Colloidal Transport in Micro-Channels
11.3 Transversal Barriers
0 100 200 300 400 500 600 700 800 100 ρ(x) [σ-2 ]
x [σ]
κDσ = 2 κDσ = 4 κDσ = 8 κDσ = 12
Figure 11.26: Density profiles along the channel for a selection of Debye screening lengthsκDof a YHC system (βV0=50). The driving force is applied only within the channel region x∈[100,700]σ and the system is periodic inx-direction. These profiles correspond to the superimposed configurations of figure11.25.
The systems with κD=8σ−1 andκD=12σ−1 show a rapid increase of the local density from about 0.5σ−2up to values greater than 0.8σ−2in the intervalx∈[600,700]σ. The particles under the influence of the constant driving force are blocked due to filling of the reservoir at the channel end. During the simulation run the particles pile up at the interface between the channel and the reservoir, because the particles of the channel are pushed into the reservoir but within the reservoir the particles diffuse almost freely due to the short range of the YHC interaction (high values ofκD).
This leads to a situation where the influx into the reservoir is greater than the particle drift within the reservoir being the reason for the sharp density gradients, which lead to the sudden onset of a layered structure with 8 layers in the figures11.25(c)–(d). ForκD=12σ−1even a layer transition to 9 layers takes place due to local density values greater than 0.9σ−2which is not observed for the other three cases. Alternatively, the particle flux can be blocked in a controlled fashion by creating so-called laser barriers perpendicular to the driving field, as we will show in the following section.
11.3 Transversal Barriers
A transparent, micron-sized colloidal particle, whose index of refraction is greater than that of the surrounding medium, can be trapped by a tightly focused laser beam. Extremely high gradients in the electric field occurring near the waist of the laser beam are associated with strong forces which drag the particle to the focal point of the laser beam. Such a setup is calledoptical tweezers[Ash86, Ash92, Gri03]. The colloidal particle acts as a lens, refracting the rays of the laser light and redirecting the momentum of their photons. The refracted rays differ in intensity over the volume of the sphere and exert a subpico-Newton force on the particle, which draws the particle towards
the region of highest intensity, i.e. the focal point of the laser beam. Optical tweezers provide a versatile tool to study soft biomaterials [She98] and can in our case be used to manipulate the particle flow along the microchannel. The strength of the optical force acting to restore the particle position to the trap center can be considered, in good approximation, to be harmonic [Roh05],i.e.
proportional to the distancerfrom the trap center. Therefore the force may be expressed by Fopt(r) =
−kr : r≤Rtrap
0 : r>Rtrap (11.6)
wherek denotes the trap stiffness perpendicular to the direction of the laser beam3 andRtrapthe interaction range of the trap.
11.3.1 Single Line Barrier
(a)
0 2 4 6 8 10
0 100 200 300 400 500 600
y [σ]
x [σ] 0
2 4 6 8 10
0 100 200 300 400 500 600
y [σ]
x [σ]
(b) Laser tweezers
Figure 11.27: (a) Full channel snapshot of a channel with a barrier of repulsive optical traps marked by the red line. Simulation parameters used: B =0.5 mT, α =0.2◦, Lx=800σ,Ly=10σ,n=0.3σ−2, and the trap parameters ˜k=−1,Rtrap=0.75σ. (b)Video microscopy configuration snapshot of stopped particles at a potential bar-rier perpendicular to the flow direction in the experiment of A. Erbe [Erba]. The barrier is created using an optical laser tweezer setup in combination with a scan-ning mirror. The superparamagnetic particles are gravitationally driven from left to right by tilt of the whole setup.
To study possible influences on the flow behavior simulations have been performed, where we additionally inserted the trap forceFoptof equation (11.6) into the overdamped Langevin equations
3In the experiment the trap stiffnesskmay be determined from the mean-square displacements of the trapped, fluc-tuating particle positions at equilibrium via the equipartition theoremkBT/2=khr2i. Good qualitative agreement between electromagnetic theory and experimentally measured trap stiffnesses was reported by Rohrbach [Roh05].
Generally, the trapping strength is proportional to the laser power; it is weaker in the beam direction than in lateral directions.
which are evaluated numerically. These simulations model optical traps of different interaction ranges, in various geometrical arrangements ranging from fixed user-defined positions to fixed random positions, and of barriers made out of traps transversal to the direction in which the particle are driven.4
Figure11.27(a) shows the stationary non-equilibrium situation of a barrier perpendicular to the particle driving direction being obtained from a BD simulation run. The particles are hindered to cross the barrier. Thus, an increasing density gradient forms in front of the barrier leading to an increase of layers. At the barrier the local density shows a sharp non-continuous drop resulting in less layers. The experimental analogue is shown in the snapshot of figure 11.27(b)taken in the experiment of Artur Erbe [Erba] where laser tweezers in combination with a scanning mirror are used to build up a potential barrier which blocks the particle movement along the channel. The transparency of the barrier for single particle penetration can be modified in the experiment by the laser intensity, the position of its focal plane and the strength of external driving force of the particles.
It is interesting to take a look on the dynamics of the time development of the local densityρ(x) which is shown in figure11.28. Not shown is the “reservoir”,i.e. the region of random particle insertion which ranges from−200σ≤x<−120σ. This region is disregarded to avoid influences due to the interaction of the channel with the “reservoir”. During the first 1000 BD simulation steps already a small increase of the local density in front of the barrier and a sharp decrease of the local density behind the barrier occurred. At the beginning of the non-equilibrium simulation part the particles were homogeneously distributed over the full channel according to a total particle density of n=0.3σ−2. The sharp drop of the local density to valuesρ(x)<0.25σ−2 after the barrier is due to the open channel end atx=600σ. Particles which dropped out of the channel are re-inserted at the beginning of the channel atx=−200σ. This results in a local density bump forx<−80σ. This bump moves in flow direction until it reaches the point of increasing density at about x=300σ. In font of the barrier the particles form more and more pronounced layers perpendicular to the direction of particle transport. This can be seen from the regular sequence of peaks and troughs left of the barrier.
For completeness, we plot in figure11.29the local lattice constants and density of the system of figure11.27. Clearly visible is the increase of the layers from 6 to 7 in front of the barrier and a controlled drop after the barrier. This system behaves contrary to the situation without a barrier.
4Simulation of optical traps with our code requires the compiler option PARABOLIC TRAP to be defined during the compilation process. The number of traps, their positions, interaction range, and trapping strength ˜k (in reduced units) can be defined in the simulation parameter file parameter.ini using the keywords N Trap, TrapPosition, TrapRangeSq, and TrapStengthrespectively. The trap positions can either be set at random position by adding the line
TrapPosition random
to the file parameter.inior by specifying thex- andy-position and the keywordfixedas in the following example, where 2 traps are located atx=500σandy=0.0σandy=5.0σrespectively which have an interaction range of 1.5σand the trapping strength ˜k=−1.0 (repulsive traps):
N Trap 2
TrapPosition fixed
500 0.0
500 5.0
TrapRangeSq 2.25 TrapStrength −1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-100 0 100 200 300 400 500 600
ρ(x) [1/σ2 ]
x [σ]
t = [0 : 1e4]
t = [1e4 : 2e4]
t = [2e4 : 3e4]
t = [4e4 : 5e4]
t = [5e5 : 1e6]
t = [1.5e5 : 2e6]
Figure 11.28: Time evolution of the change of the local density in front of the barrier atx=500σ for the system of figure11.27(a). The histograms are obtained from thex-positions of the particles during the intervals of simulation steps as given in the legend. The size of the bins is∆x=2σ. For clarity reason consecutive graphs are depicted with an offset of∆ρ(x) =0.05σ−2.
1.5 2 2.5 3 3.5 4
0 100 200 300 400 500
distance [σ]
x-position [σ]
dx dy
0 0.1 0.2 0.3 0.4 0.5
ρ [1/σ2 ]
ρ
Figure 11.29: Local lattice constants and density for a dipolar system with barrier atx=500σ as shown in figure11.27.
Instead of layer reductions we find an increase of layers at fixedx-positions.
11.3.2 Two Line Barriers
One can proceed a step further by adding two line barriers transverse to the direction of particle transport, which hinder the particle flow along the channel. Therewith one might build the mi-croscopic analogue of a so-calledsingle electron transistor(SET) [Kas93,Kas92,Kas00,Wee88, Bee91,Joh92,Mei96,Ash96]. These are nanoscale electrical devices where the electrons are con-fined to a small region (“island”) separated from the environment (transistor’s leads of source and drain) by quantum mechanical tunneling barriers.5 The energy, which is required to add a charge q through the tunneling barriers to the island is determined by the capacityC of the island with respect to the leads and is given byEC=q2/2C. By scaling down the size of the device a situa-tion arises, in which the thermal energykBT is comparable to or smaller than the charging energy ECB≡e2/2C, which is required to add a single electron onto the island. This energy barrierECB is often calledCoulomb blockade. Charge quantization (not less than one electron can be added) leads to an energy gap in the spectrum of states for tunneling.6 For an electron to tunnel onto the island, its energy must exceed the Fermi energy of the contact byECB, and for a hole to tunnel its energy must be below the Fermi energy by the same amount. Consequently the energy gap has the width 2ECB. If the temperatureT is low enough such thatkBT <ECBneither electrons nor holes can flow across the island from the “source” to the “drain”. The energy required to add one charge to the island can be altered by the magnitude of the gate voltageVg.7 In this case the conductance Goscillates as a function of the Fermi energy, which can be modified by the gate voltageVg, with periodicity e2/C [Mei96, Ash96, Kas00]. The single electron transistor turns on and off again every time a single electron is added to the isolated region in contrast to a usual transistor which only turns on when many electrons are added to it. This effect, although it is only reported for nanoscale devices, can be explained by purely classical physics.
barriers created by laser tweezers
tilt direction
Figure 11.30: Sketch of a magnetoblockade experiment: Two barriers defined by laser tweezers enclose a small region. A stable number of particles will occupy this region, pre-venting additional particles from entering the area. Only if a particle leaves this central region, a new particle can enter. The figure was created by A. Erbe [Erba].
5Such devices can be obtained either by employing material properties (a small metal region surrounded with an insulator) or by the use of electrical fields, which confine electrons to a small region within a semiconductor.
6The charging energy plays no role if the island (quantum dot) is strongly coupled to the reservoirs, but we discuss only the case of weak coupling. Strong or weak coupling is determined whether the broadening of the energy levels in the quantum dot is large or small compared to their spacing. Coulomb repulsion is important if the conductance obeysG.e2/h[Bee91,Mei96].
7Only a small voltage bias is applied across the junction.
It is therefore suggesting to think of a model system on the micron length scale, in which the electric charge is simply replaced by the magnetization of the colloidal particles. The driving forceFext (or the inclinationαin the experiment) has the same role as the bias voltage across the SET, and the effect of the gate voltage is similar to the effect of the strength of the two barriers. A sketch of such a system is shown in figure11.30.
Figure 11.31: Channel snapshot of a channel with two barriers of repulsive optical traps of strength k˜=−6 at the positionsx=300σ andx=400σ. The barriers are marked by the red lines, and the simulation parameters are: Lx=800σ, Ly =10σ, n=0.4σ−2, B=0.25 mT, andΓ=133.4.
0 0.2 0.4 0.6 0.8 1
0 100 200 300 400 500 ρ [1/σ2 ]
x [σ]
(a) k = -6k = -4 k = -2
0.4 0.5 0.6 0.7
0 50 100 150 200 250 n[300, 400]σ [1/σ2 ]
time [τB] (b) k = -6k = -4 k = -2
Figure 11.32: (a) Local particle density along the channel in non-equilibrium steady state of a channel with two barriers atx=300σ andx=400σ for three trapping strengths ˜k.
(b)Time dependency of the average particle densitynin the intervalx∈[300,400]σ between the two barriers.
In figure 11.31we depict a typical snapshot of a channel with two transversal barriers made of repulsive optical traps of strength ˜k=−6 separated by ∆x=100σ. In front of both barriers we find an increase of the number of layers. An abrupt change in the number of layers, which depends on the barrier strength ˜k, can be observed at the barrier positions. Complementary, we show in figure11.32(a)the local density along the channel in the stationary non-equilibrium situation for three different barrier strengths ˜k. In front of the barrier the local density is large enough for layering transverse to the particle flow direction to occur. After the barrier a depletion region
can be identified in all cases. The number of particles trapped between the two barriers stays constant after about ∆t =100τB, where all three systems have reached the state of stationary non-equilibrium. This is clear from figure 11.32(b) where the time-dependency of the particle density for the particles withinx∈[300,400]σ is plotted. For timest<100τBthe number density increases continuously and it stays constant thereafter. For such a large island size no correlation is found between a particle leaving the island and another particle entering onto it.
The situation changes drastically, when we reduce the size of the island between the two barriers.
In the following we will discuss the phenomena found for a channel of widthLy=2.0σ and two line barriers made of parabolic traps which are located atx=700σ andx=704σ. The overall particle density is kept fixed ton=0.4σ−2which corresponds toN=640 particles. The strength of the repulsive barriers is varied in the interval ˜k= [−0.5,−6.0].
(d) (a)
(b)
(c)
1
2
1 2
1
Figure 11.33: Snapshots taken every∆t=0.0375τB (≡500 BD time steps) of the magnetoblock-ade effect for a channel of widthLy =2σ for a system with α =0.2◦. The traps, which are marked by the orange discs, have the strength ˜k=−4 and the interaction rangeRtrap=0.75σ.
In figure11.33we present consecutive snapshots for ˜k=−4 which are taken every∆t=0.0375τB (≡500 BD time steps). Between the two barriers, being marked by the orange discs, we see seven particles on average. Due to the channel inclination of α =0.2◦ a small constant driving force of strength ˜Fext =0.0304 is applied across the two barrier region. In front of the first barrier the particles accumulate. Every now and then a particle in front of the first barrier is pushed across the barrier and thus enters the island region. Such a particle is marked by the number 1 in figure11.33.
This additional particle brings the particles on the island into an “excited state” which lasts until another particle being marked by the number 2 leaves the island. This general behavior which is found can be described as follows. First, an additional particle brings the island region into an
“excited state” which shortly afterwards “decays” back to the metastable number of particles. Very seldom we observe that two particles leave the island and thus enable another particle to enter the island.
To get an impression of the dependency of the stationary non-equilibrium particle configuration we show in figure11.34superimposed snapshots for the trap strengths ˜k=−0.8,−2.8, and−5.6.
The positions and ranges of the repulsive parabolic traps which form the two barriers are marked
(c)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
660 665 670 675 680 685 690 695 700 705 710 715 720
ρ(x) [arb. units]
x [σ]
(a)
(b)
k=−5.6 k=−0.8
k=−2.8
Figure 11.34: Superimposed snapshots taken during 5·106BD time steps near the two transversal barriers for a selection of three barrier strengths ˜k. Additionally, we plot in subfigure (b)for ˜k=−2.8 a representative stationary non-equilibrium density profile along the channel near the two barriers located atx=700σ andx=704σ.
by the red circles. Without the barriers the particles are arranged in two layers. At low barrier strengths (cf. figure 11.34(a)) these two layers do not get perturbed by the two barriers. Only a slight re-organization of the particles after the second barrier takes place. For ˜k=−2.8 (cf.
figure11.34(b)) three layers have formed in front of the barrier. Now, the energy barrier is higher than the sum of kinetic and thermal energy. Therefore the particles accumulate in front of the barrier and from time to time a particle is pushed across the barrier. Additionally, we show in figure11.34(b)the density profile along the channel in the region near the two barriers, which is also representative for the other configurations. On the island,i.e.forx∈[700,704]σ, three sharp peaks form. After the second barrier (x>740σ) a small depletion zone followed by a small peak can be identified. For x>708σ most particle re-arrangements took place and the local density stays constant. Also, in front of the first barrier the density is constant up tox=675σ. At higher xvalues we observe particle layering in front of the first barrier which gives rise to oscillations of the local density. With increasing barrier strength the density increase in front of the first barrier becomes more pronounced and the particle fluctuations become smaller (cf. figure11.34(c)). After the second barrier the particles re-arrange over a longer distance∆x. The particles are now pushed
again through the centers of the traps, whereas for ˜k=−2.8 a significant number of particles avoids the trap centers.
0 50 100 150 200 250 300 350 400
0 1 2 3 4 5 6 7 8 9 10
j(x=704σ) [1/τB]
BD time step [106] k=-0.75
k=-1.0 k=-1.5
k=-2.0 k=-2.5 k=-3.0
k=-3.5 k=-4.0 k=-4.5
k=-5.0 k=-5.5 k=-6.0
Figure 11.35: Flow rate across the second barrier of a magnetoblockade as function of the simula-tion time for a selecsimula-tion of trap strengths ˜k.
After about 4·106 BD time steps all simulated systems with different barrier strengths ˜k have reached the stationary non-equilibrium where the particle flow across the barriers stays constant.
This can be seen in figure11.35, where we plotted the particle flux at the position x=704σ as a function of time. As expected the stationary flux jdecreases with increasing barrier strength ˜k.
For higher ˜k-values it takes longer until the first particles are pushed across the first barrier and thus induce a flow across the second barrier.
The result of a systematic analysis of the average number of particles hNIslanditrapped between the two barriers as a function of ˜k is shown in figure11.36(a)(red curve). Four different plateau regions can be identified with the average numberhNIslandi=4, 6, 8, and 9 respectively. With increasing trap stiffness plateaus form at even numbers of particles trapped between the two barri-ers. Only for very high repulsive trap strengths this situation changes. Additionally, we plot in the same figure (blue curve) the corresponding average stationary particle flux. Therefore, we counted the number of particles crossing the second barrier atx=704σover 5·105BD steps and averaged over several time windows. Clearly, the curve forhji(k)˜ shows deviations from a pure linear de-crease (cf. green curve) with increasing trap strength ˜k. In figure11.36(b)we plot the difference of the data points from the linear decrease, which is given by the linear fit to the simulation results in the interval ˜k∈[−6,−4]. From the ˜k-values of the peak positions in figure 11.36(b)we can conclude, that the average flux is higher when a new particle is added to the island between the two barriers. The particle flux is smaller within the plateau regions ofhNIslandi. This situation is very
3 4 5 6 7 8 9 10
-6 -5 -4 -3 -2 -1 0
< NIsland >
trap strength k [red. units]
(a)
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
< j >(x=704σ) [1/τB]
-0.2 -0.1 0 0.1
-6 -5 -4 -3 -2 -1 0
(b)
Figure 11.36: (a)Average particle numberNIsland between the two barriers as a function of the trap strength ˜k. Corresponding particle fluxhjiat the position of the second barrier atx=704σ. The averages are evaluated for 5·106BD time steps. The green line is a linear fit tohjifor ˜k∈[−6,−4]. (b)Shows the deviation of the average particle fluxhjifrom the linear fit.
similar to the situation of single-electron transistors, where conductance oscillations are related to the addition of a single electron to the quantum dot.
similar to the situation of single-electron transistors, where conductance oscillations are related to the addition of a single electron to the quantum dot.