Stephan Ulrich
Tel.: +49 (0)151 50103308

Current Stress Level

Research Interests

Soft Matter

Random media, like polymer networks, covalent network glasses, or grains under pressure can be viewed as networks of springs and balls. The shear modulus of these type of materials typically vanishes as the network connectivity z approaches a critical value.

In this work, we employed a simple viscoelastic model of the elasticity of random networks and corroborate it by molecular dynamics (MD) simulations of corresponding networks. We apply a shear deformation to the network such that the left side of the material is sheared upwards with a constant velocity v0, while the right side is held fixed.

A snapshot of the simulation is shown in the illustration on the left. The color of each point in the material indicates its upwards velocity (orange = moving upwardsblue = at rest). We observe a shear wave which is approaching a constant velocity vf and its width (transition region from yellow to light blue) keeps increasing. The inset shows a small patch of the material, where the network comprises nodes (blue) connected with springs (red).

We showed analytically that shear strains propagate as diffusive fronts, whose width diverges and whose transverse speed of sound vanishes, as the transition is approached. Consequently, in this regime linear theory breaks down, giving rise to (nonlinear) shock waves. Comparison of the analytical wave profile to MD simulations allows the extraction of material constants of the network, which can be compared to previous predictions. Interestingly, even an undamped network yields a diverging effective viscosity caused by leaking of energy into non-affine degrees of freedom.

Granular Materials – A Short Introduction

Having a large number of macroscopic particles (>100 µm), one typically talks about granular materials. They are important for understanding soil or snow avalanches, planetary dust rings, as well as for industrial applications like processing sand, powders, nuts, cereals, wheat grains, etc.

There are two key features associated with granular materials:

  • They only interact on collision. (Different from atomic interactions which – relative to their own diameter – usually have long-range potentials.)

  • Collisions are typically dissipative. (Kinetic energy of the grains is transferred to their internal degrees of freedom and cracks.)
    The degree of dissipation is normally specified by the coefficient of restitution ε, which is the ratio of the normal component of the velocity before and after the collision. So ε = 1 corresponds to the limit of totally elastic grains, and ε = 0 to totally inelastic (often referred to as sticky) grains.

Granulates are often characterized by their granular temperature T, defined in analogy to kinetic gas theory: 

Here m is the particle mass, v the particle velocity, and d the spatial dimension.

Wet Granular Materials

When a small amount of liquid is added to the granulate, one talks about wet granular matter. Then each particle is covered with a thin film of the liquid. When two particles collide, these films will merge and create a capillary liquid bridge between the particles. It causes an attractive force F(s), which depends on the separation s of the particles, and breaks at a critical distance dcrit.

The energy needed to break a bond is calculated by , and is in good approximation independent of the velocities of the grains.

This collision dynamics is very different from dry granular materials, causing quantitative new behavior. For example, if the average kinetic energy of the grains is not sufficient to overcome Ecb and break the bond, clusters of particles will form (as seen in the image on the left). As it turns out, the cluster size distribution develops towards a self-preserving scaling form and the clusters themselves are fractal. Cooling dynamics can be described in similar fashion to Haff's law, however, yielding cooling laws very different from Haff's law. In the early cooling phase, one finds T(t) ~ (1 − t/t0)2 , and a very slow (logarithmic) time decay is found in the late stage (where bond ruptures become exponentially unlikely). If you are interested in more detailed results, you are invited to see publications on [PRL] or [PRE], or contact me!

For a rotating version of a full size resultant fractal cluster, click the play button below. The video on the left side shows a snapshot at the end of the flocculation regime, where small clusters are still present (indicated by different colors), however, a system spanning cluster (grey) has developed. The video on the right side shows a snapshot in the percolation regime, where the system spanning (percolating) cluster has ingested almost all particles. 

Rotating Cluster (φ=1.95% N=10^6)

Rotating Cluster ‎‎‎‎‎‎‎‎‎‎‎‎‎(φ=0.98%, N=262144)‎‎‎‎‎‎‎‎‎‎‎‎‎.avi

For a better resolution, use these links to download the video on the left and on the right, and use the download button.

Granular Streams

When turning on the faucet in your bathroom, a straight stream of water flows out. When you continuously decrease the flow rate, however, you will notice a transition to a state of droplet formation. This finding is explained by the Rayleigh-Plateau instability, which says that a long cylinder of water can decrease its surface by applying longitudinal undulations to the radius. Clearly, this mechanism requires surface tension in order to work, and hence would not be expected in granular materials, which are considered purely repulsive. 

This made a recent experimental observation rather surprising, which found droplet formation also in granular streams. By now, we believe that granular materials indeed are very weakly attractive, attributed to capillary bridges forming between particles (due to a non-zero humidity) and van-der-Waals interactions, whenever the particles’ surfaces are at contact. These observations have been well confirmed by large scale molecular dynamics simulations (see video):

2×10^6 cohesive particles

Granular Segregation and Brazil-nut effect

f = 15 Hz, Max accel. = 3.5 g

When shaking an ordinary pack of cereals from the supermarket, you will realize that the big nuts go to the top. This effect that large particles go to the top after injection of energy (shaking) is known as Brazil-nut effect and is still subject of research. After its discovery, also the reverse Brazil-nut effect was found, where the large particles go to the bottom, when the systems parameters are chosen appropriately.

In the picture on the left, you see a mixture of polystyrene and brass balls after shaking. Even though the density of brass is roughly 8 times higher, the large brass particles went to the top. On the right hand side, you can see a shaken container with a mixture of small and large brass spheres inside. Start the movie to see the image detection algorithm identify and track the small and large spheres, marked as blue () and red () arrows, respectively.

There are many (about 10) competing mechanisms suggested to explain this segregation effect. For a system with large and small particles of the same material and in an evacuated container, the most important mechanisms are found to be:

  • Void filling: small particles fit more easily into voids (resulting from shaking) between particles and can percolate downwards.
  • Convection: Convection rolls induced by the side walls cause a downwards stream close to the side walls. Large particles, however, might be too large to enter this thin downstream area and will be trapped on the top.
  • Thermal diffusion: Particles have a tendency to go to regions of low granular temperature. This tendency is stronger for particles with higher mass. If the temperature is not homogeneous in the system (which normally isn't the case), this mechanism leads to segregation.

This project was done at the University of Texas at Austin at the Center for Nonlinear Dynamics with Matthias SchröterJennifer KreftJack Swift, and Harry Swinney.

Further information can also be found in our main publication [Journal URL], and on [Matthias Schröter's Segregation Page]

Structure of Spider Silk

Spider silk is in my opinion one of the most amazing materials in nature! Evolution has optimized it for tensile strength (comparable to that of steel), extensibility and toughness (roughly 30 times higher than steel). Nevertheless it is a real light-weight (density about 1/6 of steel).

The reason for its high strength is still subject of investigation. Of particular interest is the so called dragline fiber, which spiders produce from essentially only two proteins to build their net's frame and radii, and also to support their own body weight after an intentional fall down during escape. It is known that the dragline fiber consists of nano-crystallites (so called β-sheets, mainly composed of alanine) which are connected by an amorphous chain network (so called amorphous matrix), however, its precise structure is still under debate (see left image).

We try to determine the structure of these nano-crystallites (unit cell composition and dimensions, overall size of the crystallites, and orientation with respect to the fiber axis). Therefore we obtain a scattering image of the fiber using wide angle x-ray scattering (WAXS), and establish a model of randomly tilted crystallites, of which the scattering function can be calculated. Adjusting the parameters of that model so that the calculated and measured images match, yields the structure of the crystallites (see right image).

As a "fall-out" of the model, one can investigate to which extent coherent scattering from different crystallites is important, or if the scattering image can be analyzed in terms of single-crystallite scattering (as it is usually assumed in the literature). A further ingredient to the model is also the possibility to allow for structural disorder. Here the alanine amino acid can be replaced by glycine with a certain probability (for which indications are found in the literature), which yields a good match to the experimental scattering image.

If you are interested in details or the precise resulting structures, see Eur. Phys. J. E 27, 229 (2008). [URL] or [arXiv].
(Note: if the calculations appear too technical or time consuming, just skip chapter 3! Only have a quick look at eq. (17), with A(q) being the scattering amplitude of a single crystallite, defined in eq. (13) ☺)

Interesting movie (avi, 9MB) of stretching a semi-crystalline material like spider silk. It's a computer simulation using Langevin-dynamics, with strong bonds (covalent, blue) and weak bonds (H-Bond, red). It never found use though...

Replica calculation of randomly cross-linked polymer networks

A very simple model for a polymer network consists of beads connected with springs (see image on the right): We consider a system of N particles (blue dots) at positions {r1,...,rN}. M pairs of these particles, {(i1,j1),...,(iM,jM)}, are connected via Hookian springs (red connections). Furthermore we introduce an excluded volume interaction affecting all pairs of particles. With these ingredients the Hamiltonian of the system becomes:

Note that in the first term we sum over all cross-links, and in the second term, over all pairs of particles.

We use statistical mechanics to calculate the partition function Z and free energy F = −ln Z, for a quenched cross-link configuration and find out structural and mechanical properties, like

  • the distribution of localization lengths, i.e. how large the fluctuations of the particles about their mean position are.
  • the gel fraction, the fraction of particles which are localized (have a finite localization length).
  • the shear modulus, the resistance of the system to a shear strain.

Obviously all these properties still depend on the cross-link configuration. To find the typical properties, we have to average the free energy F with an appropriate distribution of the cross-link configuration. This is done with the Deam-Edwards distribution (which only sets the average cross-link density) and replica theory.

It turns out, that there is a transition from a liquid (zero gel fraction => particles are not localized) to a gel (non-zero gel fraction). This is the so called gelation-transition. While most calculations of cross-linked networks are expansions valid close to the gelation-transition, in this work, the behavior for arbitrary cross-link densities is accessible.

See Europhys. Lett. 76, 677 (2006). [Journal URL] or [arXiv].


See the complete list including citations on [Google Scholar]


  1. Shear shocks in fragile networks
    Stephan Ulrich, Nitin Upadhyaya, Bas van Opheusden, and Vincenzo Vitelli
    PNAS 110, 20929 (Dec 24, 2013), [URL], [arXiv]

  2. Jamming of frictional tetrahedra
    Max Neudecker, Stephan Ulrich, Stephan Herminghaus, Matthias Schröter
    Phys. Rev. Lett. 111, 028001 (2013), [URL], [arXiv]

  3. Stability of freely falling granular streams [pdf]
    Stephan Ulrich and Annette Zippelius
    Phys. Rev. Lett. 109, 166001 (2012), [URL] [arXiv]
    Supplementary material: [movie version of Fig. 1] [supplementary calculations]

  4. Random networks of cross-linked directed polymers
    Stephan Ulrich, Annette Zippelius, and Panayotis Benetatos
    Phys. Rev. E 81, 021802 (2010), [URL], [arXiv]
  5. Dilute Wet Granulates: Nonequilibrium Dynamics and Structure Formation
    Stephan Ulrich, Timo Aspelmeier, Annette Zippelius, Klaus Roeller, Axel Fingerle, Stephan Herminghaus
    Phys. Rev. E 80, 031306 (2009), [URL], [arXiv]
  6. Cooling and aggregation in wet granulates
    Stephan Ulrich, Timo Aspelmeier, Klaus Roeller, Axel Fingerle, Stephan Herminghaus, Annette Zippelius
    Phys. Rev. Lett. 102, 148002 (2009), [URL], [EPAPS Supplementary Material], [arXiv]
  7. Diffraction from the β-sheet crystallites in spider silk
    Stephan Ulrich, Anja Glišović, Tim Salditt and Annette Zippelius
    Eur. Phys. J. E 27, 229 (2008) [URL], [arXiv]
  8. Influence of friction on granular segregation
    Stephan Ulrich, Matthias Schröter, and Harry L. Swinney
    Physical Review E 76, 042301 (2007) [URL], [arXiv]
  9. Granulare Medien: Der Paranuss-Effekt
    Stephan Ulrich and Matthias Schröter
    Physik in unserer Zeit 38, 266 (2007) [URL]  (German)
  10. Elasticity of highly cross-linked random networks
    Stephan Ulrich, Xiaoming Mao, Paul M. Goldbart and Annette Zippelius
    Europhys. Lett. 76, 677 (2006) [URL], [arXiv]
  11. Mechanisms in the size segregation of a binary granular mixture
    Matthias Schröter, Stephan Ulrich, Jennifer Kreft, Jack B. Swift, and Harry L. Swinney
    Physical Review E 74, 011307 (2006) [URL], [arXiv]

Curriculum Vitae

2000 German “Abitur” (grade 1.0), (≈ high school degree)
2000 – 2001 Distance learning program (FiPS) at the University of Kaiserslautern
meanwhile: Zivildienst (compulsory paid community service)
(substitute for army service)
2001 – 2003 Study of physics at the University of Würzburg
2002 Vordiplom (grade 1.2)
2003 – 2004 Graduate program at the University of Texas at Austin
Working on segregation of granular matter at the
Center for Nonlinear Dynamics
Dec. 2004 Master's degree (GPA: 3.8)
2005 – 2010 Working on a PhD at the University of Göttingen
Advisor: Annette Zippelius
Thesis title: Aggregation and Gelation in Random Networks
2010 – 2011 Post-Doc at the Institute for Theoretical Physics
at Göttingen University with Annette Zippelius
since 2011 Post-Doc at the Lorentz Institute for Theoretical Physics
at Leiden University with Vincenzo Vitelli

PhD Thesis

Title: Aggregation and Gelation in Random Networks
Download link: [PhDThesis_Stephan_Ulrich.pdf]
or [View in Browser]
or... go to [Google Books]

Further Research Interests

which doesn't necessarily mean I'm an expert...

  • Classical many-body problems (granulates, colloidal systems, glasses, polymers, jamming)
  • Optimization problems
  • Image processing
  • Fluid dynamics, turbulence
  • Chaos, fractals
  • Neural networks
  • Non-equilibrium statistical physics
  • Biophyics: biomolecular structures, self-organization, membranes
  • Simulations
  • Nonlinear dynamics, pattern formation
  • Social networks and related problems, e.g. spreading of diseases, traffic flow

Computer Skills

Important Links

Google Scholar profile /

ResearchGate profile

Theory page of Göttingen University

Statistical Physics and Complex Systems Group

Center for Nonlinear Dynamics, University of Texas at Austin

PhD Comics

last (serious) update 23.08.2014