Colloidal Particles close to a Wall: Dy- namics and Interaction Potentials
P. Holmqvist, D. Kleshchanok, P. R. Lang
Institut f ¨ur Festk ¨orperforschung, Weiche Materie
The influence of the hydrodynamic drag effect on the near wall dynamics of colloidal particles is studied by evanescent wave dynamic light scat- tering (EWDLS). Spherical particles as small as 27 nm in radius are shown to follow theoretical predictions for for the creeping flow limit with stick boundary conditions. The presence a sec- ond component to the colloid suspension causes an additional slowing down of the near wall dy- namics, which can not bee explained by the stan- dard mean field approach.
If a particle suspended in a quiescent solvent ap- proaches a hard wall the so called viscous drag effect causes the particles mobility to decrease and to be- come ansiotropic. The theoretical description of this effect has a history of almost hundred years [1] and was cast into it’s widely accepted final form in the six- ties of the last century by Brenner and coworkers [2].
These authors showed that the diffusion coefficient of the particle close to the wall is related to the bulk diffusion coefficient ,D0, by the direction dependent hydrodynamic friction terms, λkand λ⊥. The latter are complicated functions of the particle radius, R, and the particle–wall separation distance,z. Differ- ently, from the theoretical achievements, experimen- tal investigations of this effect are rather new. For macroscopic, i. e. non–Brownian particles, the the- oretical predictions have been proven valid in sedi- mentation measurements of large spheres in highly viscous liquids [3]. With modern microscopic and op- tical trapping techniques it was shown that colloidal particles with a radius,R, larger than about 100 nm also behave as expected [4, 5].
rj
zj
ζ=2/κ
FIG. 1:Upon total reflection of a laser beam an evanescent wave is created, which has an exponentially decaying elec- tric field strength with a penetration depthκ/2. The evanes- cent wave is used as the radiation source for a dynamic scattering experiment in EWDLS.
We have recently improved the experimental and the- oretical basis of evanescent wave dynamic light scat- tering (EWDLS) [6, 7], in a way that allows us to study the near wall dynamics of particles as small as ca. 25 nm in radius. In this method the evanescent wave, which is created upon total reflection of a laser beam is used as a the incident radiation for a dynamic light scattering experiment (see sketch in Figure 1.
With our novel instrumental design it is possible to vary the normal, Q⊥, and the parallel, Qk, compo- nent, of the scattering vector (that is the difference between the wave vector of the scattered wave and the evanescent wave,Q=kS−ke) independently.
By this means it becomes possible to distinguish be- tween the particle mobility parallel and normal to the interface. For a thorough data analysis, it was fur- ther necessary to derive a theoretical expression for the initial decay rate;Γ, of the time auto correlation function of the scattered field,g1(t), which takes into account hydrodynamic and static interaction between the colloidal particles and the wall. Combining the new set-up and the expression forΓit is possible to determine normalized mean particle diffusivities par- allel and normal to the wall
hDk,⊥i D0
= (1)
R
z>Rdzexp{−βφ(z)}exp{−(z−R)κ/2}λ−1k,⊥(z) R
z>Rdzexp{−βφ(z)}exp{−(z−R)κ/2}
Hereβφ(z)is the static particle–wall interaction po-
tential in units of the thermal energy. The integra- tions are performed inherently in the experiments, i. e. hDk,⊥iare the mean values of the diffusivities, which are averaged over the entirez–range that is illuminated by the evanescent wave.
5 10 15 20
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
<D||,⊥⊥⊥⊥>/D0
ζ
FIG. 2: Normalized mean diffusivities vs. scaled penetra- tion depth. Different colors refer to the diffusivity parallel (black) and normal (red) to the wall. Symbols are experi- mental data from spherical particles withR = 85nm (full symbols) andR= 27nm (open symbols) in pure salt solu- tion (squares) and in the presence of fd-virus (bullets). Full lines represent theoretical predictions for different contact values of the depletion potential according to eq.2: full line:
α= 0, dashed lineα= 2kBTand dotted lineα= 5kBT
In Figure 2 we show experimental data ofhDk,⊥i/D0
vs. the scaled penetration depthξ = κ/(2R)from dilute solutions of two colloidal samples with different radii in a solution of 10mM additional electrolyte. Un- der these conditions the particles may be regarded as hard spheres. Accordingly the experimental data coincide well with the theoretical curves according to eq. 1 withβφ(z) = 0forz > R(see square symbols and full lines in Figure 2).
0 200 400 600
-4 -3 -2 -1 0 1 2 3 4
φ(z)/kBT
z / nm
FIG. 3: Interaction potentials between a wall and a Polystyrene sphere withR= 2.7µm. Symbols are exper- imental data and full lines are parameter free calculations according to eqs. 2 and 3. Different colors represent the electrostatic (black) and the depletion (red) contribution to the total potential.
If an additional polymer or colloidal component is added to such a solution, this leads to an effective attractive potential between the wall an the particles, the so–called depletion interaction. In the present case we studied the influence of the depletion inter- action induced by rod–shaped colloids (fd-viruses) on the near wall dynamics of colloidal spheres. The de-
pletion interaction induced by a rod of lengthLhas the form
βφdepl(z) =−α
“ 1− z
L
”3
(2) in the Derjaguin–approximation, i. e.if the radius of the spherical particle is much larger than the rod length. Here the contact value α is an increasing function of the rod number density, the rod length and the particle radius.
Using Total Internal Reflection Microscopy (TIRM) [8]
we are able to measure directly the interaction po- tential between a single sphere and a wall [9]. In Figure 3 we show experimentally determined con- tributions to the total interaction potential between a charge stabilized sphere with radiusR= 2.7µm and a like charged wall. The electrostatic contribution can be well described in the Debye–H ¨uckel approxima- tion
βφDH(z) =Bexp{−κDHz} (3) where the amplitude,B, is related to the charge den- sity on the particle and the wall surface andκ−1DH is the Debye screening length, which can by tuned by the addition of electrolyte to the colloid solution. The experimental data for the depletion interaction fit well with the expression of eq. 2 whereαwas calculated without any adjustable parameter.
The influence of the depletion potential induced by fd- virus on the near wall dynamics of colloidal spheres with R = 85nm is displayed also in Figure 2. As compared to the fd–free case the mean diffusivities decrease significantly and the anisotropy of the mo- bility increases. A comparison of the experimen- tal data with the theoretical predictions according to eq. 1 however shows that the theory grossly underes- timates the additional slowing down especially at low penetration depths and for the case of the diffusivity normal to the interface.
This implies that it is not possible to describe the in- fluence of the a second co–solute on the near–wall dynamics of colloidal spheres by an effective mean field depletion potential potential. We conjecture that the rods have to be treated as a second component which couples hydrodynamically to the mobility of the spheres. We plan to do Brownian Dynamics com- puter simulations in the future to clarify this point.
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[6] P. Holmqvist, J. K. G. Dhont, and P. R. Lang, Phys. Rev. E 74, 021402 (2006). DOI:
10.1103/PhysRevE.74.021402
[7] P. Holmqvist, J. K. G. Dhont, and P. R. Lang, J.
Chem. Phys. accepted for publication Decem- ber 2006.
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[9] D. Kleshchanok, R. Tuinier, and P. R.
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