1 Introduction

Surface roughness is important in many fields of engineering and natural sciences. It is well established that roughness influences the wetting nature of materials and surfaces [e.g., 1,2,3,4,5]. Processes such as ink-jet printing [6], painting, coating, biofouling, drying, lubrication, and waterproofing [7,8,9] all exemplify wetting principles. Similarly, previous studies of flow in porous media have reported correlations between grain roughness and capillary trapping [10, 11], capillary pressure-saturation curves [11], fracture conductivity [12, 13] and, most recently, pore-scale distribution in contact angle [14].

Surface texturing methods for flat materials (metals/alloys, polymers, and glass) have evolved rapidly over the years. Techniques such as abrasive [15], chemical [16], and laser etching [17], as well as abrasive polishing [18, 19], blasting [20], mechanical press [21], electrical discharge machining [22], electron beam lithography [23], photolithography [24], and plasma nano-texturing [25] have all been applied to produce either random or structured rough surfaces. In contrast, established techniques for roughening batches of mm-scale particles is limited. In particular, the most widely used technique for roughening glass particles is chemical etching with hydrofluoric acid [e.g., 11, 26, 27], which is highly corrosive and particularly hazardous. This paper presents a new method for surface texturing of glass particles that is safe, effective, and cheap. Treated particles are visualized using scanning electron microscopy (SEM), and their surface topography is characterized by its two-point spatial correlation function. The permeability of packed columns of particles was measured and its dependence on surface roughness discussed.

2 Materials and Methods

Table 1 Mean permeability and median integral length scale for each treatment condition
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2.1 Particle Texturing

Fig. 1
figure 1

a Stone tumbler. The plastic barrel (black) is 103 mm in diameter and 97 mm in length. During operation, the barrel is sealed with the yellow lids and rotates at 66 rpm. b Experimental setup for permeability measurement using the falling-head method. \(h_0 \, (=580\,\hbox {mm})\) and \(h_1 \,(= 80\,\hbox {mm})\) represent the initial and final water levels, respectively (Color figure online)

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Soda-lime glass spheres (Mo-Sci Corporation, USA) of mean diameter \(966\pm 34\,\upmu \hbox {m}\) (standard dev., 35 samples) were used in this study. The particles were textured using an off-the-shelf stone tumbler (UKGE, UK; Fig. 1a), a piece of household equipment commonly used to polish rock or stones. Here, we used the equipment to either smoothen or roughen glass surfaces by tuning the grit size of the sandpaper (Klingspor Abrasives, Germany) that lines the tumbler and the duration of treatment. In total, five grit sizes, \(d_{\mathrm{g}} = 201\), 58.5, 18.3, 12.6, and \(8.4\,\upmu \hbox {m}\), where \(d_{\mathrm{g}}\) is the average grit diameter, were considered. The tumbling duration was kept constant at 12 h, except for one texturing regime comprising \(d_{\mathrm{g}} = 12.6\,\upmu \hbox {m}\) sandpaper and a shorter tumbling time of 4 h (Table 1).

For each \(d_{\mathrm{g}}\), approximately \(20\,\hbox {cm}^3\) of untreated particles were placed in a 97 mm-long, 103 mm-diameter barrel at a time and tumbled at 66 rpm for the selected tumbling duration. After tumbling, the samples were first soaked in a 2% aqueous solution of Decon 90 for 2 to 24 h to remove the dust generated by the abrasive wear process. Subsequently, the samples were gently agitated by swirling, rinsed with warm water, and oven-dried at 100 °C. Additionally, the samples were cleaned sequentially with toluene, propanol, and water, and then oven-dried again prior to surface roughness characterization.

2.1.1 Correlation Length

Fig. 2
figure 2

Selected SEM images of R0 (a, d), R6 (b), R5 (c), R1 (e), R2 (f), R3 (g), and R4 (h) samples. The autocorrelation functions for images (b–h) are included in Fig. 3; the corresponding \(\xi = 6.6\) (b), 7.9 (c), 14.7 (d), 24.3 (e), 29.3 (f), 33.5 (g), and 33.9 (h) nm, respectively

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To characterize the surface features, 8–10 particles of each treatment condition were imaged at 16,000× magnification (corresponding image resolution 6.9 nm/pix) using a Carl Zeiss GeminiSEM 300 (e.g., Fig. 2). The two-point autocorrelation function, Z(m, n), where (m, n) are the lags in pixels in the two orthogonal directions, was computed for each image using the Wiener–Khintchine theorem [e.g., 28] and MATLAB’s fft2.m and ifft2.m functions [29]. The tumbling was isotropic and, accordingly, Z(m, n) was averaged at fixed radii, \(r \,\left( = \sqrt{m^2+n^2}\right)\), following the approach of Berryman [30] to calculate the 1D, isotropic autocorrelation function, \(\hat{Z}(r)\). Finally, the integral length scale, \(\xi\), was evaluated as \(\xi = \int _{0}^{r_0} \hat{Z}(r)\, \mathrm {d}r\), where \(r = r_0\) is the first zero-crossing. The computed values of \(\xi\) are summarised in Table 1; all \(\hat{Z}(r)\) are presented in Fig. 3.

Fig. 3
figure 3

Autocorrelation functions for ten samples of R0, nine samples of R1, and eight samples of R2–R6. Axis limits are identical in each plot; the abscissa axes are truncated at 50 pixels to reveal finer details of the plots

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2.2 Permeability

Falling-head permeability tests were performed to further explore the effect of the proposed treatment. Glass spheres were wet-packed into a vertical column (ID 21 mm, Fig. 1b). X-ray micro-computed tomography images of R0, R3, and R4 samples, acquired at voxel resolutions of \(10\,\upmu \hbox {m}\), yielded consistent porosity estimates of \(35.6\pm 0.2\%\) (standard dev., 6 packed columns), confirming packing reproducibility. The upstream end of the packed column was connected to a burette (ID 11 mm) through a hose (ID 4 mm), while the outlet at the downstream end was open to the atmosphere. A test consisted of measuring time it takes for the water level in the burette to lower from the initial position, \(h_0\), to another one, \(h_1\), using a stopwatch. Both heights were measured from the datum located in the middle of the outlet. The experiment was repeated ten times to determine average values. The experiment was also carried out with the empty column to measure the flow resistance of the system without any particles. The falling head equation was modified by subtracting the resistance of the empty column in order to calculate corrected values of the absolute permeability.

3 Results

Fig. 4
figure 4

The integral length scale (a) and permeability (b) at each treatment condition and the correlation between the two (c). Each open circle represents a different particle (a, b); solid circles represent the mean k and median \(\xi\) for each treatment condition. Vertical bars in (c) depict the standard deviations across ten replicate measurements in the same packed column. The values of \((\xi , k)\) are summarized in Table 1

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Figure 4 presents the integral length scale (Fig. 4a) and permeability (b) for each treatment condition. \(\xi = 14.3\pm 6.3\,\hbox {nm}\) (standard error) and \(k = 460\pm 27\,\upmu \hbox {m}^2\) (standard dev.) for untreated samples. This value of \(\xi\) is one order of magnitude smaller than the vertical roughness length scale of untreated soda-lime glass spheres from the same supplier, but two orders of magnitude smaller than the lateral length scale [27], suggesting that \(\xi\) as measured presently is a proxy for the size of features rather than their lateral spacing. The smallest \(k\, (= 360\pm 10\,\upmu \hbox {m}^{2})\) coincides with the largest \(\xi \, (=34\,\hbox {nm})\), and was achieved using \(d_{\mathrm{g}} = 12.6\,\upmu \hbox {m}\) sandpaper and the shorter tumbling time of 4 h (treatment R4, Table 1).

Particles tumbled for 12 h display two regimes. As the grit size decreased from \(d_{\mathrm{g}} = 201\) (R1) to \(18.3\,\upmu \hbox {m}\) (R3), the roughness size increased from \(\xi = 24.3\pm 1.7\) to \(31.4\pm 2.4\,\hbox {nm}\) and permeability decreased from \(k = 440\) to \(400\,\upmu \hbox {m}^{2}\). As the grit size decreased further, however, \(\xi\) decreased with \(d_{\mathrm{g}}\) until \(\xi = 6.4\pm 1.2\,\hbox {nm}\) at the finest grit size (R6), while permeability increased to \(k = 500\,\upmu \hbox {m}^{2}\) at \(d_{\mathrm{g}}=12.6\,\upmu\hbox{m}\) (R5) beyond which it remained constant. This asymptotic limit is consistent with \(k\sim 670\,\upmu \hbox {m}^{2}\) predicted by Ergun [31]’s equation for packed columns of spheres.

Combined, k decreased with increasing integral length scale (Fig. 4c). We infer from the negative correlation that, under conditions considered presently, permeability decreases with increasing roughness even under laminar conditions and \(\xi\) is a measure of the vertical dimension of nm-scale surface features. Plausible explanations for the negative correlation include the formation of recirculation zones within troughs and an increase in the particle surface area per particle with increasing \(\xi\). Further insight requires comprehensive 3D topography measurements using sophisticated surface metrology techniques.

Treatment conditions R5 and R6 established \(\xi\) smaller than those of untreated particles, indicating that, for a given tumbling time, fine grit smoothens the surface while coarse grit roughens it. The critical grit size between the two regimes, \(d_{\mathrm{g}} \approx 15\,\upmu \hbox {m}\), corresponds to 1.5% of the particle diameter. However, the critical grit size—and the properties of the treated surface in general—are expected to be a function of the hardness of the particle relative to the grit, the concentration of particles in the tumbler, tumbling speed, and particle diameter in addition to the parameters varied in the present study, namely the grit size and tumbling time. A comprehensive exploration of the impact of these parameters on texturing remains a topic for future work.

4 Conclusions

A new protocol for nm-scale surface texturing of glass particles using a sandpaper-lined tumbler was presented. The method is cheap, effective, can be readily applied to particles of other rigid materials such as plastic and metal, and safer than the standard approach of acid etching applied to glass particles. Depending on the sandpaper grit size, tumbling was found to create or remove nm-scale roughness. The integral length scale of the radial autocorrelation function computed from 7 nm/pix resolution SEM images of the particles is negatively correlated with permeability, indicating that roughness affects permeability even under laminar conditions and that the integral length scale is an effective proxy for roughness size.

Packed beds of glass and plastic spheres are widely used as laboratory models for porous media [e.g., 11, 27, 32, 33]. The protocol presented here facilitates laboratory investigation of the impact of grain roughness on single phase and multiphase flow in porous media, a research area that has yet to be fully explored.