Abstract
Theoretical studies of nearly spherical vesicles and microemulsion droplets, that present typical examples for thermally-excited systems that are subject to constraints, are reviewed. We consider the shape fluctuations of such systems constrained by fixed area \(A\) and fixed volume \(V\), whose geometry is presented in terms of scalar spherical harmonics. These constraints can be incorporated in the theory in different ways. After an introductory review of the two approaches: with an exactly fixed by delta-function membrane area \(A\) [Seifert, Z. Phys. B, 97, 299 (1995)] or approximatively by means of a Lagrange multiplier \(\sigma \) conjugated to \(A\) [Milner and Safran, Phys. Rev. A, 36, 4371 (1987)], we discuss the determined role of the stretching effects, that has been announced in the framework of a model containing stretching energy term, expressed via the membrane vesicle tension [Bivas an d Tonchev, Phys. Rev. E, 100, 022416 (2019)]. Since the fluctuation spectrum for the used Hamiltonian is not exactly solvable an approximating method based on the Bogoliubov inequalities for the free energy has been developed. The area constraint in the last approach appears as a self-consistent equation for the membrane tension. In the general case this equation is intractable analytically. However, much insight into the physics behind can be obtained either imposing some restrictions on the values of the model parameters, or studying limiting cases, in which the self-consistent equation is solved. Implications for the equivalence of ensembles have been discussed as well.
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REFERENCES
J. S. Singer and G. L. Nicolson, “The fluid mosaic model of the structure of cell membranes,” Science 175, 720–731 (1972).
A. G. Petrov, The Lyotropic State of Matter: Molecular Physics and Living Matter Physics (Gordon Breach Sci. Publ., 1999).
C. G. Siontorou, G. P. Nikoleli, D. P. Nikolelis, and S. K. Karapetis, “Artificial lipid membranes: Past, present, and future,” Membranes (Basel) 7 (3), 38 (2017).
S. A. Safran, “Statistical thermodynamics of surfaces, interfaces, and membranes,” in Frontiers in Physics (Taylor and Francis Group, 2003).
W. Helfrich, “Elastic properties of lipid bilayers: Theory and possible experiments,” Z. Naturforsch. C 28, 693–703 (1974).
P. B. Canham, “The minimum energy of bending as a possible explanation of the biconcave shape of human red blood cell,” J. Theor. Biol. 26, 61–81 (1970).
E. A. Evans, “Bending resistance and chemically induced moments in membrane bilayers,” Biophys. J. 14, 923–931 (1974).
W. Helfrich, “Size distributions of vesicles: The role of the effective rigidity of membranes,” J. Phys. (France) 47, 321–329 (1986).
L. Miao, U. Seifert, M. Wortis, and H.-G. Döbereinert, “Budding transitions of fluid-bilayer vesicles: The effect of area-difference elasticity,” Phys. Rev. A 49, 5389–5407 (1984).
M. Deserno, “Fluid lipid membranes: From differential geometry to curvature stresses,” Chem. Phys. Lipids 185, 11–45 (2015).
S. Libler, “Equilibrium statistical mechanics of fluctuating films and membranes,” in Statistical Mechanics of Membranes and Surfaces, Ed. by D. Nelson, T. Piran, and S. Weinberg (World Scientific, 2004), pp. 49–102.
S. T. Milner and S. A. Safran, “Dynamical fluctuations of droplet microemulsions and vesicles,” Phys. Rev. A 36, 4371–4379 (1987).
U. Seifert, “The concept of effective tension for fluctuating vesicles,” Z. Phys. B 97, 299–309 (1995).
U. Seifert, Habilitation Thesis (Ludwlq-Maximllians-Unlversitat, Munchen, 1994).
F. Brochard and J. F. Lenon, “Frequency spectrum of flicker phenomenon in erythrocytes,” J. Phys. (Paris) 36, 1035–1047 (1975).
M. B. Schneider, J. R. Jenkins, and W. W. Webb, “Thermal fluctuations of large quasi-spherical bimolecular phospholipid vesicles,” J. Phys. (Paris) 45, 1457–1472 (1984).
I. Bivas, P. Hanusse, P. Bothorel, J. Lalanne, and O. Aguerre-Chariol, “An application of the optical microscopy to the determination of the curvature elastic modulus of biological and model membranes,” J. Phys. II 48, 855–867 (1987).
J. F. Faucon, M. D. Mitov, P. Méléard, I. Bivas, and P. Bothorel, “Bending elasticity and thermal fluctuations of lipid membranes. Theoretical and experimental requirements,” J. Phys. (Paris) 50, 2389–2414 (1989).
P. Meleard, C. Gerbeaud, T. Pott, L. Fernandez-Puente, I. Bivas, M. D. Mitov, J. Dufourcq, and P. Bothorel, “Bending elasticities of modified membranes: Influences of temperature and sterol content, ” Biophys. J. 72, 2616–2629 (1997).
J. Pécréaux, H.-G. Döbereiner, J. Prost, J.-F. Joanny, and P. Bassereau, “Refined contour analysis of giant unilamellar vesicles,” Eur. Phys. J. E 13, 277–290 (2004).
J. Genova, V. Vitkova, and I. Bivas, “Registration and analysis of shape fluctuations of nearly spherical lipid vesicles,” Phys. Rev. E 88, 022707 (2013).
J. Genova, “Marin Mitov lectures: Measuring and bending elasticity of lipid bilayer,” Adv. Planar Lipid Bilayers Liposomes 17, 1–27 (2013).
V. Vitkova and A. G. Petrov, “Lipid bilayers and membranes: Material properties,” Adv. Planar Lipid Bilayers Liposomes 17, 89–138 (2013).
C. Monzel and K. Sengupta, “Measuring shape fluctuations in biological membranes,” J. Phys. D: Appl. Phys. 49, 2430002 (2016).
S. A. Rautu, D. Orsi, L. Di Michele, G. Rowlands, P. Cicuta, and M. S. Turner, “The role of optical projection in the analysis of membrane fluctuations, ” Soft Matter 13, 3480–3483 (2017).
Z. C. Ou-Yang and W. Helfrich, “Bending energy of vesicle membranes: General expressions for the first, second and third variation of the shape energy and application to spheres and cylinders,” Phys. Rev. A 39, 5280–5288 (1989).
V. Heinrich, M. Brumen, R. Heinrich, S. Svetina, and B. Žekš, “Nearly spherical vesicle shapes calculated by use of spherilcal harmonics: axisymmetric and nonaxisimmetric shapes and their stability,” J. Phys. II 2, 1081–1108 (1992).
F. Sevšek, “Membrane elasticity from shape fluctuations of phospholipid vesicles,” Adv. Planar Lipid Bilayers Liposomes 12, 1–19 (2010).
S. Komura and K. Seki, “Dynamical fluctuations of spherically closed fluid membranes,” Physica A 192, 27–46 (1993).
C. Barbetta, A. Imparato, and J. B. Fournier, “On the surface tension of fluctuating quasi-spherical vesicles,” Eur. Phys. J. E 31, 333–342 (2010).
G. Gueguen, N. Destanville, and M. Manghi, “Fluctuation tension and shape transition of vesicles: Renormalisation calculations and Monte Carlo simulations,” Soft Matter 84, 6100 (2017).
G. Gomper and D. M. Kroll, “Random surface discretizations and the renormalization of the bending rigidity,” J. Phys. II France 6, 1305–1320 (1996).
W. Cai, T. C. Lubensky, P. Nelson, and T. Powers, “Measure factors, tension and correlations of fluid membranes,” J. Phys. II France 4, 931–949 (1994).
F. David, “Geometry and field theory of random surfaces and membranes,” in Statistical Mechanics of Membranes and Surfaces, Ed. by D. Nelson, T. Piran, and S. Weinberg (World Scientific, 2004), pp. 149–209.
I. Bivas, L. Bivolarski, M. Mitov, and A. Derzhanski, “Correlations between the form fluctuations modes of flaccid quasi-spherical lipid vesicles and their role in the calculation of the curvature elastic modulus of the vesicle membrane. Numerical results,” J. Phys. II 2, 1423–1438 (1992).
J. B. Fournier, A. Ajdari, and L. Peliti, “Effective-area elasticity and tension of micromanipulated membranes,” Phys. Rev. Lett. 86, 4970–4973 (2001); arXiv: 0103.495 [cond-mat].
U. Seifert, “Configurations of fluid membranes and vesicles,” Adv. Phys. 46, 13–137 (1997).
G. S. Joyce, “Critical properties of the spherical model,” in Phase Transitions and Critical Phenomena, Ed. by C. Domb and N. S. Green (World Scientific, 1972), Vol. 2, p. 375.
J. D. Brankov, D. M. Danchev, and N. S. Tonchev, Theory of Critical Phenomena in Finite-Size Systems: Scaling and Quantum Effects (World Scientific, Singapore, 2000).
O. Farago, “Mechanical surface tension governs membrane thermal fluctuations,” Phys. Rev. E 84, 051914 (2011).
H. Shiba, H. Noguchi, and J. B. Fournier, “Monte Carlo study of the frame, fluctuation and internal tensions of fluctuating membranes with fixed area,” Soft Matter 12, 2373–2380 (2016).
I. Bivas and N. S. Tonchev, “Membrane stretching elasticity and thermal shape fluctuations of nearly spherical lipid vesicles,” Phys. Rev. E 100, 022416 (2019).
J. R. Henriksen and J. H. Ipsen, “Measurement of membrane elasticity by micro-pipette aspiration,” Eur. Phys. J. E 14, 149–167 (2004).
J. W. Gibbs, Elementary Principles in Statistical Mechanics with Especial Reference to the Rational Foundation of Thermodynamics (Yale Univ. Press, 1902; Dover, New York, 1960).
S. Adams, “Lectures on mathematical statistical mechanics,” in Communications of the Dublin Institute for Advanced Studies Series A (Theoretical Physics) (Dublin Inst. Adv. Stud., 2006).
H. Touchette, “Equivalence and nonequivalence of ensembles: Thermodynamic, macrostate, and measure levels,” J. Stat. Phys. 159, 987–1016 (2015).
F. Schmid, “Are stress-free membranes really ‘tensionless’?,” Eur. Phys. Lett. 95, 28008 (2011).
F. Schmid, “Fluctuations in lipid bilayers: Are they understood?,” Biophys. Rev. Lett. 8, 1–20 (2013).
F. Brochard, P. G. De Gennes, and J. Pfeuty, “Surface tension and deformations of membrane structures: Relation to two dimensional phase transitions,” J. Phys. (Paris) 37, 1099–1104 (1976).
D. Marsh, “Renormalization of the tension and area expansion modulus in fluid membranes,” Biophys. J. 73, 865–869 (1997).
J. F. Nagle, “Introductory lecture: Basic quantities in model biomembranes,” Faraday Discuss. 161, 11–29 (2013).
M. A. Lamholt, B. Loubet, and J. H. Ipsen, “Elastic moderation of intrinsically applied tension in lipid membranes,” Phys. Rev. E 83, 011913 (2011).
I. Bivas, “Elasticity and shape fluctuation of a lipid membrane,” Eur. Phys. J. B 29, 317–322 (2002).
I. Bivas, “Shape fluctuation of nearly spherical lipid vesicles and emulsion droplets,” Phys. Rev E 81, 061911 (2010).
O. Farago and P. Pincus, “The effect of thermal fluctuation on Schulman area elasticity,” Eur. Phys. J. E 11, 399–408 (2003).
M. Fošnarič, S. Penič, A. Iglič, and I. Bivas, “Thermal fluctuations of phospholipid vesicles studied by Monte Carlo simulations,” Adv. Planar Lipid Bilayers Liposomes 17, 331–357 (2013).
J. Shapiro and J. Rudnick, “The fully finite spherical model,” Phys. Rev. E 43, 51–83 (1986).
D. C. Morse and S. T. Milner, “Fluctuations and phase behavior of fluid membrane vesicles,” Europhys. Lett. 26, 565–570 (1994).
R. Lipowsky, “Coupling of bending and stretching deformation in vesicle membranes,” Adv. Colloid Interf. Sci. 208, 14–24 (2014).
V. A. Zagrebnov, “Gibbs semigroups,” in Operator Theory: Advances and Applications (Birkhäuser, 2019), Vol. 273.
N. N. Bogolyubov, Jr., J. G. Brankov, V. A. Zagrebnov, A. M. Kurbatov, and N. S. Tonchev, The Approximating Hamiltonian Method in Statistical Physics (Publ. House Bulg. Acad. Sci., Sofia, 1981) [in Russian].
N. N. Bogolyubov, Jr., J. G. Brankov, V. A. Zagrebnov, A. M. Kurbatov, and N. S. Tonchev, “Some class of exactly soluble models of problems in quantum statistical mechanics,” Russ. Math. Surv. 39 (6), 1–50 (1984).
N. N. Bogolyubov, Jr., A Method for Studying Model Hamiltonians: A Minimax Principle for Problems in Statistical Physics (Pergamon, 2013).
S. V. Tiablikov, Methods in Quantum Theory of Magnetism (Plenum Press, New York, 1967).
F. Ahmadpor and P. Sharma, “Thermal fluctuations of vesicles and nonlinear curvature elasticity-implications for size-dependent renormalized bending rigidity and vesicle size distribution,” Soft Matter 12, 2523–2536 (2016).
I. Bivas and N. S. Tonchev, “On the statistical mechanics of shape fluctuations of nearly spherical lipid vesicle,” J. Phys.: Conf. Ser. 558, 012020 (2014); arXiv: 1409.37091 [cond-mat].
E. Evance, W. Rawicz, and B. A. Smith, “Concluding remarks. Back to the future: Mechanics and thermodynamics of lipid biomembranes,” Faraday Discuss. 161, 591–611 (2013).
M. Mell, L. H. Moleiro, Y. Hertle, I. López-Montero, F. J. Cao, P. Fouquet, T. Hellweg, and F. Monroy, “Fluctuation dynamics of bilayer vesicles with intermonolayer sliding: Experiment and theory,” Chem. Phys. Lipids 185, 61–77 (2015).
A. Yeung and E. Evance, “Unexpected dynamics in shape fluctuations of bilayer vesicles,” J. Phys. (Paris) 5, 1501–1523 (1995).
I. Bivas, P. Meleard, I. Mircheva, and P. Bothorel, “Thermal shape fluctuations of a quasi-spherical vesicle when the mutual shape fluctuations are taken into account,” Colloids Surf. A 157, 21–33 (1999).
L. Miao, M. A. Lomholt, and J. Kleis, “Dynamics of shape fluctuations of quasi-spherical vesicle revisited,” Eur. Phys. J. E 9, 143–162 (2002).
T. V. Sachin Krishnan, R. Okamoto, and S. Komura, “Relaxation dynamics of a compressible bilayer vesicle containing highly viscous fluid,” Phys. Rev. E 94, 062414 (2016).
S. Svetina, M. Brumen, and B. Žekš, “Lipid bilayer elasticity and the bilayer couple interpretation of red cell shape transformations and lysis,” Stud. Biophys. 110, 177–184 (1985).
U. Seifert and S. A. Langer, “Viscous modes of fluid bilayer membranes,” Europhys. Lett. 23, 71–76 (1993).
L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon Press, Oxford, 1980).
R. M. Corless, G. Gonnet, D. Jeffrey, and D. E. Knuthi, “On the Lambert W function,” Adv. Comput. Math. 5, 329–360 (1996).
I. Chatzigeorgiou, “Bounds on the Lambert function and their application to the outage analysis of user cooperation,” IEEE Commun. Lett. 17, 1505–1508 (2013).
S. G. Kazakova and E. S. Pisanova, “Some applications of the Lambert W function to theoretical physics education,” AIP Conf. Proc. 1203, 1354–1359 (2010).
E. S. Pisanova and S. I. Ivanov, “On the critical behavior of the inverse susceptibility of a model of structural phase transitions,” Bulg. J. Phys. 40, 159–164 (2013).
E. S. Pisanova and S. I. Ivanov, “Non-universal critical properties of the ferromagnetic mean spherical model with long-range interaction,” Bulg. Chem. Commun. 43 (B), 269–274 (2015).
R. B. Griffits, “A proof that the free energy of a spin system is extensive,” J. Math. Phys. 9, 1215 (1964).
M. E. Fisher, “Correlation function and coexistence of phases,” J. Math. Phys. 6, 1643–1653 (1965).
ACKNOWLEDGMENTS
This review is based on my lecture at the Conference in memory of Vyatcheslav Borisovich Priezzhev held in the Bogoliubov Laboratory of Theoretical Physics (JINR-Dubna, 10 September 2019). I am grateful to the Organizing Committee and especially to V.P. Spiridonov and A.M. Povolotsky, for the invitation and hospitality.
I am grateful to I. Bivas for numerous stimulating discussions on the physics of vesicles concerning theory and experiment. Many of the ideas presented in this review are based on our previous common works. I would like to thank A.G. Petrov for his useful comments on the manuscript.
Funding
This work is partly supported by the JINR (Dubna)–ISSP-BAS (Bulgaria) collaborative Grant “Investigation of the influence of nanoparticles on the properties of biologically relevant systems” 2019/2021.
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This paper is dedicated to the memory of my friend and colleague Slava Priezzhev, an outstanding scientist and man, whose contribution to statistical mechanics and probability theory is original and remarkable in many ways. His comments on physicsand things of life were always nontrivial and impactful.
Appendices
APPENDIX A
LIMIT CASE ANALYTICAL SOLUTIONS OF EQS. (46) AND (123)
Equations (46), (123) and (154) could be studied analytically by replacing the sum in its r.h.s. with an integral. In order to validate the corresponding approximation we shall use the Euler–McLaurin summation formula
where
with \(\Sigma = {{\bar {\Sigma }}_{{{\text{MS}}}}}\) in Eq. (46) and \(\Sigma = {{\bar {\Sigma }}_{{{\text{app}}}}}\) in Eq. (123), and
with \(\Sigma = {{\bar {\Sigma }}_{{{\text{app}}}}}\) in Eq. (154).
Let us ignore: (i) the higher order terms in Eq. (A.1), and ii.) approximate \(F(x) \approx F(0) + xF{\kern 1pt} '(0)\) in the interval \([0,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}]\). The approximations made are consistent only for large \(\mathop {\overline \Sigma }\nolimits_{{\text{app}}} \gg 1\) since then the relative change of \(F(x)\) is small when \(n \to n + 1\). With these approximations the Euler–Maclaurin formula Eq. (A.1) reduces to
(c.f. with Eq. (59.10), p. 173 [75]). Using Eq. (A.4) the summation in Eqs. (46) and (123) can be performed easily. The result is
and from where it follows
in the case \({\text{of}}\) Eqs. (46) and (123), and
in the case of Eq. (157). In the above expressions it is used that \({{n}_{{{\text{max}}}}} \approx \sqrt N \).
Solution of Eq. (46)
Let us introduce the notation
With the help of Eq. (A.6) and the definition of \({{\bar {\sigma }}_{0}}\) (see Eq. (71)), Eq. (46) may be presented (up to the used approximations) in the form:
Eq. (A.9) can be solved in terms of the Lambert function \({\mathbf{W}}(x)\). A review of its mathematical properties and physical applications can be found in [76–80] and refs. therein. Recall that by definition
The Lambert function can take two possible real values for \( - \tfrac{1}{e} \leqslant x \leqslant 0\). Values satisfying \({\mathbf{W}}(x) \geqslant - 1\) belong to the principal branch denoted as \({{{\mathbf{W}}}_{0}}(x)\), while values satisfying \({\mathbf{W}}(x) \leqslant - 1\) belong to the \({{{\mathbf{W}}}_{1}}(x)\) branch. The two branches meet at the branch point for \(x = - \tfrac{1}{e}\), where \({{{\mathbf{W}}}_{0}}\left( { - \tfrac{1}{e}} \right) = {{{\mathbf{W}}}_{{ - 1}}}\left( { - \tfrac{1}{e}} \right)\). All values of \({\mathbf{W}}\) for \(x \geqslant 0\) belong to the principal branch \({{{\mathbf{W}}}_{0}}(x)\).
The solution of Eq. (A.9) now reads
or finally
In the interval \( - {{e}^{{ - 1}}} \leqslant - {{e}^{{ - \tfrac{\Delta }{\gamma }}}} < 0\) the equation has two solutions given by \({{{\mathbf{W}}}_{0}}\) and \({{{\mathbf{W}}}_{{ - 1}}}\), respectively.
For large \(x\), the function \({\mathbf{W}}(x)\) is approximated by
For small \(x\), the Taylor series around \(x = 0\) is given by
The first few terms of the series expansion of \({\mathbf{W}}(x)\) near the branching point are
where \(p = \pm \sqrt {2(e.x + 1)} \) for \({\mathbf{W}}{{(x)}_{{0,1}}}.\)
Thus, using Eq. (A.14) for \(x = {{e}^{{ - \tfrac{\Delta }{\gamma }}}} \ll 1\), one gets Eq. (47):
Using the expansion near the branching point of the Lambert function, i.e. \(x = {{e}^{{ - \tfrac{\Delta }{\gamma }}}} \approx {{e}^{{ - 1}}}\), one obtains:
Solution of Eq. (123)
For \(\mathop {\overline \Sigma }\nolimits_{{\text{app}}} \gg 1\), Eq. (123) can be treated in the same way. Let us introduce the notation
Using Eq. (A.6) the self-consistent Eq. (123) may be presented (up to the used approximations) in the form:
In terms of the Lambert \({\mathbf{W}}(x)\) function the solution reads:
Thus, if \(\tfrac{1}{{\overline C }} - \tfrac{1}{N} < 0\) there will be two solutions or none (or only one solution if the argument of W is exactly \( - \tfrac{1}{e}\)). If \(\tfrac{1}{{\overline C }} - \tfrac{1}{N} > 0\) there will be one solution.
With the help of bout expansions the Lambert \({\mathbf{W}}(x)\) function (A.13) and (A.14) one easily obtains Eqs. (127) and (128).
Not that if
combining Eqs. (A.20) and (A.9), the more general relation takes place
From the above result, if \({{K}_{s}} \to \infty \), immediately follows Eq. (144) where use has been made of definitions (125) and (A.10).
APPENDIX B
THE GRIFFITS–FISHER LEMMA
There is a mathematical statement known as Griffits–Fisher lemma [81, 82], which asserts that if a sequence of convex function converges pointwise to a limit function, then the sequence of its derivatives converges to the derivative of the limit function at the points of its continuous differentiabilty. More precisely, if all functions \({\text{\{ }}{{f}_{n}}(x){\text{\} }}\) and the limit function \({{f}_{\infty }}(x)\) are differentiable at a point \({{x}_{0}} \in I \subset R\), then
More general result due to Fisher consider the case of non-differentiable functions with left and right derivatives at any point \(x \in I\). The latter is relevant if the systems undergo thermodynamic phase transitions with spontaneously symmetry breaking. These statements are useful in proving the asymptotic closeness of certain average values in the model and approximating system, see e.g. [39]. In our case, we consider both Hamiltonian \(H\), Eq. (110), and \({{H}_{{{\text{app}}}}}\) (Eq. (111)), and introduce the following auxiliary Hamiltonians
and
where \(h\) is an auxiliary real parameter which at the end of the calculations will sent to zero. Further, we obtain that
and
In the limit when the analog of the correlator Eq. (157) in the r.h.s. of the Bogoliubov inequalities with Hamiltonians (B.2) and (B.3) tends to zero as a function of its parameters:
Since \(F[{{\mathcal{H}}_{{{\text{app}}}}}(h)]\) and \(F[H(h)]\) are convex differentiable functions of \(h\) from the lemma follows that,
For the above proof to be correct the definition of the thermodynamic (or other) limit should be scrutinized.
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Tonchev, N.S. Statistical Mechanics of Thermal Fluctuations of Nearly Spherical Membranes: the Influence of Bending and Stretching Elasticities. Phys. Part. Nuclei 52, 290–314 (2021). https://doi.org/10.1134/S1063779621020064
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DOI: https://doi.org/10.1134/S1063779621020064