J. Chem. Phys. 152, 244121 (2020); https://doi.org/10.1063/5.0009734 152, 244121

© 2020 Author(s).

Elastic properties of self-assembled bilayermembranes: Analytic expressions viaasymptotic expansionCite as: J. Chem. Phys. 152, 244121 (2020); https://doi.org/10.1063/5.0009734Submitted: 02 April 2020 . Accepted: 09 June 2020 . Published Online: 26 June 2020

Yongqiang Cai , Sirui Li , and An-Chang Shi

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Elastic properties of self-assembled bilayermembranes: Analytic expressions viaasymptotic expansion

Cite as: J. Chem. Phys. 152, 244121 (2020); doi: 10.1063/5.0009734Submitted: 2 April 2020 • Accepted: 9 June 2020 •Published Online: 26 June 2020

Yongqiang Cai,1,a) Sirui Li,2,3 and An-Chang Shi4

AFFILIATIONS1Department of Mathematics, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 119077, Singapore2School of Mathematics and Statistics, Guizhou University, Huaxi District, Guiyang 550025, People’s Republic of China3School of Mathematical Sciences, Zhejiang University, 886 Yuhang Road Xihu District, Hangzhou 310027,People’s Republic of China

4Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada

a)Author to whom correspondence should be addressed: caiyq.math@foxmail.com

ABSTRACTBilayer membranes self-assembled from amphiphilic molecules are ubiquitous in biological and soft matter systems. The elastic propertiesof bilayer membranes are essential in determining the shape and structure of bilayers. A novel method to calculate the elastic moduli of theself-assembled bilayers within the framework of the self-consistent field theory is developed based on an asymptotic expansion of the orderparameters in terms of the bilayer curvature. In particular, the asymptotic expansion method is used to derive analytic expressions of theelastic moduli, which allows us to design more efficient numerical schemes. The efficiency of the proposed method is illustrated by a modelsystem composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents.

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I. INTRODUCTION

The self-assembly of amphiphilic molecules to form cellu-lar membranes is a fundamental process in biological systems.At the same time, self-assembled bilayer membranes and poly-mersomes1,2 from amphiphilic macromolecules are abundant insoft matter systems. In particular, the self-assembled bilayers fromamphiphilic polymers such as block copolymers could act as mimet-ics to biological membranes. Because there are virtually no lim-its to the combinations of monomers, polymeric hybrid materialsshow great promise for applications in biomedicine and biotech-nology.3 Phenomenologically, a membrane could be modeled asa two-dimensional surface S, whose mechanical properties couldbe used to understand the formation and stability of membranemorphologies. When the curvature of the membrane is small, itsdeformation energy could be described by Helfrich’s linear elasticitytheory.4,5 However, when the curvature is large, higher-ordercontributions to the membrane’s free energy become significant.

Specifically, the elastic energy includes fourth-order contributionsof a closed membrane and is given by

F = ∫S[γ + 2κM(M − c0)

2 + κGG + κ1M4 + κ2M2G + κ3G2]dA, (1)

where M = (c + c′)/2 and G = cc′ are the local mean and Gaussiancurvatures of a deformed bilayer, respectively (c and c′ are the twoprincipal curvatures). The linear elastic energy of a bilayer is spec-ified by the elastic constants, γ, c0, κM , and κG, corresponding tothe surface tension, spontaneous curvature, bending modulus, andGaussian modulus, respectively. In addition, κ1, κ2, and κ3 are thefourth-order curvature moduli. It should be noted that the sponta-neous curvature c0 is zero and the third-order terms are not includedin Eq. (1) due to the symmetry of the bilayers. Usually, the expansionof free energy is truncated to second-order in curvatures, which isthe so-called Helfrich model.4,5 In contrast, the fourth-order termsare less considered since they are relevant to each other, and in turn,it is more difficult to determine their coefficients, κ1, κ2,and κ3.

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In the last few decades, a number of experimental techniques,6,7

simulation methods,8–11 and theoretical methods12–14 have beendeveloped to obtain the elastic constants of bilayer membranes. The-oretically, the elastic constants can be obtained by studying bilay-ers in different geometries (e.g., planes, cylinders, and spheres) andthen comparing the free energy of membranes with the correspond-ing expressions derived from elasticity theories.13 The elastic con-stants could then be extracted by fitting the theoretical expressionsto the computed free energy curves. This approach can be imple-mented quite naturally in theoretical studies. However, this numeri-cal approach requires an accurate computation of the free energy ofcurved membranes.

Among the different theoretical frameworks developed foramphiphilic molecules, the self-consistent field theory (SCFT) pro-vides a versatile platform for the study of the self-assembled bilayermembranes. Several studies using the SCFT have been carried outto study the elastic properties of bilayers,13,15–17 where the polymersare assumed to be flexible, and the Gaussian polymer model is usedto describe the blocks. Using similar theoretical approaches, Ref. 18compared the elastic properties between the triblock membranesand the diblock membranes, and Ref. 19 extended this approachto the semiflexible chain model described by the wormlike-chainto study the elastic properties of semiflexible bilayers. These pre-vious studies firmly established that the SCFT provides a flexibleand accurate framework for the study of inhomogeneous poly-meric systems, including different micelle structures20,21 and bilayermembranes.17–19

In previous studies,17–19 the excess free energy of the bilayermembrane in different geometries, such as infinite planar, cylindri-cal and spherical bilayers with different curvatures, was computed.The excess free energy of the three geometries, which is denotedby F0, FC, and FS for the planar, cylindrical, and spherical bilay-ers, respectively, was then used to extract the elastic constants ofthe membranes. Specifically, the expressions of the free energy ofthe membranes with different shapes are obtained from the Hel-frich model [Eq. (1)]. In particular, the free energies, FC and FS,are polynomials with respect to the principal curvature c, where thecoefficients are combinations of the elastic constants. As a result, theelastic moduli, such as κM and κG, could be obtained by quadraticpolynomial fitting FC and FS as functions of c. However, there aretwo drawbacks of this strategy: (i) the fitting range of c ∈ (0, cmax)should be carefully chosen since higher-order energies are not neg-ligible when c is large and (ii) accurate fitting requires the compu-tation of the bilayer energy as a function of curvature, which, inturn, requires solving the self-consistent field (SCF) equations manytimes. In order to overcome these drawbacks, new methods to obtainelastic moduli are desirable.

In this paper, we propose an asymptotic expansion methodwithin the SCFT framework to calculate the elastic moduli of theself-assembled bilayer membranes. It should be noted that general-ization of the expansion method to other free energy functionals isstraightforward. Similar expansions had been derived for free energyfunctionals based on the packing constraints,22 Ginzburg–Landaumodel,23 and van der Waals theory.24 Specifically, we treat the cur-vature of cylindrical and spherical bilayers as a small parameter andcarry out an asymptotic expansion of the free energy in terms of thecurvature. Analytical expressions of the free energy at each orderare obtained, so they could be computed separately. In particular,

computation of the elastic moduli only requires the solution of afew SCF equations. Besides, more accurate numerical schemes couldbe developed to solve the SCF equations. Numerically, the elasticmoduli obtained by using the asymptotic expansion method areconsistent with those obtained by the fitting method. From the the-oretical viewpoint, the asymptotic expansion results directly relatethe elastic moduli to the equilibrium distribution of the amphiphilicmolecules, thus providing an understanding of the dependence ofelastic properties on molecular details.

The remainder of this paper is organized as follows: Section IIdescribes the SCFT model of bilayer membranes and the geometricconstraints used in this study. Section III gives the asymptotic expan-sion method where the corresponding modified diffusion equations(MDEs) and SCF equations are given. Section IV gives the numericalmethod for solving the whole SCFT model. Our results on the elas-tic properties of the membranes are presented in Sec. V, includingthe influence of microscopic parameters of the flexible–flexible sys-tem on the elastic properties. Finally, Sec. VI concludes with a briefsummary.

II. THEORETICAL MODEL AND GEOMETRYCONSTRAINTSA. Molecular model

The molecular model used in this study is a binary mixtureof AB-diblock copolymers and A-homopolymers. This is a genericmodel in which the amphiphilic molecules are modeled by theAB-diblock copolymers, whereas the amphiphilic solvent moleculesare modeled by the A-homopolymers. Furthermore, the polymericspecies are modeled as flexible Gaussian chains. We assume thatthe A/AB blend is incompressible, and both monomers (A and B)have the same monomeric density ρ0 (or the hardcore volume permonomer is ρ−1

0 ). The diblock copolymers and homopolymers arecharacterized by their degrees of polymerization, N and Nh ≡ f hN,respectively. The volume fraction of the A- and B-blocks in thecopolymers is denoted by f A and f B = 1 − f A, respectively. Theradius of gyration of A and B blocks is denoted by RA

g =√

fANl2a/6

and RBg =

√

fBNl2b/6, where la and lb are the statistical segment

length of A and B monomers, respectively. We set Rg ∶=√

Nl2a/6as the unit spatial length in nondimensionalization and definea = la/la = 1 as a constant and b = lb/la as the geometrical parame-ter describing the conformational asymmetry of the A and B blocks.The interaction between the A and B monomers is described by aFlory–Huggins parameter25 χ. Finally, the chemical potential of thecopolymers μc, or the corresponding activity zc = exp(μc), is used tocontrol the average concentration of the diblock copolymers in theblends.

Within the SCFT framework formulated in the grand canonicalensemble,17,20,26 the free energy of the binary mixture is given by

NF

kBTρ0= ∫ dr[χNϕA(r)ϕB(r) − ωA(r)ϕA(r) − ωB(r)ϕB(r)

− ξ(r)(ϕA(r) + ϕB(r) − 1) + ψGε(r − r1)(ϕA(r) − ϕB(r))]− zcQc −Qh, (2)

J. Chem. Phys. 152, 244121 (2020); doi: 10.1063/5.0009734 152, 244121-2

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where ϕα(r) and ωα(r) are the local concentration and the mean fieldof the α-type monomers (α = A, B), respectively. The local pressureξ(r) is a Lagrange multiplier introduced to enforce incompressibilityof the system. A second Lagrange multiplier, ψ, is used to ensure theconstraints for stabilizing the bilayer in different geometries, wherea sharp Gaussian function, Gε(r − r1), is used to ensure that theψ field only operates near the interface at a prescribed position r1.The last two terms in Eq. (2) are the contributions from the single-chain partition functions of the two polymers, Qc and Qh. Note that,in previous studies,17,27 a constraint term, δ(r − r1)(ϕA(r) − ϕB(r)),has been included in the free energy functional to stabilize a bilayerin different geometries. In this study, we replace the delta func-tion δ(r) by the Gaussian function, resulting in the constraint term,Gε(r − r1)(ϕA(r) − ϕB(r)), in Eq. (2). This modification could makenumerical schedules more robust and have negligible effects on thenumerical results.

The fundamental quantities to be calculated in the SCFT are theprobability distribution functions (or the propagators) of the poly-mers, i.e.,qhA(r, s) for the A-homopolymers and q±A(r, s), q±B(r, s) forthe AB-diblock copolymers. These propagators are obtained as solu-tions of the modified diffusion equations (MDEs)20 in the presenceof the mean fields (ωA and ωB),

∂

∂sqhA(r, s) = (a

2∇

2r − ωA(r))qhA(r, s), s ∈ (0, fh), (3)

∂

∂sq±A(r, s) = (a

2∇

2r − ωA(r))q±A(r, s), s ∈ (0, fA), (4)

∂

∂sq±B(r, s) = (b

2∇

2r − ωB(r))q±B(r, s), s ∈ (0, fB) (5)

with the initial value conditions,

qhA(r, 0) = q−A(r, 0) = q−B(r, 0) = 1,

q+A(r, 0) = q−B(r, fB),

q+B(r, 0) = q−A(r, fA).

In terms of the chain propagators, the single-chain partition func-tions are given by

Qc = ∫ drq+A(r, fA), (6)

Qh = ∫ drqhA(r, fh). (7)

Furthermore, the local concentrations of the A and B monomers areobtained from the propagators as

ϕA(r) = ϕhA + ϕcA = ∫fh

0dsqhA(r, s)q

hA(r, fh − s)

+ zc ∫fA

0dsq−A(r, s)q

+A(r, fA − s), (8)

ϕB(r) = zc ∫fB

0dsq−B(r, s)q

+B(r, fB − s). (9)

The rest of the SCF equations concerning the mean fields to the localconcentrations are

FIG. 1. Profile of the bilayer structure. The computation domain is truncated tomake sure that the order near the boundary is close to the bulk phase.

ωA(r) = χNϕB(r) − ξ(r) + ψGε(r − r1), (10)

ωB(r) = χNϕA(r) − ξ(r) − ψGε(r − r1), (11)

ϕA(r) + ϕB(r) = 1, (12)

∫ drGε(r − r1)(ϕA(r) − ϕB(r)) = 0. (13)

The set of Eqs. (2)–(13) form the self-consistent theory of the model.It should be noted that the SCF equations are highly nonlinear andnonlocal; thus, they could have many solutions. In this study, wewill focus on the solutions corresponding to a bilayer whose profileis shown in Fig. 1. Furthermore, we are interested in the free energyof a system containing a bilayer membrane compared to that of thehomogeneous bulk phase Fbulk, which can be computed analyticallyusing the constant solution of the SCF equations.17,26 The free energydifference (F −Fbulk) is proportional to the area of the membraneA; therefore, we can define an excess free energy density by

Fex =N(F −Fbulk)

kBTρ0A. (14)

The corresponding free energy density for the bulk phase is given by

fbulk ∶=NFbulk

kBTρ0V=

1fh(ln(

1 − ϕbulkfh

) + ϕbulk − 1) + χNf 2Bϕ

2bulk − ϕbulk,

(15)where the bulk copolymer concentration ϕbulk is determined by thefollowing equation:

μc = lnϕbulk −1fh

ln(1 − ϕbulk

fh) + χNfB(1 − 2fBϕbulk). (16)

Equation (16) for ϕbulk has at least one solution. In cases where it hasmore than one solution, the one with the lowest free energy densityis chosen as the bulk phase.

B. Geometric constraints and the polynomialfitting method

In order to extract the elastic constants, such as the bendingmodulus and Gaussian modulus, of the self-assembled bilayers, one

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could calculate the excess free energy of a bilayer membrane in threegeometries:13,17 an infinite planar bilayer, a cylindrical bilayer witha radius r (or a mean curvature M = c/2 = 1/2r), which is extendedto infinity in the cylindrical direction, and a spherical bilayer with aradius r (or a mean curvature M = c = 1/r). In the cylindrical andspherical geometries, the constraint ψGε(r − r1) in Eq. (2) is appliedto the outer monolayer where r1 is set to control the curvature of theself-assembled bilayer. One advantage of those geometries is that thesimulation of bilayers with these geometries can be reduced to one-dimensional problems in their corresponding coordinate systems.Specifically, the MDEs (3)–(5) in one-dimensional planar, cylindri-cal, and spherical coordinate systems could be written in a unifiedform,

∂

∂sq(r, s) = a2

(∂2

∂r2 +nr∂

∂r)q(r, s) − ω(r)q(r, s), (17)

where n = 0, 1, and 2 for planar, cylindrical, and spherical coordinatesystems, respectively.

The next step of the calculation is to compare the excess freeenergies, F0, FC(c), and FS(c), of the bilayers with the expressions ofthe Helfrich model in the corresponding geometries,

F0= γ + 2κMc2

0,

FC(c) = F0

− 2κMc0c +κM2c2 + BCc4,

FS(c) = F0

− 4κMc0c + (2κM + κG)c2 + BSc4,

where the parameters BC ∶= κ1/16 and BS ∶= κ1 + κ2 + κ3 are combina-tions of the fourth-order moduli. In previous studies,17–19 the bend-ing modulus κM , Gaussian modulus κG, and higher order moduli BCand BS are obtained by a polynomial fitting method. In detail, a seriesof curvatures ci are chosen and then the SCF equations are solvedfor each curvature ci to get the energies FC(ci) and FS(ci). Afterthat, the polynomial fitting method is used to obtain the polynomialcoefficients, which, in turn, gives the elastic moduli.

While the polynomial fitting method is fine for determiningthe second-order moduli, it becomes inaccurate when trying toextract higher-order moduli. Specifically, during the polynomial fit-ting process, two factors will affect the results. The first factor isthe size of the set ci. Since the energies, FC(ci) and FS(ci), arenumerically calculated, which must contain inaccuracies, one needsto choose a large number of ci (for example, 50 different ci) tomake sure that the fitted higher-order moduli are accurate. This, inturn, requires that one must solve the SCF equations many times(for example, 2 × 50 times). The second factor is in the range ofci ⊂ (0, cmax). The numerical errors prevent one from choosingsmall cmax since that is where the energies are very close. However,if cmax is large, the higher-order terms are not negligible. There-fore, it is not straightforward to choose cmax and the number ofci to make the fitting results reasonable. Those drawbacks of thepolynomial fitting method motivated us to develop new schemesto obtain the elastic moduli, especially the higher-order moduli. InSec. III, we propose a new scheme, based on the asymptotic expan-sion theory, which not only overcomes the drawbacks of the fit-ting method but also gives the polynomial coefficients directly andefficiently.

III. ASYMPTOTIC EXPANSION METHODSince the curvature of cylindrical and spherical bilayers is

assumed to be small, we can regard it as a small parameter for theapplication of the asymptotic expansion theory.28,29 In this section,we show that analytic expressions of the MDE, SCF equations, andfree energy at each order could be obtained separately.

A. MDE at each orderWhen the curvature c is small, i.e., r is large, we can perform

asymptotic expansions of the MDE with respect to the bilayer cur-vature. The expansion starts from the coordinate transformation,r = x − 1

c . With this transformation, we can formally write downthe functions ω(r), q(r, s), and 1

r as expansions in terms of thecurvature c,

ω(r) = ω0(x) + cω1(x) + c2ω2(x) +⋯, v

q(r, s) = q0(x, s) + cq1(x, s) + c2q2(x, s) +⋯,

1r=

1x + 1/c

=c

1 + cx= c − xc2 + x2c3 +⋯.

Substitute these expressions in the MDE [Eq. (17)], we obtain

(a2∇

2− ω(r))q(r, s)

= (a2∂rr +na2

r∂r − ω0(x) − cω1(x) − c2ω2(x) +⋯)

× (q0(x, s) + cq1(x, s) + c2q2(x, s) +⋯)

= (a2∂2xx − ω0)q0 + [(a2∂2

xx − ω0)q1 + (na2∂x − ω1)q0]c

+ [(a2∂2xx − ω0)q2 + (na2∂x − ω1)q1

+(−na2x∂x − ω2)q0]c2 +⋯.

Comparing the terms with different order of c, we can write downthe MDE at each order as

∂sqi(x, s) =i

∑

j=0Ljqi−j(x, s), i = 0, 1, 2, . . . , (18)

where Lj are derivative operators defined by

L0 = a2∂2xx − ω0, Lj = na2

(−x)j−1∂x − ωj, j ≥ 1. (19)

In particular, the MDEs up to the second-order in c are given by

∂sq0 = (a2∂2xx − ω0)q0,

∂sq1 = (a2∂2xx − ω0)q1 + (na2∂x − ω1)q0,

∂sq2 = (a2∂2xx − ω0)q2 + (na2∂x − ω1)q1 + (−na2x∂x − ω2)q0.

B. SCF equations at each orderOperating the MDE on the propagators and employing the

following initial conditions:

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qhAi(x, 0) = q−Ai(x, 0) = q−Bi(x, 0) = δ0i,

q+Ai(x, 0) = q−Bi(x, fB),

q+Bi(x, 0) = q−Ai(x, fA), i = 0, 1, 2, . . . ,

we can get the expansion for the propagators qhA, q±A , and q±B ,

qhA(x, s) =∑i≥0

ciqhAi(x, s),

q±A(x, s) =∑i≥0

ciq±Ai(x, s),

q±B(x, s) =∑i≥0

ciq±Bi(x, s).

Here, δij is the Kronecker delta function, i.e., δ0i = 1 for i = 0 andδ0i = 0 for i ≥ 1. These expressions can then be used to calculate theorder parameters expansion at each order,

ϕhA(x) =∑i≥0

ciϕhAi(x)

=∑

i≥0ci ∫

fh

0ds

i

∑

j=0qhAj(x, s)qhA,i−j(x, fh − s), (20)

ϕcA(x) =∑i≥0

ciϕcAi(x)

=∑

i≥0cizc ∫

fA

0ds

i

∑

j=0q−Aj(x, s)q+

A,i−j(x, fA − s), (21)

ϕB(x) =∑i≥0

ciϕBi(x)

=∑

i≥0cizc ∫

fB

0ds

i

∑

j=0q−Bj(x, s)q+

B,i−j(x, fB − s). (22)

In addition, ϕAi(x) = ϕhAi(x) + ϕcAi(x).The expansion of the SCF equations can be carried out sim-

ilarly. In the cylindrical and spherical geometries, the constraintψGε(r − r1) is applied to the outer monolayer only. It allows thebilayer to optimize its thickness or position h and indirectly set thecurvature radius of the membrane. The constraint position r1 can becharacterized by two quantities: one is the curvature radius 1

c andanother is the relative constraint position h ∶= r1 −

1c , which is a

function of c. Let h = ∑i≥0cihi, then using the Taylor expansion, wehave

Gε(r − r1) = Gε(x − h) =∑k≥0

(h0 − h)k

k!G(k)ε (x − h0)

=∑

k≥0

(−∑i≥1 cihi)k

k!G(k)ε (x − h0)

=:∑i≥0

ciGε,i(x − h0),

where the leading terms are

Gε,0(x − h0) = Gε(x − h0),

Gε,1(x − h0) = −h1G(1)ε (x − h0),

Gε,2(x − h0) = −h2G(1)ε (x − h0) +h2

1

2G(2)ε (x − h0),

Gε,3(x − h0) = −h3G(1)ε (x − h0) + h1h2G(2)ε (x − h0)

−h3

1

6G(3)ε (x − h0),

Gε,4(x − h0) = −h4G(1)ε (x − h0) +2h1h3 + h2

2

2G(2)ε (x − h0)

−3h2

1h2

6G(3)ε (x − h0) +

h41

24G(4)ε (x − h0).

Considering the expansion of ξ(r) = ∑i≥0ciξi(x), ψ = ∑i≥0ciψi andthen comparing the SCF equations (10)–(13) at each order, we getthe SCF equations at each order,

ωAi(x) = χNϕBi(x) − ξi(x) +i

∑

j=0ψjGε,i−j(x − h0), (23)

ωBi(x) = χNϕAi(x) − ξi(x) −i

∑

j=0ψjGε,i−j(x − h0), (24)

ϕAi(x) + ϕBi(x) = δ0i, (25)

∫

∞

−∞

i

∑

j=0Gε,j(x−h0)

i−j∑

k=0pk(x)(ϕA,i−j−k(x)−ϕB,i−j−k(x))dx = 0. (26)

Finally, the SCF equations are closed after we specify a crite-rion for the measurement of the bilayer curvature c. For a curvedbilayer with a finite thickness, the definition of the interface positioninvolves a certain degree of arbitrariness. In previous works,17–19 thebilayer interface, r, is defined to be the midpoint of the two positionswhere A- and B-segment concentrations are equal, i.e., ϕA = ϕB. Inthis paper, for mathematical simplicity, we define the curvature ofcylindrical and spherical bilayers as c, which satisfies the followingequation:

∫ (∣r∣ −1c)ρ(r)dr = ∫

∞

0rn(r −

1c)ρ(r)dr = 0, (27)

where ρ(r) is regarded as a probability density function whose expec-tation is 1/c. The choice of ρ(r) is not unique. In this paper, we choseρ(r) ∝ ϕB(r) − f Bϕbulk. It is worth noting that the definition of theinterface affects the energy curve FC(c) and FS(c). We will discussthis effect in Sec. V A. Expanding ρ(r) as ρ(r) =∑i≥0ciρi(x), we havethe following constraints:

Iψ,i ∶=i

∑

j=0∫

∞

−∞xpj(x)ρi−j(x)dx = 0, i = 0, 1, 2, . . . , (28)

where p0(x) = 1, p1(x) = nx, p2(x) = n(n−1)2 x2, pi>2(x) = 0.

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In summary, the self-consistent field equations at ith order con-tains (23)–(26) and (28). It is worth noting that there is no need toapply a constraint to stabilize the planar bilayers; hence, we haveψ0 = 0. Consequently, h0 can be directly determined by Eq. (26), i.e.,∫∞−∞ Gε,0(x − h0)(ϕA,0(x) − ϕB,0(x))dx = 0. Furthermore, the value

of hi for i ≥ 1 can be directly determined by (26) once the SCF equa-tions at jth order, j < i, are solved. The numerical method to solvethe SCF equations will be introduced in Sec. IV.

C. Asymptotic expansion of the free energyOnce we solved the SCF equations at each order, the free energy

of a bilayer can be calculated directly. Denote the semi-local excessenergy density by f (r),

f (r) = fbilayer(r) − fbulk =∑i≥0

cifi(x),

where f i(x) can be expressed as

fi(x) =i

∑

k=0[χNϕA,k(x)ϕA,i−k(x)

− ωA,k(x)ϕA,i−k(x) − ωB,k(x)ϕB,i−k(x)

− ξk(x)(ϕA,i−k(x) + ϕB,i−k(x) − δ0,i−k)]

− zcq+A,i(x, fA) − qhA,i(x, fh) − δ0ifbulk,

then the excess energy density Fex becomes

Fex =1A ∫V

f (r)dr = cn ∫∞

0rnf (r)dr

=∫

∞

−1/c(1 + cx)n(f0(x) + cf1(x) + c2f2(x) +⋯)dx

=∑

i≥0ci ∫

∞

−∞dx

i

∑

j=0pj(x)fi−j(x) =:∑

i≥0ciFi.

Here, the integration domain is changed from (−1/c,∞) to (−∞,∞)since c is small, the energy density decays very fast, and this modi-fication only introduces negligible errors. In particular, we have thefirst three terms from the free energy expansion,

F0 = ∫

∞

−∞f0(x)dx,

F1 = ∫

∞

−∞(f1(x) + nxf0(x))dx,

F2 = ∫

∞

−∞(f2(x) + nxf1(x) +

n(n − 1)2

x2f0(x))dx.

Denote the free energies of cylindrical and spherical bilayers(corresponding to n = 1 and n = 2, respectively) at ith order as FC

i andFSi , respectively, then the elastic moduli can be directly calculated,

κM = 2FC2 , κG = FS

2 − 4FC2 , BC = FC

4 , BS = FS4 , (29)

c0 = −FC1 /(2κM) = −F

S1/(4κM), γ = F0 − 2κMc2

0. (30)

It is worth noting that F0= FC

0 = FS0 since they all correspond to the

same SCF system related to the planar bilayers and 2FC1 = FS

1 sincethe corresponding SCF systems only have a difference on the valuen. However, there are no simple relations between the free ener-gies at higher-orders, Fi, i ≥ 2, because of nonlinearity. In addition,since we are considering bilayers consisting of two identical leaflets,the odd-order free energies are expected to be zeros, implying thespontaneous curvature c0 = 0, due to the symmetry of the bilay-ers. Although the introduced constraint term Gε(r − r1) will breakthis symmetry, its influence is limited, i.e., the odd-order energies Fiare small and negligible (less than 10−5 for the numerical results inSec. V).

IV. NUMERICAL METHODSchematics of the numerical procedure to solve the SCF equa-

tions are shown in Fig. 2. For a given set of control parameters, theSCF equations are solved sequentially to obtain the structure andfree energy of the bilayers at each order. Numerically, the SCF equa-tions can be solved by iteration (Sec. IV B) during which one mustcompute the propagators by solving the modified diffusion equa-tions (Sec. IV A) and store those propagators for more higher-ordercalculations.

Specifically, the computation domain is one-dimensional andlimited to x ∈ [−L, L], where the size L is determined by ensuringthat the concentration profiles at the boundaries approach to thevalues of the bulk phase. We determine the length L via the decaylength of ϕcA(x) as shown in Fig. 1, where L0 denotes the growthlength, L1 is the half-value length, and L2 is the decay length atwhich the profile approaches its bulk value. The computation boxsize is then L = L0 + L2, which is large enough such that L2 ≥ λL1,where λ = 4.5 is chosen for our numerical examples. Besides, theMDEs are supplemented with reflecting boundary conditions. Thenumber of spatial grid points is changed over 500–1000, and thenumber of time grid points is changed over 800–2000 under dif-ferent model parameters to make sure that the free energy is con-verged in the order of 10−6 and the fields are self-consistent withL2-norm error [the L2-norm of the difference between the left-handand right-hand sides of Eqs. (23)–(28)] less than 10−8. We noted thatthe calculated elastic moduli (up to four orders) are not sensitive tothe specific choices of the constraint width ε. The variances are lessthan 10−4 for ε changing from 0 Ω to 0.5 Ω, where Ω is the thick-ness of the bilayers. Therefore, in most of the calculations, we fixε = 0.25 Ω.

A. Solve the MDEsThe MDEs [Eq. (18)] to be solved for the asymptotic expansion

method are in the following generic form:

∂

∂sq(x, s) = a2 ∂2

∂x2 q(x, s) − ω(x)q(x, s) + g(x, s), (31)

q(x, 0) = q0(x),

∂

∂xq(±L, s) = 0, (32)

where g(x, s) is a given source term. Since the fields and order-parameters change quickly near the bilayer interface and slowly near

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FIG. 2. Flowchart of the asymptoticexpansion method.

the domain boundary, non-uniform grids for x are used to capturethe details of the interfaces. The employed non-uniform grids cor-respond to a given transformation x = t(ξ). With a proper change invariables, the MDE is changed but has the same form as Eq. (31),except with variable coefficients and the source term. In fact, lett(±L) = ±L, t′′(±L) = 0 and q(x, s) =

√

t′(ξ)u(ξ, s), we have

∂

∂su(ξ, s) = a2

(ξ)∂2

∂ξ2 u(ξ, s) − ω(ξ)u(ξ, s) + g(ξ, s), (33)

u(ξ, 0) = u0(ξ),

∂

∂ξu(±L, s) = 0, (34)

where the new variables are

a2(ξ) =

a2

t′2(ξ),

ω(ξ) = ω(t(ξ)) −a2

2t′4(ξ)(t′(ξ)t′′′(ξ) −

32t′′2(ξ)),

g(ξ, s) =

¿

ÁÁÀ

t(ξ)t′(ξ)

g(t(ξ), s).

A valid transformation t(ξ) is

t(ξ) = ξ + kL(1

2πsin(2πξ/L) −

4π

cos(πξ0/L) sin(πξ/L)),

where ξ0 corresponds to the position of the bilayer interface, and k isa small positive number, such that k ≤ 1/(2 + 4 cos2(πξ0/L)), which

controls the non-uniform degree of the grid. A larger k results in alarger non-uniform degree, which means that more nodes are nearthe interface.

Using the Strang split method,30 one can obtain the followingtwo-order semi-discrete scheme for the differential equation (33):

um+ 12 (ξ) = exp(−ω(ξ)Δs2 )[u

m(ξ) + Δs

8 (3g(ξ, sm) + g(ξ, sm+1))],

u∗(ξ) = exp(−a2(ξ)Δs∂2

ξ )um+1/2

(ξ),

um+1(ξ) = exp(−ω(ξ)Δs2 )u

∗(ξ) + Δs

8 (g(ξ, sm) + 3g(ξ, sm+1)),

where um(ξ) is the numerical approximation of u(ξ, sm), sm= mΔs, and u∗(ξ) can be solved by the well-known Crank–Nicolsonscheme.31

It should be noted that the differential equations (31) and (33)could also be solved by using higher-order numerical schemes suchas the compact finite-difference schemes.32 In particular, if the par-tial derivative operators about x or ξ are treated by fourth-ordercompact finite-difference schemes, and the s dependence is dealtwith by fourth-order Runge–Kutta methods,33 then one could obtainfourth-order schemes finally.

B. Solve the SCF equationsGenerally, the SCFT equations could be solved by using the

Picard-type iteration,20,34 where the fields, ωA ,i and ωB ,i, are updatedaccording to Eqs. (23) and (24), provided the Lagrange fields ξiand ψi are given. In detail, with old fields ωold

A,i(x),ωoldB,i (x) [and its

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corresponding concentrations ϕoldA,i(x),ϕoldB,i (x)] in hand, one canupdate the fields by the iteration,

ωnewA,i (x) = (1 − α)ω

oldA,i(x)

+ α⎡⎢⎢⎢⎢⎣

χNϕoldBi (x) − ξoldi (x) +

i

∑

j=0ψoldj Gε,i−j(x − h0)

⎤⎥⎥⎥⎥⎦

,

ωnewB,i (x) = (1 − α)ω

oldB,i (x)

+ α⎡⎢⎢⎢⎢⎣

χNϕoldAi (x) − ξoldi (x) −

i

∑

j=0ψoldj Gε,i−j(x − h0)

⎤⎥⎥⎥⎥⎦

,

where α is an update ratio that was chosen as α = 0.01 in our calcula-tions. The Lagrange fields ξoldi (x) and ψold

i are given by, omitted thesuperscript,

ξi(x) = 12(χN(ϕAi(x) + ϕBi(x)) − (ωAi(x) + ωBi(x)))

− γχN(ϕAi(x) + ϕBi(x) − δ0i),

ψi =1

2 ∫ xGε(x − h0)dx

×

⎡⎢⎢⎢⎢⎣

∫ x(ωAi(x) − ωBi(x) − χN(ϕBi(x) − ϕAi(x))

− 2i−1

∑

j=0ψjGε,i−j(x − h0))dx − βχNIψ,i

⎤⎥⎥⎥⎥⎦

,

where γ, β are numerical parameters, which are chosen as γ = 0.6,β = 0.4. Note that the term Iψ i appeared in the iteration since theconstraint (28) corresponds to the Lagrange multiplier ψi.

Although the Picard-type iteration is robust, it convergesslowly. Accelerated methods such as the Anderson iteration35 for thegeneral fixed point problem could be used to solve the SCF equa-tions,36 resulting in the Anderson mixing technique. We start theAnderson iteration after the fields have been updated by Picard-type, with iterations reaching the point where the L2-error of the SCFequations is less than 0.1.

V. RESULTS AND DISCUSSIONFor simplicity, we focus on tensionless bilayer membranes

where the activity or the chemical potential of the copolymers, zc,is adjusted such that the excess free energy of a planar bilayer iszero, i.e., F0

= γ + 2κMc20 = 0, which is equivalent to γ = 0 since the

spontaneous curvature c0 is zero. Section V A presents numericalexamples illustrating the two advantages of the asymptotic expan-sion method: (i) the consistency and efficiency compared to thepolynomial fitting method and (ii) the analytic decompositions ofthe order-parameters. The analytic expressions of the asymptoticmethod permit us to access the elastic constants directly. Therefore,the proposed method provides an efficient method for the study ofthe dependence of the elastic properties on the molecular details,which is given in Sec. V B. Notably, in Sec. V C, we present a localapproximation result for the Gaussian modulus κG.

A. The asymptotic expansion method

As the first step of our study, we compare the polynomial fit-ting method with the asymptotic expansion method proposed in thispaper. Instead of directly comparing the elastic moduli, we comparethe free energy curves of the cylindrical and spherical bilayers sincethe results of the polynomial fitting method will be affected by themaximal curvature used. Figure 3(a) shows an example of the freeenergy of the self-assembled tensionless cylindrical and sphericalbilayers as functions of the curvature c. For the asymptotic expan-sion method, the second-order approximation F2c2 and the fourth-order approximation F2c2 + F4c4 of the free energies are given. Thesecond-order approximation is not enough when c is large (such asc = 0.25), where the fourth-order approximation is more close to thefree energy Fex. It is worth noting that the fourth-order contributionto FS(c) is negative, while the overall contribution is small since thereis an intrinsic limit due to the bilayer thickness on how high the cur-vature could be. In Fig. 3(b), we compare the fourth-order energyto the second-order energy for cylindrical and spherical bilayerswith curvature c = 0.15. It indicates that for cylindrical bilayers withthis mild curvature c, the fourth-order energy could be larger than5% of the second-order energy, and this ratio is increasing withχN increasing or f A decreasing. However, this ratio for sphericalbilayers is relatively small. The consistency between the free energyFex and the approximations given by the asymptotic expansionmethod implies the efficiency of the asymptotic expansion methodbecause one only needs to solve the SCF equations five times (threetimes memory) for the second-order free energies and additionalfour times (additional two times memory) for the fourth-order freeenergies.

Next, we illustrate another advantage of the asymptotic expan-sion method by decomposing the concentrations of cylindrical orspherical bilayers to each order. Figure 4 shows the profiles of theconcentrations of a spherical bilayer with curvature c ≈ 1.4 where theasymptotic expansion at each order is compared. Within the spher-ical bilayer, since the area of the inner interface is smaller than theouter one, the hydrophilic monomers (A-blocks) in the inner leaflethave to pack more closely and have a higher local concentration incomparison to the outer leaflet. This mechanism can be revealed bythe trend of ϕcA,1(x), which is almost an odd function and attainsits maximum at the left part. Furthermore, the numerical resultsindicate that the odd-order terms are almost odd functions and theeven-order terms are almost even functions. As a result, the odd-order energies such as F1 and F3 are close to zeros since terms inthose integral are almost odd functions. For the example given inFig. 4, both |F1| and |F3| are less than 10−6.

It is worth noting that, in this paper, we adopt Eq. (27) todetermine the bilayer interface, which is different from previousworks.17–19 Using the definition in Refs. 17–19, the curvature ofcylindrical and spherical bilayers will have different values, denotedby c. Our numerical results indicate that the difference c−c has orderc3 or c3, i.e., c − c ≈ σc3

≈ σc3, where α is a constant that can be esti-mated by remeasuring the interface. For example, for the sphericalbilayers in Fig. 3, the estimated constant is σ ≈ 0.84. In the asymptoticexpansion method, remeasuring the interface can be done by restor-ing ϕA(r) and ϕB(r) from Eqs. (20)–(22). As a result, the second- andfourth-order energies of tensionless bilayers corresponding to c areF2 = F2, F4 = F4−2σF2, which indicate that the second-order moduli

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FIG. 3. [(a) and (b)] Comparison of thefree energy of cylindrical and sphericalbilayers. Fex are calculated by the orig-inal constraint method, and F2, F4 arecalculated by the method in this paper. Cand S denote the cylindrical and spheri-cal bilayers, respectively. The molecularparameters are χN = 20, f A = 0.5, f h = 1,and b = 1.

are independent of the definition of interface, but the fourth-ordermoduli are not.

B. Influence of interactions and amphiphilicarchitecture on the elastic moduli

Used the polynomial fitting method, previous studies17 exam-ined the effect of χN and f A on the bending modulus κM and

Gaussian modulus κG for the cases where both the copolymers andthe homopolymers are assumed to have an equal chain length char-acterized by the same degree of polymerization, and both monomers(A and B) have the same statistical segment length, i.e., f h = 1,b = 1. Those effects are reproduced in Fig. 5(a) where not only thesecond-order moduli (κM and κG) but also the fourth-order mod-uli (BC and BS) are given as functions of the hydrophilic fraction f A.The main results are are as follows: (i) The bending modulus κM is

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FIG. 4. (a) The order parameters of a spherical bilayer with curvature c ≈ 0.14. (b)Decomposition of the order parameter ϕA. The molecular parameters are χN = 20,f A = 0.5, f h = 1, and b = 1.

not very sensitive to the amphiphilic architecture specified by f A, andκM exhibits a weak symmetry to f A = 0.5 (we will see later that thisweak symmetry could be attributed to the assumption of equal sta-tistical segment lengths of A and B). (ii) The Gaussian modulus κGis a monotonically decreasing function of f A, and its value changesfrom positive to negative at around f A = 0.41. (iii) The fourth-ordermodulus BC is positive, but BS is negative, and their magnitudes(or absolute value) are decreasing functions of f A. (iv) The interac-tion χN mainly affects the magnitude of the moduli, where strongerinteractions result in larger moduli, and this effect is almost linearto χN.

Figure 5(b) illustrates the influence of f h on the elastic modulias functions of f A. The results are focused on four different val-ues of f h ∈ 0.6, 0.8, 1, 2. It could be concluded that the moduliare less sensitive to the change in f h. When f h increases, the bend-ing modulus slightly decreases and the Gaussian modulus increases.This weak influence of f h is expected because the solubility of theamphiphilic molecules does not heavily depend on the length of the

solvent homopolymers unless f h is very small. In particular, shortersolvent homopolymers could help the amphiphilic molecules to bedissolved, resulting in a higher excess at the interface. The influenceof increasing f h on the fourth-order moduli is also small where theweak trend is that BC decreases and BS increases.

The effect of varying the geometrical asymmetry parameter bon the qualitative behavior of the elastic moduli is significant, asshown in Fig. 5(c) for different values of b ∈ 0.6, 0.8, 1, 1.2. Firstof all, when b deviates from 1, the weak symmetry of κM to f A = 0.5is broken. When b increases, the magnitude of κM and κG increasesand the f A corresponding to the maximum of κM decreases. On theother hand, the increase in b will increase the magnitude of thefourth-order moduli and change the monotonicity of the fourth-order moduli with respect to f A. For large b, both BC and |BS| aremonotonically decreasing with respect to f A. However, when b isrelatively small, the trend is similar to a unimodal function.

C. A local approximation for κGAn interesting observation is that the Gaussian modulus κG is

zero at f A ≈ 0.41, independent of the value of χN, as illustrated inFig. 5(a). We found that this phenomenon occurred in the cases ofother parameters, as shown in Fig. 6. Specifically, the data shown inFig. 6 by the gray thin curves revealed that, for a given set of param-eters (f h and b), the Gaussian modulus as a function of f A crossedat a common point independent of the value of χN. Considering theGaussian modulus κG as a function of f A and χN, i.e., κG(f A, χN),and assuming that f A is close to the crossover point f ∗A , one obtainsthe Taylor expansion with respect to f A,

κG(fA, χN) ≈ α(χN)(fA − f ∗A ) + κG(f ∗A , χN),

where α(χN) is the slope at the point that is a function of χN. Theresults shown in Fig. 6 suggest that at the cross point specified by f ∗A ,the Gaussian modulus κG(f ∗A , χN) is a constant κ∗G that is indepen-dent of χN. Therefore, when f A is close to f ∗A , one has the followingapproximation:

κG(fA, χN) ≈ α(χN)(fA − f ∗A ) + κ∗G. (35)

We call (f ∗A , κ∗G) the cross point and show the influence of f h andb on (f ∗A , κ∗G) in Fig. 6. The circles in Fig. 6 are the cross pointscorresponding to different parameters f h and b and are joined tocurves with the same f h = 1 or b = 1. When f h increases, bothf ∗A and κ∗G increase slightly. When b increases, f ∗A decreases but κ∗Gincreases. The range of κ∗G is large, which covers values from −1.1to 6.2 in Fig. 6 where f h changes from 0.3 to 5 and b changes from0.3 to 1.8.

It is worth noting that f A, f h, and b are molecular parame-ters that do not depend on the temperature of the system, while theFlory–Huggins parameter χ does. Therefore, the approximation (35)suggests that we can control the molecular parameters to design abilayer membrane whose Gaussian modulus is given and insensitiveto the temperature. However, the planar bilayers are unstable if theirGaussian moduli are positive.17 Therefore, Fig. 6 also indicates thatlarge f h and large b are harmful to this property because of theirpositive κ∗G.

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FIG. 5. The bending modulus κM , Gaussian modulus κG, and fourth-order moduli BC and BS of bilayer membranes as functions of the hydrophilic fraction f A. The effect of(a) χN, (b) f h, and (c) b on the moduli with a reference parameter χN = 30, f h = 1, and b = 1.

FIG. 6. Cross points of different parameters. The gray thin lines are the Gaussianmodulus as functions of f A for different χN ∈ 20, 25, 30, 35. The circles are crosspoints corresponding to parameters b ∈ 0.3, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8 andf h ∈ 0.3, 0.5, 0.7, 1, 2, 5.

VI. CONCLUSIONIn this paper, we proposed a novel method based on the asymp-

totic expansion theory using the curvature of the self-assembledbilayers as the expansion parameter, resulting in more accurate andefficient methods to calculate the elastic moduli of bilayer mem-branes. The method is presented within the framework of the self-consistent field theory, which is capable of accurately predicting themechanical parameters of the self-assembled bilayer membranes.The curvature of the cylindrical and spherical bilayers is regardedas the small parameter in the asymptotic expansion methodology.The new method allows us to obtain analytic expressions of theelastic moduli of the self-assembled bilayer membranes. Comparedwith the existing polynomial fitting method, the asymptotic expan-sion method could be used to directly compute the elastic moduliby solving a series of self-consistent field equations. This allowsus to design efficient numerical methods and pave the way to fur-ther understanding of the dependence of the elastic properties onthe molecular architecture and microscopic parameters. Numericalexamples verify the validity and efficiency of the proposed method.

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Although the examined model is restricted to a coil–coildiblock system, it is straightforward to extend our method to othermolecular architectures of flexible polymers characterized by theGaussian-chain model, as well as non-polymeric systems. Futureresearch will explore the extension to non-Gaussian models suchas the wormlike-chain, which takes the orientational order intoaccount.

ACKNOWLEDGMENTSWe thank Professor Pingwen Zhang for helpful discussions and

anonymous reviewers for their valuable comments and useful sug-gestions. This research was supported by the NSF of China (GrantNo. 11601099), the Science and Technology Foundation of GuizhouProvince of China (Grant No. [2017]1032), and the Natural Scienceand Engineering Research Council (NSERC) of Canada.

DATA AVAILABILITY

The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.

REFERENCES1B. M. Discher, Y. Y. Won, D. S. Ege, J. C. Lee, F. S. Bates, D. E. Discher, andD. A. Hammer, “Polymersomes: Tough vesicles made from diblock copolymers,”Science 284, 1143–1146 (1999).2D. E. Discher and F. Ahmed, “Polymersomes,” Annu. Rev. Biomed. Eng. 8, 323–341 (2006).3A. Taubert, A. Napoli, and W. Meier, “Self-assembly of reactive amphiphilicblock copolymers as mimetics for biological membranes,” Curr. Opin. Chem. Biol.8, 598–603 (2004).4W. Helfrich et al., “Elastic properties of lipid bilayers: Theory and possibleexperiments,” Z. Naturforsch. C 28, 693–703 (1973).5Z. C. Tu and Z. C. Ou-Yang, “Recent theoretical advances in elasticity of mem-branes following Helfrich’s spontaneous curvature model,” Adv. Colloid InterfaceSci. 208, 66–75 (2014).6J. F. Nagle, “Introductory lecture: Basic quantities in model biomembranes,”Faraday Discuss. 161, 11–29 (2013).7R. Dimova, “Recent developments in the field of bending rigidity measurementson membranes,” Adv. Colloid Interface Sci. 208, 225–234 (2014).8F. M. Thakkar, P. K. Maiti, V. Kumaran, and K. G. Ayappa, “Verifying scalingsfor bending rigidity of bilayer membranes using mesoscale models,” Soft Matter7, 3963–3966 (2011).9A. J. Sodt and R. W. Pastor, “Bending free energy from simulation: Correspon-dence of planar and inverse hexagonal lipid phases,” Biophys. J. 104, 2202–2211(2013).10M. Hu, J. J. Briguglio, and M. Deserno, “Determining the Gaussian curva-ture modulus of lipid membranes in simulations,” Biophys. J. 102, 1403–1410(2012).11A. Berthault, M. Werner, and V. A. Baulin, “Bridging molecular simulationmodels and elastic theories for amphiphilic membranes,” J. Chem. Phys. 149,014902 (2018).12D. Marsh, “Elastic curvature constants of lipid monolayers and bilayers,” Chem.Phys. Lipids 144, 146–159 (2006).

13P.-W. Zhang and A.-C. Shi, “Application of self-consistent field theory to self-assembled bilayer membranes,” Chin. Phys. B 24, 128707 (2015).14B. Kheyfets, T. Galimzyanov, A. Drozdova, and S. Mukhin, “Analytical cal-culation of the lipid bilayer bending modulus,” Phys. Rev. E 94, 042415(2016).15M. Müller and G. Gompper, “Elastic properties of polymer interfaces: Aggrega-tion of pure diblock, mixed diblock, and triblock copolymers,” Phys. Rev. E 66,041805 (2002).16K. Katsov, M. Müller, and M. Schick, “Field theoretic study of bilayer membranefusion. I. Hemifusion mechanism,” Biophys. J. 87, 3277–3290 (2004).17J. Li, K. Pastor, A.-C. Shi, F. Schmid, and J. Zhou, “Elastic properties andline tension of self-assembled bilayer membranes,” Phys. Rev. E 88, 012718(2013).18R. Xu, A. Dehghan, A.-C. Shi, and J. Zhou, “Elastic property of membranes self-assembled from diblock and triblock copolymers,” Chem. Phys. Lipids 221, 83–92(2019).19Y. Cai, P. Zhang, and A.-C. Shi, “Elastic properties of liquid-crystalline bilay-ers self-assembled from semiflexible–flexible diblock copolymers,” Soft Matter 15,9215–9223 (2019).20G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers (OxfordUniversity Press, Oxford, 2006).21J. Zhou and A.-C. Shi, “Critical micelle concentration of micelles with differ-ent geometries in diblock copolymer/homopolymer blends,” Macromol. TheorySimul. 20, 690–699 (2011).22I. Szleifer, D. Kramer, A. Ben-Shaul, W. M. Gelbart, and S. A. Safran, “Moleculartheory of curvature elasticity in surfactant films,” J. Chem. Phys. 92, 6800–6817(1990).23G. Gompper and S. Zschocke, “Ginzburg-Landau theory of oil-water-surfactantmixtures,” Phys. Rev. A 46, 4836–4851 (1992).24E. M. Blokhuis and D. Bedeaux, “Van der Waals theory of curved surfaces,” Mol.Phys. 80, 705–720 (1993).25P. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, NY,1953).26M. Laradji and R. C. Desai, “Elastic properties of homopolymer-homopolymerinterfaces containing diblock copolymers,” J. Chem. Phys. 108, 4662–4674(1998).27M. W. Matsen, “Elastic properties of a diblock copolymer monolayer andtheir relevance to bicontinuous microemulsion,” J. Chem. Phys. 110, 4658–4667(1999).28M. H. Holmes, Introduction to Perturbation Methods (Springer Science &Business Media, 2012), Vol. 20.29J. C. Neu, Singular Perturbation in the Physical Sciences (American Mathemati-cal Society, 2015), Vol. 167.30G. Strang, “On the construction and comparison of difference schemes,” SIAMJ. Numer. Anal. 5, 506–517 (1968).31K. W. Morton and D. F. Mayers, Numerical Solution of Partial DifferentialEquations: An Introduction, 2nd ed. (Cambridge University Press, 2005).32S. K. Lele, “Compact finite difference schemes with spectral-like resolution,”J. Comput. Phys. 103, 16–42 (1992).33J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed.(Springer-Verlag, 2008).34A.-C. Shi and J. Noolandi, “Effects of short diblocks at interfaces of stronglysegregated long diblocks,” Macromolecules 27, 2936–2944 (1994).35H. F. Walker and P. Ni, “Anderson acceleration for fixed-point iterations,”SIAM J. Numer. Anal. 49, 1715–1735 (2011).36R. B. Thompson, K. O. Rasmussen, and T. Lookman, “Improved convergence inblock copolymer self-consistent field theory by Anderson mixing,” J. Chem. Phys.120, 31–34 (2004).

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