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Effect of particle roughness on the rheology of suspensions of hollow glass microsphere particles
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Effect of particle roughness on the rheology of suspensions of hollow glass microsphere particles Christopher Hoyle, Shaocong Dai, Roger Tanner, Ahmad Jabbarzadeh∗ Faculty of Engineering, School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia
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Keywords: Noncolloidal suspensions Smooth surface Effect of surface roughness Hollow glass microspheres Suspension rheology Wet chemical etching
a b s t r a c t The rheology of high volume fraction (30–40%) suspensions with hollow glass microspheres (radius r = 15 μm) in a matrix of silicone oil has been studied experimentally. The surface of the hollow glass microspheres was roughened by wet chemical processing, using hydrochloric (HCl) acid and also sodium hydroxide etching. We obtained particles of various roughnesses in the range of Ra = 62–655 nm. The effect of particle roughness on the overall rheology is investigated, demonstrating its clear impact. The effect depends on the volume fraction, the magnitude of roughness and shear rate. At a 30% volume fraction, increasing the roughness ratio (Ra /r) from 0.42% to 4.44% does not result in a measurable change in viscosity. At a higher volume fraction of 40%, even a modest increase of the roughness ratio to 1.5% causes a significant increase in the viscosity of 117%, 97%, 50% and 39%, respectively at shear rates of 0.47, 1, 10 and 100 s−1 . Increasing the roughness to 4.44% results in an increase of 272%, 105%, 41%, 23% in viscosity over the same range of shear rates. In all cases, we show that the effect of roughness diminished by increasing the shear rate. Some estimates of the average interparticle coefficient of friction are given- showing roughness increases friction. The friction coefficient is shown to increase with the volume fraction. Friction coefficient shows a power-law dependence on shear rate and decreases with the shear rate. The rate of decrease with shear rate is smaller for higher volume fraction of 40%.
1. Introduction The rheology of non-colloidal suspension systems where the hydrodynamic forces are much larger than Brownian forces is of great importance in many applications. For systems with volume fractions larger than 𝜑 = 0.25, the chance of particle-particle interactions increases significantly. In such situations, the surface roughness of the suspended particles may become an essential factor in the overall rheological behaviour of the systems. More can be found in a recent review [17] about various aspects of suspension rheology. The effect of particle roughness for colloidal suspensions has been studied by a few groups. Castle et al. [3] have shown roughness increased viscosity of latex suspensions. Hsiao et al. [5] also studied the effect of roughness of 2 μm colloidal PMMA particles. Hsu et al. [7] have studied the effect of roughness on shear thickening in colloidal suspensions. During the fabrication process of PMMA colloidal particles, they varied cross-linking agent concentration to introduce heterogeneous roughness. However, the SEM images showed some deviation from a spherical shape. Previous work by Lootens et al. [11] has also studied the effect of roughness on colloidal silica particles of 1 μm diameter for volume fractions 41–48%.
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On the other hand, the effect of roughness on non-colloidal particles, which is our concern here, has been studied by Tanner and Dai [18] and Moon et al. [14]. They have previously shown that roughening a system of polystyrene microparticles in a matrix of Newtonian silicone oil or non-Newtonian Boger fluid results in a significant increase of up to 35% in viscosity. They have also shown that the results agreed reasonably well with mathematical models which took into account interparticle friction. In that work, mechanical roughening was used as an easy and effective way to generate surface roughness on smooth microparticles. In this process, cross-linking grinding and milling are used, as the autogenous roughening by other microparticles causes particle damage on the micro or nanoscale. This method was used on 40–80 μm sized particles, and roughness ratios (the ratio of average roughness to sphere radius, Ra /r) of 0.15% to 5.3% were achieved by varying the amount of time the particles spent in the grinding machine (0 to 10 h respectively). Although this roughening process is simple, there is evidence of some change in the shape of the particles, whereas for a satisfactory analysis of the results we need to keep an approximately spherical shape of the particles. Tanner and Dai [18] had confidence that by keeping the roughness ratio around 5% this objective would be fulfilled. To make sure this deviation from a spherical shape in previous experiments does
Corresponding author. E-mail address: [email protected] (A. Jabbarzadeh).
https://doi.org/10.1016/j.jnnfm.2020.104235 Received 1 November 2019; Received in revised form 7 January 2020; Accepted 7 January 2020 Available online xxx 0377-0257/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: C. Hoyle, S. Dai and R. Tanner et al., Effect of particle roughness on the rheology of suspensions of hollow glass microsphere particles, Journal of Non-Newtonian Fluid Mechanics, https://doi.org/10.1016/j.jnnfm.2020.104235
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not have an impact on the results, we decided to repeat the experiments by using a different roughening method that would better preserve the spherical shape of the particles. A review by Hsiao and Pradeep [6] offers various methods used for roughening colloidal and non-colloidal particles. For the work in this paper, the wet chemical process offers a suitable approach to roughening the surface of microparticles. This method has been used widely in the industry on glass particles, and in dentistry to etch enamel parts. Zogheib et al. [20] used hydrochloric (HCl) acid to change the surface roughness and flexural strength of a lithium disilicate glass-ceramic. It was found that etching increased surface roughness and this increase was more significant for longer etching durations. Jang et al. [8,9] investigated the effects of hydrochloric acid and sulfuric acid on the surface of glass slides showing the RMS roughness could be increased by almost a factor of 10 depending on the type of the acid, the duration and processing methods. Shellenberger and Logan [15] used wet chemical etching to change the surface roughness of soda-lime glass beads. They used both sodium hydroxide (NaOH) and HCl and showed the effectiveness of the method to alter the surface roughness. Different wet chemical etching methods such as phosphoric acid were used for roughening enamel [4] where the RMS roughness could be increased by an order of magnitude. In this work, we study the effect of particles surface roughness in suspensions containing rigid hollow glass microspheres. We have also used a roughening method different from that used by Tanner and Dai [18]. Here wet chemical etching is used to affect only the surface of the microspheres, and there is no deformation or shape change in the spherical form of the particles. We have done the experiments for suspensions of hollow glass microspherical particles (HGMP) in a matrix of silicone oil of 1.1 Pa.s viscosity. 2. Methodology and experimental procedure 2.1. Wet chemical etching of glass particles The particles used in this study were hollow glass microspheres (HGMS) made from Soda-Lime glass which were supplied by Cospheric LLC, CA, with nominal diameters between 27 μm and 32 μm, and a density of 0.51 g/cc, as shown in Fig. 1a. The wet chemical etching was used to roughen the HGMS, and then 30% and 40% volume fraction suspensions with silicone oil were prepared. Before etching, a quantity of HGMS was weighed and placed into a beaker and submerged in 37% Hydrochloric acid solution. The HGMS HCl solution was boiled at 110 °C for 30 min and then diluted with water. It was poured through filter papers and then dried in an oven at 60 °C for 24 h to recover the microspheres. The etched HGMS particles were scraped off the filter papers and passed through a 32 μm sieve with a brush to remove any clumps of particles. Fig. 1b shows the Scanning Tunneling Microscopy (STM) images of HCl treated HGMS particles. The method was repeated using a 12.5 M NaOH solution. A 12.5 M solution of NaOH contains 50% by mass of pure NaOH. To prepare a solution, an equal mass of NaOH pellets and distilled water were mixed until a solution was formed. Instead of boiling the solution, the HGMS was only soaked for 30 min in the NaOH solution. Fig. 1c shows a Scanning Tunneling Microscopy (STM) image of NaOH treated HGMS at 2780 magnification. An important point to note is that acid etching often works by selectively attacking certain species in a sample, and sometimes chemically reacting with others. This limits its application to homogenous materials (such as PMMA). Additionally, a change in the chemical composition of a microsphere surface could, in turn, change the rheological properties of a suspension made up of these particles. This would prevent the effects of surface roughness being completely isolated. Hence, care must be taken when considering this method; samples should be thoroughly washed before drying to remove any products of the acid or alkaline reaction.
Fig. 1. STM images of hollow glass microspheres (HGMS) with nominal diameters between ∅27 μm and ∅32 μm silicone for a) untreated (b) HCl treated and (c) NaOH treated particles.
2.2. Characterizing roughness The STM images in Fig. 1 show the treated and untreated HGMS particles. While treatment of particles by HCl brings some change on the surface, a more significant change appears when NaOH is used to treat the particles. The surface modification for HCl treated particles is achieved by etching some material away, whereas for NaOH treated particles, the surface is modified by the formation of spiky deposits of sodium silicate. In all cases, the basic spherical shape of the particles is preserved. To quantify physical change on the surface, we have measured the surface roughness of untreated and acid and alkaline treated HGMS. The surface roughness of the particles was measured on a nanotriboindentation device (TI900, Hysitron Inc., USA) using a diamond
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Table 1 Average roughness (Ra ) calculated from several batches for untreated and treated hollow glass microspheres (HGMS). Particle
Batch Averaged Roughness Ra (nm)
Untreated HGMS HCl treated HGMS NaOH treated HGMS
62 ± 4 223 ± 19 655 ± 82
Roughness Ratio (Ra /r) 0.42% 1.5% 4.44%
Berkovich indenter. Before testing, the z-axis, tip-optics and tip-area were carefully calibrated. The measured area for a single particle was visually identified using the optics. Then the probe was moved in a raster scan pattern across the surface of the particle using a three-axis piezo positioner. The scan rate for all the tests was 0.1 Hz, with a setpoint of 2 μN. It was necessary to scan a very small area to avoid the curvature of the sphere being counted as roughness. (The average roughness is found by taking the absolute value of the deviation from the spherical shape) Roughness measurements of several particles taken from various batches of treated and untreated particles were conducted and the average roughness Ra determined for each batch. Table 1 shows the measured surface roughness of spheres for two batches of untreated HGMS, 4 batches treated by HCl, and 3 batches treated by NaOH. The batch averaged values for untreated, HCl treated, and NaOH treated, HGMS particles are also shown in this table. The roughness is about 62 ± 4.00 nm for the untreated glass particles, showing a relatively smooth surface. For HCl treated HGMS batches, the average roughness is 223 ± 19 nm, and 655 ± 82 nm for NaOH treated HGMS. The results show that both treatment methods produce consistent and repeatable levels of roughness. However, the magnitude of roughness is much higher for NaOH treated samples. The systems treated by HCl increased the surface roughness by a factor of 3.6, whereas the systems treated by NaOH showed an order of magnitude increase in the average roughness. Adopting the previously defined roughness ratio [17] as the percentile ratio of average roughness to particle radius, we obtain values of 0.42% for the untreated particles, and 1.5% and 4.44% for particles treated respectively by HCl and NaOH. The roughness ratio 1.5% for HCl treated particles is one-third of that obtained (5.1%–5.3%) by Dai and Tanner [18] for PMMA and PS particles using the mechanical roughening method. The roughness ratio of 4.44% for the NaOH treated system is also lower than these values. The results confirm that the spherical shape of the particles is well preserved after roughening by both chemical etching methods. 2.3. Rheometry An Anton Paar MCR302 rheometer was used to conduct all rheological measurements in our lab. Parallel plate geometry was used with 25 mm diameter plates in the rheometer. We gently mixed the samples before testing them to homogenize the material. The rheological measurements were conducted at a 0.6 mm gap and 24 °C. The selected gap was more than an order of magnitude larger than the size of particles, ensuring reliable measurements of the rheological properties. The small gap also minimizes the possibility of the material flowing out of the gap. A sudden drop in viscosity usually occurs if the material comes out of the gap. It also leaves marks on the lower plate. In all cases, we conducted a visual inspection and monitoring of the viscosity and made sure this did not happen. We follow a similar experimental procedure to that described in previous works [17,18]. To avoid particle sedimentation, a modified Shields number greater than 3 is needed [17]. The modified Shields number is given by Eq. (1), 𝑠ℎ∗ =
𝜂0 𝛾̇ Δ𝜌𝑔𝐷
(1)
Fig. 2. Viscosity as a function of shear rate for silicone oil matrix, and 30% (open symbols) and 40% (filled symbols) HGMS silicone oil suspensions. For each volume fraction, the results are shown for suspensions made from untreated HGMS particles and those roughened by HCl and NaOH treatments.
where 𝜂 0 is the suspending medium viscosity (𝜂 0 = 1.1 Pa.s for 1000 cSt silicone oil), 𝛾̇ is the shear rate, Δ𝜌 is the difference in densities between the suspending medium (silicone oil = 970 kg/m3) and the suspended particles (hollow glass = 510 kg/m3 ), g = 9.81 m/s2 is the gravitational acceleration, and D is the diameter of the suspended particles taken to be 29.5 μm. The calculations show for 𝛾̇ > 0.36 s−1 the Sh∗ >3 and no sedimentation is expected. We have done the rheological measurements for all samples for a range of shear rates between 0.01–100 s−1 , but here we only show the results for a range of 0.1–100 s−1 to avoid sedimentation effects. 3. Results and discussion Silicone oil was used as a Newtonian matrix. The measured viscosity of the silicone oil was 1.10 PaS at 24 °C. This fluid has an almost constant viscosity independent of shear rate between 0.1 and 100 s−1 . Fig. 2 shows the viscosity of the untreated HGMS suspension system at 30% volume fraction. The results here are the average of two separate measurements. The results for suspension made with HCl and NaOH treated particles are also shown for comparison. There was a slight decrease in viscosity at low shear rates. If we consider shear rates > 0.36 s−1 where we expect no sedimentation, for the 30% untreated systems the viscosity remains relatively constant. However, for the 40% untreated system, there is a slight decrease at shear rates higher than 30 s−1 (see Fig. 2). The normal force measured for these systems was small for the 30% systems; however, for the 40% system, it was small at lower shear rates and started to increase for shear rates above 10 s−1 . Considering the average viscosity of the 30% systems, the relative viscosity at shear rates of 1–10 s−1 for the 30% untreated system is 𝜂 r = 3.46. The relative viscosity of the 40% untreated HGMS system for the same range of shear rates is 𝜂 r = 9.36. These values are very close to those found by Tanner and Dai [18] for “smooth” (actual roughness ratio ~0.15–0.45%) 40 μm Polystyrene and PMMA particles at the same volume fractions (3.8 and 9.7). These values are higher than the computational values which do not take into account the particle-particle friction forces. Computations by Mari et al. [10] give 𝜂 r = 3.7 and 6.2 for 30% and 40% volume fraction for frictionless systems, whereas the computations by Sierou and
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Table 2 Relative viscosity for various shear rates and particle roughness for HGMS suspensions of two different volume fractions 0.3 and 0.4. Shear rate 𝛾, ̇ s−1 𝝋 = 0.3 0.47
1
10
100
𝝋 = 0.4 0.47
1
10
100
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Roughness (nm)
Relative viscosity 𝜂 r
62 223 655 62 223 655 62 223 655 62 223 655
3.73 4.18 4.07 3.53 4.00 3.76 3.41 3.78 3.24 3.3 3.55 3.06
62 223 655 62 223 655 62 223 655 62 223 655
9.64 20.91 25.27 9.36 18.5 20.14 9.36 14 13.23 8.03 11.18 9.91
Brady [16] reported 𝜂 r = 3.1 and 6.4 for 𝜑 = 0.30 and 𝜑 = 0.40 respectively. From the results in Fig. 2 for the viscosity of 30% volume fraction, HCl treated HGMS, we can also see some shear-thinning effects at a shear rate lower than 1. This can also be observed for the NaOH treated system. We believe this is most likely due to sedimentation which is expected at 𝛾̇ < 0.36 s−1 . The results shown here are the average of the viscosity of the three samples and show for 𝛾̇ = 1–10 s−1 that the viscosity values are relatively constant. The average viscosity for this range of shear rates is 𝜂 = 4.27 Pa.S which gives relative viscosity of 𝜂 r = 3.88, which is 11% higher than those for the untreated HGMS at the same volume fraction. However, the difference is within the margin of error we calculated for the measurements for these two systems (see Table 2). The average normal force for the 30% HCl treated suspension system is very small and is not reported here. The viscosity of 30% volume fraction NaOH treated HGMS suspensions were also measured over shear rates 0.1–100 s−1 . The average results of the two samples are shown in Fig. 2. Working with NaOH treated HGMS suspension samples were, in particular, difficult, and care should have been taken that the particles were sieved before making the suspension. Otherwise, particular agglomeration would lead to large viscosity. The measurements made for the two different samples were very close. In this case, all systems showed strong shear thinning up to a shear rate of 10 s−1 . Beyond this shear rate, the shear-thinning is much weaker. At a shear rate of 1 s−1 the viscosity is 𝜂 = 4.14 Pa.s. In comparison at the same 30% volume fraction, for the HCl treated and untreated systems 𝜂 = 4.39 Pa.s and 𝜂 = 3.89 Pa.s. This shows only a moderate increase of ~6% in viscosity when comparing the untreated HGMS, and NaOH treated suspension system. The differences in the results, however, were within the statistical uncertainty, therefore at this 30% volume fraction, we concluded there was no measurable effect of roughness on the viscosity. The results in Fig. 2 also show the viscosity for 40% volume fraction suspension fluids made by HCl treated, NaOH treated and untreated HGMS particles. Increasing the volume fraction to 40%, for the HCl and NaOH treated suspensions results in a significant increase in the viscos-
ity. We can see while for untreated HGMS system the viscosity is relatively stable, for the HCl and NaOH treated HGMS there is an evident shear thinning that persists across the imposed shear rates. The viscosity for 40% volume fraction HCl treated HGMS significantly decreases with increasing the shear rate, from 20.3 Pa.s at 1.0 s−1 to 12.3 Pa.s at 100 s−1 . For the NaOH treated HGMS suspension viscosity decreases from 22.15 to 10.9 PaS over the same range of shear rates. Considering the shear-thinning persists at 𝛾̇ > 0.36, the effect is not related to sedimentation. The overall viscosity is also much higher (see Table 2) for the treated HGMS systems. The average shear viscosity in the range of 𝛾̇ = 1–10 s−1 for untreated and HCl treated systems are 𝜂 = 10.3 Pa.s, and 𝜂 = 17.6 Pa.s respectively which give relative viscosities of 𝜂 r = 9.20 and 𝜂 r = 16 showing a significant increase in viscosity due to roughening of the HGMS particles by HCl treatment. For the NaOH treated suspensions, the average viscosity over the same range of shear rates is 22.15 PaS (𝜂 r = 20.13) showing an even higher increase in the viscosity. From the results above we deduce that the effect of roughness on the viscosity of a suspension depends on the magnitude of roughness, volume fraction, and shear rate. To put this in perspective, we have plotted viscosity as a function of HGMS particles roughness for four different shear rates of 0.47, 1, 10, 100 s−1 , where we do not expect sedimentation of particles. The results are presented in Fig. 3a and b for 30% and 40% volume fractions. From Fig. 3a, for 𝜑 = 0.3 we can see that increasing the roughness from 62 nm to 223 nm has no measurable effect. Increasing the roughness further to 655 nm causes a small increase in the viscosity. This increase, however, is smaller than the uncertainty in our measurements for HCL and NaOH treated systems Therefore we can conclude for 30% volume fraction systems a 4.44% roughness ratio (Ra = 655 nm) results in no measurable increase in the viscosity. The result presented in Fig. 3b for the higher volume fraction, 𝜑 = 0.4 show a moderate increase of roughness from 62 nm to 223 nm caused the viscosity to increase across the entire range of shear rates (0.47– 100 s−1 ). At shear rates 0.47, 1, 10 and 100 s−1 the increased roughness causes respectively 117%, 97%, 50% and 39% rise in the viscosity. For the suspensions made from NaOH treated HGMS of 4.44% roughness ratio (Ra = 655 nm), the viscosity also increases by 272%, 105%, 41%, 23% for shear rates of respectively, 0.47, 1, 10 and 100 s−1 . The results show that increasing roughness from 223 nm to 655 nm causes a further increase in viscosity for a shear rate of 0.47 and 1 s−1 . However, at higher shear rates, the viscosity is slightly lower for the NaOH treated system. Table 1 Relative viscosity for various shear rates and particle roughness for HGMS suspensions of two different volume fractions 0.3 and 0.4. 4. Discussion The tests presented above show that increased roughness generally increases the relative viscosity above the value for the smoothest spheres. One can estimate the average friction coefficient (μ∗ ) for any roughness by using the bootstrap correlation [19]. It was shown there that the ratio of the frictionless relative viscosity (𝜂 r ∗ ) to the frictional relative viscosity was, approximately, 𝜂𝑟 =
𝜂𝑟∗
{1-𝑘𝜇 ∗ 𝑃 ∕𝜏}
(2)
where k = 1.75 and P/𝜏 is the ratio of the interparticle pressure P to the shear stress; in Tanner et al. [19] the relation between P/𝜏 and μ∗ is given for 𝜙 = 0.3–0.5. Knowing 𝜂 r and 𝜂 r ∗ (calculated), one has { } 1 − 𝜂𝑟∗ ∕𝜂𝑟 ∗ 𝜇 = (3) 𝑘(𝑃 ∕𝜏) Here 𝜂𝑟∗ = 3.26 and 𝜂𝑟∗ = 6.12, respectively, for 𝜑 = 0.3 and 𝜑 = 0.4 are used based on the average values from several computational studies [19]. Hence, bearing in mind that P/𝜏 is a function of μ∗ and 𝜙, using
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Fig. 3. Average viscosity is plotted against particle roughness at different shear rates for a) 30% and (b) 40% volume fraction HGMS-Silicone oil suspensions.
ness ratios 0.42%, 1.5% and 4.44%. Showing parameter A increases with the roughness ratio. The exponent 𝛼 also increases with the roughness ratio. For 30% suspensions with particles of roughness ratios 0.42%, 1.5% A = 0.257, 0.4, and 𝛼 = 0.384, 0.181. Therefore, for higher volume fraction we conclude, A values are larger, however 𝛼 values are smaller. This indicates that the loss of frictional effects with shear rate is less severe for concentrated suspensions. The goodness of fits (R2 = 0.79) showed weaker power-law dependence for the 30% NaOH suspension system over the examined shear rates. However it still showed a reduction of friction coefficient with increased shear rate. Interparticle distance between spheres in an idealised homogenous monodisperse suspension system can be found from Eq. (5) [12,13]: (√ ) 𝜋 𝐷𝑝𝑝 = 𝐷 3 (5) −1 6𝜙
Fig. 4. Estimated average friction coefficients (μ∗ ) as a function of shear rate for suspensions made from HGMS of different roughness ratio. The dash-dot lines are the power fit to data for the 40% volume fraction suspensions. The goodness of the fit (R2 ) for each curve is shown in the legend.
P/𝜏 = 0.2 and 0.52 [19] respectively for 𝜑 = 0.3 and 𝜑 = 0.4, one can calculate the friction coefficient using the measured relative viscosity values 𝜂 r . The friction coefficient is plotted versus the shear rate for all suspensions in Fig. 4. One sees that μ∗ decreases as 𝛾̇ increases and is large for larger roughness ratios and larger volume fractions, as in Tanner and Dai [18] and Moon et al. [14]. Since the dry friction coefficient of ‘smooth’ glass on glass is about 0.9–1.0 [2] the results in Fig. 4 are acceptable. The results of μ∗ versus 𝛾̇ for the 30% and 40% volume fraction suspentions are fitted into power lines. That gives the following power-law relationship between the friction coefficient and shear rate: 𝜇 = 𝐴𝛾̇ ∗
−𝛼
(4)
We find from the fits to the 40% volume fraction suspensions data, A = 0.38, 0.4, 0.77, and 𝛼 = 0.073, 0.083, 0.126 respectively for rough-
where Dpp is the average surface-to-surface interparticle distance and D is particle diameter. The calculated interparticle distance for 𝜑 = 0.3 and 0.4 respectively are Dpp = 6.1 and 2.8 μm. That shows for the higher volume fraction of 40% the interparticle distance is less than half that of 30% system. Therefore, the particles are more likely to come into direct contact, and the effective frictional forces would be larger. The reduction in effective friction coefficient with the shear rates is due to the contribution of hydrodynamic lubrication forces that kicks in at higher shear rates as the relative velocity of the impinging particle increase. However, at lower shear rates, the particles are more likely to interact under boundary and mixed lubrication regime. Therefore, the effective friction coefficient approaches to that of the dry contact regime. The effective friction coefficient μ∗ in Fig. 4 is seen to approach ~1 as one moves to lower shear rates and higher concentrations. One expects that the static friction coefficient would be larger than ~1 for the etched glass particles. Blair et al. [1] have shown that the friction coefficient of HCl treated soda-lime glass beads (~3 mm size) increases threefold in comparison to an untreated smooth surface. Therefore these larger friction coefficients are likely to appear at higher concentrations. 5. Conclusions From the results obtained here, we can conclude that the roughness increases viscosity. Here the methodologies for roughening the surface of the particles were such that there was minimal effect on the overall spherical shape of the particles. The results at the lower volume fraction of 30% for roughness ratios of 1.5% and 4.44%% showed increased
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viscosity only at the lowest shear rate due to sedimentation. There was no measurable effect on viscosity for shear rates above 1.0 s−1 . However, for 40% volume fraction, even a roughness ratio of 1.5% caused a significant rise in the viscosity. The increase in viscosity was shear rate dependant and, varied from 97% increase at a shear rate of 1 s−1 to 39% for a shear rate of 100 s−1 . For 40% volume fraction, increasing the roughness ratio to 4.44% results in further ~155% and 8% increase in viscosity at lower shear rates of 0.47 and 1 s−1 , however at higher shear rates the viscosity did not show any further increase. It is interesting to see that these increases are much higher than those observed for PMMA and PS particles [Tanner and Dai [18]] where a roughness ratio of 5.1–5.3% caused a modest increase in the viscosity of about 4–6% at 𝜑 = 0.3 and 5–14% at 𝜑 = 0.4. This may be explained by different friction coefficients for the glass particles than those for polystyrene. The coefficient of friction for unlubricated smooth glassglass contact is 0.9–1 whereas for PS-PS contact it is half that value ~ 0.45–0.5. The friction coefficient values calculated here at low shear rates also approaches that of glass and confirms this hypothesis. Declaration of Competing Interest We as the authors of the manuscript entitled “Effect of Particle Roughness on the Rheology of Suspensions of Hollow Glass Microsphere Particles” submitted to Journal of Non-Newtonian Fluid Mechanics for publication hereby declare that this manuscript is an original work, and it has not been submitted to nor published in any other journal. We declare no conflict of interest. Acknowledgment We thank Mr Hesamodin Jami for help in obtaining the STM images. References [1] D.L. Blair, N.W. Mueggenburg, A.H. Marshall, H.M. Jaeger, S.R. Nagel, Force distributions in three-dimensional granular assemblies: effects of packing order and interparticle friction, Phys. Rev. E 63 (4) (2001) 041304. [2] F.P. Bowden, D. Tabor, The Friction and Lubrication of Solids, Oxford U. Press, 1954.
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[3] J. Castle, A. Farid, L.V. Woodcock, The effect of surface friction on the rheology of hard-sphere colloids, Trends in Colloid and Interface Science X, Steinkopff, 1996, pp. 259–265. [4] B.B. Cerci, L.S. Roman, O. Guariza-Filho, E.S. Camargo, O.M. Tanaka, Dental enamel roughness with different acid etching times: atomic force microscopy study, Eur. J. Gen. Dent. 1 (3) (2012) 187. [5] L.C. Hsiao, S. Jamali, E. Glynos, P.F. Green, R.G. Larson, M.J. Solomon, Rheological state diagrams for rough colloids in shear flow, Phys. Rev. Lett. 119 (15) (2017) 158001. [6] L.C. Hsiao, S. Pradeep, Experimental synthesis and characterization of rough particles for colloidal and granular rheology, Curr. Opin. Colloid Interface Sci. 43 (2019) 94–112. [7] C.P. Hsu, S.N. Ramakrishna, M. Zanini, N.D. Spencer, L. Isa, Roughness-dependent tribology effects on discontinuous shear thickening, Proc. Natl. Acad. Sci. 115 (20) (2018) 5117–5122. [8] H.K. Jang, Y.D. Chung, S.W. Whangbo, I.W. Lyo, C.N. Whang, S.J. Lee, S. Lee, Effects of chemical etching with hydrochloric acid on a glass surface, J. Vac. Sci. Technol. A 18 (5) (2000) 2563–2567. [9] H.K. Jang, Y.D. Chung, S.W. Whangbo, Y.S. Lee, I.W. Lyo, C.N. Whang, ..., G. Kim, Effects of chemical etching with sulfuric acid on glass surface, J. Vac. Sci. Technol. A 18 (2) (2000) 401–404. [10] R. Mari, R. Seto, J.F. Morris, M.M. Denn, Shear thickening, frictionless and frictional rheologies in non-Brownian suspensions, J. Rheol. 58 (6) (2014) 1693–1724. [11] D. Lootens, H. Van Damme, Y. Hémar, P. Hébraud, Dilatant flow of concentrated suspensions of rough particles, Phys. Rev. Lett. 95 (26) (2005) 268302. [12] A. Jabbarzadeh, The origins of enhanced and retarded crystallization in nanocomposite polymers, Nanomaterials 9 (10) (2019) 1472. [13] A. Jabbarzadeh, B. Halfina, Unravelling the effects of size, volume fraction and shape of nanoparticle additives on crystallization of nanocomposite polymers, Nanoscale Adv. 1 (2019) 4704–4721. [14] J.Y. Moon, S. Dai, L. Chang, J.S. Lee, R.I. Tanner, The effect of sphere roughness on the rheology of concentrated suspensions, J. Non-Newtonian Fluid Mech. 223 (2015) 233–239. [15] K. Shellenberger, B.E. Logan, Effect of molecular scale roughness of glass beads on colland bacterial deposition, Environ. Sci. Technol. 36 (2) (2002) 184–189. [16] A. Sierou, J.F. Brady, Rheology and microstructure in concentrated noncolloidal suspensions, J. Rheol. 46 (5) (2002) 1031–1056. [17] R.I. Tanner, Aspects of non-colloidal suspension rheology, Phys. Fluids 30 (10) (2018) 101301. [18] R.I. Tanner, S. Dai, Particle roughness and rheology in noncolloidal suspensions, J. Rheol. 60 (4) (2016) 809–818. [19] R.I. Tanner, C. Ness, A. Mahmud, S.C. Dai, J. Moon, A bootstrap mechanism for non-colloidal suspension viscosity, Rheol. Acta 57 (2018) 635–643. [20] L.V. Zogheib, A.D. Bona, E.T. Kimpara, J.F. Mccabe, Effect of hydrofluoric acid etching duration on the roughness and flexural strength of a lithium disilicate-based glass ceramic, Braz. Dent. J. 22 (1) (2011) 45–50.
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Effect of particle roughness on the rheology of suspensions of hollow glass microsphere particles Christopher Hoyle, Shaocong Dai, Roger Tanner, Ahmad Jabbarzadeh∗ Faculty of Engineering, School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Keywords: Noncolloidal suspensions Smooth surface Effect of surface roughness Hollow glass microspheres Suspension rheology Wet chemical etching
a b s t r a c t The rheology of high volume fraction (30–40%) suspensions with hollow glass microspheres (radius r = 15 μm) in a matrix of silicone oil has been studied experimentally. The surface of the hollow glass microspheres was roughened by wet chemical processing, using hydrochloric (HCl) acid and also sodium hydroxide etching. We obtained particles of various roughnesses in the range of Ra = 62–655 nm. The effect of particle roughness on the overall rheology is investigated, demonstrating its clear impact. The effect depends on the volume fraction, the magnitude of roughness and shear rate. At a 30% volume fraction, increasing the roughness ratio (Ra /r) from 0.42% to 4.44% does not result in a measurable change in viscosity. At a higher volume fraction of 40%, even a modest increase of the roughness ratio to 1.5% causes a significant increase in the viscosity of 117%, 97%, 50% and 39%, respectively at shear rates of 0.47, 1, 10 and 100 s−1 . Increasing the roughness to 4.44% results in an increase of 272%, 105%, 41%, 23% in viscosity over the same range of shear rates. In all cases, we show that the effect of roughness diminished by increasing the shear rate. Some estimates of the average interparticle coefficient of friction are given- showing roughness increases friction. The friction coefficient is shown to increase with the volume fraction. Friction coefficient shows a power-law dependence on shear rate and decreases with the shear rate. The rate of decrease with shear rate is smaller for higher volume fraction of 40%.
1. Introduction The rheology of non-colloidal suspension systems where the hydrodynamic forces are much larger than Brownian forces is of great importance in many applications. For systems with volume fractions larger than 𝜑 = 0.25, the chance of particle-particle interactions increases significantly. In such situations, the surface roughness of the suspended particles may become an essential factor in the overall rheological behaviour of the systems. More can be found in a recent review [17] about various aspects of suspension rheology. The effect of particle roughness for colloidal suspensions has been studied by a few groups. Castle et al. [3] have shown roughness increased viscosity of latex suspensions. Hsiao et al. [5] also studied the effect of roughness of 2 μm colloidal PMMA particles. Hsu et al. [7] have studied the effect of roughness on shear thickening in colloidal suspensions. During the fabrication process of PMMA colloidal particles, they varied cross-linking agent concentration to introduce heterogeneous roughness. However, the SEM images showed some deviation from a spherical shape. Previous work by Lootens et al. [11] has also studied the effect of roughness on colloidal silica particles of 1 μm diameter for volume fractions 41–48%.
∗
On the other hand, the effect of roughness on non-colloidal particles, which is our concern here, has been studied by Tanner and Dai [18] and Moon et al. [14]. They have previously shown that roughening a system of polystyrene microparticles in a matrix of Newtonian silicone oil or non-Newtonian Boger fluid results in a significant increase of up to 35% in viscosity. They have also shown that the results agreed reasonably well with mathematical models which took into account interparticle friction. In that work, mechanical roughening was used as an easy and effective way to generate surface roughness on smooth microparticles. In this process, cross-linking grinding and milling are used, as the autogenous roughening by other microparticles causes particle damage on the micro or nanoscale. This method was used on 40–80 μm sized particles, and roughness ratios (the ratio of average roughness to sphere radius, Ra /r) of 0.15% to 5.3% were achieved by varying the amount of time the particles spent in the grinding machine (0 to 10 h respectively). Although this roughening process is simple, there is evidence of some change in the shape of the particles, whereas for a satisfactory analysis of the results we need to keep an approximately spherical shape of the particles. Tanner and Dai [18] had confidence that by keeping the roughness ratio around 5% this objective would be fulfilled. To make sure this deviation from a spherical shape in previous experiments does
Corresponding author. E-mail address: [email protected] (A. Jabbarzadeh).
https://doi.org/10.1016/j.jnnfm.2020.104235 Received 1 November 2019; Received in revised form 7 January 2020; Accepted 7 January 2020 Available online xxx 0377-0257/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: C. Hoyle, S. Dai and R. Tanner et al., Effect of particle roughness on the rheology of suspensions of hollow glass microsphere particles, Journal of Non-Newtonian Fluid Mechanics, https://doi.org/10.1016/j.jnnfm.2020.104235
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not have an impact on the results, we decided to repeat the experiments by using a different roughening method that would better preserve the spherical shape of the particles. A review by Hsiao and Pradeep [6] offers various methods used for roughening colloidal and non-colloidal particles. For the work in this paper, the wet chemical process offers a suitable approach to roughening the surface of microparticles. This method has been used widely in the industry on glass particles, and in dentistry to etch enamel parts. Zogheib et al. [20] used hydrochloric (HCl) acid to change the surface roughness and flexural strength of a lithium disilicate glass-ceramic. It was found that etching increased surface roughness and this increase was more significant for longer etching durations. Jang et al. [8,9] investigated the effects of hydrochloric acid and sulfuric acid on the surface of glass slides showing the RMS roughness could be increased by almost a factor of 10 depending on the type of the acid, the duration and processing methods. Shellenberger and Logan [15] used wet chemical etching to change the surface roughness of soda-lime glass beads. They used both sodium hydroxide (NaOH) and HCl and showed the effectiveness of the method to alter the surface roughness. Different wet chemical etching methods such as phosphoric acid were used for roughening enamel [4] where the RMS roughness could be increased by an order of magnitude. In this work, we study the effect of particles surface roughness in suspensions containing rigid hollow glass microspheres. We have also used a roughening method different from that used by Tanner and Dai [18]. Here wet chemical etching is used to affect only the surface of the microspheres, and there is no deformation or shape change in the spherical form of the particles. We have done the experiments for suspensions of hollow glass microspherical particles (HGMP) in a matrix of silicone oil of 1.1 Pa.s viscosity. 2. Methodology and experimental procedure 2.1. Wet chemical etching of glass particles The particles used in this study were hollow glass microspheres (HGMS) made from Soda-Lime glass which were supplied by Cospheric LLC, CA, with nominal diameters between 27 μm and 32 μm, and a density of 0.51 g/cc, as shown in Fig. 1a. The wet chemical etching was used to roughen the HGMS, and then 30% and 40% volume fraction suspensions with silicone oil were prepared. Before etching, a quantity of HGMS was weighed and placed into a beaker and submerged in 37% Hydrochloric acid solution. The HGMS HCl solution was boiled at 110 °C for 30 min and then diluted with water. It was poured through filter papers and then dried in an oven at 60 °C for 24 h to recover the microspheres. The etched HGMS particles were scraped off the filter papers and passed through a 32 μm sieve with a brush to remove any clumps of particles. Fig. 1b shows the Scanning Tunneling Microscopy (STM) images of HCl treated HGMS particles. The method was repeated using a 12.5 M NaOH solution. A 12.5 M solution of NaOH contains 50% by mass of pure NaOH. To prepare a solution, an equal mass of NaOH pellets and distilled water were mixed until a solution was formed. Instead of boiling the solution, the HGMS was only soaked for 30 min in the NaOH solution. Fig. 1c shows a Scanning Tunneling Microscopy (STM) image of NaOH treated HGMS at 2780 magnification. An important point to note is that acid etching often works by selectively attacking certain species in a sample, and sometimes chemically reacting with others. This limits its application to homogenous materials (such as PMMA). Additionally, a change in the chemical composition of a microsphere surface could, in turn, change the rheological properties of a suspension made up of these particles. This would prevent the effects of surface roughness being completely isolated. Hence, care must be taken when considering this method; samples should be thoroughly washed before drying to remove any products of the acid or alkaline reaction.
Fig. 1. STM images of hollow glass microspheres (HGMS) with nominal diameters between ∅27 μm and ∅32 μm silicone for a) untreated (b) HCl treated and (c) NaOH treated particles.
2.2. Characterizing roughness The STM images in Fig. 1 show the treated and untreated HGMS particles. While treatment of particles by HCl brings some change on the surface, a more significant change appears when NaOH is used to treat the particles. The surface modification for HCl treated particles is achieved by etching some material away, whereas for NaOH treated particles, the surface is modified by the formation of spiky deposits of sodium silicate. In all cases, the basic spherical shape of the particles is preserved. To quantify physical change on the surface, we have measured the surface roughness of untreated and acid and alkaline treated HGMS. The surface roughness of the particles was measured on a nanotriboindentation device (TI900, Hysitron Inc., USA) using a diamond
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Table 1 Average roughness (Ra ) calculated from several batches for untreated and treated hollow glass microspheres (HGMS). Particle
Batch Averaged Roughness Ra (nm)
Untreated HGMS HCl treated HGMS NaOH treated HGMS
62 ± 4 223 ± 19 655 ± 82
Roughness Ratio (Ra /r) 0.42% 1.5% 4.44%
Berkovich indenter. Before testing, the z-axis, tip-optics and tip-area were carefully calibrated. The measured area for a single particle was visually identified using the optics. Then the probe was moved in a raster scan pattern across the surface of the particle using a three-axis piezo positioner. The scan rate for all the tests was 0.1 Hz, with a setpoint of 2 μN. It was necessary to scan a very small area to avoid the curvature of the sphere being counted as roughness. (The average roughness is found by taking the absolute value of the deviation from the spherical shape) Roughness measurements of several particles taken from various batches of treated and untreated particles were conducted and the average roughness Ra determined for each batch. Table 1 shows the measured surface roughness of spheres for two batches of untreated HGMS, 4 batches treated by HCl, and 3 batches treated by NaOH. The batch averaged values for untreated, HCl treated, and NaOH treated, HGMS particles are also shown in this table. The roughness is about 62 ± 4.00 nm for the untreated glass particles, showing a relatively smooth surface. For HCl treated HGMS batches, the average roughness is 223 ± 19 nm, and 655 ± 82 nm for NaOH treated HGMS. The results show that both treatment methods produce consistent and repeatable levels of roughness. However, the magnitude of roughness is much higher for NaOH treated samples. The systems treated by HCl increased the surface roughness by a factor of 3.6, whereas the systems treated by NaOH showed an order of magnitude increase in the average roughness. Adopting the previously defined roughness ratio [17] as the percentile ratio of average roughness to particle radius, we obtain values of 0.42% for the untreated particles, and 1.5% and 4.44% for particles treated respectively by HCl and NaOH. The roughness ratio 1.5% for HCl treated particles is one-third of that obtained (5.1%–5.3%) by Dai and Tanner [18] for PMMA and PS particles using the mechanical roughening method. The roughness ratio of 4.44% for the NaOH treated system is also lower than these values. The results confirm that the spherical shape of the particles is well preserved after roughening by both chemical etching methods. 2.3. Rheometry An Anton Paar MCR302 rheometer was used to conduct all rheological measurements in our lab. Parallel plate geometry was used with 25 mm diameter plates in the rheometer. We gently mixed the samples before testing them to homogenize the material. The rheological measurements were conducted at a 0.6 mm gap and 24 °C. The selected gap was more than an order of magnitude larger than the size of particles, ensuring reliable measurements of the rheological properties. The small gap also minimizes the possibility of the material flowing out of the gap. A sudden drop in viscosity usually occurs if the material comes out of the gap. It also leaves marks on the lower plate. In all cases, we conducted a visual inspection and monitoring of the viscosity and made sure this did not happen. We follow a similar experimental procedure to that described in previous works [17,18]. To avoid particle sedimentation, a modified Shields number greater than 3 is needed [17]. The modified Shields number is given by Eq. (1), 𝑠ℎ∗ =
𝜂0 𝛾̇ Δ𝜌𝑔𝐷
(1)
Fig. 2. Viscosity as a function of shear rate for silicone oil matrix, and 30% (open symbols) and 40% (filled symbols) HGMS silicone oil suspensions. For each volume fraction, the results are shown for suspensions made from untreated HGMS particles and those roughened by HCl and NaOH treatments.
where 𝜂 0 is the suspending medium viscosity (𝜂 0 = 1.1 Pa.s for 1000 cSt silicone oil), 𝛾̇ is the shear rate, Δ𝜌 is the difference in densities between the suspending medium (silicone oil = 970 kg/m3) and the suspended particles (hollow glass = 510 kg/m3 ), g = 9.81 m/s2 is the gravitational acceleration, and D is the diameter of the suspended particles taken to be 29.5 μm. The calculations show for 𝛾̇ > 0.36 s−1 the Sh∗ >3 and no sedimentation is expected. We have done the rheological measurements for all samples for a range of shear rates between 0.01–100 s−1 , but here we only show the results for a range of 0.1–100 s−1 to avoid sedimentation effects. 3. Results and discussion Silicone oil was used as a Newtonian matrix. The measured viscosity of the silicone oil was 1.10 PaS at 24 °C. This fluid has an almost constant viscosity independent of shear rate between 0.1 and 100 s−1 . Fig. 2 shows the viscosity of the untreated HGMS suspension system at 30% volume fraction. The results here are the average of two separate measurements. The results for suspension made with HCl and NaOH treated particles are also shown for comparison. There was a slight decrease in viscosity at low shear rates. If we consider shear rates > 0.36 s−1 where we expect no sedimentation, for the 30% untreated systems the viscosity remains relatively constant. However, for the 40% untreated system, there is a slight decrease at shear rates higher than 30 s−1 (see Fig. 2). The normal force measured for these systems was small for the 30% systems; however, for the 40% system, it was small at lower shear rates and started to increase for shear rates above 10 s−1 . Considering the average viscosity of the 30% systems, the relative viscosity at shear rates of 1–10 s−1 for the 30% untreated system is 𝜂 r = 3.46. The relative viscosity of the 40% untreated HGMS system for the same range of shear rates is 𝜂 r = 9.36. These values are very close to those found by Tanner and Dai [18] for “smooth” (actual roughness ratio ~0.15–0.45%) 40 μm Polystyrene and PMMA particles at the same volume fractions (3.8 and 9.7). These values are higher than the computational values which do not take into account the particle-particle friction forces. Computations by Mari et al. [10] give 𝜂 r = 3.7 and 6.2 for 30% and 40% volume fraction for frictionless systems, whereas the computations by Sierou and
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Table 2 Relative viscosity for various shear rates and particle roughness for HGMS suspensions of two different volume fractions 0.3 and 0.4. Shear rate 𝛾, ̇ s−1 𝝋 = 0.3 0.47
1
10
100
𝝋 = 0.4 0.47
1
10
100
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Roughness (nm)
Relative viscosity 𝜂 r
62 223 655 62 223 655 62 223 655 62 223 655
3.73 4.18 4.07 3.53 4.00 3.76 3.41 3.78 3.24 3.3 3.55 3.06
62 223 655 62 223 655 62 223 655 62 223 655
9.64 20.91 25.27 9.36 18.5 20.14 9.36 14 13.23 8.03 11.18 9.91
Brady [16] reported 𝜂 r = 3.1 and 6.4 for 𝜑 = 0.30 and 𝜑 = 0.40 respectively. From the results in Fig. 2 for the viscosity of 30% volume fraction, HCl treated HGMS, we can also see some shear-thinning effects at a shear rate lower than 1. This can also be observed for the NaOH treated system. We believe this is most likely due to sedimentation which is expected at 𝛾̇ < 0.36 s−1 . The results shown here are the average of the viscosity of the three samples and show for 𝛾̇ = 1–10 s−1 that the viscosity values are relatively constant. The average viscosity for this range of shear rates is 𝜂 = 4.27 Pa.S which gives relative viscosity of 𝜂 r = 3.88, which is 11% higher than those for the untreated HGMS at the same volume fraction. However, the difference is within the margin of error we calculated for the measurements for these two systems (see Table 2). The average normal force for the 30% HCl treated suspension system is very small and is not reported here. The viscosity of 30% volume fraction NaOH treated HGMS suspensions were also measured over shear rates 0.1–100 s−1 . The average results of the two samples are shown in Fig. 2. Working with NaOH treated HGMS suspension samples were, in particular, difficult, and care should have been taken that the particles were sieved before making the suspension. Otherwise, particular agglomeration would lead to large viscosity. The measurements made for the two different samples were very close. In this case, all systems showed strong shear thinning up to a shear rate of 10 s−1 . Beyond this shear rate, the shear-thinning is much weaker. At a shear rate of 1 s−1 the viscosity is 𝜂 = 4.14 Pa.s. In comparison at the same 30% volume fraction, for the HCl treated and untreated systems 𝜂 = 4.39 Pa.s and 𝜂 = 3.89 Pa.s. This shows only a moderate increase of ~6% in viscosity when comparing the untreated HGMS, and NaOH treated suspension system. The differences in the results, however, were within the statistical uncertainty, therefore at this 30% volume fraction, we concluded there was no measurable effect of roughness on the viscosity. The results in Fig. 2 also show the viscosity for 40% volume fraction suspension fluids made by HCl treated, NaOH treated and untreated HGMS particles. Increasing the volume fraction to 40%, for the HCl and NaOH treated suspensions results in a significant increase in the viscos-
ity. We can see while for untreated HGMS system the viscosity is relatively stable, for the HCl and NaOH treated HGMS there is an evident shear thinning that persists across the imposed shear rates. The viscosity for 40% volume fraction HCl treated HGMS significantly decreases with increasing the shear rate, from 20.3 Pa.s at 1.0 s−1 to 12.3 Pa.s at 100 s−1 . For the NaOH treated HGMS suspension viscosity decreases from 22.15 to 10.9 PaS over the same range of shear rates. Considering the shear-thinning persists at 𝛾̇ > 0.36, the effect is not related to sedimentation. The overall viscosity is also much higher (see Table 2) for the treated HGMS systems. The average shear viscosity in the range of 𝛾̇ = 1–10 s−1 for untreated and HCl treated systems are 𝜂 = 10.3 Pa.s, and 𝜂 = 17.6 Pa.s respectively which give relative viscosities of 𝜂 r = 9.20 and 𝜂 r = 16 showing a significant increase in viscosity due to roughening of the HGMS particles by HCl treatment. For the NaOH treated suspensions, the average viscosity over the same range of shear rates is 22.15 PaS (𝜂 r = 20.13) showing an even higher increase in the viscosity. From the results above we deduce that the effect of roughness on the viscosity of a suspension depends on the magnitude of roughness, volume fraction, and shear rate. To put this in perspective, we have plotted viscosity as a function of HGMS particles roughness for four different shear rates of 0.47, 1, 10, 100 s−1 , where we do not expect sedimentation of particles. The results are presented in Fig. 3a and b for 30% and 40% volume fractions. From Fig. 3a, for 𝜑 = 0.3 we can see that increasing the roughness from 62 nm to 223 nm has no measurable effect. Increasing the roughness further to 655 nm causes a small increase in the viscosity. This increase, however, is smaller than the uncertainty in our measurements for HCL and NaOH treated systems Therefore we can conclude for 30% volume fraction systems a 4.44% roughness ratio (Ra = 655 nm) results in no measurable increase in the viscosity. The result presented in Fig. 3b for the higher volume fraction, 𝜑 = 0.4 show a moderate increase of roughness from 62 nm to 223 nm caused the viscosity to increase across the entire range of shear rates (0.47– 100 s−1 ). At shear rates 0.47, 1, 10 and 100 s−1 the increased roughness causes respectively 117%, 97%, 50% and 39% rise in the viscosity. For the suspensions made from NaOH treated HGMS of 4.44% roughness ratio (Ra = 655 nm), the viscosity also increases by 272%, 105%, 41%, 23% for shear rates of respectively, 0.47, 1, 10 and 100 s−1 . The results show that increasing roughness from 223 nm to 655 nm causes a further increase in viscosity for a shear rate of 0.47 and 1 s−1 . However, at higher shear rates, the viscosity is slightly lower for the NaOH treated system. Table 1 Relative viscosity for various shear rates and particle roughness for HGMS suspensions of two different volume fractions 0.3 and 0.4. 4. Discussion The tests presented above show that increased roughness generally increases the relative viscosity above the value for the smoothest spheres. One can estimate the average friction coefficient (μ∗ ) for any roughness by using the bootstrap correlation [19]. It was shown there that the ratio of the frictionless relative viscosity (𝜂 r ∗ ) to the frictional relative viscosity was, approximately, 𝜂𝑟 =
𝜂𝑟∗
{1-𝑘𝜇 ∗ 𝑃 ∕𝜏}
(2)
where k = 1.75 and P/𝜏 is the ratio of the interparticle pressure P to the shear stress; in Tanner et al. [19] the relation between P/𝜏 and μ∗ is given for 𝜙 = 0.3–0.5. Knowing 𝜂 r and 𝜂 r ∗ (calculated), one has { } 1 − 𝜂𝑟∗ ∕𝜂𝑟 ∗ 𝜇 = (3) 𝑘(𝑃 ∕𝜏) Here 𝜂𝑟∗ = 3.26 and 𝜂𝑟∗ = 6.12, respectively, for 𝜑 = 0.3 and 𝜑 = 0.4 are used based on the average values from several computational studies [19]. Hence, bearing in mind that P/𝜏 is a function of μ∗ and 𝜙, using
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Fig. 3. Average viscosity is plotted against particle roughness at different shear rates for a) 30% and (b) 40% volume fraction HGMS-Silicone oil suspensions.
ness ratios 0.42%, 1.5% and 4.44%. Showing parameter A increases with the roughness ratio. The exponent 𝛼 also increases with the roughness ratio. For 30% suspensions with particles of roughness ratios 0.42%, 1.5% A = 0.257, 0.4, and 𝛼 = 0.384, 0.181. Therefore, for higher volume fraction we conclude, A values are larger, however 𝛼 values are smaller. This indicates that the loss of frictional effects with shear rate is less severe for concentrated suspensions. The goodness of fits (R2 = 0.79) showed weaker power-law dependence for the 30% NaOH suspension system over the examined shear rates. However it still showed a reduction of friction coefficient with increased shear rate. Interparticle distance between spheres in an idealised homogenous monodisperse suspension system can be found from Eq. (5) [12,13]: (√ ) 𝜋 𝐷𝑝𝑝 = 𝐷 3 (5) −1 6𝜙
Fig. 4. Estimated average friction coefficients (μ∗ ) as a function of shear rate for suspensions made from HGMS of different roughness ratio. The dash-dot lines are the power fit to data for the 40% volume fraction suspensions. The goodness of the fit (R2 ) for each curve is shown in the legend.
P/𝜏 = 0.2 and 0.52 [19] respectively for 𝜑 = 0.3 and 𝜑 = 0.4, one can calculate the friction coefficient using the measured relative viscosity values 𝜂 r . The friction coefficient is plotted versus the shear rate for all suspensions in Fig. 4. One sees that μ∗ decreases as 𝛾̇ increases and is large for larger roughness ratios and larger volume fractions, as in Tanner and Dai [18] and Moon et al. [14]. Since the dry friction coefficient of ‘smooth’ glass on glass is about 0.9–1.0 [2] the results in Fig. 4 are acceptable. The results of μ∗ versus 𝛾̇ for the 30% and 40% volume fraction suspentions are fitted into power lines. That gives the following power-law relationship between the friction coefficient and shear rate: 𝜇 = 𝐴𝛾̇ ∗
−𝛼
(4)
We find from the fits to the 40% volume fraction suspensions data, A = 0.38, 0.4, 0.77, and 𝛼 = 0.073, 0.083, 0.126 respectively for rough-
where Dpp is the average surface-to-surface interparticle distance and D is particle diameter. The calculated interparticle distance for 𝜑 = 0.3 and 0.4 respectively are Dpp = 6.1 and 2.8 μm. That shows for the higher volume fraction of 40% the interparticle distance is less than half that of 30% system. Therefore, the particles are more likely to come into direct contact, and the effective frictional forces would be larger. The reduction in effective friction coefficient with the shear rates is due to the contribution of hydrodynamic lubrication forces that kicks in at higher shear rates as the relative velocity of the impinging particle increase. However, at lower shear rates, the particles are more likely to interact under boundary and mixed lubrication regime. Therefore, the effective friction coefficient approaches to that of the dry contact regime. The effective friction coefficient μ∗ in Fig. 4 is seen to approach ~1 as one moves to lower shear rates and higher concentrations. One expects that the static friction coefficient would be larger than ~1 for the etched glass particles. Blair et al. [1] have shown that the friction coefficient of HCl treated soda-lime glass beads (~3 mm size) increases threefold in comparison to an untreated smooth surface. Therefore these larger friction coefficients are likely to appear at higher concentrations. 5. Conclusions From the results obtained here, we can conclude that the roughness increases viscosity. Here the methodologies for roughening the surface of the particles were such that there was minimal effect on the overall spherical shape of the particles. The results at the lower volume fraction of 30% for roughness ratios of 1.5% and 4.44%% showed increased
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C. Hoyle, S. Dai and R. Tanner et al.
viscosity only at the lowest shear rate due to sedimentation. There was no measurable effect on viscosity for shear rates above 1.0 s−1 . However, for 40% volume fraction, even a roughness ratio of 1.5% caused a significant rise in the viscosity. The increase in viscosity was shear rate dependant and, varied from 97% increase at a shear rate of 1 s−1 to 39% for a shear rate of 100 s−1 . For 40% volume fraction, increasing the roughness ratio to 4.44% results in further ~155% and 8% increase in viscosity at lower shear rates of 0.47 and 1 s−1 , however at higher shear rates the viscosity did not show any further increase. It is interesting to see that these increases are much higher than those observed for PMMA and PS particles [Tanner and Dai [18]] where a roughness ratio of 5.1–5.3% caused a modest increase in the viscosity of about 4–6% at 𝜑 = 0.3 and 5–14% at 𝜑 = 0.4. This may be explained by different friction coefficients for the glass particles than those for polystyrene. The coefficient of friction for unlubricated smooth glassglass contact is 0.9–1 whereas for PS-PS contact it is half that value ~ 0.45–0.5. The friction coefficient values calculated here at low shear rates also approaches that of glass and confirms this hypothesis. Declaration of Competing Interest We as the authors of the manuscript entitled “Effect of Particle Roughness on the Rheology of Suspensions of Hollow Glass Microsphere Particles” submitted to Journal of Non-Newtonian Fluid Mechanics for publication hereby declare that this manuscript is an original work, and it has not been submitted to nor published in any other journal. We declare no conflict of interest. Acknowledgment We thank Mr Hesamodin Jami for help in obtaining the STM images. References [1] D.L. Blair, N.W. Mueggenburg, A.H. Marshall, H.M. Jaeger, S.R. Nagel, Force distributions in three-dimensional granular assemblies: effects of packing order and interparticle friction, Phys. Rev. E 63 (4) (2001) 041304. [2] F.P. Bowden, D. Tabor, The Friction and Lubrication of Solids, Oxford U. Press, 1954.
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