Measuring a Colloidal Particle's Interaction With a Flat Surface Under Nonequilibrium Conditions

Sven H. Behrens
Present address: Department of Polymer Physics, BASF Aktiengesellschaft, 67056 Ludwigshafen, Germany
Joseph Plewa
David G. Grier
Dept. of Physics, James Franck Institute, and Institute for Biophysical Dynamics, The University of Chicago, Chicago, IL 60637

A new and general approach is proposed to analyze the dynamics of a colloidal particle interacting with a nearby wall. This analysis can be used to determine the acting forces even when the system is non-stationary. As an illustration, we use total internal reflection microscopy to investigate the forces acting on a polystyrene sulfate latex particle as it is receding from a charged glass surface.

§ I. Introduction

Solution-mediated forces between colloidal surfaces govern the structure and phase behavior of virtually all colloidal systems; in the case of unstable dispersions they also determine the kinetics of aggregation and deposition (1). The most common methods to measure such forces employ the surface force apparatus (SFA) (2), the atomic force microscope (AFM) (3), or total internal reflection microscopy (TIRM)(4). These techniques are complementary with regard to their force resolution: while the SFA measures the large forces acting between macroscopic surfaces, the AFM is suitable for measurements of forces down to a few pN (5). The typical forces acting on a Brownian particle can be smaller still (10^{{-14}}-10^{{-11}} N), and are best resolved by TIRM (4); (6). This sensitive technique uses evanescent wave light scattering by a single particle near a flat surface to determine the equilibrium distribution of particle-surface separations and the associated interaction energy.

Standard TIRM probes the forces that keep a heavy Brownian particle levitated above a glass substrate. In equilibrium, the particle elevations are distributed according to Boltzmann statistics; histograms of the height can thus be used for mapping the particle's potential energy profile. A drawback of this method is that it provides disproportionately poor sampling of the energetically unfavorable particle positions. In order to ensure good sampling throughout the range of investigated particle-substrate separations, several strategies have been developed to vary the equilibrium particle height; these include the use of radiation pressure (7); (8); (9), optical trapping (10); (11), and electrophoresis (12). All of these techniques require further calibration or rely on theoretical assumptions regarding the applied external force on the particle and local changes in the particle's environment.

Alternatively, we propose to “turn TIRM upside down”, i.e., to create an evanescent wave at the top surface of a sample cell and study the repeated sedimentation of a heavy colloidal sphere released from a trap near the surface. Every time the particle “drops down” from the surface, it probes all the particle-surface separations that TIRM can resolve, without the need for externally applied forces.

These advantages of the proposed experimental geometry come at the price of a more detailed analysis of particle trajectories than is commonly necessary; a sedimenting particle is obviously not in equilibrium, and forces acting on the particle can no longer be inferred from a static distribution of its positions but only from the particle dynamics. Moreover, one has to account for the hydrodynamic coupling of the particle to the surface, which results in a position-dependent particle mobility and diffusion constant. The benefits of this more involved analysis include an absolute calibration of the particle's height relative to the surface and an accurate determination of the particle's size.

In this paper, we discuss two methods of evaluating particle-surface forces that are amenable to equilibrium as well as nonequilibrium situations like the one described above. One of these approaches uses an asymptotically exact analytical expression for the propagation of particle positions at short observation intervals. This description resolves a long-standing quandary regarding the interpretation of Brownian dynamics near a bounding surface as witnessed in TIRM experiments. In a different approach, we propose to reconstruct equilibrium particle distributions from transition probabilities measured in nonequilibrium situations. Both procedures are used to extract the force on a polystyrene sulfate particle receding from a charged glass wall.

§ II. Theory

In equilibrium, the elevation z of a Brownian particle with potential energy U(z) has a distribution \rho^{{\rm eq}}(z), given by

\rho^{{\rm eq}}(z)=\rho^{{\rm eq}}(z_{0})\exp[-U(z)/k_{B}T] (1)

where z_{0} is a reference position corresponding to zero energy and k_{B}T is the thermal energy. Equation (1) then makes measurements of the energy profile U(z) a matter of experimentally finding this equilibrium distribution \rho^{{\rm eq}}(z).

§ Short Time Dynamics

Nonequilibrium conditions, on the other hand, warrant a description of the particle dynamics and are usually treated in the Langevin or Fokker-Planck formalism (13). Different Langevin equations have been proposed to describe the overdamped motion of a Brownian particle with position-dependent friction (14); (15). While the description in terms of stochastic equations of motion is by no means unique, the corresponding Fokker-Planck equations reduce, in the Smoluchowski limit, to the well-known form (for one dimension)

\frac{\partial\rho(z,t)}{\partial t}=L_{S}\:\rho(z,t), (2)

where \rho is the position distribution after adiabatic elimination of the momentum variable, and the Liouville operator L_{S} is given by

\displaystyle L_{S} \displaystyle= \displaystyle\frac{\partial}{\partial z}\:\frac{1}{\gamma(z)}\left[U^{{\prime}}(z)+k_{B}T\frac{\partial}{\partial z}\right] (3)
\displaystyle= \displaystyle-\frac{\partial}{\partial z}v(z)+\frac{\partial^{2}}{\partial z^{2}}D(z)

with a diffusion term

D(z)=\frac{k_{B}T}{\gamma(z)} (4)

and a drift term

v(z)=-\frac{U^{{\prime}}(z)}{\gamma(z)}+D^{{\prime}}(z). (5)

Here \gamma(z) is the position-dependent friction coefficient, and the primes denote derivatives. Note especially the second term on the right hand side of equation (5), the so-called “spurious” or “noise-induced” drift (13); this term is zero in the more familiar case of constant friction and has not been considered in previous treatments of bounded particle dynamics (16); (17). Experimental evidence of noise-induced drift has recently been reported for a system without any driving force (18). Neglecting the extra noise term in a dynamic analysis of TIRM measurements leads to inconsistencies that have been acknowledged before (17), and we believe that our approach is the first to explicitly account for the complete particle drift.

While the general solution \rho(z,t) to equation (2) is not known, its time evolution can be written as (13)

\rho(z+\Delta,t+\tau)=\int P(z+\Delta,t+\tau|z,t)\rho(z,t)dz (6)

where the propagator P(z+\Delta,t+\tau|z,t) (in the Smoluchowski approximation) does not depend on t but only on the instantaneous position z, on the propagation time \tau, and on the associated step length \Delta. It is given formally by

P(z+\Delta,t+\tau|z,0)=\exp(\tau L_{S})\:\delta(\Delta), (7)

where \delta(\cdot) is Dirac's \delta-distribution.

For short times (up to quadratic order in \tau) the solution to equation (2) is known explicitly (13):

P(z+\Delta,\tau|z,0)=[4\pi D(z)\tau]^{{-1/2}}\exp\left(-\frac{\left[\Delta-v(z)\tau\right]^{2}}{4D(z)\tau}\right), (8)

which is formally identical to the solution for constant friction but contains the full drift term, equation (5).

The above result is immediately useful for comparison with measurements. The propagator on the left hand side of equation (8) is the experimentally available probability of observing a particle displacement \Delta in the propagation time \tau given the original position z. Equation (8) states that (for small \tau) this distribution of displacements is a Gaussian of variance 2D(z)\tau whose mean drifts downward at speed v(z).

For a sphere moving perpendicularly to a flat surface, the hydrodynamic corrections to the bulk diffusion constant have been calculated exactly and expressed in terms of an infinite series (19). The result for D(z) can be well approximated at all separations by the expression (17)

D(z)\approx\frac{k_{B}T}{6\pi\eta a}\:\frac{6z^{2}+2az}{6z^{2}+9az+2a^{2}}\:. (9)

Here, \eta is the dynamic viscosity of the solution, a is the sphere's radius, and z stands for the surface-to-surface separation between the sphere and the wall. The separation-dependent correction to the classical Stokes formula was obtained by a regression of the exact results following a method introduced by Honig for the two sphere geometry (20). Since D(z) is monotonic, it can be inverted (numerically) to yield the absolute sphere-wall separation z as a function of the (measured) diffusion constant D. This method provides a new solution to the traditionally difficult task of assigning absolute separations to the intensities measured in TIRM. Like a previously proposed hydrodynamic evaluation (17), it requires rather accurate knowledge of the particle size. Once the absolute separations are found, \gamma(z) and D^{{\prime}}(z) can be computed from equation (9), and the measured total drift yields the force -U^{{\prime}}(z) on the particle via equation (5). On the other hand, if this force were known independently, equation (8) could be used to assess the hydrodynamic interaction between the particle and the wall. In either case, the propagation time \tau has to be chosen very large compared to the Brownian relaxation time, yet small enough to ensure that both the drift and the diffusion of the particle are essentially constant within \tau. The latter implies that typical displacements \Delta must be small compared to the length scale over which v(z) and D(z) vary appreciably.

§ Markovian Dynamics Extrapolation

The potential energy of the particle and hence the net conservative force acting on it can also be extracted from the measured propagator P(z+\Delta,\tau|z,0) in a different way that does not require a hydrodynamic model and is not restricted to short delay times. This method, which was first proposed by Crocker and Grier in the context of particle-particle interactions (21); (22), is based on equation (6). By construction, this equation holds very generally, for equilibrium as well as for nonequilibrium situations. Since the Smoluchowski propagator P is assumed to be history independent, it has to be the same for all distributions \rho(z,t), whether they describe a transient or a stationary state. If there are no further stationary states for the system than the equilibrium distribution \rho^{{\rm eq}}, then \rho^{{\rm eq}} can be obtained – even under nonequilibrium conditions – as the stationary solution of equation (6) with the measured P(z+\Delta,\tau|z,0). In other words, equation (6) is interpreted as an eigenvalue equation for \rho^{{\rm eq}}; once it is solved, the potential profile U(z) immediately follows from equation (1). This procedure, referred to as Markovian Dynamics Extrapolation (MDE) in the following, is in fact applicable in a more straightforward way to TIRM data than it is to the analysis of sphere-sphere interactions it was originally devised for, because the stochastic process described by the sphere-wall separation is only one-dimensional.

§ III. The Experiment

To demonstrate the use of the proposed methods, we have studied the interaction of polystyrene sulfate spheres (8 \mathrm{\upmu}\mathrm{m}, IDC Portland, Inc.) with a glass wall in KCl solutions of different ionic strength. The experimental setup is sketched in Fig. 1 and has been described in more detail elsewhere (23).

Figure 1. The experimental setup

A green laser beam (\lambda=532 nm CW, Nd:YAG) is reflected off the top wall of our sample cell at an angle \Theta greater than the critical angle of total reflection, thus creating an evanescent wave in the solution. The intensity of light scattered by a spherical particle in the evanescent wave depends exponentially on the particle's separation z from the interface (24),

I(z)=I(0)\exp(-\alpha z) (10)

with a decay length \alpha^{{-1}} given by

\alpha=4\pi\lambda^{{-1}}\sqrt{(n_{{\rm solvent}}\sin\Theta)^{2}-n_{{\rm glass}}^{2}}\:, (11)

where n_{{\rm glass}}, n_{{\rm solvent}} are the refractive indices of the two media. The exact angle of reflection is measured at the end of each experiment; typical values of 65-66^{0} correspond to a decay length around 110 nm and allow for good spatial resolution of particle elevations. The particle under consideration is positioned near the interface with an optical tweezer (25) consisting of an infrared diode laser beam (780 nm, 50 mW) that is focussed by the 100\times, 1.4 NA oil immersion objective of an inverted microscope (Olympus IMT-2). Once the particle is in position, the tweezer light is switched off via a computer-controlled shutter, and the particle is allowed to fall freely. While the released particle follows its Brownian path and slowly sinks away from the interface, the light it scatters off the evanescent field is collected from below by the objective lens and monitored simultaneously by a photodiode and a CCD camera. The photodiode signal is sent to a current preamplifier and then sampled by a 12 bit data acquisition board (National Instruments) at a rate of 100 Hz and saved on a computer. After 5 seconds, when the particle has typically dropped so far that the photodiode signal has reached the background level, data recording is interrupted, and the tweezer shutter is opened in order to reposition the particle. In each run 400-800 such “drop-and-reposition” cycles were recorded. Fig. 2 shows the intensities recorded during nine such cycles. The larger initial intensities of each cycle correspond to a small particle-wall separation at the beginning of the sedimentation event. Their variation is due to the small position fluctuations of the particle in the trap and a finite delay time between the tweezer shutoff and the beginning of data acquisition. Note also that small variations in the particle elevation translate into large variations of the scattered intensity when the particle is close to the interface. The signal diminishes as the particle moves away from the surface and eventually reaches the background level seen at the end of a typical sedimentation cycle (Fig. 2).

Figure 2. The TIRM signal of a particle receding from the glass-solvent interface

§ IV. Data Evaluation and Results

We write the signal intensity as

I_{{\rm s}}(t)=I_{{\rm ref}}\exp[-\alpha(z(t)-z_{{\rm ref}})]+I_{{\rm bgnd}}(t), (12)

arbitrarily chosing the reference signal to be I_{{\rm ref}}=\max(I_{{\rm s}})-\min(I_{{\rm s}}). The background signal I_{{\rm bgnd}} is considered constant within each sedimentation event and is determined by averaging I_{{\rm s}} over the last 500 ms of each cycle and then taking the minimum of such averages from 11 subsequent cycles, the one considered, 5 earlier and 5 later cycles.

Figure 3. (a): The experimental propagator P(\tilde{z}+\Delta,\tau|\tilde{z},0) as a function of the sphere-wall separation \tilde{z} and the jump length \Delta, measured for a 6.9 \mathrm{\upmu}\mathrm{m}polystyrene latex sphere in 0.2 mM KCl. (b): The same propagator as expected in the limit of short delay times (equation (8)) with the force in the drift term and the particle size derived from Markovian Dynamics Extrapolation, and z_{{\rm ref}} determined by a fit to the experimental propagator.

This procedure makes it possible to monitor slow background fluctuations and avoid misinterpretation of rare cycles during which the signal did not drop all the way into the background. Using equation (12), we convert the measured intensities into sphere-wall separations \tilde{z}(t)=z(t)-z_{{\rm ref}} with an as yet unknown constant offset z_{{\rm ref}}. Next we bin up the encountered separations and make a histogram of all the jumps \Delta occuring within one time step \tau=10 ms and originating from the same separation \tilde{z}. By normalization with the number of hits in each separation bin we obtain the “measured” propagator P(\tilde{z}+\Delta,\tau|\tilde{z},0), an example of which is shown in Fig. 3(a).

§ The Potential Energy

For further analysis by Markovian Dynamics Extrapolation we express the propagator P in terms of the initial and final sphere-wall separations. Applying equation (6) to the equilibrium probability density with a discretized spatial dependence, we can write

\rho^{{\rm eq}}_{{j}}=\sum _{k}P_{{jk}}\rho^{{\rm eq}}_{k}\:, (13)

where the index k denotes the k-th bin of initial positions \tilde{z} and j is an index for the bins of final positions \tilde{z}+\Delta. As the largest separations are deduced from a TIRM signal near the background intensity and are correspondingly unreliable, we limit the range of separations considered to a maximum value \tilde{z}_{{\rm max}}. Special care must then be taken of the incidents where, due to diffusion and gravity, spheres drop out of the accepted separation window. In order to compensate for the resulting tendency of \rho _{{j}}^{{\rm eq}} to vanish, we implement a reflecting boundary condition by making the replacement \Delta\to 2(\tilde{z}_{{\rm max}}-\tilde{z})-\Delta whenever \tilde{z}+\Delta>\tilde{z}_{{\rm max}}. With an accordingly modified propagator P_{{jk}}, we solve equation (13) and then use the obtained \rho^{{\rm eq}} to compute the potential energy U via equation (1). It was checked that the result did not depend on the choice of the boundary \tilde{z}_{{\rm max}} as long as the latter was not chosen too large.(26)

Figure 4. The potential energy U(z) obtained from Markovian Dynamics Extrapolation for (a) a 7.5 \mathrm{\upmu}\mathrm{m}sphere in 0.1 mM KCl, (b) a 6.9 \mathrm{\upmu}\mathrm{m}sphere in 0.2 mM KCl, and (c) a 7.7 \mathrm{\upmu}\mathrm{m}sphere in 0.5 mM KCl. Separations were combined in bins of 3 nm width; an arbitrary potential offset was chosen for plotting the curves. The diameter of the individual spheres was determined from the indicated slope at large z.

Fig. 4 shows the potential energy found for three different particles in KCl solutions of different ionic strengths. The absolute location of each curve with respect to the energy axis was chosen arbitrarily. The slope of the curves at large separations along with the known polystyrene density of 1.055 g/ml was used to calculate the actual size of the investigated spheres. We occasionally found significant deviations from their nominal size of 8 \mathrm{\upmu}\mathrm{m}.

§ The Separation Offset

In order to determine the absolute separation z=\tilde{z}+z_{{\rm ref}}, we compare the experimental propagators P(\tilde{z}+\Delta,\tau|\tilde{z},0) with their theoretical limit for small \tau (equation (8)) using the corrected particle size. As outlined in the paragraph on short time dynamics, we use the variance of P(\tilde{z}+\Delta,\tau|\tilde{z},0) in the jump length \Delta to deduce the diffusion coefficient D(\tilde{z}) and invert equation (9) numerically to get the absolute separation z=z(D(\tilde{z})). Deviations from the expected linear behavior

z(\tilde{z})=\tilde{z}+z_{{\rm ref}} (14)

are observed both for very low \tilde{z}, where sampling statistics are inferior, and for large \tilde{z} where the accuracy is diminished by a poorer signal-to-background ratio. Rather than using z(D(\tilde{z})) directly, we fit equation (14) (with z_{{\rm ref}} as a single free parameter) to the experimental z(D(\tilde{z})) in a range of separations around \tilde{z}=100 where the best accuracy can be expected. Note that a procedure using the variance \langle(\Delta(\tilde{z})-\langle\Delta(\tilde{z})\rangle)^{2}\rangle warrants such a restricted data selection as it requires much better statistics than is needed e.g. to evaluate the mean \langle\Delta\rangle or the difference between individual separations z_{1}-z_{2} with corresponding intensities well above the background level.

§ The Force Profiles

With the correct particle radius a and the absolute separations z thus determined, we use equation (9) to compute the friction and diffusion coefficient as well as the spatial derivative D^{{\prime}}(z). Inserting these and the experimental drift velocity v(z)=\langle\Delta(z)\rangle/\tau into equation (5), we can solve for the force F(z)=-U^{{\prime}}(z). The result is shown by the data points in Fig. 5. Although the curves could easily be smoothed by spatial integration or by further averaging over adjacent separation bins, no such effort has been made, as the observed scatter gives an impression of the statistical error. For comparison with Markovian Dynamics Extrapolation (MDE) we have taken the numerical derivative of the potential energies shown in Fig. 4.(27) The corresponding force is represented by the solid curves in Fig. 5.

Figure 5. The acting force -U^{{\prime}}(z) as obtained in the Gaussian approximation of short delay times (equation (8), markers) or by numerical differentiation of the potential energies from Markovian Dynamics Extrapolation (Fig. 4, lines). Sphere size and salinity (a)-(c) as before. The curves are set off by 0.5 pN for clarity.

Except for the case of curve (c) the agreement between the short time approximation and the MDE result is very good. Although gratifying, this agreement also is a little surprising because while the mean particle displacement \langle\Delta\rangle is always small, some of the typical displacements (i.e. \Delta within one standard deviation from the mean) were up to 15 nm in the experiments shown – a distance over which the diffusion coefficient and drift speed vary substantially. Consequently, the experimental delay time \tau=10 ms would appear to have been too large for the Gaussian approximation to be fully justified.

Figure 6. The experimental force -U^{{\prime}}(z) obtained from Markovian Dynamics Extrapolation (solid lines as in the previous figure) versus a prediction from double layer theory with no free parameters (dashed line). The values for the effective electrostatic potential on the glass (polystyrene) surface used in the calculation (equation (15)) are 63 mV (101 mV) in (a), 59 mV (100 mV) in (b), and 54 mV (98 mV) in (c). A constant gravitational force has been included in the theoretical curves.

Note also that the two approaches represented in Fig. 5 are not strictly independent since the short time dynamics approach still uses the particle size derived from MDE, while the location of the MDE curves along the z-axis is fixed by the reference separation z_{{\rm ref}} found in the short \tau limit. However, the shape of the MDE curves does not depend on any assumptions regarding the delay time and therefore supports the results of Fig. 5 despite the aforementioned considerations.

This shape information can also be used in combination with the Gaussian approximation for an alternative way of finding the reference separation. Equations (8) and (9) with the force and particle radius derived from the MDE results (Fig. 4) define a semi-theoretical propagator which can be fitted to the directly measured propagator with z_{{\rm ref}} as a single fit parameter. An example of such a fit is shown in Fig. 3(b). The values of z_{{\rm ref}} found this way vary a little depending on the chosen data ranges and weights, but the results are stable within roughly 10 nm. They agree, to this precision, with the results for z_{{\rm ref}} based on the experimental variance of \Delta alone.

While a quantitative theoretical discussion of the measured forces is beyond the scope of this paper, we would like to point out that the results of Fig. 5 are not consistent with an electrostatic and van der Waals interaction between the particle and the glass wall along the lines of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory (28). Fig. 6 compares the results of the Markovian Dynamics Extrapolation with the electric double layer force calculated in the Derjaguin and superposition approximations (29):

F=4\pi\varepsilon\varepsilon _{0}\kappa a\psi _{{\rm glass}}^{{\rm eff}}\psi _{{\rm PS}}^{{\rm eff}}\exp(-\kappa z), (15)

where \varepsilon\varepsilon _{0} is the total electric permittivity, \kappa^{{-1}} is the Debye screening length, and

\psi _{{\rm glass,PS}}^{{\rm eff}}=(4k_{B}T/e_{0})\tanh\left(e_{0}\psi _{{\rm glass,PS}}/4k_{B}T\right) (16)

denotes the effective electrostatic potential of either the glass or the polystyrene (PS) surface (e_{0} is the protonic charge). The values for \psi _{{\rm glass,PS}}^{{\rm eff}} given in the figure caption are consistent with a surface charge density of 56 mC/m{}^{2} quoted for the polystyrene spheres by the manufacturer, and with the charge of glass expected from the dissociation of silanol groups at pH 6 given the experimental ionic strengths (30). This calculation does not involve any free parameters. Van der Waals forces have not been considered here, because they do not seem to be resolved in our experiments. Indeed, based on published measurements of the van der Waals attraction between a polystyrene sphere and a glass wall (31), we do not expect a significant force contribution due this type of interaction under the given experimental conditions.

While theory and experiment agree remarkably well for the lowest ionic strength, this good agreement does not extend to higher salt concentrations. In fact, only the force measured in experiment (a) corresponds to an exponential with a decay length identical to the Debye length of the electrolyte solution (30.4 nm in this case). As the ionic strength is increased, the interaction becomes softer, in contradiction with DLVO theory, and can be characterized for 0.5 mM KCl solution by a decay length of 41 nm. At this salt concentration, many particles deposited irreversibly onto the glass surfaces, particularly under the influence of the optical tweezer. Even so, while they were still mobile, these particles seemed to be repelled more strongly from the wall than particles in 0.2 mM KCl, which never got stuck. Repeating the measurements with freshly prepared solutions (analytical grade KCl in deionized water from a Barnstead Nanopure water purifier) confirmed the softening of the interaction with increasing salinity. In one instance, a sphere in 0.5 mM KCl became tethered to the glass surface, i.e. instead of sinking, it stayed near the surface, but retained some mobility and showed evidence of a spring-like restoring force when moved laterally with the tweezer (23). The corresponding spring constant could be shown to agree well with a literature value for polystyrene (32). We therefore conjecture that the soft repulsive sphere-wall interaction reported in the present paper are also mediated by polymer strands protruding from the latex particles. The presence of charged groups on such strands might explain both their extension into the solution despite the hydrophobicity of polystyrene and an increased compressibility upon the addition of screening ions.

§ V. Conclusions

We have presented a new method to measure the total force on a colloidal particle near a glass wall by means of Total Internal Reflection Microscopy. Unlike well established techniques, the present type of analysis is applicable to equilibrium as well as nonequilibrium situations. It was used to measure the potential energy of a polystyrene sulfate latex particle receding from an glass wall. To our knowledge, this is the first TIRM study of a non-stationary system. While the interaction force measured at low ionic strength (0.1 mM) agrees very well with DLVO theory, measurements for higher ionic strength indicate additional types of interaction.

This work was supported by the National Science Foundation through Grant Number DMR-9730189, by the Deutsche Forschungsgemeinschaft, and by the W. M. Keck Foundation.


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