Comparing Different Light Models for Virtual Electrodes in Optoelectronic Tweezers

ABSTRACT

Optoelectronic tweezers (OET) allow for the physical manipulation of particles of interest via dielectrophoresis (DEP) in microfluidic devices. To produce the nonuniform electric field required to enable DEP, light is used to expose a photoconductive film and create a so‐called virtual electrode (VE). Several attempts have been made to model the light profile used to excite the photoconductive layer and produce the VE. However, no comparison of the models has been presented in the literature. Here, we present a comparative study among the rectangular, Gaussian, and saturated‐Gaussian models in mapping to light profiles obtained experimentally. These models were then used to predict the activation of a VE and the distribution of the electric field in an OET system. From this comparison, it is possible to conclude that the saturated‐Gaussian model should be the preferred choice to study these systems. Moreover, VEs were also compared numerically to conventional gold electrodes used regularly in DEP applications, concluding that very relevant differences exist between the electric fields produced by these two types of electrodes.

Abbreviations

LIDEP

light‐induced dielectrophoresis

ODEP

optically induced dielectrophoresis

OET

optoelectronic tweezers

VE

virtual electrode

1. Introduction

Dielectrophoresis (DEP) is an electrokinetic technique employed to exert motion on particles of interest by exploiting their dielectric polarization in the presence of nonuniform electric fields. Overall, applications requiring non‐destructive or contactless manipulation of inert or biological particles are heavily benefited by DEP [1] and include sorting and enrichment for healthcare diagnostics [2, 3, 4, 5], microassembly for manufacturing and tissue engineering [6], cell origami [7], and many more [8, 9]. Regarding the movement of particles that can be approximated as spheres, DEP is often described by a frequency‐dependent force, ${\vec{F}}_{\mathrm{DEP}}(\omega )$, acting over a particle of radius $r_p$ suspended within a liquid medium of permittivity ${\epsilon }_m$ as ${\vec{F}}_{\mathrm{DEP}}(\omega )=2\pi {\epsilon }_m{r}_p^3\mathrm{Re}\{K(\omega )\}\nabla E^2$ [10]. In this expression, $\mathrm{Re}\{K(\omega )\}$ is the real part of the frequency‐dependent Clausius–Mossotti factor that describes the difference between the polarizability of the particle and the suspending medium, and $\nabla E^2$ is the gradient of the electric field square.

It is important to note that a nonuniform electric field is necessary to induce a DEP force and that different techniques exist to implement it. As previously described by Martinez‐Duarte [1], some of them are based on metal electrodes, others on carbon electrodes [11], and even different geometries of electrical insulators that distort an otherwise uniform electric field [12, 13]. Herein we are uniquely interested in the use of light and photoconductive films to enable the sought nonuniform electric field, an approach commonly reported in the literature as light‐induced DEP (LIDEP) [14], optoelectronic tweezers (OET) [15], and optically induced DEP [16]. Crucial elements in this technique are optically transparent electrodes, commonly indium tin oxide (ITO), and the coating of one of them with a photoconductive layer, usually amorphous silicon, such that electrical coupling between the electrodes only happens in the regions of the photoconductive layer that are illuminated with light [15, 17]. In this way, one can easily dictate the spatial distribution of the electric field by leveraging patterns of light. In contrast to optical tweezers, particle manipulation in OET occurs due to the electric field gradient enabled by light and not due to the light itself [18]. In this work, we refer to the regions of the photoconductive layer excited with the patterns of light as virtual electrodes (VEs). As recently reviewed [19], OET has proven useful in the manipulation of multiple targets, particularly carbon nanotubes [20], DNA [21], microbial cells [22], and tumor cells [23]. For example, Chiu et al. [24] programmed the displacement and rotation speed of a pattern of VEs to physically manipulate tumor cells and facilitate their isolation from the background. A commercial application of OET is the Beacon Optofluidic System by Bruker, a benchtop system that enables the automatic positioning of targeted cells in wells [25].

A key element for the characterization of |∇𝐸²| in an OET system is the modeling of the light profile used to originate a VE. Lasers [20], liquid crystal displays (LCD) [26], and digital micromirror devices (DMD) [17] are common approaches to generate the light pattern that is used to excite the photoconductive layer. The model of the light profile is important because its output defines the VE, from which the gradient of the electric field square is also modeled. Several light models have been used in attempts to characterize |$\nabla E^2$|, with the rectangular one being mostly reported [27, 28, 29, 30]. Unfortunately, this light model does not consider the gradient of light intensity that arises from using a light source to illuminate the photoconductive layer. Hence, few authors have recently reported the use of a Gaussian model [21, 31, 32, 33] but without further discussing its improvement. An even less common kind of model reported in the literature is that of a uniform region followed by Gaussian decay as a piecewise function [34, 35]. In the saturated Gaussian model by Neal et al., the authors point out that light patterns from projector sources are different from Gaussian profiles but resemble a version of it with a constant intensity region or saturated region. However, these have been individual reports, and the lack of a methodical comparison among the different models prevents a clear determination of the advantages of each.

Hence, there is a need for a comparative study of different models that can be used to characterize the properties of VEs in an OET system. To this end, we first present a comparison of three different light models: rectangular, Gaussian, and saturated Gaussian, in terms of how well each of them maps to light profiles that were experimentally obtained using an OET system enabled by a light engine featuring a DMD; and then present the comparison between the electric field gradient that is expected to result from physical electrodes and when using the saturated Gaussian light model. The results are presented in a manner to facilitate the design of VEs in OET platforms, particularly those using DMD.

2. Materials and Methods

2.1. Experimental Devices and Platform

The experimental setup is shown in Figure 1A and includes three main modules: the experimental device, the light engine and optics to generate the light pattern, and the visualization of the light pattern. Of note, the experimental setup was mounted on an anti‐vibration table. The experimental devices used for this work featured a stacked layer of ITO, hydrogenated amorphous silicon (a‐Si:H), and silicon nitride (SiN) on top of a fused silica substrate. The 100 nm‐thick ITO layer was deposited using a PVD 75 RF sputterer (Kurt J Lesker, Jefferson Hills, PA, USA), followed by the use of a Unaxis PECVD machine (St. Petersburg, FL, USA) to deposit 1 µm of a‐Si. Lastly, an Oxford Instruments PECVD PlasmaLab 80 plus machine (Bristol, UK) was used to deposit a 10 nm‐thick passivating layer of Silicon Nitride (SiN) on top of the a‐Si:H. The light patterns were generated using a DLP4750LC Light Engine (EKB Technologies, Tel‐Aviv, Israel) and focused onto the chip through an optical setup built in‐house, which featured a plano‐convex lens of 50 mm focal length (Nikon, Tokyo, Japan), an NDUV20A (Thorlabs, New Jersey, USA) neutral density filter with 2.0 optical density, and a 40X objective with NA = 0.65 and 3.71 mm working distance (Nikon, Tokyo, Japan). The experiment was visualized through a 50X Nikon CFI60 objective lens (NA = 0.65) mounted on an LV100 Eclipse Nikon microscope (Nikon, Tokyo, Japan). Images were recorded using an AM7025X Dino‐Eye Edge eyepiece camera (Torrance, CA, USA).

FIGURE 1.

(A) Schematic of the experimental setup used to project and measure light patterns on an OET device. DMD stands for digital micromirror device, see text for further details. (B) Example of a 16 × 16 pixel or $N$ = 16 image illustrating how the length of VE was measured. (C) Example of how VE length, or VEL, was determined: the intensity of the light was plotted, and VEL was measured at a value of 0.5 of the normalized pixel intensity. (D) Tukey's boxplots illustrate the mean and deviation of the VEL depending on $N$ where the blue line is the linear tendency, and the open circles are the outliers. Inset shows the detail in the region $N$ = 1–8. (E) Geometric configuration implemented in COMSOL Multiphysics to predict the $|\nabla E^2|$ generated by a VE with input conductivity $\sigma (x).$ Voltage is modeled as applied across the entire bottom boundary. The light source is solely for representation, and no optical modules were included in the simulation. (F) Geometry employed to predict the $|\nabla E^2|$ of an interdigitated gold electrode. OET, optoelectronic tweezers; VE, virtual electrode.

The intensity profile of the light projected by the white and black colors used in this work was measured with a spectrometer (USB400, Ocean Optics, Ostfildern, Germany) by shining an entire field of view, that is, a full PowerPoint slide, of either black (RGB = 0, 0, 0) or white (RGB = 255, 255, 255) color directly onto the detector from a constant distance. Results are shown as Section S1.

2.2. Experimental Protocol

Digital images of white squares, that is, Figure 1B, were produced by varying the number $N$ of pixels per side, where$N=$1, 2, 3, 4, 5, 6, 7, 8, 12, 16, 24, or 32 represents the number of micromirrors used in the DMD. Of note, the open‐source computer image viewer program Geeqie v1.7.3 [36] was used to drive the light engine and turn the anti‐aliasing function off to prevent distortion of the smallest squares generated in this work. Anti‐aliasing is a function that smoothens the pixels in the borders by averaging color and intensity and is a common feature in modern electronic displays. Two hundred seventy measurements were done for each value of $N$ across three different locations of a given chip and across three different chips. Thirty measurements were taken in each location of a given chip for a given $N$. Pixel value saturation was avoided by adjusting the camera's exposure to the maximum level where red pixel values (the highest measured value) remained within the dynamic range based on previous measurements. Measurements were then stored in the uncompressed BMP format, ensuring that all captured image data was preserved without any loss due to compression.

2.3. Data Analysis

The data from 30 measurements corresponding to one randomly chosen location per $N$ value were averaged using the open‐source software Fiji [37] and taken as the pixel value profile for that $N$. Each profile was then normalized and centered around $x$ = 0 using a Python algorithm (see Section S2 for complete details) to derive the normalized amplitude $I$ in function of the projection space $x$ for each $N$ used in this work, that is, Figure 1C. The length of the VE (VEL) for each $N$ value was derived from its $I(x)$ profile using Equation (1) where VEL was defined as the width of the profile recovered from $I(x)$ within an arbitrary threshold intensity $I_{\mathrm{thresh}}$ set to 0.5 (Figure 1C):

$$\mathrm{VEL}=x_2-x_1\mathrm{where}I\left(x_2\right)=I\left(x_1\right)=I_{\mathrm{thresh}}$$ (1)

Ultimately, $I_{\mathrm{thresh}}$ will be given by the electrical properties of a given photoconductive film fabricated using a specific process. Finally, the variation of VEL when considering the 270 measurements taken across three different locations in three different chips was obtained (Figure 1D). The value of VEL is practically important as it relates $N$, a design parameter, to the light intensity profile.

The normalized amplitudes $r$, $g$, and $s$ in function of the projection space $x$ for the rectangular, Gaussian, and saturated Gaussian, respectively, were derived as follows. The rectangular model was represented by the following equation, where $l$ was the rectangle length:

$$r\left(x\right)=\{1\mathrm{if}\left|x\right|\le \frac{l}{2}0\mathrm{if}\left|x\right|\ge \frac{l}{2}$$ (2)

The Gaussian model was represented by the following equation, where $\beta$ described the width of the Gaussian, that is, standard deviation:

$$g\left(x\right)=\frac{1}{\beta \sqrt{2\pi }}\exp \left(-\frac{1}{2}\frac{x^2}{{\beta }^2}\right)$$ (3)

And the saturated Gaussian model was composed of a piecewise function that took a rectangular form with length $l$ and Gaussian decays as described by the following equation:

$$s\left(x\right)=\left\{1\begin{array}{c}\frac{g\left(x\right)}{g\left(\frac{l}{2}\right)}ifx\le -\frac{l}{2}\\ if\left|x\right|\le \frac{l}{2}\\ \frac{g\left(x\right)}{g\left(\frac{l}{2}\right)}ifx\ge \frac{l}{2}\end{array}\right.$$ (4)

The root mean square error (RMSE), Equation (5), was used to visualize the difference among each of the light models $r$, $g$, and $s$ with the experimental data $I$. For a given $N$, $n=m$ were the number of data points compared, that is, m = 71 for $N$ = 1 and m = 725 for $N$ = 32, and $y_i$ and ${\widehat{y}}_i$ were the values from the experimental data and the model prediction, respectively, for a datapoint:

$$\mathrm{RMSE}=\sqrt{\frac{1}{m}{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$$ (5)

Leveraging previous reports that light intensity is linearly related to induced conductivity [18], the light models were then transformed into a conductivity profile as follows:

$$\sigma \left(x\right)=\left({\sigma }_{\mathrm{light}}-{\sigma }_{\mathrm{dark}}\right)U\left(x\right)+{\sigma }_{\mathrm{dark}}$$ (6)

where $U$ could take the form of $I$, $r$, $g$, and $s$; and ${\sigma }_{\mathrm{light}}$ was the maximum conductivity of the illuminated region, and ${\sigma }_{\mathrm{dark}}$ the conductivity of the dark region of the a‐Si:H so that for all three models $\sigma (0)={\sigma }_{\mathrm{light}}$.

2.4. Computer Modeling

The conductivity profiles obtained using Equation (6) were used to generate ${\sigma }_{\mathrm{Substrate}}$ on the photoconductive film for different light models and computationally determine their corresponding |$\nabla E^2$| using the model shown in Figure 1E. We used the AC/DC electric currents package in COMSOL Multiphysics 6.0 (Burlington, MA) and the stationary equation of continuity $\nabla ·\vec{J}=-\nabla ·(\sigma \nabla \mathrm{\Phi }-{\vec{J}}_e)=0$, which was solved for the electric potential, $\mathrm{\Phi }$, with external current ${\vec{J}}_e=0$ while constraining by the boundary conditions and parameters depicted in Figure 1E and detailed in Table 1. Because the central interest in this work falls in studying the spatial distribution of the electric field within the channel and not its time variation or the polarization of suspended particles, a DC voltage was used in the model for simplicity. Assuming everything else remains constant, the spatial distribution of the field does not depend on the frequency of the input voltage. The 100 µm width $w$ of the modeled domain was chosen to ensure that boundary effects do not interfere with the regions of interest, that is, the illuminated zone for the VEs (Figure 1D) and the gold electrode ($w$ _Au = 19 µm). Further increase of $w$ was not considered necessary because no effects were observed at the boundaries of the modeled domain in any of the cases studied here. A similar methodology was used to determine |$\nabla E^2$| for the case of gold electrodes as illustrated in Figure 1F and using the VEL of a specific light model as the width $w_{\mathrm{Au}}$ of the gold electrode. The values for ${\sigma }_{\mathrm{light}}$ and ${\sigma }_{\mathrm{dark}}$ were selected within the range of those reported in the literature [17, 33, 35] as they depended on the fabrication parameters of a specific film. It must be stressed that the continuity equation, automatically imposed by COMSOL at the electrode–water interfaces, was not substituted by an electrical double layer because of the very low electrical conductivity of DI water modeled in this work.

TABLE 1.

Conditions, definitions, and values used for different parameters of the computational model.

Module	Definition	Geometry elements
Current conservation	$\nabla ·\vec{J}=Q_j$ $\vec{J}=\sigma \vec{E}+\overrightarrow{J_e}$ $\vec{E}=-\nabla \mathrm{\Phi }$	Full domain
Electric insulation	$\vec{n}·\vec{J}=0$	External boundaries
Initial values	${\mathrm{\Phi }}_0=0$	Full domain
Ground	$\mathrm{\Phi }=0$	Top boundary
Electric potential	$\mathrm{\Phi }={\mathrm{\Phi }}_{\mathrm{in}}$	Bottom boundary
Values used for the different parameters shown in Figure 1E,F
$w$	100 µm	${\sigma }_{\mathrm{Au}}$	45.6 × 106 S/m
$h$	50 µm	${\sigma }_{{\mathrm{H}}_2\mathrm{O}}$	5.5 × 10−6 S/m
$h_{\mathrm{Si}}$	1 µm	${\sigma }_{\mathrm{light}}$	2 × 10−2 S/m
$h_{\mathrm{Au}}$	60 nm	${\sigma }_{\mathrm{dark}}$	1 × 10−6 S/m
${\epsilon }_r$	1 (Au)	${\mathrm{\Phi }}_{\mathrm{in}}$	5 V
	80 (H2O)	${\mathrm{\Phi }}_0$	0 V
	11.8 (Si)

Note: $\vec{J}$ is the electric current, $Q_j$ is the current source, and ${\vec{J}}_e$ is the external electric current (equal to zero in this work).

Three characteristic values were directly extracted from all the models solved: the max|$\nabla E^2$| in the modeled domain, the separation between the points of max|$\nabla E^2$|, and the distribution of |$\nabla E^2$|, that is, the area of the domain featuring specific values of |$\nabla E^2$|. For max|$\nabla E^2$|, we first extracted the |$\nabla E^2$| profile along a horizontal line arbitrarily set at 0.1 µm above the silicon–water interface and then used a Python script written in‐house (see Section S3) to extract max|$\nabla E^2$|. The separation between peaks was derived from the same |$\nabla E^2$| profile. Lastly, we derived the intensity distribution by using surface integration in COMSOL for defined values of |$\nabla E^2$|. Results are shown as a Python‐generated histogram. Of note, figures are generated with Matplotlib [38] and Plotly [39] Python libraries.

3. Results and Discussion

3.1. Comparing the Rectangular, Gaussian, and Saturated Gaussian Models

When designing VEs in OET, choosing the model that generates the profile of |∇𝐸²| that is closest to the experimental scenario is of utmost importance. The results from comparing the data from the projection to each of the three models studied in this work for three representative values of $N$ (1, 16, and 32) are shown in Figure 2A–C. For the rectangular model, it is evident how poorly it represents the experimental data across all values of $N$. At $N$ values of 1 (i.e., a single pixel), both the Gaussian and the saturated Gaussian light models represent the data accurately. As $N$ increases to 16, the projection data crest becomes broader and flatter, and the saturated Gaussian becomes the closest representation. This effect is more pronounced at $N$ = 32, with the data now shaped as a broader crest with Gaussian‐like decay. The RMSE between each of the models and the projection data as the length of the VE increases is shown in Figure 2D to quantify the differences between the models. The saturated Gaussian clearly yields the least error at all lengths studied in this work. Hence, the saturated Gaussian model is the best to represent the projected data among the models studied here. From a modeling perspective, the saturated Gaussian is a truncated and normalized Gaussian (Equation 4), whose key advantage is capturing constant regions followed by a smooth Gaussian decay, unlike the other two models, which can only account for one or the other. However, a small limitation remains as its completely flat crest fails to capture the slight curvature of the data crest. We attribute this curvature to the continuous nature of light profiles, that is, the absence of abrupt transitions, and the finite number of DMD mirrors leading to a greater overlap of mirror projections at the center of the projection. Thus, such a curvature would be expected at an increased degree as $N$ increases. Although we attempted to address this issue by designing a model consisting of $N$ superimposed Gaussian profiles based on the intuition that every micromirror could represent a single Gaussian; the accuracy of such a model still closely resembled the one of the saturated Gaussian model (Section S4). Further analysis in this current work will thus focus on the saturated Gaussian.

FIGURE 2.

Comparison of the normalized amplitude between projection experimental data and that obtained using different models for (A) $N$ = 1, (B) $N$ = 16, and (C) $N$ = 32. Note that the light models were generated using a VE depending on $N$ according to Figure 1D, that is, for $N$ = 16 the VE was 10 µm. The rectangular model (blue) has an observable adjustment error in each sample, whereas the saturated Gaussian (red) shows an advantage over the Gaussian (green) as the VEL increases. (D) The RMSE score quantifies the adjustment error for each model per VEL (lower is better), showing better results for the saturated Gaussian model. A further consideration is that the profiles for projected data used in A–C were derived from one single measurement location and the measurements for different locations varied slightly (Figure 1D). RMSE, root mean square error; VE, virtual electrode; VEL, length of the VE.

3.2. Comparing Physical Electrodes Against Virtual Electrodes

We then compared |$\nabla E^2$| when using a saturated Gaussian light model and a physical electrode made of gold. We selected the saturated Gaussian model to represent the VE because of its higher accuracy as previously illustrated. The results from modeling in COMSOL are shown in Figure 3A,B, respectively. The difference between the saturated Gaussian and a physical electrode is immediately noticeable. Further analysis of |$\nabla E^2$| at a horizontal line set 0.1 µm above the substrate, as detailed in materials and methods, as well as the conductivity of the channel floor defined here as ${\sigma }_{\mathrm{substrate}}$ is presented in Figure 3C. The physical electrode is characterized by a binary transition in the conductivity of the substrate, that is, either the electrically insulative substrate or the gold electrode, that leads to a step in ${\sigma }_{\mathrm{substrate}}$. This is not the case with the saturated Gaussian light model, and hence, the profiles of $\nabla E^2$ yield differences of practical importance. Although a physical electrode leads to sharp increases in |$\nabla E^2$| at its edges, a VE yields a softly curved profile that has its peak aligned to the region when ${\sigma }_{\mathrm{substrate}}$ drops to a value ${\sigma }_{\mathrm{transition}}$. We found such transition value to be an order of magnitude greater than ${\sigma }_{\mathrm{dark}}$ and constant for every VEL simulation (Section S5). Hence, the effective length of a VEL seems different than that expected from a physical electrode of the same dimensions (Figure 3D), and this should be taken into account in the process of designing DEP‐active areas. Moreover, although the max value of |$\nabla E^2$| remains relatively constant as the size of the physical electrode increases, this value decreases exponentially with an increasing VEL (Figure 3E). Lastly, the areal distribution of different ranges of |$\nabla E^2$| is presented in Figure 3F. This is important to further quantify the differences observed in Figure 3A,B. Given its sharp geometry, the physical electrode yields a concentrated profile of |$\nabla E^2$| around the electrode edges. This is reflected in the low area covered by magnitudes above ${10}^{13}$ ${\mathrm{V}}^2/{\mathrm{m}}^3$ and the greater areas at other lower field gradient magnitudes, peaking at values in the range ${10}^{11}-{10}^{11.5}$ ${\mathrm{V}}^2/{\mathrm{m}}^3$. In contrast, a VE yields a wider dispersion and a higher area featuring magnitudes at high values, that is, in the range beyond ${10}^{12}-{10}^{13}$ ${\mathrm{V}}^2/{\mathrm{m}}^3.$ Note that both models made use of 5 V as the input voltage. Practical considerations thus include the capacity of the intended DEP‐active areas, that is, a VE appears to yield higher capacity; the sharpness of the DEP area, that is, a physical electrode will be preferred; and the sensitivity of the target to the field gradient as well as the intended functionality, that is, a higher gradient magnitude will lead to a more rapid and stronger DEP force.

FIGURE 3.

(A) Model of $\nabla E^2$ for a saturated Gaussian model of a virtual electrode of 14.5 µm length and (B) for a physical gold electrode of the same length. (C) Conductivity profiles (dotted line) for both electrodes and their predicted |∇𝐸²| (solid line) measured along a horizontal line 0.1 µm above the liquid bottom boundary. (D) The predicted |∇𝐸²| peaks separation linearly increases for both electrodes across several lengths, whereas (E) for the physical electrode remains constant and for the virtual electrode it decays. (F) Areal distribution of different ranges of $|\nabla E^2|$ for electrodes of 14.5 µm, quantifying ranges of |∇𝐸²| among the observed area in A and B.

4. Concluding Remarks

OET can be exploited for particle manipulation in microfluidic platforms, benefiting from versatility and cost‐effectiveness. Here, we contributed a needed methodical comparison between light models reported in the literature related to OET. Our results from comparing the profile of each model and its accuracy to our dataset present a strong argument for not using the rectangular model and preferring the saturated Gaussian model over Gaussian models, especially as the length of VEs increases. Additionally, we discussed how the VE model that best represented our dataset, the saturated Gaussian, compares in its |$\nabla E^2$|against the one produced by an electrode modeled as made of gold on top of an electrically insulative substrate. We report that the |$\nabla E^2$| of each electrode type differs mainly because of its conductivity profile. The binary transition of conductivity in the gold electrode generates a sharp |$\nabla E^2$| profile concentrated around the electrode edges, whereas the soft conductivity transitions in the VE lead to a softly curved |∇𝐸²| distributed more widely among the liquid media. We also pointed out that an increase in electrode length implies a decrease in max|$\nabla E^2$| for VEs, unlike physical electrodes for which it remains constant. Although these results focus solely on the variations of |$\nabla E^2$| through several VELs, it is also of practical importance to contemplate variations in parameters such as ${\sigma }_{\mathrm{light}}$ because this can change due to the intensity of the light used to expose the photoconductive film as well as the process used to fabricate it. Indeed, although we present a rigorous characterization for our specific illumination system, we acknowledge that other types of illumination setups can derive in different light intensity profiles. For instance, a lower numerical aperture in the projection lens would lead to a lower resolution and the potential inability to project low values of N. Moreover, replacing a DMD‐based light engine with a liquid‐crystal display will most likely lead to a slight difference in the light intensity profiles. Previous results have shown that an increase in light intensity implies an increase in peak separations [14]. Although this is correct when saturation of the photoconductor has not occurred, a more precise statement (drawn from our results) is that an increase in ${\sigma }_{\mathrm{light}}$ would lead to a proportional increase in the amplitude of the profile, thus displacing the values of ${\sigma }_{\mathrm{transition}}$ further from the center, that is, $x$ = 0, and increasing the separation between peaks. Additionally, ${\sigma }_{\mathrm{substrate}}$ would have a steeper decay, leading to an increase in max|$\nabla E^2$|. In this work, the VEL was derived through an arbitrary definition of a threshold $I_{\mathrm{thresh}}$ = 0.5. However, the trends observed here should remain the same as the threshold changes according to the performance of the photoconductive film that is fabricated. An opportunity to improve the computational analysis in this work is running simulations in a tridimensional domain and considering the vertical profile of light transmitted through the amorphous silicon.

Conflicts of Interest

The authors declare no conflict of interest.

Supporting information

Supporting Information

Guzman‐Saleh E., Perez‐Gonzalez V. H., and Martinez‐Duarte R., “Comparing Different Light Models for Virtual Electrodes in Optoelectronic Tweezers.” ELECTROPHORESIS 46, no. 17 (2025): 1333–1340. 10.1002/elps.8131

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.