pubs.acs.org/Langmuir © 2009 American Chemical Society
Single-Colloidal Particle Impedance Spectroscopy: Complete Equivalent Circuit Analysis of Polyelectrolyte Microcapsules Tao Sun,* Catia Bernabini, and Hywel Morgan Nano Research Group, School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K. Received August 28, 2009. Revised Manuscript Received October 2, 2009 We present a high-speed microfluidic technique for characterizing the dielectric properties of individual polyelectrolyte microcapsules with different shell thicknesses using single-particle electrical impedance spectroscopy. Complete equivalent circuit analysis is developed to describe the electrical behavior of solid homogeneous microparticles and shelled microcapsules in suspension. The complete circuit model, which includes the resistance of the shell layer and the capacitance of the inner core, has been used to determine the permittivity and conductivity in the shell of single capsules. The PSpice circuit simulations, based on the developed complete circuit models, are used to analyze the experimental data. The relative permittivity of the polyelectrolyte capsule shell is determined to be 50, and the conductivities of the shells of six- and nine-layer microcapsules are estimated to be 28 ( 6 and 3.3 ( 1.7 mS m-1, respectively.
1. Introduction The application of electric fields to suspensions of colloidal particles (i.e., dielectric microspheres and biological cells) leads to a number of interesting phenomena, including manipulation tools such as dielectrophoresis (DEP)1-3 or particle characterization methods such as electrorotation (ROT)4,5 and electrical impedance spectroscopy (EIS).6-8 The earliest electrical impedance measurements on biological cells in suspension can be traced back to the 1910s, from H€ober’s work,9-11 in which he pointed out the difference in the conductivities of erythrocytes at low and high frequencies, due to the existence of the cellular membrane, and laid the foundation for the discovery of the β-dispersion. Following that, dielectric measurements were performed over an increasing range of frequencies and biological materials by Fricke12-14 and Cole.15,16 From an electrical perspective, colloidal particles can be approximated to a solid dielectric sphere (i.e., a bead) or a sphere enclosed in a single layer or multiple layers of shells (i.e., a cell or microcapsule). When exposed to an electric field, the particle *To whom correspondence should be addressed. E-mail:
[email protected]. (1) Pohl, H. A. Dielectrophoresis; Cambridge University Press: New York, 1978. (2) Pethig, R. Dielectric and electronic properties of biological materials; John Wiley & Sons Ltd.: London, 1979. (3) Morgan, H.; Green, N. G. AC Electrokinetics: Colloids and Nanoparticles; Research Studies Press, Ltd.: Baldock, Hertfordshire, England, 2003. (4) Arnold, W. M.; Zimmerman, U. Z. Naturforsch. 1982, 37c, 908–915. (5) Jones, T. B. Electromechanics of particles; Cambridge University Press: Cambridge, U.K., 1995. (6) Foster, K. R.; Schwan, H. P. Crit. Rev. Biomed. Eng. 1989, 17, 25–104. (7) Schwan, H. P. Ann. N.Y. Acad. Sci. 1999, 83, 1–12. (8) Morgan, H.; Sun, T.; Holmes, D.; Gawad, S.; Green, N. G. J. Phys. D: Appl. Phys. 2007, 40, 61–70. (9) H€ober, R. Pfluegers Arch. Gesamte Physiol. Menschen Tiere 1910, 133, 237– 259. (10) H€ober, R. Pfluegers Arch. Gesamte Physiol. Menschen Tiere 1912, 148, 189– 221. (11) H€ober, R. Pfluegers Arch. Gesamte Physiol. Menschen Tiere 1913, 150, 15–45. (12) Fricke, H. J. Gen. Physiol. 1924, 6, 375–384. (13) Fricke, H. J. Gen. Physiol. 1925, 9, 137–152. (14) Fricke, H. J. Appl. Phys. 1931, 1, 106–115. (15) Cole, K. S. J. Gen. Physiol. 1928, 12, 29–36. (16) Cole, K. S. J. Gen. Physiol. 1928, 12, 37–54. (17) Maxwell, J. C. A Treatise on Electricity and Magnetism; Dover Press: New York, 1954.
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becomes polarized. Interestingly, for a single-shell particle in suspension, charge accumulates at the interfaces between the suspending medium and shell and between the shell and the interior. The degree of polarization depends on the frequency of the applied electric field and the complex polarizability of the particle and the suspending medium. The dielectric behavior of particles in suspension is generally described by Maxwell’s mixture theory.17 This relates the complex permittivity of the suspension to the complex permittivity of the particle, the suspending medium, and the volume fraction. While Maxwell’s mixture theory14-17 has been widely used in the analysis of the dielectric properties of biological particles, it is often convenient to be able to analyze the properties of the particle and medium as individual electrical circuit elements. In Foster and Schwan’s critical review,6 a classical equivalent circuit model was presented to describe the electrical response of a single-shell particle (i.e., a cell) in suspension, approximating the cell to a resistor and capacitor in series. The cell membrane is represented by a capacitor and the cytoplasm by a resistor. In this model, the cell membrane resistance is ignored (it is generally much greater than the reactance of the membrane) and the reactance of the cell cytoplasm is ignored compared to the cell resistance. Foster and Schwan’s simplified circuit model has been used for single-cell impedance analysis in microfluidic cytometers,8,18-20 showing good agreement with the experimental results. However, in many cases, both the cell membrane and cytoplasm properties can change drastically, for example, during electroporation21 or cell lysis.22 Many nonbiological objects can also be modeled as single-shell spheres, where the resistance of the shell layer and the capacitance of the inside cannot be ignored, (18) Gawad, S.; Schild, L.; Renaud, Ph. Lab Chip 2001, 1, 76–82. (19) Gawad, S.; Cheung, K.; Seger, U.; Bertsch, A.; Renaud, Ph. Lab Chip 2004, 4, 241–251. (20) Sun, T.; Gawad, S.; Bernabini, C.; Green, N. G.; Morgan, H. Meas. Sci. Technol. 2007, 18, 2859–2868. (21) Tsong, T. Y. Biophys. J. 1991, 60, 297–306. (22) Lu, H.; Schmidt, M. A.; Jensen, K. F. Lab Chip 2005, 5, 23–29. (23) Zhang, H. Z.; Sekine, K.; Hainai, T.; Koizumi, N. Colloid Polym. Sci. 1983, 261, 381–389.
Published on Web 10/21/2009
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i.e., polyelectrolyte (PE) microcapsules,23,24 which have recently attracted attention as novel microscopic carriers that can be used for the encapsulation and immobilization of a variety of substances, such as proteins and drugs.25 PE capsules are formed by the “layer-by-layer” adsorption of oppositely charged polyelectrolytes onto the surface of a sacrificial template core. The method allows the use of a wide variety of core materials (inorganic and polymer beads, cells, etc.) and shell constituents, including synthetic and natural polyelectrolytes, nanoparticles, and biomacromolecules. The properties of the multilayer structures, such as size, porosity, stability, composition, and surface functionality, can be precisely tuned by the appropriate selection of components and assembly conditions. The shell of the PE microcapsules is fabricated by alternate adsorption of oppositely charged PEs onto the surface of sacrificial colloidal templates (i.e., resin particles and erythrocytes).26-30 The shell thickness of the capsules can be controlled by defining the number of deposited layers of PEs. The shell is an analogue of the cell membrane and selectively allows the passage of molecules and ions. Measurements of the permeability of PE capsules are of interest and have been quantitatively studied by Georgieva et al.28-30 using electrorotation (ROT) techniques with analysis based on a singleshell or multishell spherical model. With this method, the ROT torque is measured by analysis of the rotational rate of a particle, and the data provide the imaginary part of the complex-induced dipole moment (Clausius-Mossotti factor) across the particle. Moreover, a typical ROT assay takes several seconds per particle (usually longer), resulting in low throughput measurements. In this paper, we present single-particle impedance data for two different sets of PE microcapsules. Two complete equivalent circuit models that describe the electrical properties of single colloidal particles in suspension are used to analyze the data. One model describes a solid spherical particle (i.e., a bead) in suspension, and the other is for a single-shell spherical particle (i.e., a microcapsule or a cell) in suspension. The complete model for a single-shell particle improves on Foster and Schwan’s earlier model6 and includes the resistance of the shell and the capacitance of the inner volume. These models provide a straightforward method for interpreting the experimental data and for evaluating the dielectric properties of the samples.
2. Materials and Methods 2.1. Materials. Poly(sodium 4-styrenesulfonate) (PSS; MW = 70000), poly(allylaminehydrochloride) (PAH; MW = 70 000), sodium chloride, tetrahydrofuran (THF), Rhodamine B, and phosphate-buffered saline (PBS) were purchased from Sigma-Aldrich and used without further purification. Polystyrene (PS) beads with an average diameter of 10.25 ( 0.09 μm were provided by Microparticles GmbH (Berlin, Germany); 6.2 μm latex beads were purchased from Bangs Laboratories. The water was purified before being used in a three-stage Millipore Milli-Q (24) Volodkin, D. V.; Petrov, A. I.; Prevot, M.; Sukhorukov, G. B. Langmuir 2004, 20, 3398–3406. (25) Sukhorukov, G. B.; Rogach, A. L.; Zebli, B.; Liedl, T.; Skirtach, A. G.; K€ohler, K.; Antipov, A. A.; Gaponik, N.; Susha, A. S.; Winterhalter, M.; Parak, M. J. Small 2005, 1, 194–200. (26) Sukhorukov, G. B.; Donath, E.; Lichtenfeld, H.; Knippel, E.; Knippel, M.; Budde, A.; Mohwald, H. Colloids Surf., A 1998, 137, 253–266. (27) Dejugnat, C.; Sukhorukov, G. B. Langmuir 2004, 20, 7265–7269. (28) Georgieva, R.; Moya, S.; Leporatti, S.; Neu, B.; B€aumler, H.; Reichle, C.; Donath, E.; M€ohwald, H. Langmuir 2000, 16, 7075–7081. (29) Georgieva, R.; Moya, S.; Donath, E.; B€aumler, H. Langmuir 2004, 20, 1895–1900. (30) Georgieva, R.; Moya, S. E.; B€aumler, H.; M€ohwald, H.; Donath, E. J. Phys. Chem. B 2005, 109, 18025–18030. (31) Holmes, D.; She, J. K.; Roach, P. L.; Morgan, H. Lab Chip 2007, 7, 1048– 1056.
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Figure 1. Diagram showing a microfluidic chip for high-throughput single-particle impedance spectroscopy. Individual particles are passed through the microfluidic channel (h, channel height) and become polarized under the ac electric field, generated from the microelectrodes (w, electrode width; l, electrode length). The size and dielectric properties of single particles can be characterized by measuring the impedance signal. Plus 185 purification system and had a resistivity higher than 18.2 MΩ cm-1. 2.2. Capsule Preparation. Hollow capsules were obtained by consecutive deposition of PAH and PSS layers onto the surface of 10.25 μm PS particles from 2 mg/mL PSS and PAH solutions in 0.5 M NaCl (starting from PAH), following the layer-by-layer deposition protocol described previously.29,30 Two types of capsules were prepared by adsorbing six or nine PAH/PSS bilayers on the template particles, below named (PAH/PSS)6 and (PAH/ PSS)9, respectively. Each deposition step lasted for 15 min and was followed by two washing steps with water. After the deposition of the layers had been completed, the template cores were dissolved to yield hollow capsules. Core dissolution was conducted by suspending the capsules in THF overnight. The capsules were then washed and resuspended in PBS, with a conductivity of 1.6 S m-1. 2.3. Confocal Laser Scanning Microscopy. Optical images of the capsules in a PBS solution were obtained with a Zeiss LSM 5 Excite system equipped with a 63/1.4 DIC oil immersion objective. The PE multilayers were visualized via addition of a 1 μM dye solution (Rhodamine B) to the suspension of capsules after core dissolution. 2.4. Microfluidic Chip. The microfluidic chip was fabricated using standard photolithography and full wafer thermal bonding. It consists of a microfluidic channel, which consists of a thick photosensitive polymide precursor, on the glass substrate. Two pairs of top and bottom parallel facing microelectrodes, made of a layer of titanium and platinum, were patterned inside the microfluidic channel for single-particle impedance detection. The dimensions of the microchannel and the microelectrodes are shown in Figure 1. Individual chips were diced from the wafer, with inlet and outlet holes drilled for fluid access using a diamond saw. Details of the chip microfabrication process have been reported previously.18-20,31,32 2.5. Single-Particle Impedance Spectroscopy. The instrumentation for single-particle electrical impedance spectroscopy has been described previously.8,18,19,34,34 Two AC signals at low and high frequencies are mixed and applied simultaneously to the top electrodes, generating electric fields in the microchannel. As a particle flows though the electric field region, the polarized (32) Sun, T.; Holmes, D.; Gawad, S.; Green, N. G.; Morgan, H. Lab Chip 2007, 7, 1034–1040. (33) Cheung, K.; Gawad, S.; Renaud, Ph. Cytometry, Part A 2005, 65A, 124– 132. (33) Cheung, K.; Gawad, S.; Renaud, Ph. Cytometry, Part A 2005, 65A, 124– 132. (35) Sun, T.; Gawad, S.; Green, N. G.; Morgan, H. J. Phys. D: Appl. Phys. 2007, 40, 1–8.
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particle alters the electric field distribution, and this perturbation was monitored by measuring the changes in the current response of the system from the bottom electrodes. Two pairs of top-bottom electrodes employ a differential impedance sensing scheme. A custom-designed electronic circuit measured the differential current changes. Two lock-in amplifiers (SR844, Stanford Research Instruments) are used to demodulate the signals and output the in-phase and out-of-phase signals at each particular frequency. A 16-bit data acquisition card (NI6034E, National Instruments) was used for real-time data recording. The captured data were processed and analyzed in custom-coded software in MATLAB (Mathworks Inc., Natick, MA). The impedance measurements were performed by sweeping the frequency in the range from 500 kHz to 30 MHz at an amplitude of 2.5 Vpp.
3. Complete Equivalent Circuit Models The equivalent circuit models are based on Maxwell’s mixture theory,17 which states that for a spherical homogeneous particle in suspension, situated in a homogeneous electric field, the equivalent complex permittivity of the mixture is ~ε mix ¼ ~ε m
1 þ 2Φf~CM 1 - Φf~
ð1Þ
CM
where ε~mix, ε~m, and ε~p are the complex permittivities of the mixture, suspending medium, and particle, respectively, Φ is the volume fraction (the volume ratio between the particle and the suspending system), and f~CM is the complex Clausius-Mossotti factor. ~ε p - ~ε m f~CM ¼ ~ε p þ 2~ε m
σ0 Δε þ jω 1 þ jωτ
ð3Þ
with 2εm þ εp - 2Φðεm - εp Þ 2εm þ εp þ Φðεm - εp Þ 2σ m þ σ p - 2Φðσm - σp Þ σ0 ¼ σm 2σm þ σp þ Φðσm - σp Þ Δε ¼ ½9ðεm σp - εp σ m Þ2 Φð1 - ΦÞ=f½2εm þ εp ε¥ ¼ εm
þ Φðεm - εp Þ½2σm þ σp þ Φðσm - σp Þ2 g and the time constant of the relaxation: 2εm þ εp þ Φðεm - εp Þ τ¼ 2σm þ σp þ Φðσm - σp Þ
ð4aÞ ð4bÞ
ð4cÞ
ð4dÞ
(36) Sun, T.; Green, N. G.; Gawad, S.; Morgan, H. IET Nanobiotechnol. 2007, 1, 69–79.
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Z~mix ¼
1 jω~ε mix Gf
ð5Þ
where Gf is the geometric constant of the system, which for an ideal parallel plate electrode system is A/g (units of meters). In many cases, the measurement system is not ideal. For example, in the microfluidic impedance chip shown in Figure 1, the electric field is no longer uniform in the entire volume and the effect of fringing fields has to be taken into account. Now the geometric constant is modified and becomes Gf = κl, where l is the length of the electrode and κ is a correction factor, describing the fringing field; κ can be derived analytically using conformal mapping methods, which is determined by the width of the electrode (w) and the height of the channel (h).37 Derivation of the equivalent circuit is as follows. From eq 5, replace jω with s and substitute eq 3 to give 1 1 1 1 ¼ Z~mix ¼ Gf s ε¥ þ σ0 þ Δε Gf sε¥ þ σ 0 þ 1sΔε þ sτ s 1 þ sτ
ð6Þ
The equivalent circuit for the suspending medium is simply that of a resistor, Rm, and capacitor, Cm, in parallel. When a particle is included in the system, the admittance is modified to include the impedance of the particle Z~p in parallel. The impedance of this complete circuit is
ð2Þ
The complex permittivities ε~m and ε~p are ε~ = ε - jσ/ω, where j2 = -1; ε is the permittivity, and σ is the conductivity. Equations 1 and 2 show that experimental measurement of the complex permittivity of a mixture of particles can be used to derive the complex permittivity of the individual particles in the suspension provided the volume fraction and properties of the suspending medium are known. 3.1. Single Solid Spherical Particle in Suspension. First consider the particle as a solid dielectric sphere (i.e., a bead). In this case, εp and σp are frequency-independent. Applying the Laplace and Fourier transform35 to eq 1, we can rewrite Maxwell’s mixture equation as a dielectric relaxation of the form ~ε mix ¼ ε¥ þ
Note that the expressions for parameters ε¥, σ0, Δε, and τ are identical to those given by Foster and Schwan.6 Consider an ideal parallel plate electrode geometry with electrode area A and gap g. The impedance of the system is given by
Z~cir ¼
1 1 sCm þ R1m þ Zp ~
ð7Þ
Comparing eq 7 with eq 6, we see that Rm and Cm are given by 1 σ 0 Gf Cm ¼ ε¥ Gf
Rm ¼
ð8aÞ ð8bÞ
The resistance of the medium (Rm) is determined by the limiting low-frequency conductivity σ0, and the capacitance of the medium (Cm) from the limiting high-frequency permittivity (ε¥). Therefore, comparing eqs 6 and 7, we see that the impedance of the particle is 1 þ sτ Z~p ¼ sΔεGf
ð9Þ
This equation shows that particle is represented as a resistor in series with a capacitor, as shown in Figure 2, with values for the individual components: Cp ¼ ΔεGf τ Rp ¼ Cp
ð10aÞ ð10bÞ
This shows that the capacitance of the particle (Cp) is related to the magnitude of the dielectric dispersion (Δε). The relaxation time constant (τ) is the time constant of the resistor-capacitor combination for the particle. As a summary, Figure 2 with eq 10 (37) Wachner, D.; Simeonova, M.; Gimsa, J. Bioelectrochemistry 2002, 56, 211– 213.
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Figure 2. Diagram showing the complete circuit model of a dielectric spherical particle (i.e., a bead) in suspension. The particle is modeled as a resistor (Rp) and capacitor (Cp) in series.
gives the complete equivalent circuit model of a solid spherical particle in suspension, which gives the first developed model in this paper. 3.2. Single-Shell Spherical Particle in Suspension. A single-shell model, shown in Figure 3a, is normally applied in studying the dielectric response of capsules28-30 or cells6,8,20,35-39 in suspension. The complex permittivity of the shelled particle is given by ~ε p ¼ ~ε sh
γ3 þ 2K γ3 - K
ð11Þ
with K¼
~ε i - ~ε sh R þd and γ ¼ ~ε i þ 2~ε sh R
where ε~sh and ε~i are the complex permittivity of the shell and inner volume, respectively. R is the inner radius of the particle, and d (d , R) is the thickness of the shell. As demonstrated in earlier work,35 the mixture equation can be rewritten in the following form: ~ε mix ¼
εm a 1 k2 τ 1 k3 τ2 k1 þ þ þ b1 1 þ jωτ1 1 þ jωτ2 jω
ð12Þ
where the coefficients a1, b1, k1, k2, and k3 and the two characteristic relaxation time constants, τ1 and τ2, are described in ref 35. A single-shell object has two interfacial dispersions, as seen from eq 12. Recasting this equation in a form similar to eqs 3 and 4, we have εm a 1 b1 σ0 ¼ k1 Δε1 ¼ k2 τ1 Δε2 ¼ k3 τ2 2b1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τ1 ¼ b2 - b2 2 - 4b1 b3 2b1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τ2 ¼ b2 þ b2 2 - 4b1 b3 ε¥ ¼
where coefficients b2 and b3 are described in ref 35. On the basis of Foster and Schwan’s simplified circuit model6 (Figure 3b), the values of the electrical components are Suspending medium: Rm ¼
ð13aÞ ð13bÞ ð13cÞ ð13dÞ ð13eÞ ð13f Þ
(38) Stewart, D. A.; Gowrishankar, T. R.; Smith, K. C.; Weaver, J. C. IEEE Trans. Biomed. Eng. 2005, 52, 1643–1653. (39) Asami, K. J. Phys. D: Appl. Phys. 2006, 39, 492–499. (40) Estrela-Lopis, I.; Leporatti, S.; Moya, S.; Brandt, A.; Donath, E.; M€ohwald, H. Langmuir 2002, 18, 7861–7866.
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Figure 3. (a) Schematic diagram of a single-shell spherical model. (b) Diagram showing Foster and Schwan’s simplified circuit model for a single-shell particle in suspension. The particle is modeled as a resistor Ri (inner) and a capacitor Csh (shell) in series. (c) Diagram showing the complete circuit model for a single-shell particle in suspension. The particle is modeled as a resistor Ri and a capacitor Ci in series (inner) in combination with a resistor Rsh and a capacitor Csh in parallel (shell).
1 σm ð1 - 3Φ=2ÞGf
ð14aÞ
Cm ¼ ε¥ Gf
ð14bÞ
Single-shell particle: Csh ¼
9ΦRCsh, 0 Gf 4
Ri ¼
4
1 2σm
þ σ1i
9ΦGf
ð14cÞ
ð14dÞ
with shell capacitance per unit area (Csh,0 = εsh/d) and the limiting high-frequency permittivity of the suspension (ε¥ = εm{1 - 3Φ[(εm - εi)/(2εm - εi)]}). As discussed above, these equations ignore the resistance of the shell and the capacitance of the inner volume. Langmuir 2010, 26(6), 3821–3828
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Figure 4. Plot of the impedance magnitude for different sizes of beads in suspension, calculated using mixture theory, and the parallel and series circuit models. The result demonstrates the inaccuracies of the previously published parallel circuit model.32
Consider a shelled particle suspended in an electrolyte; again the suspending medium and particle circuit elements are in parallel. If the impedance of the particle is defined as Z~p, the impedance of the circuit is given by Z~cir ¼
1 1 sCm þ R1m þ Zp ~
ð15aÞ
From mixture theory, the impedance of the system is (eqs 5 and 12) 1 Z~mix ¼ σ0 Δε1 2 s ε¥ þ s þ 1 þ sτ1 þ 1 Δε þ sτ2 Gf
ð15bÞ
The impedances of the circuit and mixture are by definition identical; therefore, mapping the corresponding terms in eqs 15a and 15b gives Rm ¼
1 σ 0 Gf
C m ¼ ε ¥ Gf τ1 τ2 s þ ðτ1 þ τ2 Þs þ 1 Z~p ¼ ½ðτ1 Δε2 þ τ2 Δε1 Þs2 þ ðΔε1 þ Δε2 ÞsGf 2
ð16aÞ ð16bÞ ð16cÞ
According to eq 16c, the complete circuit model of a single-shell particle comprises a resistor and capacitor in parallel representing the shell, and a series resistor-capacitor combinations for the inner volume, as shown in Figure 3c. From this circuit, the impedance of the particle is Z~p ¼ ½Rsh Ri Csh Ci s2 þ ðRsh Ci þ Rsh Csh þ Ri Ci Þs þ 1=ðRsh Csh Ci s2 þ Ci sÞ
ð17Þ
Matching terms in eqs 16c and 17 gives the value of the electrical components as 1 τ1 þ τ2 1 τ1 τ2 ðk2 þ k3 Þ ð18aÞ Rsh ¼ Gf Δε1 þ Δε2 k2 þ k3 ðΔε1 þ Δε2 Þ2 τ1 τ2 ðk2 þ k3 Þ ðΔε1 þ Δε2 ÞRsh 1 Ri ¼ ðk2 þ k3 ÞGf
Csh ¼
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ð18bÞ ð18cÞ
Ci ¼ ðΔε1 þ Δε2 ÞGf
ð18dÞ
where k2 and k3 are defined in ref 35. Figure 3c with eqs 18a-18d gives the complete equivalent circuit model for a single-shell particle. This is the second and also the most significant model proposed in this paper.
4. Results and Discussion 4.1. Theoretical Modeling on Single Colloidal Particles. 4.1.1. Single Solid Spherical Particle in Suspension: A Bead. In ref 32, we described the equivalent circuit for a bead in suspension in the form of a resistor and capacitor in parallel. Although this correctly models the particle when capacitance effects dominate (as in most cases), it leads to errors when the particle conductivity approaches that of the suspending medium. Figure 4 shows how the magnitude of the impedance varies with frequency for two different particle conductivities. We assume Gf = 1 m for nondimensional calculations and the particle volume fraction Φ = 0.1, and εp = 2.55ε0. The suspending medium is phosphate-buffered saline (εm = 78ε0, and σm = 1.6 S m-1). The solid lines are calculated using classical mixture theory, and the points are for the two different circuit models. For a lowconductivity particle (σp = 0.1 S m-1), the series and parallel circuits agree with each other. However, when the particle conductivity (σp = 1 S m-1) approaches that of the medium, inaccuracies occur; the series circuit model gives the correct impedance. This is because in the parallel model, the contribution from the capacitance of the particle is underestimated when the resistance of the particle becomes small, resulting in a lower impedance magnitude compared with that from mixture theory (Figure 4). [In ref 32, experimental data for 5.49 μm latex beads are presented. The conductivity of the particle was 0.87 mS m-1, much lower than the conductivity of the medium (1.6 S m-1), so that either model could have been used.] 4.1.2. Single-Shell Spherical Particle in Suspension: A Cell. To demonstrate the difference between Foster and Schwan’s simplified circuit model6 and our complete circuit model for single-shell particles in suspension, we simulated the impedance spectrum of a typical cell in suspension with different values of membrane (shell) conductivity and cytoplasm (inner) permittivity, using both mixture theory and the circuit models. The parameters are as follows: ε0 = 8.854 10-12 F m-1, R = 5 μm, d = 5 nm, εm = 78ε0, σm = 1.6 S m-1, εsh = 11.3ε0, σsh = 1 10-8 S m-1, εi = 60ε0, and σi = 0.6 S m-1, with volume DOI: 10.1021/la903609u
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Figure 5. Plot of the variation in the impedance (magnitude and phase) for different values of shell conductivity (a and b) and for different values of inner permittivity (c and d).
fraction Φ = 0.1 and Gf = 1 m. Panels a and b of Figure 5 show the magnitude of the impedance and the phase for different values of σsh, respectively. When σsh = 1 10-8 S m-1 (typical for a cell), the difference between the simplified circuit model and Maxwell’s mixture theory is small but visible (as demonstrated in ref 8), while the complete circuit model agrees perfectly with mixture theory. When the cell membrane conductivity increases, the error in the simplified circuit model becomes more apparent, and there is a large discrepancy for a σsh of 1 10-3 S m-1. The membrane resistance is in parallel with the membrane capacitance. As the membrane resistance decreases, it dominates the impedance of the system, leading to a decrease in the magnitude of the impedance of the system at low frequencies. Panels c and d of Figure 5 show the magnitude of the impedance and the phase for different values of inner permittivity (εi), respectively. The cell cytoplasm influences the response at high frequencies (>1 MHz), so the spectra are plotted over the frequency range from 100 kHz to 100 MHz. These figures show that including the cytoplasm permittivity produces small changes, which can be observed only above 20 MHz. Since cytoplasm capacitance is in series with the cytoplasm resistance, as the cytoplasm permittivity increases, the cytoplasm reactance decreases, leading to a decrease in the impedance magnitude of the system at high frequencies. For the hypothetical case in which the permittivity and conductivity of the cell membrane are identical to those in the cytoplasm, the cell should behave exactly the same as the bead. In this situation, the complete circuit model shows that the resistance of the cell membrane goes to zero and the capacitance of the cell membrane to infinity. This means that the cell membrane is short-circuited leaving the cell cytoplasm, which has the same format as the circuit model for the bead, which is a resistor and capacitor in series. 4.2. Experimental Characterization on PE Microcapsules. Two sets of PE capsules were prepared, one with six and 3826 DOI: 10.1021/la903609u
the other with nine PAH/PSS layers. The diameter and thickness of the walls were measured with confocal microscopy. Thirty individual capsules of each type were imaged for the measurements, resulting in an average diameter of 10.38 ( 0.24 μm for (PAH/PSS)6 capsules and a diameter of 10.78 ( 0.18 μm for (PAH/PSS)9 capsules. The shell thickness was estimated to be 24 and 36 nm for (PAH/PSS)6 capsules and (PAH/PSS)9 capsules, respectively. This is consistent with the literature29,30 estimate of approximately 4 nm per PSS/PAH double layer for colloidal templates. Similar values for thickness of single layers were reported by Estrela-Lopis et al.40,41 using the small-angle neutron scattering method. Images of both types are shown in Figure 6. The electric properties of the capsules were measured by passing a mixture of the two capsule types and 6.2 μm diameter latex beads at a ratio of 2:2:1 through the microfluidic single-particle impedance chip. All particles were suspended in phosphate-buffered saline (PBS) with a conductivity of 1.6 S m-1. The final concentration of the mixture was 500000 particles/mL. Figure 7a shows the average value of the impedance magnitude for the three different particles measured at six discrete frequencies between 300 kHz and 20 MHz. Approximately 2000 events were recorded at each frequency. The dots show the mean values of the impedance data, while the bar gives the standard deviation in that population at the corresponding frequencies. The data show that the magnitude of the impedance for the nine-layer capsules is highest at nearly all frequencies. The six-layer capsules have the lowest impedance of the samples. For these particles, the variation in impedance is dominated by the electrical double layer that forms on the electrodes at low frequencies (20 MHz). Langmuir 2010, 26(6), 3821–3828
The complete circuit models were used to analyze the data. The geometric constant, Gf, was calculated using conformal mapping37 and found to be 64.97 μm for the geometrical parameters of the chip: w = 30 μm, h = 22 μm, and l = 36 μm. The system was simulated using PSpice circuit simulations (Orcad, Cadence Inc.). Models for the active elements in the circuit (operational amplifier, etc.) were obtained from the manufacturers. The impedance data for the beads were used to estimate the value of the electrical double-layer capacitance (100 pF) and stray capacitance (0.5 pF) in the system. The dielectric properties of latex beads were taken from published data with permittivity εp of 2.55ε0 and a surface conductance Ks of 1.2 nS.32 Analysis of the capsules was performed using the complete shell model. Because the PE shell layer is permeable to ions and the capsules were suspended in PBS for a long period of time, the dielectric properties of the capsule interior were set equal to those of the suspending PBS. The only unknown parameters are the permittivity and the conductivity of the shell. According to the best fits from the PSpice circuit simulations, as shown in Figure 7a, the relative permittivity of the PE shell is determined to be 50. This value matches the value predicated on annealing measurements on capsules.42 The conductivities of the PE shells for six- and nine-layer capsules are found to be 28 ( 6 and 3.3 ( 1.7 mS m-1, respectively. The shell of the nine-layer capsules is less conductive than that of six-layer capsules, leading to higher impedance magnitude, as seen in the experimental data. This result is consistent with the conclusion in ref 28 that the permeability of the capsules decreases as the number of layers increases; the extra layers effectively reduce the area of the pores in the shell. From the electrorotation measurements,28 the conductivity of the PE shell was estimated to be 1 S m-1 for 10-layer capsules, and a value of 60 was determined for the relative permittivity of the shell. However, these measurements were performed with capsules templated on (43) Estrela-Lopis, I.; Leporatti, S.; Typlt, E.; Clemens, D.; Donath, E. Langmuir 2007, 23, 7209–7215.
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erythrocytes, in which the shell thickness after core removal was substantially smaller than shell thickness of PE capsules templated on latex particles.43 Figure 7b shows a scatter plot of the impedance data. The total numbers of recorded events (n) for each type of particle are 851 for six-layer capsules, 730 for nine-layer capsules, and 436 for 6.2 μm beads. This is consistent with the ratio (2:2:1) of the mixture. The x-axis is impedance at 300 kHz, which measures particle size, while the y-axis is the ratio of high-frequency (5 MHz) to lowfrequency (503 kHz) impedance (opacity19,33) which normalizes the data for particle size. The scatter plot clearly shows three different populations in the mixed samples. The impedance signal from each population was confirmed by separately running pure samples through the chip. The low-frequency impedance data for the nine-layer capsules have a wider distribution than for the beads or the six-layer capsules. This may be because some of these capsules are damaged or deformed during the experiments.
5. Conclusions To conclude, the dielectric properties of PE microcapsules were measured using electrical impedance spectroscopy in a microfluidic chip. This method is superior over the previously reported
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Sun et al.
electrorotation technique in terms of measurement speed. To extract the dielectric properties of the samples from the impedance data, two complete equivalent circuit models have been derived, one for a spherical solid homogeneous dielectric particle in suspension and a further development applicable to a spherical single-shell particle in suspension. Compared to Foster and Schwan’s classical model,6 our model incorporates the resistance of the shell layer and the capacitance of the inside of the single-shell particle. For the PE capsules, the model gives a relative permittivity for the shell of 50 and conductivities of the six- and nine-layer microcapsule shells of 28 ( 6 and 3.3 ( 1.7 mS m-1, respectively. The method presented in this paper can further be extended to yield the complete circuit model for multishell particles without loss of generality. This model can be used for characterizing the dielectric properties of colloidal particles in suspension. Acknowledgment. T.S. acknowledges the postdoctoral fellowship from National Centre for the Replacement, Refinement and Reduction of Animals in Research (NC3Rs), UK. We acknowledge Prof. G. B. Sukhorukov and Mr. M. Bedard for valuable assistance with microcapsule preparation and Dr. D. Holmes for help in the impedance experiments.
Langmuir 2010, 26(6), 3821–3828
