Result  12

Appendix 19

# Abstract

As research of active Janus colloids is still in its infancy, much remains unexplained or undiscovered in this field. Research has shown that Janus particles can be controlled to a certain degree. It is not yet clear where in society this will be used. Many papers have covered the medical and self-assembly possibilities, but before we can exploit any of the potential of Janus particles, we need to fully understand how they can be controlled. In this paper, we explain how we control asymmetrically gold coated Janus colloid particles. We have successfully, to a certain degree, controlled translational velocity, angular velocity, and inter-particle interactions of Janus particles. To observe the behaviour of the colloids, we limit ourselves to a two-dimensional field in which the particles can move freely. The propulsion of the particles is created by an alternating electric field in which ionic flows and dielectrophoresis are the main forces that cause the particles to propel. The electric field lines cause the particles to move in a two-dimensional field. Since the forces which cause the propulsion and the strong head to tail attraction of the particles are controllable with the AC frequency, we realize that a single parameter can control translational velocity, angular velocity, and the capability of Janus particles to create chains. In this paper, we show how we use a controllable alternating electric field frequency to control the behaviour of asymmetrically coated Janus colloid particles at different particle densities.

# Introduction

Active Janus colloids have been studied in various ways, but there is still much to be discovered. Asymmetrically coated active Janus colloid particles are an example of this. This paper studies movement and interaction between asymmetrically coated active Janus colloid particles.

Colloids are composed of particles that are still visible through an optical microscope, but not to the naked eye. The particles used in this paper are made by polymerization of propylene and have a diameter between 3 and 5 μm.

Janus particles have two surfaces which have distinct properties. There are multiple ways to create active Janus particles. The one used in the experiments is to coat parts of a polypropylene particle with a noble metal. In this paper, we use gold. The particles will be coated after they have been crystallized into a monolayer crystal. This is necessary to ensure every particle is coated equally. The Janus particles used in this paper are asymmetrically coated. In this context, asymmetrically coated refers to the coating not being hemispherical. To achieve this, a monolayer of Janus particles is coated at angle θ, as seen in figure 1. An asymmetrical coating is created, due to the neighboring particles shadowing the line of coating.

The artmachining china provide CNC machining and turning of rapid prototype parts and low to mid volume CNC rapid production parts.

Coated Janus particles are made active by putting them in a Pluronic solution, and placing this in a homogenous alternating electric field. The noble metal layer interacts with this field which causes the Janus particles to become a dipole. This causes the hydroxide and oxonium in the solution to form ion-dipole interactions with the Janus particles. The alternating electric field causes the poles of the dipole particles to switch at a controllable AC frequency, making the ions in the solution constantly move from pole to pole. This ionic flow causes the Janus particles to self-propel with the coated side following the uncoated side of the particle, as seen in figure 2.[1] At higher AC frequencies of the electric field, the force exerted on the particles by ionic flows increases.

However, there is another major force working on the Janus particles. The interaction between the dipole and the electric field generates a force. When there is a field gradient, the forces on the Janus particles are unequal, resulting in a net movement. This is called dielectrophoresis (DEP). The particles move to the area of the highest electric field strength, because they are more polarisable than the medium.[2] This cannot happen without a field gradient, and the Janus colloid is in a homogeneous electric field. However, the Janus particles create a local electric field in response to the alternating electric field from the electrodes, see figure 3.[3] Self-dielectrophoresis occurs (sDEP), causing the particle to move forwards, with the coated side in front. The dielectrophoretic force acting on a spherical body is given by:

where r is the radius of the homogenous sphere, is the permittivity of the medium, and is the real part of the Clausius-Mossotti function. The latter decides the AC frequency dependence of the dielectrophoresis force.

It is given by:

where is the particle’s complex permittivity and is the medium’s complex permittivity. is given by:

where is the permittivity, is the conductivity, and is the AC frequency of the alternating electric field. When is positive, the Janus particles will move towards the highest electric field strength. When it is negative, the particles will move away from these regions.

The alternating electric field forces the Janus particles to position themselves so that the dipole is along the longest axis that aligns with the electric field. As this division is perpendicular to the particles’ movement, they move on a virtually two-dimensional field perpendicular to the electric field lines. In this two-dimensional field, the translational and angular velocity can be measured. Research has shown that translational velocity can be controlled by altering the AC frequency as described in Demirörs et al. (2018).[3] At a higher AC frequency, the translational velocity of Janus particles will decrease.

The also has an effect on the angular velocity. The dielectrophoretic velocity, deriving from the , acting on an asymmetrically coated spherical body can be split into two exponents, as described in Squires and Bazant (2006).[4] The exponent in the direction of propulsion is given by:

and the exponent perpendicular to the direction of propulsion is given by:

where is the is the dielectrophoretic velocity, θ is the angle at which the Janus particles are coated, and is the angle of the electric field. This shows that the dielectrophoretic velocity in the direction of propulsion and the angular velocity, determined by, of Janus particles have a positive correlation with , and thus have a positive correlation with .

The Janus particles also interact with each other in certain ways if the particle density is high enough. At higher frequencies, the sDEP force increases, which creates stronger head to tail attraction between particles. The strength of the head to tail attraction determines how particles interact with each other. At 1-20 kHz, particle interactions are negligible, because the sDEP force is very low. At 20 kHz, the particles start moving in the other direction, with the coated side leading, because the DEP force becomes stronger than the force exerted by the ionic flows. At frequencies above 20 kHz, the particles dipole is strong enough for the particles to link together to form chains because of the strong head to tail attraction between Janus particles caused by sDEP (Yan et al., 2016).[5]

A single parameter, the alternating electric-field frequency, can therefore simultaneously control the translational velocity, the angular velocity, and the interactions between particles.

In this paper, we will be answering the following question: How does the AC frequency of the alternating electric field and the density of the asymmetrically coated Janus colloid particles affect their behaviour? Specifically: how does it affect the translational and angular velocity of the asymmetrically coated Janus colloid particles, and the interaction between Janus colloid particles?

We expect that a higher AC frequency will cause the translational velocity of our asymmetrically coated Janus colloid particles to decrease, because at a higher AC frequency the sDEP force increases, causing the particles to slow down until they stop moving at 20 kHz, as described in Yan et al. (2016).[5] We believe that the angular velocity of the Janus particles will increase at a higher AC frequency, because it has a positive correlation with the , as described in Squires and Bazant (2006).[3] As stated earlier, the angular velocity is dependent on , the angle of coating, and the angle of the AC electric field. In these experiments the AC electric field angle and the coating angle stay the same, meaning that the angular velocity only depends on . This means that the angular velocity will have a positive correlation with the AC frequency. We also predict that the Janus particles will have a lower translational and angular velocity at higher particle densities, because of interparticle attractions and collisions between Janus particles.

# Materials and Methods

To study the behaviour of asymmetrically coated active Janus colloid particles, a colloid sample had to be created that was appropriate for our research. A series of colloid samples was created and subsequently tested in order to find one that best fit our research. To best observe the behaviour of asymmetrically coated active colloid particles, the particles need to be free to move and they need to turn in circles with a diameter of about 20 μm at an AC frequency of 3 kHz and a current of 18 volts. The colloid samples were prepared and tested according to the following method:

The particles were delivered in a suspension. The colloid substrate was dried using a nitrogen gas pump directed at the liquid containing Janus colloid particles. The powder that remained after 11 hours of drying was used to create a single crystal monolayer of particles. (Figure 4) To get reproducible results, a unidirectional rubbing technique of the powder on a rubbery surface was used as described in Park et al. (2015).[6]

The sample of crystal monolayer particles was then brought to a sputter coating machine where the monolayer was coated with gold at an angle of 60°. Multiple monolayers were coated with different diameters.

The coated monolayer was then put into a 0,1 g/L Pluronic solution and placed in an ultrasonic bath for 30 seconds to get all the particles in the Pluronic solution. Lastly, the solution with the particles was injected into a cell. A cell consists of two ITO pieces separated by glass and glued to a slide (Figure 5). The Pluronic ensures the particles in the solution do not stick to the ITO slides, and ITO is used so that light can travel through the cell to the optical microscope.

Figure 5: A cell consists of two ITO pieces separated

The ITO slides in the cell were connected to an oscilloscope with an amplifier that delivered a sinus wave alternating current to create a homogeneous electric field. The behaviour of the particles was then observed using an optical microscope with 20X magnification.

The colloid sample that worked best was used in our research. It moved freely, and had a turning radius of about 18 μm at an AC frequency of 3 kHz and a current of 18 volts. The other colloid samples did not fit the requirements for our research. Some got stuck to the ITO slides. Others moved too fast, so that the radius of the circular motion was too large to analyse. The sample that was used for our research was made up of heavier particles which reduced these problems. The monolayer used for this research was coated with a 12 nm layer of gold.

To obtain the results, the particles were filmed at different frequencies and densities using the optical microscope.

See the appendix for a full step by step overview of our materials and methods.

# Data Processing

We can determine the translational and angular velocity of the particles using the Mean Squared Displacement, or MSD. This is a measure of the deviation of the particle from the source over time. It is used to identify to what degree the particles are moving in a circular motion. When the line touches the x-axis, the particle is at its starting point. The parabolic graph is an indication of the particle’s circular motion. By creating a function fit of this graph, the translational and angular velocity of the particle can be determined.

To process the videos, they were first turned into 8-bit tiff files using ImageJ. These files could then be processed using MATLAB and Mathematica (see appendix for code). Mathematica plotted a Mean Squared Displacement graph. At three different frequencies, at three different densities, the translational and angular velocity of twenty particles were plotted into graphs.

Figure 6: The MSD(Mean Squared Displacement) graph of a Janus particle in an AC electric field at 3 kHz

# Result

This graph shows the movement of three particles at different frequencies. The particles appear to be moving in circles and the diameter decreases at a higher AC frequency. At 1 kHz the circle has a diameter of about 44 μm. At 3 kHz, the circle has a diameter of about 24 μm. At 5 kHz the circle has a diameter of about 10 μm.(figure 7)

Figure 7: The movement of three particles at different frequencies

In figures 8a and 8b, the translational velocity and angular velocity are plotted against density at different frequencies. Low density corresponds to 48 particles/mm2. Medium density corresponds to 467 particles/mm2 and high density correspond to 2327 particles/mm2.

Figure 8a

Figure 8b

At higher frequencies, the particles decreased in speed, and at 16 kHz the particles stopped moving. Above this AC frequency, the particles started to move in the opposite direction. At even higher frequencies, some of the particles started forming chains. Figure 9 shows particles at medium density at 60 kHz.

At even higher AC frequency and density, the particles began to pile up. Figure 10 shows particles at high density at 100 kHz.

Figure 10: Janus particles at high density at 100 kHz

# Conclusion

The asymmetrically coated Janus particles move in a circular motion. They move either in a clockwise or in a counterclockwise rotation. The direction in which Janus particles rotate is arbitrary. The circular motion is not perfect, as shown in figure 7.

There is a strong negative correlation between the AC frequency of the alternating electric field and the translational velocity of the particles, as shown in figure 8a. At all densities, the translational velocity decreases as the AC frequency increases. For the angular velocity the opposite is true, it has a strong positive correlation with the AC frequency. (figure 8b)

According to our research, both the translational and angular velocity generally negatively correlated to the density. At 3 kHz, however, the translational velocity was higher at medium density than it was at low density. The angular velocity negatively correlates to the density at all frequencies.

At frequencies between 1 and 16 kHz, the Janus particles move with their uncoated side in front. At 16 kHz, the particles stop moving. At frequencies higher than 16 kHz, the Janus particles start moving with the coated side in front. At these higher frequencies, chains begin to form. These chains grow longer as the AC frequency increases. However, this stops at high density at 100 kHz, because the particles begin to form piles.

# Discussion

The Janus colloid particles mostly behaved how we expected them to behave. At frequencies 1-16 kHz, the translational velocity of the particles decreased, until they stopped moving at 16 kHz. Above 16 kHz, the translational velocity of the particles increased, while they were moving in the opposite direction. We had expected the particles to stop moving at 20 kHz. Yan et al. (2016) [5] explain how the force created by the ionic flows and the dielectrophoresis force are opposite forces which are at balance at 20 kHz. Our research shows the particles stopping at 16 kHz. This may be because of the larger radius of the particles we used. This causes a larger , as shown in the dielectrophoresis force equation:

This equation shows that the larger particles experience more , because of the increased radius . This means that the will match the force of the ionic flows, and the particles will stop moving, at a lower AC frequency. This also means chains form at a lower AC frequency in our experiments.

The angular velocity shows a positive correlation with the AC frequency at 1-5 kHz. We had expected this, because the velocity of the exponent perpendicular to the direction of propulsion of a Janus particle, which causes angular velocity, positively correlates to the dielectrophoretic velocity, which is caused by . This is illustrated in equation:

The radius of the circular trajectory of the Janus particle decreased as the AC frequency increased, as shown in figure 7. This can be explained with equation:

where R is the radius of the circular trajectory of the Janus particle, is its translational velocity, and is the angular velocity of the Janus particle. As AC frequency increases, angular velocity increases, and translational velocity decreases. As positively correlates to , and negatively correlates to , this results in a decrease of .

At higher densities, the translational and angular velocity of the particles decreased. We believe that this is because of inter-particle interactions. This is because the inter-particle attractions and collisions between Janus particles act out a force on the particles, which can decrease the net force. Figure 8a shows that the decrease in translational velocity between low and medium densities at 1 kHz is greater than at 3 and 5 kHz. This may be due to the fact that faster particles experience more collisions compared to slower moving particles at the same densities, therefore decreasing in translational and angular velocity faster.

At higher density, the deviation in the results is higher. Inter-particle interactions may be able to explain this. These inter-particle interactions cause the movement of the particles to become seemingly more random, because we did not have the software to observe how nearby particles affected each other’s movements. Further research could look into inter-particle interactions and how they affect the movement of particles.

Figure 8a shows a positive correlation between the translational velocity and particle density between low and medium densities at 3 kHz. This is not what we expected, especially since the dispersion of this graph was low compared to the dispersion of the other results. Looking into this is beyond the scope of this research, however.

At high density at 100 kHz, the particles started to form piles as shown in figure 10. This caused a major problem for our research, as we had limited ourselves to two dimensions, and these piles did not work with our two-dimensional model. We believe that these piles started to form because the head to head and tail to tail repulsion between particles became so strong that the way of least resistance for the particles became to stack up. We had hoped to observe the particles at frequencies above 100 kHz as well, but unfortunately this was impossible at high particle density. Our advice for further research would be to use heavier particles. Heavier particles would experience more gravitational force which could prevent the particles from forming piles.

# Appendix

## Step by Step Materials and Methods:

Tuesday 10-9-2019

Active particle sample A40:

A piece of monolayer PMMASpheres17 was coated with gold (15nm) at 60° and a pressure of 5 x 10^-4 and 2 x 10^-2 was used. The monolayer was sonicated briefly in 20 ml 0,1 g/l Pluronic F127 water. The particles were injected into the red cell (at 10 Volt and 1 kHz). The particles did not propel but were also not stuck to the glass. The particles were diluted by a factor of 10. Half the particles swam in circles as expected, the other half stuck to the glass.

1g/l Pluronic F127solution:

Dissolved 0,501 g Pluronic F127 in 500 ml mq-water and left on a stirring plate overnight.

Wednesday 11-9-2019

Active particles  sample A41:

Coated a monolayer of PMMAspheres17 with 12nm of gold. The particles circled well at 10kHz and 20 V on the 120 µm red cell. The particles circled decently at 5 kHz and 100 V in the 1mm cell.

Active particles sample A42:

Coated PMMMASpheres17 with 10 nm of gold. Tested the particles in the red cell. At 5 kHz they appeared to rotate in tight circles.

Drying of the colloids:

Samples of PS42, PSGMA9 and PS33 were dried overnight in a vial.

Thursday12-9-2019

Monolayer Formation:

Prepared monolayers of PS42, PSGMA9 and PS33 on PDMS using the unidirectional rubbing method. Viewed the monolayer using optical microscope. Coated the monolayers with 10 nm of gold.

Swimmers PS42 A43:

Injected the particles into the red cell. At 1kHz they propel in a straight trajectory. At 5 kHz their speed greatly diminishes. At 8 kHz they were stationary.

Swimmers PS33 A45:

Injected the particles into the red cell. At 1kHz they circled in wide trajectories. At 3 kHz they circled in tight trajectories.

Swimmers PSGMA9 A44:

Injected the particles into the red cell. At 3 kHz and 16 V they circled quickly in wide circles. At 5  kHz and 16 V they circled slower in shorter trajectories.

Monolayer formation PS33:

Prepared a monolayer of PS33 on PDMS using the Lego tool. Crystal orientation was neat.

Active particles samples A46 and A47:

Coated the monolayer with 10 nm and 12 nm of gold respectively at 60°.

Swimmers A46:

Injected the particles into the red cell. At 1 kHz and 16 V the particles circled quite fast in quite wide circles.

Swimmers A47:

Injected the particles into the red cell. At each frequency circling motion is observed. They switch direction at 18-20 kHz. Circle at frequencies 1-5 kHz nicely at a decent speed at 18 V.

Friday 13-9-2019

Sample A48:

Coated a piece of PS33 monolayer with 12 nm of gold at 60°. Tested the particles at 1 kHz, 3 kHz, and 5 kHz at an 18 V sine wave. Observed various amounts of rotational motion. Centrifuged the particles at 1000 rpm for 5 minutes to increase the concentration of particles by a factor of 5 to 10. Took various videos at various particle concentrations and frequencies.

## Results with Deviation

MATLAB Code

% Use gaussian blobs (1) or hough transform (0) to find particle

gaussfinder=1;

%indicate name of tiffstack (format = Path/name.tif)

fname = ‘\\client\d$\Casper_en_Jaap\A48\20x_10fps_18Vsi_5kHz_1,0pl_240.tif’; %indicate name to save output text file (format = Path/name.tif) fsave = ‘\\client\d$\Casper_en_Jaap\Verwerken_tot_txt\20x_10fps_18Vsi_5kHz_1,0pl_240.txt’;

%set parameters for image analysis software

rlower=20; %lower bound initial particle size

rupper=20; %upper bound initial particle size

scale=1; %scale for image in pixel per micrometer

sensitivity=0.0001; %sensitivity for particle find: 1 finds every contrast difference, 0 finds no contrast difference

%set preferences for printing image, inverting contrast and number of images used

invert=1;           %’1′ is invert contrast, ‘0’ is not invert contrast

% invert if particle is darker than surrounding

imageshow=1;        %’1′ is show image, ‘0’ is not show image

num_images=100;      %’0′ is use all images in stack. ‘n’ (is not 0) uses first n images from stack

%values below

%If using gaussian blob finder (pkfind), set parameters for analysis

threshold=65;

diameter=50;

%set parameters for trackingsoftware

maxdisp=15; % maximum step size

param.mem=10; % how long particle can go missing

param.dim=2; % dimensionality of data

param.good=50; % how long track needs be to be accepted

param.quiet=1; % 0 = text, 1 = no text

%assert good value for num_images

info = imfinfo(fname);

if num_images > numel(info)

error(‘num_images is larger than stacksize: use a smaller value or “0” to use entire stack’)

end

%extract numer of images from stack

if num_images==0

num_images = numel(info)

end

%% Find particles

%define matrices

A=[];

centers=[];

pos=[];

check=0.1;

%open new figure

figure

%for all images in stack

for k = 1:num_images

if k/num_images>check

%print progress in percentage

[‘Particle finding progress: ‘ num2str(check*100) ‘ %.’]

check=check+0.1;

end

%Invert Contrast if invert=1

if invert==1

A=255-A;

end

%Distinguish between two ways of finding particle

if gaussfinder==1

A=bpass(A,2,diameter);

%A=smooth(A,50);

pk = pkfnd(A,threshold,diameter);

%cnt = cntrd(b,pk,15);

pk(:,3)=k;

pos=[pos ; pk];

pkl=length(pk(:,1));

%Plot image with found particles if imageshow=1

if imageshow==1

hold on

colormap(‘gray’),imagesc(A);

hold off

end

else

%Find centers and radii of droplets

centers(:,3)=k;

%append pos matrix with new found centers

pos=[pos ; centers];

%Plot image with found particles if imageshow=1

if imageshow==1

hold on

colormap(‘gray’),imagesc(A);

hold off

end

%if only one particle in image

%and if a particle was found at all

%update bounds for radii (if droplet changes size in time)

end

end

end

end

%% Track

%run trackingcode:

%this code connects the locations of the same particle and gives an ID number

tr=track(pos,maxdisp,param);

%save [tr] as textfile

textfile=[tr];

fileID = fopen(fsave,’w’);

fprintf(fileID , ‘%6.4f %6.4f %6.4f %6.4f \n’ , textfile’);

fclose(fileID);

## Mathematica Code

SetDirectory[

“\\\\client\\d$\\Casper_en_Jaap\\Verwerken_tot_txt”]; (* Hier de locatie van de map invullen *) RuweData = Drop[Import[ “\\\\client\\d$\\Casper_en_Jaap\\Verwerken_tot_txt\\20x_10fps_\

18Vsi_3kHz_1,0pl_239.txt”, “Table”]];                     (* Hier de naam van het bestand invullen *)

TransposedRuweData = Transpose[RuweData];

FPS = 10;                                         (* Frames per seconde van het filmpje *)

R = 0.4;                                                        (* Resolutie van de microscoop in um/pixel (niet \

veranderen) *)

VerwerktX = TransposedRuweData[[1]]*R;

VerwerktY = TransposedRuweData[[2]]*R;

VerwerktT = (TransposedRuweData[[3]] – 1)/FPS;

VerwerktP = TransposedRuweData[[4]];

VerwerktXY = Transpose[{VerwerktX, VerwerktY}];

VerwerktXYTP = Transpose[{VerwerktX, VerwerktY, VerwerktT, VerwerktP}];

P = 2;                                                                                 (* Rugnummer van het deeltje *)

idx = {}; For[i = 0, i < Length[VerwerktXYTP], i++,

If[VerwerktXYTP[[i + 1, 4]] != P,

AppendTo[idx, {i + 1}]]]; VerwerktXYT1 = Delete[VerwerktXYTP, idx];

Max[VerwerktP]                     (* Probeer in Matlab de parameters aan te passen om deze waarde zo klein mogelijk te houden (niet opgesplitst) *)

Verwerkt1X = Transpose[VerwerktXYT1][[1]];

Verwerkt1Y = Transpose[VerwerktXYT1][[2]];

Verwerkt1T = Transpose[VerwerktXYT1][[3]];

Length[Verwerkt1X]

Verwerkt1XY = Drop[Drop[Transpose[{Verwerkt1X, Verwerkt1Y}], 0], 0];

PlotXY = ListLinePlot[{VerwerktXY, Verwerkt1XY},

PlotStyle -> {{Green, AbsoluteThickness[2]}, {Red,

AbsoluteThickness[2]}}, Frame -> True,

FrameLabel -> {“x (um)”, “y (um)”},

LabelStyle -> {Black, Medium, FontSize -> 16}, ImageSize -> 800,

AspectRatio -> Automatic]

PlotXY1 =

ListLinePlot[{Verwerkt1XY}, PlotStyle -> {Red, AbsoluteThickness[2]},

Frame -> True, FrameLabel -> {“x (um)”, “y (um)”},

LabelStyle -> {Black, Medium, FontSize -> 16}, ImageSize -> 800,

AspectRatio -> Automatic]

SnelheidsTabel =

Table[(((Verwerkt1X[[i + 1]] –

Verwerkt1X[[i]])^2 + (Verwerkt1Y[[i + 1]] –

Verwerkt1Y[[i]])^2)^0.5)*FPS, {i,

1, (Length[Verwerkt1X] – 1)}];

GemSnelheid = Mean[SnelheidsTabel]

SnelheidT = Transpose[{Drop[Verwerkt1T, 1], SnelheidsTabel}];

Plot1TS =

ListLinePlot[SnelheidT, PlotStyle -> {Red, AbsoluteThickness[2]},

AxesOrigin -> {0, 0}, Frame -> True,

FrameLabel -> {“t (s)”, “Speed (um/s)”},

LabelStyle -> {Black, Medium, FontSize -> 16}, ImageSize -> 800]

Mean[Table[(((Verwerkt1X[[i + DT]] –

Verwerkt1X[[i]])^2 + (Verwerkt1Y[[i + DT]] –

Verwerkt1Y[[i]])^2)^0.5)*FPS/DT, {i,

1, (Length[Verwerkt1X] –

DT)}]]; (* Snelheidsverschil met tijdstap DTmaalFPS *)

Tmax = 4.5;                                    (* Lengte van filmpjein seconden gedeeld door deze Tmax is het aantal onafhankelijke meetwaarden, hoe hoger deze waarde, des te beter de dataset *)

MSD = Mean[

Table[Total[(Verwerkt1XY[[i + j]] – Verwerkt1XY[[i]])^2], {i, 1,

Length[Verwerkt1XY] – 1 – Tmax*FPS}, {j, 0, Tmax*FPS}]];

TableT = Table[i/FPS, {i, 0, Length[MSD] – 1}];

TMSD = Transpose[{TableT, MSD}];

PlotTMSD =

ListLinePlot[TMSD, PlotStyle -> {Red, AbsoluteThickness[2]},

AxesOrigin -> {0, 0}, Frame -> True, FrameLabel -> {“\!$$\* StyleBox[\”t\”,\nFontSlant->\”Italic\”]$$ (s)”,

“MSD (\!$$\*SuperscriptBox[\(\[Mu]m$$, $$2$$]\))”},

LabelStyle -> {Black, Medium, FontSize -> 16}, ImageSize -> 800]

model = (4 DT*t) + (2*v^2*DR*t/(DR^2 + w^2)) + (2*

v^2*(w^2 – DR^2)/((DR^2 + w^2)^2)) + ((2*v^2*

Exp[-DR*t]/((DR^2 + w^2)^2)))*(((DR^2 – w^2)*Cos[w*t]) – (2*w*

DR*Sin[w*t]));

fit = NonlinearModelFit[

TMSD, {model, v > 0, w > 0, DT > 0,

DR > 0}, {{v, 25}, {w, 0.5}, {DT, 0.01}, {DR, 1}}, t]

Show[

ListPlot[TMSD, PlotRange -> All, PlotMarkers -> {Black, 8},

AxesOrigin -> {0, 0}, Frame -> True, FrameLabel -> {“\!$$\* StyleBox[\”t\”,\nFontSlant->\”Italic\”]$$ (s)”,

“MSD (\!$$\*SuperscriptBox[\(\[Mu]m$$, $$2$$]\))”},

LabelStyle -> {Black, Medium, FontSize -> 16}, ImageSize -> 800],

Plot[model /. fit[“BestFitParameters”], {t, 0, 5}, PlotRange -> All,

PlotStyle -> {Red, AbsoluteThickness[2]}, AxesOrigin -> {0, 0},

Frame -> True, FrameLabel -> {“\!$$\* StyleBox[\”t\”,\nFontSlant->\”Italic\”]$$ (s)”,

“MSD (\!$$\*SuperscriptBox[\(\[Mu]m$$, $$2$$]\))”},

LabelStyle -> {Black, Medium, FontSize -> 16}, ImageSize -> 800]]

fit[“BestFitParameters”]

model = (4 DT*t) + (2*v^2*DR*t/(DR^2 + w^2)) + (2*

v^2*(w^2 – DR^2)/((DR^2 + w^2)^2)) + ((2*v^2*

Exp[-DR*t]/((DR^2 + w^2)^2)))*(((DR^2 – w^2)*Cos[w*t]) – (2*w*

DR*Sin[w*t]));

fit = NonlinearModelFit[

TMSD, {model, v > 0, w > 0, DT > 0,

DR > 0}, {{v, 25}, {w, 0.5}, {DT, 0.01}, {DR, 1}}, t]

Show[

ListPlot[TMSD, PlotRange -> All, PlotMarkers -> {Black, 8},

AxesOrigin -> {0, 0}, Frame -> True, FrameLabel -> {“\!$$\* StyleBox[\”t\”,\nFontSlant->\”Italic\”]$$ (s)”,

“MSD (\!$$\*SuperscriptBox[\(\[Mu]m$$, $$2$$]\))”},

LabelStyle -> {Black, Medium, FontSize -> 16}, ImageSize -> 800],

Plot[model /. fit[“BestFitParameters”], {t, 0, 5}, PlotRange -> All,

PlotStyle -> {Red, AbsoluteThickness[2]}, AxesOrigin -> {0, 0},

Frame -> True, FrameLabel -> {“\!$$\* StyleBox[\”t\”,\nFontSlant->\”Italic\”]$$ (s)”,

“MSD (\!$$\*SuperscriptBox[\(\[Mu]m$$, $$2$$]\))”},

LabelStyle -> {Black, Medium, FontSize -> 16}, ImageSize -> 800]]

fit[“BestFitParameters”]

# Bibliography

[1] Velev, Orlin D., Sumit Gangwal, and Dimiter N. Petsev. 2009. “Particle-Localized AC and DC Manipulation and Electrokinetics.” Annual Reports on the Progress of Chemistry – Section C 105: 213–46. https://doi.org/10.1039/b803015b.

[2] Hughes, Michael Pycraft. 2000. “AC Electrokinetics: Applications for Nanotechnology.” Nanotechnology 11 (2): 124–32. https://doi.org/10.1088/0957-4484/11/2/314.

[3] Demirörs, Ahmet F., Mehmet Tolga Akan, Erik Poloni, and André R. Studart. 2018. “Active Cargo Transport with Janus Colloidal Shuttles Using Electric and Magnetic Fields.” Soft Matter 14 (23): 4741–49. https://doi.org/10.1039/c8sm00513c.

[4] Squires, Todd M., and Martin Z. Bazant. 2006. Breaking Symmetries in Induced-Charge Electro-Osmosis and Electrophoresis. Journal of Fluid Mechanics. Vol. 560. https://doi.org/10.1017/S0022112006000371.

[5] Yan, Jing, Ming Han, Jie Zhang, Cong Xu, Erik Luijten, and Steve Granick. 2016. “Reconfiguring Active Particles by Electrostatic Imbalance.” Nature Materials 15 (10): 1095–99. https://doi.org/10.1038/nmat4696.

[6] Park, Choojin, Kunsuk Koh, and Unyong Jeong. 2015. “Structural Color Painting by Rubbing Particle Powder.” Scientific Reports 5: 1–5. https://doi.org/10.1038/srep08340.