Skip to main content
SpringerLink
Log in

Dynamic structure factor of undulating vesicles: finite-size and spherical geometry effects with application to neutron spin echo experiments

  • Regular Article - Soft Matter
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract

We consider the dynamic structure factor (DSF) of quasi-spherical vesicles and present a generalization of an expression that was originally formulated by Zilman and Granek (ZG) for scattering from isotropically oriented quasi-flat membrane plaquettes. The expression is obtained in the form of a multi-dimensional integral over the undulating membrane surface. The new expression reduces to the original stretched exponential form in the limit of sufficiently large vesicles, i.e., in the micron range or larger. For much smaller unilamellar vesicles, deviations from the asymptotic, stretched exponential equation are noticeable even if one assumes that the Seifert-Langer leaflet density mode is completely relaxed and membrane viscosity is neglected. To avoid the need for an exhaustive numerical integration while fitting to neutron spin echo (NSE) data, we provide a useful approximation for polydisperse systems that tests well against the numerical integration of the complete expression. To validate the new expression, we performed NSE experiments on variable-size vesicles made of a POPC/POPS lipid mixture and demonstrate an advantage over the original stretched exponential form or other manipulations of the original ZG expression that have been deployed over the years to fit the NSE data. In particular, values of the membrane bending rigidity extracted from the NSE data using the new approximations were insensitive to the vesicle radii and scattering wavenumber and compared very well with expected values of the effective bending modulus (\(\tilde{\kappa }\)) calculated from results in the literature. Moreover, the generalized scattering theory presented here for an undulating quasi-spherical shell can be easily extended to other models for the membrane undulation dynamics beyond the Helfrich Hamiltonian and thereby provides the foundation for the study of the nanoscale dynamics in more complex and biologically relevant model membrane systems.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

Raw data of the experiment are available at doi.ill.fr/10.5291/ILL-DATA.DIR-277.

Notes

  1. Following the theoretical soft-matter literature [8, 10,11,12, 14,15,16], the term “dynamic structure factor” is used here instead of the common equivalent experimental term “intermediate scattering function”, even though in the experimental community, DSF typically refers to the energy domain.

  2. The term vesicle “static structure factor” used here is equivalent to the term vesicle “form factor” commonly used in the small angle scattering community.

References

  1. É. Freyssingeas, D. Roux, F. Nallet, Quasi-elastic light scattering study of highly swollen lamellar and “sponge’’ phases. J. Phys. II 7(6), 913–929 (1997)

    CAS  Google Scholar 

  2. R. Hirn, T.M. Bayerl, J.O. Rädler, E. Sackmann, Collective membrane motions of high and low amplitude, studied by dynamic light scattering and micro-interferometry. Faraday Discuss. 111, 17–30 (1999). https://doi.org/10.1039/A807883A

    Article  ADS  Google Scholar 

  3. P. Falus, M. Borthwick, S. Mochrie, Fluctuation dynamics of block copolymer vesicles. Phys. Rev. Lett. 94(1), 016105 (2005)

    Article  ADS  CAS  PubMed  Google Scholar 

  4. P. Falus, M. Borthwick, S. Narayanan, A. Sandy, S. Mochrie, Crossover from stretched to compressed exponential relaxations in a polymer-based sponge phase. Phys. Rev. Lett. 97(6), 066102 (2006)

    Article  ADS  CAS  PubMed  Google Scholar 

  5. S. Gupta, R. Ashkar, The dynamic face of lipid membranes. Soft Matter 17(29), 6910–6928 (2021)

    Article  ADS  CAS  PubMed  Google Scholar 

  6. V. Sharma, E. Mamontov, Multiscale lipid membrane dynamics as revealed by neutron spectroscopy. Prog. Lipid Res. 101179 (2022)

  7. M. Nagao, H. Seto, Neutron scattering studies on dynamics of lipid membranes. Biophys. Rev. 4(2) (2023)

  8. M.C. Watson, Y. Peng, Y. Zheng, F.L.H. Brown, The intermediate scattering function for lipid bilayer membranes: from nanometers to microns. J. Chem. Phys. 135(19), 194701 (2011). https://doi.org/10.1063/1.3657857

    Article  ADS  CAS  PubMed  Google Scholar 

  9. M.C. Watson, E.S. Penev, P.M. Welch, F.L.H. Brown, Thermal fluctuations in shape, thickness, and molecular orientation in lipid bilayers. J. Chem. Phys. 135(24), 244701 (2011). https://doi.org/10.1063/1.3660673

    Article  ADS  CAS  PubMed  Google Scholar 

  10. S.T. Milner, S.A. Safran, Dynamical fluctuations of droplet microemulsions and vesicles. Phys. Rev. A 36, 4371–4379 (1987). https://doi.org/10.1103/PhysRevA.36.4371

    Article  ADS  CAS  Google Scholar 

  11. A.G. Zilman, R. Granek, Undulations and dynamic structure factor of membranes. Phys. Rev. Lett. 77(23), 4788–4791 (1996). https://doi.org/10.1103/PhysRevLett.77.4788

    Article  ADS  CAS  PubMed  Google Scholar 

  12. A.G. Zilman, R. Granek, Membrane dynamics and structure factor. Chem. Phys. 284(1), 195–204 (2002). https://doi.org/10.1016/S0301-0104(02)00548-7

    Article  CAS  Google Scholar 

  13. S. Lovesey, P. Schofield, Inelastic coherent neutron scattering by small particles. J. Phys. C Solid State Phys. 9(15), 2843 (1976)

    Article  ADS  CAS  Google Scholar 

  14. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, vol. 73 (Oxford University Press, Oxford, 1988)

    Google Scholar 

  15. P.-G. De Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, London, 1979)

    Google Scholar 

  16. M.C. Watson, F.H. Brown, Interpreting membrane scattering experiments at the mesoscale: the contribution of dissipation within the bilayer. Biophys. J . 98(6), 9–11 (2010). https://doi.org/10.1016/j.bpj.2009.11.026

    Article  CAS  Google Scholar 

  17. I. Hoffmann, Background subtraction and dada analysis in neutron spin echo spectroscopy. Front. Phys. (2021). https://doi.org/10.3389/fphy.2020.620082

    Article  Google Scholar 

  18. E.G. Kelley, E.E. Blick, V.M. Prabhu, P.D. Butler, M. Nagao, Interactions, diffusion, and membrane fluctuations in concentrated unilamellar lipid vesicle solutions. Front. Phys. 10, 288 (2022)

    Article  Google Scholar 

  19. M. Monkenbusch, O. Holderer, H. Frielinghaus, D. Byelov, J. Allgaier, D. Richter, Bending moduli of microemulsions; comparison of results from small angle neutron scattering and neutron spin-echo spectroscopy. J. Phys. Condens. Matter 17(31), 2903–2909 (2005). https://doi.org/10.1088/0953-8984/17/31/017

    Article  ADS  CAS  Google Scholar 

  20. M. Ohl, M. Monkenbusch, N. Arend, T. Kozielewski, G. Vehres, C. Tiemann, M. Butzek, H. Soltner, U. Giesen, R. Achten, H. Stelzer, B. Lindenau, A. Budwig, H. Kleines, M. Drochner, P. Kaemmerling, M. Wagener, R. Möller, E.B. Iverson, M. Sharp, D. Richter, The spin-echo spectrometer at the spallation neutron source (sns). Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 696, 85–99 (2012). https://doi.org/10.1016/j.nima.2012.08.059

    Article  ADS  CAS  Google Scholar 

  21. O. Holderer, O. Ivanova, J-nse: neutron spin echo spectrometer. J. Large-scale Res. Facil. JLSRF 1, 11–11 (2015)

    Article  Google Scholar 

  22. B. Farago, P. Falus, I. Hoffmann, M. Gradzielski, F. Thomas, C. Gomez, The in15 upgrade. Neutron News 26(3), 15–17 (2015). https://doi.org/10.1080/10448632.2015.1057052

    Article  Google Scholar 

  23. W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C 28, 693–703 (1973)

    Article  CAS  PubMed  Google Scholar 

  24. U. Seifert, K. Berndl, R. Lipowsky, Shape transformations of vesicles: Phase diagram for spontaneous-curvature and bilayer-coupling models. Phys. Rev. A 44(2), 1182 (1991)

    Article  ADS  CAS  PubMed  Google Scholar 

  25. U. Seifert, S.A. Langer, Viscous modes of fluid bilayer-membranes. Europhys. Lett. 23(1), 71–76 (1993). https://doi.org/10.1209/0295-5075/23/1/012

    Article  ADS  Google Scholar 

  26. U. Seifert, S.A. Langer, Hydrodynamics of membranes: the bilayer aspect and adhesion. Biophys. Chem. 49(1), 13–22 (1994). https://doi.org/10.1016/0301-4622(93)E0077-I

    Article  CAS  Google Scholar 

  27. L.R. Arriaga, R. Rodríguez-García, B. López-Montero, T. Farago, F. Monroy. Hellweg, Dissipative curvature fluctuations in bilayer vesicles: coexistence of pure-bending and hybrid curvature-compression modes. Eur. Phys. J. E 31(1), 105–113 (2010). https://doi.org/10.1140/epje/i2010-10551-1

    Article  CAS  PubMed  Google Scholar 

  28. H.A. Faizi, R. Granek, P.M. Vlahovska, Membrane viscosity signature in thermal undulations of curved fluid bilayers (2022)

  29. A.C. Woodka, P.D. Butler, L. Porcar, B. Farago, M. Nagao, Lipid bilayers and membrane dynamics: insight into thickness fluctuations. Phys. Rev. Lett. 109(5), 058102 (2012)

    Article  ADS  PubMed  Google Scholar 

  30. M. Nagao, E.G. Kelley, R. Ashkar, R. Bradbury, P.D. Butler, Probing elastic and viscous properties of phospholipid bilayers using neutron spin echo spectroscopy. J. Phys. Chem. Lett. 8(19), 4679–4684 (2017). https://doi.org/10.1021/acs.jpclett.7b01830

    Article  CAS  PubMed  Google Scholar 

  31. R.J. Bingham, S.W. Smye, P.D. Olmsted, Dynamics of an asymmetric bilayer lipid membrane in a viscous solvent. EPL 111(1), 18004 (2015). https://doi.org/10.1209/0295-5075/111/18004

    Article  ADS  CAS  Google Scholar 

  32. R. Granek, Comment on “dynamics of phospholipid membranes beyond thermal undulations’’. J. Phys. Chem. B 123(26), 5665–5666 (2019). https://doi.org/10.1021/acs.jpcb.9b03049

    Article  CAS  PubMed  Google Scholar 

  33. L. Moleiro, M. Mell, R. Bocanegra, I. López-Montero, P. Fouquet, T. Hellweg, J. Carrascosa, F. Monroy, Permeability modes in fluctuating lipid membranes with DNA-translocating pores. Adv. Coll. Interface. Sci. 247, 543–554 (2017)

    Article  CAS  Google Scholar 

  34. R. Granek, Membrane surrounded by viscoelastic continuous media: anomalous diffusion and linear response to force. Soft Matter 7(11), 5281–5289 (2011)

    Article  ADS  CAS  Google Scholar 

  35. R. Granek, H. Diamant, Membrane undulations in a structured fluid: universal dynamics at intermediate length and time scales. Eur. Phys. J. E 41, 1–10 (2018)

    Article  CAS  PubMed  Google Scholar 

  36. M. Schneider, J. Jenkins, W. Webb, Thermal fluctuations of large quasi-spherical bimolecular phospholipid vesicles. J. Phys. 45(9), 1457–1472 (1984)

    Article  CAS  Google Scholar 

  37. K. Seki, S. Komura, Viscoelasticity of vesicle dispersions. Phys. A 219(3–4), 253–289 (1995)

    Article  CAS  Google Scholar 

  38. P. Olla, The behavior of closed inextensible membranes in linear and quadratic shear flows. Phys. A 278(1–2), 87–106 (2000)

    Article  MathSciNet  CAS  Google Scholar 

  39. S. Rochal, V. Lorman, G. Mennessier, Viscoelastic dynamics of spherical composite vesicles. Phys. Rev. E 71(2), 021905 (2005)

    Article  ADS  CAS  Google Scholar 

  40. P.M. Vlahovska, Electrohydrodynamics of drops and vesicles. Annu. Rev. Fluid Mech. 51, 305–330 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  41. J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1998)

    Google Scholar 

  42. M. Hope, M. Bally, G. Webb, P. Cullis, Production of large unilamellar vesicles by a rapid extrusion procedure characterization of size distribution, trapped volume and ability to maintain a membrane potential. Biochim. Biophys. Acta (BBA) Biomembr. 812(1), 55–65 (1985)

    Article  CAS  Google Scholar 

  43. A.H. Kunding, M.W. Mortensen, S.M. Christensen, D. Stamou, A fluorescence-based technique to construct size distributions from single-object measurements: application to the extrusion of lipid vesicles. Biophys. J. 95(3), 1176–1188 (2008)

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  44. N. Kuečrka, M.-P. Nieh, J. Katsaras, Fluid phase lipid areas and bilayer thicknesses of commonly used phosphatidylcholines as a function of temperature. Biochim. Biophys. Acta (BBA) Biomembr. 1808(11), 2761–2771 (2011). https://doi.org/10.1016/j.bbamem.2011.07.022

    Article  CAS  Google Scholar 

  45. R. Dimova, Recent developments in the field of bending rigidity measurements on membranes. Adv. Colloid Interface Sci. 208, 225–234 (2014). https://doi.org/10.1016/j.cis.2014.03.003

    Article  CAS  PubMed  Google Scholar 

  46. J.F. Nagle, Experimentally determined tilt and bending moduli of single-component lipid bilayers. Chem. Phys. Lipids 205, 18–24 (2017). https://doi.org/10.1016/j.chemphyslip.2017.04.006

    Article  CAS  PubMed  Google Scholar 

  47. H.A. Faizi, S.L. Frey, J. Steinkühler, R. Dimova, P.M. Vlahovska, Bending rigidity of charged lipid bilayer membranes. Soft Matter 15(29), 6006–6013 (2019). https://doi.org/10.1039/C9SM00772E

    Article  ADS  CAS  PubMed  Google Scholar 

  48. H.L. Scott, A. Skinkle, E.G. Kelley, M.N. Waxham, I. Levental, F.A. Heberle, On the mechanism of bilayer separation by extrusion, or why your luvs are not really unilamellar. Biophys. J. 117(8), 1381–1386 (2019). https://doi.org/10.1016/j.bpj.2019.09.006

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  49. H. Seto, N.L. Yamada, M. Nagao, M. Hishida, T. Takeda, Bending modulus of lipid bilayers in a liquid-crystalline phase including an anomalous swelling regime estimated by neutron spin echo experiments. Eur. Phys. J. E 26(1–2), 217–223 (2008). https://doi.org/10.1140/epje/i2007-10315-0

    Article  CAS  PubMed  Google Scholar 

  50. L. Chiappisi, I. Hoffmann, M. Gradzielski, Membrane stiffening in chitosan mediated multilamellar vesicles of alkyl ether carboxylates. J. Colloid Interface Sci. 627, 160–167 (2022)

    Article  ADS  CAS  PubMed  Google Scholar 

  51. W. Helfrich, Steric interaction of fluid membranes in multilayer systems. Zeitschrift für Naturforschung A 33(3), 305–315 (1978)

    Article  ADS  Google Scholar 

  52. J.B. Hayter, J. Penfold, Self-consistent structural and dynamic study of concentrated micelle solutions. J. Chem. Soc. Faraday Trans. 1 Phys. Chem. Condens. Phases 77(8), 1851–1863 (1981)

    CAS  Google Scholar 

  53. M. Heinen, P. Holmqvist, A.J. Banchio, G. Nägele, Short-time diffusion of charge-stabilized colloidal particles: generic features. J. Appl. Crystallogr. 43(5 Part 1), 970–980 (2010). https://doi.org/10.1107/S002188981002724X

    Article  ADS  CAS  Google Scholar 

  54. C. Haro-Perez, M. Quesada-Pérez, J. Callejas-Fernandez, E. Casals, J. Estelrich, R. Hidalgo-Alvarez, Interplay between hydrodynamic and direct interactions using liposomes. J. Chem. Phys. 119(1), 628–634 (2003)

    Article  ADS  CAS  Google Scholar 

  55. J.F. Nagle, Introductory lecture: Basic quantities in model biomembranes. Faraday Discuss. 161, 11–150 (2013)

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  56. D. Marsh, Elastic curvature constants of lipid monolayers and bilayers. Chem. Phys. Lipid. 144(2), 146–159 (2006)

    Article  CAS  Google Scholar 

  57. G. Niggemann, M. Kummrow, W. Helfrich, The bending rigidity of phosphatidylcholine bilayers: dependences on experimental method, sample cell sealing and temperature. J. Phys. II 5(3), 413–425 (1995)

    CAS  Google Scholar 

  58. M. Doktorova, D. Harries, G. Khelashvili, Determination of bending rigidity and tilt modulus of lipid membranes from real-space fluctuation analysis of molecular dynamics simulations. Phys. Chem. Chem. Phys. 19(25), 16806–16818 (2017). https://doi.org/10.1039/C7CP01921A

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  59. J. Henriksen, A.C. Rowat, E. Brief, Y. Hsueh, J. Thewalt, M. Zuckermann, J.H. Ipsen, Universal behavior of membranes with sterols. Biophys. J . 90(5), 1639–1649 (2006)

    Article  CAS  PubMed  Google Scholar 

  60. R.M. Venable, F.L.H. Brown, R.W. Pastor, Mechanical properties of lipid bilayers from molecular dynamics simulation. Chem. Phys. Lipids 192, 60–74 (2015). https://doi.org/10.1016/j.chemphyslip.2015.07.014

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  61. H. Bouvrais, L. Duelund, J.H. Ipsen, Buffers affect the bending rigidity of model lipid membranes. Langmuir 30(1), 13–16 (2014)

    Article  CAS  PubMed  Google Scholar 

  62. J.R. Henriksen, J.H. Ipsen, Measurement of membrane elasticity by micro-pipette aspiration. Eur. Phys. J. E 14(2), 149–167 (2004). https://doi.org/10.1140/epje/i2003-10146-y

    Article  CAS  PubMed  Google Scholar 

  63. Y. Liu, Intermediate scattering function for macromolecules in solutions probed by neutron spin echo. Phys. Rev. E 95(2), 020501 (2017)

    Article  ADS  PubMed  Google Scholar 

  64. B. Brüning, R. Stehle, P. Falus, B. Farago, Influence of charge density on bilayer bending rigidity in lipid vesicles: a combined dynamic light scattering and neutron spin-echo study. Eur. Phys. J. E 36, 1–8 (2013)

    Article  Google Scholar 

  65. J.L. Cascales, S.O. Costa, A. Garro, R.D. Enriz, Mechanical properties of binary DPPC/DPPS bilayers. RSC Adv. 2(31), 11743–11750 (2012)

  66. A.-F. Bitbol, J.-B. Fournier, M.I. Angelova, N. Puff, Dynamical membrane curvature instability controlled by intermonolayer friction. J. Phys. Condens. Matter 23(28), 284102 (2011). https://doi.org/10.1088/0953-8984/23/28/284102

    Article  CAS  PubMed  Google Scholar 

  67. F. Campelo, C. Arnarez, S.J. Marrink, M.M. Kozlov, Helfrich model of membrane bending: from Gibbs theory of liquid interfaces to membranes as thick anisotropic elastic layers. Adv. Colloid Interface Sci. 208, 25–33 (2014). https://doi.org/10.1016/j.cis.2014.01.018

    Article  CAS  PubMed  Google Scholar 

  68. J.-H. Lee, S.-M. Choi, C. Doe, A. Faraone, P.A. Pincus, S.R. Kline, Thermal fluctuation and elasticity of lipid vesicles interacting with pore-forming peptides. Phys. Rev. Lett. 105(3), 038101 (2010)

    Article  ADS  PubMed  Google Scholar 

  69. A.-F. Bitbol, D. Constantin, J.-B. Fournier, Bilayer elasticity at the nanoscale: the need for new terms. PLoS ONE 7(11), 48306 (2012). https://doi.org/10.1371/journal.pone.0048306

    Article  ADS  CAS  Google Scholar 

  70. M. Hu, D.H. Jong, S.J. Marrink, M. Deserno, Gaussian curvature elasticity determined from global shape transformations and local stress distributions: a comparative study using the martini model. Faraday Discuss. 161, 365–382 (2013). https://doi.org/10.1039/C2FD20087B

    Article  ADS  CAS  PubMed  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge Ryan Murphy and Lionel Porcar for assistance with the SANS data collection, and John Nagle for useful discussions. The authors also acknowledge the Partnership for Soft Condensed Matter (PSCM) for providing access to the DLS instrument and laboratory infrastructure used for sample preparation. EGK and MN acknowledge support from the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology (NIST) and the National Science Foundation under Agreement No. DMR-2010792. The identification of any commercial products does not imply endorsement or recommendation by NIST. This work benefited from the use of the SasView application, originally developed under NSF award DMR-0520547. SasView contains code developed with funding from the European Union’s Horizon 2002 research and innovation programme under the SINE2020 project, grant agreement No 654000. This research was also supported in part by the National Science Foundation under Grant No. NSF PHY-1748958

Author information

Authors and Affiliations

  1. Avram and Stella Goldstein-Goren Department of Biotechnology Engineering, and Ilse Katz Institute for Nanoscale Science and Technology, Ben-Gurion University of the Negev, 84105, Beer Sheva, Israel

    Rony Granek

  2. Institut Laue-Langevin (ILL), 71 Avenue des Martys, 38042, Grenoble, CEDEX 9, France

    Ingo Hoffmann

  3. Center for Neutron Research, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, MD, 20899, USA

    Elizabeth G. Kelley & Michihiro Nagao

  4. Department of Materials Science and Engineering, University of Maryland, College Park, MD, 20742, USA

    Michihiro Nagao

  5. Department of Physics and Astronomy, University of Delaware, Newark, DE, 19716, USA

    Michihiro Nagao

  6. Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Petia M. Vlahovska

  7. Department of Physics, University of Toronto, 60 St George St, Toronto, ON, M5S 1A7, Canada

    Anton Zilman

Authors
  1. Rony Granek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ingo Hoffmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Elizabeth G. Kelley
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michihiro Nagao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Petia M. Vlahovska
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Anton Zilman
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

RG, AZ, and PMV developed the theory. IH, EGK, and MN performed the experiment. RG, IH, EGK, and MN wrote the article.

Corresponding authors

Correspondence to Rony Granek, Ingo Hoffmann or Elizabeth G. Kelley.

Additional information

This article is dedicated to Fyl Pincus whose pioneering achievements, inspiring approach, and revolutionary ideas in soft condensed matter and biological physics have had a great impact on the entire community in general, and especially on the present authors, including on RG, EGK, MN, and AZ.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (pdf 1326 KB)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Granek, R., Hoffmann, I., Kelley, E.G. et al. Dynamic structure factor of undulating vesicles: finite-size and spherical geometry effects with application to neutron spin echo experiments. Eur. Phys. J. E 47, 12 (2024). https://doi.org/10.1140/epje/s10189-023-00400-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/s10189-023-00400-9

Search

Navigation