Confinement effects on phase separation of a polyelectrolyte solution

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Polymer 110 (2017) 49e61

Contents lists available at ScienceDirect

Polymer journal homepage: www.elsevier.com/locate/polymer

Conﬁnement effects on phase separation of a polyelectrolyte solution Jie Fu a, Bing Miao b, *, Dadong Yan a, * a b

Department of Physics, Beijing Normal University, Beijing 100875, China College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 September 2016 Received in revised form 28 November 2016 Accepted 22 December 2016 Available online 29 December 2016

Formulating an analytical theory, we study phase separation of a polyelectrolyte solution under poor solvent condition conﬁned in three types of ﬁnite geometric spaces: slab, cylinder, and sphere. Divided by a Lifshitz line, bulk polyelectrolyte solution undergoes either micro- or macro-phase separation. Conﬁnement effects for both scenarios are studied. Composition ﬂuctuations inducing phase separation are classiﬁed in terms of eigenmodes of the inverse structure factor operator in the corresponding geometric spaces. Tracking each eigenmode, the instability lines under conﬁnement effects are derived in closed forms. For the conﬁned microphase separation, we ﬁnd a decaying oscillatory dependence of the spinodal point on the conﬁnement size, which represents the commensurability between the ﬁnite period of the soft mode and the conﬁning boundary size. For the conﬁned macrophase separation, a typical mean-ﬁeld ﬁnite size scaling of the Ising universality class is observed under the strong screening condition. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Conﬁnement Self-assembly Theory

1. Introduction The short-range repulsions between polymer monomers and solvents drive a polyelectrolyte solution under a poor solvent condition to phase separate. However, due to the long-range Coulomb repulsive interaction between polyion charges, domains in a macro-scale cannot be formed. Instead, phase separation proceeds by forming micro-scale domains. In other words, the Coulomb interaction provides a stabilizer for long range composition ﬂuctuations and leads to a microphase separation in a polyelectrolyte-poor solvent system [1]. From this reasoning, it is clear that the system will recover the macrophase separation behavior, which is usual in a polymer solution, once ionic strength is increased to screen the Coulomb interaction to be short-range. These features render a polyelectrolyte solution to be a fascinating system of studying phase separation behaviors. Theoretical studies of phase separation in a polyelectrolyte solution have been carried out during past decades [2e8]. Borue and Erukhimovich formulated a microscopic statistical theory for the weakly charged polyelectrolyte solution within the framework of the random phase approximation (RPA) [3]. Through analyzing the correlation functions, a microphase separation was reported in

* Corresponding authors. E-mail addresses: [email protected] (B. Miao), [email protected] (D. Yan). http://dx.doi.org/10.1016/j.polymer.2016.12.061 0032-3861/© 2016 Elsevier Ltd. All rights reserved.

their paper. Later, this work has been developed in terms of the weak crystallization theory and phase diagrams have been constructed for this system [4,5]. The correlation functions of different components have been derived and analyzed carefully [6e8]. On the experimental side, the microphase separation of a polyelectrolyte solution has also been observed by several groups [9e11]. Obtaining ordered structures through the microphase separation of block copolymer system has been an active research area in polymer community for decades [12e14]. Besides the classic morphologies formed in a bulk block copolymer system, it has been established that conﬁnements generate signiﬁcant effects on phase behavior of block copolymer self-assembly; novel structures, which are absent in a bulk system, can be created by conﬁning bulk system into ﬁnite geometric spaces. As the result, nanolithography, which is to utilize conﬁnement effects for self assembling, has become a promising and fast developing research area in recent years [15]. Compared to this situation, the microphase separation of a polyelectrolyte solution, which provides a new possible route to create nanostructures, has not been widely noticed and systematically investigated up to now. In particular, how does conﬁnement affect phase separation of a polyelectrolyte solution? What types of novel morphologies can be induced through introducing conﬁnements into a polyelectrolyte solution? These remain to be important yet unanswered questions. In addition to the importance from the viewpoint of nanolithography, clarifying these questions about

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J. Fu et al. / Polymer 110 (2017) 49e61

polyelectrolyte phase behaviors within conﬁned spaces will shed lights on the understanding of bio-systems, considering that biomacromolecules, which are essentially polyelectrolytes, are under a conﬁned environment within a cell. Furthermore, the conﬁnement effect on phase transition has been a fascinating research topic in statistical mechanics [16e25]. Studies have shown that the critical temperature is shifted relative to the bulk value adopting a scaling form in the conﬁned binary small molecule and polymer thin ﬁlms [18e20]. Due to the fact that two types of transitions, micro- and macro-phase separation, can occur in one model of a polyelectrolyte solution by adjusting screening strength, studying the conﬁned polyelectrolyte solution can provide important insights for the fundamental research of phase transitions in conﬁned spaces. Motivated by these factors, we address phase separation of a polyelectrolyte solution within conﬁned spaces theoretically in this manuscript. To our knowledge, this work is the ﬁrst theoretical investigation towards the understanding of conﬁnement effects on self-assembly in a polyelectrolyte solution. Speciﬁcally, we formulate an analytical model, which originates from our theoretical methods developed for studying the block copolymer melts within conﬁned spaces [26,27]. This calculational framework has also been developed by Erukhimovich et al. independently [28,29]. Three types of conﬁnement scenarios have been invoked in this paper. Namely, the polyelectrolyte solution is conﬁned between two inﬁnite parallel plates (slab), within an inﬁnitely long cylindrical pore (cylinder), and within a spherical pore (sphere), respectively. As the main results, the instability lines are derived in closed forms by tracking eigenmodes of ﬂuctuations. For the slab and the cylinder conﬁnements, on one hand, the spinodal point experiences no difference from the bulk correspondence, which is attributed to the existence of free dimensions in these two set-ups of conﬁnements, wherefrom the perpendicular mode free of conﬁnements induces the instability of the homogeneous phase; on the other hand, through analyzing eigenmodes, a degenerate soft-mode set is identiﬁed for the conﬁned microphase separation which, due to the competition among the degenerate modes, leads to complex morphologies in the conﬁned space. For the sphere conﬁnement, the spinodal of the microphase separation behaves a decaying oscillatory dependence on the conﬁning size. This represents the commensurability between the ﬁnite size of the spherical pore and the natural period of the soft mode. A re-entrance of the disordered phase is observed in the phase diagram with increasing the pore

. 0 S1 ðqÞ þ y þ uf þ 4pl f 2 q2 B 0 0 . B S1 ðqÞ ¼ @ 4plB f q2 . 4plB f q2

. 4plB f q2 . 2 f1 0;c þ 4plB q . 4plB q2

2. Phase separation in bulk Consider a polyelectrolyte solution. There are four types of components in the system: polyions, counterions, salts, and solvents. Polyions are weakly charged and treated as ﬂexible polymer chains with a homogeneous charge distribution along each chain contour; for each polyion chain, the degree of polymerization is N, the Kuhn length is b, the fraction of charged monomers with charge e is f, the excluded volume parameter of monomers is y, and the third virial coefﬁcient is u, which is a positive constant. Here e denotes the elementary charge unit. Monovalent counterions and salts move around in the system volume V. The average densities of polymer monomers, counterions, and salts are f0 ; fc;0 ; and n0 , respectively. The global electroneutrality of the system enforces the constraint of f f0 þ n0 ¼ fc;0 . Note that counterions include positively charged ions dissociated from both polyions and salts. Solvents are treated in an implicit way by introducing a homogeneous background dielectric constant ε. In the calculation, all the lengths are scaled to be dimensionless by the Kuhn length b. To study the phase separation of a homogeneous polyelectrolyte solution, we formulate a Gaussian ﬂuctuation theory here. Start with the homogenous phase, we deﬁne a vector ﬁeld, ! f ðrÞ ¼ ffðrÞ; fc ðrÞ; nðrÞg, to characterize the density ﬂuctuations from the homogeneous state for polymer monomers, counterions, and salts, respectively. In the Gaussian level, this order parameter is subject to a Hamiltonian

bH ¼

1 X! ! f ðqÞS1 ðqÞ f ðqÞT 2V q

which has been written down in the reciprocal space to utilize the translational invariance of the homogenous solution. Here b ¼ 1=kB T with kB being the Boltzmann constant and T being the ! absolute temperature; f ðqÞ stands for the Fourier transform of ! f ðrÞ; the coefﬁcient matrix S1 ðqÞ captures the energy spectrum of ﬂuctuation modes labelled by the wave vector q. In the Gaussian ﬂuctuation theory, the coefﬁcient matrix is 1 constructed as S1 ðqÞ ¼ S1 0 ðqÞ þ V0 ðqÞ, where S0 ðqÞ and V 0 ðqÞ stand for the inverse bare correlation matrix and the bare interaction matrix, respectively [30,31]. As exposed in detail in the Appendix, substituting related quantities, we have

. 4plB f q2 1 . C 4plB q2 A . 2 n1 0 þ 4plB q

size around the bulk spinodal temperature. On the macrophase separation side, conﬁnement leads the instability point shift, relative to the bulk value, to behave a typical mean-ﬁeld ﬁnite size scaling of the Ising universality class under the strong screening condition. The paper is organized as follows. In Section 2, we derive the correlation matrix for the polyelectrolyte system through a general way. Section 3 introduces conﬁnements into the system, ﬂuctuation modes are classiﬁed in terms of eigenmodes, instability lines are derived. In Section 4, we discuss the results. Section 5 gives a conclusion.

(1)

(2)

where S0 ðqÞ is the structure factor of the polyion in the reference interaction-free system, which is taken to be the approximate Debye function, S0 ðqÞ ¼ S0 ðqÞ ¼ ðNf0 Þ=½1 þ q2 R2g =2 with R2g ¼ N=6 being the mean-square radius of gyration for an ideal chain. Note that, by choosing this form, we are dealing with ﬂexible chains for the weakly charged polyelectrolytes. lB ¼ e2 =ð4pεkB TÞ is the Bjeruum length characterizing the strength of the electrostatic interaction. y ¼ 1 2c is the excluded volume parameter between the polyelectrolyte monomers with c being the Flory-Huggins parameter dependent on the temperature.

J. Fu et al. / Polymer 110 (2017) 49e61

Inverse the above matrix, we obtain the correlation matrix, namely, S ¼ ½S1 1 . To analyze the phase separation, we focus on the correlation function of the polyelectrolyte monomer density ﬂuctuation, SðqÞ ¼ 〈fðqÞfðqÞ〉. Finishing the matrix inversion straightforwardly, we obtain

SðqÞ ¼

1 2 S1 0 ðqÞ þ y þ uf0 þ 4plB f

.

q2 þ k2

(3)

where k2 ¼ 4plB ðfc;0 þ n0 Þ ¼ 4plB ðf f0 þ 2n0 Þ is the Debye-Hückel screening strength. For convenience of discussion, we take the inverse of the correlation function to have

S1 ðqÞ ¼

. 1 þ q2 R2g 2 Nf0

þ y þ uf0 þ

4plB f 2 q2 þ k2

(4)

where we have substituted S1 0 ðqÞ. Note that due to the rotational symmetry of the homogenous phase, the inverse correlation function, also called the inverse structure factor, only depends on the magnitude of the wave vector. The inverse correlation function characterizes the energy cost of the polyion density ﬂuctuation mode of wave vector q. For a stable phase, S1 ðqÞ > 0 holds for all the modes. Namely, all ﬂuctuation modes upon the homogeneous phase cost energy, which deﬁnes the stability of the homogeneous phase. With decreasing the value of y, the energy cost S1 ðqÞ decreases. When there appears a mode with a negative energy cost, the phase under study loses its linear stability by the development of this negative-energy mode. From this analysis, the most important mode for the stability of a given phase is the one with the lowest energy cost and we shall call it the soft mode of the phase. Tracking the variation of the soft mode with the change of parameters, the spinodal point can be determined as the point where the soft-mode energy cost vanishes; at the spinodal point, the scattering function diverges at the soft mode, namely, Sðq0 Þ ¼ ∞, where we labelled the soft mode as q0. We note that, within the present Gaussian ﬁeld theory level, this method of determining the spinodal gives the same result as that from requiring the deter minant of the coefﬁcient matrix to vanish, namely, S1 ¼ 0, for the multi-component system. We analyze the inverse correlation function here. The ﬁrst three terms are identiﬁed with the inverse correlation function in the traditional Landau theory, which describes the macrophase separation due to the long range ﬂuctuation in a neutral polymer solution. The last term comes from the screened Coulomb interaction in the polyelectrolyte solution, which increases with decreasing the wave number q and provides a stabilizer for the long range ﬂuctuation, leading to a microphase separation induced by a soft mode of ﬁnite wavelength. Let us consider two limiting situations. First, take k2 /0, corresponding to a weakly screened system. The inverse structure factor reads

S1 ðqÞ ¼

. 1 þ q2 R2g 2 Nf0

51

determining the energy spectrum of ﬂuctuations, which is to select out a soft mode of ﬁnite wavelength. Second, let k2 /∞, corresponding to a strongly screened system. The inverse structure factor becomes

S1 ðqÞ ¼

. 1 þ q2 R2g 2 Nf0

þ y þ uf0 þ

4plB f 2

(6)

k2

This falls into the same form of S1 ðqÞ ¼ t þ gq2 , with t, g being constants, as that of homopolymer blends, belonging to the Ising universality class. Because there exists no a stabilizing factor for the long range ﬂuctuation of q ¼ 0, the short-range repulsive interaction between incompatible components drives the system to a macrophase-separated state. The boundary between the micro- and macro-phase separation mechanism for the system is given by the Lifshitz line. It can be determined by checking the curvature at q ¼ 0 of the function S1 ðqÞ. Speciﬁcally, the positive curvature, v2q S1 ðqÞ > 0, means q¼0

the long-wavelength mode of q ¼ 0 is the soft mode of the system and leads to a macrophase separation; while, the negative curva < 0, means a ﬁnite-wavelength mode begins to ture, v2q S1 ðqÞ q¼0

dominate the phase behavior and leads to a microphase separation. ¼ 0 deﬁnes the Lifshitz line, Therefore, the condition v2q S1 ðqÞ q¼0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ which can be derived as ðk2 ÞL ¼ 8plB f 2 Nf0 =R2g ¼ 48plB f 2 f0 . Above the Lifshitz line, namely, k2 > ðk2 ÞL , system undergoes macrophase separation; below the Lifshitz line, namely, k2 < ðk2 ÞL , system undergoes microphase separation. For convenience of presentation, we shall call the region of k2 > ðk2 ÞL the screening regime stressing that the strong electrostatic screening leads system to macrophase separation; call the region of k2 < ðk2 ÞL the osmotic regime stressing that the long-range Coulomb interaction stabilizes the long-wavelength ﬂuctuation and leads to microphase separation. Now we determine the soft mode explicitly by solving vq S1 ðqÞ ¼ 0 and v2 S1 ðqÞ > 0. In the osmotic regime, the soft q0

q

q0

is the ﬂuctuation mode with wave vector pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q0 ¼ ð 48plB f 2 f0 k2 Þ1=2 ¼ ½ðk2 ÞL k2 1=2 . In the screening regime, the soft mode is the long-wavelength mode of q0 ¼ 0. mode

3. Introducing conﬁnements The conﬁnement set up is illustrated in Fig. 1 schematically. The polyelectrolyte solution is restricted into three types of geometric spaces of ﬁnite sizes: a parallel slab in Fig. 1(a), a cylindrical pore in Fig. 1(b), and a spherical pore in Fig. 1(c). To enforce conﬁnements, we employ the reﬂecting boundary condition. Namely, the component of the gradient of the order parameter ﬁeld along the conﬁned direction vanishes at the boundary. 3.1. Fluctuation modes

þ y þ uf0 þ

4plB q2

f2

(5)

This bears the same form of S1 ðqÞ ¼ Aq2 þ Bq2 þ C, with A, B, C being constants, as that of diblock copolymer melts, belonging to the Brazovskii universality class. Phase separation in this limit occurs at a length scale set by a ﬁnite wave vector q0. For the polyelectrolyte solution, the long-range Coulomb interaction plays the same role as the diblock copolymer chain connectivity does when

To analyze ﬂuctuation modes, we recast the Gaussian ﬂuctuation theory in an operator form. The energy cost of the order parameter ﬂuctuation over a mean-ﬁeld structure takes a quadratic form

bH ¼

1 D b1 E 1 X f S f ¼ En jfn j2 2 2 n

(7)

52

J. Fu et al. / Polymer 110 (2017) 49e61

Fig. 1. Schematic illustration for the conﬁned polyelectrolyte solution. (a) Slab conﬁnement. (b) Cylinder conﬁnement. (c) Sphere conﬁnement.

1 where jf > stands for the order parameter ﬂuctuation, b S is the inverse structure factor operator whose eigen-equation reads 1 b S jn i ¼ En jjn i. Projecting a ﬂuctuation into the eigenvector

space, the quadratic form has been written as a diagonal form, where fn ¼ hjn jfi, by which order parameter ﬂuctuations are classiﬁed in terms of the eigenvectors jjn > with a corresponding energy cost En . In the positive deﬁnite case, where En > 0 for all n, the mean-ﬁeld structure is stable. Once there appears a mode with negative eigenvalue, this mode will destroy the stability of the phase under study. In the present study for phase separation, the homogenous structure gives the mean-ﬁeld state over which ﬂuctuations develop. For this phase, the inverse structure factor operator takes the following form:

1 1 1 1 b 1 R2g V2 þ y þ uf0 þ 4plB f 2 V2 þ k2 S ¼ Nf0 2

(8)

where the ﬁrst part, representing the energetics of a reference system with interactions switched off, corresponds to the inverse structure factor of a multi-Gaussian-chain system; the second part is given by the energy cost due to interactions between components, where y and u characterize interactions for a neutral system, while the last term represents the Coulomb interaction between charges under the screening from small mobile ions. Density ﬂuctuations shall be classiﬁed in terms of the eigen1

modes jn ðrÞ of the operator b S , with n characterising the general mode index. From Eq. (8), it is found that, in the present problem, 1 are identical with those of the Laplace the eigenmodes of b S operator, V2 jn ðrÞ ¼ ln jn ðrÞ, and the eigenvalues read

En ¼

pﬃﬃﬃﬃﬃ 1 1 4plB f 2 1 þ R2g ln þ y þ uf0 þ ¼ y þ uf 0 þ F ln Nf0 2 ln þ k2 (9)

ðNf0 Þ þ ð4plB f 2 Þ=ðx2 þ k2 Þ. where we deﬁne FðxÞ ¼ 1 þ 12R2g x2 For the bulk system, the eigenmodes are plane waves indexed by continuous wave vectors, as shown in the last section. Now we determine the eigenmodes of ﬂuctuations for the three types of conﬁned geometric spaces. For the slab conﬁnement shown in Fig. 1(a), two parallel plates of inﬁnite extensions in the x-plane, where x ¼ ðx; yÞ, apart from each other d in the b z -direction, are enforced. The polyelectrolyte solution locates between the two plates. By this set-up, the symmetry of the three-dimensional space is broken identifying the

b z -direction as the dimension under conﬁnements and the x-plane as the conﬁnement-free dimensions. The rotational symmetry in the x-plane is still kept in this set-up. Choosing the co-ordinate origin in the center of the bottom plate, the eigen-function is written as

jnq⊥ ðrÞ ¼

1 cosðkn zÞeiq⊥ $x N n

(10)

with the corresponding eigenvalue of the Laplace operator

lnq⊥ ¼ k2n þ q2⊥

(11)

where N n denotes the normalization constant, n ¼ 0; 1; 2; …, and kn ¼ an =d with an ¼ np for the reﬂecting boundary condition. q⊥ ¼ ðqx ; qy Þ is a two-dimensional vector taking continuous values. The eigenmodes indexed by ðn; q⊥ Þ correspond to a cosine wave in b z -direction and a plane wave in x-plane. For the cylinder conﬁnement shown in Fig. 1(b), the eigenfunction is

jmnqz ðrÞ ¼

1 N

mn

im4 iq z Jm km e z nr e

(12)

with the eigenvalue

lmnqz ¼ km n

2

þ q2z

(13)

where N mn denotes the normalization constant, Jm ðxÞ is the m-th order Bessel function, m ¼ 0; ±1; ±2; …, n ¼ 1; 2; 3; …, and m m km n ¼ an =r0 with an standing for the n-th root of the boundary 0 ðxÞ ¼ 0. r denotes the radius of the cylindrical pore. q condition Jm z 0 is a continuous one-dimensional wave vector. The eigenmodes indexed by ðm; n; qz Þ correspond to a cylindrical wave in ðr; 4Þ-plane and a plane wave in b z -direction. For the sphere conﬁnement shown in Fig. 1(c), the eigenfunction reads

jlmn ðrÞ ¼

1 N

lmn

jl kln r Plm ðcos qÞeim4

(14)

with the eigenvalue

lln ¼ kln

2

(15)

where N lmn denotes the normalization constant, jl ðxÞ is the l-th order spherical Bessel function, Plm ðxÞ is the associated Legendre function, l ¼ 0; 1; 2; …, m ¼ l; … ; l, and kln ¼ aln =R with aln being

J. Fu et al. / Polymer 110 (2017) 49e61

the n-th root of the boundary condition j0l ðxÞ ¼ 0. R denotes the radius of the spherical pore. The eigenmodes are spherical waves indexed by ðl; m; nÞ. The eigenmodes of ﬂuctuations in the conﬁned spaces can be classiﬁed as the parallel, tilt, and perpendicular modes according to their propagating directions relative to the conﬁnement dimension. The parallel mode is deﬁned to be the ﬂuctuation propagating along the conﬁnement direction and still respecting the symmetry in the remaining conﬁnement-free dimensions. For the slab conﬁnement, the parallel mode is ðns0; q⊥ ¼ 0Þ, the tilt mode is ðns0; q⊥ s0Þ, the perpendicular mode is ðn ¼ 0; q⊥ Þ. For the cylinder conﬁnement, the parallel mode, which is independent of the ð4; zÞ degrees of freedom, behaves as the concentric cylindrical wave in the ðr; 4Þ-plane characterized by ðm ¼ 0; n; qz ¼ 0Þ. The perpendicular mode is ðm ¼ 0; n ¼ 1; qz Þ, where k01 ¼ 0, which represents a plane wave propagating along the b z -direction. Others correspond to the tilt modes. For the sphere conﬁnement, the parallel mode is ðl ¼ 0; m ¼ 0; nÞ, which is invariant for the ðq; 4Þ degrees of freedom. The tilt mode is ðls0; m; nÞ. There are no perpendicular modes in the sphere case.

3.2. Instability line and spinodal The instability line of each mode is calculated by requiring the corresponding eigenvalue to be zero. Namely, En ¼ 0 deﬁnes the instability line of mode n. From Eq. (9), the instability line is solved pﬃﬃﬃﬃﬃ by yn þ uf0 ¼ Fð ln Þ. For the bulk system, the spinodal of the homogeneous solution is derived as y0s þ uf0 ¼ Fðq0 Þ, where q0 is the wave number of the soft mode. We deﬁne the conﬁnement-induced shift of the instability line of mode n relative to the bulk spinodal as dn ¼ cn c0s , and it is easy to obtain

dn ¼

F

pﬃﬃﬃﬃﬃ

ln Fðq0 Þ 2

(16)

where we have used y ¼ 1 2c. Because the wave number q0 corresponds to the soft mode, Fðq0 Þ is the minimum of the function FðxÞ. Therefore we immediately arrive at dn 0. This means that the energy cost of ﬂuctuations in a conﬁned system is no less than that in the bulk; conﬁnements stabilize the homogeneous phase and shift up the instability line deﬁned by cn relative to the bulk spinodal. Furthermore, the spinodal shift is given by pﬃﬃﬃﬃﬃﬃﬃ d ¼ ½dn min ¼ dn0 ¼ ½Fð ln0 Þ Fðq0 Þ=2, where n0 denotes the soft mode of the conﬁned system. At a given condition, if ln0 ¼ q20 can be realized, the spinodal of the conﬁned system will be the same to that of the bulk; if not, the spinodal will be shifted up.

3.3. Approximation theory Around the spinodal point, the soft mode ﬂuctuation is dominant in the system and the structure factor SðqÞ is peaked around the soft mode q ¼ q0 . Therefore, we can expand the structure factor around q ¼ q0 to the Gaussian order. Namely, approximating the structure factor by a Lorentz shape

S1 ðqÞxS1 ðq0 Þ þ

1 1 cðq0 Þðq q0 Þ2 ¼ 2t þ cðq0 Þðq q0 Þ2 2 2

cðq0 Þ ¼ v2q S1 ðqÞ

where, we have deﬁned t ¼ S1 ðq0 Þ=2 ¼ c0s c and the curvature

q0

¼

32plB f 2 q20 1 8plB f 2 þ 3 2 6f0 q2 þ k2 q2 þ k2 0

(18)

0

With the introduction of conﬁnements, the eigenvalue of the Laplace operator changes from q2 to ln . As the result, the energy cost of ﬂuctuation in the conﬁned system reads

En x2t þ

2 pﬃﬃﬃﬃﬃ 1 cðq0 Þ ln q0 2

(19)

The instability line of mode n is deﬁned by requiring En ¼ 0, which leads to the approximation for the instability line shift

1 4

dn x cðq0 Þ

pﬃﬃﬃﬃﬃ

ln q0

2

(20)

4. Results and discussions The concentration of salts in a polyelectrolyte solution is an important factor to determine the phase behaviors. In the present theoretical model, the effect of salt concentration, n0 , is characterized by the Debye-Hückel screening strength k2 ¼ 4plB ðf f0 þ 2n0 Þ. With the increase of the salt concentration, the electrostatic interactions between charges are screened more strongly and this will lead to a phase behavior similar to that of the neutral polymer solution. Speciﬁcally, we can investigate the effect of salt concentration quantitatively through analyzing the wave number of the soft mode q0. As derived, the polyelectrolyte solution is in the osmotic regime and undergoes a microphase separation induced by the mode q0 ¼ ½ðk2 ÞL k2 1=2 for k2 < ðk2 ÞL, while system is in the screening regime and a macrophase separation occurs due to the soft mode of q0 ¼ 0 for k2 > ðk2 ÞL. Therefore, given the set of parameters ðf ; lB ; f0 Þ, a critical concentration of salt, n0 , can be deﬁned by requiring k2 ¼ ðk2 ÞL . After the substitution of k2 and ðk2 ÞL and a straightforward calculation, one can arrive at ! qﬃﬃﬃﬃﬃﬃﬃ ðk2 Þ 3f0 f0 n0 ¼ 8pl L f f20 ¼ f 2 . When the salt concentration is 4pl B

B

n0 > n0 ,

one has q0 ¼ 0 inducing a macrophase sepaincreased to ration, while for low salt concentration of n0 < n0 , a microphase separation is induced by a ﬁnite q0. From the formula, one ﬁnds that, for lB f0 < 3=p, n0 > 0. A scaling relation n0 f holds for the effect of the charge fraction, which means that the critical salt concentration increases with the charge fraction f linearly. Therefore, system admits a larger osmotic regime for polyelectrolytes with higher charge fractions. This trend can be understood by considering the entropy of counterions dissociated from polyelectrolytes. Namely, to keep the charge neutrality in a macro-length-scale, counterions are localized in the region of concentrated polyelectrolyte chains after macrophase separation, which leads to a free energy penalty due to the lowering of the translational entropy of counterions. Hence, a system with more counterions, corresponding to the increase of f, tends to undergo a microphase separation. qﬃﬃﬃﬃﬃﬃ 0 For the effect of lB , it is found that I ¼ f f0 þ 2n0 ¼ f 3f pl , where B

I denotes the ionic strength of the polyelectrolyte solution and I corresponds to the critical ionic strength. Then, one has the scaling 1=2

(17)

53

I lB . This indicates that system has a smaller osmotic regime with increasing lB . Therefore, a solution with a smaller dielectric constant, corresponding to a larger lB, is favorable to undergo a macrophase separation. In Fig. 2(a), we plot the critical salt

54

J. Fu et al. / Polymer 110 (2017) 49e61

v2 n0 =vf20 ¼ 4f

qﬃﬃﬃﬃﬃﬃﬃ

3 f3=2 4plB 0

< 0 for f0 2½0; 1, which means that the

critical line is convex up. The maximum point of the curve is determined by solving vn0 =vf0 ¼ 0, which leads to f0 ¼ 4p3l , where B

f0 denotes the concentration where the maximum of n0 is taken. Lines in Fig. 2(b) correspond to the cases of lB ¼ 0:2; 0:5; 3=p; 2:0, respectively, which represent different scenarios of the concentration dependence. In the case of lB ¼ 0:2 < 43p, f0 > 1, therefore, n0 is a monotonically increasing function of f0 , as shown by the solid line in Fig. 2(b). For lB ¼ 0:52 43p; p3 , f0 < 1 and n0 ðf0 ¼ 1:0Þ > 0,

the curve is shown by the dashed line. At lB ¼ p3, one has f0 < 1 and n0 ðf0 ¼ 1:0Þ ¼ 0, which behaves as shown by the dotted line. When lB ¼ 2:0 > p3, f0 < 1 and n0 ðf0 ¼ 1:0Þ < 0, as shown by the dash-dotted line. Considering the salt-free system with n0 ¼ 0, one can notice that, for lB < p3, n0 > 0 in the whole concentration range, and n0 ¼ 0 < n0 , system is always in the osmotic regime. In the case

of lB > p3, the salt-free system is in the osmotic regime for f0 < p3l and B

it is in the screening regime for p3l < f0 < 1. B From this investigation for the effect of salt concentration, we ﬁnd that, given other parameters, the critical salt concentration n0 is small. This means a macrophase separation can always be induced by putting a ﬁnite amount of salts into the polyelectrolyte solution. Based on this observation, in order to study both types of phase separations, we set the salt concentration n0 ¼ 0 hereafter and analyze the effects of other parameters on the two types of phase separations. The period of the structure to be formed after phase separation is determined by 2p=q0. Clearly, with the soft mode q0 ¼ 0 corresponding to the macrophase separation, a microstructure is characterized by a ﬁnite q0. We can study the transition of a salt-free polyelectrolyte solution from a macrophase separation to a microphase separation with the variation of parameters by looking into the behavior of the soft mode q0. From the formula,

Fig. 2. The critical concentration of salt. Above the line, n0 > n0 , system is in the screening regime and a macrophase separation occurs. Below the line, system is in the osmotic regime and a microphase separation occurs. The charge fraction f ¼ 0:1. The line of n0 ¼ 0 is included to indicate a salt-free solution. (a) the critical concentration n0 as a function of the Bjeruum length lB . (b) the critical concentration n0 as a function of the polyelectrolyte concentration f0 . The solid, dashed, dotted, and dash-dotted lines correspond to lB ¼ 0:2; 0:5; 3=p; 2:0, respectively.

q0 ¼ ½ðk2 ÞL k2 1=2 , setting n0 ¼ 0, it is ready to have pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q0 ¼ f 1=2 ð 48plB f0 4plB f0 Þ1=2 . First, we consider the effect of the charge fraction f. A scaling relation, q0 f 1=2 , is found. As expected, for a neutral solution of

concentration n0 as a function of lB with a given set ðf ¼ 0:1; f0 ¼ 0:2Þ. Above the critical line, system is in the screening regime undergoing a macrophase separation; below the critical line, system is in the osmotic regime undergoing a microphase separation. The critical line of n0 decreases with increasing lB monotonically. This trend is also due to the translational entropy of counterions. Namely, with increasing lB , counterions are more bounded to the polyelectrolye chains due to the stronger electrostatic attraction, which leads counterions to have a smaller entropy. This weakening of the effect of counterion entropy renders the system to favor a macrophase separation. In particular, it can be derived that, independent of the value of f, (as long as f s0), when lB f0 > p3, one has n0 < 0. Therefore, given f0 , when lB > p3f , a salt-free 0

polyelectrolyte solution with n0 ¼ 0 is in the screening regime and it will undergo a macrophase separation. In Fig. 2(b), we investigate the effect of the polyelectrolyte concentration by plotting n0 with respect to the concentration f0 . one can ﬁnd that From the formula of n0 ,

Fig. 3. The soft mode q0 of a salt-free polyelectrolyte solution as a function of the polyelectrolyte concentration f0 for different lB . The charge fraction f ¼ 0:1. The solid, dashed, dotted, and dash-dotted lines correspond to lB ¼ 0:2; 0:5; 3=p; 2:0, respectively.

J. Fu et al. / Polymer 110 (2017) 49e61

f ¼ 0, one always has q0 ¼ 0 corresponding to a macrophase separation. For a polyelectrolyte solution which undergoes a microphase separation corresponding to a ﬁnite q0, with increasing f, q0 increases following the above scaling relation. This trend is due to the translational entropy of counterions as explained above. The larger f is, the more counterions exist leading to a larger entropy cost during a phase separation in a larger length scale. As the result, a microstructure with a shorter period, corresponding to a larger q0, is formed through microphase separation. Second, we consider the effects of f0 and lB . In Fig. 3, we plot q0 with respect to f0 for a salt-free polyelectrolyte solution for different lB . From the formula of a ﬁnite q0 , one can ﬁnd that v2 q0 =vf20 < 0, which indicates a convex up line. Requiring vq0 =vf0 ¼ 0, it is found that the maximum occurs at f0 ¼ 4p3l , B which is identiﬁed with the concentration corresponding to the maximum of the critical salt concentration. Also, the maximum

value q0 ¼ q0 ðf0 Þ ¼ ð3f Þ1=2 independent of lB . The Lifshitz line, where a microphase separation transits to a macrophase

55

separation, is determined by requiring q0 ¼ 0, which leads to fL0 ¼ p3l . Therefore, the soft mode q0 transforms from a ﬁnite value B

to zero continuously when the concentration f0 is increased through fL0 . In the case of fL0 > 1, realized by lB < 3=p, microphase separation occurs in the whole concentration range. For lB ¼ 0:2, f0 > 1, therefore, the soft mode q0 increases monotonically with increasing concentration. For lB ¼ 0:52 43p; p3 , f0 < 1 and fL0 > 1, the line is as shown in Fig. 3. At lB ¼ p3, fL0 ¼ 1, q0 transforms to zero at f0 ¼ 1. For lB ¼ 2 > p3, fL0 ¼ 23p ¼ 0:48, therefore q0 is reduced to zero continuously at f0 ¼ 0:48, accompanying the transit from a microphase separation in f0 < 0:48 to a macrophase separation in f0 > 0:48. In Fig. 4(a), we have plotted the Lifshitz line for the bulk salt-free polyelectrolyte solution. As stated, the line is determined by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðk2 ÞL ¼ 48plB f 2 f0 , leading to ðlB f0 ÞL ¼ 3=p for the Lifshitz line in the case of n0 ¼ 0. In terms of lB and f0 , Fig. 4(a) is constructed. Above the line, system is in the screening regime where a macrophase separation is expected. Below the line, system is in the osmotic regime and a microphase separation shall occur. Based on this line, in the following Figures, we take the parameters for calculation: ðf ; lB ; f0 ; n0 Þ ¼ ð0:1; 1:0; 0:2; 0:0Þ for the osmotic regime; ðf ; lB ; f0 ; n0 Þ ¼ ð0:1; 5:0; 0:2; 0:0Þ for the screening regime. We note that the ﬁnite size effect on the phase behavior is essentially represented through a relative compatibility between the conﬁning size and the soft mode q0. Adjusting the parameter set ðf ; lB ; f0 ; n0 Þ, q0 will be modiﬁed accordingly as discussed above, which, in turn, will lead to corresponding modiﬁcations for the instability lines. However, the general trends of the ﬁnite size effect can be captured by a thorough study choosing these two parameter sets. The energetics of ﬂuctuations in bulk is characterized by S1 ðqÞ ¼ y þ uf0 þ FðqÞ. In Fig. 4(b), we plot FðqÞ with respect to the wave number q, representing the energy cost spectrum of ﬂuctuations in the bulk system. It is clearly demonstrated that, the minimum occurs at q ¼ 0 in the screening regime above the Lifshitz line. On the other hand, below the Lifshitz line, the soft mode changes to a wave number of ﬁnite value representing a microphase separation. In the following, we present results for conﬁnement effects within three geometric spaces in the osmotic regime and the screening regime, respectively.

4.1. Osmotic regime In the osmotic regime, system admits an inﬁnite number of soft modes whose wave numbers lie in a spherical surface of radius q0, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where q20 þ k2 ¼ 48plB f 2 f0 . Under conﬁnements, the ﬂuctuation spectrum is modiﬁed. From Eq. (16), we derive the instability line shift of mode n relative to the bulk spinodal

dn ¼

Fig. 4. Bulk phase behavior. (a) Lifshitz line, in terms of the Bjeruum length lB and the monomer concentration f0 , dividing macro- and micro-phase separation in a salt-free polyelectrolyte solution. (b) The spectrum of ﬂuctuation energy cost. The dashed line represents the spectrum in the screening regime where the soft mode is given by q0 ¼ 0. The solid line represents that in the osmotic regime where a mode of ﬁnite q0 becomes the soft mode.

R2g 4Nf0

2

2

ln q20 ln q20 1 ¼ 2 24f0 ln þ k2 ln þ k

(21)

As claimed before, one has dn 0. The unshifted case is realized when ln ¼ q20 can be met. From the formula, it is clear that, for the conﬁned system, the closer the eigenvalue ln can approach q20 , the smaller the conﬁnement-induced shift is. This represents the commensurability between the size of the conﬁning boundary and the natural microdomain size due to microphase separation.

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J. Fu et al. / Polymer 110 (2017) 49e61

4.1.1. Slab conﬁnement For the slab conﬁnement, one has lnq⊥ ¼ k2n þ q2⊥ for the eigenmode ðn; q⊥ Þ, where kn ¼ np=d. The perpendicular modes ð0; q⊥ Þ propagate along directions of q⊥ parallel to the slabs. Because the conﬁnements are enforced along the b z -direction, these modes are free of conﬁnement effects. Apparently, the soft modes for the perpendicular ﬂuctuations are ð0; jq⊥ j ¼ q0 Þ, leading to the formation of ordered structures of periods 2p=q0 , same to the natural periods in bulk, parallel to the slabs. Since one has l0;jq⊥ j¼q0 ¼ q20 for

the soft modes, from Eq. (21), the spinodal shift d0;jq⊥ j¼q0 ¼ 0, namely, the spinodal of a slab conﬁnement bears no difference from that of the bulk due to the perpendicular soft modes free of conﬁnements. It shall be noted that the perpendicular soft modes are degenerate with wave vectors lying in the circle of radius q0 in the two-dimensional x-plane. This degeneracy leads to the result that it is hard to achieve a long-range orientational order for the perpendicular structures, which motivates a variety of studies in the lithography utilizing guiding surface interaction to orient the to-be-formed morphology [15]. For the modes ðns0; q⊥ Þ, the ﬂuctuations are under conﬁnement effects. One can deﬁne a critical slab depth d ðnÞ ¼ an =q0 for the mode ðn; q⊥ Þ, where an ¼ np. For d < d ðnÞ, one has kn ¼ np=d > q0 . So for a given n, from Eq. (21), it is found that the parallel mode ðn; jq⊥ j ¼ 0Þ, which propagates along the conﬁned b z -direction, corresponds to the mode with the lowest energy cost,

which then induces instability of the homogeneous phase. The instability shift reads

dn0 ¼

1 24f0

k2n q20

2

k2n þ k2

(22)

With increasing d, kn approaches q0, leading to the decrease of dn0 . At d ¼ d ðnÞ, kn ¼ q0 and dn0 reaches zero. Further increasing the slab depth to d > d ðnÞ, then kn < q0 . In this circumstance, ln;q*⊥;n ¼ q20 can always be realized for the ﬂuctuation by developing qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a component with q*⊥;n ¼ q20 k2n . The mode ðn; q*⊥;n Þ becomes the lowest-energy-cost mode for d > d ðnÞ and we have dn;q*⊥;n ¼ 0, namely, the instability line of ðn; q*⊥;n Þ is the same to the bulk spinodal. In Fig. 5(a), we plot the conﬁnement-induced shift for the instability line of the lowest-energy-cost mode for n ¼ 1 and n ¼ 2. For d < d ðnÞ, we plot dn0 with respect to the slab depth d by using the exact formula from Eq. (22) and the approximation formula

dn0 xð1=4Þcðq0 Þðkn q0 Þ2 from Eq. (20). These two lines agree with

each other well near d ðnÞ, while the relatively large deviations away from d ðnÞ are due to the fact that the approximation theory is valid around q0 . For d > d ðnÞ, dn;q*⊥;n ¼ 0. In Fig. 5(b), we demonstrate the variation of the lowest-energycost mode with increasing slab depth schematically by plotting the wave vector of the lowest mode at a given n. One ﬁnds that, below the critical depth d ðnÞ, parallel orders will be formed due to the instabilities of parallel modes. While for d > d ðnÞ, the wave vector of the lowest mode tilts and this will lead to interesting morphologies. There arises another feature in the case of d > d ð1Þ. Namely, the perpendicular modes and the non-perpendicular modes (including the parallel mode) compete to be the soft mode, because these modes become degenerate at certain values. In the thin slab of d < d ð1Þ, the soft modes can only be the perpendicular modes ð0; jq⊥ j ¼ q0 Þ, due to the fact that all non-perpendicular modes cost more energies. At the critical depth d ¼ d ð1Þ, the parallel mode ð1; jq⊥ j ¼ 0Þ costs the same energy as the perpendicular soft modes. In the range of d2ðd ð1Þ; d ð2ÞÞ, the tilt modes ð1; q*⊥;1 Þ compete for

the soft mode. When d ¼ d ð2Þ, the parallel mode ð2; jq⊥ j ¼ 0Þ appears as another candidate for the soft mode. Keeping increasing the slab depth, the soft mode becomes a set with more degenerate ﬂuctuation modes. We illustrate this in Fig. 6 schematically, where we have chosen d ¼ d ðnÞ, the degenerate soft-mode set includes the perpendicular modes ð0; jq⊥ j ¼ q0 Þ, the tilt modes ðm; q*⊥;m Þ with m2½1; n 1, and the parallel mode ðn; jq⊥ j ¼ 0Þ. As the result of the increasing degeneracy of the soft-mode set, a prediction for the equilibrium morphology at the given parameter is beyond the capability of the linear stability analysis. The morphology that

Fig. 5. (a) Illustrating the conﬁnement effects in the slab case by plotting the instability line of the lowest-energy-cost mode for n ¼ 1 and n ¼ 2. The solid and dashed lines correspond to results from the exact and approximation formula, respectively. Below the critical slab depth d ðnÞ ¼ np=q0 , the lowest mode is ðn; jq⊥ j ¼ 0Þ; above qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d ðnÞ, the lowest mode is ðn; q*⊥;n Þ, with q*⊥;n ¼ q20 k2n . The slab depth has been rescaled to q0 d=p, therefore, d ðnÞ rescaled to n. (b) Schematic illustration for variation of the lowest mode of ns0 with increasing the slab depth. Below d ðnÞ, parallel modes induce instability. Above d ðnÞ, tilt modes dominate.

Fig. 6. Schematic illustration for a degenerate set of soft mode. Competition of these degenerate modes leads to the formation of complex morphologies.

J. Fu et al. / Polymer 110 (2017) 49e61

forms eventually will be determined by the dynamics of structure formation, the external stimuli etc. On the other hand, this also provides a playground to generate more intricate morphologies from the homogeneous polyelectrolyte solution by exciting a chosen soft mode in a suitable manner. 4.1.2. Cylinder conﬁnement Conﬁning the homogeneous polyelectrolyte solution within an inﬁnitely long cylindrical pore of radius r0 , the eigenvalue of the Laplace

operator

becomes

2 2 lmnqz ¼ ðkm n Þ þ qz .

The

ðm; n; qz Þ ¼ ð0; 1; qz Þ, with ¼ 0, is the perpendicular mode whose propagating direction is along the axis of the cylindrical pore. Since this mode is free of conﬁnements, it is expected to lead to no spinodal shift from the bulk situation. This is in fact the case in our theory, where the soft mode is ð0; 1; q0 Þ leading to d01q0 ¼ 0 from Eq. (21). The modes with km n s0 experience conﬁnement effects. Similar to the slab case, we can identify a critical cylinder radius m r ðm; nÞ ¼ am n =q0 . For r0 < r ðm; nÞ, kn > q0 ; for this given m and n, the lowest-energy-cost mode corresponds to the mode ðm; n; qz ¼ 0Þ, which admits no perpendicular component. From Eq. (21), the instability line shift for the mode ðm; n; qz ¼ 0Þ reads

dmn0 ¼

1 24f0

(23)

With the increase of the pore radius, the instability point shift decreases and it reaches zero at r0 ¼ r ðm; nÞ. For r0 > r ðm; nÞ, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 km q20 ðkm n Þ can n < q0 and a perpendicular component qz;mn ¼ be developed leading to lmnqz;mn ¼ q20 and dmnqz;mn ¼ 0. In Fig. 7, we plot the conﬁnement-induced shift for the instability line of the lowest-energy-cost mode for ðm ¼ 1; n ¼ 1Þ and ðm ¼ 2; n ¼ 1Þ. For r0 < r, we plot dmn0 with respect to the radius of the cylindrical pore r0 using the exact formula from Eq. (23) and the 2 approximation formula dmn0 xð1=4Þcðq0 Þðkm n q0 Þ from Eq. (20). For r0 > r , dmnqz;mn ¼ 0. The situation is similar to the case of the slab conﬁnement.

Fig. 7. Illustrating the conﬁnement effects in the cylinder case by plotting the instability line of the lowest-energy-cost mode for ðm ¼ 1; n ¼ 1Þ and ðm ¼ 2; n ¼ 1Þ. The solid and dashed lines correspond to results from the exact and approximation form mula, respectively. Below the critical cylinder radius r ðm; nÞ ¼ am n =q0 , where an is the 0 ðxÞ, the lowest mode is ðm; n; q ¼ 0Þ; above r ðm; nÞ, the lowest n-th zero point of Jm z qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 mode is ðm; n; qz;mn Þ, with qz;mn ¼ q20 ðkm n Þ . The cylinder radius has been rescaled to q0 r0 , therefore, r ðm; nÞ rescaled to am n.

The degeneracy of soft modes, as shown in Fig. 4, also arises in the cylinder conﬁnement. At the sizes of r0 < r ð1; 1Þ, the perpendicular mode ð0; 1; q0 Þ is the only soft mode. When r0 ¼ r ð1; 1Þ, the mode ð1; 1; 0Þ competes with ð0; 1; q0 Þ as the soft mode. For r0 > r ð1; 1Þ, the modes ð1; 1; qz;11 Þ appear as candidate soft modes. With the increase of the cylinder radius, the system admits a larger soft-mode set. The ﬁnal structure formed after the homogeneous polyelectrolyte solution loses its stability is beyond the prediction of the present linear stability analysis.

mode

k01

h i2 2 q20 km n 2 þ k2 km n

57

4.1.3. Sphere conﬁnement For a sphere conﬁnement, because all directions are under conﬁnements, there exist no perpendicular modes which are free of conﬁnement effects. The eigenvalue lln ¼ ðkln Þ2 . The instability line shift of the homogeneous solution conﬁned within a spherical pore of radius R is

dln ¼

1 24f0

kln

2

kln

2

2 q20 (24) þ k2

Clearly, dln 0. The closer the value of kln approaches q0 , the smaller the shift dln is. The minimum dln ¼ 0 can be achieved at some chosen pore radius, Rln ¼ aln =q0 , where kln ¼ q0 meaning that the period of the ﬂuctuation mode at this radius is the same to the bulk correspondence. This undeformed mode leads the spinodal point to be the same as the bulk value. Except these special radii, the soft mode must be under deformation, either stretched or compressed compared to the bulk case, which raises the spinodal point and stabilizes the disordered phase. This represents the commensurability effect which is known for a system under frustration. In our case, the frustration is due to the interplay between the boundary condition and the natural period of the microdomain for a microphase separation system. We note that the commensurability effect is absent in the slab and the cylinder conﬁnement cases, due to the fact that the frustration can always be relaxed through the

Fig. 8. Illustrating the commensurability effect within a spherical pore in terms of the spinodal line: the shift of spinodal point, relative to the bulk correspondence, d versus the sphere radius R. The solid and dashed lines correspond to results from the exact and approximation formula, respectively. The inset focuses on the re-entrance of the disordered phase near the bulk spinodal point with the increase of the pore size. Above dt11 , the re-entrance disappears. Two triple points, ðRt11 ; dt11 Þ and ðRt21 ; dt21 Þ are speciﬁed.

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J. Fu et al. / Polymer 110 (2017) 49e61

dimensions free of conﬁnements. In Fig. 8, we plot the phase diagram in terms of d and the pore radius R. Here d ¼ ½dln min at a given radius R. Therefore, d stands for the shift of the spinodal point relative to the bulk value at a given pore size. We also plot the spinodal line using the approximation theory, where dln xð1=4Þcðq0 Þðkln q0 Þ2 from Eq. (20). The two lines agree with each other very well. With the increase of the pore radius, the spinodal line behaves a decaying oscillation. The oscillation represents the commensurability effect clearly. With the increase of the pore radius, the soft mode changes from one to another due to the commensurability effect. We calculate the crossover radii between soft modes here. Let us consider the soft mode jump from ðl1 ; n1 Þ to ðl2 ; n2 Þ at R ¼ Rtl1 n1 , one has dl1 n1 ¼ dl2 n2 . Using the approximation formula in Eq. (20), we can derive

Rtl1 n1 ¼

dtl1 n1

aln11 þ aln22 2q0

(25)

2 cðq0 Þ aln11 aln22 cðq0 Þq20 ¼ ¼ 2 4 16 Rtl n 1

aln11 aln22 aln11 þ aln22

!2 (26)

1

The point ðRtl1 n1 ; dtl1 n1 Þ denotes a triple point, where the ðl1 ; n1 Þ phase, the ðl2 ; n2 Þ phase, and the disordered phase meet. From Eq. (26), one ﬁnds that dtl1 n1 ðRtl1 n1 Þ2 , therefore, the spinodal shift

dtl1 n1 at the triple point gets smaller in a larger pore size. This explains the decay of the amplitude of d in Fig. 8. It is found that, for q0 R > 6, dx0 without oscillation. Because q0 ¼ 2p=l with l denoting the natural period of the microdomain in bulk, this observation means that the commensurability effect is relaxed effectively when the diameter of the spherical pore is larger than about twice of the natural microdomain period. As shown in the inset of Fig. 8, ﬁxing c to be a little higher than the bulk c0s , namely, around the bulk spinodal point, with increasing the spherical pore size, system undergoes a sequence of morphology transitions: Dis/ð1; 1Þ/Dis/ð2; 1Þ/Dis//. Therefore, a re-entrance of the disordered phase with increasing the pore size is observed. Because dtl1 n1 ðRtl1 n1 Þ2 , dt11 , corresponding to the smallest Rt11 , gives the largest spinodal shift at a triple point, which means that, for c c0s dt11, the re-entrance of the disordered phase disappears, as observed in the inset of Fig. 8.

4.2. Screening regime For k2 > k2L , the electrostatic screening renders the long-range Coulomb interaction to be short-range. The absence of a longrange stabilizer leads the system to undergo a macrophase separation induced by the instable q0 ¼ 0 mode. Conﬁning this system within ﬁnite spaces, the instability line shift reads

"

dn ¼ ln

1 2plB f 2 24f0 ln þ k2 k2

# (27)

Note that, under the condition of k2 > k2L , it can be found that dn 0, meaning that the instability line of mode n is shifted up relative to the bulk spinodal. The soft mode in bulk is the long-wavelength ﬂuctuation q0 ¼ 0. In the conﬁned system, the modes with ln s0 lead to dn > 0, while the mode with ln ¼ 0 will result in a same

Fig. 9. Finite size effect of conﬁned macrophase separation in the screening regime. Lengths are scaled to be dimensionless by the Kuhn length. (a) The instability lines of modes ðn; q⊥ Þ ¼ ð1; 0Þ and ðn; q⊥ Þ ¼ ð2; 0Þ for the slab conﬁnement. (b) The instability lines of modes ðm; n; qz Þ ¼ ð1; 1; 0Þ and ðm; n; qz Þ ¼ ð2; 1; 0Þ for the cylinder conﬁnement. (c) The instability lines of modes ðl; nÞ ¼ ð1; 1Þ and ðl; nÞ ¼ ð2; 1Þ for the sphere conﬁnement.

spinodal temperature as the bulk system.

J. Fu et al. / Polymer 110 (2017) 49e61

4.2.1. Slab conﬁnement For the slab conﬁnement, there are two free dimensions lying in the x-plane parallel to the plates. The perpendicular modes of ð0; q⊥ Þ feels no conﬁnements and the soft modes of ð0; q⊥ ¼ 0Þ can be developed to lead to a phase separation in a macroscopic scale along the free dimensions. In this case, one has l00 ¼ 0 and the spinodal is the same as that of the bulk case. If external constraints, such as surface interactions, are enforced to the system and a parallel mode is excited as a result, the instability line of the parallel mode must be shifted up compared to the bulk long-wavelength instability. One arrives at the instability line by substituting ln0 ¼ k2n ¼ ðnp=dÞ2 into Eq. (27). The size dependence has been shown in Fig. 9(a), where we have chosen n ¼ 1 and n ¼ 2 for illustration. Starting with a small slab depth, the instability line is highly shifted up due to the ﬁnite size effect. Increasing the slab depth, the instability point decreases monotonically, reaching zero at a large slab depth where the ﬁnite size effect is effectively relaxed in the system. In the limit of strong screening, k2 /∞, from Eq. (27) we have dn0 xln0 =24f0 d2 . This represents the typical mean-ﬁeld ﬁnite size scaling of the Ising universality class. 4.2.2. Cylinder conﬁnement For the cylinder conﬁnement, the dimension along the axis of the cylindrical pore is a free direction where the soft mode can be developed free of conﬁnements and leads to a macrophase separation. Therefore, the spinodal due to this perpendicular soft mode is the same to the bulk value. For the modes developed within the circular plane in the presence of a conﬁning boundary, the ﬁnite size effect leads to a shifted-up instability point of the disordered phase. By substituting 2 2 m lmn0 ¼ ðkm n Þ ¼ ðan =r0 Þ into Eq. (27), one arrives at the formula for

the instability lines. In Fig. 9(b), we illustrate this ﬁnite size effect by choosing the modes ð1; 1; 0Þ and ð2; 1; 0Þ. Similar to the parallel mode instability in the slab case, the instability point decreases monotonically with increasing the pore size and recovers the bulk value at a large pore. In the strong screening limit of k2 /∞, we have dmn0 xlmn0 =24f0 r2 0 , which represents the typical meanﬁeld ﬁnite size scaling of the Ising universality class. 4.2.3. Sphere conﬁnement For the sphere conﬁnement, there exist no perpendicular modes since there are no free dimensions in the system. Therefore, the ﬁnite size effect raises the spinodal relative to that of the macrophase separation in bulk. The formula for the instability line is obtained by substituting lln ¼ ðkln Þ2 ¼ ðaln =RÞ2 into Eq. (27). This is plotted in Fig. 9(c) by choosing two modes ðl ¼ 1; n ¼ 1Þ and ðl ¼ 2; n ¼ 1Þ, where the instability line of ðl ¼ 1; n ¼ 1Þ corresponds to the spinodal line. In the strong screening limit, one ﬁnds that dln xlln =24f0 R2 , representing the mean-ﬁeld ﬁnite size scaling of the Ising universality class. 5. Conclusions We study conﬁnement effects on the phase separation in a polyelectrolyte solution utilizing a Gaussian ﬂuctuation theory. The existence of a conﬁning boundary of ﬁnite size renders a discrete spectrum of density ﬂuctuations, which, depending on the phase transition type in bulk, leads to two types of conﬁnement effects: commensurability for the bulk-microphase-separation and ﬁnite size scaling for the bulk-macrophase separation. This ﬁnding characterizes, in general, the ﬁnite size effect for two types of statistical-mechanics-models: the Brazovskii and the Ising

59

universality class. Speciﬁcally, conﬁning the Brazovskii model leads to a decaying oscillatory dependence of instability point on the conﬁning size, which is attributed to the commensurability between the ﬁnite wavelength of soft mode and the conﬁning boundary size. On the other hand, the instability point of the conﬁned Ising model decreases to the bulk value monotonically, behaving the typical mean-ﬁeld ﬁnite size scaling of the Ising model. These ﬁndings can serve as a guide for experimentalists to investigate the structure formation through conﬁning a polyelectrolyte solution within ﬁnite geometric spaces. Furthermore, the results on conﬁnement-induced instability line shifts and responsible ﬂuctuation modes can also provide an insightful perspective to analyze the related structures formed by biomacromolecules in a cell. For a system under conﬁnements, in general, conﬁnement effect can be classiﬁed as ﬁnite size effect and surface effect. Finite size effect is due to the ﬁnite extension of the system under consideration. Surface effect is due to the interactions of different types between the conﬁning surfaces and components of the system. The surface effect has been studied extensively in the polymer community [32,33]. Essentially, the surface interaction preferential to one component breaks the translational symmetry of the system. As the result, non-homogeneous structures are induced by the surface interaction to be the equilibrium morphology even at good solvent conditions for a polymer solution. For example, the adsorption of polyelectrolye chains onto charged surfaces due to the electrostatic interaction leads the system to form a nonhomogeneous structure [32]. To construct a theoretical model including the surface effect, a term which couples the surface ﬁeld and the order parameter ﬁeld needs to be included into the model Hamiltonian. Through minimizing with respect to the order parameter ﬁeld utilizing the new model Hamiltonian, nonhomogeneous morphologies can be found as equilibrium states. However, in the present work, our focus is on the ﬁnite size effect. Our treatment is to perform a linear stability analysis of the homogeneous structure, which is the equilibrium state of the system without surface interaction. Thanks to the symmetric properties of the homogeneous state, the eigen-problem of the inverse structure factor operator is identiﬁed with that of the Laplace operator in given geometric spaces, which are ready to be solved analytically. If the surface interaction is included into the problem, instead of the homogeneous state, a non-homogeneous structure will appear as the equilibrium state. The eigen-problem of the inverse structure factor operator for the non-homogenous structure usually needs to be solved in a numerical or approximation manner. Therefore, further works need to be done in order to clarify the surface effect on the phase behaviors which includes two respects: (1) surfaceinduced structures; (2) stabilities of these structures, which are out of scope of the present study. It is worth comparing the arising structures in the present work, which are characterized by the ﬂuctuation modes, to the conﬁnement-induced morphologies observed in the previous works. The set up of a slab conﬁnement has been studied extensively [12]. The parallel modes ðns0; q⊥ ¼ 0Þ and the perpendicular modes ðn ¼ 0; q⊥ s0Þ in our work correspond to the parallel and the perpendicular lamellae, respectively. The only soft mode in a thin slab of d < d ð1Þ is the perpendicular mode, which is consistent with the result that the equilibrium structure is the perpendicular lamellae for a thin slab [12]. For a thick slab, a degenerate set of soft modes appear in our theory, which corresponds to the fact that the perpendicular, the tilt, and the parallel morphologies compete to be the equilibrium structures. The cylinder conﬁnement corresponds to a conﬁned polymer system in narrow capillaries. In this case, the ﬂuctuation modes are classiﬁed

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J. Fu et al. / Polymer 110 (2017) 49e61

by the “quantum numbers” ðm; n; qz Þ. Speciﬁcally, m characterizes the angular modulation of the arising morphology. In particular, m ¼ 0; 1; 2 correspond to rotationally symmetric, Janus-cylinder, and Janus-dumbbell structures, respectively, which have been found in previous studies for conﬁned diblock copolymers and twocomponent cylindrical bottlebrush polymers [28,29]. n deﬁnes the radial modulation of the morphology along the conﬁned direction. The parallel mode ðm ¼ 0; n; qz ¼ 0Þ in the cylindrically conﬁned system is identiﬁed with the concentric lamellae in a tube. qz depicts the modulation along the axis of the cylinder. A ﬁnite qz means that an axial order is induced in the conﬁned system, corresponding to the lamellar-like morphology in reference [29]. The modes ðm; n; qz Þ can also describe the helical structures which are induced as equilibrium morphologies in a cylindrical pore [12,28,29]. For the sphere conﬁnement, the three “quantum numbers” are ðl; m; nÞ, which describe the modulations of arising structures along the polar angle, the azimuthal angle, and the radial direction, respectively. These modes characterized by ðl; m; nÞ can provide a quantitative way to classify the conﬁnement-induced morphologies observed in a spherical nano-pore [12]. We would like to note several cautions for the theoretical model developed here. First, the model is based on an analysis of ﬂuctuation spectrum in the Gaussian level, which provides linear stability lines of the disordered polyelectrolyte solution. Therefore, the non-linear phase behaviors are beyond the power of the present model. To name an example, the counterion condensation of a polyelectrolyte system cannot be captured by a direct application of the present model. To take counterion condensation into account, speciﬁc methods, such as two-phase model, have to be built into the present model. Second, the Gaussian ﬁeld model corresponds to the RPA approximation for the correlation between density ﬂuctuations. Therefore, strong charge correlation related phenomena cannot be studied by the present Gaussian model. To capture the correlation effect, within the coarse-grained ﬁeld theoretic method, appropriate renormalization schemes have to be devised. Another methodology is to treat the sizes of ions explicitly in the model. These are out of scope of the present work and will be studied in our future works. Third, while, the analysis on the ﬂuctuation modes can provide insights on the conﬁnement-induced structures in the nano-space, the equilibrium morphologies formed through phase separations in the system cannot be predicted in the present work, due to the nature of a linear stability analysis. In order to obtain a detailed phase diagram for the morphologies, free energies corresponding to different morphologies have to be calculated resorting to, for example, the self-consistent ﬁeld theory, or the Landau theory up to the fourth order. With these noted, the present model sets a ﬁrst step to understand the phase behavior of a conﬁned polyelectrolyte solution. Acknowledgments This work is supported by the National Natural Science Foundation of China Grant Nos. 21544007, 21104094, 21434001, and 21374011. B. M. acknowledges support from the Youth Innovation Promotion Association, CAS (No. 2012316). Appendix In this Appendix, we give a detailed exposition on the derivation of Eq. (2) for the inverse correlation matrix of the polyelectrolyte solution. The system is composed of polyions, counterions, salts, and solvents. Treating solvents implicitly by introducing a homogeneous dielectric constant ε, we write down the bare matrices for the

system in order. The bare correlation matrix reads

0

S0p

B S0 ðqÞ ¼ @ 0 0

0 S0c 0

1

0

S0 ðqÞ C 0 A¼@ 0 0 S0 0

n

0 f0;c 0

1 0 0 A n0

(28)

where, S0p ¼ S0 ðqÞ ¼ ðNf0 Þ=½1 þ q2 R2g =2, S0c ¼ f0;c , and S0n ¼ n0 denote the correlation functions of polyions, counterions, and salts in the reference system free of interactions, respectively. Note that the bare correlation matrix is diagonal due to the absence of correlations between different components in an interaction-free system. The bare interaction matrix reads

0

0 Vpp

B 0 V0 ðqÞ ¼ @ Vcp

0 Vpc

0 Vpn

1

0 0 C Vcc Vcn A 0 0 Vnc Vnn . 0 4plB f 2 q2 þ y þ uf0 . ¼@ 4plB f q2 . 4plB f q2 0 Vnp

. 4plB f q2 . 4plB q2 . 4plB q2

. 4plB f q2 1 . 4plB q2 A . 4plB q2 (29)

0 , with a; b ¼ p; c; n, stand for the bare interaction bewhere Vab

tween components and the speciﬁc forms have been substituted. Within the framework of the Gaussian ﬁeld theory, the inverse correlation matrix of the system is constructed as: S1 ðqÞ ¼ S1 0 ðqÞ þ V 0 ðqÞ. Inverse the bare correlation matrix and substitute into this formula, we obtain Eq. (2). References [1] M. Seul, D. Andelman, Domain shapes and patterns: the phenomenology of modulated phases, Science 267 (1995) 476e483. [2] J.-L. Barrat, J.-F. Joanny, Theory of polyelectrolyte solutions, Adv. Chem. Phys. 94 (1996) 1e66. [3] V.Yu. Borue, I.Ya. Erukhimovich, A statistical theory of weakly charged polyelectrolytes: ﬂuctuations, equation of state, and microphase separation, Macromolecules 21 (1988) 3240e3249. [4] E.E. Dormidontova, I.Ya. Erukhimovich, A.R. Khokhlov, Microphase separation in poor-solvent polyelectrolyte solutions: phase diagram, Macromol. Theory Simul. 3 (1994) 661e675. [5] I.A. Nyrkova, A.R. Khokhlov, M. Doi, Microdomain structures in polyelectrolyte systems: calculation of the phase diagrams by direct minimization of the free energy, Macromolecules 27 (1994) 4220e4230. [6] J.-F. Joanny, L. Leibler, Weakly charged polyelectrolytes in a poor solvent, J. Phys. Fr. 51 (1990) 545e557. [7] T.A. Vilgis, R. Borsali, Mean-ﬁeld theory of concentrated polyelectrolyte solutions: statics and dynamics, Phys. Rev. A 43 (1991) 6857e6874. [8] K.B. Zeldovich, E.E. Dormidontova, A.R. Khokhlov, T.A. Vilgis, Microphase separation transition for polyelectrolyte gels in poor solvents, J. Phys. II Fr. 7 (1997) 627e635. [9] F. Schosseler, A. Moussaid, J.P. Munch, S.J. Candau, Weakly charged polyelectrolyte gels: temperature and salt effects on the statics and the dynamics, J. Phys. II Fr. 1 (1991) 1197e1219. [10] A. Moussaid, F. Schosseler, J.P. Munch, S.J. Candau, Structure of polyacrylic acid and polymethacrylic acid solutions: a small-angle neutron-scattering study, J. Phys. II Fr. 3 (1993) 573e594. [11] M. Shibayama, T. Tanaka, C.C. Han, Small-angle neutron scattering study on weakly charged temperature sensitive polymer gels, J. Chem. Phys. 97 (1992) 6842e6854. [12] A.-C. Shi, B. Li, Self-assembly of diblock copolymers under conﬁnement, Soft Matter 9 (2013) 1398e1413. [13] L.-T. Yan, X.-M. Xie, Computational modeling and simulation of nanoparticle self-assembly in polymeric systems: structures, properties and external ﬁeld effects, Prog. Polym. Sci. 38 (2013) 369e405. [14] S. Yang, A. Vishnyakov, A.V. Neimark, Self-assembly in block polyelectrolytes, J. Chem. Phys. 134 (2011) 054104. [15] S. Kim, P.F. Nealey, F.S. Bates, Directed assembly of lamellae forming block copolymer thin ﬁlms near the order-disorder transition, Nano Lett. 14 (2014) 148e152.

J. Fu et al. / Polymer 110 (2017) 49e61 [16] L.D. Gelb, K.E. Gubbins, R. Radhakrishnan, M. Sliwinska-Bartkowiak, Phase separation in conﬁned systems, Rep. Prog. Phys. 62 (1999) 1573e1659. [17] Y. Rouault, J. Baschnagel, K. Binder, Phase-separation of symmetrical polymer mixtures in thin-ﬁlm geometry, J. Stat. Phys. 80 (1995) 1009e1031. [18] H. Nakanishi, M.E. Fisher, Critical-point shifts in ﬁlms, J. Chem. Phys. 78 (1983) 3279e3293. [19] A.O. Parry, R. Evans, Inﬂuence of wetting on phase-equilibria: a novel mechanism for critical-point shifts in ﬁlms, Phys. Rev. Lett. 64 (1990) 439e442. [20] H. Tang, I. Szleifer, S.K. Kumar, Critical-temperature shifts in thin polymer blend ﬁlms, J. Chem. Phys. 100 (1994) 5367e5371. [21] J.E. Pye, C.B. Roth, Two simultaneous mechanisms causing glass transition temperature reductions in high molecular weight freestanding polymer ﬁlms as measured by transmission ellipsometry, Phys. Rev. Lett. 107 (2011) 235701. €nhals, R. Zorn, B. Frick, Inelastic neutron spectroscopy as a tool to [22] A. Scho investigate nanoconﬁned polymer systems, Polymer 105 (2016) 393e406. [23] L. Chen, J.M. Torkelson, Tuning the Tg-conﬁnement effect in thin polymer ﬁlms via minute levels of residual surfactant which ”cap” the free surface, Polymer 87 (2016) 226e235. [24] A.W. Tan, J.M. Torkelson, Poly(methyl methacrylate) nanotubes in AAO templates: designing nanotube thickness and characterizing the Tg-conﬁnement effect by DSC, Polymer 82 (2016) 327e336. [25] S.-J. Xie, H.-J. Qian, Z.-Y. Lu, Hard and soft conﬁnement effects on the glass

61

transition of polymers conﬁned to nanopores, Polymer 56 (2015) 545e552. [26] B. Miao, D. Yan, C.C. Han, A.-C. Shi, Effects of conﬁnement on the orderdisorder transition of diblock copolymer melts, J. Chem. Phys. 124 (2006) 144902. [27] B. Miao, D. Yan, R.A. Wickham, A.-C. Shi, The nature of phase transitions of symmetric diblock copolymer melts under conﬁnement, Polymer 48 (2007) 4278e4287. [28] I.Ya. Erukhimovich, A. Johner, Helical, angular and radial ordering in narrow capillaries, EPL 79 (2007) 56004. [29] I.Ya. Erukhimovich, P.E. Theodorakis, W. Paul, K. Binder, Mesophase formation in two-component cylindrical bottlebrush polymers, J. Chem. Phys. 134 (2011) 054906. [30] B. Miao, On the structure of statistical ﬁeld theory of polymers, Int. J. Mod. Phys. B 27 (2013) 1361009. [31] T.A. Vilgis, Polymer theory: path integrals and scaling, Phys. Rep. 336 (2000) 167e254. [32] A.G. Cherstvy, R.G. Winkler, Polyelectrolyte adsorption onto oppositely charged interfaces: image-charge repulsion and surface curvature, J. Phys. Chem. B 116 (2012) 9838e9845. [33] P.G. de Gennes, Polymer solutions near an interface: 1. Adsorption and depletion layers, Macromolecules 14 (1981) 1637e1644.

Confinement effects on phase separation of a polyelectrolyte solution

Confinement effects on phase separation of a polyelectrolyte solution

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