Article pubs.acs.org/JPCC
Cohesion of Granular Matter in Subzero Humidity E. M. Culligan and H. K. Christenson* School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom S Supporting Information *
ABSTRACT: Atmospheric humidity has an important effect on granular media by increasing the cohesion of powders and grains and is hence of significance for the industrial processing of finely dispersed materials. Water bridges around the contact points of hydrophilic particles lead to an increased cohesion at higher humidities, and this has been studied at ambient temperatures in model systems. Because liquid water bridges may form below the freezing point of water, we decided to study the cohesion of millimetersize glass beads in humid atmospheres at subzero temperatures. We find that the angle of repose of a pile of glass beads in a saturated atmosphere of water vapor below 0 °C decreases as the temperature falls, and closely mirrors the behavior as the humidity at room temperature is reduced from saturation; that is, the angle of repose of the glass beads is similar for water bridges that are of a similar size. These results show that both temperature and moisture determine the cohesion of granular media in subzero environments, and confirm that surface roughness acts to decrease the area over which capillary forces between particles act. Our work also illustrates how interfacial curvature reduces both the vapor pressure and the melting temperature of a substance.
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INTRODUCTION Granular media, which consist of particles with diameters of several micrometers or more, are nonequilibrium systems where gravity dominates over thermal effects. van der Waals forces are unimportant and do not influence the interparticle cohesion,1 but the large effect of moisture on the cohesion and thereby the behavior of granular matter is well-known to most of us. Small particles will adhere strongly to each other due to microscopic water bridges between the grains,1 causing salt shakers to clog and sand to cake. Whether the moisture enters via adsorption and condensation from the atmosphere or from absorption of bulk water is unimportant. The net effect is to lead to a greatly increased cohesion as long as the total amount of water is small. Amounts of water large enough to fill the voids between the particles will reduce the cohesion and result in a slurry of particles in water. A stable sandcastle requires sand with just the right amount of water. Systematic investigations of the behavior of wet granular media have been carried out using techniques such as the rotating drum,2,3 or the avalanche chamber (draining crater).4 With these the stability of an inclined surface of granular matter is quantified in terms of two angles. When the slope of the surface is increased beyond the angle of maximal stability θm, the grains begin to flow and an avalanche occurs until the angle decreases to θr, the angle of repose. For a dry (nonadhesive) assembly of smooth, spherical particles θm and θr are related in a complex manner to the friction between the spheres.5,6 Not surprisingly, both θ values increase with surface roughness5 and interparticle adhesion,5 and a good correlation between the cohesion as quantified by θ, and the relative amount of liquid has been obtained.2,4,5,7,8 Such a correlation has also been found with nonaqueous liquids, albeit with a smaller cohesion due to the lower surface tension of the liquid.2,8 © 2014 American Chemical Society
The simplest situation to consider is when liquid has been added by equilibration through vapor.2,7,8 This avoids uncertainty about the location of the liquid4,6 as capillary condensation occurs only around the points of contact between the particles, and the cohesive force is due to the negative Laplace pressure Δp in this liquid. For smooth surfaces in contact there is a single annulus1,9,10 (Figure 1a) whose interface has two principal radii of curvature r1, r2, where r2 is the radius of the annulus and r1 is the radius of curvature in a plane perpendicular to the plane of the annulus. By definition, r1 is negative, and for small annuli, |r2| ≫ |r1|, and11 ⎛1 γ 1⎞ γ Δp = γLV ⎜ + ⎟ ≅ LV = − LV r2 ⎠ r1 r ⎝ r1
(1)
where γLV is the surface tension of the liquid and r is the magnitude of the total radius of curvature. The r increases with the amount of liquid, but the area over which the negative pressure acts also increases, and the two effects exactly cancel each other for two identical spheres of radius R, giving a force: F = −2πRγLV cos α
(2)
where α is the contact angle of the liquid on the particle surface, and the minus sign indicates an attractive force. As there is no rdependence in eq 2, the increase in cohesion of the particles with the relative humidity2,8 must be due to other factors. Real particles are invariably rough with contact between them often occurring at many points, each with its own capillary bridge.7 The R in eq 2 is then not the macroscopic radius of the particle, Received: May 28, 2014 Revised: July 4, 2014 Published: July 7, 2014 15929
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a study of the cohesion of a model granular matter system, glass beads, in water vapor. We have compared the angle of repose θr measured in an avalanche chamber as a function of relative humidity at room temperature, with the θr measured in an atmosphere saturated with respect to ice as a function of temperature below zero. We find a remarkably good correlation between θr and the size of the capillary bridge, whether due to temperature or humidity effects.
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EXPERIMENTAL SECTION Roughly spherical glass beads, of diameter 1.00−1.25 mm (G. Kisker - Products for Biotechnology), were used as a model system, and scanning electron micrographs of a representative bead are shown in Supporting Information Figure SI1. The avalanche chamber (Supporting Information Figure SI2) consisted of a crystallization dish fitted with a Teflon chamber as a lid, enclosed in an airtight Perspex chamber (Supporting Information Figure SI3). The aperture (diameter 12 mm) of the avalanche chamber could be unblocked by means of a sliding cover operated from outside the chamber by a magnetic release mechanism. The beads drained though the aperture into the crystallization dish (inner diameter 92 mm, drop height 60 mm) to form a pile whose θr was measured from photographs. The chamber atmosphere was equilibrated either with a reservoir of salt solution (CaCl2) of known molality at room temperature (20−23 °C) or with ice (below 0 °C). The chamber was housed in a table-top freezer whose temperature could be varied between −35 and 0 °C, or turned off and left open for room temperature experiments. The Teflon chamber was filled with loosely packed beads, and trial runs showed that a bead volume of 65 cm3 was necessary to ensure that θr was independent of the dimensions of the container into which the beads drained. Porous tubing from a membrane air pump inside the Perspex chamber circulated air through the beads to assist in equilibration. With the pump on, θr did not change significantly after equilibration times exceeding 72 h, so this was taken as sufficient to ensure equilibrium between the beads and the atmosphere. Without assisted circulation, θr had not reached a maximum value even after 196 h. Natural sand (B&Q Natural Sharp Building Sand, manuf. part no.: SMBQSHS40) containing grains of diameters ranging from ca. 0.1−0.5 mm and baked for 30 min at 100 °C was also studied at room temperature and ambient humidity for comparison.
Figure 1. (a) Schematic depiction of contact between two smooth glass beads of radius R in a humid atmosphere, with a bridging capillary condensate, where R ≫ r2 ≫ r1. (b) Contact between rough particles at a low relative humidity, where contact at many asperities with much reduced local radii of curvature (R1, R2, etc.) leads to a smaller net force of adhesion. (c) Contact between rough particles at a high relative humidity where the macroscopic R determines the adhesion force.
but rather a smaller, local radius of curvature Rn of the contacting asperities (Figure 1b). As the relative humidity increases, the size of the bridges increases and covers an increasing number of contact points between the particles, making the effective Rn’s larger. The increase in cohesion with relative humidity is hence due to an increase in the area over which the negative Laplace pressure acts.7,12 Near saturation, there is little difference in the adhesion between smooth and rough surfaces (Figure 1c). Just as a bulk liquid usually does not freeze when the temperature is reduced below the bulk melting temperature Tm, capillary condensates will remain liquid below Tm. Using the crossed mica cylinders (equivalent to a sphere on a flat) of the surface force apparatus,13 it has been shown that the equilibrium size of capillary condensates in saturated vapor below Tm is inversely proportional to the temperature depression ΔT below Tm.11,13,14 While the inner part of the condensates (closest to the contact point between the crossed cylinders) is thermodynamically supercooled and cannot freeze due to surface energy constraints,11 the outer part may freeze if the surface tension between the solid and vapor is more than twice that of the interfacial tension between liquid and solid. This is certainly true for water (and most other liquids), so the lack of freezing must be due to a free energy barrier to nucleation.15,16 The size of a water condensate may hence be reduced below its value in saturated water vapor (where it is effectively infinite) either by a reduction in humidity or by a reduction in temperature. Accordingly, one might expect a correspondence between the effect of temperature and humidity on the cohesion of granular matter. We therefore decided to carry out
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RESULTS
The θr of both glass beads in dry nitrogen or laboratory air, whether previously baked in an oven at 100 °C or not, was the same at 23 ± 2°, in good agreement with theory5 and other experiments.5 The measured θr was the same for 400−600 μm beads, also in line with theoretical expectations17 (these smaller beads were often found to clog the equipment at high humidities and were not used for systematic studies). The sand gave a θr of 38° ± 2° in laboratory air of 40% approximate relative humidity (r.h.) at room temperature, in agreement with the expected greater θr of rougher particles. The error in θr of ±2° is based on an estimated error in determining the angle from photographs, and the error in humidity is estimated to be at most ±2%. Figure 2 shows θr as a function of r.h. for the 1 mm beads at room temperature. There is no significant change between 0 and 50% r.h., but at higher r.h. θr increases gradually to 38° ± 15930
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attributed to the much smaller size of the beads used in those studies, about 200 μm in both cases, which makes gravitational forces that scale with R3 (R = particle radius) and reduce θr less important as compared to capillary forces that scale with R (eq 2) and increase θr, consequently increasing the angle of repose. Above Tm the relationship between the relative vapor pressure (relative humidity) p/ps and the magnitude of the total radius of curvature r of the condensate−water interface is given by the Kelvin equation: r=
VMLγLV RT ln[ps /p]
(3)
where VML is the molar volume (of water). At saturation ps = p and r goes to infinity, which corresponds to a planar liquid− vapor interface. At a temperature T below Tm the magnitude of r of a capillary condensate at saturation (p/ps = 1) is given by11,14,18
Figure 2. Measured angle of repose for d = 1 mm glass beads as a function of relative humidity at room temperature (T = 20−23 °C).
r=−
2° at 98%. At saturation (100% r.h.), the beads clogged the aperture and water was observed in the holding chamber. Figure 3 shows θr of the beads in water vapor in equilibrium with ice (100% r.h.) as a function of ΔT. Clearly, the cohesion
VMLγLVTm ΔHfusΔT
(4)
where ΔHfus is the enthalpy of fusion, and ΔT = T − Tm is negative. Equating the magnitude of r at some temperature TR above zero (e.g., room temperature) from eqs 3 and 4 gives ΔT =
TmRTR ln[p /ps ] ΔHfus
(5)
In eq 5, ΔT is the temperature depression at which a capillary condensate is the same size as it is at an undersaturation p/ps at a temperature TR. In deriving eq 5, the temperature dependence of γLV and ΔHfus has been neglected, at most a 10% error, less than the combined uncertainty in ΔT and p/ps. Expansion of the logarithm in eq 5 gives ln[p /ps ] = ln[(ps + Δp)/ps ] = ln[1 + (Δp/ps )] ≈ Δp/ps (6)
to give ΔT =
TmRTR [Δp /ps ] ΔHfus
(7)
where both Δp and ΔT are negative. With ΔHfus = 6.01 kJ mol−1and TR = 22 °C, one finds
Figure 3. Measured angle of repose for d = 1 mm glass beads as a function of temperature relative to the melting point of ice (ΔT = 0) in an atmosphere saturated with respect to ice.
ΔT ≈ 112 ×
decreases from its value near 0 °C and falls as the temperature decreases (toward the left on the plot). The error in the temperature was estimated to be at most ±2 K.
Δp ps
(8)
So a 1 K temperature depression close to zero is approximately equivalent to a 1% reduction in r.h. near saturation at room temperature. The r of capillary condensates at room temperature was calculated using eq 3 for the r.h. values in Figure 2, and the r values of the subzero condensates for the experimental temperatures in Figure 3 were calculated with eq 4. Figure 4 shows the measured θr as a function of the 1/r values of the capillary condensates for both the humidity and the temperature experiments. The errors are, as before, ±2° for θ, and the errors in 1/r are based on errors in T and r.h. of ±2 K and ±2%, respectively. The correlation between the decrease in the angle of repose due to a reduction in relative humidity at room temperature and that due to a reduction in temperature below zero at 100% r.h. is remarkably good, although there are no data for small condensates at subzero temperatures because we
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DISCUSSION The θr measured for the glass beads in the dry state agrees well with theoretical expectations for uniform and smooth spheres in the absence of cohesive or frictional forces (23.4°),5 suggesting that our glass beads are a good model system for studying the change in cohesion with r.h. θr is also independent of the size of the beads, as expected as long as they are small as compared to the size of the container.17 The significantly higher angle of repose found with sand also supports the validity of our method. Previous studies of the cohesion of glass beads in humid atmospheres have yielded similar angles in the dry state, which have increased to much larger values at higher r.h.,2,8 causing the beads to clog at r.h. ∼90%. This difference can be 15931
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relevant to the behavior of granular matter in ambient atmospheres. However, there are instances when water would enter the system via direct capillary imbibition, as in natural sands and soils. Capillary migration of supercooled water below zero leads to frost heave,25 and because less liquid water would end up around the contact points of particles as the temperature falls, one might expect that the cohesion in this case too would decrease with temperature below freezing, provided that the relative humidity is constant.
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ASSOCIATED CONTENT
S Supporting Information *
Supplementary Figures SI1−SI3. This material is available free of charge via the Internet at http://pubs.acs.org.
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Figure 4. Angle of repose plotted against the inverse of the radius of curvature of a vapor−water interface calculated from eq 3 (equilibrium with water vapor at room temperature, □) and eq 7 (equilibrium with vapor saturated with respect to ice at different subzero temperatures, ■).
AUTHOR INFORMATION
Corresponding Author
*Tel.: 441133433879. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
could not carry out measurements below −30 °C. The agreement supports the idea that surface roughness increases the cohesion of the glass beads with humidity at room temperature because of an increase in the area covered by capillary bridges.7 The cohesion increases as Tm is approached for the subzero experiments because the area covered by capillary bridges also increases in this case. Note that a reduction in r.h. from 100% below 0 °C would give an additional decrease in r as quantified previously,11,14 and hence should further reduce the cohesion and θr. It is known that ions and silica may leach into water from glass surfaces, and that restricted volumes such as capillary condensates become solutions with properties different from those of pure water.19−22 In particular, capillary condensates would become much larger than expected for pure liquids,23,24 and the 1/r values in Figure 4 would not be correct. However, as the leaching of solute into a capillary bridge should depend mainly on the surface area it covers, which would be the same in both cases, the equivalence between temperature and humidity effects on θr and the cohesion should not be affected. From eq 6, Δp/ps = p/ps − 1, and with Raoult’s law, p/ps = x1, we can write Δp/ps = x1 − 1 = −x2, where x1, x2 are the mole fractions of solvent (here water) and solute. Hence ΔT = −
TmRTR x 2 RT 2 x ≈− m 2 ΔHfus ΔHfus
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ACKNOWLEDGMENTS Invaluable knowledge was gained from preliminary experiments by T. Smith, J. Krzyzosiak, and H. Geever as part of their B.Sc. projects in Physics at the University of Leeds. F. C. Meldrum and J. M. Campbell provided useful comments on the manuscript.
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REFERENCES
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(9)
which is the approximate relationship for melting-point (freezing-point) depression by solute (with ΔT = T − Tm). This is because melting-point depression compares reductions in vapor pressure (or chemical potential) and melting temperature due to the presence of solute, whereas the capillary condensation experiment compares reductions in vapor pressure (chemical potential) and melting point due to interface curvature. In both cases, one ends up with expressions comparing temperature reduction with a concentrationdependent term−vapor pressure reduction or mole fraction of solute. Varying the water content of granular matter by equilibration with vapor rather than by addition of liquid water leads to a more well-defined system and allows the comparison between humidity and temperature effects made here. Furthermore, the properties of model systems in such conditions are highly 15932
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