C09 - Dynamics of rock dehydration on multiple scales

Head(s): Dr. Marita Thomas (WIAS), Prof. Dr. Timm John (FU Berlin)
Project member(s): Konstantin Huber, Dr. Dirk Peschka, Dr. Johannes Vrijmoed, Andrea Zafferi
Participating institution(s): FU Berlin, WIAS

Project Summary

This project deals with the dynamic formation of dehydration-related fluid flow structures within rocks. Field observations of natural occurrences along with thermodynamic calculations reveal that rock dehydration is characterised by three stages:
(i) the initial formation of porosity caused by fluid liberation during dehydration reactions,
(ii) the intermediate stages of fluid pooling and vein network formation, and
(iii) the final stages of fluid release from the dehydrating system.
While the initial stage is primarily induced by chemical processes, the later stages are dominated by mechanical interactions of solid and fluid. In particular, an increase of fluid pressure causes mechanical stresses that ultimately lead to fracturing. While dehydration-associated mineral reactions are mainly driven by slow conductive heat transfer and occur on grain boundaries, hence on  µm-scales, fracture-related fluid release may occur within the time scale of seismic events and up to  km-scales. The goal of this project is to decipher the hierarchical structure of the interacting processes on multiple time and length scales. We will develop a thermomechanical multiscale model and validate it with geological field observations. Our approach joins mathematical analysis and fieldand laboratory-based data coupled via numerical simulations. To achieve this goal we will first focus on the scales where the transition from a chemically controlled to a deformation-controlled system can be observed. The research is structured in four workpackages (WP 1 to 4):
WP1: “From Observation to Model” will set up a prototype model, based on published data and concepts, featuring the coupling of fluid and heat transport with chemical reactions and with deformations and fracturing of the solid in a thermodynamically consistent way.
WP2: “Measurements and Field Data” aims to gather comprehensive field data on multiple scales from natural observations on serpentinite, a hydrous rock that has not experienced dehydration. This dataset will cover domain sizes ranging from  µm2,  m2  up to tens of  m2.
WP3: “Mathematical Analysis” studies the well-posedness of the prototype model. Modifications of this model will be obtained by incorporating scaling parameters or by restricting certain processes to spatial, scale-dependent subdomains. Via Γ-convergence and homogenisation methods the scaled prototype model will be mathematically rigorously carried over to different scales. This will ultimately lead from the (modified) prototype model (root model) to a hierarchical multiscale model that encodes the aggregated dynamics.
WP4: “From Model to Observation” aims to implement the prototype model and its extensions in a numerical code. We will apply the numerical model to simulate the dehydration network pattern formation using the natural data. To validate our model, simulation output will be compared to observed network patterns developed within the same rock type that has undergone dehydration.

Project publications

to be filled

C08 - Stochastic spatial coagulation particle processes

Head(s): Dr. Robert Patterson (WIAS)
Project member(s): Dr. Luisa Andreis, Dr. Michiel Renger
Participating institution(s): WIAS

Project Summary

We use large deviations principles for analysing large systems of stochastic interacting particles and then performing further scaling limits. Our particle systems are models for (bio-)chemical reactions, aggregate coagulation and the evolution of convective towers in the atmosphere. In each case some discrete entities, which we term particles, experience binary interactions when they are ‘close’ in an appropriate sense. These entities have internal structure of some kind, such as mass or chemical composition or the height of a convective tower, which play an important role in determining the interaction rates. We consider three scenarios where a parameter   N   proportional to the number of ‘particles’ becomes large: Firstly, where the particles are contained in a single, small and well-mixed volume, with size independent of   N.  Secondly, where the particles are contained in many small volumes, whose sizes are bounded and possibly vanishing, and thirdly, where the volume containing the particles is of size proportional to  N.
Each of our motivating applications has, in addition to   N   and the scales it generates, a second small parameter  ε,  which generates an additional cascade of scales: In the first two applications it is possible to describe systems by the concentration   C(N) (t, x, y)   of the particles of type   y   at time   t   at the site   x ∈ IRd   defined as the quotient of their number with   N  (here interpreted as a kind of volume). Application 1 is a family of chemical reactions with sub-families accelerated by different negative powers of ε to model biochemical processes on very different time scales. Application 2 deals with accelerated bulk diffusion and slow diffusion in a vanishing membrane separating two compartments. We expect to interpret an existing reduced model and associated numerical method for the nucleus and cytoplasm of a cell as a limiting case of a more general approach.
In the third application concentrations are not an appropriate mathematical description, because we have to work in the thermodynamic limit   N → ∞   of  N  coagulating Brownian motions in a large container with volume   V (N)  N.  Here in the simple setting of spheres that merge on reaction/coagulation we seek to understand the statistics of the coagulation events now that we model the positions of the particles explicitly. We use a novel mathematical approach with marked point processes. A simplified form of the point process model will allow us to model the evolution of convective towers, which are triggered due to surface layer turbulence in the atmosphere and play an important role in tropical storm formation as studied in C06.
In all three applications large scale effects arise from small scale randomness in the presence of additional scales. In this project we seek to use these applications to guide the development of tools from the theory of large deviations principles (LDPs) and G-convergence to rigorously understand the micro–macro transition in conjunction with additional scaling limits.


Project publications

Andreis, L. and König, W. and Patterson, R. I. A (2019) A large-deviations approach to gelation. SFB 1114 Preprint in arXiv:1901.01876 . pp. 1-22. (Unpublished)

Heida, M. and Patterson, R. I. A and Renger, M. (2018) Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space. J. Evol. Equ. . pp. 1-42. ISSN Online: 1424-3202 Print: 1424-3199

Renger, M. (2018) Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory. J. Stat. Phys., 172 (5). pp. 1261-1326. ISSN 0022-4715

Koltai, P. and Renger, M. (2018) From Large Deviations to Semidistances of Transport and Mixing: Coherence Analysis for Finite Lagrangian Data. Journal of Nonlinear Science, 28 (5). pp. 1915-1957. ISSN 1432-1467 (online)

Renger, M. (2018) Gradient and Generic systems in the space of fluxes, applied to reacting particle systems. SFB 1114 Preprint in arXiv:1806.10461 . pp. 1-29. (Unpublished)

Patterson, R. I. A and Renger, M. (2018) Large deviations of reaction fluxes. SFB 1114 Preprint in arXiv:1802.02512 . pp. 1-21. (Unpublished)

Mielke, A. and Patterson, R. I. A and Peletier, M. A. and Renger, M. (2017) Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics. SIAM Journal on Applied Mathematics, 77 (4). pp. 1562-1585. ISSN 1095-712X (online)

Liero, M. and Mielke, A. and Peletier, M. A. and Renger, M. (2017) On microscopic origins of generalized gradient structures. Discrete and Continuous Dynamical Systems - Series S, 10 (1).

Mielke, A. and Peletier, M. A. and Renger, M. (2016) A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility. Journal of Non-Equilibrium Thermodynamics, 41 (2).

Erbar, M. and Maas, J. and Renger, M. (2015) From large deviations to Wasserstein gradient flows in multiple dimensions. Electronic Communications in Probability, 20 (89).

Project surrounding publications

Patterson, R. I. A (2016) Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries. Journal of Evolution Equations, 16 . pp. 261-291.

Yapp, E.K.Y. and Patterson, R. I. A and Akroyd, J. and Mosbach, S. and Adkins, E.M. and Miller, J.H. and Kraft, M. (2016) Numerical simulation and parametric sensitivity study of optical band gap in a laminar co-flow ethylene diffusion flame. Combustion and Flame, 167 . pp. 320-334.

C03 - Multiscale modelling and simulation for spatiotemporal master equations

Head(s): Prof. Dr. Felix Höfling (FU Berlin), Prof. Dr. Frank Noé (FU Berlin), Prof. Dr. Christof Schütte (FU Berlin)
Project member(s): Dr. Rainald Ehrig, Christoph Fröhner, Dr. Daniela Frömberg, Luigi Sbailò, Dr. Arthur Straube, Dr. Stefanie Winkelmann
Participating institution(s): FU Berlin, ZIB

Project Summary

Chemical reactions cover many length scales from the formation of chemical bonds at the atomistic scale up to the physiological response of a whole cell or the size of a chemical reactor. Concomitantly, copy numbers may vary drastically between molecular species and as a function of time, covering many orders of magnitude. In biological and technological applications, complex geometrical landscapes may arise from cellular crowding or nanoporous grains. Simulations of such situations are thus facing a number of challenges, which need to be overcome by efficient modelling.
In C03 we want to push the methodological boundaries of modelling reaction kinetics in different regimes of the scaling cascades (low and high particle concentrations, fast and slow diffusion), and the scaling transitions between them. In the first funding period, significant progress has been achieved regarding models for different levels of resolution (mathematical foundations, new hybrid models coupling different levels, and their algorithmic realisation and practical use).
In the subsequent funding period, we aim at (i) a seamless integration of models for specific scaling regimes into hybrid models for real-life systems where population numbers may vary dramatically; (ii) developing rigorous particle-based reaction dynamics schemes with particle interaction forces and characterising the effect of molecular crowding on reaction kinetics; (iii) derive effective particle-based reaction dynamics schemes that account for crowded media and establish the coarse-graining transition to the spatiotemporal Master equation; (iv) bridging particle-based reaction dynamics with molecular dynamics (MD), both in order to obtain kinetic parameters for the reaction dynamics simulation, and to form a multiscale simulation method.

Project publications

Scherer, M. K. and Husic, B.E. and Hoffmann, M. and Paul, F. and Wu, H. and Noé, F. (2018) Variational Selection of Features for Molecular Kinetics. SFB 1114 Preprint in arXiv:1811.11714 . pp. 1-12. (Unpublished)

Wehmeyer, C. and Scherer, M. K. and Hempel, T. and Husic, B.E. and Olsson, S. and Noé, F. (2018) Introduction to Markov state modeling with the PyEMMA software — v1.0. LiveCoMS, 1 (1). pp. 1-12. ISSN E-ISSN: 2575-6524 (Unpublished)

del Razo, M.J. and Qian, H. and Noé, F. (2018) Grand canonical diffusion-influenced reactions: a stochastic theory with applications to multiscale reaction-diffusion simulations. J. Chem. Phys., 149 (4). ISSN 0021-9606, ESSN: 1089-7690

Sadeghi, M. and Weikl, T. and Noé, F. (2018) Particle-based membrane model for mesoscopic simulation of cellular dynamics. J. Chem. Phys., 148 (4). 044901.

Paul, F. and Wehmeyer, C. and Abualrous, E. T. and Wu, H. and Crabtree, M. D. and Schöneberg, J. and Clarke, J. and Freund, C. and Weikl, T. and Noé, F. (2017) Protein-peptide association kinetics beyond the seconds timescale from atomistic simulations. Nat. Comm., 8 (1095).

Winkelmann, S. and Schütte, Ch. (2017) Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems. Journal of Chemical Physics, 147 (11). pp. 1-18.

Olsson, Simon and Wu, H. and Paul, F. and Clementi, C. and Noé, F. (2017) Combining experimental and simulation data of molecular processes via augmented Markov models. Proc. Natl. Acad. Sci. USA, 114 . pp. 8265-8270.

Pinamonti, G. and Zhao, J. and Condon, D. and Paul, F. and Noé, F. and Turner, D. and Bussi, G. (2017) Predicting the kinetics of RNA oligonucleotides using Markov state models. J. Chem. Theory Comput., 13 (2). pp. 926-934.

Schöneberg, J. and Lehmann, M. and Ullrich, A. and Posor, Y. and Lo, W.-T. and Lichtner, G. and Schmoranzer, J. and Haucke, V. and Noé, F. (2017) Lipid-mediated PX-BAR domain recruitment couples local membrane constriction to endocytic vesicle fission. Nat. Comm., 8 . p. 15873.

Winkelmann, S. and Schütte, Ch. (2016) The spatiotemporal master equation: approximation of reaction-diffusion dynamics via Markov state modeling. Journal of Chemical Physics, 145 (21). p. 214107.

Albrecht, D. and Winterflood, C. M. and Sadeghi, M. and Tschager, T. and Noé, F. and Ewers, H. (2016) Nanoscopic compartmentalization of membrane protein motion at the axon initial segment. J. Cell Biol., 215 (1). pp. 37-46.

Wu, H. and Paul, F. and Wehmeyer, C. and Noé, F. (2016) Multiensemble Markov models of molecular thermodynamics and kinetics. Proceedings of the National Academy of Sciences, 113 (23). E3221-E3230 . ISSN 0027-8424

Trendelkamp-Schroer, B. and Wu, H. and Paul, F. and Noé, F. (2015) Estimation and uncertainty of reversible Markov models. J. Chem. Phys., 143 (17). p. 174101.

Scherer, M. K. and Trendelkamp-Schroer, B. and Paul, F. and Pérez-Hernández, G. and Hoffmann, M. and Plattner, N. and Wehmeyer, C. and Prinz, J.-H. and Noé, F. (2015) PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models. J. Chem. Theory Comput., 11 (11). pp. 5525-5542.

C05 - Effective models for materials and interfaces with multiple scales

Head(s): Prof. Dr. Alexander Mielke (WIAS)
Project member(s): Thomas Frenzel, Dr. Martin Heida, Markus Mittnenzweig, Artur Stephan
Participating institution(s): WIAS

Project Summary

This project provides analytical techniques for discrete or continuous material models that depend on one or several small parameters. Special emphasis is given to systems that have a variational structure such as static minimisation problems or gradient-flow equations systems. Methods of static or evolutionary Gamma convergence are employed and further investigated, in particular EDP-convergence (i.e., in the sense of the energy dissipation principle). The small parameter may determine material properties via small layers or periodic, fractal, or stochastic material properties with a small correlation length. Applications involve diffusion in strongly heterogeneous media, elastic bulk materials with embedded interfaces along which Coulomb friction and other processes may occur.
More precisely we consider the topics:
- Gradient systems and evolutionary Gamma convergence, which provide tools for deriving effective models for systems with many scales. The connection between large-deviation principles for microscopic stochastic models and the gradient structures for the macroscopic deterministic models.
- Homogenisation of discrete elliptic operators (linear or non-linear) on regular or random graphs with random coefficients. Fractal homogenisation of elliptic problems with transmission on a fractal set of interfaces.
- Mathematical and thermodynamical modeling of evolutionary processes in bulk materials and in materials with interfaces. Periodic, fractal and stochastic homogenisation of evolutionary systems with variational structure.
- Rate-independent and rate-and-state friction between elastic bodies and its justification via dimension reduction.
- Connections between discrete chemical master equations and continuum descriptions like the reaction-rate equation. Hybrid models for reaction kinetics and reaction-diffusion systems.


Project publications

Donati, L. and Heida, M. and Weber, M. and Keller, B. (2018) Estimation of the infinitesimal generator by square-root approximation. Journal of Physics: Condensed Matter, 30 (42). p. 425201. ISSN 0953-8984, ESSN: 1361-648X

Heida, M. and Patterson, R. I. A and Renger, M. (2018) Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space. J. Evol. Equ. . pp. 1-42. ISSN Online: 1424-3202 Print: 1424-3199

Mielke, A. and Rossi, R. and Savaré, G. (2018) Global existence results for viscoplasticity at finite strain. Archive for Rational Mechanics and Analysis, 227 (1). pp. 423-475. ISSN Print: 0003-9527; Online: 1432-0673

Heida, M. and Kornhuber, R. and Podlesny, J. (2017) Fractal homogenization of multiscale interface problems. SFB 1114 Preprint in arXiv . pp. 1-17. (Submitted)

Heida, M. and Neukamm, S. and Varga, M. (2017) Stochastic unfolding and homogenization. SFB 1114 Preprint at WIAS 12/2017 . pp. 1-45. (Unpublished)

Liero, M. and Mielke, A. and Savaré, G. (2017) Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures. Invent. math. . pp. 1-149. ISSN 1432-1297 (online)

Mielke, A. and Patterson, R. I. A and Peletier, M. A. and Renger, M. (2017) Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics. SIAM Journal on Applied Mathematics, 77 (4). pp. 1562-1585. ISSN 1095-712X (online)

Heida, M. (2017) On Convergences of square-root approximation scheme to Fokker--Planck operator. SFB 1114 Preprint at WIAS 05/2017 . pp. 1-30. (Unpublished)

Heida, M. (2017) Stochastic homogenization of rate-independent systems and applications. Continuum Mech. Thermodyn., 29 (3). pp. 853-894. ISSN 1432-0959 (online) 0935-1175 (print)

Gussmann, P. and Mielke, A. (2017) Linearized elasticity as Mosco-limit of finite elasticity in the presence of cracks. Adv. Calc. Var. . (Submitted)

Flegel, F. and Heida, M. and Slowik, M. (2017) Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps. SFB 1114 Preprint in arXiv:1702.02860 . (Unpublished)

Heida, M. and Schweizer, B. (2017) Stochastic homogenization of plasticity equations. ESAIM: Control, Optimisation and Calculus of Variations . pp. 1-30. (Submitted)

Liero, M. and Mielke, A. and Peletier, M. A. and Renger, M. (2017) On microscopic origins of generalized gradient structures. Discrete and Continuous Dynamical Systems - Series S, 10 (1).

Heida, M. and Nesenenko, S. (2017) Stochastic homogenization of rate-dependent models of monotone type in plasticity. SFB 1114 Preprint in arXiv:1701.03505 . pp. 1-26. (Unpublished)

Heida, M. and Mielke, A. (2017) Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete and Continuous Dynamical Systems - Series S, 10 (6). pp. 1303-1327.

Mielke, A. (2017) Three examples concerning the interaction of dry friction and oscillations. In: Trends on Application of Mathematics to Mechanics. Springer INdAM series. (In Press)

Mielke, A. and Mittnenzweig, M. (2017) Convergence to Equilibrium in Energy-Reaction–Diffusion Systems Using Vector-Valued Functional Inequalities. Journal of Nonlinear Science . pp. 1-42. ISSN 1432-1467 (online)

Bonetti, E. and Rocca, E. and Rossi, R. and Thomas, M. (2016) A rate-independent gradient system in damage coupled with plasticity via structured strains. ESAIM: Proceedings and Surveys, 54 . pp. 54-69.

Liero, M. and Mielke, A. and Savaré, G. (2016) Optimal Transport in Competition with Reaction: The Hellinger--Kantorovich Distance and Geodesic Curves. SIAM J. Math. Anal., 48 (2). pp. 2869-2911. ISSN 1095-7154 (online)

Mielke, A. and Peletier, M. A. and Renger, M. (2016) A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility. Journal of Non-Equilibrium Thermodynamics, 41 (2).

Mielke, A. and Rossi, R. and Savaré, G. (2016) Balanced-Viscosity solutions for multi-rate systems. Journal of Physics: Conference Series, 727 . pp. 1-27.

C02 - Interface dynamics: Bridging stochastic and hydrodynamic descriptions

Head(s): Prof. Dr. Roland Netz (FU Berlin), Prof. Dr. Alexander Mielke (WIAS)
Project member(s): Cihan Ayaz, Laura Lavacchi
Participating institution(s): FU Berlin, WIAS

Project Summary

The properties of liquid water are relevant for a broad range of processes in biology, chemistry, and physics. The goal of this project is to relate macroscopic water properties in bulk and at interfaces to the microscopic structure and thus to the hydrogen bonding pattern between individual water molecules. We are striving at combining three different viewpoints on water dynamics, namely the large scale hydrodynamic description, the mid-scale diffusive description, and the microscopic viewpoint where the hydrogen bond network fluctuates on the pico-second scale. Interfaces play a dominant role in this project since they are abundant in nature and give rise to unique and intereresting phenomena.
Using Markov-state modeling as a diagnostic tool to understand the most elementary kinetic process of breaking a single H-bond between two water molecules that are embedded in the bulk liquid matrix, we could by transition path analysis show that there are different kinetic pathways that compete with each other. Only one of these pathways corresponds to Stillinger’s original switching-of-allegiance description of the local water dynamics, according to which after an H-bond breaking event a new H-bond forms right after. The other dominant pathways involve an intermediate state where no new H-bond is formed, which is a process that has been overlooked in previous MD simulation analysis due to the overwhelmingly large conformational space. These results and in particular the methodological advance will be helpful towards a full understanding of the kinetics of water molecules at interfaces and in particular in the hydration layer around proteins, membranes and other hydrated objects, which will be studied in the next funding period.
On an intermediate length scale and time scale we investigated how the diffusive description of molecular motion is modified in interfacial boundary layers. As an explicit system we analyzed experimental data of drug diffusion through human skin layers and developed a robust method to extract diffusivity and free-energy profiles from experimental concentration profiles recorded at subsequent different times. Multiscale behavior in the molecular penetration dynamics arises because human skin is a multilayered biological barrier and presents competing diffusional and free energetic barriers. Our modeling reveals that the first cell layers in the stratum corneum, which is the 10 micrometer thick top skin layer that consists of dead skin cells, are invaded on the second time scale, while the dermis, which is located underneath the roughly 100 micrometer thick epidermis, is reached only after weeks of diffusion. It would be interesting to add another length and time scale to the problem by including the sub-cellular skin structure by resolving water diffusion in and through cell membranes and organelles.
Lastly, we investigated how interfaces modify the hydrodynamic description, using two different geometries. First, we considered the case of a single interface and studied surface-localised wave phenomena in the presence of viscoelastic interfacial layers. The coupling between the viscous water bulk and the viscoelastic surface creates surface waves with fractional dispersion relations, which is a clear signature of dynamic multiscale behavior. Motivated by experimental results on waves traveling in lipid monolayers at the air-water interface, we derived a non-linear fractional wave equation for surface waves. We will further develop our model to include wave guiding phenomena in cylindrical membrane structures including the coupling to viscoelastic tissue embeddings, which has interesting applications to action potentials in nerve cells. Second, we studied how the hydrodynamic description has to be modified when the water slab between two surfaces reaches sub nanometer thicknesses. In particular we were interested in the transition from hydrodynamic friction between two surfaces that are far apart, to the dry friction limit when there is no water left between the surfaces. Here we could explain the simulation results by a three-scale model that includes the slip between water and the surfaces, the presence of nanometer thick interfacial hydration layers with modified viscous properties, and the bulk water layer. This research direction shall be enlarged to also include frequency-dependent viscosity effect, thereby combining the spatial multiscale model we have developed in the previous funding period with temporal multiscale effects.

Project publications

Schulz, R. and von Hansen, Y. and Daldrop, J.O. and Kappler, J. and Noé, F. and Netz, R.R. (2018) Collective hydrogen-bond rearrangement dynamics in liquid water. J. Chem. Phys., 149 (24). -244504. ISSN 0021-9606, ESSN: 1089-7690

Schulz, R. and Hansen, Y. von and Daldrop, J.O. and Kappler, J. and Noé, F. and Netz, R.R. (2018) Markov state modeling reveals competing collective hydrogen bond rearrangements in liquid water. SFB 1114 Preprint 02/2018 . (Unpublished)

Kappler, J. and Shrivastava, S. and Schneider, M.F. and Netz, R.R. (2017) Nonlinear fractional waves at elastic interfaces. Phys. Rev. Fluids, 2 (11). p. 114804.

Kappler, J. and Netz, R.R. (2017) Pulse propagation at interfaces and their possible relevance for biology. Journal Club for Condensed Matter Physics .

Schlaich, A. and Kappler, J. and Netz, R.R. (2017) Hydration Friction in Nanoconfinement: From Bulk via Interfacial to Dry Friction. Nano Lett., 17 (10). pp. 5969-5976.

Schulz, R. and Yamamoto, K. and Klossek, A. and Flesch, R. and Hönzke, S. and Rancan, F. and Vogt, A. and Blume-Peytavi, U. and Hedtrich, S. and Schäfer-Korting, M. and Rühl, E. and Netz, R.R. (2017) Data-based modeling of drug penetration relates human skin barrier function to the interplay of diffusivity and free-energy profiles. PNAS, 114 (14). pp. 3631-3636. ISSN 1091-6490 (online)

Kappler, J. and Netz, R.R. (2015) Multiple surface wave solutions on linear viscoelastic media. EPL, 112 (1). p. 19002.

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