Investigation of apparent correlated motion of Brownian particles

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 55, Issue 2, 1997, Pages 1998-2000

  Durand, Richard V ^a   Franck, Carl ^a  

^a Laboratory of Atomic and Solid State Physics and Sibley School of Mechanical and Aerospace Engineering   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0010114157     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.56.1998     Document Type: Article

References (36)

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2. D. Cannell (in a private communication relating his own direct observations) and B. Ackerson (in a private communication describing video microscopy data due to N. Ise).
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4. Scientists have been interested for many years in the visual interpretation of data, albeit in a different context than in the present work. High-energy experimentalists employed human “scanners” to search through large numbers of bubble chamber photographs for decay events [A. G. Frodesen, O. Skjeggestad, and H. Tofte, Probability and Statistics in Particle Physics (Universitetsforlaget, Bergen, 1979)]. Such methods have a long tradition in biology.For example, in a recent paper, naive observers were used to determine the size distribution of certain fluorescently labeled cellular structures [S. Henderson, R. Allsopp, D. Spector, S.-S. Wang, and C. Harley, J. Cell Biol. 134, 1 (1996)].
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9. Central Scientific has been known to change the specifications on its latex spheres while keeping the same stock number.
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11. The controlling software was written by Rolf Ragnarsson, Physics Dept., Cornell University.
12. Brendan Plapp, Cornell University, discovered that for unknown reasons, the digitized pixels produced by the imaging system were generally smaller in the horizontal direction by 2.5%. He included instructions in the digitization software that expanded the horizontal scale of the pixels by 2.5% to correct for this. However, after the experiment had already been completed, it was found that the anomalous digitization behavior had inexplicably not occurred with our images, and therefore the correction resulted in slightly distorted images. This led to a slightly different pixel conversion scale in the horizontal and vertical directions, but is not expected to affect the results in any way.
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17. The filter used here is very similar to the Schmidt trigger of electronic circuitry. The Schmidt trigger responds at a certain threshold level, and stays high until the input voltage drops below a certain level that is lower than the triggering voltage [J. Millman and A. Grabel, Microelectronics (McGraw-Hill, New York, 1987), pp. 679–683]. This hysteresis gives a certain level of noise immunity. What we are employing is a “noncausal” Schmidt trigger, as it uses information from the entire time-varying signal.
18. An “unreasonable” choice for these lingering criteria can lead to assumptions (3) and (4) becoming invalid. For example, one could make the criteria to be that the particles must be initially less than 1000 radii apart and are still so after ten frames. Here, (Formula presented) and (Formula presented) because the vast majority of pairs will satisfy these criteria, and the range of the interactions is much less than 1000 particle radii. This would give (Formula presented) and then Eq. (3) is equivalent to the statement that (Formula presented) which is not something we want to assume a priori. If very strict lingering criteria are chosen so that (Formula presented) then analogous reasoning shows that Eq. (4) leads to a contradiction. Since the observers seem to have chosen a lingering criteria for which (Formula presented) Eqs. (3) and (4) are probably reasonable assumptions.
19. The six data points shown in Fig. 55 and their uncertainties were computed as follows. For each of the six animations given in Table I, we averaged together the eight event counts (four observers with RD’s and CF’s counts for each observer). The preliminary uncertainty associated with the average event count in a given animation was determined by first averaging together RD’s and CF’s event counts for each observer, then finding the standard deviation of the mean of these average event counts associated with the four observers of the animation. The final uncertainty was computed by averaging together each of the six standard deviations associated with the mean event counts of the six animations. This uncertainty was used for each of the data points.
20. The uncertainties in (Formula presented) were computed as follows. For each data set, we obtained values for the excess event statistic by averaging (Formula presented) over the four volunteers. The standard deviations of the two means independently derived from RD’s and CF’s scorings were then averaged in quadrature to obtain a conservative estimate of the random error in (Formula presented) The systematic error was simply taken to be half the difference between CF’s and RD’s values for the excess event statistic.
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24. 8they obtained a Debye length of 320 nm, but did not specify the height of their sample cell.
25. Thus it appears that the ionicity is somewhat independent of the dimensions of the system.
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29. After the present work was completed, Crocker found that the Batchelor diffusivity tensor [Eq. (25)] is quantitatively consistent with the variance of the distribution of displacements that two 0.97-μm-diameter spheres in water undergo during a 16.7-ms interval [J. C. Crocker, J. Chem. Phys. 106, 2837 (1997)]. Due to this short time scale, the spheres in Crocker’s experiment have an effectively constant diffusivity during the taking of each data point. In the present paper, we consider displacements that occur over much larger time scales, and thus the particles sample a larger domain of separation vectors during the diffusion process.JCPSA6
30. The uncertainties in (Formula presented) were computed as follows. (Formula presented) and (Formula presented) are strongly correlated, but (Formula presented) seems to be only weakly dependent on (Formula presented) for the parameters relevant to this work. Thus we treated (Formula presented) and (Formula presented) as the independent variables, and calculated the random and systematic errors in (Formula presented) according to the usual error propagation techniques. The same method was used to compute the uncertainties in (Formula presented) which is discussed later in the text.
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32. For the case of two hard spheres, the probability flux should also vanish at contact (Formula presented) or (Formula presented) where n is the surface normal. This requirement was neglected, because for the length scales of interest, it would not change the probability distribution significantly, as can be seen from Fig. 77.
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36. found experimentally that, for certain sample cell geometries, pairs of colloidal particles experience an attractive force. We do not feel that either of these possible attractive forces was present in our system, but this does not mean that other workers that have noticed lingering among colloidal particles are not seeing some evidence of these attractive interactions. JCPSA6