Theory of microphase separation in bidisperse chiral membranes

Raunak Sakhardande,1 Stefan Stanojeviea,1 Arvind Baskaran,2

Aparna Baskaran,1 Michael F. Hagan,1 and Bulbul Chakraborty1

1Martin Fisher School of Physics, Brandeis University, Waltham, MA 02453, USA2Department of Chemistry and Biochemsitry, University of Maryland, College Park, MD 20742

(Dated: September 13, 2018)

We present a Ginzburg-Landau theory of micro phase separation in a bidisperse chiral membraneconsisting of rods of opposite handendness. This model system undergoes a phase transition froman equilibrium state where the two components are completely phase separated to a microphaseseparated state composed of domains of a finite size comparable to the twist penetration depth.Characterizing the phenomenology using linear stability analysis and numerical studies, we tracethe origin of the discontinuous change in domain size that occurs during this to a competitionbetween the cost of creating an interface and the gain in twist energy for small domains in whichthe twist penetrates deep into the center of the domain.

Introduction: When two immiscible fluids are mixed,they typically undergo bulk phase separation. Applica-tions ranging from food science, catalysis, and the func-tion of cell membranes require the arrest of this phaseseparation to form microstructures. A common pathwayto accomplish this is the introduction of a third compo-nent, such as a surfactant, that stabilizes interfaces be-tween the two fluids [1]. Here, we theoretically demon-strate a novel mechanism for microphase separation influid membranes that is mediated by the chirality of theconstituent entities themselves, and hence does not re-quire the introduction of a third component. In additionto identifying a design principle to engineer nano struc-tured materials, this work could shed light on the role ofchirality in compositional fluctuations and raft formationin biomembranes [2–7].

Our theory is motivated by a recently developedcolloidal-scale model system of fluid membranes, com-posed of fd-virus particles [8–13]. The system containstwo species of virus particles that have opposite chiralityand different lengths (Fig 1). In the presence of a de-pletant, they self-assemble into a monolayer membranethat is one rod length thick. The competition betweendepletant entropy, mixing entropy of the two species, andmolecular packing forces leads to a rich phase behaviorwithin a membrane, including bulk phase separation ofthe two species, microdomain formation, and homoge-neous mixing. In particular, the experiments find thatin the regime where a single species forms a macroscopicmembrane, limited only by the amount of material, amixture of two species leads to the formation of circularmonodisperse microdomains (rafts) of one species in abackground of the other.

In this work, to understand the mechanisms controllingthis raft formation, we develop a continuum Ginzburg-Landau theory that captures the physics of chirality andcompositional fluctuation in a 2D binary mixture of rodswith opposing chiralities. The primary physics that weincorporate into the theory is a coupling between thetwist of the director field and the compositional fluctua-

tions [14]. By using linear stability analysis and numer-ical solutions of the time-dependent Ginzburg-Landauequations, we show that the tendency of the molecules totwist arrests the phase separation of the two species, andstabilizes a droplet phase whose phenomenology closelymimics that observed in experiments. In particular, thetheory shows a discontinuous jump in the droplet ra-dius as the system transitions from a microphase sepa-rated state to bulk separation, a phenomenon observed inthe experiments as well. In contrast, previously studiedmechanisms of microphase separation lead to a dropletsize that continuously diverges as the system approachesbulk phase separation [15–18].Model: The Ginzburg-Landau (GL) model involves

two fields: a director field n (r) that characterizes theorientation of the rods with respect to the membrane nor-mal and a scalar field ψ (r), which characterizes the localcomposition of the membrane in terms of the two species.We choose a coordinate system in which the layer normalof the membrane lies along the z axis and normalize theorder parameter ψ such that ψ = ±1 correspond to thehomogeneous one-component phases. The GL functionalis taken to be of the form:

F =

∫d2r

[1

2K1(∇ · n)2 +

1

2K2 (n · ∇ × n− q (ψ))

2

+1

2K3 (n×∇× n)

2+C

2sin2 θ − ψ2

2+ψ4

4+λψ2

(∇ψ)2

]The physics incorporated in the GL functional can be

summarized as follows : i) The first three terms arisefrom the Frank elasticity associated with director distor-tion, with K1, K2 and K3 being the elastic constantsassociated with splay, twist and bend respectively [19].The twist term involves a pitch q(ψ (r)) that encodes thechirality and hence the associated tendency of the rodsto develop a spontaneous non-zero twist. In a mixture ofleft and right handed rods, q is naturally a function of thecomposition, which introduces a coupling between n (r)and ψ (r). ii) The term C

2 sin2 θ encodes the fact that the

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rods in the membrane tend to align with the layer normal[19], and gives rise to the standard mechanism of twistexpulsion seen in Smectic C systems. When ψ = ±1, theterms discussed in (i) and (ii) reduce to the theoreticaldescription used successfully to describe single compo-nent chiral membranes in earlier works [20–22]. iii) Thecompositional fluctuations encoded in the field ψ are de-scribed by a standard ψ4 theory below the critical pointthat leads to bulk phase separation, with an energeticcost to forming interfaces controlled by the parameterλψ. Thus, the difference in the length of the rods thatleads to phase separation in the experimental system isrepresented as an effective interaction, and our 2D modeldoes not include information about the spatial variationof the membrane in the third dimension.

In the following, we work in the single elastic constantapproximation of the Frank elasticity: K1 = K2 = K3 ≡K. We model the variation of q with composition througha minimal linear coupling, q(ψ) = q0 + aψ, which definesthe coupling parameter a. We nondimensionalize the GL

functional using the twist penetration depth λt ≡√

KC

as the characteristic length scale. Defining dimension-

less parameters: q′0 = λtq0, a′ = λta(1− Ca2

)1/2,

λ′ψ =λψ/λ

2t

(1−Ca2) ,ψ′ = ψ

(1−Ca2)1/2 and C ′ = C(1−Ca2)2 the

GL functional becomes:

F =

∫d2r′ [fLC + fψ + fCross]

fLC =C ′

2

[(∇′ · n)

2+(q′20 − q′0n · ∇′ × n

)]fψ =

[−ψ′2

2+ψ′4

4+λ′ψ2

(∇′ψ′)2]

fCross = [−C ′a′ψ′ (n · ∇′ × n) + C ′a′q′0ψ′] (1)

This nondimensionalized GL functional is used in all ofour subsequent analysis and the ′’s are dropped for com-pactness of notation.

We model the dynamics by the time-dependent GLequations with a conserved composition field ψ: ∂tψ =∇2 δF

δψ (Model B dynamics), and a non-conserved direc-

tor field ∂tn = − (I− nn) · δFδn (Model A dynamics)[23].The n dynamics accounts explicitly for the fact that itis a unit vector. The time constants for the relaxationdynamics of ψ and n have been chosen to be same andset equal to 1. The resulting equations are :

∂tψ = ∇2(−ψ + ψ3 − λψ∇2ψ − Ca (n · ∇ × n) + Caq0

)(2)

∂tn = − (I− nn) ·(−C∇2n− 2Cq∇× n + Cn×∇q

+C(nxx+ ny y)) (3)

Linear Stability Analysis: Eqs.(2-3) admit homoge-nous steady states of the form ψ = ±1 and n = z. As

FIG. 1: a) Schematic of the experimental system. b) Theprimary results of this work summarized in a phase diagramas a function of the smectic alignment parameter C and thetwist-composition coupling parameter a. The lines indicatethe phase boundary between the microphase separated andbulk phase separated states as obtained from linear stabilityanalysis (black/solid) and numerical integration of Eqs. 2-3(green/dashed). The snapshots show configurations at steadystate obtained from numerics at the indicated parameter val-ues (o). c) Illustration of the evolution to steady state for twoparameter sets. The results shown here and in the rest of thepaper are for a 60-40 mixture with λψ = 0.1, and q0 = 0.1

a first step in understanding the dynamics of phase sep-aration, we analyze the instability of the homogeneousstate to small fluctuations of the form ψ = 1 + δψ andn = z + δn. We introduce Fourier transformed variablesX (k, t) =

∫d2reik·rX (r, t). Without loss of generality,

we choose a coordinate system in the plane of the mem-brane such that the x axis lies along the spatial gradientdirection. We find that the longitudinal fluctuations inthe director δnx decouple from the other variables ([24])and we obtain the linearized equations

∂t

(δψδny

)=

(−2k2 − λψk4 ik3Ca−ikCa −C − Ck2

)(δψδny

)(4)

The homogeneous state is found to be linearly unstableto modes k that satisfy

λψk6 − (Ca2 − 2− λψ)k4 + 2k2 < 0 . (5)

We see from Eq. 5 that the k = 0 mode is alwaysmarginally stable, and that the linear instability is con-trolled only by the combination Ca2 and does not dependindividually on the strengths of the smectic alignmentand the twist-composition coupling. At a critical value ofCa2 determined by (Ca2−2−λψ)2 = 8λψ, the mode withkmax = (2/λψ)1/4 becomes unstable[24]. Fig. 2 shows the

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FIG. 2: (color online) a) The largest eigenvalue ω(k) of thelinear stability matrix in Eq.(4) as a function of the wavevec-tor k for indicated values of the alignment strength C, witha = 0.8 and λψ = 0.1. b) Dependence of the optimal domainsize on Ca2 obtained by three different analysis methods: thesteady-state mean radius of domains obtained by numericalintegration, the radius which minimizes the GL free energy(calculated as described in the text), and wavelength corre-sponding to the fastest-growing mode calculated by linear sta-bility analysis, for λψ = 0.1

largest eigenvalue ω(k) of the linear stability matrix inEq.(4) for different parameters. For any non-zero value ofλψ, the instability thus occurs at a finite k, which demon-strates that the instability of the homogeneous phase is tomicrophase domains of a finite size. The transition froma macroscopically phase separated state (infinite domainsize, k = 0) to a microphase separated state should thusbe accompanied by a discontinuity in the domain size[24].

Numerical analysis of Eqs.(2-3) verifies this discontin-uous change in the domain size. We solve Eqs.(2-3) nu-merically by using an implicit convex splitting scheme toevolve the equation for ψ and the forward Euler methodto evolve the director field [24]. We initialize the systemwith random compositional fluctuations around a homo-geneous mixture with ψ = 0.2 and we explore the phasespace spanned by C and a. For most of the results shownhere, we choose λψ = 0.1, as the interface width in theexperiments is found to be much smaller than the twistpenetration length [12]. Also, we set q0 = 0.1 as thepreferred chiral twists of the two species of rods in theexperimental system are not equal. The phase diagramobtained from numerics are shown in Fig. 1. It is evidentthat linear stability analysis captures all qualitative as-pects of the numerically determined phase diagram. Thesteady state domain sizes obtained from numerics areshown in Fig. 2 and clearly demonstrate the discontinu-ous change accompanying the phase transition.

The formation of finite size domains is controlled by acompetition between chirality and interfacial tension. Asimilar competition exists even in a chiral membrane ofa single species, where the interfacial tension exists be-tween the membrane edge and the bulk polymer suspen-sion. A theoretical analysis of this system [20] showeda transition between membranes of finite size and un-

bounded macroscopic membranes. Within such a mem-brane, the twist is expelled to the edge, decaying overa length λt, and the membrane size grows continuouslyas the transition is approached. Here we see that intro-ducing a second species with opposite handedness intosuch a membrane provides a mechanism for the twist topenetrate the interior of the membrane. As shown inFig. 1, the director twists at the edge of each domain,and then untwists (twists in the opposite direction) intothe background. This twist is confined to within approx-imately λt of a domain edge. The ability of the inter-face to accommodate twist is the mechanism that leadsto the formation of microdomains in the region of pa-rameter space where each species by itself would form amacroscopic membrane.

To quantitatively unfold this mechanism, and to un-derstand the discontinuous change in domain size thatoccurs at the transition to bulk phase separation, we ex-amine how the spatial variations in ψ and n influencethe free energy Eq. (1). To this end, we calculate thefree energy of a domain of radius R of one species in abackground of the other. We do so by assuming profilesfor ψ and n that are consistent with the results obtainedfrom numerical integration [[24] section 2]. The optimaldomain size is then determined by the value of R at eachC0, a, q0, λψ, for which the free energy is minimized (Fig.3). The resulting domain sizes are consistent with thoseobtained from linear stability analysis and numerical in-tegration (Fig. 2).

FIG. 3: (color online) a) The free energy density in a domainis shown as a function of its radius R for C = 30 and a =0.7 calculated using a quasi-static approximation along withthe different contributions to it, fLC, fψ and fCross. Theradius of the droplet is varied from 0.2 to 20 in terms of thetwist penetration length. b) The same free energy densitycontributions with a focus on small R.

The origin of discontinuity in domain size is revealedby examining the variations in different contributions tothe free energy density (fLC, fψ and fCross) as the dropletsize changes. Fig. 3 shows these variations for a parame-ter set in the microphase separation regime. Note that inan extensive system with clear scale separation betweenbulk and interface, the interfacial contribution to a free

4

energy density decays with increasing domain size, whilethe bulk contribution remains constant. In contrast, wesee that fLC and fCross are super-extensive for small do-main sizes, only becoming extensive asymptotically. Thissuperextensivity is significant only for domain sizes of theorder of the twist penetration length (R ∼ 5λt). Thus,finite-sized domains appear only when the increase infLC and fCross with R is sufficient to outcompete fψ atthese small domain sizes. As Ca2 decreases, the super-extensive behavior diminishes, forcing the critical domainsize (at which fLC and fCross dominate over fψ) to largerR. At the threshold value of Ca2, fLC and fCross becomeextensive before dominating over the interfacial tension,and macrophase separation sets in.

FIG. 4: (color online) a) The theoretical twist profile obtainedthrough dynamics in our study is compared with the one ob-tained from the experiments[12]. b) The free energy densityprofile as a function of radius of droplet is compared betweentheory and experiments.

The source of the super-extensive growth in fLC andfCross can be understood from the dependence of twistprofiles on R. For large R, twist decays exponentiallyfrom the domain edge (Fig. 2 in [24]); thus ensuringscale separation between the bulk and the interface. Onthe other hand, such a separation does not exist for smalldomains where the twist penetrates to the center of thedomain.

In conclusion, we have presented a theory of mi-crophase separation in membranes, which is driven bychirality of its constituent entities. The underlying mech-anism of microphase separation can be traced to the thegain in twist energy in these structures, which can accom-modate twist at the boundaries of domains. We haveprovided quantitative analysis that unfolds the precisefactors leading to the appearance of microdomains. Wehave also shown that the microdomains have a naturallength scale determined by the twist penetration depth,and therefore the domain size does not increase continu-ously as the system transitions to the macrophase sepa-rated state. Domains that are much larger than the twistpenetration depth fail to gain enough free energy fromthe twisting at the interface to compensate for the freeenergy cost of creating an interface where the composi-tion changes. By reducing λψ, this limiting length can be

made larger, however the transition is discontinuous forall finite values of λψ. This feature of the microdomainsis appealing from the perspective of creating nanostruc-tures since the domain size can be tightly controlled.

Acknowledgment: This work was supported by theBrandeis University NSF MRSEC, DMR-1420382. Com-putational resources were provided by the NSF throughXSEDE computing resources (Stampede) and the Bran-deis HPCC which is partially supported by DMR-1420382. We gratefully acknowledge Robert Meyer,Robert Pelcovits, Zvonimir Dogic, and Prerna Sharmafor helpful discussions; we also thank Prerna Sharma forproviding her experimental data.

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