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Total Internal Reflection Microscopy: Distortion Caused by Additive Noise Paul C. Odiachi† and Dennis C. Prieve* Center for Complex Fluids Engineering and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Total internal reflection microscopy (TIRM) is an optical technique for monitoring Brownian fluctuations in the separation between a single microscopic sphere and a flat plate in aqueous media. The sphere is levitated above the plate by colloidal forces such as double-layer or steric repulsion. Changes in elevation as small as 1 nm can be detected by measuring the light scattered by a single sphere when illuminated by an evanescent wave. From the Boltzmann distribution of elevations sampled by the sphere over a long time, the potential energy (PE) profile can be determined with a resolution of about 0.1kT. By corrupting clean data (having a mean scattering intensity of hIs and a standard deviation of σs) obtained by Brownian dynamics simulations with various levels of additive background noise (having a mean background intensity of hIb and a standard deviation of σb), we show how the PE profiles obtained from TIRM are distorted. Increasing hIb narrows the profile and shifts it toward smaller elevations; conversely, increasing σb broadens the PE profile. Subtracting hIb from the measured total intensities before analysis removes much of the distortion. The extent of remaining distortion (broadening) depends primarily on the ratio σb/σs and can be neglected when this ratio is less than 0.15. Introduction Light scattered from an evanescent wave by a microscopic sphere is exponentially sensitive to the distance between the sphere and the plate at which the evanescent wave is generated.1 The exponential decay length corresponds to the penetration depth of the evanescent wave. Total internal reflection microscopy (TIRM) exploits this sensitivity to monitor the Brownian fluctuations in the separation distance. Changes in distance as small as 1 nm can be detected. When the Brownian sphere is prevented from sticking to the wall (e.g., by double-layer or steric repulsion), the histogram of elevations sampled over a long time converges to a Boltzmann distribution. This histogram can then be converted into the potential energy (PE) profile of the sphere. Owing to the molecular gauge for energy (kT) appearing in Boltzmann’s equation, TIRM is able to detect interactions that are much weaker than those measured by mechanical instruments such as an atomic force microscope or a surface force apparatus. For charged spheres of a few microns in diameter in aqueous solutions of less than 1 mM in ionic strength interacting with a like-charge plate, the PE profile has contributions from double-layer repulsion and gravity.2 Other forces measured with TIRM include retarded van der Waals attractions,3 steric repulsions due to adsorbed polymers,4 depletion attractions due to nonadsorbing clay particles,5 long-range attractions between receptor-ligand pairs,6 and optical forces.7 These and other applications of TIRM were recently reviewed by Prieve.8 These past applications of TIRM employed optically ideal materials such as glass, polystyrene (PS), and clear * Author to whom correspondence should be sent. Phone 412-268-2247, fax 412-268-7139, e-mail
[email protected]. † Present address: Shell International E&P, 3737 Bellaire Blvd., Houston, TX 77025.
aqueous fluids. Their transparency and the difference in refractive index between glass or PS and water was sufficient to produce good contrast and strong scattering. In the absence of a particle, background scattering intensity readings from the glass slide are usually very low. In recent work, we have been attempting to extend the technique beyond these model systems. For example, we have employed fluids made cloudy by the addition of clay9 or glass slides coated with thin bitumen9 or metal10 films (which scatter light), and we have used biological cells10 (which scatter little light in water) as the particle. Either reducing the scattering of the particle or increasing the scattering from the plate makes the background more important relative to the foreground and (if ignored) can lead to serious distortion in the PE profile deduced from TIRM. In this manuscript, we start with a clean set of TIRM data obtained from Brownian dynamics simulation11 and corrupt it with various distributions of additive background intensities. We then analyze the corrupted data in the same way as the experimental data to determine the distortion in the PE profiles thus produced. In addition, we test the efficacy of subtracting the mean background intensity, which is found to remove most of the distortion provided that the standard deviation of the background fluctuations is small compared to the standard deviation of the foreground intensities. Theory Analysis with Negligible Background. In the usual analysis of TIRM results, the measured intensity I is assumed to arise solely from scattering (Is) of the evanescent wave by the sphere, which, in turn, is related to the elevation h of the sphere according to the relation1
I ) Is ) I0e-βh
10.1021/ie010377t CCC: $22.00 © 2002 American Chemical Society Published on Web 10/04/2001
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where β-1 is the penetration depth of the evanescent wave and I0 is taken as the scattering intensity of a particle stuck to the plate (h ) 0). Then the probability of measuring a particular intensity between I and I + dI is the same as the probability of finding the sphere within the corresponding interval of elevations [i.e., between h and h + dh where I(h + dh) ) I(h) + dI]
P(I) dI ) ps(Is) dIs ) p(h) (-dh) The minus sign in the last expression was inserted to force both probabilities to be positive (when dIs > 0, eq 1 implies that dh < 0). Solving for P(I), one obtains
P(I) ) ps(Is) )
p(h(I)) p(h) ) -dI/dh βI
(2)
where the last equality was obtained with the help of eq 1. The probability of finding the sphere at any elevation h is related to the potential energy φ(h) of the sphere at that elevation
p(h) ) Aeφ(h)/kT
(3)
where A is a normalization constant chosen so that ∫p(h) dh ) 1. Expected Potential Energy Profile. When the separation distance is several Debye lengths, we expect van der Waals attractions to be severely retarded and screened and double-layer repulsions to be well modeled using linear superposition and Derjaguin’s approximations. Then, for a 1:1 electrolyte, the total PE profile is expected to obey
φ(h) ) B exp(-κh) + Gh
( ) kT e
2
tanh
( ) ( ) eψ1 eψ2 tanh 4kT 4kT
(5)
is the dielectric permittivity of water; a is the radius of the sphere; e is the elemental charge; ψ1 and ψ2 are the Stern potentials of the sphere and the plate, respectively
κ)
x
8πCe2 kT
(6)
is the Debye parameter; C is the total ionic strength; and
4 G ) πa3(Fs - Ff)g 3
(7)
is the net weight of the sphere. Equation 4 has a single minimum at
κhm ) ln
κB G
In other words, the shape of the PE profile is not affected by B. Increasing the charge on either the sphere or the plate will shift the minimum PE to larger separation distances hm according to eq 8, but it will not affect the shape of the PE curve. Analysis with Fluctuating Background. In the absence of a particle, background intensity readings can be on the order of 0.1-1% of the stuck-particle intensity I0. With less than ideal optical conditions, the background can be comparable to I0. This background intensity Ib adds to the intensity of scattering Is of the evanescent wave by the sphere
I ) Is + Ib Suppose that the distribution of background intensities is given by pb(Ib) and that the fluctuations in the background intensities are completely uncorrelated with the fluctuations in Is. How will these fluctuations in the background intensity distort the distribution of total intensity P(I) measured with the photomultiplier tube (PMT)? First, consider how a constant background intensity would distort P(I). If Ib is constant during the period in which the total intensity I is being sampled, then the shape of the histogram of I will be the same as the shape of the histogram of Is, except that the histogram will be shifted by an amount equal to the background Ib
P(I) ) ps(I - Ib)
(4)
where
B ) 16a
φ(h) - φ(hm) G ) {exp[-κ(h - hm)] - 1} + kT κkT G (h - hm) (9) kT
(8)
The charge parameter B is difficult to determine independently. Fortunately, we can eliminate B using eqs 4 and 8 to obtain the relative PE in terms of the relative separation distance h - hm
where ps(Is) is still related to p(h) by eq 2. Next, suppose that the background intensity assumes a value Ib1 for a long time t1 and then changes to Ib2 for a second long time t2. If both t1 and t2 are long enough that an equilibrium distribution of scattering intensities is sampled during each period, then P(I) is given by ps(I - Ib1) during period t1 and by ps(I - Ib2) during period t2. Adding these two histograms and renormalizing leads to the composite histogram
t1 t2 + ps(I - Ib2) (10) P(I) ) ps(I - Ib1) t1 + t2 t1 + t2 where ti/(t1 + t2) is the fraction of time the background intensity is Ibi. Note that the same distribution of total intensities would be expected even if the background intensity switched at irregular intervals between Ib1 and Ib2 instead of remaining fixed at one value for a long time, followed by remaining fixed at another value for a long time, provided that the fraction of the total time spent at each value was the same. Equation 10 can be generalized to a large number of different background values by replacing the fraction ti/(t1 + t2) of time spent at background intensity Ibi by pb(Ib) dIb and replacing the sum in eq 10 by an integral over the range of backgrounds intensities
P(I) )
∫-∞∞ ps(I - Ib) pb(Ib) dIb
(11)
Note that the resulting probability density for total intensity is the convolution of the probability densities
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Figure 1. Three typical histograms of background intensities obtained on different days. Solid curves are Gaussian fits to the experimental data.
for the scattering of the evanescent wave and for the background. From Intensity Histogram to Potential Energy Profile. Typically 100 000 readings of scattering intensity I are sampled at intervals of 5 or 10 ms. A histogram N(I) is constructed by dividing the range of intensities into a set of intervals or bins of equal width ∆I. Then, N(I) is the number of readings of intensity that fall in the bin assigned to intensity I. In the limit of an infinite number of readings taken over an infinite time, the histogram N(I) is proportional to the probability density P(I), the two functions differing by the normalization constant. In turn, this probability density for intensity P(I) can be converted into the probability density for elevation p(h) using eq 2, which is related to the potential energy profile φ(h) by Boltzmann’s equation (eq 3). Thus, we construct the relative potential energy profile from the histogram of intensities using
{
}
φ(h) - φ(hm) N[I(hm)] I(hm) ) ln kT N[I(h)] I(h)
(12)
where hm is the most probable elevation. This analysis assumes that all fluctuations in the measured intensity are due to fluctuations in elevation. Other assumptions are discussed in the review by Prieve.8 When fluctuations in the background are significant but ignored, this procedure yields the distorted PE profiles discussed below. Results Typical Background Intensities. Figure 1 shows some typical histograms of experimental TIRM intensity readings obtained with a clean glass slide without any particle in view. The solid curves are best-fitting Gaussians, which can be seen to represent well the shapes of all three histograms. Background intensities usually display a histogram with a Gaussian shape. The one notable exception occurs with bitumen-coated slides in which the histogram of background intensities resembles a Gaussian with its top flattened. This distorted shape arises because the mean background intensity tends to drift upward with time as the bitumen absorbs energy from the incident light. However, if a histogram could be obtained over a short enough time (compared to the time required for the mean to drift), that histogram would still be expected to be Gaussian.
Figure 2. Relevant dimensions in the field of view as seen through the microscope objective. The solid square is the observation window in which the microsphere resides while data to be used in constructing the energy profile is being collected. To obtain background intensity readings, the particle is moved out of the window. The broken lines represent the edge of the reticule in the eyepiece, and the solid circle represents the full field of vision through the eyepiece. Table 1. System Noise in the Photomultiplier Tube (PMT) room lights off room lights on
σb
hIb
0.023 0.021
0.024 0.021
Figure 2 is a schematic of the view through the microscope that has been annotated to show the relevant dimensions. The photometry system (which is part of our microscope) includes an adjustable rectangular window to control the subset of the total microscope field from which light is admitted to the photomultiplier tube. This observation window is shown as the solid square in Figure 2. To record intensities due to the Brownian motion of a sphere, a single particle is kept resident in the window (for the duration of the experiment), and the light scattered from the evanescent wave is measured.8 While background intensities are being measured, no particle is present in the window. The amount of light measured during any experiment is dependent on the size of the observation window and on the neutral density filters inserted before the photomultiplier tube (PMT). For this work, the size of the window was kept constant at approximately 36 µm × 36 µm, and the same neutral density filters were used. The square represented by the dashed line in Figure 2 represents the edge of a rectangular grid superimposed by the eyepiece. Its dimensions are 120 µm × 120 µm. The circle (approximately 216 µm in diameter) represents the full view of the eyepiece. Table 1 shows the statistics (mean and standard deviation) of the system noise with the room lights off and on. The measurements were made with no particle in the window and no evanescent wave propagating through the window. This system noise should include both shot and thermal noises and light from the surroundings. The statistics are essentially the same, which indicates that the lighting in the room does not contribute significantly to the background noise. The values are less than 1 because most of the intensity readings were 0. Note that PMT readings are discretized in the range 0, 1, ..., 1000.
Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 481 Table 2. Total and Background Intensity Data Statistics from a Good Experimenta total
background
experiment
[PAANa] (ppm)
σ
hI
σb
hIb
A1 A2 A3 A4 A5
2 5 10 15 stuck
3.5 7.8 6.7 7.2
9.0 37.4 55.0 287.8
0.5 0.5 0.7 0.8 0.7
0.8 0.9 1.6 2.5 2.3
a
Nominal 6.24-µm PS sphere interacting with a glass slide in varying polyacrylic acid salt (PAANa) concentration. Table 3. Total and Background Intensity Data Statistics from a Good Experimenta total
background
experiment
[CaCl2] (mM)
σ
hI
σb
hIb
B1 B2 B3 B4 B5
0.025 0.05 0.1 stuck stuck
12.2 21.4 30.9 10.2 10.2
29.6 54.3 80.7 258.8 258.8
0.5 0.6 0.6 0.8 1.0
1.0 1.5 1.5 2.9 4.6
a Nominal 6.24-µm PS sphere interacting with a glass slide in varying CaCl2 concentration.
Tables 2 and 3 show the corresponding statistics for the total scattering (foreground plus background) from two different sets of experiments performed under “good” experimental background conditions. Also shown in the tables are the means and standard deviations of the background. For both tables, the foreground data are for nominal 6.24-µm polystyrene spheres interacting with a glass slide. The mean of the experimental intensities increases because the ionic strength of the fluid is being increased by adding 20 kDa MW poly(acrylic acid) (Table 2) or CaCl2 (Table 3). This draws the microsphere closer to the slide in both experiments and causes the scattering to increase according to eq 1. The background data were collected after each corresponding experiment. For the background readings associated with A1-A5 and B1-B4, the microsphere was located on the outer circle in Figure 2. This circle is about 108 µm from the center of the window that represents the edge of view through the eyepiece. There were no other microspheres in the field of view of the eyepiece. The background readings under these conditions have standard deviations of less than 1.0 and means of less than 3.0. The background readings associated with B4 and B5 show the effect of having the sphere outside the window but at different distances from the window. In B5, the sphere has been moved to a corner of the reticule grid (about 85 µm from the center of the window). The mean and standard deviation of B5 are greater than those of B4. The increase in the mean and standard deviation of the background as the sphere approaches the wall is due to the sphere’s contribution to the background. It is apparent that, even though the sphere is far from the window, it can contribute to the recorded background intensity. This suggests that any microsphere or particle within the field of view of the eyepiece will contribute to the background, even if it is outside the window admitting light to the PMT. Thus, to obtain experimental data with the least amount of corruption, only the sphere of interest should be in the field of view of the eyepiece. Tables 4 and 5 show the statistics of the background intensities (with no particle in the window) recorded for “nonideal” experiments. When the background readings
Table 4. Background Intensity Statistics for Experiments that Show a Higher Degree of Noise (Nonideal Experiments) with Varying Synthetic Clay Concentration clay conc (ppm)
σb
hIb
50 200 500
0.6 1.9 3.3
1.2 9.3 16.2
Table 5. Background Intensity Statistics for Experiments that Show a Higher Degree of Noise (Nonideal Experiments) with Different Spin-Coated Bitumen Slides σb
hIb
9.2 12.0 43.4
451.8 431.6 63.6
were recorded, the sphere was located at the edge of view of the eyepiece. These experiments are termed nonideal because of the relatively higher background intensities recorded. The data in Table 4 were obtained with synthetic flouro-hectorite clay (∼1 µm in size and thus larger than the Laponite clay we have used previously).5 The data show that the standard deviation and mean of the background increase with increasing clay concentration. The ranges of the standard deviations and means are greater than those for the good experiments. Table 5 summarizes readings for three different glass microscope slides spin-coated with bitumen. They have statistical values that are higher than the good background data or the synthetic clay data. Simulating the Effect of Additive Noise. During a TIRM experiment, observations of intensity are made at fixed intervals of time ∆t (usually 5 or 10 ms) over a long time period T (usually 8-20 min). In the presence of background scattering, the corrupted intensity is measured and can be expressed (using the notation already developed) as
I(ti) ) Is(ti) + Ib(ti) i ) 1, 2, ..., T/∆t
(13)
The Brownian dynamics simulation (BDS) of Sholl et al.11 was used to generate “noise-free” results for a 7-µm polystyrene sphere interacting with a wall in a 0.1 mM 1:1 electrolyte solution. The mean of the intensity data hIs generated was 123 (arbitrary units), and the standard deviation σs was 42. The histogram of intensities generated from the data is shown in Figure 3. Using this data set (containing 100 000 Is values) and an equal number of randomly generated uncorrelated noise values Ib, 100 000 corrupted intensity data points I were obtained. In this work, varying levels of corruption were achieved by adding randomly generated noise with different means and standard deviations randomly sampled from the Gaussian distribution
pb(Ib) )
1
x2πσb2
[
exp -
]
1 (Ib - hIb)2 2σb2
where hIb and σb are the mean and standard deviation, respectively, of the background intensities (noise). Figure 4 shows a histogram of the 100 000 background intensities thus obtained with hIb ) 123 and σb ) 24.7. The histogram of total intensities obtained by pointby-point addition of the scattering and background intensities according to eq 13 can now be compared to
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Figure 3. Probability histogram of light scattered by a 7-µm polystyrene sphere interacting with a wall across a fluid containing a 0.1 mM 1:1 electrolyte. The data were obtained using the Brownian dynamics simulation of Sholl et al.11
Figure 4. Probability histogram of background intensities (noise). A Gaussian distribution with hIb ) 123 and σb ) 24.7 (20% of the mean) was assumed.
the histogram obtained by the convolution formula of eq 11. Discretization of eq 11 was carried out using circular convolution, which is defined by12 n-1
P(Ii) )
ps(Ii-j)pb(Ij) ∑ j)0
for j ) 0, 1, ..., n - 1 (14)
where n is the number of bins in each histogram and Ik is the intensity corresponding to the kth bin in each histogram. Histograms or PE profiles generated by the application of eq 13 will be referred to as “additivecorrupted”, whereas those generated by eq 14 will be referred to as “convolved-corrupted”. Figure 5 compares the histograms obtained from the two methods. The two histograms virtually overlay each other. The only distinguishable difference between the two histograms lies in the fact that the convolved-corrupted histogram is smoother than the additive-corrupted histogram. This is not a surprise as it is well-known that integration is a smoothing operation and thus smoothes out the sampling noise present in all three histograms. Figure 6 compares the interaction energy profiles deduced from the clean BDS histogram of Figure 3 and the additive-corrupted histogram of Figure 5. The stuck intensity I0 has been chosen to be 500 and thus I0b ) 500 + 123 ) 623. Clearly, ignoring the noise leads to distortion in the profile. Fitting the PE profiles to the expected form given by eq 9, we obtained regressed
Figure 5. Comparison of the additive-corrupted histogram from eq 13 (open symbols) and the convolved-corrupted histogram calculated from eq 11 (curve). The clean histogram from Figure 3 (solid symbols) is included for comparison.
Figure 6. Distortion of the PE profile caused by the background scattering of Figure 4. The PE profiles are obtained from the additive-corrupted histogram (open circles) in Figure 5 and from the clean BDS histogram (solid circles) in Figure 3. The solid curve represents the PE profile input to the BDS. Table 6. True and Regressed Values of Parameters Deduced from PE Profiles of Figures 6 and 8 true BDS corrupted corrected
κ-1 (nm)
G (fN)
30.4 27.9 ( 0.6 32.4 ( 0.8 33.7 ( 0.6
96.9 105.5 ( 2.0 286 ( 6 100.2 ( 1.8
values for the parameters κ-1 and G. Table 6 shows how the distortion affects these parameters. Note that even the parameters deduced from the clean BDS differ somewhat from those values input to the BDS (solid curve in Figure 6). This difference arises from truncation error associated with taking a finite number of observations.11 At higher ionic strengths, the particle samples elevations closer to the wall where van der Waals attraction might be evident. Figure 7 shows that noise can completely mask van der Waals attractions. The clean profile was generated for a 6-µm polystyrene sphere in 1 mM 1:1 electrolyte using the BDS. The empirical equation used for the van der Waals component is3
h a φvdw(h) ) -10kT exp 40 nm 3.12 µm
(
)
Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 483
Figure 7. Distortion of the PE profile of a particle (6-µm PS in 1 mM 1:1 electrolyte) experiencing van der Waals attraction. The corrupted PE profile (open circles) was obtained by adding noise with hIb ) 300 and σb ) 100 to the clean data (solid circles). The solid curve represents the data input to the BDS.
Figure 8. Effect of applying a simple correction techniques subtracting the mean background noisesto the corrupted data in Figure 6. The open circles represent the BDS, whereas the solid circles represent the “corrected” profile. The solid curve represents the data input to the BDS.
where a is the radius of the sphere. The corrupted profile was generated by adding noise with hIb ) 300 and σb ) 100 to the clean data. The corrupted profile appears to be made up of just double-layer repulsion and gravity components (the van der Waals attraction has been masked). The distorted profile appears closer to the surface and has values of κ-1 ) 27.0 ( 0.3 nm and G ) 232.4 ( 1.8 fN, quite different from the true values of 9.6 nm and 61.0 fN, respectively. A Simple Correction. In deducing φ(h) for Figures 6 and 7, no effort was made to account for noise. In the past, we corrected for noise by subtracting the mean background noise from the corrupted intensities. Figure 8 shows the results obtained when this approach was applied to the additive-corrupted profile in Figure 6. It is obvious that the profile obtained after subtracting the mean background intensity looks much more like the clean profile (in fact, the most probable separations are the same). However, this profile is still wider than the clean profile. The values of Debye length and buoyant weight obtained from this corrected profile, although closer to the true values, are incorrect (see last entry in Table 6). The underlying problem with this simple
Figure 9. Effect of the variance in the background intensity on the energy profile obtained from Figure 3. The solid circle represents the clean energy profile, whereas the open circles and triangles represent the profiles obtained after the addition of noises with σb ) 12.3 and σb ) 49.1, respectively. The curve represents data input to the BDS.
Figure 10. Effect of the mean of the noise on the energy profile obtained from Figure 3. The solid circle represents the clean energy profile, whereas the open circles and triangles represent the profiles obtained after the addition of constant noises (zero standard deviation) of 247 and 62, respectively. The solid line represents data input to the BDS.
correction is best illustrated by considering separately the effects of the mean and variance of the noise. Figure 9 shows the effects of adding noise with hIb ) 0 (for both) and σb ) 12.3 and σb ) 49.1 to the BDS shown in Figure 3. In Figure 10, the effect of adding constant noise (σb ) 0 for both) of 62 and 247 is shown. The conclusion that can be drawn from these figures is that the effect of the variance in the noise is to broaden the profile, whereas the effect of the mean of the noise is to narrow the profile and shift it to lower separations. Discussion and Conclusions To understand the opposite effects of the mean and the variance, recall how the separation distance h is inferred from the measured intensities. Ignoring the distorting effect of noise, elevation is deduced from eq 1 by comparing the instantaneous intensity I of a levitated particle with its intensity when stuck to the plate I0
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h ) β-1 ln
I0 I
Background noise (with a nonzero mean) adds equal amounts to both I and I0, thereby decreasing their ratio I0/I and leading to a smaller inferred value for h. For this reason, the minima of the curves in Figure 10 are shifted to the left by an amount that increases with the mean background. The widths of the profiles in Figure 10 also decrease with the mean background for the same reason. On the other hand, Figure 9 shows that any variance in the noise (having zero mean) tends to increase the width of the profiles. Fluctuations in the background intensity add to fluctuations in the total intensity. If all fluctuations in intensity are assumed to arise from fluctuations in elevation, then those fluctuations in elevation are overestimated, and the inferred PE profile is broadened. The mean of any background noise is easy to measure and subtract from the measured total intensities before analysis of the TIRM data. Thus, distortions such as those in Figure 10 can be easily avoided. Distortion caused by variance in the background noise is more difficult to remove. How small does the standard deviation in noise σb have to be to cause negligible distortion in the PE profiles? First, we will show that being small compared to the mean scattered intensity (i.e., σb , hIs) is not always sufficient. Suppose that the histogram of true scattered intensity ps(Is) is very narrow (compared to the background) and can be treated as a delta function
ps(Is) ) δ(Is - hIs) Then, eq 11 yields
P(I) ) pb(I - hIs) Thus, in the extreme case in which σs , σb, the histogram of total intensity P(I) takes on the distorted shape of the background histogram pb(Ib) instead of the true scattering histogram ps(Is), regardless of how big or small hIs is. In the opposite extreme in which σb , σs, the histogram of background values can be treated as a delta function
pb(Ib) ) δ(Ib - hIb) Then, eq 11 yields
P(I) ) ps(I - hIb) and the histogram of total intensity P(I) takes on the shape of the true scattering histogram ps(Is), except that the intensity values are shifted by the mean background hIb. When we subtract the mean background value as suggested above, this shift is eliminated. This discussion suggests that the extent of distortion caused by fluctuations in background depends primarily on the ratio σb/σs. Table 7 summarizes the effect of this ratio on the distortion as reflected in the values of κ-1 and G deduced from the potential energy profiles of Figures 8 and 9 by a regression against eq 9. Figure 11 shows how these parameters (after normalization) depend on the ratio σb/σs. Generally, fluctuations in the background broaden the potential energy profile on either side of the minimum. With gravity dominating the right-hand side of the profile, this means reducing
Table 7. Effect of σb/σs on the Regressed Parameters after Correction σb
σb/σs
κ-1 (nm)
G (fN)
true 0 12.3 24.7 49.1
0 0.293 0.588 1.169
30.4 27.9 ( 0.6 29.8 ( 0.5 33.7 ( 0.6 38.5 ( 0.8
96.9 105.5 ( 2.0 100.4 ( 0.7 100.2 ( 1.8 76.9 ( 1.4
Figure 11. Distortion in best-fit parameters after correction for the mean noise level.
the apparent weight G; with double-layer repulsion dominating the left-hand side, this means increasing the apparent decay length κ-1. Taking an error of 5% in either parameter as tolerable, this means that the ratio σb/σs must be kept below about 0.15. This criterion is met for all of the good experiments described in Tables 2 and 3, but could easily be violated in other cases. In a subsequent paper, we will present a low-pass filtering method for removing the distortion in cases in which this ratio is as large as unity. Acknowledgment We acknowledge financial support from Syncrude Canada Ltd. and the National Science Foundation (Grant CTS97-11214). P.C.O. also acknowledges a fellowship from PPG Industries. Literature Cited (1) Prieve, D. C.; Walz, J. Y. The Scattering of an Evanescent Surface Wave by a Dielectric Sphere in Total Internal Reflection Microscopy. Appl. Opt. 1993, 32, 1629. (2) Prieve, D. C.; Frej, N. A. Total Internal Reflection Microscopy: A Quantitative Tool for the Measurement of Colloidal Forces. Langmuir 1990, 6, 396. (3) Bevan, M. A.; Prieve, D. C. Direct Measurement of Retarded van der Waals Attraction. Langmuir 1999, 15, 7925. (4) Bevan, M. A.; Prieve, D. C. Forces and Hydrodynamic Interactions between Polystyrene Surfaces with Adsorbed PEOPPO-PEO. Langmuir 2000, 16, 9274. (5) Odiachi, P. C.; Prieve, D. C. Effect of Added Salt on the Depletion Attraction Caused by Nonadsorbing Clay Particles. Colloids Surf. A 1999, 146, 315. (6) Liebert, R. B.; Prieve, D. C. Species-Specific Long-Range Interactions between Receptor/Ligand Pairs. Biophys. J. 1995, 69, 66. (7) Walz, J. Y.; Prieve, D. C. Prediction and Measurement of the Optical Trapping Forces on a Microscopic Dielectric Sphere. Langmuir 1992, 8, 3043. (8) Prieve, D. C. Measurement of Colloidal Forces with TIRM. Adv. Colloid Interface Sci. 1999, 82, 93.
Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 485 (9) Odiachi, P. C. Effect of Clay Platelets on Long-Range Interparticle Interactions. Ph.D. Dissertation, Carnegie Mellon University, Pittsburgh, PA, 2001. (10) Atman, E. S. Refinements Needed to Measure with TIRM the Colloidal Forces Acting on a Yeast Cell. Ph.D. Dissertation, Carnegie Mellon University, Pittsburgh, PA, 2001. (11) Sholl, D. S.; Fenwick, M.; Atman, E. S.; Prieve, D. C. Brownian Dynamic Simulation of the Motion of a Rigid Sphere in a Viscous Fluid Very Near a Wall. J. Chem. Phys. 2000, 113, 9268.
(12) Heideman, M. T. Multiplicative Complexity, Convolution and the DFT; Springer-Verlag: New York, 1988.
Received for review April 27, 2001 Revised manuscript received July 20, 2001 Accepted July 24, 2001 IE010377T
