Theoretical modeling of dendrite growth from conductive wire electro-polymerization

Electropolymerization is a bottom-up materials engineering process of micro/nano-scale that utilizes electrical signals to deposit conducting dendrites morphologies by a redox reaction in the liquid phase. It resembles synaptogenesis in the brain, in which the electrical stimulation in the brain causes the formation of synapses from the cellular neural composites. The strategy has been recently explored for neuromorphic engineering by establishing link between the electrical signals and the dendrites’ shapes. Since the geometry of these structures determines their electrochemical properties, understanding the mechanisms that regulate polymer assembly under electrically programmed conditions is an important aspect. In this manuscript, we simulate this phenomenon using mesoscale simulations, taking into account the important features of spatial–temporal potential mapping based on the time-varying signal, the motion of charged particles in the liquid due to the electric field, and the attachment of particles on the electrode. The study helps in visualizing the motion of the charged particles in different electrical conditions, which is not possible to probe experimentally. Consistent with the experiments, the higher AC frequency of electrical activities favors linear wire-like growth, while lower frequency leads to more dense and fractal dendrites’ growth, and voltage offset leads to asymmetrical growth. We find that dendrites' shape and growth process systematically depend on particle concentration and random scattering. We discover that the different dendrites’ architectures are associated with different Laplace and diffusion fields, which govern the monomers’ trajectory and subsequent dendrites’ growth. Such unconventional engineering routes could have a variety of applications from neuromorphic engineering to bottom-up computing strategies.


Results and discussion
Proposed modeling methodology. Figure 1a represents the experimental setup of the electropolymerization technique, with a drop of aqueous electrolyte containing EDOT monomers, NaPSS electrolyte and benzoquinone oxidizing agent. Dendrites were grown by the bipolar alternating current with one electrode connected to time-varying signal and the other connected to the ground. Figure 1b depicts the morphology of PEDOT, with dendritic branches extending towards the opposite electrodes. Furthermore, imaging of the growing process reveals that particular particles may be traveling between the electrodes during the process. The characteristics of time-varying signals can be used to modulate the geometry of dendrites. One can achieve wire-like, fractal, engulfing nature of dendrites by playing with electrical parameters such as applied voltage, frequency, duty cycle and offset etc. 14 (see Fig. 1c,d for a few examples). From the experimental perspective, one can only perform the imaging of the deposited conducting polymer morphology occurring on the electrodes. In order to model the experimental phenomena, herein, we simulate a simplified version of the electrodeposition problem on the electrode considering the necessary ingredients. In the electropolymerization process, one can expect monomers to become oligomers and higher sizes, which can be carrying the charge and hence driven by the electric field. Alternatively, one can consider the motion of concentration limited PSSmoving with the electric field, activating the growth of PEDOT on the electrodes. Irrespective of the identity of charge carrier (which is not clearly known), one can consider generic charge carriers represented as charged moving particles in the simulations. Further, the arguments of the model would be valid for systems in which the nature of charge is opposite; in such a scenario, the deposition would occur on the opposite electrodes, maintaining similar morphologies. The advantage of the generic nature of the particles in the simulations makes the modeling applicable  14 . (e) Simulation geometry for electro-polymerization by considering motion and attachment of particles on the electrodes. The complete box represents the liquid, gray lines represent the electrodes connected to a specific signal, the white points represent the moving particles, and black points represent the polymerized particles on the electrodes. The spatiotemporal potential map is evaluated based on the applied waveform and modified electrode, simulation parameters are listed in Table 1. (f) The motion of a particle is driven by combined contribution of scattering and electric field motion. The particle is made to move in any direction with the probability based on effective forces. (g) Parameters of the signal applied on the electrodes. (h) With the voltage signal (shown in gray), the voltage experienced at the dielectric surface is calculated based on capacitive-resistance time scales (shown in blue and red). www.nature.com/scientificreports/ to electro-polymerized materials grown by different species. The parameters of the applied signals applied on the electrodes can control the potential in the liquid. Further, the particles in the liquid phase can be expected to have several particle-particle and particle-fluid forces affecting the motion. Thus, one needs to consider the following ingredients in the model (1) potential distribution variation in the liquid due to time varying signal, (2) motion of particles governed by electrical parameters and various fluid interactions, and (3) electro polymerization of the conducting polymer on the electrode. For the electro-polymerization, the two electrodes are biased with AC signal, with 2V P being peak to peak voltage, D being duty cycle (0.5 for symmetrical pulse) and V off being DC voltage offset. The potential map in the liquid can be calculated based on Laplace equation ∇ 2 V = 0 by defining the voltage signal as the boundary conditions on the electrodes. The particle motion located at x,y,z can be written in terms of electric drift, viscous stress and stochastic scattering as: Here, q and m are respectively the charge and mass of the individual agglomerated particle. Based on Stock's relation, the mobility of the particle ( µ ) depends on particle radius, r and viscosity of liquid, ν which are material specific parameters as µ = 1 6πνr . The ƞ(t) has a Gaussian probability distribution with correlation function η i η j = 2 µk B T p δ i,j δ t − t ′ with i and j being components, k B as Boltzmann constant and T p as the temperature 35 . The above equation can be used to compute the concentration C(x,y,z, t) of particles based on electric field and stochastic scattering. With NaPSS as electrolyte, electrochemical double layer (EDLA) is formed on the electrode with ions of PSSon the anode and Na + on the cathode with assumed R-C time constant of τ and voltage across double layer as V D . Consider V A and V B being the potential on the surface of dielectric double layer on which further deposition takes place. The variation in potential during a pulse, for different t ′ ∈ (0,T) can be written as: For t ′ Consider a charged particle at location P(x,y,z) close to electrode A. The oxidation of this charge carrier depends on the voltage difference between the charge carrier, V x, y, z, t and the EDLA, V A considering threshold voltage, V T . The sticking probability (S) in this case is represented as sigmoidal function with α a material-electro polymerization specific parameter, controlling the increase in sticking per unit voltage difference. Further, the rate of sticking depends on the local concentration of charged EDOT oligomer particles near the electrodes.
The sticking probability,S A (t ′ ) per particle can be written as Rate of growth, R A (t) depends on the concentration of EDOT in the neighborhood of electrode C A (t) and the sticking probability, S A (t ′ ) as With similar arguments, for t ′ > D the rate of deposition on electrode B, R B (t) depends on potential at the dielectric layer. www.nature.com/scientificreports/ As discussed, the formation of dendritic patterns is influenced by several parameters, including particle concentration, dielectric and charge properties, fluid-fluid and fluid-particle interactions, and chemical and thermodynamics influencing the electro-polymerization process. The majority of these important properties, however, cannot be measured directly during the experiments, and so their values cannot be plugged into the model. Furthermore, the experiments span a range of time and distance scales involving a growth time of ~ 100 s with a pulse width of ~ 1 ms and electrode spacing of ~ 240 µm with particle sizes of few nanometers. To have one to one mapping of the experimental problems, one might need to model the potential map of 3D box of mm dimension, containing Avogadro number of particles (10 mM monomer concentration), and studying the motion and sticking of individual particles for an order of 10 6 (1000 Hz for 500 s) duty cycles, which is not computationally feasible for parameters study. As a result, we evaluated a simplified version of the problem by inserting relevant components into the model and studying the evolution of dendritic morphology by altering the electrical and related parameters one at a time. For better readability, 100-simulation time steps is treated as 1 s, yielding (100 × Simulation time steps) -1 as f 0 . Thus, if the signal's time-period is α simulation time units, the frequency of the signal is 100 α f 0 . For further miniaturized experimental settings including submicrometer electrode spacing, which has yet to be experimentally explored, the modeling can be one to one mapped. As a result, the given generic model will aid in understanding the generality of the process as well as the qualitative nature of the type of morphological variation with individual parameter change. Figure 1e shows the proposed model with two bipolar metal electrodes shown in gray, and charge particles moving in the liquid shown in white. The codes are designed on python for convenience with the computational expensive function of potential calculation written in Fortran language utilizing F2PY interface. The electrodes are biased with the AC signal and spatiotemporal potential map based on the iterative solution of Laplace equation: With voltage of electrodes as boundary conditions defined by the signal. The particles' evolution is simulated based on a model that considers collective contributions of both field-effect electrodynamic drift and scattering (see Fig. 1f). Consider a particle is at O and it would be experiencing unequal electrical fields in all directions and stochastic force. At a short time scale, water can no longer be treated as a continuous fluid, and molecular motion due to thermal fluctuations influences the particle trajectory. To introduce this random value of force, stochastic term ƞ(t) is introduced into the treatment. The velocity-dependent viscosity drift is ignored for simplicity. For the modeling, we have used the noise values of arbitrary scale, accounting for the contribution of stochastic force on particles in contrast to electric field-based deterministic force. The force experienced by a particle at O towards X can be written as: The constant, k is assumed unity as it depends on mass, charge, and spacing between the pixels which are constant throughout the experiments. Thus the particle would move in the direction where the collective drift based on both these effects (electrical-field assisted and thermally activated) would be higher. The probability of the motion is made proportional to the normalized value of drift along the direction with c being proportionality of constant.
Since particles would be attracted towards the anode, and repelled by the cathode, thus their transience between both electrodes is ensured by the time varying nature of the voltage signal, according to the time constant of the monitored phenomenon and the frequency range of the applied signal (see Fig. 1g). During the experimental conditions, 10 mM EDOT is utilized for 25 µL of solution accounting for ~ 35 µg of available EDOT, while the EDOT consumption (for dendrites obtained at 5 V) is ~ 9 µ g. For simplicity, we have assumed constant particle density. In the simulations, particles that happen to leave the boundaries are made to enter from the opposite direction at random positions to maintain constant concentration conditions ensured in the experimental setup. During the motion of particles with the passage of time, the moving particles can happen to come near the electrode and once it comes near the electrode, it can polymerize and permanently attach to the electrode. Further, the potential value incorporates the drop in voltage at the double layer of the electrode with specific time constant as shown in Fig. 1h, with red and blue colors in contrast to the square signal shown with gray. The probability of particle-electrode attachment is made finite at a location where the particle electrode distance is < 2 pixels, enabling the contact from the straight and diagonal directions. Without enabling the diagonal direction motion, the morphology is seen to be having rectangular artifacts. Further, since the conducting polymer polymerization is enabled by oxidation only if the anode reach the given oxidation potential, particle's potential shall be taken into account in the probability of attachment to the anode. This is experimentally evidenced, with dendritic growth occurring above 3.5 V as voltage amplitude, and the density of the morphology increases with the amplitude. To introduce voltage dependent oxidation, the attachment probability is made sigmoidal function, considering the voltage difference between the location of the particle and potential of double layer, higher the difference more is the attachment probability (representing oxidation). The stuck particles are made permanently attached to the electrode, as the polymerization process is irreversible. Since the polymerized particles on the electrode are conducting by nature, the deposited particles are considered integral part of the electrode with further oxidation  Figure S1) To quantify the dendritic growth, the density of the morphology is evaluated based on deposited area on both the electrodes. The completion time of the process is defined to quantify the kinetics of the growth and are considered when two dendrites are very close to each other. A term asymmetry is defined to compare dendrites' morphologies originating from both the electrodes, with the value corresponding to the ratio of dendrite with high density to lower density. The minimum possible value of 1 corresponds to complete symmetrical structure. The error bars are calculated based on multiple simulations utilizing similar conditions and random seed values. The different random seed values changes the initial distribution of particles and random variable in the stochastic process. It should be noted that though in a real situation, the charge particles might be having negligible size as compared to the electrode spacing, and particle size may be variable, however, due to computational simplicity and cost restrictions, we assume the monomer or charge particle to have size of 1 pixel (1/20th of electrode spacing = 12 µm), and other length and time scale also based on simulation parameters. The present modeling parameters might be closer to the experimental system if one would be performing these experiments at much miniaturized conditions. Herein, we qualitatively, compare the model prediction with the experimental findings and predict the growth process in unexplored experimental conditions.

Comparison of modeling results with experimental observations. Figure 2a and Supplementary
Video S1 show the growth process observed from modeling prospective with a low frequency of 2.5 f 0 and 50% duty cycle and zero offset. The morphology advances towards the opposite electrode with several branches making a fractal structure. Figure 2b and Supplementary Video S2 represent a typical growth process obtained at experimental conditions of 5 V, 20 Hz, 50% duty cycle and zero voltage offset. The growth process takes 600 s, with initiation of growth in both the electrodes, followed by diameter growth and fractal-like morphology growing towards the counter electrode. Furthermore, we see particle motions between the electrodes in the experimental video, which is consistent with the modeling assumption of particles moving between the www.nature.com/scientificreports/ electrodes. However keeping the conditions same, and only increasing the signal frequency one ends up having wire like growth for increased signal frequency (25 f 0 ). Growth starts from one end followed by growth from other electrode and moving towards each other. In the modeling, (Fig. 2c and Supplementary Video S3) we find that at increasing frequency, the particles tend to localize in the center of the electrodes vibrating minimally with low time period (high frequency) signal. The restricted particles' motion forces the growth only at the extreme tip of the dendrite, rather than in any other location, offering wire like morphology with the passage of time. Figure 2d and Supplementary Video S4 show the experimental growth of dendrites at higher frequency (850 Hz) resembling the modeling growth process. Note that, the drastic change in fractal-to-wire-like growth cannot be trivially investigated in the experiments, given the distribution of moving particles in the liquid phase. Figure 2e illustrates the images of dendrites obtained for a range of frequencies (f 0 -50 f 0 ), the morphology is fractal and dense for f 0 , the morphology becomes thin and less dense with increasing value of frequency, and becomes wire like at very high frequencies. The simulation well matches the experimental observations as shown in Fig. 2f, as morphologies are fractal-like and very dense at 20 Hz, become less branchy and less dense with increasing frequency till 300 Hz, and become wire-like thereafter (see 850 Hz and 900 Hz). In Fig. 2g, we plot the variation in the density of modeled dendrite morphologies obtained at different frequencies. One observes two regimes: (i) the decrease in density with increasing frequency from f 0 to 4 f 0 , and (ii) saturation of density thereafter.
In the experiments (Fig. 2h) as well, it is observed that density drops systematically from 20 to 300 Hz, while it saturates at a specific value from 300 to 900 Hz. It is observed that the completion time in modeling and experiments (Fig. 2i,j) both show similar nature; the completion time decreases with increase in frequency in regime I, while it increases in regime II. From the modeling we see that, in regime I, the increase in frequency helps in concentrating the particles near the center of electrodes. The high particle concentration increases the probability of dendrite growth and in turn decreases the completion time. However, in regime (ii) i.e. at very high frequency, the particles are dragged almost at the complete center of electrodes, vibrating minimally at low time-periods (high frequency). The restricted motion of these particles reduces the growth probability, and in turn slows down the process. Next, we studied the effect of voltage offset, defined as a time invariant (DC) voltage component introduced in the applied periodic signal. Figure 3a shows the dendritic growth process with voltage offset of 2 units on the top electrode. An increased dendrite density is observed at the electrode that experience a positive voltage contribution by the voltage offset. The reason for this behavior can be explained based on increased density of particles and higher probability of attachment. Figure 3b,c compares the morphologies obtained from the modeling and the experimental methodologies at variable voltage offsets. The voltage offset increases asymmetry in both experiments and modeling. Figure 3d represents the systematic variation in the asymmetry with the increase in voltage offset for modeling images for electrode with offset (shown in orange) and counter electrode (shown in magenta). The similar trend is observed in the experimental studies 14 (Fig. 3e), wherein the density on www.nature.com/scientificreports/ the electrode with offset (blue) increases while the density on counter electrode (green) decreases. In this way, the simplified model with few parameters can explain the morphologies obtained for different electrical signals.
Modeling predictions. Next, we studied parameters others than the voltage signal affecting the growth process of dendrites, which have not yet been systematically studied by experimentalists so far, and could be important points while designing the future experiments. The monomer concentration is an important aspect in experiments that affects dendritic growth. Figure 4a shows the dendritic growth at increasing particle densities, demonstrating very distinct morphologies for different particle densities: (i) wire-like for N = 70, (ii) fractal and symmetrical at N = 120, and (iii) asymmetrical at N = 250. The findings are consistent with the experimental observations of Ohira et al. wherein, wire like dendrites were obtained in the reduced concentration by glass covering the region during bipolarization process 36 . According to the simulation, this is due to more attachment events at large particle densities. However, at very high particle density, the structure becomes asymmetrical, despite having similar applied electrical conditions on both electrodes. Because even minor noises initiate growth in any of the electrodes, which operate as the nucleating center, and generate a cascade effect that triggers attachments of surrounding particles for additional growth. Many particles can adhere to the nucleating centers in a short time interval with increased particle density. As a result, excessive particle density might result in uneven growth and a winner-takes-all situation caused by noise.We have tried several initial distribution of particles, the error bar in Fig. 4h corresponds to multiple simulations. The error bar is minimal for lower particle densities, indicating their repeatability for initial distribution and stochastic process. The error bar, on the other hand, is substantially larger for simulations with greater particle densities, demonstrating the sensitive role of initial particle distribution and stochastic effects similar to butterfly effect. The time evolution of the growth process shown in Fig. 4b strengthens the claim as the growth is uniform for lower particle densities, while the growth process becomes fast, erratic and steps like for higher particle densities, manifesting the growth of several particles in a small interval. As shown in Fig. 4c, increasing particle density increases the possibility of attachment and hence density, which rises from N = 70 to N = 200 and saturates further. The growth rate can be quantified in terms of completion time as shown in Fig. 4d, it drops systematically from N 70 to N = 175 and nearly becomes constant further. It is observed that the asymmetry value (a parameter used to compare dendrites' morphologies originating from both the electrodes) is www.nature.com/scientificreports/ below 2 for dendrites N = 70 to N = 150, while the value rises significantly further with an asymmetry value above 4 for N = 250. (Fig. 4e) Thus, without any asymmetry in electrical parameters, one can end up in asymmetrical structures based on intrinsic growth processes with higher particle densities. The motion of the particles is regulated by the electric field and thermally activated random motion (scattering). To understand the impact of scattering as a random contribution in the dendritic growth, the growth has been studied at different values of noise (ƞ) under constant signal conditions. If we see the effect of noise on the morphologies (Fig. 4f), at very low noise (ƞ < 0.2), there is no growth on any of the electrodes, with the increasing noise the growth turns to wire like (ƞ = 0.2), and with medium noise the growth is bulk fractal structure (ƞ = 0.5). On the other hand, for very high noise (ƞ = 0.7), the growth starts occurring on the electrode wire in the transversal direction. The reasons for these effects can be explained based on the difference in particle distribution at variable noise levels. Due to the relatively higher value of the electric field at low noise levels, the particles are solely governed by the field, since the signal changes the polarity based on the periodic signal, hence the particles concentrate near the center. At very low noise, the particles are frozen in the mid of electrodes (ƞ < 0.2), leading to no growth. With increasing contribution of the noise, particles can move from the center towards the electrode. At ƞ = 0.2, only a few particles can reach the electrode, it is similar to the case of less concentration (say N = 70) studied previously, and hence wire-like growth is manifested. The increase in noise (ƞ = 0.5) increases the particle density and hence fractal structure is obtained. However, at very high noise, the motion is not controlled by the field, and high density of particles get available throughout the system, increasing the transversal growth on the electrode wires. The growth process of wire (ƞ = 0.2), plotted in Fig. 4g shows that the wire growth takes a certain threshold time to begin with. The dendrite density and completion time plots (Fig. 4h,i) also show distinct regimes: with no growth for low noise (ƞ < 0.2), high variability in the growth time and asymmetry for medium noise = 0.2, a reduction in completion time for medium noise (ƞ = 0.2 to 0.3) because the particles can easily approach the electrodes, and an increase in completion time for high noise (ƞ = 0.3-0.7). Figure 4j demonstrates that asymmetry is notably high at ƞ = 0.2 which is the lowest noise required for growth, then reduces and saturates more as the value of noise increases.
As a result, variations in the value of noise (which, as previously discussed, is dependent on particle velocity and temperature) can result in a variety of states. Because particle mobility is influenced by particle shape and particle-solvent interaction, different monomers may result in different morphological morphologies, which is an important consideration for experiments throughout the optimization process. At high concentration, the particle concentration can also play a role in converting wire-like growth to bulk fractal-like growth and asymmetrical growth, which should be considered while experimenting with concentration in the optimization process. Furthermore, the information on the error bars can be used to determine the reproducibility of pattern formation under certain environments. Apart from the neuromorphic engineering community, the investigation can be an important contribution to various other domains wherein such interconnections and morphological shapes control their functionalities [23][24][25][26][27][28][29][30] . The approach can also be explored for pattering applications 37 and locomotion of conducting objects with similar techniques 38,39 . The modeling can also be translated to problems involving wireless electro-polymerization 16,40 . Since we used generic charged particles in the modeling, the model and its findings would also be important to inorganic electrochemical depositions 34,35,41 . Future work can be done with non-equal mesh size for potential determination to have the fine resolution in potential map near the dendrites. Currently, due to different time scales and length scale of simulations as compared to experiments, we are not able to provide one-to-one mapping. Future work could attempt in relating the modeling parameters to the experimental units. The modeling can also be extended for network-based simulations by introducing multiple electrodes in the modeling. Currently, we have shown the simulations for periodic signals, the approach can also be applied to non-periodic time series for neuromorphic applications. We anticipate that the current modeling underpinnings and inferences from the modeling studies will be important in analyzing and optimizing electrode designs and electrical signals for future neuromorphic devices.

Conclusion
In conclusion, we have developed a mesoscale model to understand and optimize the electro polymerization technique as a bottom-up strategy for neuromorphic computation and other applications. The model involves consideration of the spatial-temporal potential mapping for the time-varying signal across the electrodes, motion of the charged particles in the electric field and, attachment of particles to the electrodes. The model attempts to explain the morphological differences for different electrical activities and experimental conditions, based on the nature of particles' distribution and their trajectories for electrical activities that have not previously been explored using modeling or microscopic techniques. The increasing frequency turns a bulky structure into a wire like morphology due to the preferential distribution of particles near the center of electrodes and tip of the growth. The voltage offset brings the asymmetry in the morphology due to asymmetry in the particle distribution due to constant potential. The effect of concentration shows different regions, which are wire like, bulky and asymmetrical at increasing concentration values due to the increased time scale of the growth process. The effect of scattering can also a have a huge impact leading to no-growth, wire like growth, symmetrical growth and transversal growth at increasing value of the scattering due to the reduction of relative electric field contribution. The model can be explored for multiple electrodes, variable signals and making quantitative experimental comparisons after further modifications.