails. Only when that activity is close to the pK of a group will that group be partially ionized and have a significant buffering effect. If the surface activity of Hþ is far from the pK of a group, then it will be either completely protonated or completely deprotonated, unable to buffer the double layer. For example, when an applied field renders the surface potential negative, the local proton activity will be relatively high and proton absorption by the positively ionizable groups dominates. The buffering action of the negatively ionizable groups is suppressed in this regime because they are fully neutralized. The fact that ionizable groups with widely separated pK values act independently of one another also makes it possible to model surface groups accurately with multiple ionization states as combinations of singly ionizable groups, as described previously. Dependence on the pK of Ionizable Groups. Because ionizable surface groups buffer the double layer against field-induced modulations, it is natural to expect the buffering range to depend on the dissociation constant. Figure 4 shows how the pK Langmuir 2010, 26(11), 8161–8173
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Figure 5. pH dependence of electrofluidic gating. The dependence of σDL on Eg is plotted for two surfaces at pH values of 5, 7, and 9 and n = 1 mM. (a) A surface with a single, negatively ionizable surface group whose pK is 7.5 and whose density, Γ, is 1017 m-2. (b) A surface with both positively and negatively ionizable group types (pK1 = 5 and pK2 = 9, respectively, and Γ = 1017 m-2 for both).
influences the buffering range and the slope of the electrofluidic gating response within it. The dependence of σDL on Eg is plotted for three surfaces with negatively ionizable groups whose densities are equal (Γ = 1 1017 m-2) but whose pK values are 5.5, 7.5, and 9.5. The electrolyte has n = 1 mM and pH = 7. For all three surfaces, a buffering regime is observed for intermediate Eg, where the slope of the curve is reduced relative to its maximum near Eg = -1 V/nm. The onset of the buffering regime occurs at higher Eg for increasing values of pK. The pK also has a significant influence on both the width of the buffering regime and the slope of the curve, which are positively related. The buffering regime is narrowest for the pK = 7.5 surface, which also exhibits the strongest buffering effect (i.e., the weakest response to the applied field). We note that the pK of this surface is closest to the pH of the solution. By contrast, the two surfaces whose pK values differ significantly from the pH have wide buffering regions. In the buffering regime, the slope of the pK = 9.5 curve is steeper than that for the pK = 5.5 curve, which is steeper than for the pK = 7.5 curve. These results indicate that whereas the pK of ionizable groups influences the range of Eg over which the double layer is buffered, the response of the surface to electrofluidic gating is also linked to the difference between the pK and the pH of the electrolyte. The most effective buffering of σDL occurs when the pK is close to the pH. Dependence of the Electrolyte on pH. The pH dependence of electrofluidic gating is of considerable practical importance because the pH is an experimental parameter that can be easily Langmuir 2010, 26(11), 8161–8173
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adjusted. The pH of the bulk electrolyte influences the chemical charge of oxide surfaces and is consequently expected to influence the magnitude and polarity of the double layer and its response to applied fields. The nanofluidic bipolar transistors realized by Kalman et al. operated successfully thanks to control over the pH of the bulk electrolyte.40 Figure 5a plots the dependence of σDL on Eg for an oxide surface with a single type of negatively ionizable group (pK = 7.5, Γ = 1017 m-2) at pH values of 5, 7, and 9. The salt concentration is taken to be n = 1 mM. The strongest buffering effect (i.e., the flattest response of σDL to Eg) is observed at pH 7. The center of the buffering range shifts toward more positive values of Eg with increasing pH and toward negative values with decreasing pH. In either direction, the slope of the response curve in the buffering region is observed to increase. The case of multiple (or ambipolar) surface groups is considered in Figure 5b. The influence of Eg on σDL is plotted for an oxide surface with equal densities (Γ1 = Γ2 = 1017 m-2) of positively and negatively ionizable groups (pK1 = 5 and pK2 = 9) at pH values of 5, 7, and 9 with n = 1 mM. When the pH of the electrolyte equals the pK of one of the groups, a strong buffering effect is observed: buffering occurs for negative Eg at pH 5 and for positive Eg at pH 9. However, when the pH falls between the two pK values (pH 7), the slope of the response curve is steeper but the buffering range extends roughly twice as far as in the previous two cases, spanning the values of Eg for which strong buffering was observed at those pH values. The pH also has the expected influence on the surface charge density in the absence of an applied field: the chemical charge at the surface becomes more negative with increasing pH and more positive with decreasing pH. For the surface modeled in Figure 5b, the highly symmetric choice of pK and Γ values leads to charge neutrality at pH 7, referred to as the point of zero charge (PZC).55 The influence of pH on electrofluidic gating can again be easily understood in terms of the buffering properties of the surface groups. Buffering is the result of partially dissociated surface groups shifting their equilibrium in response to an applied field. Effective buffering results from a steep change in σchem in response to a change in Eg. Starting with the expression for a particular chemical group’s contribution to σchem (eq 17), it is straightforward to show that ∂σchem/∂Eg is maximal when pH = pK. Groups whose pK is far from the pH of the electrolyte, however, do not contribute strongly to buffering because these remain mostly protonated (pH < pK) or deprotonated (pH > pK). The efficacy with which surface groups buffer the double layer during electrofluidic gating is consequently related to how close their pK is to the pH. In developing our model of electrofluidic gating, we noted that pH played an analogous role to the applied gate field in eq 17. This observation, together with the modeling results presented in Figure 5, suggests that by controlling the pH the electrofluidic gate can be chemically biased. The practical implications of controlling the pH are clear. An electrofluidic gate can be chemically biased to a useful range (e.g., close to the PZC) without applying impractically high gate fields. Dependence on the Ionic Strength of the Electrolyte. The concentration of salt in an electrolyte can be easily adjusted over several orders of magnitude in experiments. The concentration of salt determines both the conductivity and the Debye screening length of a solution, and it affects dissociation reactions. For these reasons, it is important to consider its impact on electrofluidic gating. (55) Parks, G. A.; Bruyn, P. L. J. Phys. Chem. 1962, 66, 967–973.
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Figure 6. Ionic-strength dependence of electrofluidic gating. The dependence of (a) σDL and (b) ψDL on Eg is plotted for n ranging from 0.1 mM to 1 M, as indicated. In this simulation, the pH of the electrolyte was 7 and the chemically reactive surface had a density of Γ = 2 1017 m-2 of negatively ionizable groups with a pK of 7.5.
The salt-dependent response of a chemically reactive surface to electrofluidic gating, presented in Figure 6, shows that the buffering effect of the surface groups is moderated by increasing the ionic strength. The dependence of σDL on Eg is plotted for n ranging from 0.1 mM to 1 M in Figure 6a. The surface was assumed to have a density of Γ = 2 1017 m-2 of negatively ionizable groups, whose pK is 7.5, and the pH of the electrolyte was 7. Under these circumstances, the buffering effect of the surface occurs primarily at positive electric fields, as previously discussed. The strongest buffering effect is observed at the lowest salt concentration of 0.1 mM, where the σDL curve shows a clear plateau within a range of Eg from 0 to 0.75 V/nm . With increasing salt concentration, the response of σDL to Eg in the buffering regime grows steeper. At n = 1 M, the σDL curve shows no sign of a buffering plateau. For very large positive or negative values of Eg, the slope of σDL grows slightly shallower with increasing n. These observations suggest that the buffering effect is distributed over a wider range of Eg with increasing ionic strength. The dependence of ψDL on Eg is plotted for the same conditions in Figure 6b. The strong buffering effect observed in the gating of σDL at low salt concentrations is reflected in a sharp reduction in the slope of ψDL in the buffering regime. The relative flattening of the slope is more pronounced at low salt concentrations. However, the absolute value of the slope of ψDL remains higher at low n than at high n. These observations can be understood by considering the differential capacitance of the double layer, ∂σDL/∂(ψDL - Vb), which is obtained by differentiating the Grahame equation (eq 9).15 The electric double layer has a lower differential capacitance at low n than at high n. A smaller change in the density of the counterions is consequently required to screen a given potential across the double layer when n is low. The flat response of σDL in the buffering regime at low n can be explained 8170 DOI: 10.1021/la9044682
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by the fact that only a small shift in the chemical charge density is needed to offset a given change in the potential at the oxide surface. The low differential capacitance of the double layer at low n also explains the enhanced response of ψDL to an applied field, even in the buffering regime. The MOE structure was modeled as a series of capacitors, as shown in Figure 1b. The voltage drop between the gate and the bulk electrolyte falls increasingly over the double layer when its capacitance is low. At high n, however, the potential falls increasingly across the oxide dielectric layer and changes in the gate potential have relatively little influence on ψDL for reasons of capacitance, rather than chemical buffering. An interesting feature of Figure 6 is that the σDL and ψDL curves all cross zero at the same value of the applied electric field. The reason is that in order to neutralize the double layer completely, ψDL must equal the bulk potential, Vb (set at 0 V in our calculations), according to eq 9. According to eq 3, the induced charge on the capacitor, σins, must then balance the chemical surface charge density, σchem, which is independent of the ionic strength when ψDL = Vb (it depends only on the pH in the absence of a double layer). The ionic-strength dependence of electrofluidic gating reveals important tradeoffs for applications based on controlling the ionic conductance of a nanofluidic device. The key is for σDL to respond sensitively to Eg, which is favored at high n, where the buffering effect of the surface is relatively weak. However, high n also implies a high background conductance of the fluid, which may obscure the desired field effect. There is no such tradeoff for applications based on controlling the Coulomb forces on charged objects in solution. ψDL is most responsive to Eg at low n, despite the strong buffering effect.
Practical Limits of Electrofluidic Gating Simulations of electrofluidic gating suggest that nearly any value of the charge density or the potential in the double layer can be obtained by simply applying the appropriate gate field. In practice, however, the influence of the field effect over fluidic systems is limited. The nonlinear charging of the double layer and the dielectric strength of oxide materials both place sharp restrictions on the maximum fields that can be transmitted in solution. Nonlinear Response of the Electrolyte. Because the doublelayer structure is determined by a competition between electrostatic and thermal forces on the ions in solution, the electrochemical potential far from a charged surface cannot significantly exceed the thermal voltage, kBT/e ≈ 25 meV. When |ψDL| . kBT/e, counterions condense in a narrow layer at the surface and reduce the potential there until it is comparable with the thermal voltage,56 which is analogous to the Manning condensation of counterions at a highly charged rod.57 Beyond this layer of counterions, increasing the surface potential has no influence.58 In the diffuse part of the double layer, where the potential is small compared with kBT/e, the potential decays exponentially with distance from the surface. The effective surface potential, ψeff, and the effective surface charge density, σeff, describe the apparent electrical state of the surface, toward which the linearized Poisson-Boltzmann equation extrapolates. These quantities are relevant in situations where electrofluidic gating seeks to modulate Coulomb forces on charged objects. (56) Manning, G. S. J. Phys. Chem. B 2007, 111, 8554–8559. (57) Manning, G. S. J. Chem. Phys. 1969, 51, 924–933. (58) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions, Cambridge Monographs on Mechanics and Applied Mathematics; Cambridge University Press: Cambridge, U.K., 1989
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Figure 7. Saturation of the effective surface charge and potential. (a) The effective surface potential, ψeff, is plotted as a function of ψDL. (b) The effective surface charge density, σeff, is plotted as a function of σDL for ionic strengths of 1, 10, and 100 mM. The shaded regions in panel a and the dotted sections of the curves in panel b indicate the strongly nonlinear regime, where ψDL > 200 mV and the Poisson-Boltzmann model of the double layer is expected to break down.
For a flat, isolated surface, ψeff is related to ψDL by26 ψeff ¼
4 βeψDL tanh βe 4
ð20Þ
and σeff is related to σDL by the Grahame equation (eq 9), eq 20, and26 σeff ¼ εεo Kψeff
ð21Þ
Figure 7a plots the relationship between ψeff and ψDL and clearly shows how ψeff tracks ψDL for low surface potentials, but its magnitude saturates at 4kBT/e ≈ 100 mV when the magnitude of ψDL exceeds ∼4kBT. The saturation in ψeff limits the electrostatic potential that can be induced by electrofluidics because only values of ψDL below ∼4kBT lead to measurable differences in long-range electrostatic forces. Similarly, Figure 7b shows a saturation in σeff for large values of σDL; however, the maximum σeff increases with salt concentration. We also note that when the magnitude of the electrochemical potential exceeds ∼8kBT, the double layer enters a very nonlinear regime where the mean-field Poisson-Boltzmann description is expected to break down.59 The structure and behavior of the double layer are not yet understood in this highly charged regime. The shaded regions in Figure 7a and the dotted sections of the curves in Figure 7b (59) Kilic, M. S.; Bazant, M. Z. Phys. Rev. E 2007, 75, 021503–021518.
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indicate this strongly nonlinear regime, where the validity of our model is dubious. Measurements of the Dielectric Strength of MOE Capacitors. The dielectric strength of the oxide insulator places another limit on the maximum fields that can be applied for electrofluidic gating. Current-voltage (I-V) characteristics of metal oxide-semiconductor (MOS) devices have been extensively studied to characterize the dielectric strength of thin insulating films.60-64 A high-quality dielectric such as thermally grown SiO2 can withstand fields as high as ∼1 V/nm in such devices. The mechanism by which breakdown is understood to occur in MOS structures is a two-step process: first, high fields generate defects in the oxide, and then a conductive path is created through the network of defects. It has been reported that the generation of defects is promoted by the presence of water vapor.65 Furthermore, the high mobility of ions such as sodium in SiO2 is known to cause problems in the semiconductor industry.30 It is therefore unclear at what electric field, or by what mechanism, dielectric breakdown will occur in an MOE capacitor. We are not aware of any previous measurements of the dielectric strength of oxides in contact with electrolytes. Here we report I-V measurements on MOE structures that tested the dielectric strength of different oxide materials. A number of oxide films were deposited on conductive substrates by different methods in preparation for I-V measurements: (1) 500 nm of thermal oxide was grown on a silicon substrate (Æ100æ, phosphorus doping, 210 Ω m, Virginia Semiconductor). (2) SiO2 (20 nm) was deposited by low-pressure chemical vapor deposition (LPCVD) onto a doped silicon substrate using SiH4 and O2 as the source gases. (3) Al2O3 (30 nm) was deposited on Cr by atomic layer deposition (ALD). The precursor gases used were water vapor and trimethylaluminum, and the deposition was performed at 300 C. (4) HfO2 (30 nm) was deposited on Cr by ALD. Water vapor and tetrakis(dimethylamido)hafnium were the precursor gases, and the growth temperature was 300 C. (5) SiO2 (30 nm) was deposited on Cr by ALD. Water vapor and SiCl4 were the precursor gases, and the growth temperature was 300 C. For all samples grown on Cr, a 30-nm-thick Cr layer was deposited onto a silicon substrate by electron-beam evaporation. The experimental setup used to perform the I-V measurements is shown in Figure 8a. Either the doped silicon substrate or the Cr layer was used as the gate electrode. Electrical contact with the gate was made by scratching the surface of the insulating film with a diamond scribe and then fixing a silver wire to the exposed area using silver paint. Nitrile O-rings (4 mm i.d., 6 mm o.d.) were glued to the oxide surface using silicone elastomer (PDMS, Corning), defining reservoirs for the electrolyte. The reservoir was filled with a 1 mM KCl þ 1 mM Tris-HCl (pH 8.2) solution, and a Ag/AgCl electrode was immersed in it. I-V curves were obtained using a computer-controlled DAQ card (National Instruments, PCIe-6251) to apply a voltage across the electrodes and to record readings of the current at 1 kHz obtained from a current-to-voltage converter (Stanford Research Instruments, SR 570). The voltage sweep was controlled by a custom Labview (60) Friedrichs, P.; Burte, E. P. Appl. Phys. Lett. 1994, 65, 1665–1667. (61) Rao, V. R.; Eisele, I.; Patrikar, R. M.; Sharma, D. K.; Vasi, J.; Grabolla, T. IEEE Electron Device Lett. 1997, 18, 84–86. (62) Kakos, J.; Mikula, M.; Harmatha, L. Microelectron. J. 2008, 39, 1626– 1628. (63) Groner, M. D.; Fabreguette, F. H.; Elam, J. W.; George, S. M. Chem. Mater. 2004, 16, 639–645. (64) Forsgren, K.; Harsta, A.; Aarik, J.; Aidla, A.; Westlinder, J.; Olsson, J. J. Electrochem. Soc. 2002, 149, F139–F144. (65) DiMaria, D. J.; Stathis, J. H. J. Appl. Phys. 2001, 89, 5015–5024.
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Figure 8. I-V measurements of the dielectric strength in MOE structures. (a) Schematic showing the MOE capacitor structure and the measurement setup. (b) Dependence of the gate current density on Eg for thermal SiO2 and LPCVD SiO2. (c) Dependence of the gate current density on Eg for Al2O3, HfO2, and SiO2 layers grown by ALD.
program. Each voltage was applied for 10 s, which turned out to be sufficient for the current to stabilize in our experimental setting. To exclude capacitive charging transients and to reduce noise, we averaged the last 2 s of the current readings. The dependence of the current density through oxide films on the applied electric field is plotted in Figure 8b,c for all of the materials tested. The nominal electric field, Eg, was obtained by dividing the voltage by the oxide thickness. The measured current was converted to the gate current density using an estimate of the contact area between the electrolyte and the dielectric film, π(i.d.)2/4 = 0.126 cm2. For all materials tested, the current density was imperceptibly small until a threshold field was reached, at which point the gate current density increased rapidly. The highest threshold of 0.63 V/nm was observed for thermal SiO2. LPCVD oxide exhibited the next highest threshold of 0.15 V/nm. All three materials grown by ALD had approximately the same threshold of 0.07 V/nm, the lowest for the materials tested. 8172 DOI: 10.1021/la9044682
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We ascribe the abrupt rise in the current density to the dielectric breakdown of the oxide films. The dielectric strength of the oxides in contact with 1 mM KCl solution was generally lower than has been reported for similar oxides in MOS structures, although the large contact area used in our tests would tend to reduce the measured dielectric strength; the dielectric strengths of thermal, LPCVD, and ALD SiO2 in MOS structures have been reported to be 1,60 0.7,61 and 0.54 V/nm,62 respectively, and strengths of 0.37-0.4463 and 0.4 V/nm64 have been reported for ALD Al2O3 and HfO2, respectively. These results represent the first experimental tests of the dielectric strength of oxides in contact with electrolytes. They suggest that the range of applied fields that can be used for electrofluidic gating is narrower than for gating comparable MOS devices, and as a consequence, understanding the dielectric breakdown in MOE structures will become important as electrofluidic applications are developed. Simulation of SiO2 and Al2O3. SiO2 and Al2O3 are among the most commonly used dielectric materials, and their surface chemistry has been extensively studied. Here we apply our model of electrofluidic gating to predict their response, using experimentally determined values for the density of ionizable sites Γ, their dissociation constants pKi, and the dielectric constant ε. For SiO2, the density of negatively ionizable silanol groups is ∼Γ = 8 1018 m-2, and the dissociation constant is pK = 7.5.26 The dielectric constant is chosen to be ε = 4.5 in our simulation, whereas the actual dielectric constant of SiO2 depends on the mode of deposition and processing and can vary from 3.9 to 4.5.66 As for Al2O3, the ambipolar hydroxyl groups have dissociation constants of pK1 = 10 for the negative ionization reaction and pK2 = 6 for the positive ionization reaction.28 The dielectric constant of Al2O3 is 9.67 The density of ionizable sites on crystalline Al2O3 ranges from 6 to 16.5 nm-2 depending on the crystal structure.68 Here, we set the density of ionizable sites to be Γ = 8 1018 m-2, equal to the density of sites on SiO2, in order to approximate an amorphous structure of Al2O3. Although the actual values of Γ, pK, and ε may vary depending on the preparation of the material, our choice of parameters here is meant to provide generic predictions for the gating of SiO2 and Al2O3 surfaces. Figure 9 shows the responses of realistic SiO2 and Al2O3 surfaces to electrofluidic gating. For both materials, three quantities are plotted against Eg: the double-layer potential, ψDL, the surface charge density screened by the double layer, σDL, and the differential capacitance per oxide thickness, dσDL/dEg. This last quantity corresponds to the slope of the σDL curve, which represents the responsiveness of the surface to gating. Simulations were performed for pH values of 5, 7, and 9 for SiO2 and for pH values of 6, 8, and 10 for Al2O3. The salt concentration was n = 1 mM in all cases. The plots also indicate ranges of Eg over which the electric field effect can be applied without inducing significant leakage currents across the oxide. These ranges, indicated by shaded bands, were determined experimentally on thin films of thermal, LPCVD, and ALD SiO2 and on ALD Al2O3. The predicted behavior of SiO2 and Al2O3 indicate how strongly electrofluidic gating depends on the chemistry of the oxide surface and on the pH of the electrolyte. Gating is also (66) Mizuno, S.; Verma, A.; Tran, H.; Lee, P.; Nguyen, B. Thin Solid Films 1996, 283, 30–36. (67) Shimada, Y.; Yamashita, Y.; Takamizawa, H. IEEE Trans. Compon., Hybrids, Manuf. Technol. 1988, 11, 163–170. (68) Sposito, G., Ed. The Environmental Chemistry of Aluminum, 2nd ed.; Lewis Publishers: Boca Raton, FL, 1995.
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modeling also suggests how the chemistry of the oxide surface should be tailored to reduce the density of ionizable groups and to neutralize the net chemical charge in order to benefit electrofluidic applications.
Conclusions
Figure 9. Electrofluidic gating simulations for SiO2 and Al2O3
surfaces. The dependences of (a) ψDL, (b) σDL, and (c) the differential capacitance per oxide thickness, dσDL/dEg, on the applied electric field, Eg, are plotted for a realistic SiO2 surface (see the text for model parameters) at n = 1 mM and for pH values of 5, 7, and 9. The range of applied fields that can be accessed without inducing significant leakage are indicated as shaded bands whose width was determined experimentally for thermal, LPCVD, and ALD SiO2 films. The dependences of (d) ψDL, (e) σDL, and (f) the differential capacitance per oxide thickness, dσDL/dEg, on the applied electric field, Eg, are plotted for Al2O3 (see the text for model parameters) at n = 1 mM and for pH values of 6, 8, and 10. The experimentally determined range of Eg that does not cause leakage in ALD Al2O3 films is indicated by the light band.
strongly constrained by the dielectric strength and by the saturation of ψeff and hence long-range electrostatic forces when ψDL exceeds 4kBT. Nevertheless, our modeling has indicated that it is possible to find a useful range of pH for electrofluidic gating, which is near the point of zero charge of a particular surface. Our
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We have presented a theoretical model for electrofluidic gating at a chemically reactive surface. Our model accounts for the presence of ionizable chemical groups at the solid-liquid interface and how the chemical equilibrium shifts in response to an applied electric field. The main effect of chemically reactive surface groups is to buffer the double layer against changes, which is a distinguishing feature of the fluidic version of the field effect. The response of a real surface to electrofluidic gating is consequently predicted to depend strongly on the chemical properties of both the surface (i.e., the density and dissociation constants of the ionizable groups) and the electrolyte (i.e., the ionic strength and the pH). We also identified the practical limits to electrofluidic gating that are imposed by the nonlinear charging behavior of the double layer and by the diminished dielectric strength of oxides in contact with electrolytes. These results provide a framework of understanding for electrofluidic gating that can help in the development of devices and applications. Interesting technological opportunities for electrofluidics should arise from the ability to modulate Coulomb forces, ionic transport, and even noise69 in nanofluidic systems. Acknowledgment. We thank Alexander Zaslavsky and John Tsakirgis for the oxide-coated wafers. We also thank Travis Del Bonis-O’Donnell for help with the manuscript. Z.J. acknowledges support from the National Science Foundation through a Rhode Island EPSCoR Graduate Student Fellowship. This work was supported by the NSF under grant DMR-0805176. Supporting Information Available: The chemical charge density from amphoteric surface groups. Comparing the simplified and exact expressions for the surface charge. A simplified model based on two independent types of surface groups. This material is available free of charge via the Internet at http://pubs.acs.org. (69) Hoogerheide, D. P.; Garaj, S.; Golovchenko, J. A. Phys. Rev. Lett. 2009, 102, 256804.
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