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Reprogrammable allosteric metamaterials fromdisordered networks†
Nidhi Pashine,

Amir Mohammadi Nasaband Rebecca Kramer-Bottiglio*

Prior works on disordered mechanical metamaterial networks—consisting of fixed nodes connected bydiscrete bonds—have shown that auxetic and allosteric responses can be achieved by pruning a specific
set of the bonds from an originally random network. However, bond pruning is irreversible and yields asingle bulk response. Using material stiﬀness as a tunable design parameter, we create metamaterial
Received 22nd September 2022,
Accepted 31st January 2023

networks where allosteric responses are achieved without bond removal. Such systems are

DOI: 10.1039/d2sm01284g

a disordered mechanical network with variable stiﬀness bonds, diﬀerent subsets of bonds can beexperimentally realized through variable stiﬀness bonds that can strengthen and weaken on-demand. In
strategically softened to achieve diﬀerent bulk responses, enabling a multiplicity of reprogrammablersc.li/soft-matter-journal

input/output allosteric responses.

1 Introduction
Mechanical metamaterials are materials with unusual mechanical properties not commonly seen in nature, which arise predominantly due to clever structural design rather than material
composition. There has been considerable interest in metamaterials that show novel material responses, including metamaterials that can exhibit more than one mechanical response.1–6
In recent years, there has been notable development ofmetamaterials made from disordered systems. In a systemcomposed of a disordered network of nodes connected bybonds, each bond contributes diﬀerently to the bulk propertiesof the system.7 This variation can be utilized by modifying
individual bonds in a way that the system evolves to have a
specific, programmed response. One example is a disorderedmechanical network wherein an output strain between two
nodes is tuned in response to an input strain applied between
two source nodes, where both sets of nodes are located along
the periphery of the network.8 A mechanical network with along-range interaction between input and output sites is calledan ‘‘allosteric’’ metamaterial, as allostery is the process by whichbiological macromolecules transmit the eﬀect of binding at onesite to another (often distal) site.
Allosteric responses have been designed in disordered mechanical networks in various ways such as bond pruning,8,9 network
evolution,10,11 as well as by using local design rules.12,13 In the caseof bond pruning (i.e., the complete removal of network bonds),

School of Engineering & Applied Science, Yale University, New Haven, CT, 06520,
USA. E-mail: rebecca.kramer@yale.edu† Electronic supplementary information (ESI) available. See DOI: https://doi.org/
10.1039/d2sm01284g

This journal is © The Royal Society of Chemistry 2023

once a programmed behavior is designed into a system, the output
response is immutable.7,8,13,14 To enable tunable and recoverable
allosteric metamaterial responses, we seek approaches to vary thestiﬀness of select bonds on-demand. With variable stiﬀness bonds,
a subset of bonds could be softened to achieve one target response,
and another set softened to achieve a completely diﬀerent output
response. This approach removes the need to prune bonds andenables any new target response to be achieved while all priorresponses remain fully recoverable.
The field of soft robotics has recently seen substantial development of novel composite materials with tunable material properties,
especially material stiﬀness. A variety of methods have been
employed to attain variable stiﬀness,15 including pneumatic
jamming,16,17
magnetorheologicalandelectrorheologicalmaterials,18–21 shape memory polymers and alloys,22–25 liquid crystal elastomers,26–28 and phase-changing materials.29–33 Typicallyderived from metallic alloys, waxes, or thermoplastic polymers,
encapsulated phase-changing materials exhibit a decrease in modulus via the transition from solid to liquid, and raise in modulus via
the reverse (solidification) transition. One phase-changing material
gaining traction in the literature is Field’s metal, a eutectic alloy ofbismuth, indium, and tin known for its low melting point of
Tm = 62 1C and non-hazardous composition.34–41
Our instantiation of a re-programmable allosteric metamaterial is realized through a disordered mechanical network
wherein we replace a predetermined set of inert bonds withvariable stiﬀness bonds, which are fabricated by incorporating
Field’s metal cores in soft silicone shells. A variable stiﬀness
bond is in the high-stiﬀness state when the Field’s metal core issolid and in the low-stiﬀness state when the core is liquefied
using an embedded copper heater. The bond network is
designed by incorporating local stiﬀness as a parameter inSoft Matter, 2023, 19, 1617–1623 | 1617

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simulations and varying the bond stiﬀness to modify themechanical input/output response between two peripheral
node sets. When diﬀerent sets of the variable stiﬀness bonds
are softened, diﬀerent allosteric responses are achieved withfull reversibility and re-programmability.

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2 Design algorithm
Our proposed mechanical network is a spring network inspired byRocks et al. 2017.8 The networks are generated from random
configurations of jammed soft disks in 2D.42 By joining the centersof the disks with an unstretched central-force spring, these random jammed packings are converted into disordered spring networks. With a network of N nodes and Nb bonds, the authors ofRocks et al. tuned the output strain eout between two nodes in
response to the input strain ein applied between two source nodes,
where both sets of nodes are located far away from each otherwithin the same network—a so-called allosteric response. In thiswork, we make allosteric networks by modifying the pruningalgorithm presented in ref. 13 to a tuning algorithm that is basedon local stress distributions in our system.
Starting with simulations of disordered networks with periodic boundaries, we randomly choose two pairs of nodes
halfway across the system to be our source and target nodes.
The success of our network design is measured in terms of theratio of strain at the target nodes to the applied input strain at
the source nodes; strain ratio, Z = eout/ein. In our initial network,each bond has an initial spring constant of kistiff ¼ K0stiff li ,where K0stiff is a constant and li is the length of bond i. Our
design algorithm identifies a set of bonds j in {Bsoften}, whichwhen softened to a spring constant of kjsoft ¼ K0soft lj create anallosteric interaction between the source and target sites. To
tune an allosteric response, we choose the source and target
sites, the desired strain ratio (Zgoal), and the stiﬀness ratio of
the bonds, R ¼ K0soft =K0stiff . Notably, the initial network contains no soft modes and any stresses from an applied strain
would dissipate locally. Therefore, the initial strain ratio ofthese networks is Z0 E 0.
The set of bonds to be softened, {Bsoften}, is determinedbased on the stress distribution in the network under externallyapplied strains. We define TSourceas the tension in bond i as airesult of an applied strain at the source site and TTarget
as theitension in bond i due to applied strain at target site. A positive
value of Ti corresponds to tension and a negative value corresponds to compression of bond i.
Our design algorithm is based on the values of TSource
andiTargetTi
for each bond i. These terms calculate how much stressis channeled into each bond when a strain is applied at aparticular site in the system. Previous work has shown that the
product of the two terms, TSource TTarget, determines thejjrelevance of each bond in linking the source to the target.13
The larger the value of TSource TTarget, for a given bond, thejjmore it hinders an allosteric interaction. By softening thesebonds, we allow them to stretch or compress easily, hence
aiding an allosteric interaction. Until this point, the design

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Soft Matterprocess is symmetric with respect to the source and the target.
However, we are designing an inherently assymetric network
where the input energy is provided at the source and an output
strain is expected at the target. In order to break the symmetry,
we change the relative weights of TSource
and TTarget
by softi
i
Sourceening the bond with the highest Tj
 (TTarget)n, where n isjan odd integer 41. Higher values of n biases the algorithm
towards the target by eﬀectively lowering the energy required tocreate an output strain at the target site. Hence, larger values ofn lead to more eﬃcient solutions with fewer numbers ofsoftened bonds. On the other hand, previous work has shownthat as the value of n increases, the discrepancy between
simulations and experiments also increases.13 In this work,
we choose n = 3 as a trade-oﬀ between these two eﬀects.
The design algorithm is summarized in Fig. 1. We measure
TSourceand TTarget
for each bond, take a product of the twoiiterms, and soften the bond with the largest value of TSourcej(TTarget)3. We repeat this process iteratively until Z reaches its
jdesired value or the process fails due to creation of a localized
low energy mode. For simplicity, the network drawings in Fig. 1
only show the magnitude of tensions. However, the sign of eachterm decides the sign of Z. Softening bonds where TSourcej3(TTarget)ispositiveleads
toapositiveZ,
while
softeningbonds
jwhere TSource (TTarget)3 is negative leads to a negative Z. In thisjjwork we focus on designing networks with Z 4 0.
The results from simulations of allosteric responses areshown in Fig. 2. Disordered networks are often characterized
by the average number of bonds coming out of each node,
known as the coordination number (hZi). The networks in our
simulation are isostatic at an average coordination number of
hZ0i = 4. We work with three sets of networks with a system sizeof 70 nodes with average coordination numbers hZi = 4.5, hZi =4.8, and hZi = 5.4. We tune each network using the design
protocol described above. Note that the design process is notdependent on the strain ratio, Z. We simply keep softening
bonds one at a time and call the design process successful assoon as the final strain ratio goes above the desired Z. As aresult, the output strain of these systems is never exactly equal
to the desired Z but always slightly higher than it.
Fig. 2A shows the success rate as a function of the stiﬀness
ratio for Z = 1. The stiﬀness ratio is given as ratio of stiﬀness ofa softened bond to the stiﬀness of a regular bond, such that it
can take a value between 0 and 1 and a lower stiﬀness ratio
corresponds to a higher contrast in bond stiﬀnesses. The
success rate rapidly declines as the stiﬀness ratio increases,
with the best results at a stiﬀness ratio of a few percent (0–2%).
This decrease in the success rate is expected because themechanism that incorporates an allosteric response relies on
creating a low energy mode in the system which is achieved bylowering the strain energy of a particular set of bonds—ones
that are softened. Increasing the stiffness ratio increases theenergy stored in the soft bonds, thereby increasing the totalenergy of the system and lowering the success rate. An interesting feature to note here is that the closer these systems are to
isostaticity (hZi = 4), the higher the success rate. We speculatethat this is a result of the vibrational response of the original

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Fig. 1 A schematic of the design process. I. Apply an external strain at the source site and measure the distribution of resulting tension in the system.
Darker bond corresponds to higher magnitude of tension. II. Apply an external strain at the target site and measure the resulting tension distribution. III.
Using I and II, get the distribution of TSource (TTarget)3. This identifies the bonds that are relevant to linking the source and target and creating an allostericjjresponse. IV. Soften the selected bond (dashed line) by changing its stiﬀness from kstiﬀ to kstiﬀ  R, and measure the new Z. This process is repeatediteratively till the desired Z is achieved.

Fig. 2 (A) Success rate of networks with diﬀerent coordination numbers
as a function of stiﬀness ratio (ratio of stiﬀness of soft bonds to that ofregular bonds). Networks were designed to have a strain ratio (ratio ofstrain at output nodes to the applied input strain) of 1. The vertical dashedline at 0.01 corresponds to the value of stiﬀness ratio used for designing
networks that were built in experiments. (B) Percentage of bonds that need
to be softened as a function of stiﬀness ratio. (C) Success rate as a functionof strain ratio for a stiﬀness ratio of 0.01. The vertical dashed line at 1
corresponds to the strain ratio used in plots A and B.

system. It has been shown that the correlation length ofvibrational modes increases as the system goes closer toisostaticity,43–45 and since allosteric interactions span thewhole system, they are easier to achieve when extended vibrational modes are already present in the system.

This journal is © The Royal Society of Chemistry 2023

Another measure of ease of designing allosteric systems isthe number of bonds that need to be softened. As mentioned,
our system achieves an allosteric response by lowering theenergy of allosteric interaction, which is realized by softening
the bonds that contribute the most to this interaction. The netchange in the interaction energy depends on the number ofbonds softened, as well as the amount by which their stiﬀness
is reduced. The higher the stiﬀness diﬀerence between stiﬀ andsoft bonds, the fewer the number of bonds that need to bemodified. This trend can be seen in Fig. 2B, where the fractionof bonds that need to be softened is plotted as a function of
stiﬀness ratio. As expected, we see that with an increase in thestiﬀness ratio, more bonds need to be softened in a network.
Moreover, this plot shows that the fraction of bonds that need
to be softened goes up with an increase in coordination
number. This finding is consistent with the data from
Fig. 2A, which shows that networks with lower coordination
numbers are better for designing an allosteric response.
Fig. 2C shows the success rate as a function of strain ratio, Z.
The success rate falls sharply as soon as Z goes over 1. We notethat one can achieve a more dramatic response in allosteric
networks if the bonds are removed instead of softened.8,13 Thisdiﬀerence is not surprising considering that softening a bondlowers the local stress in the region whereas pruning a bondgets rid of it completely. Achieving high strain ratios requires
certain soft bonds to have high strains, but the residual stressin soft bonds increases rapidly with an increase in strain, thus
preventing them from showing a high output response.

3 Variable stiﬀness bonds
Experimental realizations of allosteric networks are built out ofsilicone rubber sheets, with the bonds that need to be softened
replaced with variable stiﬀness bonds. By actuating thesevariable bonds, we can switch between two diﬀerent states ofSoft Matter, 2023, 19, 1617–1623 | 1619

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the allosteric network. The variable bonds are made out ofField’s metal and silicone, as shown in Fig. 3, and are passively
stiﬀ. By sending an electric current through co-located copper
heaters, the Field’s metal inside the variable bonds can bemelted via Joule heating, resulting in a soft bond. In the stiﬀ
state, the stiﬀness of variable bonds is comparable to theregular silicone–rubber bonds but in their soft state, the variable bonds are orders of magnitude softer than regular bonds.
Details about the construction of variable bonds, regularbonds, and experimental networks containing the bonds areincluded in ESI† S1.
We estimate the spring constant of the bonds by measuring
the force response of both regular and variable bonds using amaterials tester (Instron 3365). The response of regular as wellas variable bonds in both stiﬀ and soft states is shown in
Fig. 4A. The same data is zoomed in at small strains (Fig. 4B),
and low forces (Fig. 4C). The response is averaged over 5
measurements of 3 diﬀerent bonds of each kind. To calculate
the spring constant of each type of bond, we fit a slope to theforce-displacement curve in the small strain region where theresponse is mostly linear. In our experiments, the stiﬀ bonds aswell as the regular bonds (apart from the ones right at the inputsite) experience negligible strain. On the other hand, when thevariable bonds are in their soft state, they get strained toaround 5%. For the present calculations, we include
strains between 0 and 1.5% and get the following spring
constants for each bond type: kregular = 4.1  0.2 N mm1,
kstiﬀ = 2.2  0.6 N mm1, ksoft = 0.036  0.004 N mm1. As
expected, the spring constant of bonds in the soft state is much
lower than the other two. Soft bonds are 0.9% as stiﬀ as theregular bonds, and as shown in the previous section, designingallosteric responses is very successful at this low stiﬀness ratio.
We note here that the bonds have a non-linear response overlarge strain ranges, as shown in Fig. 4. In particular, thevariable bond in the stiﬀ state gets substantially softer at larger
strains. However, the stiﬀ bonds usually do not experience large
strains and therefore this non-linearity does not have a

Fig. 3 (A) Schematic of a variable stiﬀness bond comprising of a Field’s
metal core and a copper heater that are sandwiched between three layersof silicone. (B) Image of a variable bond.

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Fig. 4 (A) Force response of regular bonds (red), variable bonds in the stiﬀ
state (blue), and variable bonds in the soft state (black). Response at (B) low
strains, and (C) small forces. Stiﬀness values (ksoft, kstiﬀ, kregular) are obtained
from the slopes of the force curves for strains between 0 and 1.5%.

significant impact on the response of our networks. The strain
distribution within a simulated network is shown in ESI† S2.

4 Allosteric networks in experiments
To test the eﬃcacy of our experimental networks, we measure
their response with the variable bonds in stiﬀ and soft states. A
linear spring model is an oversimplification of real material
networks, and previous studies have shown that materials haveother interactions such as angle bending forces and nonlinear
stress–strain response present in them.14 Therefore, only afraction of allosteric networks designed in simulations show
a comparable response in experiments.13
To evaluate the behavior of incorporating variable stiﬀness
composites in a way that is decoupled from other eﬀects in oursystem, we make two copies of each network that we want totest. In the first copy, C1, we take the set of selected bonds,
Bsoften, and replace them with variable stiﬀness bonds. In thesecond copy, C2, we prune the set of bonds, Bsoftened. Then wecompare the response, Z(C1) to Z(C2). We test 3 such networks
and find that Z(C1) = 90  5% of Z(C2). Since there are multiple
mechanisms that lead to a loss in output response, the output
response in experiments often ends up being substantiallysmaller than the designed response.
The designed allosteric networks show an output strain only
when the correct set of variable bonds is softened. We can build
multiple input and output responses in a single network andour design process guarantees that each incorporated response
works independently when the corresponding set of variablebonds is softened. However, as long as diﬀerent responses haveallosteric pathways that do not overlap spatially, we often seethat multiple allosteric responses can coexist in experiments.
One such example is shown in Fig. 5(A): bonds in blue correspond to output 1, and bonds in green correspond to output 2.
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without detriment to either of the outputs. Fig. 5(C) shows theoutput strain responses as a function of input strain, when both
the responses are activated simultaneously. The output strain
responses at each of the output sites are slightly diﬀerent than
before, with Z1 = 0.60  0.02, and Z2 = 0.81  0.01. Fig. 5(D)shows the strains at input and output sites as a function oftime, where both sets of bonds have been softened and both
output sites simultaneously respond to the input strain.
Although multiple responses can be designed to work
simultaneously, this is not how this particular network was
designed. Our design process finds independent solutions foreach output which often turn out to work concurrently. Despitesome experimental losses in the designed networks, our resultshows that this method of designing allosteric responses withvariable stiﬀness materials is successful and feasible.

5 Conclusions

Fig. 5 (A) Allosteric network with multiple responses. Variable stiﬀness
bonds are highlighted in blue and green. Softening blue bonds creates a
response at output 1, softening green bonds shows a response at output 2,
and softening both sets of bonds shows responses at output sites 1 and 2.
Responses can be seen in Movie S3 (ESI†). (B) Strain at output site as afunction of input strain when each response is activated individually.
Output 1 is blue (solid) line and output 2 is green (dashed) line. (C) Output
responses when both outputs are activated simultaneously. (D) Strain
response as a function of time with both outputs activated. Cyclic input
strain (black) results in output strains at output 1 (blue) and output 2(green). Data has been smoothed, using a Savitzky–Golay filter with awindow length of 300 ms, for clarity.

stiﬀ, an applied strain at the input site results in a strain at
output 1 but not at output 2. Similarly, softening just the green
bonds results in a strain at output 2 but not at output 1.
Fig. 5(B) shows the measured output strain responses as afunction of input strain at sites 1 and 2, when each response isactivated individually. The responses are linear in the small
strain regime, with output at site 1, Z1 = 0.64  0.01, and output
at site 2, Z2 = 0.65  0.01. Note that both responses are smaller
than the designed response of Z = 1. These two sets do not haveany bonds in common and the pathways are spatially separated
from each other. In such cases, when both the sets of bonds aresoftened, we see a response at both the output sites 1 and 2This journal is © The Royal Society of Chemistry 2023

The ability to actively tune the behavior of a mechanicalmetamaterial is essential for developing the next generationof materials with increased adaptability and functionality. In
this work, we have demonstrated designing and building disordered allosteric metamaterials with multiple input and output responses incorporated in them that can each beindividually activated on demand. To achieve this, we designed
and fabricated variable stiﬀness bonds whose stiﬀness changes
by two orders of magnitude between their soft and stiﬀ states.
This work brings together the active tunability of material
properties and the diversity of responses that can be achievedin disordered systems to create a new kind of allostericmetamaterial.
Our work opens further avenues in both theoretical andexperimental directions. We introduced material stiﬀness as adesign parameter in disordered networks, but eventually our
variable bonds exist in one of the two possible states. Further
work is needed to create metamaterial systems that truly utilize
the range of material properties that can be achieved throughvariable stiﬀness composites. Such a system might be able tosustain multiple responses that are not spatially separated.
Additionally, our current simulation uses a simplified linear
spring model which deviates from experiments, especiallywhen extended to large system sizes or systems with a higher
number of responses. More realistic simulations would give us
a platform to design and build networks that can exhibit morediverse and complex responses.
A distinct feature of our design algorithm is that we use alocal rule, as opposed to a global optimization. Local design
rules provide an ideal framework to build adaptable materials
because the system changes its properties based on local
stimuli. Our success with using a variable stiﬀness material
inspires us to consider other materials that respond to external
stimuli in diﬀerent ways, such as bonds that can connect anddisconnect adjacent nodes or expand or shrink in length. Usinga variety of materials that respond to stimuli in diﬀerent ways

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will substantially expand our design space and allow us to build
systems with adaptable mechanical responses.
Relatedly, there has been much recent interest in mechanical memory, computing, and supervised learning in physicalsystems.12,46–51 We believe that variable bonds, and moregenerally the concept of locally tunable properties in material
systems,50–53 could be applied beyond the case of allostery inmaterials. For example, we envision such local stiffness control
may contribute toward metamaterials with adaptable force
paths, bulk moduli, and computing functions.54,55 This paper
lays groundwork for the development of adaptable smart
materials that can modify their properties based on external
stimuli.

Conflicts of interest
There are no conflicts to declare.

Acknowledgements
We thank F. Arceri, T. Buckner, and S. Eristoﬀ for helpful discussions. This material is based upon work supported by the National
Science Foundation under Grant No. DMREF-2118988.

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