## Soft Matter Physics

#### Isostaticitiy and the verge of mechanical instability

As J. C. Maxwell pointed out in 1864, when the total numbers of *degrees of freedom* and *constraints* of a system become equal, the system is
at the verge of instability, and this state is named as the "isostatic point".

The majority of materials grouped under the name "soft matter" share the common feature that they are *soft*, or, more specifically,
the vibrational spectrum of these materials contains a large number of low-energy modes that can make the materials easily deformable.
Examples of these low-energy modes can be found in *jammed solids*, taking the form of anomalous vibrational modes;
in *fiber networks*, in the form of filament bending modes;
and in *framework structure crystals* such as cristobalites and zeolites, in the form of rigid-unit-modes.
Even in the case of *glasses*, which have elastic moduli similar to those of crystals,
there are excess low-frequency modes comprising the so called boson peak, which significantly affect low temperature transport properties of glasses.
Many aspects of these low-energy vibrational modes can be understood by tracing back to the "isostatic points" of these solids,
and relating them to the zero-frequency floppy modes at that point.

My ongoing research in soft matter physics focuses on a *unified understanding* of real systems near the isostatic point
via *lattice models*, where rigorous theory can be constructed using well-controlled approximations.
These lattice models show rich varieties of mechanical properties and floppy modes.

##### From isostatic lattices to jammed packings

Jamming is a zero-temperature version of the glass transition, and is believed to capture lots of the essential physics in more complicated
glassy materials. Previous numerical and experimental studies demonstrated that *jamming occurs at the isostatic point*.
Part of my recent research concerns the understanding of jammed solids using lattice models.
Central force isostatic lattices, including the square and the kagome lattices in 2D, are characterized by coordination number z=2d,
and exhibit sub-extensive numbers of zero-frequency floppy modes when the lattice is of finite size.
These floppy modes can be lifted to finite frequency by the addition of random additional constraints,
a process similar to what occur in jammed solids when packing fraction is increased.
Our methods of study include mean field theory (coherent potential approximation, or CPA), numerical simulations, and exact solutions.
Using these lattice models my collaborators and I derived scaling relations near the isostatic point that agree with observations in jammed solids.

Related publications:

- Xiaoming Mao, Ning Xu, and T. C. Lubensky,
Physical Review Letters
**104**, 085504 (2010). - Xiaoming Mao and T. C. Lubensky,
Physical Review E
**83**, 011111 (2011). - Wouter Ellenbroek and Xiaoming Mao,
Europhysics Letters
**96**, 54002 (2011).

##### From isostatic lattices to fiber networks

Understanding the mechanical properties of fiber networks is very important for various areas ranging from biology to materials science and engineering.
One key feature of these fiber networks is that their mechanical stability relies on the *bending stiffness of the fibers*,
which lifts zero-energy floppy modes of the corresponding central-force lattices to nonzero energy.
Isostaticity is a natural concept to capture how these floppy modes gain stability.
My collaborators and I investigated the elasticity of fiber networks in triangular and kagome lattice model networks using both the CPA and
numerical simulations. We discovered that the central-force isostatic point exhibit rich zero-temperature critical behavior,
including a crossover between various mechanical regimes along with diverging strain fluctuations.
Our theoretical and numerical results provide important guidelines to experimental study and engineering designs of fiber networks.

Related publications:

- Chase P. Broedersz, Xiaoming Mao, F.C. MacKintosh, and T. C. Lubensky,
Nature Physics
**7**, 983 (2011). - Xiaoming Mao, Olaf Stenull, and T. C. Lubensky,
arXiv:1111.1751v1, submitted to
*Physical Review E*.

##### Topological properties and emergent conformal symmetry in isostatic lattices

In addition to improving our understanding about known experimental systems, isostaticity also offers a playground to discover new exotic phenomena and to study fundamental physics. A good example of this type is the holographic isostatic system my collaborators and I recentaly discovered. On the quantum side of condensed matter physics, topological states of matter have been one of the central focuses of recent research. These topological states are generally associated with quantum phenomena. However, we discovered a family of classical elastic systems which contain large numbers of floppy edge modes, strongly resembling the edge states in topological insulators and some other topological states of matter. Interestingly, we further discovered that the two keys to understanding this phenomenon are an emergent conformal symmetry and the holographic principle, both of which have been widely used by high-energy physicists and string theorists in the search for the fundamental laws of our universe.

Related publications:

- Kai Sun, Anton Souslov, Xiaoming Mao and T. C. Lubensky,

*"Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices"*,

Proceedings of the National Academy of Sciences of the United States of America**109**, 12369 (2012).

##### Future plans

- Mechanical instability in the presence of thermal fluctuations
- Novel materials based on isostatic structures
- General classification near the isostatic point
- More to come...

#### Entropic elasticity in heterogeneous polymer networks

##### Vulcanization theory and elastic heterogeneities in soft random solids

Related publications:- Xiaoming Mao, Paul M. Goldbart, Xiangjun Xing and Annette Zippelius,
Physical Review E
**80**, 031140 (2009).. - Xiaoming Mao, Paul M. Goldbart, Xiangjun Xing and Annette Zippelius,
Europhysics Letters
**80**, 26004 (2007).. - Stephan Ulrich, Xiaoming Mao, Paul M. Goldbart and Annette Zippelius,
Europhysics Letters
**76**, 677 (2006). - Xiaoming Mao, Paul M. Goldbart, Marc Mézard and Martin Weigt,
Physical Review Letters
**95**, 148302 (2005).

##### Nonaffine displacements in flexible polymer networks

Related publications:- Anindita Basu, Qi Wen, Xiaoming Mao, T. C. Lubensky, Paul A. Janmey, and Arjun G. Yodh,
Macromolecules
**44**, 1671 (2011).