Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
  • Open Access

Linear viscoelastic response of the vertex model with internal and external dissipation: Normal modes analysis

Sijie Tong, Rastko Sknepnek, and Andrej Košmrlj
Phys. Rev. Research 5, 013143 – Published 24 February 2023
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • NORMAL MODES ANALYSIS
  • APPLICATION TO THE VERTEX MODEL WITH…
  • DISCUSSION AND CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We use the normal mode formalism to study the shear rheology of the vertex model for epithelial tissue mechanics in the overdamped linear response regime. We consider systems with external (e.g., cell-substrate) and internal (e.g., cell-cell) dissipation mechanisms, and derive expressions for stresses on cells due to mechanical and dissipative forces. The semi-analytical method developed here is, however, general and can be directly applied to study the linear response of a broad class of soft matter systems with internal and external dissipation. It involves normal mode decomposition to calculate linear loss and storage moduli of the system. Specifically, displacements along each normal mode produce stresses due to elastic deformation and internal dissipation, which are in force balance with loads due to external dissipation. Each normal mode responds with a characteristic relaxation timescale, and its rheological behavior can be described as a combination of a standard linear solid element due to elastic stresses and a Jeffreys model element due to the internal dissipative stresses. The total response of the system is then fully determined by connecting in parallel all the viscoelastic elements corresponding to individual normal modes. This allows full characterization of the potentially complex linear rheological response of the system at all driving frequencies and identification of collective excitations. We show that internal and external dissipation mechanisms lead to qualitatively different rheological behaviors due to the presence or absence of Jeffreys elements, which is particularly pronounced at high driving frequencies. Our findings, therefore, underscore the importance of microscopic dissipation mechanisms in understanding the rheological behavior of soft materials and tissues, in particular.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    3 More
    • Received 2 October 2022
    • Accepted 3 February 2023

    DOI:https://doi.org/10.1103/PhysRevResearch.5.013143

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Applications of soft matterBiomechanicsMechanical & acoustical propertiesRheologyStress
    1. Physical Systems
    Biological materials
    1. Techniques
    Coarse grainingMaterials modeling
    Physics of Living SystemsPolymers & Soft MatterCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Sijie Tong1, Rastko Sknepnek2,3,*, and Andrej Košmrlj1,4,†

    • 1Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
    • 2School of Science and Engineering, University of Dundee, Dundee DD1 4HN, United Kingdom
    • 3School of Life Sciences, University of Dundee, Dundee DD1 5EH, United Kingdom
    • 4Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544, USA

    • *r.sknepnek@dundee.ac.uk
    • †andrej@princeton.edu

    Article Text

    Click to Expand

    References

    Click to Expand
    Issue

    Vol. 5, Iss. 1 — February - April 2023

    Subject Areas
    Reuse & Permissions
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      A schematic of the decomposition of the motion and the response stress along normal modes. The top left panel depicts the deformation of cells crossing the periodic boundary due to the relative motion of image boxes that are displaced according to the applied shear. The remaining panels show three representative normal modes for the vertex model. The grey mesh is the equilibrium configuration, the red arrows indicate displacements associated with the normal mode ξk, and the blue mesh is the configuration after displacing vertices in the direction of the normal mode. Similarly, in the top left panel polygons outlined in blue indicate the distorted cells due to the relative movement of image boxes under applied shear. Perturbation from the equilibrium state δr(t) can be written as a linear superposition of displacements along the normal modes ξk. The stress response of the system [see Eq. (18)] due to shear deformation can be represented as a linear superposition of stresses ε(t)σ̂pbe,lin and ε̇(t)σ̂pbid due to the elastic deformation and internal dissipation of cells crossing the periodic boundary, respectively, and stresses ak(t)σ̂ke,lin and ȧk(t)σ̂kid due to the elastic deformation and internal dissipation of the kth normal mode, respectively. The rheological response of the system due to shear deformation [see Eq. (21)] can thus be represented as a parallel sequence of a spring and a dashpot due to shear of cells crossing the periodic boundary and standard linear solid (SLS) model elements and Jeffreys model elements, where each SLS model element and Jeffreys model element describe the shear response of a normal mode. Expressions for spring constants and dashpot viscosities are given in the text.

      Reuse & Permissions
    • Figure 2
      Figure 2

      Schematics of the three types of dissipation mechanisms in the vertex model considered in this paper. Velocity vectors are shown in blue and friction forces are red. (a) Dissipation due to the relative motion between vertices and a solid substrate. Vertex i experiences a frictional force proportional to its velocity Vi, with friction coefficient γ. (b) Friction force on the vertex i is due to its motion relative to neighboring vertices. Si is the set of all vertices connected to vertex i by a cell-cell junction. (c) Dissipation is due to the relative motion of neighboring cell centers. NC includes all neighboring cells of cell C. Ni includes all cells that share vertex i. fCid is the friction force that cell C experiences due to relative motion with respect to its neighboring cells. fied and fiid are the total friction forces applied at vertex i due to external and internal dissipation, respectively. NC is the number of vertices that belong to cell C.

      Reuse & Permissions
    • Figure 3
      Figure 3

      Shear rheology of hexagonal tilings with dissipation due to friction with the solid substrate. Results for two representative values of the cell shape parameter p0=3.5 in the solid phase (top row) and p0=3.73 in the fluid phase (bottom row) are shown. [(a),(b)] Storage and loss moduli from the simulations (symbols) compared with the predictions based on the normal mode analysis (lines). [(c),(d)] Nonzero eigenvalues λk vs the mode number k; eigenvalues are sorted in the ascending order. [(e),(f)] Absolute values of normalized coefficients |α¯kG¯ke|=|αkGkeγ/(KA0)2| and |β¯kG¯ke|=|βkGke/(KA0)| [see Eqs. (14) and (20)]. Also note that the ordinate covers ∼25 decades. The inset in (c) shows a schematic of the normal mode ξD that dominates the shear rheology of hexagonal tilings in the solid phase. This dominant normal mode is a linear combination of degenerate normal modes within the shaded region in panels (c) and (e). Labeled arrows in panel (b) denote peaks that correspond to the characteristic timescales from the normal modes in the shaded regions in panels (d) and (f). The inset in (d) shows the density of states ρ(λ).

      Reuse & Permissions
    • Figure 4
      Figure 4

      Shear rheology of disordered tilings with dissipation due to friction with the substrate for different values of the area elastic moduli K. Results for two representative values of the cell shape parameter, p0=3.71 (top row) and p0=3.87 (bottom row), are shown. [(a),(b)] Storage and loss moduli from the simulations (symbols) compared with the predictions of the normal mode analysis (lines). Different colors represent the results from different values of the ratio KA0/Γ. [(c),(d)] Nonzero eigenvalues λk in ascending order for different values of KA0/Γ. [(e),(f)] Absolute values of normalized coefficients |α¯kG¯ke|=|αkGkeγ/(KA0)2| and |β¯kG¯ke|=|βkGke/(KA0)| at one representative value of KA0/Γ=34.641.

      Reuse & Permissions
    • Figure 5
      Figure 5

      Shear rheology of hexagonal tilings with internal dissipation due to the relative motion of neighboring vertices in addition to the vertex-substrate friction. Results for two representative values of the cell shape parameter, p0=3.5 in the solid phase (top row) and p0=3.78 in the fluid phase (bottom row) are shown. [(a),(b)] Storage and loss moduli from the simulations (symbols) compared with the predictions from normal modes (lines) for different values of the internal friction coefficient ζV (see colorbar). [(c),(d)] Nonzero eigenvalues λk in ascending order for different values of ζV. [(e),(f)] Normalized coefficients α¯kG¯ke=αkGkeγ/(KA0)2, β¯kG¯ke=βkGke/(KA0), α¯kG¯kid=αkGkid/(KA0), and β¯kG¯kid=βkGkid/γ [see Eqs. (14) and (20)] for a representative value of ζV/γ=10. In the solid phase, the rheological response is dominated by the single normal mode ξD marked by the arrow in panel (e), which corresponds to the highest value of coefficients α¯kG¯ke and β¯kG¯ke. Note very different values used for the ordinate axes in the top and bottom panels in (e).

      Reuse & Permissions
    • Figure 6
      Figure 6

      Shear rheology for disordered tilings with internal dissipation due to the relative motion of neighboring vertices in addition to cell-substrate friction. Results for three representative values of the cell shape parameter, p0=3.06 deep in the solid phase (first row), p0=3.87 close to the solid-fluid transition point on the solid side (second row), p0=3.99 in the fluid phase (bottom row) are shown. [(a)–(c)] Storage and loss moduli from the simulations (symbols) compared with the predictions from normal modes (lines) for different values of the internal friction coefficient ζV (see colorbar). [(d)–(f)] Nonzero eigenvalues λk in ascending order for different values of ζV. [(g)–(i)] Normalized coefficients α¯kG¯ke=αkGkeγ/(KA0)2, β¯kG¯ke=βkGk/(KA0), α¯kG¯kid=αkGkid/(KA0), and β¯kG¯kid=βkGkid/γ for a representative value of ζV/γ=10.

      Reuse & Permissions
    • Figure 7
      Figure 7

      Shear rheology of hexagonal tilings with internal dissipation due to relative motion of neighboring cell centers and external cell-substrate friction. Results for two representative values of the cell shape parameter, p0=3.5 in the solid phase (top row) and p0=3.78 in the fluid phase (bottom row) are shown. [(a),(b)] Storage and loss moduli from the simulations (symbols) compared with the predictions of the normal mode analysis (lines) for different values of ζC/γ (see colorbar). [(c),(d)] Nonzero eigenvalues λk in ascending order for different values of ζC/γ. [(e),(f)] Normalized coefficients α¯kG¯ke=αkGkeγ/(KA0)2, β¯kG¯ke=βkGk/(KA0), α¯kG¯kid=αkGkid/(KA0), and β¯kG¯kid=βkGkid/γ for a representative value of ζC/γ=10. In the solid phase, the rheological response is dominated by the single normal mode ξD marked by the arrow in panel (e). This mode corresponds to the highest value of coefficients α¯kG¯ke and β¯kG¯ke. Note very different numerical values on ordinate axes in top and bottom panels in (e).

      Reuse & Permissions
    • Figure 8
      Figure 8

      Shear rheology for disordered tilings with internal dissipation due to the relative motion of neighboring cell centers and external cell-substrate friction. Results are shown for three representative values of the cell shape parameter, p0=3.06 deep in the solid phase (top row), p0=3.87 close to the solid-fluid transition point on the solid side (middle row), and p0=3.99 in the fluid phase (bottom row). [(a)–(c)] Storage and loss moduli from the simulations (symbols) compared with the predictions of the normal mode analysis (lines) for different values of ζC/γ (see colorbar). [(d)–(f)] Nonzero eigenvalues λk in ascending order for different values of ζC/γ. [(g)–(i)] Normalized coefficients α¯kG¯ke=αkGkeγ/(KA0)2, β¯kG¯ke=βkGK/(KA0), α¯kG¯kid=αkGkid/(KA0), and β¯kG¯kid=βkGkid/γ at one representative value of ζC/γ=10.

      Reuse & Permissions
    • Figure 9
      Figure 9

      An efficient way of calculating the elastic force on vertex i, fie, is to loop over all cell-cell junctions that originate at i in the counterclockwise direction. For each junction, the two cells that share it contribute to the total force. These contributions are the two terms in the sum in Eq. (A10). For consistency, we adopt a convention that when looking along the junction away from the vertex i, the cell to the right (blue) is labeled as C1, and the cell to the left (red) is labeled as C2. Note that since vertices within each cell are ordered counterclockwise, the endpoint of the junction, i.e., the vertex j appears in cell 1 (2) as i−1 (i+1) in the cell's internal labeling.

      Reuse & Permissions
    • Figure 10
      Figure 10

      The total force on a vertex is a sum of the mechanical forces from surrounding cells (blue, orange, and green) and internal (black) and external (red) dissipative forces. In order to compute the stress on each cell, it is convenient to make a virtual split of the vertex between all cells sharing it, as shown in the right panel. The force balance is then used for each subvertex to compute the reaction forces (grey) that develop due to the interactions between subvertices.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review Research

    Reuse & Permissions

    It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

    ×

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×