# Trajectories and Forces in Four-Electrode Chambers Operated in Object-Shift, Dielectrophoresis and Field-Cage Modes—Considerations from the System’s Point of View

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Materials and Methods

^{®}routine, resulting in “total conductance matrices” (hereafter referred to as “conductance matrices”) with 160 × 160 elements [11]. The conductance matrices were used to derive “conductance fields”, using the matrix values as interpolation points for the MatLab

^{®}quiver line function. The conductance fields are available in Tables S1–S6 in the Supplementary Materials. More details about the software used for the calculations can be found in the previous publications. For better visibility of the inhomogeneous polarization of the sphere in the respective figures, equipotential line and current line plots were calculated with a MatLab

^{®}routine. Within the conductance fields, the sphere’s center follows trajectories along the conductance gradient, i.e., each step increases the conductance of the DEP system and hence the dissipation of electric field energy.

## 4. Results and Discussion

#### 4.1. General

^{®}routine for calculating the field distributions.

#### 4.2. (++−−)—Drive Mode (Object-Shift Mode)

#### 4.2.1. Field Distribution and Chamber Conductance

#### 4.2.2. Trajectories and Forces

_{1}, E

_{2}, E

_{3,}and E

_{4,}near the electrodes. These endpoints are located slightly away from the electrode tips because the sphere experiences a lateral bias generated mainly by the attraction of the nearest counter electrode. This is evident from the current lines in Figure 2A, where the sphere at the left electrode is displaced toward the upper counter electrode. Three unstable endpoints (E

_{5}, E

_{6}and E

_{7}) are saddle points located on the inverted mirror plane. The trajectories within the mirror planes and the triangular planes have one sibling (a

_{1}, c

_{1}, c

_{2}and d

_{1}) and three siblings (b, e, f, g, h, i, and j), respectively. The trajectories a

_{2}, c

_{3}and d

_{2}are three of four siblings.

_{1}, c

_{2}, and c

_{3}. Section c

_{1}starts precisely in the corner and ends at the unstable saddle point E

_{5}. A slight disturbance may cause it to run along the vertical watershed to one of the unstable saddle points E

_{6}or E

_{7}(here c

_{2}toward E

_{6}), where the sphere can be deflected almost perpendicularly to either side (c

_{3}or a

_{2}) and hit the chamber wall near the electrode, creating a small force peak (Figure 4C). A higher force peak is generated before the sphere reaches the end point. The terminal steps lead to negligible changes in the chamber’s conductance, resulting in minimal forces, according to Equation (2). As noted in previous work [1], the highest force peaks are generated when the electrode is hit directly, which is almost the case with trajectory e. The DEP force is zero at the unstable saddle points E

_{5}, E

_{6,}and E

_{7}, but not at the stable endpoints E

_{1}, E

_{2}, E

_{3,}and E

_{4}, where the sphere’s motion stops. Experimentally, the DEP force at the stable endpoints is compensated for by the wall pressure.

_{2}, b

_{3}, d, h

_{1}and h

_{2}. However, the four stable endpoints are located distant from the electrodes in the corners of the chamber, and there is no diagonal watershed from low left to top right. On the other diagonal, a watershed runs only between the two unstable saddle points E

_{6}and E

_{7}, with the third unstable saddle point E

_{5,}halfway corresponding to the center of the chamber. The bifurcations at both ends of the diagonal watershed make E

_{6}and E

_{7}triple points, each with three catchment areas, i.e., an instability at E

_{6}(E

_{7}) can deflect the object to one of the three endpoints E

_{1}, E

_{2}, or E

_{4}(E

_{2}, E

_{3}, or E

_{4}). In either case, it is theoretically possible for the sphere to move along the watershed and stop at E

_{5}or to pass E

_{5}and continue to E

_{2}or E

_{4}. In the chamber plane, the three watersheds form two pairs of mirror symmetric catchment areas, one pair of mutually distant small triangular-like areas with the endpoints E

_{1}and E

_{3}and another neighboring pair of large pentagon-like areas with the endpoints E

_{2}and E

_{4}.

_{2}and b

_{3}in Figure 5, which correspond to the mirror plane in Figure 4. Another example is the trajectories d and h

_{1}in Figure 5, which correspond to the trajectories a

_{1}and d

_{1}, respectively, in Figure 4. In Figure 5, the trajectories h

_{1}and d start at the unstable saddle point E

_{7}in opposite directions, h

_{1}along the watershed and with d as the central trajectory of a bundle of trajectories with the end point E

_{3}(cf. trajectories c and f with end point E

_{1}).

_{2}and E

_{4}, e.g., a, e and g, have larger arcs than trajectories between counter electrodes bounded by the curved watersheds. Along the watersheds, the trajectory b is divided into sections b

_{1}, b

_{2}, and b

_{3}. Section b

_{1}originates at the top edge of the chamber and runs along the curved watershed. For numerical reasons, b

_{1}deviates slightly from the watershed before E

_{6}, so it turns left at E

_{6}and continues as b

_{2}to E

_{5}, where it turns left again toward E

_{2}. The trajectory h is divided into sections h

_{1}and h

_{2}, which are siblings of b

_{2}and b

_{3}, respectively. Section h

_{1}starts from the saddle point E

_{7}and turns left at E

_{5}toward E

_{4}.

#### 4.3. (+++−)—Drive Mode (DEP Mode)

#### 4.3.1. Field Distribution and Chamber Conductance

#### 4.3.2. Trajectories and Forces

_{1}and E

_{3}) or near the electrodes (E

_{2}and E

_{4}) (Figure 8A). The latter two are located somewhat away from the electrode tips because the sphere experiences a lateral bias generated mainly by the attraction of the single negative electrode. Each watershed has one saddle point (E

_{5}, E

_{6}, and E

_{7}). E

_{5}is an unstable saddle point located at the intersection of the mirror plane and the curved vertical watershed.

_{5}on the symmetry line into opposite directions, straight to E

_{3}and E

_{1}, respectively. Except for these trajectories, the sphere does not move along the shortest possible trajectory to the endpoints. From the corner regions, the sphere is deflected to the center of the chamber before the trajectories are redirected toward one of the nearby electrodes, e.g., trajectories c, d, f, i, and k. Trajectories starting in the catchment areas of E

_{2}and E

_{4}can pass the endpoint and then move along the chamber wall back to the endpoint, e.g., f and i. E

_{3}at the single counter electrode has the largest catchment area, and the trajectories ending at E

_{3}can be long, e.g., e, j, and l.

_{5}(Figure 6H), E

_{6}, and E

_{7}, but not at the stable endpoints E

_{1}, E

_{2}, E

_{3}, and E

_{4}, where the motion of the sphere comes to a stop, and the DEP force is (experimentally) compensated by the wall pressure.

_{5}, E

_{6}, and E

_{7}but not at the stable endpoints E

_{1}, E

_{2}, E

_{3}, and E

_{4}.

_{1}, E

_{2}, E

_{3}and E

_{4}at the corners of the chamber. While the straight watershed is the borderline between the two large catchment areas with stable endpoints E

_{1}and E

_{4}, the two curved watersheds bound two catchment areas with stable endpoints E

_{2}and E

_{3}. Each watershed has one saddle point (E

_{5}, E

_{6}, and E

_{7}).

_{6}and E

_{7}are in slightly different locations than for the high-conductance sphere in Figure 8. E

_{5}is at the same location for both conductance scenarios. The trajectories a and b along the mirror symmetry line have no sibling, and all other trajectories have one sibling. From areas near an electrode, the sphere moves within the respective catchment area of the start point before it is deflected toward the corresponding stable endpoint in one of the near corners (e.g., trajectory c). In general, the sphere does not move along the shortest possible path to an endpoint. Only trajectories a and b on the mirror symmetry line are straight and can theoretically end at E

_{5}.

#### 4.3.3. Relevance for DEP Measurements

#### 4.4. (+−+−)—Drive Mode (Field-Cage Mode)

#### 4.4.1. Field Distribution and Chamber Conductance

#### 4.4.2. Trajectories and Forces

_{2}, E

_{3}, E

_{4}and E

_{5}at the electrodes. Both diagonals have the common unstable endpoint E

_{1}at their intersection in the chamber’s center. On both diagonals, there are two additional unstable saddle points near the intersections of the diagonals, with the lines connecting adjacent electrode tips (E

_{7}, E

_{9}and E

_{6}, E

_{8}). From E

_{1}, eight straight trajectories run either to one of the four endpoints at the electrodes or along the diagonals to the unstable saddle points E

_{6}, E

_{7}, E

_{8,}or E

_{9}. Trajectories along one of the four mirror symmetry lines have three siblings, e.g., b, n, j. Trajectories starting in the chamber volume (apart from mirror symmetry lines) are curved and have seven siblings. They all hit the chamber wall before running to the endpoints at the nearest electrode., e.g., a, c, d, e, f, g, h, i, k, l, m. Approaching the endpoint E

_{6}, the sphere moving along trajectories j or n may be deflected perpendicularly, depending on the nature of a possible slight perturbation, and then continue along either trajectory a or l, hitting the chamber wall near an electrode and generating a small force peak (Figure 13C).

_{6}, E

_{7}, E

_{8}, E

_{9}, but not at the stable endpoints E

_{2}, E

_{3}, E

_{4}, E

_{5}. Again, except for the direct electrode hit (trajectory b), from Equation (2) only minimal forces result for the end steps within the conductance field. Experimentally, when the sphere’s motion stops, the DEP force at the terminal point is compensated for by the pressure on the electrode.

_{1}is surrounded by four separate, nearly triangular catchment areas with stable endpoints E

_{2}–E

_{5}at the corners of the chamber. Nearly straight watersheds separate the five catchment areas. The four watersheds have unstable saddle points (E

_{6}–E

_{9}) located at the intersection between the watersheds and the diagonals of the chamber. The effective DEP forces are zero at E

_{1}and at the unstable saddle points E

_{6}–E

_{9}, but not at the stable endpoints E

_{2}–E

_{5}. As with the high-conductance sphere, trajectories along one of the four mirror symmetry lines have three siblings, for example, c, d, h, and i. In-plane trajectories have seven siblings, e.g., a, b, e, f, g. Within a radius of about 250 µm, the sphere is deflected straight or almost straight to E

_{1}, the central endpoint of the field cage.

## 5. General Discussion

#### 5.1. Conductance Fields, Electric Work and Dissipation

#### 5.2. Thermodynamic Aspects

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Potential and current line distributions in the 1 × 1-mm chamber without the object. The four-pointed electrodes were energized with 0.5 V (marked by “+”; in AC-drive, this corresponds to the 180°-phase) versus −0.5 V at the counter electrodes (marked by “−“; in AC-drive, this corresponds to the −180°-phase). For a clearer presentation of the current and potential distributions, equidistant current lines were used at x = 0 in A and x = −250 µm in B. For C, two distributions with equidistant current lines at x = −250 µm and at x = 250 µm were combined. The calculated basic sheet conductances ${L}_{Basic}^{2D}$ for the 100 mS//1 S media are (

**A**): 42.31 mS//422.9 mS, (

**B**): 32.75 mS//327.4 mS and (

**C**): 46.67 mS//466.5 mS, corresponding to cell constants ${k}^{2D}$ of approx. 0.423, 0.327 and 0.466, respectively.

**Figure 2.**Potential and current line distributions for different positions of the 1.0-S sphere in 0.1-S medium for the (++−−)—drive mode. The position of the electrodes is sketched in G only. The overall conductances of the chamber are (

**A**): 51.62 mS, (

**B**): 42.50 mS, (

**C**): 42.31 mS, (

**D**): 43.09 mS, (

**E**): 43.07 mS, (

**F**): 42.46 mS and (

**G**): 42.94 mS (Table 1). The current lines were chosen to be equidistant at x = 0.

**Figure 3.**Potential and current line distributions for different positions of the 0.1-S sphere in 1.0-S medium for the (++−−)—drive mode. The position of the electrodes is sketched in G only. The conductances of the chamber are (

**A**): 173.0 mS, (

**B**): 420.1 mS, (

**C**): 422.9 mS, (

**D**): 414.6 mS, (

**E**): 414.3 mS, (

**F**): 421.2 mS and (

**G**): 416.4 mS (Table 1). The current lines were chosen to be equidistant at x = 0.

**Figure 4.**Single 200-µm, 1.0-S sphere (reddish circles in (

**A**)) in the chamber of Figure 1A with 0.1-S medium. (

**A**): Conductance field plot with trajectories (a–j). Two watersheds (two diagonal red lines) separate the four catchment areas of the stable endpoints E

_{1}, E

_{2}, E

_{3}and E

_{4}. E

_{5}, E

_{6}and E

_{7}are unstable saddle points on one of the watersheds. (

**B**): Chamber conductance along the trajectories. The basic, minimum, mean, and maximum conductances are 42.31 mS (w/o sphere), 42.31 mS (Figure 2C), 42.03 mS, and 51.75 mS (E

_{1}–E

_{4}, Figure 2A), respectively (Table 1). Trajectories c

_{1}, c

_{2}, and d

_{1}run on watersheds and through unstable endpoints. (

**C**): Normalized DEP forces calculated from the conductance values in (

**B**).

**Figure 5.**Single 200-µm 2D sphere of 0.1-S (reddish circles in (

**A**)) in the chamber of Figure 1 with 1.0-S medium. (

**A**): Conductance field plot with trajectories (a–h). Three watersheds (the diagonal white line between E

_{6}and E

_{7}and two curved lines through E

_{6}and E

_{7}at the end of the diagonal watershed) separate the four catchment areas of the stable endpoints E

_{1}, E

_{2}, E

_{3}and E

_{4}. E

_{5}, E

_{6,}and E

_{7}are unstable saddle points. (

**B**): Chamber conductance along the trajectories. The basic, minimum, mean, and maximum conductances are 422.9 mS (w/o sphere), 169.7 mS, 414.1 mS, and 422.9 mS (E

_{1}, E

_{2}, E

_{3}and E

_{4}, Figure 3C), respectively (Table 1). Trajectories b

_{1}, b

_{2}, and h

_{1}follow watersheds and run across unstable endpoints. (

**C**): Normalized DEP forces calculated from the conductance values in (

**B**). The initial force peak of 75.35 for trajectory a (green) was truncated, and the ordinate was shortened to increase the resolution for all other trajectories.

**Figure 6.**Potential and current line distributions for different positions of the 1.0-S sphere in 0.1-S medium for the (+++−)—drive mode. The position of the electrodes is sketched in (

**G**) only. The conductances of the chamber are (

**A**): 32.76 mS, (

**B**): 35.03 mS, (

**C**): 72.19 mS, (

**D**): 34.90 mS, (

**E**): 32.82 mS, (

**F**): 33.34 mS, (

**G**): 33.09 mS and (

**H**): 32.96 mS (Table 1). The current lines were chosen to be equidistant at x = −300 µm (

**A**), x = −250 µm (

**E**), x = −200 µm (

**H**), x = 50 µm (

**B**,

**D**,

**G**), x = 250 µm (

**F**) and x = 300 µm (

**C**).

**Figure 7.**Potential and current line distributions for different positions of the 0.1-S sphere in 1.0-S medium for the (+++−)—drive mode. The position of the electrodes is sketched only in (

**G**). The conductances of the chamber are (

**A**): 327.3 mS, (

**B**): 217.9 mS, (

**C**): 60.02 mS, (

**D**): 218.8 mS, (

**E**): 326.6 mS, (

**F**): 321.0 mS, (

**G**): 323.8 mS and (

**H**): 325.2 mS (Table 1). The current lines were chosen to be equidistant at x = −300 µm (

**A**), x = −250 µm (

**E**), x = −200 µm (

**H**), x = 50 µm (

**B**,

**D**,

**G**), x = 250 µm (

**F**) and x = 300 µm (

**C**).

**Figure 8.**Single 200-µm, 1.0-S sphere (reddish circles in (

**A**)) in the chamber of Figure 1B with 0.1-S medium. (

**A**): Conductance field plot with trajectories (a–k). Three watersheds (curved white lines) with unstable saddle points E

_{5}, E

_{6}, and E

_{7}separate four catchment areas for the stable endpoints E

_{1}, E

_{2}, E

_{3}, and E

_{4}. (

**B**): Chamber conductance along the trajectories. The basic, minimum, mean, and maximum conductances are 32.75 mS (w/o sphere), 32.76 mS (Figure 6A), 33.27 mS, and 72.19 mS (E

_{3}; Figure 6C), respectively (Table 1). (

**C**): Normalized DEP forces calculated from the conductance values in (

**B**). The arrows mark the ends of trajectories.

**Figure 9.**Single 200-µm 2D sphere of 0.1-S (reddish circles in (

**A**)) in the chamber of Figure 1B with 1.0-S medium. (

**A**): Conductance field plot with trajectories (a–k). Three watersheds (horizontal symmetry line, two curved white lines) with the unstable saddle points E

_{5}, E

_{6}, and E

_{7}separate four catchment areas for the stable endpoints E

_{1}, E

_{2}, E

_{3}, and E

_{4}. (

**B**): Chamber conductance along the trajectories. The basic, minimum, mean, and maximum conductances are 327.4 mS (w/o sphere), 58.45 mS (Figure 7C), 321.4 mS, and 327.3 mS (E

_{1}, E

_{4}; Figure 7A), respectively (Table 1). (

**C**): Normalized DEP forces calculated from the conductance values in (

**B**). The arrows mark the ends of trajectories.

**Figure 10.**Normalized DEP forces and DEP force reversibility along the chamber’s symmetry line in the DEP mode. (

**A**): Normalized DEP forces acting on the 1.0-S (full line, corresponding to trajectories a and b in Figure 8) and 0.1-S spheres (dashed line, corresponding to trajectories a and b in Figure 9). (

**B**): Zoom of A. The forces vanish for x = −201 µm for both spheres at the bifurcation points E

_{5}(Figure 8 and Figure 9). (

**C**): DEP force reversibility as calculated from the quotient of the normalized DEP forces on the 1.0-S and 0.1-S spheres using the data of Tables S3 and S4, which are summarized in Table S7, Supplementary Materials). The horizontal line marks the force ratio of −1, i.e., ideal reversibility, which can be assumed to be within the “reversibility range” −60 µm < x < 270 µm.

**Figure 11.**Potential and current line distributions for different positions of the 1.0-S sphere in 0.1-S medium. The location of the electrodes is sketched only in (

**G**). The conductances of the chamber are (

**A**): 46.78 mS, (

**B**): 47.04 mS, (

**C**): 46.84 mS, (

**D**): 57.85 mS, (

**E**): 47.10 mS, (

**F**): 46.77 mS and (

**G**): 46.69 mS (Table 1). Equipotential and current lines were combined from separate calculations for the left and right halves of the chamber, with current lines chosen to be equidistant at x = −40 µm and x = 40 µm, respectively.

**Figure 12.**Potential and current line distributions for different positions of the 0.1-S sphere in 1.0-S medium. The position of the electrodes is sketched only in (

**G**). The conductances of the chamber are (

**A**): 464.8 mS, (

**B**): 462.3 mS, (

**C**): 464.7 mS, (

**D**): 180.5 mS, (

**E**): 461.8 mS, (

**F**): 465.4 mS and (

**G**): 466.2 mS (Table 1). Equipotential and current lines were combined from separate calculations for the left and right halves of the chamber, with current lines chosen to be equidistant at x = −40 µm and x = 40 µm, respectively.

**Figure 13.**Single 200-µm, 1.0-S sphere (reddish circles in (

**A**)) in the chamber of Figure 1C with 0.1-S medium. (

**A**): Conductance field plot with trajectories (a–n). Two diagonal watersheds separate the four catchment areas of the stable endpoints E

_{2}, E

_{3}, E

_{4}and E

_{5}. E

_{1}is a single unstable point precisely in the chamber’s center. (

**B**): Chamber conductance along the trajectories. The basic, minimum, mean, and maximum conductances are 46.67 mS (w/o sphere), 46.69 mS (Figure 11G), 47.21, and 57.85 mS (E

_{2}, E

_{3}, E

_{4}, E

_{5}; Figure 11D). (

**C**): Normalized DEP forces calculated from the conductance values in (

**B**). The arrows mark the end of the trajectories.

**Figure 14.**Single 200-µm 2D sphere of 0.1-S (reddish circles in (

**A**)) in the chamber of Figure 1C with 1.0-S medium. (

**A**): Conductance field plot with trajectories (a–i). Four watersheds (white lines cutting off the chamber’s corners) separate the five catchment areas of the stable endpoints E

_{1}–E

_{5}. E

_{6}–E

_{9}are unstable saddle points in the middle of the watersheds. (

**B**): Chamber conductance along the trajectories. The basic (w/o sphere), minimum, mean, and maximum conductances are 466.5 mS, 176.9 mS (Figure 12D), 459.3 mS, and 466.2 mS (E

_{1}; Figure 12G), respectively. (

**C**): Normalized DEP forces calculated from the conductance values in (

**B**).

**Table 1.**Conductances of the chambers for the drive modes (++−−), (+++−), and (+−+−) (Figure 1) without (w/o sphere) and with the 2D sphere at different positions in the chamber (A–H) in mS. The conductances of the spheres and the outside chamber combine either 1.0 S for the sphere with 0.1 S for the medium (1.0 in 0.1) or 0.1 S for the sphere with 1.0 S for the medium (0.1 in 1.0).

Mode | ++−− | +++− | +−+− | |||
---|---|---|---|---|---|---|

Conductance | 1.0-in-0.1 | 0.1-in-1.0 | 1.0-in-0.1 | 0.1-in-1.0 | 1.0-in-0.1 | 0.1-in-1.0 |

Basic (w/o sphere) | 42.3103 | 422.9419 | 32.7518 | 327.3895 | 46.6665 | 466.4686 |

Minimum | 42.3132 | 169.6693 | 32.7567 | 58.4465 | 46.6926 | 176.8777 |

Mean | 43.0311 | 414.1074 | 33.2723 | 321.3707 | 47.2079 | 459.3097 |

Maximum | 51.7546 | 422.9101 | 72.1644 | 327.3172 | 57.8512 | 466.1858 |

A | 51.6242 | 172.9551 | 32.7572 | 327.3137 | 46.7797 | 464.7503 |

B | 42.4985 | 420.1077 | 35.0269 | 217.9206 | 47.0445 | 462.2856 |

C | 42.3139 | 422.9059 | 72.1854 | 60.0215 | 46.8354 | 464.6568 |

D | 43.0865 | 414.6169 | 34.9014 | 218.8037 | 57.8507 | 180.4669 |

E | 43.0702 | 414.2677 | 32.8186 | 326.6342 | 47.0987 | 461.7775 |

F | 42.4621 | 421.2087 | 33.3379 | 321.0329 | 46.7698 | 465.3648 |

G | 42.9407 | 416.3622 | 33.0919 | 323.8072 | 46.6926 | 466.1859 |

H | -- | -- | 32.9595 | 325.1618 | -- | -- |

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**MDPI and ACS Style**

Gimsa, J.; Radai, M.M.
Trajectories and Forces in Four-Electrode Chambers Operated in Object-Shift, Dielectrophoresis and Field-Cage Modes—Considerations from the System’s Point of View. *Micromachines* **2023**, *14*, 2042.
https://doi.org/10.3390/mi14112042

**AMA Style**

Gimsa J, Radai MM.
Trajectories and Forces in Four-Electrode Chambers Operated in Object-Shift, Dielectrophoresis and Field-Cage Modes—Considerations from the System’s Point of View. *Micromachines*. 2023; 14(11):2042.
https://doi.org/10.3390/mi14112042

**Chicago/Turabian Style**

Gimsa, Jan, and Michal M. Radai.
2023. "Trajectories and Forces in Four-Electrode Chambers Operated in Object-Shift, Dielectrophoresis and Field-Cage Modes—Considerations from the System’s Point of View" *Micromachines* 14, no. 11: 2042.
https://doi.org/10.3390/mi14112042