Conducting polymer nanowires for control of local protein concentration in solution

Abstract

Interfacing devices with cells and tissues requires new nanoscale tools that are both flexible and electrically active. We demonstrate the use of PEDOT:PSS conducting polymer nanowires for the local control of protein concentration in water and biological media. We use fluorescence microscopy to compare the localization of serum albumin in response to electric fields generated by narrow (760 nm) and wide (1.5 μm) nanowires. We show that proteins in deionized water can be manipulated over a surprisingly large micron length scale and that this distance is a function of nanowire diameter. In addition, white noise can be introduced during the electrochemical synthesis of the nanowire to induce branches into the nanowire allowing a single device to control multiple nanowires. An analysis of growth speed and current density suggests that branching is due to the Mullins-Sekerka instability, ultimately controlled by the roughness of the nanowire surface. These small, flexible, conductive, and biologically compatible PEDOT:PSS nanowires provide a new tool for the electrical control of biological systems.

1. Introduction

The basic components of biological systems are small, ranging from nanometer-scale proteins to micron-sized cells, and soft. For example, the Young’s modulus of neural tissue is 100 kPa-1000 kPa.^1,2 The small size and soft materials of human biology provide a challenge for the use of implantable bioelectric devices such as neural electrodes.^3–5 The mismatch between the stiffness of implanted materials and the softness of cells and tissues leads to cellular damage and elicits an immune response. Soft materials, such as polymers and hydrogels, are more biocompatible with a Young’s modulus comparable to tissue. However, materials used at the bioelectric interface need to be electrically conductive, as well as small and flexible.

Electrically conductive polymer nanowires, described previously,^6–8 provide a small, flexible, electrically active material for the bioelectric interface. Poly(3,4-ethylenedioxythiophene):polystyrene sulfonate (PEDOT:PSS) nanowires are of specific interest due to the extensive characterization and known biocompatibility of PEDOT:PSS.^9–14 These nanowires have been electrochemically synthesized with average diameters of 340 nm, a Young’s modulus of ~1 GPa, and conductivity of ~8.0 S/cm.^6–8,15 Although the PEDOT:PSS nanowires are still stiffer than cells or tissue, they are two orders of magnitude more flexible than current state-of-the-art carbon fiber neural electrodes with a diameter of 4.5 μm and a Young’s modulus of 380 GPa.¹⁶

Conducting polymer nanowires are promising tools for controlling the local concentrations of charged molecules, including proteins, at cellular and subcellular length scales. Previous work has demonstrated the use of conductive polymer films to control the concentration of proteins in solution.¹⁷ In this work, we demonstrate the use PEDOT:PSS nanowires, rather than bulk films, to control the local concentration of proteins in solution. We compare localization of charged proteins in response to electric fields generated by narrow (760 nm) and wide (1.5 μm) nanowires. We show that proteins in deionized water can be manipulated over a surprisingly large micron length scale through the application of an electric field. We then compare this to an electric field applied in a high salt biological media, phosphate buffered saline (PBS). For future biological applications, which are likely to require multiple nanowires rather than a single nanowire, we demonstrate the synthesis of branched nanowires, allowing a single device to control multiple nanowires.

2. Experimental details

2.1. Electrochemical synthesis of PEDOT:PSS nanowires

Conducting polymer nanowires were synthesized using directed electrochemical nanowire assembly in which nanowires are electropolymerized between two sharp gold electrodes.^6–8,18 Sharpened gold electrodes were fabricated by adapting methods used to etch scanning tunneling microscope electrodes.¹⁹ Briefly, solid gold wire (0.2 mm diameter, 99.9 %, Alfa Aesar, 10195-G1) was secured to 20 gauge stranded wire using parafilm. Gold wires were submersed ~1 mm in high-concentration hydrochloric acid (6 M). Coiled platinum wire (0.3 mm diameter, 99.9%, Alfa Aesar, 43014-BU) served as the counter-electrode. A function generator (Agilent 33120A) provided a 10 Hz full square wave, ±5 V amplitude. The square wave was rectified using a diode to deliver positive 5 V square pulses to the gold anode to initiate the reduction of gold into solution. Etching was terminated after ~90 s to yield tip diameters < 100 nm. After etching, electrodes were rinsed with ethanol, then water, and dried under nitrogen. Electrodes were plasma cleaned (Harrick) for 20 seconds before use.

PEDOT:PSS nanowires were synthesized in an aqueous solution containing 10 mM 3,4-ethylenedioxythiophene (EDOT, Sigma-Aldrich, 483028) monomer and 20 mM polystyrene sulfonate (PSS, Sigma-Aldrich, 243051) as a counterion. PEDOT:PSS nanowires were grown using a function generator (Agilent 33120A) supplying an alternating, square-wave voltage (2–100 kHz) across two sharp gold electrodes. The length of the nanowires is controlled by the spacing of the gold electrodes. The diameter of the nanowires is controlled by the frequency of the voltage used for the electrochemical synthesis. The wider nanowires (1.50 ± 0.55 μm diameter) were grown at 2 kHz and the thinner nanowires (760 ± 220 nm diameter) were grown at 10 kHz. Nanowire diameter was measured using a scanning electron microscope (SEM, Hitachi SU8230) and are the average of 4 different nanowires. Nanowire length was measured using brightfield microscopy (Olympus IX71, 60x objective, Andor iXon CCD camera). Nanowires were grown and imaged in a custom made flow cell to facilitate media exchange between the monomer solution and protein solution. Electrodes were spaced 50 μm apart (tip-to-tip). A −100 mV DC offset voltage was applied to promote PEDOT:PSS nanowire growth from a single electrode. The gap between the counter-electrode and the end of the growing nanowire was held constant by manually adjusting one of the micromanipulators. A Raman microscope (iHR550 Horriba-Jobin Yvon spectrometer fiber-coupled to a BX-41 Olympus microscope) was used to confirm nanowire composition.

Branched nanowires were fabricated by adding a white noise voltage signal to the square wave during nanowire growth. The white noise signal was produced by a second function generator (Agilent, 3220A). A custom-designed high bandwidth summing amplifier was used to combine this wave with the square wave. To prevent attenuation of the noise signal, the amplifier bandwidth should exceed that of the white noise generator (20.0 MHz). The voltage-dependent current density, $\langle \widehat{\text{J}}\rangle$, (Equation (1)), is required to understand the branching of the PEDOT:PSS nanowires. Average growth velocities, 〈v〉, of the PEDOT:PSS nanowires at different voltage amplitudes were determined by analyzing movies of nanowire growth. The movies are collected at a rate of 10 Hz and were typically 100–400 s in length. Mass conservation implies that growth at velocity 〈v〉 requires an average current density of;

$$\langle \widehat{\text{J}}\rangle \cong ze{\rho }_{\mathit{\text{PEDOT}}}\langle v\rangle ,$$ (1)

where ρ_PEDOT = 6.22 × 10²⁷ m⁻³ is the estimated number density of EDOT monomers in the PEDOT:PSS nanowire and ze (where z = 1) is the charge that is transferred during the addition of one EDOT monomer to the nanowire.

The Butler-Volmer equation is commonly used to analyze the current density associated with electrochemical deposition, as occurs during PEDOT:PSS nanowire growth. This Arrhenius-type rate law characterizes the rate-limiting step of the overall chemical mechanism by which the deposition process occurs. We assume that this process is rate-limited by a single electron step of the form R⁰ =O^z + e⁻, where R⁰ is the EDOT monomer and O^z is the rate-limited, oxidized EDOT species in the z = 1 oxidation state. The Butler-Volmer equation relates the current density collected by an electrode of curvature κ to the overpotential η:²⁰

$$\widehat{\text{J}}={\widehat{\text{J}}}_0\left[-e^{-\alpha \beta \left(e\eta +\frac{1}{\beta }ln\frac{c}{c_0}-{\rho }_{\mathit{\text{PEDOT}}}^{-1}\widehat{\gamma }\kappa \right)}+\frac{c}{c_0}{e}^{\left(1-\alpha \right)\beta \left(e\eta +\frac{1}{\beta }ln\frac{c}{c_0}-{\rho }_{\mathit{\text{PEDOT}}}^{-1}\widehat{\gamma }\kappa \right)}\right],$$ (2)

where ĵ₀ is the exchange current density, η = V_App − V_Eq. V_App is the applied potential, and V_Eq is the equilibrium potential at which no current flows. Only the oxidation term is significant during the positive half cycles when deposition occurs. In Equation (2), a positive current density corresponds to deposition and implies a net flow of negative charge into the working electrode. α describes the symmetry of the activation energy barrier along the reaction coordinate and β is the inverse thermal energy (k_BT)⁻¹. c (c₀) is the interfacial (bulk) concentration of EDOT. The Gibbs-Thomson factor ${\rho }_{\mathit{\text{PEDOT}}}^{-1}\gamma \kappa$ accounts for curvature effects²¹ and γ is the surface tension of the polymer-solution interface. κ is the local curvature of the interface and is defined as $\kappa \cong -\raisebox{1ex}{${\partial }^{2}h$}\!\left/ \!\raisebox{-1ex}{$\partial x^2$}\right.$. A protrusion (depression) has a positive (negative) curvature and a retarded (enhanced) growth rate. The Gibbs-Thomson effect is due to the decreased (increased) degree of bonding between neighboring surface molecules on the protrusion (depression). The Gibbs-Thomson effect is counteracted by the Mullins-Sekerka instability, which enhances the deposition rate on sharp protrusions, as described in the Appendix. The competition between these two effects results in branching at sufficiently high voltage amplitude, V_App.

2.2. PEDOT:PSS nanowire conductivity

The conductivity of the nanowire was determined using a two-point probe resistance measurement. A peristaltic pump was used to rinse the nanowire with 25 mL of deionized water to remove residual monomer. A 2 V, 10 kHz square wave was applied between the nanowire and the counter-electrode to fuse the nanowire across the electrodes. A Keithley 2400 source meter was controlled using a custom Igor Pro script to measure resistance. Voltage was swept between −1 and +1 V while measuring current. The resistance of the wire was determined by the inverse slope of the linear best fit line. Conductivity of the nanowire was calculated using the formula:

$$\sigma =\frac{L}{AR}$$ (3)

where σ is conductivity (S cm⁻¹), L is nanowire length, A is nanowire cross-sectional area, and R is electrical resistance. Two-point probe measurement were carried out on 13 nanowires.

2.3. Imaging protein concentration

To image protein localization, bovine serum albumin (BSA, Thermo Fischer Scientific, BP1600–100) was labeled with AlexaFluor546 (Thermo Fischer Scientific, A20002) according to the manufacturer’s instructions. After the growth of a nanowire, the EDOT and PSS solution was exchanged for ultrapure deionized water (EASYpure II, Barnstead) or phosphate buffered saline (PBS, Gibco, 14040). Fluorescently-labeled BSA was then added to the solution for a final volume of 1.5 mL. Fluorescence images were taken with an EMCCD camera (Andor iXon CCD camera) coupled to an inverted microscope (Olympus IX71, 40x objective). Profile plots of fluorescence intensity within a 25 μm × 10 μm box, centered on the nanowire, were used to quantify the amount of protein as a function of distance. The plots are an average of 25 line profiles taken perpendicular to the nanowire, averaging the fluorescence intensity for 20 ms at each pixel. This area was chosen to cover the full distance over which protein concentration was affected by the nanowires and to maximize the number of lines available for averaging. An average, rather than a single line profile, was used to minimize noise. The plots were generated using Igor Pro’s image processing package.

3. Results

3.1. Use of PEDOT:PSS nanowires to control local protein concentration

The electrochemical synthesis of conducting polymer nanowires has been described previously.^6–8 SEM was used to measure the diameter of the nanowires (Figure 1a and b, Supporting Figure 1) and nanowire composition was confirmed by comparing Raman spectra of the nanowires with a PEDOT:PSS film (Figure 1c). Conductivity of the nanowires was measured with two-point probe resistance measurements. The average PEDOT:PSS nanowire conductivity was found to be 22 ± 10 S/cm (n=13 wires). Surface roughness of the nanowires, measured as an integrated area normalized by the length of the segment measured, was 17.6 ± 8.3 and 23.4 ± 7.8 μm for nanowires synthesized at 10 and 2 kHz, respectively (non-significant difference, p-value = 0.35, n = 4 nanowires at each diameter, Supporting Figure 1).

Figure 1.

PEDOT:PSS nanowires. (a) SEM of a PEDOT:PSS nanowire grown with an AC frequency of 2 kHz. (b) SEM of a PEDOT:PSS nanowire grown with an AC frequency of 10 kHz. (c) Raman spectrum of a PEDOT:PSS nanowire (black), compared to a PEDOT:PSS film (red).

We first used relatively thick PEDOT:PSS nanowires with a diameter 1.5 μm (length = 25 μm) to control local protein concentration. Fluorescently-tagged BSA protein was added to the solution after nanowire growth and an AC field (2 V, 1 Hz, square wave) was applied. Protein concentration, measured as fluorescence intensity, increased in the region of the PEDOT:PSS nanowire while a positive bias was applied (Figure 2a). A decrease in concentration was observed at negative biases (Figure 2b). This behavior is consistent with expectations given the net negative charge of BSA.^22,23 To quantify the change in protein concentration as a function of distance from the nanowire, fluorescence intensity within a 25 μm × 10 μm box, centered on the nanowire, was averaged over 20 ms (Figure 2c). An oscillation in fluorescence intensity, from a high value of 182.0–188.5 a.u. to a low value of 103.9–122.9 a.u., was observed within this region in response to the electric field, applied at t = 4 seconds. To confirm that the oscillation in protein concentration is due to the nanowire, and not the gold electrode, we took line profiles of the fluorescence intensity roughly 12 μm away from the gold electrode surface, similar to the distance used for the nanowire analysis (Figure 2). There was no significant change in fluorescence with changes in bias indicating that it is the PEDOT:PSS nanowire, not the gold electrode, responsible for altering the BSA concentration (Supporting Figure 2).

Figure 2.

Representative fluorescence microscopy imaging of a PEDOT:PSS nanowire with fluorescently-labeled BSA (~100 nM) in the surrounding solution. The protein responds to an applied AC field (2V, 1Hz) from the nanowire. (a) Image of a nanowire and protein (green); +1V with respect to the gold counter-electrode. (b) Image of a nanowire and protein (green); −1V with respect to the gold counter-electrode. (c) Protein concentration, measured as fluorescence intensity, as a function of voltage in a 25 μm × 10 μm region of interest surrounding the PEDOT:PSS nanowire.

3.2. Comparison of protein modulation as a function of nanowire diameter

To determine the importance of nanowire diameter for controlling local protein concentration, we repeated experiments using a thinner PEDOT:PSS nanowire with a diameter of 760 nm. In order to compare the local control of BSA, we again used profile plots of fluorescence intensity as a function of distance, centered on the nanowire (Figure 3). A positive bias shows less of a drop in fluorescence intensity at the nanowire compared to a negative bias due to an accumulation of BSA at the PEDOT:PSS nanowire surface. The distance over which protein concentration was modulated was found to be 29.6 μm ± 8.6 μm and 16.7 μm ± 2.5 μm for the wide (d = 1.5 μm) and narrow (d = 760 nm) nanowires, respectively (Figure 3a and b). Averages and standard deviations were determined from measurements using three separate nanowires. We next quantified the amount of protein modulated for each diameter of nanowire. The relative quantity of protein modulated by the nanowire can be obtained by integrating the difference in area under the profiles at each bias. This integration indicates that the thin nanowire manipulates 22.1% ± 7.3% of the quantity of protein displaced by the wider nanowire.

Figure 3.

Representative profile plots of BSA concentration, measured as fluorescence intensity, as a function of nanowire diameter. (a) Charged (red, +1V) and discharged (black, −1V) PEDOT:PSS nanowire (d = 1.5 μm). The inset (50 μm × 25 μm) shows the cross section of the nanowire used to generate profile plots. Profile plots are generated from an average of 25 pixel lines perpendicular to the nanowire. (b) Charged (red, +1V) and discharged (black, −1V) PEDOT:PSS nanowire (d = 760 nm).

3.3. Comparison of nanowire activity in water and biological media

The deionized water used in the experiments described above (Figures 2 and 3) provides an effective model environment for studying the modulation of protein concentration with PEDOT:PSS nanowires. However, in an actual biological environment, either cellular experiments or in vivo applications, salts will be present. For this reason, we investigated the impact of biological media on the ability to control local protein concentration using PBS, a saline solution containing sodium chloride, potassium phosphate, and sodium phosphate (150 mM total salt concentration). Similar to the experiments described above, we applied an AC field (2 V, 1 Hz, square wave) to a PEDOT:PSS nanowire (d = 1.5 μm, l = 25 μm) in the presence of BSA (300 nM) in PBS. We again used profile plots to compare the spatial extent of control and find that the thickness of our electrostatic double layer drops below the resolution of our microscope (Supporting Figure 4).

3.4. Electropolymerization of branched PEDOT:PSS nanowires

For future biological applications, there are advantages to using multiple nanowires controlled by a single device. To this end, we have developed a method for growing multiple PEDOT:PSS nanowires from a single gold electrode. The addition of electrical noise during growth causes branches to form on the nanowire (Figure 4). The number and length of the branches scales with the amplitude of the additional white noise. For example, ~5 μm long branches and an average of 5 branches per nanowire at 800 mV (Figure 4e–h) and 14 μm long branches and 7 branches per nanowire at 1.5 V (Figure 4i–l). No growth was observed when a noise signal was applied in the absence of the square wave.

Figure 4.

Brightfield images of PEDOT:PSS nanowires grown with only a square wave signal (a-d, 940 mV, 10.0 kHz), with addition of white noise (800 mV) to the square wave (e-h), and an increased amplitude of white noise (i-l, 1.5 V). Scale bar = 20 μm.

To gain insight into the branching process, we have estimated the average current density associated with the oxidative deposition of EDOT during nanowire growth. Average growth velocities, 〈v〉, of the PEDOT:PSS nanowires were measured at different amplitudes (Figure 5). Velocity values were determined by analyzing movies of nanowire growth. There is a ~750 mV threshold to nanowire growth, below which no growth was observed. As the amplitude is increased above this threshold, the growth velocity nonlinearly increases. Based on mass conservation (Equation (1)), we convert the velocity measurements to current density values, also plotted in Figure 5. Current density data are fit to the Butler-Volmer equation (Equation 2). An exchange current density of ĵ₀ ~ 10 A m⁻² and a symmetry factor α = 0.145 are needed to account for the ~0.75 V threshold to charge flow. The other fitting parameter values are c_R/c*_R = 0.5, γ = 400 J m⁻² and κ = R⁻¹ where R is the 380 nm radius of the nanowire. The fit quality suggests that the Butler-Volmer model accurately describes the AC (10 kHz) EDOT oxidation that occurs during wire growth.

Figure 5.

Growth speed (〈v〉, unfilled circles) and mean current density ($\langle \widehat{J}\rangle$, filled circles) as a function of the square wave voltage amplitude. Error bars in the current density are the propagated uncertainties from the velocity for 3–5 measurements at each amplitude value. The solid line is a best-fit of Equation (2) to the estimated current density data.

4. Discussion

4.1. Use of PEDOT:PSS nanowires to control local protein concentration

PEDOT:PSS nanowires have been reported with diameter of <500 nm.^6–8 Within our lab, diameters of 500 nm – 1.5 μm are typical (Figure 1a and b, Supporting Figure 1). The length of the nanowire is controlled by the position of the two gold electrodes, typically 800 nm to 10 mm. On our size scales, conductivity was found to be independent of nanowire diameter and length. Surface roughness of the nanowires, which determines the electrochemical surface area in contact with the electrolyte solution, was similar for all of the nanowires used in these experiments (Supporting Figure 1).

The results described above show that nanowires can be used to control the local protein concentration (Figure 2 and Figure 3). The profile plots used to quantify the amount of protein as a function of distance are representative of the behavior in the xy plane around the nanowire. The fluorescence intensity at each pixel is the sum of fluorescence in the z direction through the depth of field. For our microscope, the depth of field (~1 μm) is on the same scale as the diameter of the nanowires, (~750 nm to 1.5 μm). Given the cylindrical symmetry of the nanowires, the dominant electric field component is perpendicular to the nanowire and the dominant motion of proteins in this 1 μm slice is in the xy plane.

The distance of activity, 29.6 μm ± 8.6 μm and 16.7 μm ± 2.5 μm for the wide (d = 1.5 μm) and narrow (d = 760 nm) nanowires, respectively, and amount of protein modulated depends on the diameter of the nanowire with the thin nanowire modulating just 22.1% ± 7.3% of the quantity of protein displaced by the wider nanowire (Figure 3, n = 3 for each nanowire diameter). At equilibrium, the small nanowire would be expected to store 50% less charge due to a 50% reduction in surface area. Since BSA acts as the negatively charged species in the electrostatic double layer, this decrease in charge storage will result in the modulation of less BSA. At our frequency (1 Hz), however, the nanowires have not yet charged to equilibrium. This likely explains the deviation from the expected 50% reduction in charge storage. The increased quantity of protein manipulated by the larger nanowire highlights how altering the diameter of a PEDOT:PSS nanowire can provide the appropriate degree of charge storage for a desired application.

PBS was used as a representative biological media. In a high concentration of electrolytes, the electrostatic double layer will be comprised primarily of salts instead of BSA. In this case, we would expect the electrostatic double layer to be dramatically reduced. We find that the thickness of our electrostatic double layer drops below the resolution of our microscope for experiments in PBS (Supporting Figure 4). This change is expected since the diffuse layer portion of the electrostatic double layer decreases as electrolyte concentration increases.²⁴

It is important to note that under our experimental conditions a large number of factors that are difficult to control can alter the spatial extent over which protein is modulated. For example, slight changes in electrode geometry, changes in nanowire roughness or branching, variations in the contact resistance between the gold electrode and the nanowire, and the exact sharpness of the gold electrodes could all cause differences in the spatial extent of protein modulation. These variations are reflected in the relatively large standard deviations of the spatial extent of nanowire activity. Regardless of these factors, the most surprising aspect of this result is the large distance over which protein is controlled for both nanowires. These distances are dramatically larger than the expected thickness of the electrostatic double layer at equilibrium, which is on the length scale of angstroms.²⁴ Previous reports have demonstrated ~250 nm control of proteins using electric fields,²⁵ also much longer than expected. This previous research used fluorescence microscopy to observe the electrostatic trapping of proteins at the surface of silica. The negative zeta potential of silica (−0.25 mV) trapped the positively charged protein concanavaline A at distances up to ~250 nm. At higher ionic strength no trapping occurred, confirming the observed control was electrostatic. We show modulation of proteins over a length scale that is 100-fold greater. Conditions unique to our experiment which may alter the equilibrium thickness of the electrostatic double layer include the high curvature of the nanowire surface²⁶ and large counter ions (BSA).²⁷ The curvature of the nanowires studied here, however, is below what is necessary to induce significant changes in the electrostatic double layer.²⁶ The large size of BSA is expected to increase the equilibrium electrostatic double layer, but only out to a few nanometers. Instead, we suggest that our profiles are not at equilibrium. This is possible if the nanowire is charging in a regime controlled by bulk diffusion. These slow charging times allow for variation of the concentration of electrolytes over much larger distances.²⁸ Experiments at lower frequencies (50 mHz) show continued charging of the PEDOT:PSS nanowire, supporting this hypothesis (Supporting Figure 3).

4.2. Electropolymerization of branched PEDOT:PSS nanowires

We have described a method for the synthesis of branched PEDOT:PSS nanowires (Figure 4). The current density associated with nanowire growth obeys the Butler-Volmer equation (Figure 5). Haataja and co-workers have recently shown how electrochemical systems that obey Equation (2) may undergo branching via the Mullins-Sekerka instability.²⁰ We outline the mechanism here with a full discussion provided in the Appendix. The surface of a PEDOT:PSS nanowire is rough on a microscopic scale (Figure 1). Its roughness profile, which we will call z(x,t), may be decomposed into a sum of Fourier modes, each of spatial frequency k (units: rad/m). The contribution of a single mode is $\text{z}+{\mathrm{\delta }\text{z}}_{\text{k}}\left(\text{x},\text{t}\right)=\text{vt}+{\mathrm{\delta }\text{z}}_{0,\text{k}}{\text{e}}^{\text{ikx}}{\text{e}}^{{\mathrm{\omega }}_{\text{k}}\text{t}}$. v is the steady-state growth velocity of the interface in the z-direction and δz_0,k is the amplitude of the k^th mode. The factor ${\text{e}}^{{\mathrm{\omega }}_{\text{k}}\text{t}}$ is the stability factor. If the surface is unstable, ω_k is non-zero. If ω_k is positive (negative), mode k will experience amplified (retarded) growth. For example, if only mode k is unstable, the nanowire profile will become wave-like (with spatial wavelength λ = 2π/k). As time passes, the crests of the wave will grow into branches. This effect occurs because a protrusion on a surface steepens the local solute concentration gradient increasing the local current density and the growth rate of the protrusion (i.e. the Mullins-Sekerka instability).²⁹ Time differentiation of $\text{z}+{\mathrm{\delta }\text{z}}_{\text{k}}$ gives the growth velocity of the k^th mode: $\text{v}+{\mathrm{\delta }\text{v}}_{\text{k}}=\text{v}+{\mathrm{\omega }}_{\text{k}}{\mathrm{\delta }\text{z}}_{\text{k}}$. Growth at velocity $\text{v}+{\mathrm{\delta }\text{v}}_{\text{k}}$ requires a current density $\widehat{\text{J}}+{\mathrm{\delta }\widehat{\text{J}}}_{\text{k}}\cong {\text{neρ}}_{\text{PEDOT}}\left(\text{v}+{\mathrm{\delta }\text{v}}_{\text{k}}\right)$, where Equation (1) was used. Substituting for ${\mathrm{\delta }\text{v}}_{\text{k}}$ (i.e. ${\mathrm{\delta }\text{v}}_{\text{k}}={\mathrm{\omega }}_{\text{k}}{\mathrm{\delta }\text{z}}_{\text{k}}$) gives the current density fluctuation, which is a measure of the electrical noise level:

$${\mathrm{\delta }\widehat{\text{J}}}_{\text{k}}\left(\text{x},\text{t}\right)={\text{neω}}_{\text{k}}{\mathrm{\rho }}_{\text{PEDOT}}{\mathrm{\delta }\text{z}}_{\text{k}}.$$ (4)

The noise in the current density ${\mathrm{\delta }\text{j}}_{\text{k}}$ and the amplification rate ω_k are proportional, suggesting that elevated noise levels induced by the applied voltage signal can increase the amplification rate and induce branching, providing a qualitative explanation for the observed noise-induced branching (Figure 4).

5. Conclusion

Conducting polymer nanowires provide a tool to control the concentration of charged molecules, such as proteins, on a biologically-relevant length scale, using an applied electric field (Figure 1). In deionized water, these nanowires can modulate protein concentration over ~30 μm for the wide, 1.5 μm diameter, nanowires and ~17 μm for the thinner, 760 nm diameter, nanowires (Figure 2 and 3). The quantity of protein moved through solution is determined by the diameter of the nanowire, which is controlled by the frequency at which the nanowires are electrochemically synthesized. In PBS, a high salt buffer representative of a biological environment, the distance over which protein concentration can be modulated drops significantly as the thickness of the electrostatic double layer decreases (Supporting Figure 4). Based on our results in deionized water, we expect that protein concentration is modulated by the nanowires in PBS, but quantification of the distance will require methods beyond standard fluorescence microscopy.

We envision the use of these nanowires for cellular applications in which modulating the concentration of ions at individual cells leads to a specific cellular response. For example, to initiate action potentials in neurons through the movement of ions across the plasma membrane. A challenge for interfacing any device with any tissue is reconciling the relative softness of tissue with the stiffness of the inserted material. This mismatch in material properties leads to inflammation and scarring at the electrode, limiting the lifetime of the device and causing tissue damage.^3–5,30 Recent work has addressed this challenge with a focus on smaller, more flexible, materials, such as ~100 nm single crystalline gold nanowires,³¹ softer or conformable materials,^32–35 and highly sophisticated nanoelectrode arrays of metal or silicon nanoneedles, nanopillars, and nanotubes that are expected to be less invasive.^36–39 Similarly, “meshes” of nanowire field-effect transistors provide both a nanoscale device for neural recording and minimize disruption during the implantation step.^40,41 We expect that PEDOT:PSS nanowires could provide an additional tool for the modulation of single cells with minimal disruption to the surrounding tissue. Additionally, the nanowires could be used as an active material, providing both topographic and ionic control for the enhanced growth and migration of excitable cells (neurons, heart cells, muscle cells),^42–46 which have been shown to respond to an applied electric field.^47–50 As with neural probes, a soft polymer substrate would provide a better interface for cells. For these cellular applications, it will be useful to have multiple nanowires to control multiple cells. Our results demonstrate that white noise can be introduced during the electrochemical synthesis of the nanowire to induce branches into the nanowire (Figure 4 and 5). We expect these small, flexible, conductive, and biologically-compatible PEDOT:PSS nanowires will provide a new tool for the electrical control of biological systems.

APPENDIX

Branching in systems that grow by dendritic solidification, as is the case for the PEDOT:PSS nanowires, is due to the Mullins-Sekerka instability.¹ Hataaja and co-workers have described how the Mullins-Sekerka instability occurs during the electrochemical reduction and deposition of metals. Here we apply this theory to PEDOT:PSS nanowire growth, which occurs by oxidative deposition of the neutral EDOT moiety. The basic concept underlying this electrochemical version of the Mullins-Sekerka instability is that the roughness of the nanowire-solution interface perturbs the EDOT concentration in solution which, in turn, perturbs the local growth rate of the interface. For this reason, the interface grows non-uniformly with the sharpest protrusions experiencing the steepest concentration gradients at their tips. These features would then undergo runaway growth, but the Gibbs-Thomson effect, which slows the growth rate of sharp protrusions, acts to balance the diffusive instability.

The basic picture is of a working electrode immersed in a salty solution containing the neutral EDOT monomers. These monomers diffuse to the electrode-solution interface where they undergo oxidative charge transfer and polymerize on the electrode surface. Conservation of the EDOT species in the solution is expressed by a continuity equation of the form:

$$\frac{\partial c}{\partial t}+\overrightarrow{\nabla }\cdot j_i=0$$ (A1)

where c is the EDOT concentration. The EDOT flux j (units: $\frac{\mathit{\text{particles}}}{m^{2}s}$) is given by

$$j=-D\nabla c$$ (A2)

where D is the EDOT diffusivity. The spatial and temporal dependence of the quantities c and j is implied. Equations A1 and A2 must be solved subject to appropriate boundary conditions in order to determine the concentration and flux profiles near an electrochemical interface. A key result of the Hataaja study (p. C702 of Ref. 2) is that for a perfectly flat (unperturbed) interface, the steady state building block concentration c^SS is found to be approximately

$$c^{SS}\left(z\right)\cong c_0-\frac{jL}{D}+\frac{jz}{D}$$ (A3)

where c₀ is the bulk cation concentration and j is the metallic cation flux onto the surface. In this steady state case, the surface will grow steadily in time in the +z direction with velocity v. We apply equation A3 to the present analysis of EDOT monomers near an interface.

Equation A3 describes the concentration profile near a perfectly flat interface. When the surface is rough, however, the surface can grow unstably. The profile of the surface may be expressed as a Fourier sum given by $h\left(x,t_0\right)=vt+{\Sigma }_k\delta z_k\left(x,t\right)$, where v is the steady-state growth velocity of the interface in the z-direction and $\delta z_k\left(x,t\right)$ is the k^th oscillatory perturbation (i.e. Fourier mode) to the flat interface. The basic question regarding how the surface changes in time may be resolved by performing a linear stability analysis. In this approach, the form of $\delta z_k\left(x,t\right)$ is $\delta \widehat{z}{e}^{ikx}{e}^{{\omega }_{k}t}$, where $\delta \widehat{z}$ is the initial amplitude of the k^th oscillatory mode. The logic of this choice is as follows. For simplicity, we will treat the surface as one dimensional. It is convenient to consider just the k^th mode $h_k=vt+\delta z_k\left(x,t\right)=vt+\delta {\widehat{z}}_k\left(t\right)e^{ikx}$. $\delta \widehat{z}\left(t\right)$ is the amplitude of the k^th mode, whose time dependence is (as yet) unknown. The time derivative of ${\dot{h}}_k$ is ${\dot{h}}_k=v+\delta {\dot{z}}_k$. We regard ${\dot{h}}_k$ as a function of h_k(x,t) and write ${\dot{h}}_k=f\left(h_k\right)=f\left(z_0+\delta z_k\right)$. To first order in the perturbation $\delta z_k$, a Taylor expansion about z₀ yields $f\left(h_k\right)\cong f\left(z_0\right)+\delta z_k{\left[\frac{df}{d{h}_k}\right]}_{ z_0}$ Equating terms between this expression and ${\dot{h}}_k=v+\delta {\dot{z}}_k$ (above) yields the differential equation $\delta {\dot{z}}_k={\omega }_k\delta z$, where ${\omega }_k\equiv {\left[\frac{df}{d{h}_k}\right]}_{ z_0}$ This equation may be solved to give $\delta z_k\left(x,t\right)=\delta \widehat{z}{e}^{ikx}{e}^{{\omega }_{k}t}$, as stated above.

Sharp protrusions on a rough surface are described by highly oscillatory, large k-modes. Here we show that such protrusions experience enhanced growth rates. Equation A1 reduces to the diffusion equation ${\nabla }^2{c}_k\cong 0$ if c_k is nearly stationary. We expect the rough surface to perturb the EDOT concentration in the adjacent solution and write the perturbed concentration as $c_k\left(x,t\right)=c^{SS}+\delta c_k\left(x,t\right)$, where $\delta c_k\left(x,t\right)$ is the roughness-induced concentration perturbation. By substituting this expression and Equation A3 for c^SS into ${\nabla }^2{c}_k\cong 0$, we obtain ${\nabla }^2\delta c_k\cong 0$. The solution $\delta c_k\left(x,t\right)=\delta \widehat{c}{e}^{ikx-kz+{\omega }_{k}t}$, where $\delta \widehat{c}$ is the (unknown) amplitude of the k^th mode of the concentration perturbation, satisfies ${\nabla }^2\delta c_k\cong 0$, so we use this form for the concentration perturbation. The EDOT flux onto the interface is given by Equation A2: $j_k\cdot \widehat{n}=-D_C\nabla c_k\cdot \widehat{n}$, where $\widehat{n}$ is the surface normal. Making the substitutions $j_k=j_0+\delta j_k$, $\nabla \equiv \widehat{i}\frac{\partial }{\partial x}+\widehat{k}\frac{\partial }{\partial z}$, $c_k=c^{SS}+\delta c_k$, and $\widehat{n}=\left(-\frac{\partial h}{\partial x},1\right)$ yields the result

$$\delta j_k\cong kD\delta c_k.$$ (A4)

This result says that modes of large k, which describe surface protrusions of large curvature, experience larger fluxes (i.e. larger $\delta j_k$ contributions), than do modes of small k, which describe surface features of small curvature. Because the flux onto the sharp features is enhanced, so is their growth rate.

The diffusive flux in solution must be equivalent to the interfacial charge density $\widehat{J}$ (units: $\frac{C}{m^{2}s}$) described by Equation (2), the Butler-Volmer equation. The relationship between j and $\widehat{J}$ is $\widehat{J}$ = ej where e is the electronic charge, for the single electron oxidation under consideration here. These two descriptions of the current flux can be combined to obtain an expression for the instability parameter ω_k, which dictates whether mode k will be unstable, as explained above. Making the substitutions ${\widehat{J}}_k=\widehat{J}+\delta {\widehat{J}}_k$ and $c_k=c^{SS}+\delta c_k$ into Equation 2, where we have kept only the oxidative term, yields:

$$\widehat{J}+\delta {\widehat{J}}_k={\widehat{J}}_0\left[{\left(\frac{c^{SS}+\delta c_k}{c_0}\right)}^{1+\alpha }{e}^{\alpha \beta \left(e\eta -\mathrm{\Omega }\gamma \kappa \right)}\right],$$ (A5)

where $\mathrm{\Omega }={\rho }_{\mathit{\text{PEDOT}}}^{-1}$. Evaluation of Equation A3 on the rough surface $c^{SS}\left(z=\delta z_k\right)=c_{\delta z_k}^{SS}$ yields $c_{\delta z_k}^{SS}\cong c_0-\frac{jL}{D}+\frac{j\delta z_k}{D}=c_0^{SS}+\frac{j\delta z_k}{D}$, where $c^{SS}\left(z=0\right)=c_0^{SS}=c_0-\frac{jL}{D}$. Substitution of this result; $\kappa \cong -\frac{{\partial }^2{h}_k}{\partial x^2}=k^2\delta z_k$; and $\widehat{\gamma }=\beta \gamma \mathrm{\Omega }$ into Equation A5 gives:

$$\widehat{J}+\delta {\widehat{J}}_k\cong {\widehat{J}}_0{\left(\frac{c_0^{SS}}{c_0}\right)}^{1+\alpha }{e}^{\alpha \beta e\eta }\left[{\left(1+\frac{j\delta z_k}{D{c}_0^{SS}}+\frac{\delta c_k}{c_0^{SS}}\right)}^{1+\alpha }\left(1-\alpha k^2\widehat{\gamma }\delta z_k\right)\right],$$ (A6)

where $\widehat{\gamma }=\beta \gamma \mathrm{\Omega }$ and the exponential factor $e^{-\mathrm{\Omega }\widehat{\gamma }\kappa }$ was linearized. Using the approximation ${\left(1+a\right)}^b\cong 1+ab$ allows:

$$\widehat{J}+\delta {\widehat{J}}_k\cong {\widehat{J}}_0{\left(\frac{c_0^{SS}}{c_0}\right)}^{1+\alpha }{e}^{\alpha \beta e\eta }\left\{\left[1+\left(1+\alpha \right)\left(\frac{j\delta z_k}{D{c}_0^{SS}}+\frac{\delta c_k}{c_0^{SS}}\right)\right]\left[1-\alpha k^2\widehat{\gamma }\delta z_k\right]\right\}.$$ (A7)

Keeping only terms up to first order in the perturbations $\delta z_k$ and $\delta c_k$ yields:

$$\widehat{J}+\delta {\widehat{J}}_k\cong \widehat{J}\left[1+\left(1+\alpha \right)\left(\frac{j\delta z_k}{D{c}_0^{SS}}+\frac{\delta c_k}{c_0^{SS}}\right)-\alpha k^2\widehat{\gamma }\delta z_k\right],$$ (A8)

where $\widehat{J}$ has been equated to ${\widehat{J}}_0{\left(\frac{c_0^{SS}}{c_0}\right)}^{1+\alpha }{e}^{\alpha \beta e\eta }$, denoting the steady state flux. The current density fluctuation $\delta {\widehat{J}}_k$ is:

$$\delta {\widehat{J}}_k\cong \widehat{J}\left[\left(1+\alpha \right)\left(\frac{j\delta z_k}{D{c}_0^{SS}}+\frac{\delta c_k}{c_0^{SS}}\right)-\alpha k^2\widehat{\gamma }\delta z_k\right],$$ (A9)

which is a function of the unknown quantities $\delta z_k$, $\delta c_k$, and $\delta {\widehat{J}}_k$. This expression for the strength of the charge transfer perturbation scales with −k², and thus decreases with increasing k. This result contrasts with that of Equation A4 for the diffusive flux perturbation whose strength is proportional to k. These counteracting effects give rise to a range of k-modes that are unstable. This competition underlies branching.

To obtain an expression for ω_k that will define the k-range of unstable surface modes, we recognize that mass conservation requires the diffusive flux and current density must be equivalent and use Equation A4 to substitute for $\delta {\widehat{J}}_k$ in equation A9. We also use $c_0^{SS}=c_0-\frac{jL}{D}$ and obtain:

$$\delta c_k\cong \frac{j\left[\left(1+\alpha \right)\frac{j}{D}-\left(c_0-jL/D\right)\alpha \widehat{\gamma }{k}^2\right]}{\left[j\left(1+\alpha \right)+D\left(c_0-jL/D\right)k\right]}\delta z_k.$$ (A10)

Mass conservation further requires that $j_0+\delta j_k=\rho \left(v+\delta j_k\right)=\rho \left(v+{\omega }_k\delta z_k\right)$, so $\delta j_k={\omega }_k\delta z_k$, and as $\delta j_k\cong kD\delta c_k$, we obtain $\delta c_k=\frac{{\omega }_k}{kD}\delta z_k$. On substituting this expression for $\delta c_k$ we obtain the recursion relation that defines the instability factor ω_k:

$${\omega }_k\cong \frac{j\left[\left(1+\alpha \right)\frac{j}{D}k-D{c}_0\left(1-jL/D{c}_0\right)\alpha \widehat{\gamma }{k}^3\right]}{\left[j\left(1+\alpha \right)+D{c}_0\left(1-jL/D{c}_0\right)k\right]}\mathrm{\Omega }.$$ (A11)

As the first term in Equation (A11) increases with k and the second term decreases, there is a finite range of k-values for which ω_k is positive, as shown in Figure A1. Therefore, modes with wavelengths of 2π/k, where the k fall in the unstable range, may develop into branches.

Figure A1.

Plots of the instability factor ω_k versus k for a small (solid line) and large (dashed line) value of j.

References

1. Y. Saito, Statistical Physics of Crystal Growth. (World Scientific, River Edge, 1996).

2. M. Haataja, D. J. Srolovitz and A. B. Bocarsly, J. Electrochem. Soc. 150 (10), C699-C707 (2003).