Reports: ND951987-ND9: Ultrasonic Studies of Jamming Transitions

Jonathan I. Katz, PhD, Washington University in St. Louis

As preparation and background for our ultrasonic studies of starch suspensions, we first investigated the rheology of cornstarch suspensions in oil and in CsCl brine (a density-matching substitute for water; brine suspensions have similar rheological properties as water suspensions, but avoid the problem of sedimentation that prevents quantitative measurements of water suspensions).  It is well known that starch suspensions in nonpolar solvents (oil, benzene, CCl_4, etc.) do not show shear stiffening.  In order to understand why aqueous and brine suspensions shear stiffen, it is necessary to understand why these other suspensions do not.  We hypothesize that the difference is the different polarity of these fluids.

We first measured the rheology of oil suspensions, obtaining apparently the first quantitative data.  Their effective viscosities vary roughly as the -1/2 power of the strain rate, over a wide range of concentration.  There is no evidence for shear thickening at any concentration, even at concentrations so high that they resemble wet pastes, with finite shear strength.

We then repeated the classical measurements of the rheometry of CsCl brine (density-matched) suspensions.  As others found before, these show discontinuous shear thickening, with apparent viscosity in the shear thickened state inversely proportional to the shear rate.  This may be explained as the effect of sliding friction in a jammed state of the grains produced by shear dilation.   We found the new result that this jammed state is hysteretic: if the shear rate is then reduced, even below the threshold of discontinuous shear thickening, the shear and normal stresses remain approximately constant.  Their ratio may be considered a coefficient of sliding friction.  This ratio was found to be approximately 0.3, ranging between 0.2 and 0.5 over more than an order of magnitude of strain rate.  Such irregularity is a familiar property of sliding friction.

It is believed that confinement, either by surface tension acting on grains that penetrate the fluid surface as a result of shear dilation or by external stiff surfaces, is required for discontinuous shear stiffening.  In order to explain the differences between the properties of shear thickening starch suspensions in water or brine, and shear thinning oil suspensions, we must appeal to differences in the interactions of their fluids with starch surfaces.  Because it is difficult to measure the surface properties of loose piles of powder, we prepared smooth surfaces of starch by gelatinizing it in warm water and then evaporating the water.  We found that water, brine and water-glycerin (80%) suspensions (water-glycerin was a control with the same viscosity as our oil) all have finite contact angles on dried gelatinized starch, while oil wets it (spreading indefinitely).  If this conclusion applies to ungelatinized starch granules, then in oil there are no three phase contact lines because the granules are completely wetted.  Confining forces are only exerted on contact lines, explaining the absence of discontinuous shear thickening in starch suspensions in nonpolar fluids.

The principal thrust of our study is the use of ultrasonic scattering to measure the structure function of starch suspensions in static, shear thinned (below the onset of shear stiffening) and shear stiffened states.  Our graduate student Ben Johnson's thesis consisted of ultrasonic measurements of static suspensions, a necessary step towards measurement of suspensions under shear.  He determined the group and phase velocities, attenuation and backscattering of ultrasound in CsCl brine suspensions of cornstarch at a range of volume fractions from 0 to 0.40.

From the sound speed we were able to evaluate the bulk modulus of dry corn starch (because of its granular nature, this cannot be determined in any other manner), finding it to be (1.1 +/- 0.1) X 10^{10} Pa.  In the 4--8 MHz band, at which our initial measurements were made, the attenuation is approximately proportional to frequency (as is the case for most composite media).  It is also approximately proportional to (volumetric) starch concentration in the range 10--30%, possibly exceeding the proportionality by about a factor of 1.2 at 40% concentration.

The phase velocity of CsCl brine suspensions of starch showed measurable dispersion, even though the velocity varies by only a few km/s (out of about 1.5--1.8 km/s, depending on the starch concentration) between 4 and 8 MHz.  It is remarkable that the dispersion is "negative", or "anomalous", with the phase velocity decreasing with increasing frequency; this is opposite to the sense of dispersion predicted by the Kramers-Kronig relations from the attenuation measured in that frequency range.  Control measurements on solids, such as lucite, showed "normal" dispersion with magnitude consistent with the Kramers-Kronig relations.  We have suggested that the explanation of the anomalous dispersion is strong absorption at frequencies higher than those measured, in analogy to the Sellmeier dispersion in optics that results from ultraviolet absorption.  This is supported by the approximately quadratic measured dependence of dispersion on frequency in a 30% suspension.

Our goal is to probe the internal structure of corn starch suspensions in the static, shear thinning (low shear rate) and shear thickening regimes.  To do this we will have to make measurements at frequencies around 100 MHz, for which attenuation lengths are very small (sub-mm) but backscattering from internal structure (such as jammed starch grains, and grains organized into sheets in shear thinning) is very strong, increasing with frequency more rapidly than attenuation.  As a step in that direction we measured backscattering in the 4--8 MHz range.  At low starch concentrations the backscatter varies approximately as the 4th power of frequency, as expected for uncorrelated particles, but the exponent decreases to about 2.3 at a 40% concentration, indicating anti-correlation of the grains, as required by their mutual hard repulsion (and consequent spatial anticorrelation).  The backscattering actually decreases with concentration for concentrations > 10% (in contrast to the low concentration limit, in which it is proportional to the concentration of scatterers), indicating that repulsion anti-correlates the grains.

These results were reported in Ben Johnson's Washington University Ph.D. thesis.  A portion of them have been published in the Journal of the Acoustical Society of America, and a further paper is in preparation.