# Robust scaling of strength and elastic constants and universal cooperativity in disordered colloidal micropillars

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Edited by John W. Hutchinson, Harvard University, Cambridge, MA, and approved November 11, 2014 (received for review July 24, 2014)

## Significance

The mechanical response of glassy materials is important in numerous technological and natural processes, yet the link between the embryonic stages of plastic deformation and macroscopic mechanical failure remains elusive. The incipient inelastic rearrangements are believed to be highly cooperative and characterized by a scaling of yield strength and elastic constants. Whether this behavior transcends the nature of bonding is still an open question. Here, we show that disordered colloidal micropillars spanning the spectrum of glassy packing also demonstrate a robust scaling of elastic and plastic properties. Our measured relationship and deduced cooperative rearrangement strains bear striking resemblance to other glassy systems with disparate bonding, implying a universal building block for macroscopic flow.

## Abstract

We study the uniaxial compressive behavior of disordered colloidal free-standing micropillars composed of a bidisperse mixture of 3- and 6-μm polystyrene particles. Mechanical annealing of confined pillars enables variation of the packing fraction across the phase space of colloidal glasses. The measured normalized strengths and elastic moduli of the annealed freestanding micropillars span almost three orders of magnitude despite similar plastic morphology governed by shear banding. We measure a robust correlation between ultimate strengths and elastic constants that is invariant to relative humidity, implying a critical strain of ∼0.01 that is strikingly similar to that observed in metallic glasses (MGs) [Johnson WL, Samwer K (2005) *Phys Rev Lett* 95:195501] and suggestive of a universal mode of cooperative plastic deformation. We estimate the characteristic strain of the underlying cooperative plastic event by considering the energy necessary to create an Eshelby-like ellipsoidal inclusion in an elastic matrix. We find that the characteristic strain is similar to that found in experiments and simulations of other disordered solids with distinct bonding and particle sizes, suggesting a universal criterion for the elastic to plastic transition in glassy materials with the capacity for finite plastic flow.

- plasticity in disordered solids
- shear transformation
- cooperative deformation
- Eshelby inclusion
- colloidal glasses

In an ideal, defect-free system, the relationship between a material’s elastic constants and yield strength is indicative of the underlying plastic event that generates macroscopic yielding. For the case of a crystalline solid, Frenkel predicted the ideal yield stress by estimating the energy necessary to cooperatively shear pristine crystallographic planes. The result estimates that the ideal shear strength, *μ*, as *d* is the grain size (thus controlling the fraction of planar defects), and Taylor strengthening (4) predicts *ρ* is the dislocation density. This ability to tailor material strength is a reflection of the large catalog of plastic events found in crystals and the associated broad range of energies necessary for their operation. In such defected crystals, the highly cooperative shear mechanism that defines the intrinsic ideal strength is superseded by mechanisms that require the motion of only a few atoms (e.g., dislocation glide or climb), rather than the coordinated motion of many atoms.

Metallic glasses (MGs)—amorphous alloys—on the other hand, exhibit a surprisingly robust relationship between yield strength and elastic constants despite their atomic heterogeneity and absence of long-range order. Various MG alloys have been synthesized with shear moduli that range from 10 to 80 GPa (5). Remarkably, an approximately universal critical shear strain limit of

In this article, we report on free-standing amorphous colloidal micropillars with compressive strengths that also exhibit a robust correlation with elastic constants and thus a universal elastic limit. By varying the packing fraction, *ϕ*, we are able to vary the maximum transmitted force, equivalent to strength, and the elastic constants over almost three orders of magnitude. We reconcile the robustness of the measured relationship by considering the energetics of the fundamental plastic event at criticality that is believed to underlie yielding in amorphous solids. This approach is based on the cooperative rearrangement first proposed by Argon following observations of sheared amorphous bubble rafts (17). The idea has since been extended in several models, including the cooperative shear model (CSM) of Johnson and Samwer (6, 18) and the shear transformation zone (STZ) theory of Falk and Langer (19, 20). Our analysis results in an estimation of the characteristic transformation strain of a cooperative rearrangement with a magnitude that bears striking resemblance to that estimated in MGs, supporting the notion of a characteristic cooperative mechanism for plasticity in amorphous solids.

We previously reported on a synthesis route for producing free-standing colloidal micropillars with cohesive particle–particle interactions (21). Briefly, capillaries are filled with colloidal suspensions, subsequently dried, and carefully extruded to produce free-standing specimens for uniaxial mechanical testing. The pillars relevant to the current work are 580 μm in diameter and composed of a bidisperse mixture of 3.00- and 6.15-μm-diameter polystyrene (PS) spheres. The mixture is prepared with a volume ratio

We developed a novel mechanical annealing procedure to alter the packing fraction *ϕ*—and consequently the mechanical response—which is described as follows. After allowing the suspension of colloidal particles to dry within the capillary tube, two steel wires with diameters slightly smaller than the capillary diameter are inserted into both ends of the tube, rendering the packing fully confined (Fig. 1*A*). A piezoelectric actuator is brought into contact with one of the wires and the opposite wire is coupled to a force transducer, enabling measurement of the axial force. Sinusoidal displacements (*ϕ*. After mechanically annealing the pillar from one side, the capillary tube orientation is reversed, and the process is repeated from the other side to promote uniform compaction. Varying the confining force in addition to the amplitude, frequency, and number of displacement cycles allows for some control of compaction. Following annealing, the average packing fraction of the confined micropillar, *D*, length, *L*, and mass,

Following our annealing steps, the micropillars were prepared for uniaxial compression. The micropillars are made free standing by extruding a desired length from the capillary using a precision drive screw (Fig. 1*B*). As the pillars are oriented with their major axes perpendicular to the force of gravity, the stability of our free-standing micropillars suggests strong cohesive particle–particle interactions. Between one and four free-standing specimens are obtained from each capillary, and thus fluctuations in

Mechanical responses for specimens with *C* and *D*. Both specimens exhibit an initial loading regime where the transmitted force, *F*, increases linearly with *SI Text* and Figs. S2 and S3); the consequences of underestimating the true elastic modulus will be discussed later.

The measured ultimate strengths and effective elastic moduli for 27 micropillar specimens are shown in Fig 2 *A* and *B*. Both

Our measurements demonstrate a strong correlation between *C*), with a slope of *C*). The robust relationship between *C*. We further note that such scaling between strength and elastic constants has been reported in atomistic simulations of nanocrystalline alloys (27), which in the limit of diminishingly small grain sizes have been shown to exhibit cooperative mechanisms of plasticity reminiscent of metallic glasses (e.g., shear banding, pressure-sensitive yield criteria) (28).

To understand the scaling relationship in our relatively athermal colloidal micropillars, we model the fundamental building block of cooperative plastic flow in the framework of Eshelby-like elasticity. Specifically, we consider the change in free energy associated with the introduction of an ellipsoidal inclusion—representing the cooperative shear transformation—in an elastic matrix subjected to an applied far-field stress. This approach is motivated by experiments (15, 17, 29) and simulations (30, 31) on the deformation of amorphous solids that suggest that the fundamental plastic event is a cooperative, shear-induced rearrangement of ∼10-100 particles (32), referred to as a shear transformation zone (STZ) (Fig. 3). After operation, in which the STZ evolves from the initial to the deformed state, an elastic strain field is generated in the STZ and the surrounding matrix owing to elastic compatibility. The corresponding change in the Gibbs free energy due to the introduction of an inclusion in a finite elastic matrix subjected to an applied stress has been analyzed using Eshelby’s method, where the confined transformation is modeled by allowing for a stress-free unconfined transformation followed by reinsertion into and elastic accommodation by the matrix (33⇓–35). In addition to the elastic energy of the confined shear transformation, the applied stress field interacts with the stress field generated by the inclusion, resulting in an extra part of the Gibbs free energy (34) (see derivation in *SI Text*). Taken together, the elastic energy and interaction terms yield a simple expression for the change in the Gibbs free energy associated with the introduction of the inclusion

Here, *a*, lying along the direction of maximum shear stress,

which can be described by the scalar dilatational strain magnitude *C* denotes the confined transformation strain and may be related to the unconfined transformation strain with superscript *T* by Eshelby’s tensor **2** reduces to

where *E* is Youngs modulus, and *ν* is Poisson's ratio (39). This expression represents the self-stress of the inclusion, which is completely defined by the material’s elastic constants, *E* and *ν*, and the transformation strain magnitudes

where **1** then reduces to

With the assumption that *SI Text*). We assume that at *SI Text* for full expression for Θ)

We assume *ν* between 0.15 (14) and 0.45 and find a best fit for the data with *ν*. Because our system is dissipative, the true elastic modulus is larger than the stiffness measured on loading. Assuming that 50% of the work done on the system during loading is stored as elastic energy (21), the true elastic modulus is underestimated by a factor of 2 (*SI Text* and Figs. S2 and S3 for load-unload measurements and evidence of quasi-linear loading). This error results in an overestimation of

The magnitude of *SI Text*, Fig. S4, and Table S1 for a table of compiled experimental values of

This simple model does not capture the complex dynamical interaction of activated and nucleating STZs that determine the ultimate deformation morphology, which likely governs the extent of plastic deformation and the spatio-temporal evolution from individual STZ operation to macroscopic shear localization. However, the robustness of the correlation between *T* is the temperature, and *ε* is a measure of the interaction energy between particles assuming Hertzian contact. This parameter is a measure of the thermal energy relative to the elastic energy stored in the particles and vanishes in the athermal limit. For our system, *τ* is the timescale associated with the nucleation of water capillaries. Assuming a capillary nucleation timescale similar to that measured on silicon surfaces (47),

## Acknowledgments

We thank the Penn Nanoscale Characterization Facility for technical support and D. J. Magagnosc for technical assistance and insightful discussions. We also thank A. Liu for critical reading of our manuscript and insightful comments. We gratefully acknowledge financial support from the National Science Foundation through University of Pennsylvania Materials Science Research and Engineering Center DMR-1120901.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: gianola{at}seas.upenn.edu or daeyeon{at}seas.upenn.edu.

Author contributions: D.J.S., D.L., and D.S.G. designed research; D.J.S. and Y.-R.H. performed research; D.J.S. and D.S.G. analyzed data; and D.J.S., D.L., and D.S.G. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1413900111/-/DCSupplemental.

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