The preparation steps in this method are relatively simple and do not limit the throughput of characterization. First, a single layer of particles was deposited at the bottom of each well on the plate. Then, triblock gel solutions of varying concentrations were heated and pipetted into each well, and then they were allowed to cool and solidify around the particles. After the samples solidified completely, the plate facing outwards was centrifuged multiple times at successively increasing speeds. After each speed, the well plate was removed from the centrifuge, photos of the plate were taken, and then it was returned to the centrifuge for the next speed. The results show that centrifugation can distinguish between materials of varying strengths, validating the feasibility of this method for measuring bulk mechanical properties (FIGS. 2A-2B).

It was observed that when one particle broke through a sample, the rest of the particles in that well would usually break through all at once. Thus, the critical centrifugal speed was defined as the midpoint between the highest speed at which a sample was intact and the speed at which most of the particles broke through the sample. Critical centrifugal speed was observed to be independent of the duration of centrifugation above ˜30 seconds. To investigate whether the number or location of particles in a well had any effect on the critical centrifugal speed, results from several particles in a flat-bottomed well were compared to results obtained from a single particle in the center of a U-shaped well. The critical centrifugation speed was consistent across both scenarios, which illustrates that the number and location of particles in each well is not significant.

To correlate the results to those from more classical mechanical tests, they were compared to related mechanical properties. A serendipitous correlation was observed between the shear modulus of the triblock gel measured by oscillatory rheology and the centrifugal acceleration at which that gel fractures (FIG. 2B). To determine the origin of this correlation, it was first hypothesized that the correct metric for the material strength is the centrifugal force at break normalized by the particle's cross-sectional area, i.e., a “centrifugal stress,”

σ

c
⁢
e
⁢
n
⁢
t

=

F

c
⁢
e
⁢
n
⁢
t

A

=

4
3

⁢
π
⁢

R
3

⁢

a

c
⁢
e
⁢
n
⁢
t

⁢
Δ
⁢
ρ

π
⁢

R
2

=

4
3

⁢

Ra
cent

⁢
Δρ

(
1
)

where R is the particle radius, Δρ is the difference between the densities of the particle and the gel, and a_centis the centrifugal acceleration. To test this hypothesis, the centrifugation experiment was repeated while varying the size and density of the particles. There was excellent agreement of critical centrifugal stress across different particle sizes and densities over almost an order of magnitude (FIGS. 3A-3B). This demonstrates that critical centrifugal stress is the correct metric for interpreting the results because it normalizes for the particle size and density, allowing for direct measurement of the material's inherent strength.

To investigate the mechanism of gel fracture in the centrifugation method, a similar experiment was set up in which a particle was physically pulled out of a gel using a thin wire (FIGS. 4A-4E). As the wire was connected to an inchworm motor, load transducer, and optical sensor, the force and displacement could be directly measured during the pulling process. A side camera also enabled visual monitoring. To estimate the forces on the particle alone, a control experiment was run consisting of a wire with no particle. After subtracting the load of this control experiment from the total load, it was found that the force on the particle increased nonlinearly from zero, when the particle was at the bottom, to a plateau value (FIG. 4B). This plateau was reached at a displacement of ˜0.5 mm for a 3 mm diameter particle, and the force eventually fell off when the gel completely fractured.

Direct quantitative comparison of pulling the particle using a wire vs. centrifugation was difficult because of a few issues. Generally, it was observed that adhesion of the gel to the wire contributed between 60% and 85% of the total measured load at fracture. Even after subtracting the wire's adhesive force, the pulling force drastically underestimated the centrifugal force that one would expect for a particle of similar size and density. To explain this, it was hypothesized that the wire generated a pathway of broken gel that the particle easily tunneled through as it moved up, while the gel above the particles was pristine in the centrifugal experiment. The presence of the wire thus limited the quantitative results that can be drawn from this experiment.

Nevertheless, the wire pulling experiment contributed an important insight into the analysis of the mechanics of centrifugation. While the plateau in pulling force did not quantitatively match the centrifugal results, it did correspond to when the particle experienced its maximum force, which only occurred when the particle was a considerable distance away from the bottom surface. This suggests that if a particle begins at the bottom surface, the critical centrifugal force at break should be the same as if the particle had begun far away from the bottom. While this specific comparison could not be performed in the centrifugal test, the thickness of the gel was instead varied above the particles on the bottom surface of the well plates. It was found that as this thickness increased above a few particle diameters, the results were generally insensitive to the thickness (until it was so large that the centrifugal force exceeded the tensile strength of the gel, and the gel itself broke out of the well plate). This is all consistent with the same picture: a particle that begins at the bottom of the well plate must eventually move up, through the entire gel, before it breaks out. The critical centrifugal stress for a particle starting at the bottom surface should thus be identical to that of a particle in an infinitely large gel.

The stress and strain fields associated with a force exerted on a solid sphere in an infinite elastic medium have a well-known analytical solution. From the frame of reference of a particle of radius R that has moved a distance U in the +z direction (up), the deformation field (FIG. 4C) in the surrounding medium is given by,

u
r

(

r
,
θ

)

=

-
U

⁢
cos
⁢

θ
⁡
(

1
-

3
2

⁢

(

R
r

)

+

1
2

⁢

(

R
r

)

3

)

(
2
)

u
θ

(

r
,
θ

)

=

U
⁢
sin
⁢

θ
⁡
(

1
-

3
4

⁢

(

R
r

)

-

1
4

⁢

(

R
r

)

3

)

,

using a cylindrical coordinate system where r represents the radial distance from the center of the particle and θ represents the angle from the +z direction (up). The force on the particle is given by,

F=6πGRU, (3)

where G is the shear modulus of the medium. Here it was assumed that the medium was incompressible, and so by the correspondence principle of viscoelasticity, this situation was identical to the familiar fluid mechanics problem of Stokes flow around a sphere. (A. C. Pipkin, Lectures on Viscoelastic Theory, Springer-Verlag; New York, 1986; E. M. Furst and T. M. Squires, Microrheology, Oxford University Press, 2017; W. M. Deen, Analysis of Transport Phenomena, Oxford University Press, 2011.) Prior to failure, the upward force on the sphere from centrifugation was exactly counterbalanced by a downward force that had contributions from the hydrostatic pressure (FIG. 4D) and shear forces at the sphere's surface. As pressure does not cause failure unless it reaches the regime of cavitation, failure was investigated by only considering the shear stress, and the maximum shear stress experienced throughout the medium was plotted, given by,

τ
max

=

1
2

⁢

❘
"[LeftBracketingBar]"

σ
max

-

σ
min

❘
"[RightBracketingBar]"

(
4
)

where σ_maxand σ_minare the largest and smallest principal values of the stress tensor (FIG. 4E). Note that this is the measure used in the familiar Tresca yield criterion. (R. M. Jones, Deformation Theory of Plasticity, Bull Ridge Publishing, 2009.) It was observed that there were local maxima above and below the particle at positions r=√{square root over (2)}R, where the gel experienced uniaxial compression and tension, respectively. Notably, the highest stress regions occurred closest to the sphere's surface at the equator (θ=90°), where the deformation mode was simple shear. Although this conclusion was reached by examining the Tresca stress measure, it was noted that the stress at these regions was much higher than anywhere else in the material, so selecting other failure criteria will ultimately result in the same conclusion. Therefore, during centrifugation the gel first broke near the equatorial surfaces.

Because the point of failure occurred at a region experiencing simple shear, the correct mechanical property corresponding to the critical centrifugal stress can be potentially determined by comparing the stress fields to a shear rheology experiment. The shear stress at the equatorial surfaces is given by,

σ

r
⁢
θ

=

2
⁢
G
⁢

ε

r
⁢
θ

=

G
⁡
(

1
r

⁢

∂

u
r

∂
θ

+

r
⁢

∂

∂
r

(

u
θ

r

)

)

❘
"[RightBracketingBar]"

r
=
R

,

θ
=

90
⁢
°

=

3
⁢
G
⁢
U

2
⁢
R

(
5
)

The failure criterion occurs when this shear stress exceeds the critical stress at fracture measured by a shear rheometer,σ_rθ>τ_crit, i.e. Bingham number is less than unity. After applying eqn. (3) and (5) to this inequality, the criterion becomes,

3
⁢
G

2
⁢
R

⁢

(

F

6
⁢
π
⁢
GR

)

>

τ
crit

(
6
)

F

π
⁢

R
2

>

4
⁢

τ

c
⁢
r
⁢
i
⁢
t

.

Note that the left side of this inequality is simply the centrifugal stress (eqn. (1)),

σ_cent>4τ_crit (7)

This centrifugal experiment is thus a quantitative measurement of the fracture stress as measured on a shear rheometer, and this inequality predicts a quantitative mapping between the two techniques.

This key prediction can be tested by performing shear rheology on triblock gels over the same range of concentrations as in the centrifugal experiment. (Furst and Squires, 2017.) The centrifugal method was interpreted as a quasi-static test, as stress was ramped up during startup until fracture occurred at maximum stress. Because of this, the centrifugal results were compared to those from a static shear stress growth rheology experiment. It was found that the critical σ_centvalues from FIGS. 3A-3B are readily superimposable with the results from the rheometer, linearly shifted by the factor of 4 from eqn. (7) (FIG. 5A). This agreement is remarkable and worth additional emphasis, as the prediction from the analysis was confirmed with no fitting parameters, and the centrifugal test indeed quantitatively measured the shear fracture stress of the gels. To further confirm this, both centrifugal and shear rheology tests were performed over a range of hydrogels composed of chemically crosslinked polyacrylamide (PAM). This new system was selected as a completely different category of soft material compared to the triblock gels, using water vs. organic solvents, crosslinked through covalent bonds vs. physical associations, and containing different microstructures and fracture mechanisms. However, the agreement between the two techniques was similarly striking in the PAM hydrogels (FIG. 5B), confirming that the centrifugal method and its analysis are extremely robust, even for quantitative measurement.

There should be no debonding between the gel samples and the bottom of the well plate when the methods are being carried out. If debonding were to occur, failure would be observed at an artificially low centrifugal speed because the particles would never penetrate through the gel. Several scenarios could potentially cause debonding or similar behavior. Larger amounts of sample in each well may result in larger centrifugal forces at the sample-plate interface, causing adhesive failure at the interface or cohesive failure in the material if this force exceeds the tensile strength of the gel. Pains were taken to avoid this in these experiments. An excessive number of particles could also cause debonding by adding to the sample mass during centrifugation and potentially introducing unnecessary defects at the sample-plate interface. This effect was investigated by confirming that the critical centrifugal stress was similar for a single particle in a well vs. several particles. Finally, debonding would be problematic if there is poor adhesion between the materials comprising the sample and the well plate. In the triblock gel system, this material naturally adhered to common polypropylene plates, and no debonding was observed.

To avoid debonding during centrifugation of the PAM hydrogels, separable glass-bottomed plates were used, which were surface-functionalized with methacrylamide groups that covalently bond with the network. This solution is widely applicable to other classes of soft materials, as other functionalization schemes could be easily adapted by choosing a silane coupling agent that can bond to the chosen material. For some systems, storing the functionalized plates in a cool, dry environment is required to ensure long-term stability of surface treatment. Using high-density particles such as stainless steel or tungsten carbide may be optimal, because the acceleration required to cause fracture is lower, so there is less occurrence of debonding. Moreover, because material fracture occurs rapidly when the critical centrifugal stress is exceeded, it is recommended to utilize short centrifugation periods (30 seconds or less) to avoid placing unnecessary excess stress on the point of adhesion to the plate. While pains were taken to avoid debonding of the PAM hydrogel at the bottom of the well plate through surface functionalization, no such effort is necessary for the surface of the particles. This is because the shear stress in the material is maximized at the interface with the spheres at the equatorial region (FIG. 4E). Therefore, adhesive failure at the particle-gel interface is indistinguishable from cohesive failure, which would also occur at the same location. Centrifugation thus removes the particles from the material through decohesion.

The serendipitous correlation with shear modulus in the triblock gels (