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Electrohydrodynamic Coupling

The hydrodynamic measurements described in the previous section were carried out under conditions for which electrostatic interactions could be ignored. By contrast, hydrodynamic interactions may well have influenced the interaction measurements described in Sec. 2. Squires and Brenner [56] pointed out that a charged wall repels a nearby charged sphere with a force proportional to the wall's surface charge density, $ \sigma_W$:

$\displaystyle F^{(1)}_{z}(h) = 4 \pi \, \frac{Ze\sigma_W}{\epsilon} \, \frac{\exp(\kappa a)}{1 + \kappa a} \, \exp(-\kappa h).$ (44)

Now consider two spheres released from optical tweezers at height $ h$ above the wall with an initial center-to-center separation $ r$. The force, $ F^{(1)}_{2z}(h)$, on sphere 2 establishes a flow which advects sphere 1 according to Faxén's first law [58]. The advection velocity's in-plane component,

$\displaystyle v_{1x}(r,h) = b^{(1)}_{1x,2z}(r,h) \, F^{(1)}_{2z}(h),$ (45)

draws the sphere toward its neighbor, a like-charge attraction of electrohydrodynamic origin. The relevant component,

$\displaystyle b^{(1)}_{1x,2z}(r,h) \approx \frac{3}{32 \pi \eta h} \, \frac{\xi^2}{(1+\xi^2)^\frac{5}{2}},$ (46)

of the mobility tensor arises from the sphere's hydrodynamic interaction with its neighbor's image in the wall. The corresponding component of the force on sphere 1,

$\displaystyle f_{1x}(r,h) = b^{(1)}_{1x,2z}(r,h) \, F^{(1)}_{2z}(h) / b^{(1)}_{1x,1x}(h),$ (47)

is purely kinematic rather than thermodynamic [56]. Nevertheless, it could be mistaken for an equilibrium attraction in optical tweezer measurements such as those described in Sec. 2.

Using the wall-corrected in-plane mobility $ b^{(1)}_{1x,1x} (h) \approx b_0 [1 - 9a/(16h)]$ (Sec. 3.3), Squires and Brenner [56] found that Eqs. (9) and (47) together account for Larsen and Grier's optical tweezer measurement of like-charge attractions near a single charged wall [27]. The polystyrene sulfate spheres in those measurements had a radius of $ a = 0.475~\ensuremath{\mathrm{\mu m}}\xspace $ and an effective charge number of $ Z = 7300$ established by interaction measurements far from walls [27]. When the same spheres were released at $ h = 2.5~\ensuremath{\mathrm{\mu m}}\xspace $ above the nearest glass wall, their measured interaction developed a minimum $ 0.5~k_BT$ deep at a separation of $ r \approx 3~\ensuremath{\mathrm{\mu m}}\xspace $. The Squires-Brenner theory reproduces these features for $ \sigma_W = 2200 \, e~\ensuremath{\mathrm{\mu m}}\xspace ^{-2}$, a value consistent with silanol dissociation in pure water.

Squires and Brenner's success at explaining Larsen and Grier's optical tweezer measurements casts the problem of like-charge colloidal attractions in a new light. Similar electrohydrodynamic coupling might explain optical tweezer measurements of attractions measured between two parallel walls [25] if the spheres were not released precisely at the midplane. However, electrohydrodynamic coupling cannot have influenced the two independent equilibrium measurements of confined colloids' interactions [23,24] which revealed attractions comparable to those measured with optical tweezers [25,27]. If charged colloids' electrostatic interactions are purely repulsive, as suggested by mean-field theory and assumed by Squires and Brenner, attractions measured with optical tweezers might be ascribed to hydrodynamics while those observed by equilibrium structural measurements would have to be in error.

The sequence of measurements in Fig. 2.4 suggests instead that the success of the Squires-Brenner theory for Larsen and Grier's measurement near one wall does not rule out an equilibrium origin for confinement-induced like-charge attractions under slightly different conditions. This interaction measurement was performed a bit further than $ h = 3~\ensuremath{\mathrm{\mu m}}\xspace $ away from two charged walls [2] and shows no sign of a long-range attraction, kinematic or otherwise. If charged walls do induce equilibrium attractions between nearby spheres, Larsen and Grier's measurements may have been performed too far from the wall for these to have been evident. In this case, the electrohydrodynamic mechanism may explain the data in Ref. [27] while leaving all of the other anomalous observations unresolved.

There would be little room for alternative interpretations if the available equilibrium interaction measurements on confined monolayers [23,24] were known to be free of artifacts. However, such measurements are subject to subtle sources of error, some of which could mimic long-ranged attractions [44]. To explore this possibility, we measured the equilibrium pair interactions in a monolayer of silica spheres near a single charged planar surface.


next up previous
Next: Electrostatics Near One Wall Up: Interactions in Colloidal Suspensions: Previous: Pair diffusion near a
David G. Grier 2001-01-16