Addition/Correction pubs.acs.org/Langmuir
Correction to “Effect of Interparticle Interactions on Agglomeration and Sedimentation Rates of Colloidal Silica Microspheres” Yung-Jih Yang, Aniruddha V. Kelkar, David S. Corti, and Elias I. Franses* Langmuir 2016, 32(20), 5111−5123. DOI: 10.1021/acs.langmuir.6b00925 S Supporting Information *
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INTRODUCTION This addition and correction contains some corrections to an article published by the authors in Langmuir in 2016.1 The theory is correct, as published in ref 1, and the conclusions remain unchanged. Nonetheless, there is a significant error in the sample calculations made with eq 22 of ref 1. Instead of including each term in each summation in the calculations described in ref 1, we erroneously used only the last term, which is the largest one in the summation. Hence, the reported values in ref 1 are about half of the correct values. These errors affect Figures 5b, 6ab, 8a, and 9 and the associated discussion in
FIRST SECTION OF REFERENCE 1 WHERE CORRECTIONS WERE MADE Corrections were made in certain sentences found in ref 1, from page 5118, right column, line 4 from the bottom to page 5120, left column, line 6 from the bottom, “At W”. The specific sentences changed from ref 1 are underlined. Sample Calculations. Monomer and Clusters Number Densities and Agglomeration Times. Some sample calculations are shown for systems similar to some of our experimental systems, with d1 = 484 nm, Δρ1 = 1.0 g/cm3, ϕ1 = 0.0025 or 0.015, η0 = 0.89 cP, T = 298 K, and L = 1.8 cm. For the loose fractal clusters, the fractal dimension of 1.8 for diffusion-limited agglomeration was used. The time evolution of the monomer, dimer, and tetramer number densities for W = 1 for the SS model is shown in Figure 5a. For ϕ1= 0.0025, in the early stage of agglomeration, the monomers form dimers, whereas the dimers remain nonagglomerating. At t = tSS a2 = 0.05 h, the monomer number density drops below 1% of its original value. Then dimers start forming tetramers. At t = tSS a4 = 0.18 h, the tetramers start forming octamers, which do not agglomerate until tSS a8 = 0.41 h. The time evolution of the monomer and the cluster number densities for the USS model at this value of ϕ1 and at ϕ1 = 0.015 for the SS and USS models are shown in Figure S6. USS/C The agglomeration times, tSS for the compact an , or tan USS/F clusters, or tan for the loose fractal clusters are shown in Figure 5b. The variation of log(tan) with log(n) is quite linear. For the SS model and the same n, the agglomeration times are the same for the compact and fractal clusters. By contrast, for the USS model, the predicted agglomeration rates for the fractal clusters are slightly faster. In general, the agglomeration times are smaller for the USS model than for the SS model. For ϕ1 = USS 0.0025, tSS a2 ≅ ta2 ≅ 0.05 h for both types of clusters. The differences among the predicted agglomeration times for the four cases (SS/compact, SS/fractal, USS/compact, and USS/ fractal) become larger with increasing n. For n = 32, tSS a32 = 1.64 h vs tUSS/C = 1.61 h and tUSS/F = 1.56 h. For n = 512, tSS a32 a32 a512 is USS/F 27.08 h vs tUSS/C = 26.46 h and t = 23.52 h. The a512 a512 agglomeration times decrease strongly with increasing ϕ1; tSS a2 (ϕ1 = 0.015) ≅ 0.01 h vs tSS an (ϕ1 = 0.0025) ≅ 0.05 h. For the USS/compact model, tUSS an increases with n more strongly than for the USS/fractal model. As mentioned earlier, for the same n, the compact clusters are smaller than the fractal clusters, and hence the time for the clusters to reach a diameter dn differs for the four models. For
Figure 5. (b) For particles with d1 = 484 nm, W = 1, and ϕ1 = 0.0025 (black) or 0.015 (gray), agglomeration times tan predicted by the SS/ compact, SS/fractal, USS/compact, and USS/fractal models. The models predict tan only for n = 1, 2, 4,..., 2Nm; hence, the values of tan for other values of n are calculated by interpolation.
the text. The corrections described here are needed for the proper understanding of the original article and for the proper interpretation of the theory. We regret the errors and any inconvenience to the readers of the journal. The next two sections contain the two sections of the text of ref 1 where the corrections made are shown. Another section contains a brief description of the Supporting Information, which has been slightly revised for the same reasons as the text. This note contains Figures 5b, 6a,b, 8a, and 9, which are the corrected versions of the figures in ref 1 with the same numbers. © 2016 American Chemical Society
Published: November 2, 2016 11993
DOI: 10.1021/acs.langmuir.6b03566 Langmuir 2016, 32, 11993−11995
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Figure 6. Agglomeration time tan and intrinsic sedimentation time tsn for ϕ1 = 0.0025 and clusters of size dn for the four models. The respective intersections, where they are equal, represent the values of dn* for n = n* and the net sedimentation half-times ts*. (a) W = 1: point A, SS/compact model; point B, USS/compact model; point C, SS/fractal model; point D, USS/fractal model. (b) USS/fractal model and various W’s: point D, W = 1; point E, W = 10; point F, W = 100; point G, W = 1000.
example, for ϕ1 = 0.0025 and dn = 2000 nm, the agglomeration times are 3.34, 3.26, 1.31, and 1.27 h for the SS/compact, USS/ compact, SS/fractal, and USS/fractal models, respectively; see Figure S7. Overall (or Net) Sedimentation Half-Times. Whereas the cluster agglomeration time tan increases with increasing n, the intrinsic sedimentation half-time tsn decreases; see Figure 6a. The overall sedimentation half times ts* and the cluster sizes for n = n* for which tan = tsn are quite different for the four cases. For the SS/compact and USS/compact cases, the times ts* are 1.72 h (point A) and 1.71 h (point B), respectively; the cluster sizes are about the same, ca. 1600 nm. For the SS/fractal and USS/fractal cases, the sedimentation times are 3.02 h (point C) and 2.93 h (point D), and the cluster sizes are about 4300 and 4500 nm. In general, the predictions of the overall sedimentation half times are in the order SS/fractal > USS/ fractal > SS/compact > USS/compact. The predictions of the cluster sizes are in the order USS/fractal > SS/fractal > USS/ compact > SS/compact. The USS/fractal model should provide the most accurate predictions and was chosen for further calculations and comparison with our experimental data. For interacting particles, as W increases, the agglomeration time increases dramatically, resulting in an increase in the overall sedimentation time; see Figure 6b. For the USS/fractal model, as W increases from 1 to 1000, ts* increases from 2.93 to 16.40 h. At W ≥ 104, t*s reaches the limit of ts1 = 17.41 h for W → ∞. Under such conditions, tan ≫ ts1, implying that monomers sediment without any significant agglomeration.
Figure 8. (a) Predicted overall (or net) sedimentation half-times are plotted with various W’s for the 350, 505, and 750 nm silica particles at w = 0.005.
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SECOND SECTION OF REFERENCE 1 WHERE CORRECTIONS WERE MADE Corrections were made from page 5121, left column, line 4 until page 5121, right column, line 4 from the bottom. Again, the specific sentences changed from ref 1 are underlined. Comparison between Modeling Predictions and Data. For the USS/fractal model, the sedimentation half times ts* for w = 0.005 (ϕ1 = 0.0025) with various values of W were calculated; see Figure 8a. At W = 1 and for the 750, 505, and 350 nm particles, ts* = 2.42, 2.93, and 3.47 h. The respective values of n were quite high; n ≅ 12, 55, and 163; dn ≅ 3000,
Figure 9. Predicted overall (or net) sedimentation half times are plotted with various W for the 505 nm silica particles at w = 0.002, 0.005, 0.01, and 0.03.
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DOI: 10.1021/acs.langmuir.6b03566 Langmuir 2016, 32, 11993−11995
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4500, and 5900 nm; Δρn = 0.19, 0.069, and 0.034 g/cm3; and ϕn ≅ 0.013, 0.036, and 0.075, respectively. When ϕn ≳ 0.05, cluster concentration effects, which are ignored here, may become important. The density differences of the large clusters are much smaller than Δρ1 because the clusters are loose and occlude a lot of solvent. The 350 nm particles agglomerate the fastest to form the largest clusters because they collide more frequently than the larger ones. Those clusters occlude the solvent the most and hence have the smallest Δρn value and the largest ϕn value. The size effects balance the density effects, and hence the t*s values differ by less than the ts1 values of 7.40, 17.45, and 33.36 h. At W = 10 and for the 750, 505, and 350 nm particles, t*s = 4.59, 6.09, and 6.99 h; n ≅ 3, 11, and 33; dn ≅ 1400, 1800, and 2600 nm; Δρn = 0.49, 0.21, and 0.096 g/cm3; ϕn ≅ 0.0051, 0.012, and 0.026, respectively. As W increases, the t*s values increase. The particles form smaller clusters with a higher density difference. When W is about 100, 1000, and 5000 for the 750, 505, and 350 nm particles, ts* values reach the limit of ts*, suggesting that a higher value of W is needed to prevent agglomeration for the smaller particles. The sedimentation half-times t*s from the USS/fractal model were compared to the experimental data in Figures 8b−d. The measured zeta potentials and several possible values of A were used. For the 505 nm particles with w = 0.005 in region I, the t*s value was fairly consistent with the experimental data of 18.3 h, and the particles remained nonagglomerated during sedimentation. For cNaBr = 0 or 50 mM, ts* = ts1 = 17.4 h and dn = d1 = 484 nm for A = 4.4, 5.2, and 5.8 zJ. In region II, ts* was quite different from the observed data, which had large error bars. The sedimentation half-times were sensitive to the NaBr concentration and to the chosen Hamaker constant value and hence to the stability ratio. For cNaBr = 200 mM and A = 4.4, 5.2, and 5.8 zJ, t*s was found to be 17.5, 11.2, and 3.5 h vs the observed data of 5.0 h. In region III, the model overpredicted the sedimentation half-times, as expected, but still indicated that the particles agglomerated rapidly, forming large clusters, and settled rapidly. For cNaBr ≥ 400 mM, ts* = 2.4 h, twice the observed value, and dn ≈ 5700 nm, much larger than d1. The A value that best fits the data in all regions is about 5.2 zJ, which is out of the range of the reported value as adjusted. The reason is unclear but may be related in part to the underestimation of the agglomeration rate in the model. The predicted trends in t*s in all regions were, however, similar to the experimental ones; namely, a constant t*s ≅ ts1 in region I, a decrease in t*s with an increase in the ionic strength in region II, and a small, nearly constant ts* in region III. Similar results were found for the 750 and 350 nm particles, with rough estimates of A ≅ 5.8 and 7.5 zJ, respectively; see Figure 8b,d. The sedimentation half-times ts* for the 505 nm particles with various values of w and W were calculated with the USS/ fractal model. As w increased, ts* decreased, and the limit of W where t*s approaches ts1 increased (Figure 9). This limit was about 300, 1000, 2000, and 5000 for w = 0.002, 0.005, 0.01, and 0.03, respectively. The results suggest that at a given W and at a higher w the particles tend to agglomerate and sediment faster, hence a higher W is needed for the particles to remain nonagglomerating at higher particle concentration. For w = 0.002, 0.005, 0.01, and 0.03 and W = 1, t*s = 3.87, 2.94, 2.40, and 1.60 h; n ≅ 28, 55, 87, and 216; dn ≅ 3200, 4500, 5800, and 9600 nm; Δρn ≅ 00.10, 0.069, 0.051, and 0.027 g/cm3; and ϕn ≅ 0.0096, 0.036, 0.098, and 0.54. In this example, the high
predicted values of ϕn, 0.098 and 0.54, indicate that the predicted ts* values may not be realistic without explicit consideration of particle volume fraction effects. The trend in ts* with w is consistent with the observed formation of the white sediment layers but does not account for the remaining particles in the turbid supernatant layer; see Figure 3. This is because the sedimentation process in the model is assumed to occur after the agglomeration process, during which the volume fraction is assumed to remain uniform. The particle volume fraction decreases, however, after some particles sediment, and then the remaining particles agglomerate and sediment more slowly.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b03566. Additional data for the characterization of the silica particles and the effect of the particle concentration on the sedimentation rate and some detailed derivations and model predictions. Certain figures from the Supporting Information of ref 1 have been modified for reasons described in the Introduction. (PDF)
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REFERENCES
(1) Yang, Y.-J.; Kelkar, A. V.; Corti, D. S.; Franses, E. I. Effect of interparticle interactions on agglomeration and sedimentation rates of colloidal silica microspheres. Langmuir 2016, 32, 5111−5123.
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DOI: 10.1021/acs.langmuir.6b03566 Langmuir 2016, 32, 11993−11995
