Tracking colloidal silica particles to evaluate their dispersion and interactions in concentrated suspensions under shear force applications

This study aimed to characterize interactions within colloidal silica particles in their concentrated suspensions, using rheo‐confocal measurements and imaging, followed by image analysis. We studied the effect of shear rate (0–500 s–1) and solution pH (6, 10) on the dispersion degree of colloidal silica particles via the determination and comparison of interparticle distances and their modeling. Images corresponding to different shear rates were analyzed to identify the coordinates of the particles. These coordinates were further analyzed to calculate the distance among the particles and then their surface‐to‐surface distance normalized by the particle diameter (H/D). It was found that the population of the particles per unit area of the image and H/D varied with increasing shear rate. The comparison between experimentally measured and theoretically calculated H/D identified that for some particles, the former was shorter than the latter, indicating the unexpected attractions among them against the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory. Then, the modification of previously reported equations for H/D was suggested and confirmed its validity. Assuming pair potential interaction and hydrodynamic interaction were the main non‐DLVO interactions, their magnitudes were calculated and confirmed the significance of pH and shear application strength on particle dispersion/coagulation.


INTRODUCTION
Although concentrated colloidal/fine particle suspensions can be found in our daily life and industry products/processes, their proper characterization and manipulation are still challenging [1][2][3].For fine particle processing, its proper dispersion/liberation plays a key role in achieving a good performance in processing plants (e.g., selective enrichment of valuable minerals from natural ores) [2,[4][5][6][7].Hence, precise characterization and manipulation of particle dispersion and coagulation in processing media are necessary [8,9].Derjaguin-Landau-Verwey-Overbeek (DLVO) theory is one of the best known theories for describing particle-particle interactions and thus particle dispersion/coagulation with the summation of the van der Waals potential and electrostatic potential.This is a straightforward theory that can explain particle dispersion/coagulation in many different colloidal systems [10][11][12].On the other hand, there are several cases that the DLVO theory cannot explain, including the cases in high solid concentrations or confined environment.They are as follows: (1) dominance of van der Waals attraction between like-charge particles within narrow interparticle distance below Debye length [2,13] and (2) long-range attraction over several hundred nanometers beyond the range of two interactions considered in the DLVO theory [14,15].As the origins of those interactions are not explained by the DLVO theory, it would be useful to develop a measurement and analysis method to quantify their interactions.
In order to study the long-range interaction between like-charge particles in free or confined situation, previous studies reported microscopic observation methods under static conditions.Vondermassen et al. [16] demonstrated their methodology to determine how effective pair potentials can be experimentally determined using dilute latex particle dispersions with microscopic observations.Crocker and Grier [17][18][19] also reported their microscopic measurement of pair interaction potential using dilute polystyrene sulfate particle dispersions.Sugimoto et al. [20] developed their optical method to measure surfaceto-surface separation distance and force between two free colloidal particles.Kepler and Fraden [21] developed a methodology to determine the pair potential of colloidal particles from the pair-correlation potential, and they confirmed that particle confinement is responsible for a greater attractive interaction than the one predicted by the DLVO theory.Larsen and Grier [22] also reported longrange attractions between like-charge latex particles in confined geometry and provided an explanation of the metastability of colloidal crystals.Carbajal-Tinoco et al. [23] studied the effective pair potential between colloidal polystyrene particles confined between two parallel glass plates under static condition.
As many processes dealing with colloidal particle suspensions are performed under dynamic environment [24,25], observation under such an environment should be useful.The combination of a confocal/optical microscope and rheometer is a good option to study concentrated colloidal systems under shear application [26,27].It was applied to Pickering emulsions [28], colloidal silica particle suspension [29], colloidal poly methyl methacrylate particle suspension [30], and biomaterials [31,32].
This study performed rheological measurements coupled with confocal microscopic imaging in order to characterize interactions within colloidal particle concentrated suspensions via analyzing changes in surface-to-surface distance normalized by the particle diameter (H/D) under different shear rates and pH values.The experimental H/D was compared with theoretically calculated H/D in order to detect and analyze the unexpected attractions among them mentioned above.

Materials
Monodisperse amorphous spherical silica particles from Whitehouse Scientific (silicon microspheres) and their average particle diameter of 7.5 µm were used in this study.Using this original silica solution, silica particle suspensions were prepared in aqueous salt solution (1 × 10 −2 M KNO 3 , Sigma-Aldrich) at particle/solid concentration set to 20 vol.%, and their pH was adjusted using HNO 3 or KOH followed by conditioning the suspensions for 30 min.
A typical ionic strength 1 × 10 −2 M been used to study particle-particle interactions in concentrated colloidal particle suspensions/dispersions [3,33,34].A tiny volume of each sample was pipetted and gently deposited on the glass plate of the rheometer before the commencement of shear load application and image acquisitions.

Confocal-rheometry
The imaging was performed on a VT-Eye confocal microscope using a 560 nm laser for the rhodamine and a 480 nm for the Nile red.An Anton Paar MCR301 rheometer was mounted over the confocal microscope frame [30].The cone with a diameter of 5 cm, 1 • angle sandblasted to a roughness average R a = 4.37 µm with the maximum profile R Z = 33.8µm was used with a truncation gap of 100 µm.
The glass bottom plate was made rough by a coating of fine sand containing (clusters of) grains between 10 and 100 µm in diameter attached with UV curing glue [28].Rheo-confocal imaging and measurements were performed to investigate colloidal particle aqueous suspensions in order to understand the effects of different parameters (e.g., solution pH and applied shear rate) on particle coagulation/dispersion behaviors under both confined and unconfined environments.The experiments with concentrated colloidal silica particle suspensions (20 vol.%) prepared at two different suspension pH values (6,10) were performed under changing applied shear rates (0-500 s −1 ).pH 6 and 10 were selected due to negligible difference in DLVO total potential energies showing shortrange strong repulsive interaction under those pH values (Figure A-1), and they were good candidates to examine the presence/absence of a long-range attractive interaction and quantify it that cannot be explained by the DLVO theory in concentrated colloidal silica particle suspensions.Our experiments can be well suited for this investigation which we find the size of particles and aggregates up to around 20 µm.Moreover, the preliminary study of confocal imaging in Figure A-2 shows the feasibility of the proposed measurements.

Image analysis
Image acquisitions were performed with the purposebuilt confocal rheoscope setup that was introduced in Section 2.2.1 and can perform rheological measurements together with image acquisition simultaneously.A total of 3500 images at pH 6 (e.g., Figure 1A-G) and 4000 images at pH 10 (e.g., Figure 1H-N) were obtained at different shear rates (e.g., 0, 0.005, 0.05, 0.5, 5, 50, and 500 s −1 ).As seen in Figure 1, particle dispersion is clearly visible above 50 s −1 for both pH 6 and 10 as indicated by the decrease in the number of particles observed per unit area of the images.Figure B-1 shows the image analysis procedure.
The obtained images were processed with Fiji/ImageJ to evaluate the dispersion of particles at these shear rates.In Fiji, x and y coordinates were determined for each particle.These coordinates were then treated in the visual studio (Appendix B) to calculate the distance of each particle from the rest of the population using the following equation: where X a , Y a are x and y coordinates of a specific particle and X n , Y n are coordinates of the rest of the particles.
This interparticle distance was then used to calculate the surface-to-surface distance H between particles using the following equation: where r is the radius of the particle.Surface-to-surface distance calculated by using Equation (2) was divided by the diameter of the particle to determine H/D.This ratio was then compared with the calculated H/D values as defined in Equations ( 3) and ( 4) that were developed for hard sphere noninteracting colloidal particles [35,36].By doing so, it was found that experimentally determined values have two population, above and below theoretically calculated value.To fit both these equations, two series of coefficient among k < 1 and k > 1 were determined by using Equations ( 5) and ( 6): where ∅ is the solid/particle fraction.By using k, Equations ( 3) and ( 4) can be modified as Equations ( 7) and ( 8) that can be used under both confined and unconfined environments: Geometric mean of H/D was calculated for subpopulations through Equation ( 9) by using the visual studio (Appendix C) and plotted against k (for k < 1 and k > 1), and the regression analysis was then performed to find the slope and R 2 with the best fitted curve for particular shear rate and pH: where x 1 and x 2 are H/D of particles 1 and 2, respectively, and n is the total number of particles.

Particle numbers and interparticle distance
Figure 2A shows the number of particles observed per unit area of the image at various shear rates for both pH values (6 and 10).For both pH values, the particle population showed very similar trend at the same shear rate (about 200 at 0-5 s −1 while about 130 at 50 and 500 s −1 ), implying that pH has no influence over the dispersion/coagulation of particles observed under a confocal microscope at least.This result does not explain the different rheological behaviors of colloidal silica particle suspensions at pH 6 and 10.As shown in Figure D-1-(A), pH 6 suspension shows shear thinning, whereas pH 10 suspension shows shear thickening behavior which partially agreed with literature [37][38][39].On the contrary, Heinze and Carastan [40] demonstrated decease in the viscosity of fumed silica suspension at 14 vol.%particle concentration with increasing shear rate.Santos et al. [41] also reported decrease in viscosity with increasing shear rate on colloidal silica prepared at 4, 5, and 6 vol.%.In the latter two cases [40,41], the solid fraction (vol.% of solid particle) might not be enough to lead to shear thickening.In order to explain the difference between the thinning and thickening behaviors at pH 6 and 10, the following critical shear stress (  ) model [42] was used: where  b is the Bjerrum length (=  2 ∕4 0  B ), a is the radius of the particle, Ψ (=  s ∕ B ) is the dimensionless surface potential, and  is the Debye-Huckel reciprocal length calculated by using Equation (A-6) [43,44]. is the elementary charge,  is the dielectric constant of water (78.2), 0 is the permittivity of free space,  B is the Boltzmann constant, and T is the room temperature.Experimental zeta potential values of silica particles obtained at the exactly same chemical conditions (i.e., pH 6 or 10 at 1 × 10 −2 M KNO 3 ) [4] were used as  s .
The calculation that  c is 1.45 Pa at pH 10 and 1.17 Pa at pH 6.Compared with our results in Figure D-1(B), at pH 10, experimentally measured shear stress is higher than  c at shear rate of around 0.5 s −1 in the middle of shear thickening, whereas, at pH 6, the shear stress is smaller than  c up to 20 s −1 where shear thinning is observed.This confirms hydroclustering formation at pH 10 while no evident hydroclustering at pH 6 and thus their difference between the thickening and thinning behaviors at pH 10 and 6.
After determining the coordinates for all the particles, the distances between each particle and the rest of the particles were calculated, as explained in Section 2. This led to a distance matrix of n × n, where n is the number of particles determined per unit area of the image under a certain shear rate.A cumulative frequency distribution with respect to surface-to-surface distance H normalized by particle diameter D for both pH is plotted as shown in Figure 2 ((B The ratio of surface-to-surface interparticle distance to particle diameter (H/D), as obtained experimentally, is shown in Figure 2B,C for pH 6 and 10, respectively.These results were compared to H/D calculated using Equations (3) and (4) [35,36], as summarized in Tables D-1 and D-2 for pH 6 and 10, respectively.These two equations are based on the assumption that particles do not interact.However, we found that two different populations of particles exist, one with values lower than those calculated by using Equations ( 3) and (4) and the other with higher value.This finding leads us to conclude that an unexpected interaction among particles not considered in these two equations exists.
Considering the interparticle distance is in the order of single µm or less, the identified interparticle interactions are categorized as long-range interactions that are not explained by the DLVO theory consisted of the van der Waals interaction and electrostatic interaction [45,46].The DLVO theory explains well the interactions between isolated and unconfined pairs of spheres [14].On the other hand, other also reports confined spheres experiencing long-range attraction not explained by the DLVO theory [15].The origin of the long-range attraction between likecharge colloid particles is poorly understood [47].The nature and strength of non-DLVO interactions will be discussed in the latter part of the results and discussion.In the following section, the degree of interaction is quantified via surface-to-surface distance and discussed by the modification of Equations ( 3) and ( 4).
Frequency distributions for the H/D ratio for both pH 6 and 10 are shown in Figures D-2 and D-3, respectively.Both pH values display similar distribution behavior under similar shear rates.From the comparison of these distributions, one cannot clearly see their pH dependency.On the other hand, the distribution is different above the shear rate of 50 s −1 in terms of the magnitude of the most frequent fraction.In general, the mean of H/D is in the range between 6 and 8, irrelevant to the shear rate and pH.

Correlation between k and interparticle distance
As shown in Figure 2B,C; Figures D-2 and D-3; and Tables D-1 and D-2, experimentally determined H/D had distributions that a single value of H/D as calculated through Equations ( 3) and ( 4) cannot fit all of the H/D values.As experimentally determined values have two sets of populations one below Equations ( 3) and (4) and another above, different coefficients are to be multiplied with those equations.After fitting the experimental curve, results indicate that different systems of linear Equations ( 7) and (8) (i.e., the modifications of Equations 3 and 4) correlate well against the geometric mean of H/D for each shear rate and for both pH 6 and 10 as shown in Figure 3  In general, regardless of k > 1 or k < 1, k varies and also depends on shear rate (i.e., 0-500 s −1 ) as shown in Figure 3 and Figure D-4 plotted using the data for the pH 6 sample.The presence of k < 1 (Figure 3) implies the presence of long-range attraction among some particles as Equation (3) assumes no interaction between particles.The modification of Equation (3) multiplied by k (i.e., Equation 6) well explains the experimentally determined H/D with a high coefficient of determination R 2 (0.9763-0.9923), and k has a linear correlation with H/D.Slight deviations from the linear correlation at GM < 0.05 indicate the presence of attractive interactions in addition to repulsive interactions.This point will be further discussed in Figures 4 and 5.
For k > 1, as with the case of k < 1, the modified Equation (3) with multiplied by k works well to fit experimental H/D, and also k and H/D have a linear correlation (Figure D-4).The slight deviations from the linear correlation at k > 1 are more evident than Figure 3 (k < 1, pH 6) but at GM < 2. This can be explained by the collective repulsive interactions against the linear decrease in the magnitude of repulsive interactions among particles.8) with the experimentally determined H/D for the samples prepared at pH 6 and 10.The general trend is the same as the use of the modified Equation (3) (i.e., Equation 7).The modification of Equation (4) multiplied by k (i.e., Equation 8) also well explains the experimentally F I G U R E 5 Sogami-Ise potential (U s−i ) normalized by thermal potential energy as a function of H/D at pH 6 and 10, respectively.The surface charge density values from (A) literature [57], (B) literature [59], or (C) literature [60] were used; summation of Sogami-Ise potential calculated using surface charge density of literature [57] and hydrodynamic potential (U hydro ) normalized by the thermal potential energy as a function of shear rate and H/D at (D) pH 6 or (E) pH 10, respectively.The number at the end of each legend indicates the shear rate ( γ).
determined H/D with a high coefficient of determination R 2 (0.9641-0.9957), and k has a linear correlation with H/D.
Table D-3 summarizes the slope values for all of the shear rates under both pH conditions.The slope values indicate the rate of change between GM and k.In comparison with the slope value without shear (i.e., shear rate of 0 s −1 ), an increase in the slope value was observed with the increase in the shear rate, and the maximum slope value was recorded at the shear rate of either 50 or 500 s −1 .This shows that the rate of change in k value is higher with a high shear rate application, meaning that larger deviation from H/D is calculated from Equation ( 3) or ( 4) with increase in GM.Comparing the pH 6 and 10 results, the slope change between the absence of shear and the presence of the high shear is more noticeable with pH 6.It can be explained by the less particle dispersive condition (i.e., less zeta potential at particle surfaces [4]) assigned by pH 6 than pH 10, and thus, particle dispersion can be more enhanced by shear force application.
The comparison between Equations ( 3) and ( 4) shows no significant difference between them in terms of fitting experimentally determined H/D by multiplying k but the fitting results well explained H/D with a high R 2 shown in Figure 3 and Figures D-4-D-10.In other words, both modified Equations of ( 3) and (4) (i.e., Equations 7 and 8) can be used to well explain the experimentally determined H/D in our case study.

Physical meaning of k values
Figure 4 shows the H/D calculated by using Equation ( 7) (modified Equation 3) as a function of solid fraction and k.The physical meaning of the k values is the nature and degree of interactions among particles present in the system of interest.In general, k > 1 indicates particle repulsion/dispersion under a certain physical/chemical condition (e.g., dominance of hydrodynamic interaction at higher shear rate in our case), whereas k < 1 indicates particle attraction/coagulation under the presence of attractive interaction (e.g., pair interaction).In other words, using k, comparing the experimentally determined H/D and theoretical H/D, we can identify the nature (i.e., repulsive and attractive) and strength of the interaction present among the particles.Conventional radial distribution functions and structure factors are useful to capture ordered structural information [48][49][50][51][52], whereas our developed method can be applicable to investigate both ordered and nonordered structures.The latter is the case for this study that focused on investigating the nature and the degree of particle-particle interactions (Figure 4).The nature and strength of the interaction were also studied by the potential energy calculations, assuming that here the nature of interaction is non-DLVO type interactions.The following contents will discuss this point.
As previously reported, the Coulombic interaction is one of the popular candidates of the long-range interaction between like-charge particles as our case in this study, and Sogami-Ise potential and its derivative have been used to evaluate the Coulombic repulsive interaction and a longrange attractive interaction that is considered similar to the depletion interaction due to the presence of small ions (or macromolecules) between neighboring macroions (colloids) [53].The Sogami-Ise potential energy (U s−i ) normalized by the thermal potential energy [54][55][56] is defined as where Z e is the effective charge on the particle and is defined by the surface charge density, that is, σ = Z e /(πd 2 ) whose literature values of silica particles obtained at similar conditions (i.e., −0.02 C/m 2 at pH 6 and 1 × 10 −2 M ionic strength, −0.12 C/m 2 at pH 10 and 1 × 10 −2 M ionic strength) [57] were used in this study,  is the dielectric constant of the solvent (i.e., water in this study),  is the inverse Debye length, d is the particle diameter, and a is the particle radius.
In order to study the sensitivity of surface charge density on the potential energy, we performed additional literature survey based on the following criteria as we have a limitation on performing experiments to obtain the surface charge density of our particles: (a) Silica particle is amorphous as our particles; (b) particle diameter is above 80 nm as literature [58] reporting that above 80 nm there is no change in surface charge density and equivalent to the flat silica surface that can thus be considered the same as our particle with the diameter of 7.5 µm.This point was also confirmed by comparing literature [57] with [59] showing very similar/same surface charge density with different particle sizes above 80 nm as summarized in Table D- D-4 in Appendix D. We selected literature [57,59] whose reported values were generated under the conditions that passed all our criteria mentioned above.For comparison purpose, additional literature source from Ref. [60] whose surface charge density value missed only our temperature criteria but performed at room temperature was also used.Other literature values [61][62][63] obtained under the conditions that do not completely match our criteria are also cited in Table D-4 as some additional examples but not selected for our calculation and comparison, whereas more others [65][66][67][68][69][70][71][72][73][74][75][76][77] are not listed in Table D-4 due to significant deviation from our criteria.
In addition, as the Sogami-Ise potential does not include the contribution of shear force application, in this study, we also considered the hydrodynamic interaction normalized by the thermal energy, also known as the Peclet number [42,78]: Figure 5A-C shows the Sogami-Ise potential energy normalized by thermal potential energy as a function of H/D, and Figure 5D,E shows the summation of Sogami-Ise potential energy and hydrodynamic normalized by the thermal potential energy as a function of shear rate and H/D.As shown in Figure 5A-C, the Sogami-Ise potential energy is weakly negative at very close surface-to-surface distance (around < 0.1 H/D at pH 10, around < 0.05 H/D at pH 6), whereas the hydrodynamic potential energy becomes strongly positive with shear rate, indicating repulsive interaction dominant.For the Sogami-Ise potential energy, it is more negative and thus more attractive between particles at pH 10 than pH 6 (Figure 5A-C).This agrees with literature [56] showed that the higher the σ, the more the negative potential energy.Moreover, the difference in the potential energies explains the higher viscosity at low shear rate (around < 0.1 s −1 ) at pH 10 than pH 6 (Figure D-1(A)) but it cannot be explained by the classical DLVO theory that indicates the higher repulsion with the higher zeta potential of silica particles at higher pH [11].By comparing Figure 5A-C, they indicate that there is no significant difference on potential energies using different surface charge densities reported in literature that used the same particle type and conditions as our experiments.
As shown in Figure 5D,E, the summation of the two potential energies can indicate the particle attraction or repulsion.It is found that the hydrodynamic potential energy strongly influences the interaction between the particles with increase in the shear rate.What if we apply the same threshold potential energy as we use the DLVO theory (15 k B T [2]), we can clearly see the difference between the potential energies between the shear rates of below 0.05 s −1 and above 0.5 s −1 .As we discussed before in this article, particle dispersion was clearly visible above 50 s −1 (Figure 2), but the hint of particle dispersion also identified at this small shear rate of 0.5 s −1 due to highly repulsive interaction between particles.

CONCLUDING REMARKS
Different industrial processes involving fine/colloidal particle processing require their effective dispersion/liberation, which is essential to maximize the overall efficiency of the process.This study aimed to characterize interactions within colloidal silica particles, using rheological-confocal measurements and imaging, followed by image analysis.
After the image analysis, the population of the particles per unit area and their surface-to-surface distance normalized by the particle diameter (H/D) were calculated.Upon the shear force application, the former decreased, whereas the latter increased.The results showed that the dispersion of particle was independent of the assigned pH (6, 10) but was dependent on shear rates (0-500 s −1 ).Shear rate of 50 s −1 was a critical shear rate for our colloidal systems in terms of particle dispersion.The shear thinning behavior at pH 6 was nicely explained by the image analysis of the particle dispersion status, whereas the shear thickening behavior at pH 10 required the shear thickening model for its explanation.
The experimentally determined H/D was compared with theoretically calculated H/D.It was identified that for some particles, the former was shorter than the latter, indicating the unexpected attractions (non-DLVO interactions) among them.The modification of the theoretical equations for H/D was proposed and confirmed its validity.In addition, assuming pair potential interaction and hydrodynamic interaction were the main non-DLVO interactions, their magnitudes were calculated and confirmed the significance of shear application strength on particle dispersion/coagulation.
The developed image analysis methods and modified equations for H/D can be applicable to not only silica particles but also many other types of colloidal particles used in different industries, such as foods, drugs, and cosmetics.

A C K N O W L E D G M E N T S
This research was partially funded by ESMI, grant number E150300690.

C O N F L I C T O F I N T E R E S T S TAT E M E N T
The authors have declared no conflict of interest.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available in the Supporting Information section of this article.
Number of particles observed per unit area of the image under different shear rates and at different pH.Cumulative distribution particle pairs as a function of surface-to-surface interparticle distances relative to particle diameter (H/D) and shear rate at (B) pH 6; (C) pH 10.
) H/D at pH 6, (C) H/D at pH 10).Irrespective of the pH values, two different behaviors of particles' dispersion were observed, below and above shear rate of 50 s −1 .This noticeable change in the frequency distributions of H/D corresponds to the results shown in Figure 2A where less particles were observed with higher shear rate indicating particle dispersion.Another point to note is that approximately 80% of particles are present in the surface-to-surface distance smaller than 70-85 µm (i.e., about H/D = 10) irrespective of the shear rates and pH.

F I G U R E 3
Geometric mean (GM) of H/D for k < 1, pH 6, and Equation (3).Shear rate (s −1 ) = (A) 0, (B) 0.005, (C) 0.05, (D) 0.5, (E) 5, (F) 50, or (G) 500.The unit of GM is µm.F I G U R E 4 H/D calculated by using Equation (7) (modified Equation 3) as a function of solid fraction and k.As with pH 6, pH 10 sample results in the same trend.Regardless of k > 1 or k < 1, k varies and also depends on shear rate (i.e., 0-500 s −1 ) as shown in Figures D-5 and D-6.For both k < 1 and k > 1, the modification of Equation (3) multiplied by k (i.e., Equation 7) well explains the experimentally determined H/D with a high coefficient of determination R 2 (0.9727-0.9961), and k has a linear correlation with H/D.Similarly, Figures D-7-D-10 (in Appendix D) show the fitting results of using Equation (4) by multiplying k (i.e., Equation 4 [57, 59-64] (Appendix D); (c) data at both pH 6 and 10, that is, the same pH values as ours, are available; (d) 1-1 salt (e.g., KCl) as our salt; (e) 1 × 10 −2 M salt concentration as ours; (f) temperature is 298.15K or similar as our experiment; and (g) experimental data or experimentally validated model data.The summary of our literature survey on surface charge density is given in Table