Nonaffine deformation under compression and decompression of a flow-stabilized solid
NNonaffine deformation under compression anddecompression of a flow-stabilized solid
Carlos P. Ortiz, Robert Riehn and Karen E. Daniels
Department of Physics, North Carolina State University, Raleigh, NC, 27695, USA1 April 2016
Abstract.
Understanding the particle-scale transition from elastic deformation toplastic flow is central to making predictions about the bulk material properties andresponse of disordered materials. To address this issue, we perform experiments onflow-stabilized solids composed of micron-scale spheres within a microfluidic channel,in a regime where particle inertia is negligible. Each solid heap exists within a stressgradient imposed by the flow, and we track the positions of particles in response tosingle impulses of fluid-driven compression or decompression. We find that the resultingdeformation field is well-decomposed into an affine field, with a constant strain profilethroughout the solid, and a non-affine field. The magnitude of this non-affine responsedecays with the distance from the free surface in the long-time limit, suggesting thatthe distance from jamming plays a significant role in controlling the length scale ofplastic flow. Finally, we observe that compressive pulses create more rearrangementsthan decompressive pulses, an effect that we quantify using the D statistic for non-affine motion. Unexpectedly, the time scale for the compression response is shorterthan for decompression at the same strain (but unequal pressure), providing insightinto the coupling between deformation and cage-breaking.
1. Introduction
Understanding how structural rearrangements in disordered solids differ from crystallinesolids is central [1, 2, 3] to achieving control of material properties such as resistance toflow [4], sound propagation [5], heat capacity [6] , and dielectric constants [7]. For largedeformations, the microscopic response differs non-perturbatively from the predictions oflinear elasticity [8]. Instead of linear deformations, phenomena such as shear banding [9],yielding and plastic rearrangements [10], and non-local effects [11] are present. Recentexperiments have explored non-affine deformations in 3D sheared colloidal glasses [12],3D emulsions [13], and 2D foams [14]. For sufficiently slow deformations, it is an openquestion whether the flow behavior [15] is controlled by the jamming transition, wheremoduli vanish as the packing approaches a critical packing fraction [16].In this paper, we present experiments quantifying the particle-scale deformation offlow-stabilized solids: particle heaps formed under controlled hydrodynamic stress (seeFig. 1). These quasi-2D heaps are assembled via the slow accumulation of micron-scale a r X i v : . [ c ond - m a t . s o f t ] M a r eformation of flow-stabilized solids (a)
25 µm xy (b) Figure 1 (a) Schematic of channel geometry, not to scale, illustrating barrier with overflow.Streamlines are scaled versions of a calculation solving Stokes’ equations on the true devicedimensions. (b) Image of central region of a heap of bidisperse particles. Dark regionsare dimly fluorescent 600 nm particles and bright regions are brightly fluorescent 710 nmparticles. White box indicates the location of the region in which deformations are analyzed. particles against a barrier within a microchannel, and are found to be stable aboveP´eclet number ∼ eformation of flow-stabilized solids
2. Experimental Setup
Our experiments begin by assembling a microsphere heap by flowing a dilute suspensionagainst a barrier (see Fig. 1a). The microchannel is fabricated to have a height H = 897 nm, higher than the height of a barrier ( h b = 694 nm), so that the fluidoverflow accumulates particles against the barrier of width W = 512 µ m. The heightsare chosen to create a quasi-2D heap, shallow enough to suppress both stacked and non-stacked bilayer phases [23]. The suspension is pumped into the channel by compressinga reservoir at the inlet using a low pressure, piezoelectrically actuated, digital regulator(AirCom PRE1-UA1), at P = 10 kPa above atmospheric pressure. After two hours ofequilibration, the heap is 154 µ m deep (30 ◦ angle of repose) and contains approximately40,000 particles, as shown in Fig. 1b. The coordinate system takes ˆ x parallel to thebarrier and ˆ y perpendicular to the barrier, with the origin at center of the barrier; thefluid flow is in the − ˆ y direction.The dilute, aqueous suspension is prepared at a concentration of ρ = 180 / (100 µ m) fluorescent microspheres. The particles are a bidisperse mixture of equal concentrationsof 600 nm and 710 nm polystyrene microspheres ( ≈
6% polydispersity, elastic modulus4 GPa from Bangs Laboratories). We use steric and electrostatic stabilization (sulfatefunctionalized surface with ζ -potential = −
60 mV and coated with Triton X-100) toprovide reversible inter-particle and channel-particle interactions. The suspending fluidis a density-matched aqueous solvent at pH 5.4, buffered by citric acid to preventcrystallization and with 17%( w/v ) sucrose to provide density-matching. The latersuppresses segregation and sedimentation effects, important both at the barrier andat the inlet reservoir. Because the total particle brightness scales approximately withthe particle volume, and we are working near the diffraction limit, the 600 nm particlesappear dimmer than the 710 nm particles. This effect aids in tracking the motion ofthe particles.We quantify the affine and non-affine deformation due to a pulse of eithercompression (∆
P >
0) or decompression (∆
P <
P/P = 19 for thecompression pulse and ∆ P/P = 0 .
95 for the decompression pulse. Imaging occursin two phases. A full view of the initial heap is accessible with a 10 × objective, while eformation of flow-stabilized solids μ m (a) (b) (c) Figure 2
Illustration of image analysis process. (a) Sample micrograph, with grayscaleindicating raw data. (b) Edge detection between bright and dark regions.(c) WienerDeconvolved Image. experiments quantifying the particle motions require visualizing a zoomed-in regionusing a 60 × objective with a 4 × beam expander. The image was recorded by a CCDcamera with 10 × µ m pixels and an exposure time 36 µs . As shown in Fig. 1b, thezoomed measurement region is of size 40 d × d and is located adjacent to the barrier.For each pulse, we first acquire an image of a region of interest at the center of the heapprior to the pressure change, allowing us to extract the initial configuration of particlepositions. Additional images, taken at 27 Hz, characterize the particle-scale responseof the heap to the change in pressure. After a wait of 100 seconds, long enough forparticles to settle onto a new, equilibrated configuration, we repeat this process for thecompression pulse.
3. Image Processing
For either compressive or decompressive pulses, we first compare the initial and finalconfigurations (separated by 100 sec), and quantify both the total deformation andthe non-affine deformation. Second, using the series of frames immediately followingthe pulse, we track individual particles to identify non-affine effects on the local scale.Below, these are referred to as “long-time” and “short-time” dynamics, and requireslightly different image-processing. For the long-time dynamics, the total distancetraveled is on the order of a few particle diameters. Therefore, we first subtract thetotal affine deformation before performing particle tracking using the Blair-Dufresneimplementation [24] of the Grier-Crocker particle tracking algorithm [25].
Particle identification:
Fig. 2 summarizes how we obtain particle positions beginningfrom a raw image. We identify the location of each particle by performing a Wienerdeconvolution on the raw image, using a Gaussian approximation to the point-spreadfunction with full-width at half maximum of 540 nm. This value is found to maximizethe contrast in the output image, as measured from the ratio of the standard deviation tothe mean intensity, but is more effective at locating the large (bright) particles than thesmall (dim) particles. The resulting deconvolved image allows us to detect the centroid eformation of flow-stabilized solids
Total deformation:
We estimate the total affine deformation ∆ y due to a single pulseby making a coarse-grained measurement of the particle displacements between an initialimage and a final image. These two images are created by averaging 10 initial images I i ( x, y ) = (cid:104) I i ( x, y, t ) (cid:105) t and 10 final images I f ( x, y ) = (cid:104) I f ( x, y, t ) (cid:105) t . We divide I f ( x, y ) intohorizontal strips of width 2 d and compute a cross-correlation with I i ( x, y ) to determineits displacement. We find that the cross-correlation is sharply-peaked function for stripsof at least this width.Due to the large total strains, we perform particle pair matching between initial andfinal configurations based on particle positions from which the total affine deformation∆ y has already been subtracted. After this adjustment, pair identification proceedsas in the one-step particle tracking [24], with the size of the search region selected tocorrespond to the estimate of the maximum non-affine displacement amplitude, plus anestimate of the error in the affine strain. Short-time particle tracking:
In order to obtain particle trajectories during the fullduration of the dynamics, we make several assumptions about the nature of validtrajectories. We limit the displacement per frame to 0 . µ m; this value is consistentwith the total affine deformation rate determined above. In addition, we consider aparticle’s identified size (brightness) in order to either split incorrect trajectories or orreconnect broken trajectories.
4. Results
In previous experiments [18], we observed that flow-stabilized solids exhibit a nonlinearstress-strain relationship in which the magnitude of the deformation of the surface ofthe flow-stabilized solid is well-described by∆ y surface y surface ∝
11 + ∆ PP − . (1)The success of this description is somewhat surprising, as Eqn. 1 does not contain anyinformation about the distribution of stresses or strains throughout the flow-stabilizedsolid. The stress field within the solid is anticipated to be similar to that in asedimentation experiment where particles “on top” of the sedimented material applysome stress on lower layers (in the limit of shallow sediments without side walls).We believe that the success in describing our experiments is due to the universalityof the van-der-Waals thermal argument. However, that argument breaks down ifnon-affine motions occur, and we anticipate that the anticipated lower stress at theupstream (“top”) surface of the flow-stabilized solid is not fully characterized by thevan-der-Waals argument. In the following, we first identify the distributions of particledisplacements in the asymptotic long-term limit, before following individual trajectories eformation of flow-stabilized solids x/d y / d (a) -1 -0.5 0 " y/d y / d (b) Figure 3
Determination of the total (long-time) affine field from particle tracks for ∆ P =+9 . σ of the positionuncertainty in initial (blue) and final (yellow) configuration. (b) The associated affinedeformation field. The dashed line is a linear regression with γ = 3 . ± . through compression and decompression. Our particular interest is in the associatedparticle-scale non-affine motions and their dependence on the sign of ∆ P . We find displacements of individual particles in the heap immediately before and 100 safter a compressive/decompressive pulse through a two-step analysis. Following thehomogeneous strain field assumption from our prior work [18], we first use the imagecross-correlation analysis of images before and after deformation to obtain a globalestimate of the affine strain field. We then use the affine transformation identified bythe cross-correlation analysis as a scaffold for the matching of particles in the imagesbefore and after deformation. In Fig. 3a, we show an example of the particle locationsafter Wiener deconvolution and prior to finding the centroids, for both I i (red, beforecompression) and I f (white, after compression). By tracking each centroid, we can plotthe local displacement ∆ y as a function of y -position within the heap.As shown in In Fig. 3b, the mean behavior is linear, confirming that the overallassumption of an affine deformation was sufficiently accurate. The best fit line to thesepoints provides a measure of the strain: ∆ y = γ ∞ y with γ ∞ = − . y -direction.To obtain the total non-affine deformation field, we subtract the local affinemotion from each displacement vector, as in [2]. Figs. 4 and 5 (both compression anddecompression) show the total, affine, and non-affine displacement fields, for comparison. eformation of flow-stabilized solids (a) Total deformation (b) Affine deformation (c) Non-affine deformation Figure 4
Deformation field for a compression given by a pressure change at the inlet of∆ P = +9 . x -position and the y -position in units ofparticle diameters. (a) Full deformation field. (b) Affine deformation field. (b) Non-affinedeformation field, magnified by a factor of two. (a) Total deformation (b) Affine deformation (c) Non-affine deformation Figure 5
Deformation field for a decompression given by a pressure change at the inlet of∆ P = − . x -position and the y -position in units ofparticle diameters. (a) Full deformation field. (b) Affine deformation field. (b) Non-affinedeformation field, magnified by a factor of two. Importantly, we observe bands of correlated motions, as expected from [21, 22]. Becausethe total deformation field is not robust in tracking individual bead pairs over long times,we next examine the short-time dynamics.
Using the estimate of strain provided by Fig. 3b, we track the fast dynamics arising froma compressive/decompressive pulse. For each frame, we first subtract the estimatedaffine deformation, based on the fraction of total strain which should have accumulatedso far (see § eformation of flow-stabilized solids (a) Compression10 20 30 10 20 30 y / d y / d x/d x/d Figure 6
Particle Tracks and Packing Fraction. (a) Particle tracks during compression.Tracks are colored to introduce contrast. (b) Particle tracks during decompression.
Sample trajectories are shown in Fig. 6. While the total deformation field is of thesimilar magnitude under compression and decompression, we find a more pronouncedscrambling of the particle trajectories under compression, as compared to decompression.Below, we quantify both the affine and non-affine contributions to these trajectories.
Affine deformations:
Data was binned within strips along the x -direction, providingensembles of particles-dynamics sampled as a function of depth. Figs. 7a and 8ashow the mean deformation field ∆ y as a function of y -position for compression anddecompression, respectively. For both deformation directions, we find an exponential-like asymptotic approach to the final displacement magnitude. The depth-dependenceof the asymptotic value of ∆ y (Fig. 7b and 8b) demonstrates the same linear relationshiporiginally shown in Fig. 3b. The resulting slope ( γ ∞ = ∆ y/y ) quantifies the dynamicsof affine reorganization. We find a marked difference between decompression (Fig. 7c)and compression (Fig. 8c) in that decompressions are far slower than compressions, andthat the strain curves for decompression collapse better onto a single dynamic curve.To quantify the difference, we make the Ansatz of a single-exponential approach tothe asymptotic deformation∆ y = γ ∞ y (cid:16) − e − ∆ tτ (cid:17) (2)where ∆ y is the particle displacement after a time interval ∆ t , τ is a characteristictime scale of particle rearrangements, and γ ∞ is the asymptotic strain. Note thatthe value γ ∞ here is a fitting parameter; we find its value to be consistent with theestimate from the long-time dynamics. As shown in both panels (d), this exponentialform is a good fit for the decompression pulses with τ affine,decompression = 0 . ± .
05 s.For compressive deformations, a single-exponential form is less consistent with theobserved dynamics. Instead, there appears to be a two-step process of compressionin which the viscous stress increase acts nearly instantaneously throughout the solid,while stresses due to particle-particle contacts propagate at a distinct speed of soundfrom the immobile barrier on which the solid is formed. Given the two-step nature of eformation of flow-stabilized solids Figure 7
Affine Deformation Time-Series Analysis During Decompression. (a) Magnitudeof correlated displacement field as a function of time at varying distances upstream of thebarrier. (b) Long-time displacement amplitude ∆ y , as a function of distance upstream ofthe barrier, both in units of particle diameters. (c) Correlated displacement amplitudenormalized by the long-time displacement amplitude. (d) Log-linear plot of growth curvesin (c). the process under compression, we establish an upper bound on the relaxation timescale of τ affine,compression = 0 . ± .
05 s.
Non-affine deformations:
We identify the non-affine contribution to the deformationfield by subtracting the affine portion associated with the best-fit instantaneous valueof the strain, which we designate γ ( t ) = γ ∞ (cid:16) − e − ∆ tτ (cid:17) . (3)To quantify the resulting non-affinity, we use the D measure [2] defined by D ( t ) ≡ (cid:88) neighbors (∆ (cid:126)r ( t ) − γ ( t )∆ (cid:126)r ( t )) . (4)Here, (cid:126)r ( t ) is the set of local displacement vectors connection nearest neighbors, and t is the time immediately before the pressure step was applied. Fig. 9 showsthe time-evolution of the non-affine displacement as a function of y -position duringcompression and decompression, respectively. In both graphs, D grows and ultimately eformation of flow-stabilized solids Figure 8
Affine Deformation Time-Series Analysis During Compression. (a) Magnitudeof correlated displacement field as a function of time at varying y -positions. (b) Long-timedisplacement amplitude ∆ y , as a function of distance upstream of the barrier, both in unitsof particle diameters. (c) Correlated displacement amplitude normalized by the long-timedisplacement amplitude. (d) Log-linear plot of growth curves in (c). Time [sec] y / d y / d D m i n / d (a) Compression (b) Decompression Figure 9
Magnitude of non-affine deformation field D as a function of time and depthfor (a) compression and (b) decompression. eformation of flow-stabilized solids Time [sec] D m i n / y Time [sec] D m i n / y (a) (b)(c) (d) Compression Decompression
Figure 10
Plots of D as a function of time, with the y -axis scaled either by the size ofthe large particles or the y -position at which the non-affine deformation is being probed.The legend in panel b applies to all panels. (a) Compression. D scaled by d . (b)Compression. D scaled by y . (c) Decompression. D scaled by d . (d) Decompression. D scaled by y . saturates. Interestingly, the magnitude of the non-affine field scales linearly with depthas demonstrated by the collapse of D /y data series shown in Fig. 10. This asurprising finding in light of the assumed constant strain throughout the flow-stabilizedsolid. Furthermore, the magnitude of non-affine deformations is approximately twice aslarge under compression than under decompression at near identical asymptotic strain γ . We observe that the growth of D with time is smooth. We are able todetermine a characteristic time for the approach to the asymptotic value of D byfitting a single exponential approach, as we did for the affine deformation field. Indoing so, we neglect the low background value of D in steady-state flow-stabilizedsolids arising from Brownian motion. We find τ non-affine,compression = 0 . ± .
08 s, and τ non-affine,decompression = 0 . ± . τ affine,decompression = 0 . ± .
05 s). For compression, wherea single time scale is less well defined, and an upper bound on the affine time scale is τ affine,compression = 0 . ± .
05 s, the non-affine field also lags the affine deformation. eformation of flow-stabilized solids
5. Discussion
We have observed particle-scale non-affine motions within flow-stabilized solids, andexamined how their spatiotemporal dynamics depend on whether the deformation iscompressive or decompressive. We observed the typical swirling regions often associatedwith non-affine deformations, arising through cooperatively rearranging regions. Themagnitude of these effects is nearly twice as large under compression than underdecompression, in spite of very similar total strains.We observe that compressive pulses (large ∆
P/P ) generate more non-affinedeformation, which is able to dissipate the effect of the pulse more quickly. Because thenon-affine fields for both compressive/decompressive deformations occur after similardelays with respect to the affine deformations, is suggests that they are triggered bythe affine deformations. In the context of caging behavior, this suggests that theaffine deformation distorts the cages provided by the neighboring particles and therebymakes Brownian cage-breaking (non-affine deformation) more likely. Remarkably, thisis the case even though the strain is approximately the same for decompression andcompression.In probing the spatial dependence to the magnitude of the non-affine deformations(Fig. 9), we observe that the degree of non-affinity increases with distance from thebarrier. This effect can be rescaled by the position to indicate a universal behavior.The form of this dependence suggests D ∝ p ∝ K , for pressure p and modulus K [18]. One interpretation is that the surface of the heap is less rigid (smaller K ),and therefore more prone to undergoing non-affine deformations (higher D ). Similareffects have been observed in numerical simulations [22], where increasingly non-affinedisplacements are present in proximity to unjamming.The significance of the above conclusions to soft-matter particle assemblies isto reinforce the centrality of understanding non-affine rearrangements to link bulkproperties of the material, such as its modulus and global stability, to local propertiesabout the typical particle geometry and rearrangement timescales. Based on theseresults, this experimental setup opens the possibility to explore this connection,by studying multiple orders of magnitude of heap sizes, under dynamically tunableinteraction potentials and heap geometry, maintaining the ability relate particle-scalerearrangement dynamics to bulk properties. By doing so, it should be possible todetermine length and time scales at which localized and collective rearrangements havethe greatest impact on bulk properties, and shed light on the general mechanisms bywhich it is feasible to control the bulk properties of soft matter systems. Acknowledgments
We are grateful for support from the National Science Foundation through anNSF Graduate Fellowship, grants DMR-0644743, DMS-0968258, DMR-1121107,MRSEC/DMR-112090, and INSPIRE/EAR-1344280. Research was also supported by eformation of flow-stabilized solids
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