Charging changes contact composition in binary sphere packings

TL;DR

This study demonstrates that tribo-charging influences contact composition in binary sphere packings, enabling control over local granular material properties without causing global segregation.

Contribution

It reveals how tribo-charging affects contact types in binary sphere packings, offering a new method to tune granular material properties.

Findings

Charge density alters contact numbers between same and opposite types.

No global segregation occurs despite charge-dependent contact changes.

Tribo-charging can be used to modify local contact composition.

Abstract

Equal volume mixtures of small and large polytetrafluorethylene (PTFE) spheres are shaken in an atmosphere of controlled humidity which allows to also control their tribo-charging. We find that the contact numbers are charge-dependent: as the charge density of the beads increases, the number of same-type contacts decreases and the number of opposite-type contacts increases. This change is not caused by a global segregation of the sample. Hence, tribo-charging can be a way to tune the local composition of a granular material.

Equations

$$\varphi =\frac{\frac{4\pi }{3}(N_L{r}_L^3+N_s{r}_s^3)}{\sum _i^{N_L}{\nu }_L^i+\sum _j^{N_s}{\nu }_s^j}$$

$$\frac{\partial Z_{Ls}}{\partial \sigma }=\frac{N_s}{N_L}\frac{\partial Z_{sL}}{\partial \sigma }\text{mbox}.$$

$$g_{AB}(d)={⟨\frac{1}{4\pi d^2\rho }\sum _{B,j}\delta (d-|{\vec{x}}_A-{\vec{x}}_{B,j}|)⟩}_A$$

$$n_Z(D_v)=\frac{Z_{AB}}{\sqrt{2\pi }\sigma }{\int }_0^{D_v}\exp \left(-\frac{{\left(D_v^{\prime }-D_{avg}\right)}^2}{2{\sigma }^2}\right)d{D}_v^{\prime }$$

$$n_{CNS}(D_v)=n_Z(D_v)+\Theta (D_v-D_{avg})m(D_v-D_{avg})$$

Full text

I Introduction

The term granular media comprises all ensembles of particles where the individual entities are large enough to be unaffected by Brownian motion. Besides gravity and contact forces, the dynamics of granular media is also controlled by forces originating from the surface of the particles: electrostatic interactions Matsusaka et al. (2010), capillary forces Herminghaus (2013), and friction Schröter (2017). Understanding the role of these forces is not only an interesting scientific problem, but also important for technological applications because many raw materials in industry come in granular form Duran (2000). Especially tribo-charging of granular particles proves to be challenging because it can lead to both repulsive and attractive interactions between the beads Lee et al. (2015); Qin et al. (2016); Chen et al. (2016a); Yoshimatsu et al. (2016); Kolehmainen et al. (2016); Yoshimatsu et al. (2017).

The simplest model system to investigate the generally poly-disperse granular materials are binary sphere mixtures. They have been widely studied with respect to their jamming behavior Hopkins et al. (2011, 2013); Chen and Torquato (2015); Koeze et al. (2016), their structural features Dijkstra et al. (1998); Kummerfeld et al. (2008) and their binary contact numbers Epstein and Young (1962); Pinson et al. (1998); Biazzo et al. (2009); Meng et al. (2014); Kumar et al. (2016). Binary sphere packings agitated vertically tend to segregate Rosato et al. (1987); Knight et al. (1993); Kudrolli (2004); Schröter et al. (2006); Garzó (2011); Brey et al. (2011). Depending on the prevailing segregation mechanism, the larger spheres either rise to the top (which is also called the Brazil nut effect), or they sink to the bottom. Segregation is a common problem in the manufacturing industry where mixing of different types of granular materials is often a crucial process Ottino and Khakhar (2000); Muzzio et al. (2002); Lu and Hsiau (2005); Cheng et al. (2014).

Tribo-charging is pervasive in the handling of granular material because every time two materials get in contact some charge will be transferred Harper (1957); McCarty and Whitesides (2008); Matsusaka et al. (2010). Tribo-charging of granular samples can lead to the formation of clusters Lee et al. (2015), de-mixing Mehrotra et al. (2007), or even prevent pore clogging Chen et al. (2016b). Recently we have shown that tribo-charging can also be used to counteract segregation Schella et al. (2017).

The amount of tribo-charging is known to depend on the humidity of the air Nieh and Nguyen (1988); Pence et al. (1994); Greason (2000); Rhodes et al. (2003); Németh et al. (2003); Vandewalle et al. (2012); Xie et al. (2016); Schella et al. (2017). Here, we use this dependence to control the amount of surface charges on the beads in a binary mixtures of Teflon spheres. At the same time we ensure that the charges are large enough to avoid global segregation. Using X-ray tomography we then investigate how the composition of small-small, small-large, and large-large contacts changes as a function of the surface charge.

II Experiment

All experiments are performed with a mixtures of approximately 10000 small and 1483 large polytetrafluorethylene (PTFE) spheres, purchased from TIS. The radius of the small spheres $r_{s}$ is 0.795  mm ($\pm$ 3.1 % according to the manufacturer), the large spheres have a radius $r_{L}$ of 1.5 mm ($\pm$ 0.8 %).

The binary mixtures are shaken sinusoidally in cylindrical containers (diameter 50 mm, made of polyamide Nylon 6-6) using an electromagnetic shaker (LDS 406). In order to assure steady state conditions, all samples are shaken for one hour at a frequency of 100 Hz and an acceleration of 2 g. To avoid the accumulation of dust, the beads and the container are cleaned with ethanol and pure water after each five measurements.

The average charge of individual beads is measured after the shaking has stopped by extracting each ten large and small beads from the sample using an antistatic tweezer. The beads are then deposited into a Faraday Cup connected to a Keithley 6514 electrometer. Because the magnitude of the charge on a dielectric particle will scale with the beads’ surface area, we consider here the surface charge density $\sigma_{L,s}=Q_{L,s}/4\pi r_{L,s}^{2}$ of large resp. small beads instead of the total magnitude of charge $Q_{L,s}$ Cademartiri et al. (2012). We note, that the sum of all charges on the beads is not necessarily zero, as the walls of the shaking container will also charge electrostatically.

In order to modify the charge accumulation on the beads, the experiments are performed under controlled relative humidity (RH). A self-built climate chamber equipped with a cooling trap and an ultrasonic transducer allows to tune the ambient humidity in the range between 10 %RH and 100 %RH Schella et al. (2017). The humidity inside the chamber is logged constantly and changes on average about 2 %RH during the course of an experiment. Humidity control is started one hour prior to the the experiment in order to equilibrate the water content on the surface of the beads and the container walls Zitzler et al. (2002).

An advantage of using PTFE beads is their high contact angle with water (108*∘* Kwok and Neumann (1999)) which prevents the formation of capillary bridges at high humidity levels. Consequentially, segregation due to capillary attraction Samadani and Kudrolli (2000); Geromichalos et al. (2003) will not affect our experiments.

Figure 1 demonstrates that under our shaking conditions large spheres charge negatively and small spheres charge positively. This observation is the opposite of what has been found in previous granular experiments Hu et al. (2012); Xie et al. (2013); Waitukaitis et al. (2014) and predicted by some models of same-material tribo-charging Kok and Lacks (2009); Lacks and Sankaran (2011). A result similar to our observation was found in experiments with spheres sliding along a plane made from the same material Lowell and Truscott (1986a, b).

To estimate the threshold for tribo-charging, we have de-ionized large PTFE beads on a grounded metal plate using an electrostatic ion gun prior to depositing them in the Faraday Cup. The residual charge density on these beads was found to be $\sigma_{th}=-1.8\,\mbox{pC}/\,\mbox{mm}^{2}$ ($Q_{th}=-52\,\mbox{pC}$), which is comparable to previous results Kaufman et al. (2009).

II.1 X-ray computed tomography

The structure of the packings created by shaking is analyzed using X-ray computed tomography. The tomography setup (Nanotom, General Electrics) is operated at 130 kV and 90 $\mu$A using a tungsten target. The side length of a voxel (which is the 3D equivalent of a pixel) is 60$\mu$m3 and data sets consist typically of $900\times 900$ voxel in horizontal direction and, depending on the expansion of the bed, 800 to 900 voxel in vertical direction.

Particle centers and radii are identified using the image processing procedure described in Ref. Schella et al. (2017). Since the structural features of the tribo-charged mixtures might be modified in the vicinity of the walls Kumar et al. (2014), we exclude all particles which are closer than three large particle diameter to the container walls from our further analysis. At the top and bottom we discard two layers of large particles. The remaining core region consists of $3090\pm 430$ particles.

Figures 2 (a) and (b) show reconstructed sphere positions from the inner part of two samples, the two panels correspond to the samples with the smallest and largest surface charge density on the large spheres. Neither packing shows signs of vertical segregation. This can also be seen in figure 2 (c): within fluctuations the contribution of the large spheres to the total volume is one half, independent of height. This result holds also for all other experiments reported here.

The X-ray tomographies allow us to compute both the average number of contacts of the spheres and the volume fraction $\phi$ of the packing. A binary mixture has four different contact numbers: first the number of contacts an average large spheres forms with other large spheres $Z_{LL}$, or with small spheres $Z_{Ls}$. Then, the number of contacts an average small sphere forms with large spheres $Z_{sL}$ (which is different from $Z_{Ls}$, cf. Sec. III) and finally the number of contacts between small spheres $Z_{ss}$. We have measured those four numbers by adapting the contact number scaling function method described in Aste et al. (2005); Schaller et al. (2013); Weis and Schröter (2016). Details can be found in the Appendix V.

In order to compute the volume fraction of the analyzed region, without the interference of any boundaries, we first perform a set Voronoi tessellation of our sample which assigns each point of the interstitial space between the particles to the sphere which surface is closest  Schröder-Turk et al. (2010); Weis and Schröter (2016). The global packing fraction is then computed as

[TABLE]

The enumerator contains the total volume of all the $N_{L}$ large and $N_{s}$ small spheres in the analyzed region and the denominator the sum of all the individual Voronoi volumes $\nu_{L}$ and $\nu_{s}$ of the large respectively small spheres.

III Charge controls the contact numbers

Figure 3 shows the main result of our study: the binary contact numbers exhibit a clear dependence on $\sigma_{L}$ and $\sigma_{s}$. The numbers of large-small and small-large contacts, $Z_{Ls}$ and $Z_{sL}$, increases linearly with increasing electrostatic charge density. At the same time the number of same type contacts, $Z_{LL}$ and (less obvious) $Z_{ss}$, decreases with increasing surface charge density. This change in contact numbers is in good agreement with a simple model assuming that like-charged large beads repel each other whereas oppositely charged particles attract each other.

The increase of opposite type contacts in charged samples is also compatible with the visual impression gathered from figures 2 (a) and (b). While neither of the two packings shows macroscopic segregation, the local structure differs in that the large particles form more string-like structures in the highly charges sample. Similar structures have been identified in simulations of charged binary colloidal aggregates Barros and Luijten (2014) and mono-disperse charged grains Chen et al. (2016a). An interesting follow-up question will be if these changes in microstructure do also alter the macroscopic mechanical behavior of the material. This would open an avenue for granular packings with tunable properties.

There is a noticeable difference in how strong $Z_{Ls}$ and $Z_{sL}$ dependent on their respective $\sigma$, i. e. $\partial Z_{Ls}/\partial\sigma_{L}>\partial Z_{sL}/\partial\sigma_{s}$. This difference can be explained using the fact that the total number of large-small contacts in a given volume is the same as the number of all small-large contacts: $N_{L}Z_{Ls}=N_{s}Z_{sL}$. Taking the derivative with respect to an average $\sigma$ we obtain

[TABLE]

where we have used the additional condition that the total number of particles in the observation volume is independent of the charge, which is indeed justified in our experiments. As we have studied equal volume mixtures, equation 2 predicts $N_{s}/N_{L}=(r_{L}/r_{s})^{3}\approx 6.7$. A linear fit to the data of Fig. 3 yields $\partial Z_{Ls}/\partial\sigma\approx 5.5\;\partial Z_{sL}/\partial\sigma$ which is in reasonable agreement with the predicted slope ratio.

We can also compare our contact numbers results with previous experimental  Pinson et al. (1998) and theoretical Biazzo et al. (2009); Meng et al. (2014) work on the contact numbers of uncharged binary mixtures. A linear regression of our data and an extrapolation to the value $\sigma\rightarrow 0\leavevmode\nobreak\ \mbox{pC}/\mbox{mm}^{2}$ yields $Z_{LL}$ = 1.9, $Z_{Ls}$ = 10.9, $Z_{sL}$ = 1.6 and $Z_{ss}=4$, which agrees well with the previously published results for packings of comparable size ratio.

IV Average contact number and global packing fraction

In the previous section we have shown that tribo-charging leads to a local rearrangement and hence changes the binary contact composition. However, tribo-charging does also affect global quantities of the binary sphere packings, as shown in figure 4. The packing fraction $\phi$ decrease approximately 1% with increasing surface charge density. This trend is in agreement with simulations of mono-disperse particles Chen et al. (2016a).

The average contact number $\left\langle Z\right\rangle$ does also decrease with increasing surface charge density. Hence, the bed expands and gets looser. Such a correlation of $\phi$ and $\left\langle Z\right\rangle$ is to be expected based on previous studies of mono-disperse sphere packings Aste et al. (2005); Schaller et al. (2015).

Qualitatively, increasing the charge density on the beads will also increase attractive interactions between large and small particles. Thus, a decreasing packing fraction with increasing charge density seems counter-intuitive at first glance. However, attractive interactions also alter the mechanical stability of granular packings since these have a stabilizing effect, causing the formation of chain-like, porous structures Barros and Luijten (2014); Chen et al. (2016a). To what extend additional many-body Cheng et al. (2014) or polarization effects of the dielectric beads Barros and Luijten (2014); Lee et al. (2015); Qin et al. (2016) contribute to our findings has to be clarified in future studies.

V Summary

Binary systems of dielectric particles have been shaken vertically at different humidity levels which allows to control the tribo-charging of the beads. Because small and large beads differ in the sign of their charge, the resulting attractive interactions inhibit macroscopic segregation of the sample. At the same time the electrostatic interactions change the local structure of the packing: the stronger the charge carried by the individual particles is, the more likely becomes the formation of contacts between small and large beads at the expense of same bead type contacts. Previous studies of binary packings stated that the composition of contacts can be changed by changing the number ratio of small to large particles. Here, we suggest an alternative route: The composition of contacts can also be altered by tribo-charging the particles.

Acknowledgements.

We thank Wolf Keiderling for repeated mechanical support.

Appendix A Contact number analysis

A contact between bead $A$ and $B$ is defined as touching beads, i.e. when the distance $d$ between the beads is equal to the particle radii $d=r_{A}+r_{B}$. Applying this definition to experimental data, as e.g. gathered by X-ray tomography, is a challenging task due to two reasons. First, errors in the image acquisition and processing add random noise to the particle coordinates and therefore distances between pairs of particles. And secondly, all granular particles are to some degree poly-disperse, hence $r_{A}+r_{B}$ is not a constant but depends on the individual particles under consideration. To mitigate these two problems we use an ensemble based fitting method which determines $Z_{AB}$ by modeling the effect of inaccuracies in the particle coordinates using the best average representations of $r_{A}$ and $r_{B}$ Aste et al. (2005); Schaller et al. (2013); Weis and Schröter (2016).

The method works in two steps. First, the average interparticles distance $\langle r_{A}+r_{B}\rangle$ is determined from the first peak of the binary radial distribution function $g_{AB}(d)$ which measures the probability to find a particle of type $B$ in a distance $d$ from a given particle of type $A$. $g_{AB}(d)$ can be computed by counting the number of particles in spherical shells around a reference particle:

[TABLE]

Here the sum over $j$ runs over all particles of type $B$ and the delta function gives only a contribution if the distance between the two particle centers $|\vec{x}_{A}-\vec{x}_{B,j}|$ is equal to $d$. The triangular brackets denote the average over all particles of type $A$. The normalisation consists of two parts: the volume of the spherical shell analyzed grows with $4\pi d^{2}$ and by dividing with the number density $\rho$ we assure that an uncorrelated system will have $g_{AB}(d)=1$.

Figure 5 (a) shows the large-large $g_{LL}(d)$, large-small $g_{Ls}(d)$ and small-small $g_{ss}(d)$ pair distributions for a mixture shaken at approximately 13 % RH. The first peak in these distributions originates from particle pairs in contact, therefore an extrapolation of the peak positions provides the best possible estimate for the three different combinations of $\langle r_{A}+r_{B}\rangle$.

To determine the four different contact numbers $Z_{AB}$ we follow an adapted version of the procedure described in Refs. Schaller et al. (2013); Weis and Schröter (2016). First we determine how the number of contacts $n_{Z_{AB}}$ (defined as touching or overlapping particles) changes if we multiply the particle radii with a scaling factor, thereby creating particles with virtual diameters $D_{v}$ ranging from 0.98 to 1.03 times $2r_{A}$ respectively $2r_{B}$. The resulting $n_{Z_{AB}}(D_{v})$ can be seen in figure 5 (b).

The idea is that errors in the particle positions due to image processing and polydispersity should be Gaussian distributed. We expect therefore that for virtual diameters smaller than $D_{avg}=\langle r_{A}+r_{B}\rangle$ the binary contact numbers $n_{Z}$ will follow a cumulative normal distribution

[TABLE]

where the experimental uncertainties are captured by the variance $\sigma$ and $Z_{AB}$ is the average contact number we try to determine.

For $D_{v}>D_{avg}$, a linear term has to be added to $n_{Z}(D_{v})$ to account for close, but non-contacting particles, i. e. particles from the right shoulder of the first peak of $g_{AB}(d)$. The full contact number scaling function is thus given by

[TABLE]

with $m$ being an unknown slope and $\Theta$ the Heaviside function.

Figure 5 (b) shows that equation 5 provides reasonable fits for all four possible combinations of binary contacts.