Colloidal Transport in Micro-Channels
11.2 Influence of Boundary Conditions
Now, we address alternative realizations of the boundary conditions and their influence on the transport behavior along the channel.
Figure 11.20: Contour plots of the potential energy surface in the presence of fixed dipolar point particles on the wall separated by∆x=4σ for the two channel widthsLy=5σand Ly=3σ. The particles on both edges have the same periodicity(a)with no offset and(b)with an offset of half the periodic length.
As alternative to ideal hard channel walls we kept the boundary particles fixed after the end of the equilibration process,i.e. their velocity was set to zero for the rest of the simulation run.2 These fixed edge particles act like a periodic wall potential which significantly influences the particle movement inside the channel. In figure 11.20 we show contour plots of the potential energy surface in the presence of fixed dipolar point particles along the wall for the two channel widths Ly=5σ andLy=3σ. The particles are separated by the distance∆x=4σ. Due to this potential energy landscape the particles no longer move with a uniform (x-position dependent) drift velocity as for channels with ideal hard walls. The particles belonging to the inner layers move faster than the particles of the layers next to the channel walls. This gives rise to shear effects between the different layers which becomes stronger for decreasing channel widthsLy.
The relative particle positions of four example particles, which are colored in green, of a driven dipolar system in a channel of widthLy=8σare shown in figure11.21(a), (b)for two instances of time, which are separated by∆t=24τB. In figure11.21(c)we show the results of Gaussian fits to the histograms of the drift velocity for the four layers of moving particles (which are color coded blue and green). The two peaks centered about hvdrifti ≈0.149µm/s belong to the two central particle layers, whereas the particles next to the layers of fixed wall particles move with the lower average drift velocityhvdrifti ≈0.142µm/s. The particles in the outer layers move slower, since they are trapped in the potential minima of the fixed edge particles most of the time. They can only overcome the potential barrier in a collective movement behavior. Whenever a neighbor particle
2The boundary particles can be kept fixed after the equilibration process by defining the compiler options FIX PARTICLEandFIX BOUNDARY PARTICLES. Alternatively, one can just set the compiler optionFIX PARTICLE and define the positions of all fixed particles in the parameter fileparameter.ini.
(a) (b)
0 5 10 15 20 25 30 35
0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18
P(vdrift) [a.u.]
vdrift [µm/s]
(c)
y = [0.5:2.5]
y = [2.5:4]
y = [4:5.5]
y = [5.5:7.5]
Figure 11.21: (a)and(b): Snapshots of a driven dipolar system with fixed boundary particles at two different times separated by∆t=24τB(≈3.2·105BD time steps). The particles in green are marked to highlight the shearing between the two inner particle layers and the outer layers. Both figures are relative to the green particle positions.(c) His-tograms and Gaussian fits to the hisHis-tograms of the drift velocity of the four different moving layers for a system with fixed edge particles. The system parameters are:
Lx=800σ,Ly=8σ,n=0.4σ−2,Γ=341.6,α =0.5◦.
in direction of the external driving force has left a potential minimum the subsequent particle can enter into this minimum and otherwise it is hindered. For the central layers this effect is less pronounced due to the diffusion of particles in the neighboring layers. Therefore, they move at a higher average drift velocity. Also, the drift velocity of these central particle layers shows smaller fluctuations compared to the particles in the layers next to the edge, as can be seen from the widths of the Gaussian distributions in figure11.21(c).
Entropic Barriers
x y
F
Figure 11.22: Schematic image of a microchannel setup with periodic entropic channel walls.
The effect of the fixed edge particles on the particle transport behavior is related to the case of periodic entropic barriers,i.e. microchannels with hard walls having a periodic boundary shape
as sketched in figure 11.22. Recently, such systems have been discussed in the following pa-pers [Reg06,Laa07,Kal05, Kal06,Yar07,Bur07a,Ai07a,Ai07c, Ai07b,Ai06]. The dynamics of these systems in presence of an external driving forceFacting along the channel can be recast by means of the so-calledFick-Jacobs(FJ)equation. The standard protocol of solving the Smolu-chowski equation with the appropriate boundary conditions imposed typically becomes a very difficult task in presence of uneven boundaries. The trick of a coarsened description by reduction of the dimensionality of the system opens up a way to circumvent this problem of the solution of a challenging boundary problem. Only the main direction of transport is considered and the irregu-lar nature of the boundaries is taken into account by means of an entropic potential. The resulting Fick-Jacobs equation is a kinetic equation for the time evolution of the one-dimensional probabil-ity distributionP(x,t)along the channel. It is similar in form to the Smoluchowski equation, but contains an entropic term. Explicitly, it reads [Reg06,Bur07a]
∂P with the effectivex-dependent diffusion coefficient
D(x) = D0
and the free energy A(x) =V−TS =−Fx−TkBlnh(x) consisting of the energy contribution V =−Fxand the entropy contributionS=kBTlnh(x)due to the periodic channel shape of peri-odicityLand the dimensionless widthh(x)≡2ω(x)/L. The reduction of the dimensionality of the system is based on the assumption, that the particle distribution in transverse direction equilibrates much faster than that in the main (unconstrained) direction of transport. The entropic contribution leads to genuine different particle dynamics than observed for transport through energy barriers as it is reported by Regueraet al.[Reg06]. These authors found that the particle current and the effective diffusion coefficient are controlled by a single parameter
f≡ FL
kBT (11.4)
that measures the relative importance of the external work done to the particle and the thermal energy. The scaling in the parameter f makes it possible to tune and control the efficiency of transport by a proper combination of the temperature and the applied driving field.
11.2.2 Influence of the Boundary Conditions in Flow Direction
The connection of the channel to the two reservoirs has great influence on the characteristics of the stationary non-equilibrium density profile along the channel. Therefore, we performed simulations with the boundary condition 2 in flow direction as being introduced in subsection9.4.3. The con-stant external driving force only acts within the intervalx∈[100,700]σ and a periodic boundary condition is applied in x-direction. Figure11.23shows the resulting stationary non-equilibrium density profiles after 3·106 BD time steps for a selection of inclinationsα of a dipolar system.
For each inclination two curves are plotted which correspond to the two channel widthsLy=8σ andLy=10σ. Obviously, the steady-state density profile along the channel does not depend on the channel width.
0.2 0.3 0.4 0.5 0.6 0.7
0 100 200 300 400 500 600 700 800 100 ρ(x) [σ-2 ]
x [σ]
α = 0.04o α = 0.08o α = 0.1o α = 0.2o
Figure 11.23: Density profiles for a selection of inclinations α of a system with the inclination (i.e. the driving force) applied only within the regionx∈[100,700]σ. The system is periodic in x-direction. Shown are histograms obtained by evaluation of 1000 configurations of the system having reached a stationary non-equilibrium situation (after≈2·106BD time steps). The applied magnetic field strength isB=0.25 mT, Γ=133.4, and the overall particle densityn=0.4σ−2. For better clarity, we repli-cate the intervalx∈[0,100]σ again on the right hand side of the diagram.
Comparison of these density profiles with those of figure11.3highlights the strong influence of the different realization of the reservoirs. All simulations are started from a homogeneous particle distribution of local densityρ =0.4σ−2. Instead of a density decrease we find in figure11.23a buildup of the local density occurring due to the filling of the reservoir at the channel end. This corresponds to the experimental situation, where the reservoir at the channel end is filled. For the small inclination α =0.04◦ a linearly increasing density profile is obtained within the channel region. Higher inclinations lead to deviation from such a linear profile. For α=0.2◦a constant profile with local densityρ≈0.275σ−2inx∈[100,400]σ is followed by a sharp increase of the local density up toρ=0.67σ−2at the channel end atx=700σ.
In the stationary non-equilibrium state the density profile in the reservoirs can be approximated by a linear gradient. The net fluxJin the reservoirs fulfills Fick’s law
J=kBT
2l0 (ρ1−ρ0) (11.5)
where ρ0≡ρ(x=100σ) andρ1≡ρ(x=700σ) are the local number densities at the channel beginning and end respectively. Due to the periodic boundary condition inx-direction this is equal to the net flux in the channel regionx∈[100,700)σ. Therefore,J may be approximated by the slope of the linear density profiles in the two reservoir regions.
Figure11.24shows the layer order parametersΨlayer,nl for a selection of inclinationsα in combi-nation with the corresponding superimposed configurations. Clearly, the layer configuration and the number of layer transitions can be tuned by the strength of the driving force for the realization of the boundary condition 2 of the flow. Increasingα leads to multiple transitions. Interestingly, the layer transitions from 7 to 8 layers occur at identicalx-positions in the figures11.24(b)–(d). As before, the particle flow across the position of the layer transition, which remains fixed in position.
0 0.2 0.4 0.6 0.8 1
0 100 200 300 400 500 600 700 800
Ψlayer, nl
(a)
nl = 9 nl = 8 nl = 7 nl = 6
0 0.2 0.4 0.6 0.8 1
0 100 200 300 400 500 600 700 800
Ψlayer, nl
(b)
0 0.2 0.4 0.6 0.8 1
0 100 200 300 400 500 600 700 800
Ψlayer, nl
(c)
0 0.2 0.4 0.6 0.8 1
0 100 200 300 400 500 600 700 800
Ψlayer, nl
x [σ] (d)
Figure 11.24: Stationary non-equilibrium situations of systems where the driving force is applied only inx∈[0,700]σ for a selection of inclinations: (a)α =0.04◦,(b)α=0.08◦, (c)α=0.1◦,(d)α=0.2◦. For every inclination we show the average local layer or-der parametersΨlayer,nl and the corresponding superpositon of 1000 snapshots. The other simulation parameters are:Lx=800σ,Ly=10σ,n=0.4σ−2,B=0.25 mT, andΓ=133.4.
Systems with Screened Coulomb Interaction
Figure 11.25: Superimposed configurations of systems with screened Coulomb pair interaction for a selection of inverse screening lengths: (a) κD =2σ−1, (b) κD =4σ−1, (c) κD=8σ−1, and (d) κD =12σ−1. The particle transport is induced for x∈ [0,700]σ by the inclinationα=0.1◦.
We discussed the equilibrium density profiles transverse to the walls of a YHC system with βV0=50 and the selection of Debye screening lengths κD= (2,4,8,12)σ−1 in the context of figure 10.4. Now, we plot in figure 11.25 the superposition of 100 configurations with a time separation of ∆t=500 BD steps after 1.4·106 BD steps for the case of boundary condition 2 in flow direction. The driving force corresponding to an inclination of α =0.1◦ acts within x∈[100,700]σ. All four superimposed configurations show the formation of layers near the channel end at x=700σ, where the particles enter the reservoir. In figure 11.25(a) the charac-teristic interaction range of the YHC pair-potential is greater than the average particle spacingR.
For this case we find multiple layer transitions from 5 layers up to 8 layers along the channel. The system behavior is similar to the situation of the dipolar systems. With increasing values of κD less layer transitions are observed. Figures11.25(b)–(d) show increasing depletion zones at the channel start atx=100σ. These depletion zones are followed by regions where the particles are
in the liquid state. Notice, that forκD=8σ−1andκD=12σ−1the systems are in the liquid state in equilibrium, too (cf. figure11.25). The corresponding density profiles inx-direction are given in figure11.25.
0 0.2 0.4 0.6 0.8