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The Quasi-One-Dimensional Colloid Fluid Revisited† Binhua Lin,*,‡ David Valley,§ Mati Meron,| Bianxiao Cui,⊥ Hau My Ho,§ and Stuart A. Rice*,§ The James Franck Institute and CARS, The UniVersity of Chicago, Chicago, Illinois 60637, The James Franck Institute and Department of Chemistry, The UniVersity of Chicago, Chicago, Illinois 60637, CARS, The UniVersity of Chicago, Chicago, Illinois 60637, and Department of Chemistry, Stanford UniVersity, Palo Alto, California 94305 ReceiVed: February 28, 2009; ReVised Manuscript ReceiVed: June 10, 2009
We report the results of studies of the pair correlation function and equation of state of a quasi-one-dimensional colloid suspension, focusing attention on the behavior in the density range near close packing. Our data show that, despite deviations from true one-dimensional geometry, the colloid fluid is well described as a hard rod Tonks fluid. In our experimental realization, the colloid suspension does not wet the confining walls, one consequence of which is a surface tension induced weak attractive interaction between the particles. The reality of this interaction is confirmed after correction of the raw experimental data for overlap of the optical images of particles that are nearly in contact and by an alternative particle location algorithm based on edge location. I. Introduction Understanding the effect of spatial confinement on the properties of matter has become a forefront issue in condensed matter chemistry and physics. In the typical experimental realization, particles are enclosed in a three-dimensional (3D) space with boundaries that severely restrict their motions along one or two of the three dimensions. In this sense, the confined systems really have only quasi-two-dimensional (q2D) or quasione-dimensional (q1D) character. Many such systems exist, typical examples of which are surfactant monolayers (q2D),1 membranes (q2D),2 channels in zeolites (q1D),3 and microfluidic devices (q1D or q2D).4 Numerous experimental studies have confirmed the expectation that the equilibrium structure and the dynamic properties of confined simple liquids, and of confined colloid suspensions, are fundamentally different from those of the corresponding unconfined systems.5-8 In support of the experimental observations, many theoretical studies have confirmed the existence of differences between the equilibrium structure and the dynamic properties of confined simple liquids, and of confined colloid suspensions, and those of the corresponding unconfined systems.9,10 However, these theoretical analyses typically constrain the particle motion to be strictly one- or two-dimensional, and they often simplify the representation of the system, e.g., by neglect of effects associated with solvent-wall interactions such as exist in a real confined colloid suspension. For these reasons, simplified theoretical analyses may require modifications before comparison to the corresponding experimentally realizable quasi-low dimensional systems can be completely satisfactory. Whether or not modifications are required turns on two issues. The first and most important theoretical issue that must be confronted is whether or not the properties of a quasi-ndimensional system deviate from those of the corresponding †
Part of the “H. Ted Davis Special Section”. * Corresponding authors. E-mail:
[email protected];
[email protected]. The James Franck Institute and CARS, The University of Chicago. § The James Franck Institute and Department of Chemistry, The University of Chicago. | CARS, The University of Chicago. ⊥ Department of Chemistry, Stanford University. ‡
true n-dimensional system in a fundamental way, for example, by virtue of symmetry differences that support qualitative changes in the predicted system properties. That is, we seek to distinguish between quasi-n-dimensional induced qualitative changes in properties and quasi-n-dimensional generated quantitative changes in system parameters that lead to small changes in the system properties. Thus, although it is well established that a 1D system of particles with finite range pair interaction will not undergo a phase transition to a condensed state at any nonzero temperature,11 the general conditions under which a q1D system can support a transition to, say, a planar zigzag or a helical ordered state are not known. For example, a q1D polypeptide undergoes the random coil-to-R-helix disorder-toorder transition.12-14 Is there a threshold to the magnitude of the deviation from 1D motion that must be passed, or must there be a nonzero contribution from multiparticle interactions for a q1D fluid-to-ordered state transition to be allowed? The second important issue that must be addressed is the quality of the matching of the properties of an experimentally realizable system with those that define the model system that is described by the theory. Bluntly put, to test theoretical predictions it is necessary to ascertain whether differences between the experimental realization of a system and the theoretical model of a system are negligible or important. For example, does the nonwetting of hydrophobic walls that confine an aqueous colloid suspension induce effects not accounted for in models that ignore the solvent and assume simple particle-wall interactions? In this paper, we focus attention on two aspects of the second issue and indirectly provide some information concerning the first issue. In the following we assess the accuracy with which the structure of a q1D colloid suspension that does not wet the confining walls is describable as a 1D nearly hard rod (sphere) fluid. The results we report represent an extension of our previously reported study of the structure of q1D colloid suspensions. That previous study revealed that, up to about 70% of linear close packing, the spatial distribution of the particles in a q1D suspension of near-hard-sphere colloid particles is well characterized by the exact solutions for the 1D nearest-neighbor
10.1021/jp9018734 CCC: $40.75 2009 American Chemical Society Published on Web 07/01/2009
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distribution function and pair correlation function of a hard rod fluid, a Tonks fluid, except for pair separations near one sphere diameter (σ). The deviation in that region from the predicted hard rod fluid behavior was traced to a weak attractive colloid-colloid interaction with a depth of ∼0.3kBT at a colloid-colloid separation of ∼1.18 σ.15 We speculated that said effective interaction was a consequence of competition between wetting of the colloid particles and a surface tension effect at the suspension-air interface that separates the nonwetting solvent from the confining walls. The notion is that, since the suspending water wets the colloid particles but not the walls and the tight confinement restricts the layer of water separating the surface of the colloid from the air/water interface to be very thin, there could be small surface dimples that, when they overlap and thereby reduce the combined dimple area relative to that of the isolated dimples, generate an effective attraction between colloid particles. The new experiments reported in this paper describe the results obtained from studies of q1D colloid suspensions using larger diameter colloid particles in wider channels than used in our previous studies. These experiments extend the q1D packing density range studied from 0.7 to nearly 1, and they show that up to the highest packing density studied there is no evidence for an ordering transition. We also reexamine carefully the inference that in our q1D colloid suspension there is an attractive component in the effective pair interaction. The motivation for this reexamination is the suggestion, originally raised by Bechinger and coworkers,16 and later also considered by other research groups,6 that an optical artifact arising from the overlap of the images of colloid particles with small separation generates an error in the measurements of the separation of the particles that results in fictitious features in the inferred pair-interaction potential. To ascertain the influence of this artifact on our inferred weak attractive colloid-colloid interaction, we have explicitly corrected our data to account for optical overlap. We find that the existence and the qualitative features of the inferred effective pair interaction are sensibly the same when computed with and without the optical artifact correction. II. Theoretical Background Consider a 1D hard-rod Tonks gas consisting of N indistinguishable rods with length σ in a domain of length L, in a thermodynamic state with number density F ) N/L and packing fraction η ) σF. The equation of state is
p(1 - η) ) FkBT
(2.1)
with p being the 1D pressure. The pair correlation function of this hard rod system, g2(x), is17 ∞
g2(x) )
∑
) (x -k!k) × η exp[(x - k)] 1-η
η 1 Θ(x - k) 1 - η k)1 1-η
(
k-1
k-1
(2.2)
with x ) |x1 - x2| and Θ(x) ) 0 or 1 for x < 0 or x g 0, respectively. The corresponding structure function, S(q), is
{
S(q) ) 1 + 2
η sin(qσ/2) η sin(qσ) + (1 - η) qσ (1 - η) qσ/2
[
]}
2 -1
(2.3)
We note that
S(0) ) (1 - η)2
(2.4)
If, in addition to the hard rod part of the potential there is a short-range weak potential, βu(x) , 1 and we define 〈u〉 ) ∫u(y)dy and ∂〈u〉/∂L ) -〈F〉/N, the equation of state takes the form
(p - 〈F〉)(1 - η) ) FkBT,
(2.5)
and S(0) is the same as displayed in eq 2.4. An exact calculation of the pair correlation function for a 1D fluid with arbitrary short-range interactions has been reported by Salzburg, Zwanzig, and Kirkwood,17,18 but we will not need that complicated expression, so do not reproduce it here. III. Experimental Details The q1D systems we have studied consist of suspensions of silica colloid spheres (mass density of 2.2 g/cm3) confined in q1D narrow grooves. The spheres were suspended in pure water. Although the colloid particles are weakly charged, determination of the effective pair interactions in pure water and in various salt solutions reveals that the charge contribution to the interaction potential is negligible. We comment further on the colloid-colloid effective pair interaction later in this paper. The q1D grooves were “printed” on the surface of a polydimethylsiloxane sheet from a master pattern fabricated lithographically on a Si wafer (Stanford Nanofabrication Facility). To within the resolution of the microscope objective used to view the samples (∼0.2 µm), the width of a groove is uniform, and the walls of a groove are smooth. Sample cells were fabricated by enclosing a drop of suspension between the polymer mold and a coverslip. Both the polymer mold and the coverslip were coated with polydimethylsiloxane, so that all of the surfaces that the aqueous colloid suspension contacted were highly hydrophobic, and these surfaces were not wetted by the suspension. To keep the colloid suspension from being squeezed out of the grooves, a spacer (∼100 µm) was placed between the polymer mold and the coverslip, leaving the top of the grooves open to a thin layer of suspension. Since the colloid spheres are denser than water, they fall into and float near the bottom of the groove with ∼0.3 µm between the surface of a sphere and the bottom surface of a groove. The colloid particles are small enough to exhibit vigorous Brownian motion. A more detailed description of the sample preparation can be found in our earlier papers.15 We used digital video microscopy to extract the timedependent trajectories of the spheres along (the x-axis) and transverse (the y-axis) to the groove, with a time resolution of 0.033 s. As part of our analysis of the measurements, we have calculated the effective colloid-colloid pair interaction from the pair correlation function; a detailed description of the data analysis can be found in our earlier papers. In carrying out this analysis, we have addressed the suggestion, originally raised by Bechinger and co-workers,16 that an optical artifact arising from the overlap of the images of colloid particles with small separation generates an error in the measurements of the separation of the particles that results in fictitious features in the pair-interaction potential. Specifically, Berchinger et al. identified a systematic deviation between the digitized (or apparent) particle separation, r˜, and the real one, r (r˜ < r), when particles are close to contact, caused by optical image overlap.
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TABLE 1: Parameters for the Three Experimental Samples sample no. 1 2 3
σ (µm)
w (µm)
h (µm)
1.58 ( 0.04 3.0 ( 0.3 3.0 ( 0.3 3.17 ( 0.10 5.0 ( 0.1 4.0 ( 0.5 5.14 ( 0.20 7.0 ( 0.1 4.0 ( 0.5
L (mm) l (µm) 2 10 10
106 220 220
ε 0.01 0.03 0.04
This error in the measurements of the separation of the particles affects the calculated pair correlation function, and the inversion of that function thereby generates fictitious features in the pairinteraction potential. We have used one of the methods introduced by Polin et al.6 that allows us to correct retrospectively the imaging data used previously to calculate the pair potential (see Appendix I). We have applied this correction to data used for the calculation of the effective colloid-colloid pair interaction. The changes in the pair correlation function generated by the correction are small enough to not affect the qualitative features of g2(x), and the inferred pair interaction is substantially the same as inferred from the uncorrected data. We have also used a method for locating particle centers, different from the IDL routine of Crocker and Grier, that is designed to eliminate the optical effects associated with image overlap.19 The reason that image overlap leads to an error in the determination of the particle separation is that there is an overall skewing of the intensity distribution along the line between the centers of the particles that leads to an apparent shift of the peak positions. We have carried out a new set of measurements of the locations of colloid particle centers determined with an edge location method (see Appendix II). Our results show that for particle separations slightly greater than the contact separation there is less than 0.5% deviation between the true separation and that inferred from a digital optical image when using the edge detection technique to locate the particle centers. We also show that the superposition of optical effects is linear in the number of near neighbor particles under our experimental conditions. With respect to the main subject of this paper, we studied three sets of samples, each over a large range of packing fraction. The parameters defining the colloid particle and groove dimensions for these samples are listed in Table 1; for convenience, we denote them as Samples 1, 2, and 3, respectively. The sphere diameters and groove widths for each sample satisfied the singlefile condition; namely, in each case the width of the groove is less than two sphere diameters. In our previous work, based on Sample 1, we could not reach very high packing fraction because when η > 0.72 the spheres frequently popped out of the groove and then re-entered it at some different location, thereby destroying the single-file condition required for our study. In the work reported in this paper, we circumvented this problem by using larger (and, therefore, heavier) spheres and were able to make measurements up to η ∼ 1 with samples 2 and 3. Figure 1shows snap shots of Samples 1, 2, and 3 at various packing fractions.
Figure 1. Typical images from the movies of Samples 1, 2, and 3.
Figure 2. Colloid density distribution functions transverse to the confining channels for Samples 1-3.
density distributions; for each sample, these are found to be the same at all packing fractions. In the single file limit, in the fluid phase, in the absence of other than an excluded volume interaction between a particle and the groove wall, we expect the transverse distribution of particle displacements from the center line of the groove to be uniform, i.e., square topped. Our results indicate that the particles in Samples 2 and 3 have access to about 60% of the available groove width, (W - σ), while the particles in Sample 1 have access to only about 10% of the available groove width. Put in different terms, these data suggest that there is a gap between the surface of the fluid colloid suspension and the wall of the groove that is of an order of 0.5-1 µm. We attribute the existence of this gap to the nonwetting of the groove walls by the aqueous colloid suspension, with consequent trapping of a layer of air between the suspension and the walls of the grooves. We note that two of the distributions displayed are asymmetric, which we attribute to small imperfections in the flatness of the floor of the groove and/or to a slight tilt of the plane of the floor of the groove from horizontal. The q1D pair correlation function for each sample, at each packing fraction, was calculated from a histogram of the pair separations obtained from the coordinates of the spheres
g2(x) )
n(x)σ 1 〈F(x + x, t)F(x, t)〉 ) 2η∆x F20
(4.1)
In eq 4.1, ∆x is the bin size of the histograms (∆x ≈ 0.01σ); n(x) is the average number of spheres in the range x - ∆x/2 to x + ∆x/2; F0 is the average particle density; and the particle number density function, F(x,t), is defined by N
IV. Results and Discussion We have mentioned that the experimental determination of particle-particle separation must be corrected for image overlap of closely spaced colloid particles. This optical artifact does not affect the measurement of the particle displacement transverse to the groove axis. A major difference between Samples 2 and 3 and Sample 1 is in the degree of spatial confinement transverse to the grooves. This is clearly shown in Figure 2, which displays the transverse
F(x, t) )
∑
1 δ[x - xi(t)] N i)1
(4.2)
To avoid the introduction of artifacts arising from the finite field of view observed, our calculations of g2(x) only used spheres that are at least a distance x from the edges of the image. Figures 3a-d display the experimentally determined pair correlation functions for Samples 1, 2, and 3 at various packing fractions and the corresponding hard-rod pair correlation func-
Quasi-One-Dimensional Colloid Fluid Revisited
J. Phys. Chem. B, Vol. 113, No. 42, 2009 13745 is softer than the hard-rod interaction. In addition, all the peaks of g2(x) are shifted slightly to larger x relative to the peaks in the hard-rod g2(x). In contrast, the first peaks of g2(x) of Samples 2 and 3 rise as steeply, and the peak positions essentially coincide with those of the hard-rod g2(x), suggesting that Samples 2 and 3 behave more like a hard-rod fluid than does Sample 1. At very high packing fraction (η ∼ 1.0), the pair distribution functions of Samples 2 and 3 are in good agreement with that of a Tonks gas, including the decay of amplitude with increasing x that is characteristic of a fluid phase, so it is evident that there is no phase transition to an ordered state in these samples in the range of packing fraction studied. The effect of spatial confinement on ordering in the q1D fluid is even more apparent in the behavior of structure factor. We calculated the q1D S(q) from the histogram of the Fourier transform of particle number density
S(q) )
1 〈Fq(t)F-q(t)〉 F20
(4.3)
Figures 4a-e show S(q) for Samples 1, 2 and 3, compared with S(q) predicted for the Tonks gas calculated from eq 2.3. The oscillations in S(q) are much more pronounced and persistent than in g2(x) at the same packing fraction. As before, the structure factors reveal that Samples 2 and 3 behave more like a Tonks gas than does Sample 1. Note also that when η ∼ 1.0 the peaks in the theoretical S(q) are much higher and somewhat narrower than that in the experimental S(q), due to the limited resolution of the experimental measurements of x/σ. A different view of the extent to which our q1D systems have the properties of a 1D Tonks gas is provided by a comparison of the experimentally determined values of S(0) and the values predicted for a Tonks gas; this comparison is shown in Figure 5. In our previous study of the equilibrium properties of Sample 1, we reported that the effective colloid-colloid pair-potential, U(x), was weakly attractive; the minimum of the potential was about 0.3kBT deep, located at a colloid-colloid separation x ≈ 1.18σ. We have used inversion of the hypernetted chain (HNC) equation20
βU(x) ) g2(x) - 1 - c(x) - ln g2(x)
(4.4)
to obtain U(x) from g2(x). In eq 4.4, c(x) is the direct correlation function, defined via the Ornstein-Zernike equation
g2(x) - 1 ≡ h(x) ) c(x) +
∫ c(|x - x′|)h(x′)dx′ (4.5)
The direct correlation function can be determined from the structure factor via the Fourier transform of eq 4.5
hˆ(q) ) Figure 3. (a) Pair correlation functions for Sample 1 at various packing fractions. (b) Pair correlation functions for Sample 2 at various packing fractions. (c) Pair correlation function for Sample 2 at η ) 0.986. (d) Pair correlation functions for Sample 3 at various packing fractions.
tions calculated using eq 2.2. For Sample 1, the first peak of g2(x) rises more slowly than that of the exact hard-rod g2(x), indicating that the repulsive part of the colloid pair-interaction
cˆ(q) S(q) - 1 ≡ 1 - F0cˆ(q) F0
(4.6)
We calculated the effective pair-potentials for Samples 2 and 3 the same way as we did for Sample 1. Figure 6 shows U(x) for Samples 1, 2, and 3 for η ) 0.39, 0.33, and 0.38, respectively. As a measure of the effectiveness of our inversion procedure, Figure 6 also displays a calculation of U(x) for a Tonks gas for η ) 0.40, calculated using eq 4.4 and the theoretical g2(x).
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Figure 4. (a) Structure factors for Sample 1 at various packing fractions. (b) Structure factors for Sample 2 at various packing fractions. (c) Structure factor for Sample 2 at η ) 0.986. (d) Structure factors for Sample 3 at various packing fractions. (e) Structure factor for Sample 3 at η ) 0.96.
Figure 5. Relative compressibility of Samples 1-3 compared with that of a Tonks gas.
Figure 6. Effective pair-potentials for Samples 1-3 compared with that of a Tonks gas.
For the Tonks gas, the potential obtained with our inversion procedure has a very small repulsive ramp in the region 1 e x/σ e 1.5, which can be taken as a measure of the error generated by use of the hypernetted chain approximation. The strongly repulsive part of the hard rod interaction is rather accurately retrieved. Overall, the inferred potential is very close to the hard rod potential. As to the inversion of the experimental data, the calculated pair interactions clearly show that repulsive parts of the effective interactions for Samples 2 and 3, though not as steep as a hard wall, are both very steep and very nearly
the same shape as that of Sample 1. The softening of the repulsive wall from the hard sphere repulsion likely arises from the nonzero width of the distribution of particle sizes (see later). The effective pair potentials for Samples 2 and 3 have very weak attractive components with minima at values of x/σ smaller than that characterizing Sample 1; the possible origin of these attractive components is discussed below. Note that the existence of an attractive component of the effective colloid-colloid interaction is consistent with the increased amplitudes of the first peaks of g2(x) at low packing fraction. What is the origin of the attractive component and the softening of the repulsion from that expected for hard spheres in the effective interactions in all of the samples? We have considered three possible sources: (1) wall-induced attraction between charged particles,6 (2) the previously mentioned attractive interaction associated with surface dimples that can overlap in the air/water interface,15 and (3) an effective attraction artificially generated by treating a polydisperse system as if it is monodisperse. As to possibility (1), in previous work we found that the salt concentration in the suspension had no influence on the inferred colloid-colloid interaction potential, so we have ruled out any contribution to the attractive interaction from the small residual charge of the colloid particle. Possibility (2) relies on the observation that the width of the displacement distribution function transverse to the channel for all of the samples is only a fraction of the width of the channel, implying that there is a water/air interface separating the colloid suspension from the walls of the groove. As to possibility (3), Frydel and Rice21 have shown that analysis of a q2D polydisperse hardsphere fluid as if it were monodisperse both softens the repulsive part of the effective pair interaction and generates an apparent attractive component in the pair interaction; at any fixed polydispersity, the effective attraction increases as the packing fraction is increased. Frydel and Rice also find that increasing
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polydispersity increases the attractive component of the interaction potential and increases the softening of the repulsive component of the interaction potential. To examine how polydispersity affects the pair correlation function and effective pair-interaction potential of a q1D fluid, we carried out Brownian dynamics simulations of two 1D nearly hard-rod fluids,15 with different pair potentials, at packing fractions that were comparable with those used in our experiments. We employed the standard Brownian dynamics algorithm. The term “nearly hard-rod” refers to the use of a very steep, differentiable, repulsive component in the pair potential. For the first simulation, the pair potential was taken to be
U1 ) (σ/x)50
Figure 7. Effect of polydispersity on the pair correlation function from the simulation using the pair-potential U1, compared with the pair distribution function of a Tonks gas.
(4.7)
and for the second simulation, the pair potential was taken to be
U2 ) 12(σ/x)20 - 4.3(σ/x)10
(4.8)
The potential U2 closely resembles the experimentally determined effective pair-potential of Sample 1. The step propagator used was
xn+1 ) xn - Dst
( ∂U ∂x )
n
+ (2Dst)1/2zn
Figure 8. Effect of polydispersity on the effective pair-potential from simulation using the pair-potential U1, compared with the effective pairpotentials of Samples 2 and 3.
(4.9)
in which zn is a Gaussian random number with zero mean and variance unity; Ds is the self-diffusion coefficient in the zero concentration limit; and U is the pair-potential given in eqs 4.7 or 4.8, for the two fluids, respectively. We used a Gaussian form for the particle size distribution to describe the polydispersity, specifically
P(σ) ) exp[(σ - σ0)/2ε2] P(σ0)
(4.10)
where σ0 is the mean particle diameter. Simulations were carried out for ε ) 0.0, 0.05, 0.10, 0.15, and 0.20, at η ) 0.17, 0.30, and 0.50, for both fluids, respectively (the results shown are for η ) 0.3 only). The length of the simulation channel was 100σ; with periodic boundary conditions, the time step was chosen to be 0.001 s; and the simulation was carried out for 5 × 106 time steps. We first examine the effect of polydispersity on the pair correlation function, the structure function, and the effective pair-potential of the fluid with potential U1(x). Figure 7 shows g2(x) at η ) 0.30 for ε ) 0.0, 0.10, and 0.20, respectively. These results are compared with the corresponding functions for a Tonks gas. Clearly, the existence of polydispersity broadens and lowers the first peak in g2(x), so that the effect of the polydispersity is equivalent to softening the repulsive part of the pair interaction. This effect manifests itself in the behavior of effective pair-potential, U(x), as shown in Figure 8. That is, the broader the size distribution, the softer the repulsive part of the effective interaction becomes. However, polydispersity in a system with pair potential U1(x) does not induce an effective attractive interaction in a 1D hard rod fluid, in contrast to the case of a q2D hard sphere fluid. A comparison of the repulsive component of U(x) obtained from the simulation with those inferred from the experimental data for Samples 2 and 3 implies
Figure 9. Effect of polydispersity on the pair correlation function from simulations using the pair-potential U2, compared with the pairdistribution function of a Tonks gas.
that the width of the size distribution is of the order of 0.15, which is larger than specified by the manufacture (0.05). We now examine the effect of polydispersity on the pair correlation function, the structure function, and the effective pair-potential of the fluid with potential U2(x). Figure 9 shows g2(x) at η ) 0.30 for ε ) 0.0, 0.10, and 0.20, respectively. These results are also compared with those for a Tonks gas. Again, the existence of polydispersity broadens and lowers the first peak in g2(x), but the effect is more drastic than for the case that the potential is U1(x). In this case the repulsive part of the effective potential becomes softer as the polydispersity is increased, as it does when the potential is U1(x). Since U2(x) has an attractive component, we expect the effective pair potential to also have an attractive component. Figure 10 shows that the attractive part of the effective potential becomes shallower as the polydispersity is increased, unlike what is predicted for the effect of polydispersity in 2D. A comparison of the repulsive component of U(x) obtained from this simulation with those inferred from the experimental data for Sample 1 implies that the width of the size distribution is of the order of 0.10. Overall, we regard it as plausible that the effective colloidcolloid interactions inferred from our data arise from a combina-
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Figure A1. Snapshot of our q1D colloid liquid at a 1D packing fraction η ) 0.17.
Figure 10. Effect of polydispersity on the effective pair-potential from simulations using the pair-potential U2, compared with the effective pair-potential of Sample 1.
tion of polydispersity (mainly influencing the shape of the repulsion) and interface deformation (dimple) induced attraction. Acknowledgment. The artifact correction described in Appendix I was carried out with the help of the IDL codes developed by Dr. Polin and Prof. Grier. We are grateful for their assistance in the application to our imaging data. The research reported in this paper was supported by a Dreyfus Foundation Mentor Grant to S.A.R. and by the NSF funded MRSEC Laboratory at The University of Chicago (NSF DMR0213745).
Figure A2. Difference between the apparent and real colloid-colloid separations as a function of the real separation.
Appendix I As reported in this paper and earlier work, we have found that in q1D and q2D colloid suspensions that do not wet the confining walls there is an effective colloid-colloid interaction with weak attractive well. We now examine corrections to the measured particle separations arising from overlap of optical images. A method introduced by Polin et al6 allows us to correct retrospectively the imaging data used previously to calculate the pair potential. Briefly, one first identifies a sphere in an image in which all particles are very well separated from each other, so there can be no image overlaps. Then the image of such a sphere is cropped, and a two-sphere image is constructed by duplicating the cropped image a designated distance r away from the original (cropped) image; this process is repeated for variable r. The apparent separation, r˜, between the two spheres in the constructed image is determined using the standard method. Finally, the difference between the apparent separation and the real separation, ∆r ) r˜ - r, is used to obtain the undistorted pair distribution function, g2(r), from the distorted pair distribution function, g2(r˜), with the relation g2(r)dr ) g2(r˜)dr˜, which can be approximated to g2(r) ) g2(r + ∆r)[1 + (d)/(dr)∆r] at low density (eq 4 in ref 6). Figure A1 shows a snap shot of our q1D colloid liquid at a q1D packing fraction η ) 0.18. The sphere in the middle of the group was selected to construct the two-sphere composite, and 100 frames of such an image were used to improve the data accuracy. Figure A2 shows the relation between the difference ∆x ) x˜ - x and the real separation x. The pair distribution function and effective pair-interaction potential, before and after the correction, are plotted in Figures A3 and A4. Clearly the process removed some seemingly nonphysical features in the g(r˜), such as the sharpness of the first peak and the lack of the first minimum. These corrections shift the interaction potential to the left, making the interaction slightly harder, but they do not change the depth of the well significantly within the precision of the inversion, the measure of which we take to be the oscillations in the curve.
Figure A3. Corrected and uncorrected pair correlation functions.
Figure A4. Corrected and uncorrected effective pair-interaction potentials.
Appendix II For our new measurements,19,22 we used silica microspheres with diameter σ ) 1.57 ( 0.04 µm, suspended in water, and confined between a coverslip and a microscope slide with separation between the coverslip and slide achieved by using as a spacer a very small amount of σ ) 2.0 µm Borosilicate glass microspheres in the suspension. The sample colloid suspensions were prepared in a fashion similar to that described in ref 23. The density of the suspension was determined to be F ) 0.26. After confining the q2D colloid suspension, the cell was sealed with epoxy glue and left to equilibrate for 12 h. The suspension was imaged using a Leitz inverted microscope equipped with an oil immersion objective (100×, N.A. ) 1.25) and a CCD camera. The digital video streams were taken by a Sony DVCAM and transferred to a PowerMac G5 computer using FinalCutPro software. The centers of the particles in these images were detected using an Obj-C++ program which employs the Canny-Edge edge detection algorithm24 combined
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Figure B1. Intensity distribution along a particle diameter, from an image of a typical single particle. The line connecting the dots is a visual aid.
Figure B2. Average intensity distribution along a particle diameter, from the images of ten particles (see text).
Figure B3. Correction curve relating real and apparent colloid-colloid separation.
J. Phys. Chem. B, Vol. 113, No. 42, 2009 13749 with the least squares fitting of circles optimized with the Levenberg-Marquardt method.25 In the first phase of the Canny-Edge algorithm, a Gaussian filter is applied to the image; such a filter averages for each pixel its intensity and those of its neighbors. These intensities are weighted according to a Gaussian function centered on the pixel. The process smoothes out the image noise which otherwise would generate random tiny edges. In the second phase, Sobel operators are applied to compute the intensity gradient of each pixel forming the image. Every pixel having a gradient intensity greater than a specified threshold is considered as an edge seed. In the third phase of the algorithm, the edges are grown out of the seeds. To do so, we follow the edges (which at the beginning of the phase are only seeds) pixel-by-pixel perpendicular to the intensity gradient direction. If the next pixel met has intensity higher than a lower threshold, we consider it as being a part of the edge that is being followed. When all the edges of the image are located, they are fit to a circle. An edge is considered to define the boundary of a particle if the best fitting circle radius falls within a specified range: if all the pixels of the edge are no further apart from the circle than 1 pixel and if the edge contains at least 20 pixels. If any of these conditions is not met, that edge is ignored. Figure B1 shows the cross-section of the intensity distribution of a single colloid particle from a typical image of the q2D suspension. Note the slight asymmetry of the distribution and the “oscillations” in the wings of the distribution. To account for variations between the intensity profiles of individual particles arising from polydispersity and small deviations from spherical shape, we constructed an average over the intensity distributions for ten isolated particles, i.e., those that are a few particle diameters away from the rest of the particles in the suspension. The average was constructed as follows: x and y axes were placed with origin at the center of each particle, and the separate partial intensity distributions in the ( x and (y directions for each particle were superposed. Then the distributions for the ten particles were averaged. The result obtained is shown in Figure B2.
Figure B4. Particle patterns used to test the linearity of the superposition correction to the particle-particle separation.
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Figure B5. Comparisons of the corrections to the particle-particle separation generated by the superposition and global corrections, for the five different patterns shown in Figure B4.
Lin et al. correction curve for the inferred particle separation. The correction curve is shown in Figure B3. To correct the positions of particles when more than two images overlap, we must assess the linearity of the separation correction with respect to number of nearest neighbors of a particle. We did this by examining the intensity distributions for four different patterns of particle triplets and one “closepacked” particle septuplet (Figure B4). One “central” particle in each pattern was fixed, and the others were moved in simple patterns, as illustrated in Figure B4. The correction to the position of the central (black) particle obtained by superposition of all the particle images was compared with the sum of the corrections to the position of the same particle arising from each of its neighbors taken individually. The quality of the fit to linearity was judged from the value of the modified Pearson coefficient. Several comparisons between the corrections to the xposition of the central particle generated by the sum of the individual corrections and by the global correction are shown in Figure B5 for all the particle patterns considered. In every case, the difference between the two corrections is negligible. We find that the Canny edge scanning method of particle center detection is less subject to overlap error than the maximum intensity method. This result is reasonable because the Canny edge method is less sensitive to intensity variation within the particle as we are using the gradient of the intensity instead of the intensity to find the center. Finally, the raison d’etre for this investigation is assessment of the reality of the inferred colloid-colloid interaction in a confined geometry. The pair correlation function was calculated and the pair potential was determined using the inverse Monte Carlo method.26 As shown in Figure B6, there is no significant difference between the colloid-colloid interactions determined from the pair correlation functions calculated with corrected and with uncorrected particle positions. We conclude that the effective colloid-colloid interaction found in confined geometries is real and not an artifact of the reduction of the experimental data. References and Notes
Figure B6. Pair potentials inferred from uncorrected (box) and corrected (diamond) particle separations vs particle separation. The connecting line is a visual guide.
We generated the intensities of multiparticle images by superposing the image of a single particle on itself once shifted by a specific amount of pixels. The addition was performed using intensities measured against the image background intensity. The multiparticle images were then fed into the ObjC++ program to generate the putative particle locations and separation. Comparisons of the particle separations generated by the Obj-C++ program and the shifts were used to define a
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