Capacitive biosensing of bacterial cells: Analytical model and numerical simulations

Sensors and Actuators B 211 (2015) 428–438

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Sensors and Actuators B: Chemical journal homepage: www.elsevier.com/locate/snb

Capacitive biosensing of bacterial cells: Analytical model and numerical simulations N. Couniot ∗ , A. Afzalian, N. Van Overstraeten-Schlögel, L.A. Francis, D. Flandre ICTEAM Institute, Université catholique de Louvain, Place du levant 3-L5.03.02, 1348 Louvain-La-Neuve, Belgium

a r t i c l e

i n f o

Article history: Received 5 November 2014 Received in revised form 14 January 2015 Accepted 27 January 2015 Available online 7 February 2015 Keywords: Bacterial detection Interdigital capacitor modeling Metal–insulator–electrolyte (MIE) Electrical double layer Planar electrodes CMOS biosensors

a b s t r a c t Impedimetric biosensors with a passivation layer, also called capacitive biosensors, have recently shown great promise towards sensitive, selective and rapid detection of pathogen bacterial cells. However, few studies focus on their modeling, yet critical for the optimization of their sensitivity. To address this issue, we propose a comprehensive framework by developing analytical models and 2D numerical simulations of passivated interdigitated microelectrodes (IDEs) with adherent bacterial cells in electrolyte. While models provide a qualitative and semi-quantitative analysis of the AC impedance spectroscopy based on the system cutoff frequencies, finite element method (FEM) simulations based on Poisson–Nernst–Planck equations enable accurate quantification of the sensitivity to bacteria versus the applied frequency thanks to modeling of complex phenomena such as ion transport, surface and space charges, multi-shell bacterial dielectric properties and sensor topology. These numerical simulations are assessed by experimental results and compared to analytical models. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Capacitive biosensors could enable compact, portable and electrical sensing of biological species [1]. They consist of non-faradaic electrodes that prevent charge transfer at the electrode interface. The application of sinusoidal voltages on electrodes enables extraction of the AC complex impedance, often dominated by the sensor capacitance. Its variation is related to the amount of biological species that specifically bind on bioreceptors grafted to the sensor surface. Using micrometer-sized interdigitated electrodes (IDEs) also called interdigitated array microelectrodes (IDAM), capacitive biosensors have demonstrated large sensitivity to bacterial cells [2,3] and capability to be easily co-integrated with a CMOS readout interface [4–8], providing miniaturization, system integration, affordability and portability, which are key features for point-ofcare (PoC) diagnosis tools [9]. The microelectrodes are typically coated by a thin insulating passivation layer. When self-assembled from biological materials [10], this insulation mainly acts as a selectivity means to bind target bacteria [2]. When grown by microfabrication (SiO2 [7], Al2 O3 [8], Ta2 O5 [11], etc.), the insulating layer mainly acts as protective coating against electrochemical corrosion of the electrode material such

∗ Corresponding author. Tel.: +32 10472174. URL: http://[email protected] (N. Couniot). http://dx.doi.org/10.1016/j.snb.2015.01.108 0925-4005/© 2015 Elsevier B.V. All rights reserved.

as aluminum in CMOS process [12]. Despite these advantages, the insulating layer can degrade the bacterial sensitivity [13]. To optimize the complex frequency-dependent dielectric system formed by discrete bacteria and metal–insulator–electrolyte (MIE) structure towards the highest sensitivity, most research works model the electrodes and bacterial cells by an equivalent electrical circuit combining lumped elements [2], providing only a qualitative analyzis of the sensor behavior in the presence of bacterial cells. Despite exact expressions of the IDE cell constant [14] and electric field penetration depth [15] in simple electrolytes, the spectral variation of the IDE impedance in presence of bacterial cells remains unformulated. Instead of analytical models, 2D or 3D electrostatic simulations have been reported to accurately quantify the sensitivity to bacterial cells depending on their random positions on the 3D electrode topology [16], their number [16,17] and the thickness of the passivation layer [13]. Such 2D or 3D simulations have also been implemented to analyze the impact of the applied frequency on the sensor impedance [18] and sensitivity [19–21] of nanometer-scale biomolecules. In this work, we propose a comprehensive analysis of the spectral impedance and sensitivity of capacitive biosensors in presence of bacterial cells based on analytical models, 2D numerical simulations and experimental measurements. In Section 2, the analytical models are first established for the MIE interface without and with bacterial cells, showing how cutoff frequencies govern the sensor and bacterial impedances. The maximal sensitivity is

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Fig. 1. Schematic cross-section (not at scale) of passivated microelectrodes with an adherent Gram-positive bacterial cell in electrolyte buffers. The bacterial cytoplasm (3) presents two outer shells: (1) the cell wall and (2) the plasma membrane. All parameters are summarized in Tables 1 and 2.

estimated and the impact of metallic access lines on the sensor impedance is investigated. In Section 3, finite element method (FEM) numerical simulations are developed in coupled electrostatic (equilibrium) and AC domains with 2D Poisson–Nernst–Planck equations. Thanks to ion transport modeling, the electrical double layers atop the oxide and around bacterial cells are accurately modeled. Furthermore, oxide and bacterial charges are considered and shown to impact the sensor impedance and sensitivity. Such simulations also account for the multi-shell dielectric representation of bacterial cells and the 2D topology of electrodes and bacteria. Finally, in Section 4, model and simulation results of the sensor impedance and sensitivity are compared with experimental data.

2.1. Electrolyte without bacterial cells

2. Analytical model

Cins 

To provide high sensitivity to bacteria, the transducer typically consists of IDEs with gap and width of similar size to bacteria [3]. These micrometer dimensions enhance the sensitivity, thanks to the larger active area for bacteria binding and the more confined electric field around adherent bacteria [3,15]. Sinusoidal voltage amplitudes applied on these microelectrodes are typically smaller than the thermal voltage (26 mV) to ensure the system linearity [23], which enables the extraction of the complex impedance by a phasorial analysis. The applied frequency and angular frequency are denoted by f [Hz] and ω = 2f [rad/s], respectively. The Section 2.1 models the system in presence of an electrolyte without bacterial cells while in Section 2.2, bacterial cells are added to the model. Fig. 1 depicts the sensor cross section used throughout this paper and its equivalent electrical model. Table 1 summarizes the used parameters and 0 denotes the vacuum permittivity.

The device consists of metal electrodes, placed atop a buried oxide and covered by a thin insulating layer (Fig. 1). The AC device behavior is represented by a system of linear capacitors and resistors, whose key part resides in the series combination of Cins , CDL , and the parallel association of Rsol and Csol . It is also usual to place Csol in parallel to the series combination of CDL and Rsol [2], but this does not change the global complex impedance at electrolyte conductivities larger than 1 mS/m. Due to the non-faradaic nature of the interface, Warburg impedance and charge-transfer resistance can be neglected [24]. Expressions of the surface capacitances can be simplified by the capacitance formula between two-parallel plate conductors:

CDL 

0 r,ins tins

0 r,sol D

where D 

· (Ne − 1)Ae

(1)

· (Ne − 1)Ae

(2)



0 r,sol kB T 2q2 Nav Cions · 103

is the Debye length (cfr Table 2 for def-





inition of all physical parameters), Ae  te + w2e · Le the electrode area and Ne the total number of electrodes. The electric field inside the double layer (DL) is assumed sufficiently small (<107 V/m) to keep the DL relative permittivity identical to r,sol [21]. Based on values mentioned in Table 1, D  24 nm and CDL /Cins 

tins D



· r,sol  12 r,ins so that the insulator capacitance Cins dominates in series with CDL . The DL is then screened by the insulator layer for electrolyte ionic strengths Cions > 0.1 mM, since D decreases at higher ionic strength making CDL /Cins larger than 12. The medium conductance −1 and capacitance Csol can be expressed as the sum of the Gsol  Rsol

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Table 1 Physical parameters used for the equivalent electrical model Symbols

Descriptions

Dependences

Typical values

Units

r,ox

Oxide relative permittivity Oxide capacitance

– ∼r,ox , de , we , Ne , Le

3.9a –

– F

Cins lat Cins

Insulator thickness Insulator relative permittivity Insulator capacitance Lateral insulator capacitance

– – ∼tins , r,ins , te , we , Ne , Le ∼tins , r,ins , de , Ne , Le

33 nma 9a – –

m – F F

Electrode

Le we te de Ne Ae

Electrode length Electrode width Electrode thickness Electrode gap Total number of electrodes Electrode surface area

– – – – – ∼te , we , Ne , Le

250 ␮ma 2 ␮ma 1 ␮ma 4 ␮ma 39a –

m m m m – m2

Electrolyte

Cions

Electrolyte ionic strength Electrolyte relative permittivity Electrolyte conductivity Debye length Electrolyte capacitance and resistance Double layer capacitance

– – ∼Cions ∼Cions , r,sol ∼r,sol , sol , de , te , we , Ne , Le ∼r,sol , D , te , we , Ne , Le

0.16 mMb 80b 1.8 mS/mb – – –

M – S/m m F- F

Bacterial diameter Cytoplasm diameter Plasma thickness Cell wall thickness Cytoplasm relative permittivity Plasma relative permittivity Cell wall relative permittivity Cytoplasm conductivity Cell wall conductivity Plasma capacitance Cytoplasm capacitance and resistance Cell wall capacitance and resistance Bacterial impedance

– – – – – – – – ∼Cions ∼tpl , r,pl ∼dcyt , r,cyt ,  cyt ∼twall , r,wall ,  wall ∼dbact , dcyt , tpl , twall , r,cyt , r,pl , r,wall ,  cyt ,  wall

1.2 ␮mc 1.2 ␮mc 8 nmc 20 nmc 70c 16c 60c 0.8c – – – – –

m m m m – – – S/m S/m F F- F- 

Box

Cox Insulator

tins

r,ins

r,sol  sol D Csol − Rsol CDL Bacterial cell

dbact dcyt tpl twall

r,cyt r,pl r,wall  cyt  wall Cpl Ccyt − Rcyt Cwall − Rwall Zbact a b c

Al/Al2 O3 microelectrodes parameters [3]. PBS 1:1000 buffer parameters from datasheet and measurements. S. epidermidis model parameters [22].

Table 2 Parameters used in the 2D numerical simulations. Symbols

Descriptions

Values

Units

q kB T Ut Nav KW D

Elementary charge Boltzmann constant Temperature Thermal voltage (= kB T/q) Avogadro number Dissociation constant of water Ionic diffusion coefficient

0.19 aC 1.38 × 10−23 300 K 26 mV 6.02 × 1023 10−14 1.52 × 10−9

C J/K K V – – m2 /s

cp cm  E Jc Je I f ω ±Va

Positively charged ion concentration Negatively charged ion concentration Electrical voltage Electric field Conduction current density Displacement current density Total current through microelectrodes Applied frequency Angular frequency AC voltage on microelectrodes

[100, 109 ] 2f ±10 mV

mol/m3 mol/m3 V V/m A/m2 A/m2 A Hz rad/s V

Eq. (33) Eq. (30) 8 × 1018 m−2 a 10−10 a 10−8 a 0 m−2 b 10−9.5 b 6 × 1016 m−2 b 10−1.8 b

S/m F/m C/m3 C/m3 C/m2 m−2 – – m−2 – m−2 –

multicolumn4l 

 v v,b s Ns Ka Kb Nsi Kai Nsj Kaj a b

Al2 O3 material parameters [29]. S. epidermidis model parameters [25].

Electrical conductivity Permittivity Space charge density Space charge density of bacteria Insulator surface charge Hydroxyl group density of oxide Acidic dissociation constants of oxide Basic dissociation constants of oxide Anionic group density in cell wall Dissociation constant of anionic groups Cationic group density in cell wall Dissociation constant of cationic groups

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capacitance between parallel conductors of thickness te and the planar capacitance between half electrodes of width we /2: Csol 

0 r,sol de

· (Ne − 1)Ae G

(3)

sol · (Ne − 1)Ae G (4) de √ t +d · K( 1−k2 )/(2K(k)) where G  e e t +w /2 is a geometric constant with

Gsol 

e

parameters k  cos

 e 2

·

we de +we



and K(k) 

 t=1 t=0

√

dt (1−t 2 )(1−k2 t 2 )

given in [14]. With the electrode geometry given in Table 1, the constant G is equal to 1.28. Based on Eqs. (1)–(4) and on the fact that the DL is screened by the insulator layer, the system is characterized by two cutoff frequencies (Fig. 2a): fc,1 =

1 sol 1 t ·G  · ins ·  de 0 r,ins 2 · Rsol (Cins /2)

(5)

fc,2 =

1 1 sol  · 2 · Rsol Csol 2 0 r,sol

(6)

In parallel to the impedance formed by Cins , CDL , Csol and Rsol , the system also comprises the oxide capacitance Cox and lateral lat : insulator capacitance Cins



Cox  0 r,ox · lat Cins 

0 r,ins de

K(

1 − k2 ) · (Ne − 1)Le 2K(k)

(7)

· (Ne − 1)tins Le

(8)

lat can be neglected in The lateral insulator capacitance Cins √ 2  K( 1−k d r,ox lat = · t e · 2K(k) ) ≈ 20.5. At f < fc,1 , the parallel, since Cox /Cins  r,ins

ins

impedance is dominated by (Cins /2) and the oxide capacitance Cox can be neglected in parallel since Cox  0.003 · Cins . At frequencies larger than fc,2 , the capacitance Csol dominates and Cox  0.03 · Csol is sufficiently small compared to Csol for being neglected in parallel. Between fc,1 and fc,2 , the impedance is governed by Rsol and Cox can again be neglected at such frequencies, since f < fc,2 = 2 · R1 C  1 . 2 · Rsol Cox

sol sol

In conclusion, Cox can always be neglected and the fre-

quency must be larger than fc,1 to sense volume properties (Rsol , Csol ) instead of surface properties (Cins ). Also, fc,1 and fc,2 both depend on the electrolyte conductivity, but not their ratio: fc,2 1 r,ins de 1  · · · 2 r,sol tins G fc,1

(9)

To achieve a predominence of the resistive level between fc,1 and fc,2 , i.e. a larger fc,2 /fc,1 ratio, the electrode gap de and the insulator thickness tins must be enlarged and reduced, respectively. For low-salt buffers such as phosphate buffered saline (PBS) diluted 1:1000 by volume in deionized (DI) water and characterized by  sol = 1.8 mS/m, typical values are fc,1 = 76 kHz, fc,2 = 405 kHz and fc,2 /fc,1 = 5.3. For high-conductive saline buffers such as PBS with  sol = 1.8 S/m, the cutoff frequencies are three orders of magnitude larger, i.e., fc,1 = 76 MHz and fc,2 = 405 MHz, but their ratio is kept unchanged. Depending on the required range of frequency set by the electrical readout and target application (surface versus volume sensing), an electrolyte with an appropriate conductivity must be chosen. Also, de and tins must be chosen for an suitable fc,1 . 2.2. Electrolyte with bacterial cells Bacterial cells are typically composed of several outer shells, two for Gram-positive bacteria (e.g. Staphylococcus spp.) and three for Gram-negative bacteria (e.g. Escherichia spp.). In this work, we mainly focus on Gram-positive bacteria since S. epidermidis was

Fig. 2. Analytical expressions of the impedance modulus and phase versus the applied frequency for (a) the electrolyte without bacteria based on lumped elements in Fig. 1 and (b) the bacterial cell based on parameters in Table 1.

used as reference strain in the experimental procedure (see Section 4). Conclusions can be extended to Gram-negative bacteria since their outer membranes have the same dielectric properties and thicknesses as plasma membranes (see Escherichia coli model in [22]), so that both membranes can be grouped together in one equivalent shell to obtain a similar morphology as Gram-positive bacteria. A multi-shell representation is considered to account for the cytoplasm, plasma membrane and cell wall (Fig. 1) whose dielectric values are given in Table 1. It is important to note that the bacterial cell wall is an ion-penetrable layer [25], and has therefore an electrical conductivity proportional to the electrolyte one, i.e.  wall  0.4 ·  sol [26]. The impedance of the bacterial cell is generally characterized by four cutoff frequencies (Fig. 2b). The first results from the series connection between Cpl and Rwall , while the second from the parallel association between Rwall and Cwall . These cutoff frequencies

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have little impact on the impedance modulus as they express the −1

∗ = (C −1 + C −1 ) , formed by slight transition between Cpl and Cout pl wall the series association of Cwall and Cpl . We can therefore simplify the bacterial spectrum with the two other cutoff frequencies:

fcb,1 =

1 1 (tpl + twall ) · cyt ∗ ∗ /2)   · d 2 · Rcyt (Cout cyt · 0 r,out

(10)

fcb,2 =

cyt 1 1  · 2 · Rcyt Ccyt 2 0 r,cyt

(11)

where the equivalent relative permittivity of outer shells is ∗r,out = ∗ · tout



t

tpl

wall r,wall + r,pl

−1

∗ t + t  34 with tout pl wall . These cutoff fre-

quencies strongly depend on the bacterial dielectric properties, and therefore on the species. For Staphylococcus spp. (Table 1), fcb,1 and fcb,2 are approximately equal to 23 MHz and 200 MHz, respectively. At relatively low frequencies (f < fcb,1 ), the bacterial cell behaves as a large capacitance formed by the cell wall and plasma membrane. However, at larger frequencies, the cytoplasm dominates either in a resistive (Rcyt ) or a capacitive (Ccyt ) way. As we consider both the bacterial cell and the electrolyte, the four main cutoff frequencies must be ordered to understand bacterial effect on the global impedance. In low-conductive buffers ( sol = 1.8 mS/m), the order is the following: fc,1 < fc,2 < fcb,1 < fcb,2 . Below fc,1 ≈ 76 kHz, the impedance is hardly affected by the bacterial cell since the insulating layer screens both the DL and volume ∗ properties. However, large values of Cpl , Rwall and Cout can still slightly modify the sensor impedance dominated by Cins . Between fc,1 and fc,2 , the conductive path through electrolyte (Rsol ) is short∗ , so that the overall ened by the large bacterial capacitance Cpl or Cout volume resistance decreases. For fc,2 < f < fcb,1 , the volume capacitance increases since the capacitive path through the electrolyte is ∗ . For f reduced by the large bacterial capacitance Cout cb,1 < f < fcb,2 , the overall capacitance increases for the same reasons, due to the short circuit behavior of the bacteria cell at these frequencies (f < fcb,2 < 2R 1 C ). Finally, at frequencies larger than fcb,2 , the cyt sol

overall capacitance decreases since r,cyt < r,sol . For high-conductive saline buffers such as PBS ( sol = 1.8 S/m), the order is different: fcb,1 < fc,1 < fcb,2 < fc,2 . Again, below fc,1 ≈ 76 MHz, the impedance is hardly impacted by bacteria. Between fc,1 and fcb,2 , bacteria have a resistive behavior that slightly increases the whole medium resistance since  cyt <  sol . At frequencies larger than fcb,2 , the bacterial capacitive behavior still increases the overall impedance, but by reducing the medium capacitance at f > fc,2 since r,cyt < r,sol . 2.3. Analytical estimation of the sensitivity In the frequency range ∈[1 kHz, 10 MHz], the sensor hardly achieves a perfect capacitive behavior since the impedance phase slightly differs from −90◦ (Fig. 2a). It is therefore recommended to describe the sensor response in terms of the normalized admittance modulus Y/ω instead of the capacitance [3]. The sensitivity to adherent bacterial cells is then defined as the relative variation of Y/ω , in percent:

Y1 (ω) − Y0 (ω)

Y (ω)

S(ω) =

(12)

0

where Y0 (ω) and Y1 (ω) are the initial and final admittances, before and after bacterial binding, respectively. The maximal sensitivity is defined as Smax = max{S(ω)} and can be used to ω

compare two different conditions independently of the applied fre2D at quency. To analytically estimate the 2D maximal sensitivity Smax  sol = 1.8 mS/m, several assumptions are used:

• The optimal frequency f where the maximal sensitivity is achieved is assumed one order of magnitude larger than fc,2 , as experimentally proven in [3]. • The sensor impedance is dominated by the initial medium capac  itance Csol,0  0 dr,sol · (Ne − 1)Ae · G at the optimal frequency, as e explained in Sections 2.1 and 2.2. Only the capacitance Cins also lat can be neglected as impacts the impedance, as Cox and Cins explained in Section 2.1. • To estimate the solution capacitance Csol,1 in presence of the bacterial cell, the Ne semi-planar electrodes are approximated to perfect parallel conductors of thickness (te + we /2) · G, gap de and length Le , thus giving the same initial capacitance as Eq. (3). Furthermore, the bacteria cell is simplified √ by a square box of side deq  (dbact · /2), giving an identical perturbed volume to the spherical bacteria. At the optimal fre∗ dominates the bacterial quency, the outer shell capacitance Cout dielectric behavior. The resulting solution capacitance is then Csol,1 

0 r,sol de

with T 

∗ tout de

·

· (Ne − 1)Ae · (G + P), with P  r,sol ∗r,out

·

de . deq

deq te +we /2

·

1−2T de /deq −1+2T

The complete development is available

in Supplementary Information. The following formula are then obtained for the initial −1 −1 + Csol,0 ) and final admittances: Y0 (ω)  jω · (2Cins −1 −1 jω · (2Cins + Csol,1 ) approximated to: 2D Smax 



−1

and Y1 (ω) 

. Consequently, the maximal sensitivity is

−1 −1 Csol,0 − Csol,1 −1 −1 2Cins + Csol,1

P 1 · G 1 + 2·

−1

G+P Q

(13)

(14)



with Q  r,ins · tde . In this work, since G  1.28, T  0.06, P  0.17 r,sol ins 2D is equal to 11.2%. In realand Q = 13.6, the maximal sensitivity Smax ity, the bacterial cytoplasm resistance never perfectly shortens the sensor capacitance, so that  sol , r,cyt ,  cyt , r,wall ,  wall , r,pl ,  pl and other metric parameters can still have a slight impact. 2.4. Impact of metallic connections to IDE Impact of passivated metallic tracks for connecting cellsubstrate impedance sensors (ECIS) have already been reported to modify the spectral impedance [27,28]. In this section, we show that similar effects arise for interdigitated microelectrodes patterned in microfluidic channel, and contacted outside the channel by electrical pads through metallic tracks (Fig. 3a). When located on the same side, the metal accesses can couple to each other and slightly perturb the global impedance modulus and phase at low frequency despite the large IDE capacitance, as experimentally shown in Fig. 3b (see Section 4 for details about experimental procedures). While the low-frequency capacitance is shown to increase up to 60% due to parasitics, the increase of the impedance phase around 1 kHz expresses the more resistive behavior of the impedance. To explain this effect, metal accesses are modeled with two equivalent electrical circuits (Fig. 3a). On one hand, the part covered with a passivation layer such as KMPR photoresist [3] for sealing and electrical isolation from the electrolyte results in a dielectric passivation capacitance Cpass . The large distance between tracks and the small relative permittivity of the passivation material makes Cpass dominating in series with Cins,p  Cpass and then being neglected in parallel with Cins , Gsol or Csol . On the other hand, the unpassivated area has a similar equivalent circuit with Cins,i , Rsol,i and Csol,i as the IDE (Fig. 3a). Because the distance between access

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• Lumped elements, i.e., resistances and capacitances, are represented by simplified expressions (see Eqs. (1)–(4) for instance). Actually, the curvature of the electric field by the bacterial sphericity and electrode topology can hardly be described accurately by analytical formula. • The impact of the bacterial impedance on the global impedance can hardly be estimated in the spectral domain from a weighted superposition of Fig. 2a and b. Indeed, the bacterial cell replaces a small part of the inter-finger volume, which completely modifies parallel and series capacitances or resistances in the system. The spherical shape of the bacterial cell further complicates this. • Despite their impact on the thickness of electrical double layers, charges at the insulator–electrolyte interface and in the bacterial cell wall are not considered in the analytical model because of their dependence on the local pH. • The junction between the electrical double layers, atop the insulating layer and around the bacterial cell, cannot be modelled analytically. • Only AC analysis is considered in the analytical model, despite the possible importance of the equilibrium point.

3. Numerical simulations To solve aforementioned limitations and accurately quantify the bacterial sensitivity versus the applied frequency, 2D finiteelement simulations of the system described in Fig. 1 were implemented with Comsol Multiphysics® . Simulation parameters are summarized in Table 2.

3.1. Assumptions Fig. 3. Impact of access lines: (a) schematic top view (not at scale) of the metal connections to the interdigitated microelectrodes (IDE), (b) comparison between experimental and analytical values in water solution of  sol  0.5 mS/m, giving fc,1i  550 Hz, fc,1  13 kHz and fc,2  112 kHz for a 20-nm thick insulator, as characterized by ellipsometry. The impedance analyzer has been calibrated with electrical probes positioned in the air (IDE + connections), or positioned on an open structure having the same access lines as the IDE structure (IDE only). Typical values listed in Table 1 are used for the analytical model.

lines is significantly larger than the electrode gap (ca. 250 ␮m [3] −1 versus 4 ␮m as mentioned in Table 1), Gsol,i  Rsol,i and Csol,i are two

−1 orders of magnitude smaller than Gsol  Rsol and Csol , respectively.

Consequently, the cutoff frequency fc,1i =

1 2Rsol,i (Cins,i /2)

is two

orders of magnitude smaller than fc,1 and thus explains the shape change of the impedance phase in Fig. 3b. Indeed, as fc,1i < f < fc,1 , the inequality ωCins < 2 · Gsol,i < ωCins,i explains why Rsol,i dominates (Cins,i /2) in series and (Cins /2) in parallel, resulting in a more resistive behavior of the impedance. Below fc,1i , the impedance is given by the parallel association of Cins,i and Cins , explaining the larger capacitance due to access lines. At frequencies larger than fc,1 , the impact of access lines is considerably reduced because Rsol and Csol largely dominates in parallel their counterparts Rsol,i and Csol,i , both two orders of magnitude smaller, respectively.

2.5. Limitations of the analytical model As shown in Sections 2.2 and 2.3, the analytical model becomes very complex when considering bacterial cells, but provides a good physical understanding of the sensor physics. However, such approach is unsuitable for accurate quantification of the spectral sensitivity to bacterial cells because of the following rough approximations:

Some hypotheses are considered to simplify numerical simulations. First, the electrolyte buffer is a H2 O solution containing a weak acid HA and its conjugate basis A− , on the form of A− B+ . In this simplest form, the electrolyte is then characterized by four ions: H+ , OH− , B+ and A− . For the considered solution of PBS 1:1000, the major species are Na− and Cl− and therefore play the role of B+ and , HPO2− and K+ can be A− , respectively. Other ions such as H2 PO− 4 4 neglected because their concentrations are one order of magnitude smaller. Second, these hydrated ions are assumed to have the same hydrated diameters and therefore the same diffusion coefficients. In this case, PBS 1:1000 electrolyte with measured conductivity of  sol = 1.8 mS/m and ionic strength of ca. 0.16 mM (cfr Table 1) is considered. The global diffusion coefficient can then be estimated sol Ut −9 by the Kohlraush law: D = m2 /s, a value 3  1.52 × 10 2qNav Cions 10

between diffusivities of the two major species Na+ and Cl− featuring 1.33 × 10−9 and 2.03 × 10−9 m2 /s, respectively [30]. As a result, positive and negative ions can each be grouped together to form single concentrations cp and cm , respectively. It is important to note that, for extremely diluted electrolytes (Cions < 1 −10 ␮M), H+ and OH− can play a non-negligible role and lead to underestimation of the electrolyte conductivity because of their high diffusion coefficients of 9.3 × 10−9 and 5.3 × 10−9 m2 /s, respectively. Third, the concentration [H+ ] is set to a fixed part of cp at each point of the 2D system. Indeed, there is no preferential attraction between H+ and K+ , because of their identical valencies and diffusion coefficients. This constant fraction of cp is estimated 100 ␮m-away from microelectrodes. Fourth, steric effects in the electrical double layer are not modeled. The small AC voltage amplitude indeed guarantees that cp and cm do not exceed their maximal physical value (∼ 200 M for a 0.2 nm-hydrated diameter), as it has been verified after each simulation. Additional terms could be included in constitutive equations

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to model steric effects, but the formulation would then become more intricate. Fifth, only one pair of electrodes are simulated (Fig. 1). Coupled effects between non-adjacent electrodes and access lines are thus neglected. These two effects have slight impact on the complex impedance (see Section 2.4), but do not constitute the key sensing part as discussed in Section 4. Another geometric feature is the upper and bottom boundary layers of the electrolyte and buried oxide, which are connected to ground and located 100 ␮m above and 50 ␮m below electrodes, respectively, to mimic experimental conditions [3]. Six, the insulator material is assumed to be an oxide and more precisely Al2 O3 as in [3]. It does not change the dielectric behavior of the system, but the surface charges of oxide materials can be modelled more easily. Finally, the system is assumed linear because of the small voltage amplitude (Va < Ut ) used in simulations [23], which is representative to those used in experimental works [3,31,32]. In this case, the electrical double layer has a linear dependence with the local potential, contrasting with non-linearity properties reported at large AC voltages [33].

electrical conductivity . For the ion-penetrable bacterial cell wall, the ionic diffusion coefficient is assumed 40% of the one of the surrounding electrolyte [26]. The ion transport is described by the Nernst–Planck equation [20,34,35]:

3.2. Constitutive equations

∇ · (−∇ 0,a ) = qNav · (cp0,a − cm0,a )

Three variables characterize the system depicted in Fig. 1: the electrical potential , the positively and negatively charged ion concentrations cp and cm , respectively. They are expressed as the sum of the electrostatic and AC terms, the latter being expressed as a phasor thanks to the system linearity and the sinusoidal voltage applied on electrodes:

For the current densities, Je has the typical formulation while Jc is obtained by combining ∇ · Jc = −jω va with Eq. (24):

 = 0 + a × R{ejωt }

(15)

cp = cp0 + cpa × R{ejωt }

(16)

Insulator–electrolyte interface. As explained in Section 3.1, the insulator material is assumed to be an oxide. Hydroxyl groups at the insulator–electrolyte interface are subject to protonation or deprotonation, depending on the local pH value, according to the following chemical reactions [29]:

cm = cm0 + cma × R{e

jωt

}

(17)

where subscripts 0 and a stand for DC and AC, respectively. Once constitutive equations are solved for , cp and cm in equilibrium and AC regimes, the total current density is extracted at each point of the 2D system. The conduction current density is given by Jc =  · E if the medium conductivity  is known, or by the continuity equation ∇ · Jc = −∂ v /∂t otherwise. For the displacement current density, the Maxwell formulation gives Je = jω · E, with  the local medium permittivity. By integrating Je on the electrode–oxide interface, the total current I through the electrode is obtained: I = Ia · R{e

jωt

}

(18)

The impedance is then computed as Z = Va /Ia , where Va is the AC voltage amplitude applied on microelectrodes. Different constitutive equations with , cp and cm must be considered for each medium and additional inputs must sometimes be considered, e.g., the surface and space charges. The following subsections summarize equations that are used for the different media in Fig. 1. Conductive and dielectric media. They are characterized by a fixed permittivity  and conductivity . The concerned media in Fig. 1 are the insulator layer, the buried oxide layer, the bacterial cytoplasm and plasma membrane. The electrical potential is described by the Poisson equation: 0 = ∇ · (( + jω)∇ )

(19)

Jc = − · ∇ 

(20)

Je = −jω · ∇ a

(21)

Electrolyte and bacterial cell wall. In this case, the mobility of positively and negatively charged ions intrinsically determines the



∂cp,m D · cp,m ∇  = ∇ · D∇ cp,m ± Ut ∂t

 (22)

where the negative sign holds only for negatively charged ions (i.e. cm ). Four equations are obtained by rewriting these equations at equilibrium (Eq. (23)) and in AC regime (Eq. (24)):



0=∇·

∇ cp0,m0 ± cp0,m0 ·

jωcpa,ma =∇· D

∇ 0





(23)

Ut

∇ cpa,ma ± cpa,ma ·

∇ 0 Ut

± cp0,m0 ·

∇ a



Ut

(24)

Thanks to the small applied voltage amplitude, the 2nd-order term cpa,ma · ∇Ua · ej2ωt can be neglected and is therefore not t included in Eq. (23). The electrical potential in DC and AC is obtained through the Poisson equation knowing that v = v0 + va · R{ejωt } with v0 = cp0 − cm0 and va = cpa − cma :



Jc = −qDNav ·

∇ (cpa − cma ) + (cpa + cma ) ·

(25)

∇ 0 Ut

+ (cp0 + cm0 ) ·

Je = −jω · ∇ a

AOH 

Ut



(26) (27)

AOH  AO− + H+ AOH+ 2

∇ a

(with −

+ OH

Ka)

(with

(28)

Kb)

(29)

− where Ka  [AO− ][H+ ]/[AOH] and Kb  [AOH+ 2 ][OH ]/[AOH] are the

corresponding dissociation constants with values for Al2 O3 given in Table 2. Affecting DC and AC ion distributions through Eqs. (23)–(25), the surface charge is thus given by [29]:



s = qNs ·



2

Kb [H+ ] − Ka KW Ka KW + KW [H+ ] + Kb [H+ ]

(30)

2

Bacterial charge. In most electrolytes, the bacterial cell wall of Gram-positive bacteria is typically negatively charged due to protonation or deprotonation of carboxyle, phosphate and amine groups in the peptidoglycan layer, as expressed by the following chemical reactions: + HA1 + H2 O  A− 1 + H3 O +

B2 + H3 O 

HB+ 2

+ H2 O

(with Ns1 , Ka1 )

(31)

(with Ns2 , Ka2 )

(32)

where HA1 andB 2 are anionic and cationic groups, respectively, Ns1 and Ns2 the corresponding group densities and Ka1 and Ka2 their respective dissociation constants. Uniformly distributed in the cell wall volume because of its ion penetrability, the bacterial charge is characterized by a space charge density impacting DC and AC ion distributions through Eqs. (23)–(24) [25]:

⎛

v,b =

1 · twall

 −qN K si ai ⎝ i

Kai + [H+ ]

+

 qN sj [H+ ] j

Kaj + [H+ ]

⎞ ⎠

(33)

N. Couniot et al. / Sensors and Actuators B 211 (2015) 428–438

Fig. 4. Convergence issue between the oxide surface charge  s and the local pH close to the electrode in the electrolyte.

The same method can be used for Gram-negative bacteria by considering their ionic groups on the outer membrane [25]. 3.3. Convergence issues The mesh is strongly refined at the insulator–electrolyte interface (∼0.1 nm), since the electrical double layer can reach very small values (∼ nm) in saline buffers. To limit the number of nodes in the micrometer-sized system, we use the “boundary layer mesh” tool provided by Comsol Multiphysics® . The mesh error on the extracted impedance and sensitivity decreased with smaller mesh sizes as expected (data not shown) and is estimated to 0.01% for the chosen mesh configuration. The consideration of the insulator surface charge  s can result in convergence difficulties. Indeed, as the local pH decreases (i.e. larger local [H+ ]),  s increases following Eq. (30), but positive ions in the electrolyte such as H+ are then more repulsed at the same time, increasing the local pH. The solver can therefore exhibit difficulties for converging. To address this problem, a bisection method was implemented according to the following steps (Fig. 4): 1. The initial pH interval is defined as [pH1 ; pH2 ] = [pHsol ; PZCins ], where pHsol the electrolyte pH and PZCins is the point zero of charge of the insulator. 2. The system is solved by imposing the  s that corresponds to pH2 (see Eq. (30)). The local pH, denoted pHloc at the insulator–electrolyte interface, is then extracted. 3. If |pH2 /pHloc − 1| < 0.1%, the system has converged. If not, a new pH value (w · pH2 +pH1 )/(w + 1) is computed by defining w as a weight factor, and attributed to pH2 if pHloc > pH2 or to pH1 otherwise. The algorithm then goes on at Eq. (2), by using the modified pH value (either pH1 or pH2 ) for  s computation. With the simulation framework now completely defined, the simulation results are compared to experimental and analytical data in the next section. 4. Matching with experimental data The experimental procedures, protocols and electrode design are detailed in our previous work [3]. Briefly, 250 ␮m-diameter circular-shape interdigitated microelectrodes consisting of 1 ␮mthick, 2 ␮m-wide and 4 ␮m-spaced Al fingers covered by 33 nm of atomic layer deposited (ALD)-Al2 O3 were fabricated in cleanroom

435

facilities. A 300 ␮m-thick microfluidic channel was used to flow solutions at a 0.5 ␮L/min flow rate. The impedance modulus and phase were recorded at 50 mV amplitude from 100 Hz to 1 MHz by an impedance analyzer (Agilent LCR4284A). An open calibration was performed with electrical probes 250 ␮m above the IDE pads to remove parasitic capacitances of BNC cables. For the preparation of the bacterial sample following the overnight culture, S. epidermidis ATCC35984 was resuspended in PBS 1:1000 using three successive centrifugation steps and the number of viable cells was subsequently estimated to 109 CFU/mL by using agar plates. During measurements of the capacitive biosensor, real-time observation of the sensor surface was performed using an inverted microscope and enabled precise estimation of the bacterial surface coverage. In the two following subsections, the normalized admittance

Y/ω and sensitivity S(ω) defined in Section 2.3 are obtained analytically, numerically and experimentally and a comparison between each case is provided. It is important to note that previously defined figures of merit obtained by 2D analytical models and simulations are not strictly representative of those experimentally obtained in 3D, since the 2D system in Fig. 1 assumes that the bacterial cell spans infinitely along the third dimension. To enable the comparison between 2D and 3D figures of merit, the normalized admittance at 1 MHz and the maximum sensitivity are equalized by using a factor K evaluated in next subsections and shown realistic with regards to theoretical approximations. 4.1. Naked sensor in sterile PBS 1:1000 In the low-conductive buffer PBS 1:1000 free of bacteria, both the analytical model and simulation of the normalized admittance

Y/ω show good fitting with experimental data, especially at frequencies larger than 10 kHz (Fig. 5a). The factor used to link the 2D with the 3D normalized admittance modulus is K 

Y/ω exp

Y/ω sim

=

7.21 mm at 1 MHz, with is very close to the theoretical value of the total electrode length Le · (Neff - 1)  7.25 mm, where Neff  (1 +

(Ne −1)/2

2

i=1



2

1 − ( Ne2i−1 ) )  30 is the effective number of elec-

trodes of length Le in the circular IDE. Since the parallel access lines are not simulated, a 28% decrease of Y/ω is observable at low frequency compared to experimental values (see Sections 2.4 for C G more explanation). The transition 2ins → ωsol → Csol , mentioned in Section 2.1, is also clearly observable and similar to that reported G C in [19] and to the 2DL → ωsol → Csol transition of unpassivated electrodes simulated in [18]. 4.2. Sensor covered by S. epidermidis in PBS 1:1000 In this case, a gentle wash with PBS 1:1000 was performed after bacterial incubation to conserve only adherent S. epidermidis on the sensor surface, at a surface coverage of 3.2 · 104 bacteria per mm2 corresponding to approximately Nbact = 2250 adherent bacteria on the 250 ␮m-diameter sensor. To link 2D with 3D sensitivity values, the factor K 

3D Smax 2D Smax

= 0.58 was used. Good

fitting between numerical and experimental data is obtained through the whole frequency range (Fig. 5b), but especially at high frequency where the largest sensitivity is achieved. As previously explained in Section 2.2, the peak sensitivity arises from the sensing of volume properties (Rsol , Csol ), instead of surface properties (Cins ). The analytical model based on Eq. (12) shows more imprecise matching, but still with a satisfying spectral tendency and intensity. Based on the above factor K = 0.58 and Eq. (14), the 3D is analytically estimated to maximal theoretical sensitivity Smax 6.5%, which is close to experimental and simulated values around 9%. It is important to note that the experimental sensitivity does

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Fig. 6. Impact of oxide surface charges  s (see Eq. (30)) and bacterial space charges v,b (see Eq. (33 ) on the bacterial sensitivity S(ω). The 2D simulations were adjusted to 3D with K = 0.58, similarly to Fig. 5. Typical values listed in Table 1 are used.

Rwall in series with Cins (Fig. 6). It is slightly compensated by oxide charges  s that increase Cins . At larger frequencies (>100 kHz), the ∗ , which is bacterial space charges v,b increase the value of Cout beneficial for the bacterial sensitivity since the medium resistive path is more perfectly shortened. This charge analysis should be identical for Gram-negative bacteria, as similar space charge density can be quantified in the outer ion-penetrable layer [25]. 5. Conclusion

Fig. 5. Spectral comparison between analytical model, numerical simulations and experimental data for (a) the modulus of the normalized admittance in PBS 1:1000 without bacterial cells with an adjusting factor K between 2D and 3D equal to 7.21 mm and (b) the sensitivity to bacterial cells in PBS 1:1000 with a bacterial surface coverage of 3.2 × 104 S. epidermidis per mm2 , with an adjusting factor K between 2D and 3D equal to 0.58. Details of this experimental procedure with bacterial cells is available in [3]. Parameter values listed in Table 1 are used for analytical models, simulations and experimental procedures.

not present the bump around 10 kHz predicted by models and simulations, which is related to the slight impedance increase caused by Rwall in series with Cins . This discrepancy can be explained by the larger bacterial space charges in reality, impacting the Rwall value and reducing the related sensitivity as detailed in the next paragraph. The impact of oxide surface charges  s and bacterial space charges v,b on the sensor impedance and bacterial sensitivity is non-negligible on figures of merit. In PBS 1:1000, Y/ω is shown to slightly increase at f < 100 kHz because of the larger Cins provided by  s and more strongly around 10 kHz because of the lower Rwall value due to v,b (see Supplementary information). The DL capacitance CDL also slightly increases since the Debye length D is shortened by larger charge repulsion. At higher frequencies, the charges do not impact Y/ω , as expected from previous works reporting the impact of DNA charges only at low frequency in [35]. When considering the sensitivity S(ω), the bump around 10 kHz is drastically lowered by v,b because of the smaller influence of

In this paper, we have proposed a comprehensive study of capacitive biosensors in presence of bacterial cells. An analytical model has first been established to qualitatively and semi-quantitatively understand how the impedance modulus and phase behaves spectrally in electrolytes with and without bacterial cells. The large sensitivity to bacterial cells was shown to be due to the low cytoplasmic resistance that shortens electric field lines and therefore increases the global capacitance between electrodes. The maximal theoretical sensitivity to bacteria and the impact of access lines has also been investigated by this method. To provide a more accurate quantitative analysis of the maximal sensitivity, 2D finite-element simulations have been developed based on the Poisson–Nernst–Planck equation, accounting for complex phenomena such as ion distribution, charge effects and dielectric repartition. The simulation results were shown to accurately fit experimental data, and highlight the impact of bacterial space charge and oxide surface charge on the impedance and sensitivity. In conclusion, this work provides the necessary analytical and numerical material to fully understand and optimize capacitive biosensing of bacterial cells. Acknowledgments N. Couniot is F.R.S. – FNRS Research Fellow. The authors thank O. Poncelet for the atomic layer deposition of Al2 O3 and T. Vanzieleghem for help with bacterial handling. Appendix A. Supplementary Data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.snb.2015.01.108. References [1] V. Tsouti, C. Boutopoulos, I. Zergioti, S. Chatzandroulis, Capacitive microsystems for biological sensing, Biosens. Bioelectron. 27 (1) (2011) 1–11.

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Biographies Numa Couniot was born in Charleroi, Belgium, in 1988. He received the Electrical Engineer degree from the Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 2011. Since then, he joined the Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM), at the UCL, where he is currently pursuing a Ph.D. degree granting by the Fond national de la recherche scientifique (FNRS). His doctoral research focuses on the modeling, optimization and characterization of fundamental bioelectronics interactions in order to optimize the design of integrated biosensor microarrays. Aryan Afzalian was born in Ottignies, Belgium in 1977. He received the Electromechanical Engineer degree and the Ph.D. degree from the Université catholique de Louvain (UCL), Louvain-La-Neuve, Belgium, in 2000, and 2006, respectively. During his Ph.D., he was working on the modeling, optimization and characterization of SOI integrated optical sensors and analog circuits. From 2006 to 2009 he was a Postdoctoral Research Fellow at Tyndall National Institute, Cork, Ireland, where he worked on modeling quantum transport (development of quantum simulators based on the NEGF formalism) in advanced Silicon nanoscale devices, such as SOI nanowires and new quantum enhanced architectures such as Resonant tunneling FETs. He is currently “Chargé de recherche” FNRS with UCL, Belgium, working on modeling quantum transport in advanced Silicon and alternative channel materials (III–V, Ge, Graphene) nanoscale devices for electronics and bio-electronics applications. Dr. Afzalian has authored or co-authored about 80 technical publications in international conferences, journals and books and holds two patents. He is the recipient of the 2001 AILV award for his master thesis work on SOI image sensors, and of the 2009 UCC Invention of the year awards for his work on Resonant Tunneling FETs. He also serves as a reviewer for various journal and conferences such as IEEE Trans. on Electron Devices, IEEE Trans. on Nanotechnology, Solid-State-Electronics or IEEE Electron Device Letters. Nancy Van Overstraeten-Schlögel was born in 1975. She lives in Belgium where she earned the degree of bioengineer in chemistry and bioindustries from the Université catholique de Louvain. She completed her education by a master and a Ph.D. in biomedical sciences from the faculty of medicine of the University of Liège. She applied the multidisciplinary approach to postdoctoral scientific projects in fundamental and applied research or in development. Her career focuses on the improvement of the human health through the integration of innovative technologies in ambitious scientific projects. Her most recent endeavour is in the development of biomedical applications to microelectronic biosensors for creating new diagnostic tools. Laurent A. Francis was born in Louvain-la-Neuve in 1978. He received the M.Eng. degree in materials science and the Ph.D. degree in applied sciences from UCL in 2001 and 2006, respectively. Since September 2007, he holds the Microsystems Chair position at UCL as associate professor. His Ph.D. thesis was related to acoustic-wave based microsystems for biosensing applications and resulted from collaboration between the department of materials science of UCL and IMEC (Interuniversitary MicroElectronics Center) in Leuven. Between 2000 and 2007 he was with IMEC as researcher, successively in the Biosensors and RF-MEMS groups. His scientific interests are related to thin films integration for microsystems components (mainly piezoelectric and diamond-like materials), acoustic sensors, bio-inspired approaches, extreme miniaturization and device packaging. He is regular member of IEEE and of the UCL Alumni. He has authored or co-authored 60 scientific publications in international journals and holds one patent. Denis Flandre was born in Charleroi, Belgium, in 1964. He received the Electrical Engineer degree, the Ph.D. degree and the Post-doctoral thesis degree from the Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1986, 1990 and 1999, respectively. His doctoral research was on the modeling of silicon-on-insulator (SOI) MOS devices for characterization and circuit simulation, and his Post-doctoral thesis on a systematic and automated synthesis methodology for MOS analog circuits. Since 2001, he is a full-time Professor at UCL. He is currently involved in the research and development of SOI MOS devices, digital and analog circuits as well as sensors and MEMS, for special applications, more specifically high-speed,

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low-voltage low-power, microwave, rad-hard and high-temperature electronics and micro-systems. He has authored or co-authored more than 800 technical papers or conference contributions. He holds 10 patents. Prof. Flandre was the recipient of the 1992 Biennial Siemens-FNRS Award for an original contribution in the fields of electricity and electronics, of a 1997 Wernaers Award for innovation in pedagogical presentation of advanced research work, and of the 1999 SCK-CEN Prof.

Roger Van Geen Prize for innovation in nuclear science instrumentation. He participated in many short courses on SOI technology in universities, industrial companies and conferences, as well as organized and lectured several Eurotraining courses on SOI technology, devices and circuits. Prof. Flandre is a co-founder of CISSOID S.A., a start-up company, which spun-off of UCL in 2000, focusing on SOI circuit design services.