Molecular Thermodynamic Modeling of the Morphology Transitions in

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J. Phys. Chem. B 2010, 114, 8846–8860

Molecular Thermodynamic Modeling of the Morphology Transitions in a Solution of a Diblock Copolymer Containing a Weak Polyelectrolyte Chain Alexey I. Victorov,*,† Nikolay V. Plotnikov,‡ and Po-Da Hong§ Department of Chemistry, St. Petersburg State UniVersity, UniVersitetsky Prosp., 26, 198504, St. Petersburg, Russia, Department of Chemistry, UniVersity of Southern California, 3620 McClintock AVenue, 418 SGM, Los Angeles, California 90089, and Department of Polymer Engineering, National Taiwan UniVersity of Science and Technology 43, Section 4, Keelung Road, Taipei 10607, Taiwan ReceiVed: February 2, 2010; ReVised Manuscript ReceiVed: June 1, 2010

For a solution of the diblock copolymer composed of a hydrophobic block and a weak polyelectrolyte block, we obtain regions of stable aggregate morphologies in pH-solution salinity plane with the aid of the selfconsistent field theory in the strong-segregation approximation. Lamellar, cylindrical, branched cylindrical, and spherical aggregates have been considered in the large interval of pH and salinity. The morphology stability maps are obtained to help control self-assembly of aggregates by variation of pH and salinity of the medium. In qualitative agreement with experiment, our calculations predict the coexistence of long wormlike micelles with branched and spherical micelles in transition zones. We compare the results of our calculations with available computer simulation and experimental data on micelles and brushes (planar and curved) formed by a diblock copolymer with one polyelectrolyte block. We show that for both weak and strong polyelectrolytes the agreement between the theory and experiment is satisfactory in most systems. 1. Introduction Controlling self-assembly in solutions is a key issue in many applications such as encapsulation of drugs, nanoreactor engineering, and fabrication of nanoporous materials. For example, gels of differing structure may be prepared from micellar solution when cross-linking is performed upon tuning the micellar characteristics by variation of salinity and pH of the medium.1 For providing guidance to control self-assembly, a reliable prognosis of aggregation behavior would be indispensable. Solutions of diblock copolymers containing weak polyelectrolyte chains show a variety of self-assembled structures1,2 including spherical, wormlike and toroidal micelles, branched wormlike aggregates, bicontinuous structures, vesicles, lamellae, and gels. This rich morphologic behavior can be manipulated by changing the length of copolymer’s blocks and also by the response of the aggregative system to external stimuli such as changing temperature, pH, and salinity of the environment. For the aggregates of lamellar, cylindrical, and spherical shape, Zhulina and Borisov3 derived an analytical version of the self-consistent field theory for the limits of crew cut (large core, thin corona) and star like (thick corona, small core) aggregates at high and low salinity of solution. With increasing salinity the morphology transitions lamellae-cylinders-spherescylinders-lamellae have been predicted from this theory for weak polyelectrolyte at pH around pK, where K is the dissociation constant of the polyelectrolyte. In this work we include branched wormlike micelles in addition to spherical, cylindrical and lamellar aggregates studied previously.3 Our model is based on the self-consistent field theory of Zhulina, Birshtein, and Borisov3,4 for the annealed polyelectrolyte brush; however, we choose a full numerical * Corresponding author. E-mail: [email protected]. † St. Petersburg State University. ‡ University of Southern California. § National Taiwan University of Science and Technology.

version of that theory to be able to include the intermediate regimes between those of crew cut and star like micelles and consider arbitrary salinities. We compare the results from two different approximations for describing the elasticity of the swollen aggregate corona: (1) the Alexander-de Gennes brush, where all polyelectrolyte chains terminate at the same distance from aggregate’s core,5 and (2) the parabolic brush that implies a radial distribution of chain ends.6 The model for predicting the equilibrium shape and structure of the aggregate in solution is derived in the next section. We then calculate the stability maps for the aggregates of different shapes in the pH-salinity coordinates for varying lengths of the hydrophobic and the polyelectrolyte subchains of the diblock copolymer molecule. For a planar polyelectrolyte brush, our model predictions are tested versus the results of computer simulation.7 The paper is concluded by comparing model calculations with experiment on the structure of spherical diblock copolymer ionic micelles and brushes. 2. The Model We consider diblock copolymer aggregates of different shapes in an aqueous solution of fixed pH and salinity. The diblock copolymer chain consists of NB hydrophobic B segments and NA polyelectrolyte A segments. Hydrophobic aggregate cores remain completely dry. Swollen polyelectrolyte brushes contain water and mobile univalent ions: H+, OH-, Na+, and Cl-. Subchains of type A carry negative charges owing to partial dissociation of weakly acidic groups. The expressions are derived here for polyacidic chains but are readily transformed for the case of weak polybases by changing the sign of fixed charge. The micelles are assumed monodisperse. In this work we do not account for the interaction between micelles and for the intermicellar-scale entropic contributions to the free energy of the system. Thus our model describes the free energy of a single aggregate of a given shape in a large external reservoir of water and ions. We model aggregates of different assumed

10.1021/jp100987h  2010 American Chemical Society Published on Web 06/18/2010

Morphology Transitions in Diblock Copolymer

J. Phys. Chem. B, Vol. 114, No. 27, 2010 8847

shapes (spherical, cylindrical, branched cylindrical, and lamellar) and determine the dimensions and the shape of the equilibrium aggregate by minimizing the free energy of the system. The free energy of an aggregate in solution includes an electrostatic part and a number of nonelectrostatic contributions resulting from chain elasticity and nonelectrostatic interactions between copolymer chains and solvent molecules. Electrostatic Contribution to the Aggregate Free Energy. Consider a swollen polyelectrolyte chain (or swollen polyelectrolyte brush) in a bulk solution of salt. For a given particular arrangement of charges fixed along the chains (quenched polyelectrolyte), the excess electrostatic free energy of polyelectrolyte over the bulk solution is given by

Felec ) Uelec - Selec /kB )

1 2

∫ Ψ(r)Fq(r) dV +

∑ ∫ {Fi[ln Λi3Fi - 1] - Fib[ln Λi3Fib - 1]} dV

Λ ) ln

Rb 0 0 0 0 b ) µHA - µΗ + - µA- - ln FH+ ) µHA 1 - Rb bulk µA0 - - µHbulk (5) + ) ln K + pH

Here Rb is the degree of dissociation of the polyacid in some hypothetical state, where Ψ(r) ) 0, e.g., outside the brush in the external bulk solution and with no effect of neighboring 0 , µA0 -, and µH0 + are the charges on the polyelectrolyte chain; µHA reference chemical potentials of the polyacid group before and + is the chemical potential of H+ ion in after dissociation, µHbulk the bulk solution, FHb + and pHbulk are the concentration of hydrogen ion and the pH of the bulk solution, respectively. Equation 5 shows that the Lagrange multiplier is determined by the pH of the environment and by the dissociation equilibrium constant K

(1)

i

0 ln K ≡ µHA - µH0 + - µA0 -

Here Uelec is the excess electrostatic energy and Selec is the excess entropy of mobile charge; the integration is performed over the entire volume of the system; Fi is the local number density of a mobile ion of kind i; Λi is the de Broglie wavelength; Ψ(r) is the local electric potential in reduced units: Ψ ) eψ/(kBT), where ψ is the electric potential, e is the electron charge, kB is Boltzman’s constant, and T is the temperature; Fq(r) is the local charge density (number of elementary charges per volume) in the swollen polyelectrolyte

Fq(r) ) FNa+(r) + FH+(r) - FCl-(r) - FOH-(r) - R(r)cA(r) (2) where R(r) is the fraction of charged monomers of subchains A (local degree of dissociation) and cA(r) is the local number density of polyelectrolyte monomers at point r. Here and below all energies are expressed in units of kBT. Minimization of Felec with respect to profiles Fi(r) leads to the Boltzmann distribution of mobile charge. For a weak polyelectrolyte, there is an additional entropic contribution to the free energy arising from possible different arrangements of charged groups along each chain at fixed degree of dissociation8,9

Hence the resulting Euler-Lagrange equation that determines the equilibrium distribution R(r) of dissociated groups is

Λ ) ln

Rb R(r) ) -Ψ(r) + ln 1 - Rb 1 - R(r)

F

)

R(r)FH+(r) )K 1 - R(r)

(8)

The equilibrium value of the thermodynamic potential is obtained using eqs 3, 5, and 7 in eq 4

∫ cA[R ln 1 -R R + ln(1 - R)] dV + 0 (µA0 - µHA + µHbulk) ∫ RCA dV ) Felec + ∫ ΨRCA dV + ∫ CA ln(1 - R) dV

elec Fion ) F +

∫ cA(r)[R(r)ln R(r) + (1 - R(r)) ln(1 - R(r))] dV

+

(9)

(3) The spatial variation of the degree of dissociation is found by minimizing the electrostatic free energy for a given total value of fixed charge, ∫R(r)cA(r) dV. Thus we minimize the thermodynamic potential

(7)

Using eq 7 and the Boltzmann distribution law for mobile hydrogen ions, ln FH+(r) ) ln FbH+ - Ψ(r), eq 7 may be expressed as a local version of the mass-action law in familiar form

-

diss

(6)

We introduce the excess semigrand potential

Ωion ≡

∫ ωiοn(r) dV ≡ Fion - ∑ µibulk ∫ [Fi(r) - Fibulk] dV i)(

Fion ≡ Felec + Fdiss - Λ

∫ R(r)cA(r) dV

(10) (4)

where Λ is the Lagrange multiplier that controls the total value of fixed charge. Performing functional differentiation of eq 4 with respect to R(r) and inserting Ψ(r) ) 0 in the result, we obtain for the Lagrange multiplier

where the summation is over all mobile ions, ω(r) is the free energy density (i.e., the free energy per unit volume) that reflects all electrostatic contributions to the free energy of a weak and Fbulk are the chemical potential polyelectrolyte brush; µbulk i i and the concentration of mobile ions, respectively, in the external reservoir. From eqs 1, 2, 9, and 10, we have

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Ωion )

∑∫ )(

1 2

{[

Fi ln

∫ Ψ(F

+

Fi Fbi

] }

Victorov et al. mix Ω ) Fsurf + Fstr + FFlory +

- 1 + Fbi dV +

∫ ΨRC dV + ∫ C ln(1 - R) dV

- F- - RCA) dV +

A

A

(11)

Performing algebra, we obtain the resulting expression for ωion(r)

1 ωion(r) ) Ψ(r)[Φion sinh Ψ(r) + R(r)cA(r)] 2 Φion(cosh Ψ(r) - 1) - cA(r) ln(1 - R(r)) (12) where Φion ) ∑iFbi is the sum of the number densities of the mobile ions in the bulk. For 1:1 electrolytes, Φion measures the ionic strength of the solution. In eq 12, the first term is the local electrostatic energy, the second term takes into account the local excess osmotic pressure of mobile ions over that in the bulk, and the last term reflects the ideal van’t Hoff entropic contribution to the free energy of a partly charged polyelectrolyte. The problem of estimating the electrostatic free energy is simplified substantially by applying the assumption of local electroneutrality.3,9 This assumption has been justified for charged brushes by experiment and computer simulation.1 When the local charge (reflected by the square brackets in eq 12) is zero the first term vanishes. Combination of the local electroneutrality condition with the Boltzmann distribution of mobile charges leads to a quadratic equation that expresses the local version of the Donnan rule. The local electrostatic potential may thus be readily found from this quadratic equation. Using this electrostatic potential in eqs 7 and 12, we find our final expressions for the local degree of dissociation and for the local free energy density of the weak polyelectrolyte brush as function of the local polyelectrolyte concentration

Rb R(r) ) 1 - R(r) 1 - Rb

((

R(r)cA(r) Φion

)

2

+1-

)

R(r)cA(r) Φion (13)

and

ωion(r) ) -Φion

(

((

cA(r) ln 1 +

) [(

R(r)cA(r) Φion

Rb 1 - Rb

2

)

+1-1 -

R(r)cA(r) Φion

)

2

+1-

R(r)cA(r) Φion

])

(14)

Equation 13 is a cubic equation with respect to R(r). Equations 13 and 14 are identical to those developed in analytical theories of annealed polyelectrolyte brushes4 and block-copolymer micelles.3 Nonelectrostatic Contributions to the Aggregate Free Energy. Apart from the electrostatic part, the aggregate free energy includes contributions from chain elasticity and from the nonelectrostatic interactions. In the limit of strong segregation of A and B subchains the total semigrand potential of the aggregate may be written as10

∫ ωion(r) dV

(15)

Here the first term is the free energy of the bare interface between the hydrophobic domain and the hydrophilic domain of the aggregate, Fstr ) FstrA + FstrB is the sum of stretching contributions mix from different parts of the diblock copolymer molecule and FFlory is the free energy of mixing of water with the copolymer segments in the swollen corona of the aggregate. In eq 15 and below the free energies are calculated per copolymer chain. We have

Fsurf ) σABσ

(16)

where σ is the area per copolymer molecule at the (sharp) A-B interface, σAB is the bare interfacial tension. For a symmetric copolymer11 σAB ) (χAB/6)1/2/aK2, where aK ≡ aKA ) aKB is the length of the Kuhn segment and χAB is the Flory interaction parameter for copolymer segments A and B. For a nonsymmetric copolymer, aKA * aKB, analytical expression for σAB is also available but is more involved.12,13 The free energy of mixing is given by mix FFlory )

ωFlory(r) )

∫ ωFlory(r) dV

(17)

1 [φ (1 - φA)χAS + (1 - φA) ln(1 - φA)] VA A (18)

where VA is the specific volume of monomer A, φA ) VAcA is the local volume fraction of segments A in the swollen brush of the aggregate, the volume of solvent molecule is taken approximately equal to VA, and χAS is the Flory parameter for interactions between solvent and monomer A. The free energy is a functional of the aggregate shape and describing aggregate of an arbitrary shape is difficult. For unidimensional morphologies (sphere, cylinder, and plane), the problem is much simpler, because only the elasticity part of the free energy and the volume element in the spatial integrals depend on the aggregate shape. For the branching portion of a micelle we use two different approximations: (1) An effective unidimensional description of the 3-D branch. The central idea of this method (developed in previous works on bicontinuous copolymer gels14-17 and micelles of classical surfactants18) is that the most essential property, specifying the molecular packing for a given morphology is how fast the area element changes with the spatial coordinate; (2) 3-D model of the micelle branch shown in Figure 1 where the aggregate is built of the elements having specified geometry (and optimal dimensions). The micellar junction (branch) connects three semiinfinite cylindrical aggregates and consists of three inner pieces of a toroid and a planar bilayer patch in the middle. As in aggregation models of classical surfactants,18 we assume additivity of contributions from these elements to the total free energy of the branch. Details of implementation of this 3-D model of branch are delegated to Appendixes A and C. For the unidimensional structures, we write geometric quantities in terms of the reduced distance from the center of the aggregate core: y ≡ x/R, where R is half the linear size of the core and x is the actual distance. The reduced surface area at point y is given by

a(y) ≡

A(y) ) yν σ

(19)



cA(r) )

dn σa(y)R dy

(22)

For the elastic free energy per chain in the A brush, we have

FstrA )

3 R2 2aKA2

dy ∫1β ( dy dn )

(23)

where β is the reduced coordinate of the outer edge of the swollen brush. Thus from FstrA ) ∫ωstrA(r) dV the local stretching free energy density is given by3

ωstrA(y) )

Figure 1. Geometry of a branch that connects three cylindrical micelles. Weak polyacid A subchains (dark) form swollen corona. Hydrophobic B subchains (light) form dry core of the aggregate.

where σ is the area per chain at the hydrophobic surface, A(y) is the actual area per chain at point y, and ν ) 0, 1, 2 for the aggregate of lamellar, cylindric, and spherical morphology, respectively. For the unidimensional model of a micelle branch we take ν ) 1/2, see Appendix A. We note that this choice of the exponent leads to a simple mathematical description; for example, it makes possible an analytical solution of the linearized Poisson-Boltzmann equation.17 The volume element per polymer chain is given by

dV ) A(y) dx ≈ σa(y)R dy

(20)

The elasticity contribution from the dry core is calculated from Semenov’s analytical theory for strongly stretched chains19

strB

F

R3 ) BBσ VB

∫0

1

[

]

2

(24)

R3 VA

∫1β φA(y)(1 - y)2yν dy

(25)

where BA ) 3π2/(8NA2aKA2). Hence

ωstrA(y) ) BA

R2 φ (y)(1 - y)2 VA A

(26)

Equilibrium Aggregates in Solution. The equilibrium profile of polymer segments in the swollen aggregate’s corona is obtained by minimizing the semigrand potential per chain, eq 15, taking into account the conservation of polymer mass

σR

∫1β cA(r)yν dy ) NA

(27)

The Euler-Lagrange equation is

(21)

where BB ) 3π /(8NB aKB ) and VB is the specific volume of monomer B. For the swollen polyelectrolyte brush the chain-stretching free energy depends on the spatial distribution of polymer segments. In this work we compare two extreme cases: (1) the Alexander-de Gennes brush where every chain stretches (uniformly) from the grafting surface to the outer end of the brush and (2) the parabolic brush where a chain end may be located at any point within the brush.20,21 Both approximations are incorrect22 because (1) always underestimates the concentration of chain ends inside the swollen brush particularly at low grafting densities, whereas (2) leads to unphysical negative density of chain ends in the regions close to convex grafting surfaces crowded with chains. When corrected for the presence of zones devoid of chain ends, approximation (2) only provides a lower boundary for the chain stretching free energy.6,15,22 In case (1) the local extension of chains, R dy/dn, is simply related to the polymer concentration3 2

FstrA ) BAσ

(1 - y) y dy R3 1 ν ν(ν - 1) - + VB 3 4 10

2φA(y)[σa(y)]2

Although this is a rough approximation (e.g., it gives constant polymer concentration within the brush for lamellae), it leads to concentration profiles for spherical micelles that agree well with the numerical self-consistent field calculations.23 For a parabolic brush (case 2), we have16

2 ν

) BBσ

3aKA

δΩ ∂ω ) )λ δφA ∂φA

2

(28)

where λ is the Lagrange multiplier, and

ω ) ωstrA + ωion + ωFlory

(29)

is the free energy density in the swollen brush. The unknowns are the local polymer concentration, cA(r), the grafting density, σ-1, and the equilibrium thickness of the swollen brush, β - 1. At a given grafting density we need one more equation to find the equilibrium thickness. For a free aggregate in equilibrium with surrounding solution, we may use the condition of vanishing osmotic pressure at the edge of the brush as suggested by Zhulina et al.3,4

(

φA(y)

)

∂ω -ω ∂φA(y)

)0 y)β

(30)

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At given β eq 30 is solved numerically with respect to φA(β), the polymer volume fraction at the edge of the brush. This gives the Lagrange multiplier λ from eq 28 and determines the profile cA(r) by solving numerically eq 28 at points of a grid between y ) 1 and y ) β. For every trial cA(r), cubic eq 13 is solved with respect to R(r). The expressions for all terms of eqs 28 and 30, are given in Appendix B. The outer loop of our calculation performs a numerical solution of eq 27 with respect to β. This gives cA(r) and β for a fixed gafting density. For simple unidimensional models of aggregates (lamella, cylinder, sphere, and branch, ν ) 1/2), specifying the grafting density also specifies the size R of the hydrophobic domain for the aggregate of every shape ν

σR ) (ν + 1)VBNB

(31)

The aggregate free energy per chain for a given grafting density is calculated from eqs 13-18, 21, and 24 or 26, depending on the model used for the elasticity of the swollen brush. Minimization of the calculated excess free energy with respect to the grafting density gives the equilibrium dimensions and local compositions of corona for unidimensional aggregates. The equilibrium aggregation number for a spherical micelle is calculated from the known optimal radius of the core by dividing the volume of the core (supposed to be dry) by the volume occupied by one hydrophobic subchain

4 R3 Nag ) π 3 VBNB

(32)

For 3-D model of a branch, Figure 1, the optimal geometry is found by minimizing its free energy as explained in Appendix C. Performing the minimization of the free energy, the micellar cell has to be maintained at constant volume. Nevertheless using the excess free energies over the bulk solution, we may no longer care about the requirement of constant volume because the regions outside the aggregate give zero contributions to the excess free energy. Model Parameters and Calculation Procedure. The input parameters of the model are as follows: the numbers of segments NB and NA in the diblock copolymer chain; the specific volumes of A and B monomers, VA and VB; the Kuhn segment lengths aKA and aKB; the Flory parameters χAB and χAS for intersegmental A-B and segment A-solvent interactions. The polyacid strength is quantified by the “bare” dissociation constant of a polyacid group, pK that describes dissociation equilibrium for the bulk solution in the absence of polyelectrolyte effects. Our model may be used both for weak and for strong polyacids/ polybases by taking corresponding pK. For specified salinity (NaCl) and pH of the external solution, we calculate the total concentration of mobile ions (the “ionic strength” Φion) taking into account the excesses of HCl and NaOH at low and high pH, respectively. We also take into account the dissociation of water molecules: pKw ) 14. For given pK and pH in the bulk, Rb is obtained from the massaction law. Within our model the electrostatic effects including those of pH and salinity of the medium are embedded in eq 13 that gives the local degree of dissociation of the brush as function of the local polymer concentration. The electrostatic contribution to the free energy, eq 14, becomes coupled to the nonelectrostatic contributions via the minimization of the total

free energy with respect to aggregate’s dimensions and concentration profiles in the swollen corona. Performing calculations for experimentally studied systems, we use experimental monomer volumes VA and VB, NA and NB, and, where possible, experimental estimates of aKA and aKB; the bare interface tension, σAB, is treated as a free adjustable parameter for the usually nonsymmetric copolymers.12 3. Calculated Results Morphology Stability Maps. For a wide range of pH and salinities of solution, we predict stable aggregate morphologies from different versions of our model. In all calculations we use two different approximations for the swollen coronae (the Alexander-de Gennes brush and the parabolic brush) and two different models of a micelle branch (the unidimensional model and the 3-D model). We first discuss the results obtained for the Alexander-de Gennes coronae and unidimensional model of the branch. The morphology stability maps are predicted for copolymers of varying molecular mass, composition and polyacid strength. Rich morphologic behavior is predicted for copolymers with relatively short polyelectrolyte block. Figure 2 shows example of such maps for copolymers of a given total number of segments as we increase the length of the weak polyelectrolyte block. As expected, our findings agree with analytical results obtained by Zhulina in the limits of low and high salinity and small aggregate curvature.3 At low pH there is no dissociation and stable morphologies are similar to those for a nonionic system. Increasing pH promotes dissociation and aggregates of larger curvatures are preferred since they allow fixed charges to be farther apart. Nevertheless high concentration of NaOH for pH substantially above 7, implies an increase of ionic strength. Enhanced screening of electrostatic repulsion between polyelectrolyte chains leads to appearance of flat structures again. When the cylindrical aggregates have the lowest free energy, there are two different scenarios. Cylindrical aggregates with spherical end-caps form when the free energy of a sphere is lower than that of a branch. Branched cylinders are preferred when the free energy of a branching portion is lower than that of the spherical end-caps.18,24 The unidimensional model predicts that the branch itself becomes the most stable structure in the regions where branched cylindrical micelles are about to transform into lamellae. The cylindrical parts of the aggregates would then tend to disappear, the branches proliferate, and the system would possibly produce highly connected sponge-like aggregates. At intermediate pH (slightly higher than pK) an increase of solution salinity results in the following sequence of stable structures: lamellae-branch-branched cylinderscylinders-spheres-cylinders-branched cylinders-branchlamellae. In the regions corresponding to branches and cylindrical aggregates, the free energies per chain in competing structures (branches, spherical end-caps and cylindrical portions of the micelles) are close, differing by much less than kT. Owing to thermal fluctuations, highly branched wormlike aggregates together with spherical micelles are likely to appear in these transition regions. Thus the model predictions imply that the diversity of branched and long wormlike micelles will actually coexist in these regions. This is confirmed by experimental TEM pictures, see Figure 6 of ref 25, where branched structures appear as a result of adding salt and are clearly seen together with the aggregates of other shapes.2 The morphology stability map depends both on the asymmetry of the block copolymer molecule, fA ) NA/(NA + NB)



Figure 2. Morphology stability maps for A(NA)-B(NB) diblock copolymer containing overall N ) 400 segments and different number NA of polyelectrolyte segments. The Kuhn length of segments of both type is 0.68 nm; VA ) VB ) aK3, the Flory parameter is χAB ) 0.0879 for polymer and χAS ) 0 for solvent, the dissociation constant is pK ) 5; σABaK2 ) (χAB/6)1/2 ) 0.1210. Notation “Singly dispersed” means that the copolymer molecules do not aggregate, Nag ) 1, but may persist as single globules. (a) NB ) 396, NA ) 4; (b) NB ) 380, NA ) 20; (c) NB ) 364, NA ) 36.

and on its total length. As expected, decreasing fA promotes formation of structures having smaller curvature (e.g., larger spherical micelles). At large fA only small spherical micelles are formed; the block copolymer eventually becomes singly dispersed upon the increase of fA. As the total length NA + NB of the block copolymer decreases the zone of flatter structures enlarges. For example, for NA + NB ) 800 and model parameters of Figure 2, the lamellae disappear from the morphology stability map for fA > 0.075, and for fA > 0.1 the spheres is the only stable morphology. For NA + NB ) 80, the lamellar morphology persists up to fA ) 0.15 and spheres become the only stable shape for fA > 0.1875. At intermediate pH the model predicts an unusual morphology sequence with increasing ionic strength in agreement with previously known theory.3 The aggregate curvature increases rapidly with salinity and then decreases gradually at high ionic strength. This is illustrated in Figure 3 that shows the predicted aggregation numbers of spherical aggregates as function of the ionic strength for different pH. This figure shows aggregation numbers of optimal spheres, even though in some intervals of salinity and pH other morphologies are preferred over spheres, see Figure 2b. At low salinity there is an abrupt transition from large to small spherical micelles. Figure 4 shows two minima of equal depths for the free energy of spheres versus area per chain. This implies the coexistence of spherical micelles having small aggregation number (larger area per chain) with spherical micelles having large aggregation number (smaller area per chain). First predicted from theory,23,26 such coexistence is confirmed in recent experiments for diblocks with a weak polybase chain.27

Figure 3. Aggregation numbers of stable and metastable spherical micelles versus salinity at different pH. Nag ) 1 corresponds to singly dispersed block copolymer molecules. The model parameters are given in Figure 2b.

The large and the small micelles are very different. Small micelles have strongly swollen, highly ionized corona. The micelles with large hydrophobic core are weakly ionized and have much drier coronas. At high salinity the micelles grow gradually with increasing ionic strength. The aggregation number scales with salinity as 0.79 at pH about 7-12 and Csalt > 1 mM. This scaling agrees with the exponent ca. 0.7-0.8 obtained from the numerical self-consistent field calculation, see Table 1 of ref 23, for starlike micelles at high salinity and high pH. For small aggregation numbers and short polyelectrolyte blocks, the model may become no longer viable and its

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Figure 4. Free energy for aggregates of different morphologies as function of grafting density of chains at the hydrophobic core. Most stable are spherical aggregates. Two equal minima correspond to the coexistence of large and small spherical micelles. The model parameters are given in Figure 2b, pH ) 6.5, Csalt ) 0.44 mM.

predictions require careful interpretation. For example, large grafting area per chain of ca. 90 nm2 in small spherical aggregates represented by the right minimum in Figure 4 implies partial contact of the hydrophobic blocks with solvent that may not be completely screened by NA ) 20 ionic monomers (aKA ) 0.68 nm) and would possibly lead to macrophase separation. The results shown in Figures 2-4 have been obtained using eq 24 for the elasticity of A chains. All calculations were repeated using eq 26 instead. Calculations were also performed

for planar charged brushes of varying grafting density studied by molecular dynamics.7 The model parameters were taken from simulation. Figure 5 shows the results. Even though the predicted profiles are very different, both versions of the model give quite reasonable results. The brush thickness from simulation, calculated as the first moment of the polymer concentration profile,7 agree well with our calculation. It has been known previously15,28,29 that the free energy of the brush is not very sensitive to the distribution of chain ends within the brush. On the scale of Figure 2, it is hard to discern the difference in morphology stability maps from the two versions of the model (though use of eq 26 always results in a somewhat lower free energy). Finer details of these maps are illustrated in Table 1 and discussed below. Coexistence of small and large spherical micelles is shifted to a somewhat higher pH: higher charge is needed to “explode” the micelle that has a parabolic corona. Table 1 shows intervals of salinity where aggregates of different morphologies are stable according to different versions of the model. Three cross sections of a morphology stability map at constant pH (5, 7, and 13) are represented. For low pH and for high pH, the stability zones of curved structures predicted from the model of a parabolic brush are narrower than that for the Alexander-de Gennes brush. For intermediate pH, this trend is only observed at high salinity, whereas at low salinity (prior to the appearance of stable spheres) the parabolic model enhances the stability of curved structures. Because fixed charges are accommodated more easily in the parabolic brush than in the Alexander-de Gennes brush, less screening is

Figure 5. (a) Predicted profiles of polyelectrolyte segments (curves) and computer simulation data (points)7 for a flat fully charged brush at different grafting densities: 0.63/d2, 0.12/d2, and 0.020/d2, (top to bottom), where d ) 0.7 nm is the diameter of the monomer hard cores in computer simulation; (b) Predicted thickness of the swollen polyelectrolyte brush (curves) and computer simulation data7 (points) as function of the reduced grafting density; NA ) 30, aK ) 0.686 nm and χAS ) 0.5.

TABLE 1: Predicted Salinity Intervals Where Structures of Different Morphologies Are Stablea model of branch lamella ν ) 1/2 3-D ν ) 1/2 3-D ν ) 1/2 3-D a

branch

branched cylinder

1 2 1 2

1413

Copolymer parameters are from Figure 2b. Results are obtained using different models of the branch and different models of micelle coronas: the unidimensional model (ν ) 1/2); the model shown in Figure 1 (3-D); the Alexander-de Gennes brush (1); the parabolic brush (2).



Figure 6. Dependence of micellar size on the fractional charge of corona and the salinity of solution for diblock copolymer (DEAEMA)94-(DMAEMA)97. Points, experiment;30 curves, calculation. Corona thickness (a) and aggregation number (b) at salinities 20-26 mM KCl; Corona thickness (c) at fractional charge ca. 0.17. Model parameters used in calculation: NA ) 97, NB ) 94, pK ) 7.2, VB ) 0.110 nm3, VA ) 0.69 nm3. Two parameters aKA ) 0.65 nm and σABaKA2 ) 3.267 are fitted from experiment.

required for a shape transition at high salinity. At the same time for a given pH about pK and low salinity, less salt is needed to cause dissociation of the parabolic brush (and shape transition). Table 1 also shows the results from the 3-D model of a micelle branch. Overall, the results are similar to those obtained with the aid of the unidimensional model. Both models predict the existence of stable branched cylindrical aggregates before cylinders transform into lamellae. Both models show very small difference of the free energies for aggregates of different shapes in these transition zones. The models give similar equilibrium characteristics of the branches (see Appendix C): the average grafting area, the thickness of swollen corona, and the linear size of the core. Nevertheless fine details of the morphology stability map are different. The 3-D model leads to wider salinity intervals of stable lamellae both at low and at high salinity. For intermediate salinity the 3-D model predicts larger interval of stable cylindrical aggregates. These trends are observed for low, intermediate and high pH. The main difference is that the 3-D model does not predict that a branch may have the lowest free energy whereas the unidimensional model does. Comparison with Experiment. Structural data for diblock copolymer micelles with one ionic block have been obtained by a number of authors. The effect of fractional ionic charge (regulated by pH of the solution) and the effect of salinity on the size of spherical micelles have been studied30 for micelles with 2-(diethylamino)ethyl methacrylate (DEAEMA) core and a weak polybase corona consisting of 2-(dimethylamino)ethyl methacrylate (DMAEMA) subchains. Figure 6 shows that the results of our calculation are in good agreement with experiment, though the model gives too strong dependence of the brush thickness on salinity. With the same model parameters we

TABLE 2: Predicted and Experimental Aggregation Numbers As Function of Salinity for Spherical Micelles DEAEMA-Q-DMAEMA Csalt, mM

Nag exptl.30

Nag calc.

9.5 50 89

16 ( 3 26 ( 4 40 ( 5

11 30 47

predicted the dependence of the aggregation number on salinity for DEAEMA-Q-DMAEMA diblock copolymer micelles studied in the same work.30 Here Q-DMAEMA is strong polyelectrolyte obtained from DMAEMA, whose side chain amine groups are selectively quaternized with benzyl chloride. Choosing pK to ensure full charge of Q-DMAEMA subchains, independently of pH, we obtain excellent prediction of aggregation numbers, Table 2. Figure 7a shows data for weakly acidic polystyrene-bpoly(acry1ic acid) block copolymer PS(20)-b-PA(85).31 These micelles have glassy core that according to experimental observation does not seem to change appreciably. We fitted the model parameters from data of Figure 7a. With these parameters we then predicted the aggregation numbers of equilibrium micelles with different length of the polysterene block. The results of this prediction shown in Figure 7b agree with experiment.32 However there is only qualitative agreement for the predicted dependence of the aggregation numbers on salinity for PS(23)-b-PA(300) and PS(6)-b-PA(180) micelles. Over substantial salinity intervals the model predicts twice smaller aggregation numbers than in experiment. We also predicted equilibrium aggregation numbers and micelle core radii for a variety of polystyrene-b-polyacry1ic acid block copolymers with long core blocks.33 The agreement with experiment is poor,

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Figure 7. (a) Micellar radii as function of fractional charge of polyelectrolyte chains PS(20)-b-PA(85);31 (b) Micellar aggregation numbers for PS(N)-b-PA(85) at 2.5 M salinity and pH 7 as function of length NB of polysterene block.32 Points, exptl.; curves, calc. pK ) 5, VA ) 0.092 nm3, VB ) 0.191 nm3; model parameters fitted from data of Figure 7a: χAS ) -1.0, aKA ) 0.280 nm, σABaKA2 ) 2.420.

Figure 8. Measured hydrodynamic radii (points) versus salinity for PMMA-DMAEMA diblocks grafted on PMMA particles of bare radius 66.2 nm at pH 7.35 Curves: calculated total radii for different grafting areas per DMAEMA subchain. NA ) 98 from the measured extension of a fully stretched chain.35 Model parameters for DMAEMA are taken from Figure 6. Different grafting areas correspond to different NB, VB ) 0.149 nm3.

probably because the micelles in experiment were not truly equilibrium; according to experimental data33 the volume per monomer in the allegedly dry micellar core varies from 0.160 nm3 to more than 0.200 nm3. Thus good results shown in Figure 7b for this type of micelles might be a fortunate coincidence. Kriz et al.34 studied the effect of fractional charge on the radius of micellar core for poly(methyl methacrylate)-b-poly(acrylic acid) PMMA(93)PAAc(149). The core radius changes from 8.0 to 6.8 nm with fractional charge increasing from 0 to 1. With model parameters from Figure 7 and VB ) 0.149 nm3, our calculations for equilibrium micelles predict a somewhat stronger dependence of the core radii on the micellar charge (calculated radii change from 11 to 4 nm). The effect of salinity on the dimensions of a curved weak polybase brush have been studied for PMMA-DMAEMA diblocks grafted on a PMMA latex particle of a bare radius 66.2 nm.35 Unfortunately the grafting density in these experiments has not been reported. Figure 8 shows experimental data and the results of our calculations for different grafting densities. The modeling has been performed by considering a metastable micelle with the core radius equal to the bare particle radius and with the swollen to equilibrium DMAEMA-brush. For the grafting area per chain about 28 nm2, the agreement between theory and experiment is good. The results in Figure 8 illustrate combined effect of electrostatic screening and grafting density on the dimensions of polyelectrolyte brush. Nonmonotonic dependence of the brush dimensions on salinity that has been predicted previously for weak

polyelectrolytes23 is clearly seen from both the experiment and calculation for intermediate and low grafting densities. The effect of curvature and ionic strength on the dimensions of swollen polyelectrolyte brush has been studied for poly(tertbutylstyrene)/poly(styrenesulfonate) block copolymers (PtBS PSSH).36 Adsorbed polyelectrolyte brushes and micelles of copolymers with longer polyelectrolyte subchains (MT3, NA ) 709 repeat units36) and shorter polyelecetrolyte subchains (MT2, NA ) 404 repeat units37) have been used in these experiments. The experiment shows approximately constant micelle aggregation numbers at different salinities and constant surface coverage for adsorbed copolymers. We performed calculations at experimental grafting densities and did not tune any model parameters. Figure 9a shows that the results of this prediction are reasonable though not entirely quantitative. However when we estimate the equilibrium aggregation number for MT2 and MT3 micelles we find that the model predicts singly dispersed copolymer molecules, except at high salinity. Other copolymers with the same strongly acidic block have been also studied.7,38 Figure 9b shows the dependence of brush thickness on the grafting density for poly(ethylethylene-bstyrenesulfonic acid (PEE PSSH) planar brushes.7 Calculated results are shown for different parameters of PSSH block: those taken for predictions of Figure 9a and those obtained fitting the experimental data of Figure 9b. The model predicts substantially larger extension of the brush (78%-84% of the fully extended chain length for PSSH83 and 74%-83% for PSSH136) than in experiment. Fitted value pK ) 4 gives the fractional charge (0.56-0.64 for PSSH83 and 0.53-0.75 for PSSH136) that appreciably deviates from experiment (0.85 for PSS83 and 0.49 for PSS136). The positive χAS reflects that water is a nonsolvent for polystyrene backbone of PSSH blocks. Unfortunately matching experiment requires too small aK ) 0.2 nm corresponding to unrealistically small length of a fully extended PSSH-chain (27 nm vs the measured value of 34 nm7); calculated relative chain extension remains too large (75%-83% for PSSH83 and 75%-80% for PSSH136). We note that for a planar brush at a given grafting density, the interfacial term of the free energy is constant. Therefore, structural characteristics of the brush at fixed grafting density are not affected by σAB and this parameter is not shown in Figure 9. Model parameters of Figure 9 have been used for calculating structural characteristics of equilibrium PEE144PSSH136 spherical micelles.38 Figure 10a-c shows the results. Directly measured in experiment are core radii, hydrodynamic radii, and interfacial polymer fraction. Other reported quantities38 were deduced from experimental data using a model, where micellar corona extends beyond the dense part of the polyelectrolyte brush. The thickness



Figure 9. (a) Dependence of the dimensions of the swollen PSS brush on salinity at different grafting densities for particles of different curvature. Circles and solid curve: experiment36 and calculation, respectively, for MT3 copolymer adsorbed on a spherical particle of 137 nm diameter with the area per chain 131.6 nm2; Triangles up and dotted curve: experiment36 and calculation for micelles of copolymer MT3 with the aggregation number 23 and 6.58 nm2 per chain; Triangles down and broken curve: experiment37 and calculation for micelles of copolymer MT2 with the aggregation number 38 and 5.44 nm2 per chain. MT3 and MT2 are PtBS(NB)-PSSH(NA) copolymers with NB ) 27, NA ) 709 and NB ) 26, NA ) 404, respectively. Model parameters: VA ) 0.197 nm3, VB ) 0.280 nm3, χAS ) 0, aKA ) 0.248 nm, pK ) 2. (b) Monolayer thickness for the diblock copolymers poly(ethylethylene)144poly(styrenesulfonic acid)136 (PEE144PSS136, degree of sulfonization 0.9) and poly(ethylethylene)144poly(styrenesulfonic acid)83 (PEE144PSS83, degree of sulfonization 0.85) ancored at the air-water interface. Points, experiment;7 curves, calculation at pH 7 and Csalt ) 0.1 mM: aKA ) 0.248 nm, pK ) 2, χAS ) 0 (solid); aKA ) 0.20 nm, pK ) 4, χAS ) 0.3 (dashed).

Figure 10. Effect of solution salinity on the structure of PEE144PSS136 spherical micelles. Points, experiment;38 curves, calculations at pH 7; VA ) 0.197 nm3, VB ) 0.107 nm3, NB ) 144, NA ) 136. (a) Experimental hydrodynamic radius (stars), core radius (solid circles and dashes), corona thickness (light circles and dotted curves), and total micellar radius (rectangles and solid curves); (b) aggregation number; (c) polymer volume fraction at the core-corona interface. Curve numbers indicate different parameter sets: (1) aKA ) 0.248 nm, pK ) 2, χAS ) 0, σABaKA2 ) 3.993; (2) aK ) 0.200 nm, pK ) 4, χAS ) 0.3, σABaKA2 ) 2.650. σAB is adjusted from data of Figure 10b, and other parameters are taken from Figure 9.

of this dense region was defined as the corona thickness.38 We use different definition of the corona thickness because in the applied version of our model all chains terminate at the corona outer edge. Figure 10a shows good agreement between calculated and experimental core radii. The calculated total micellar radii agree well with experimental hydrodynamic radii (cf. similar results of ref 39). The calculated thickness of corona is much larger than that deduced from experiment38 but both have similar dependence on salinity. The model gives reasonable estimates of the aggregation numbers over a wide range of the ionic strength using one adjustable parameter, σAB, see Figure 10b. Large error of predicted polymer volume fraction at high salinity, Figure 10c, can be attributed to approximating mobile ions by point charges.

4. Conclusion For solution of the diblock copolymer composed of a hydrophobic block and a weak polyelectrolyte block, we obtain the regions of stable aggregate morphologies in pH-salinity plane with the aid of the self-consistent field theory in the strongsegregation approximation. Our model is based on previous theoretical work3,4 that gives analytical expressions for the crew cut and star like micelles in the limits of high and low salinity. However, we choose a full numerical version of the theory to include the intermediate regimes between those of crew cut and star like micelles and to consider arbitrary salinities. We also extended the theory to branched wormlike micelles. We compare

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the results from two different approximations for describing the elasticity of the swollen aggregate corona: (1) the Alexander-de Gennes brush, where all polyelectrolyte chains terminate at the same distance from aggregate’s core,5 and (2) the parabolic brush that implies a radial distribution of chain ends.6 We obtained similar morphology stability maps using the two versions of our model (although the parabolic brush always gives a somewhat lower free energy). We also compared different models of a branching portion that connects three wormlike micelles: (1) a unidimensional model of the branch, ν ) 1/2, and (2) a 3-D model where the micelle junction is constructed of three pieces of toroid and a bilayer planar patch. Our major conclusion is that all models show a substantial tendency of the wormlike micelles toward branching. The morphology stability maps demonstrate structural response of the aggregates to changing pH and salinity of the environment. These maps span the whole interval of pH and salinity that may be important for most practical needs. Lamellar, cylindrical, branched cylindrical and spherical aggregates and also sponge-like aggregated structures are included in these maps. For a weak polyelectrolyte at intermediate pH (slightly higher than pK), an increase of solution salinity results in a sequence of aggregate’s morphologies: lamellae-branched cylinders-cylinders-spheres-cylinders-branched cylinderslamellae. Different models of the micellar branch predict different details of the morphology stability map in the zones between the stable lamellae and branched cylinders. The unidimensional model predicts that the branch itself becomes the most stable structure. This may lead to formation of highly connected sponge-like aggregates. The 3-D model does not predict stable branches contrary to the previous result obtained from a similar model for classical ionic surfactants.18 Unfortunately with the existing structural experimental data it does not seem possible to check which model is more adequate in this respect; although the trend for branch-like structures (that have saddle-shaped surface elements) to proliferate is known for more concentrated systems of amphiphiles and leads to bicontinuous morphology in microemulsions and gels. Nevertheless according to all versions of the model the free energies per chain in different competing structures differ by much less than kT in the transition zones, implying a likely coexistence of long wormlike micelles with branched and spherical micelles, as indeed observed in experiment.2,25 The stability diagram depends on both the length and the composition of a diblock copolymer molecule. When the polyelectrolyte block is long enough relative to the hydrophobic block, sphere is the only stable aggregate shape. To test quantitative performance of the model we compare our calculations with computer simulation and experimental data on micelles and brushes (planar and curved) formed by diblock copolymers where one block is a polyelectrolyte. Both weak and strong polyelectrolytes are included. We show that adjusting only one or two parameters, leads to a fair agreement between the theory and experiment. To the best of our knowledge this is the first systematic comparison of the self-consistent field theory with structural experimental data on micelles formed by diblock copolymer containing an ionic block. Our results may be helpful for controlling self-assembly of diblock copolymers in solution because these results show how aggregate’s characteristics may be tuned by variation of acidity and salinity of the environment. Acknowledgment. For financial support, the authors are grateful to the Russian Foundation for Basic Research (Project No. 09-03-00746-a), to the program “Leading Scientific Schools

Victorov et al. of Russia” (Project No. NSch-165.2008.3), and to National Taiwan University of Science and Technology (Project No. RP07-1). Appendix A. 1-D and 3-D Models of a Micelle Branch: Geometry and Elasticity Figure 1 shows a specific model of an aggregate that mimics typical cryo-TEM images of the micelle Y-shaped junction. The surface area and the volume of the junction are obtained by adding contributions from its toroidal and planar parts

(

π + πb(πc - 2b) 2

(

πc π 2b + πb2 2 2 3

Ajun ) 2c2 √3 Vjun ) 2rplc2 √3 -

)

)

(

(A1)

)

(A2)

where b and c are the minor and the major radii of the torus and 2rpl is the thickness of the bilayer patch in the middle of the junction. For torus, the elementary surface area depends on angle θ. The θ-averaged surface area for the inner part of the torus of a minor radius r is given by

c y π - 2y b ator(y) ) c π -2 b

( (

)

)

(A3)

where y ) r/b. The grafting area is calculated by dividing the area of the toroid element by the number of chains

c π - 2) ( b σ) N V πc 2 b( - ) 2b 3

B B

(A4)

Figure 11 shows the reduced area for the junction and for other morphologies. For c f ∞ toroid transforms into a cylinder;

Figure 11. Reduced area vs reduced distance for different morphologies. Light curves (black) are calculated for the junctions of geometry depicted in Figure 1. The c/b ratio is indicated for every curve. Line for ν ) 1/2 (magenta) is our abstract unidimensional model of the micelle branch.



TABLE 3: Geometrical Factors in the Elasticity Free Energy Cg, eq A6

sphere cylinder structures representing branch

c/bf0 3 c/bf∞ 5 toroid inner part c/b ) 1.01 6.45 c/b ) 2 5.53 c/b ) 4 5.23 c/b ) 10 5.09 c/b ) 50 5.02 c/b ) 100 5.01 10

plane

the junction as a whole transforms into an infinite planar bilayer. Because for r g c the hole inside toroid disappears, we are only interested in the behavior of a junction at y < c/b. The behavior of a junction is intermediate between that of a cylinder and a plane. Our abstract unidimensional model of a branch, where we set ν ) 1/2 in eq 19 roughly follows the behavior of the junction. Strongest deviations between the unidimensional and 3-D models are observed for large c/b in the vicinity of the center of the micelle core where we expect significant contribution from the chain elasticity. This contribution for the junction is calculated below. For toroid element of a junction, the elasticity free energy per chain in the dry parabolic brush is given by

FstrB ) BB

σb3 VB

∫01 (1 - y)2ator(y) dy

(A5)

Integration in eq A5 using eq A3 gives

FstrB ) Cg

( )

π2 b 80NB aKB

2

(A6)

where Cg ) (15πc - 12b)/(3πc - 4b) is the geometrydependent coefficient. For c f 0 and c f ∞, we obtain Cg ) 3 and 5, respectively, and eq A6 reproduces the correct results for spheres and cylinders, where b is equal to rsph or to rcyl. To reproduce the result known for plane, we must set Cg ) 10 and b ) rpl. We assume additivity of free energies and incompressibility of the aggregate core. For the micelle junction, the free energy is estimated as combination of contributions from its toroid and planar parts

Fjun ) ηtorFtor + ηplFpl

C˜g(ν), eq A10

Cjun , eq A8 g

morphology

3D-junction, Figure 1 6.81 6.17 6.31 7.08 8.84 9.34

FstrB ) BB

3 5 8.25

ν)0

10

( )

σb 1 b 2 π2 ν ν(ν - 1) 2 b ) C˜g(ν) - + VB 3 4 10 80NB aKB (A9)

[

]

where

1 ν ν(ν - 1) C˜g(ν) ) 30(ν + 1) - + 3 4 10

[

]

(A10)

is a geometrical factor. Table 3 shows geometrical factors calculated for different morphologies from eqs A6 and A8 and those from eq A10. We see that the elasticity of core for the morphology defined by setting ν ) 1/2 in eq 19 is intermediate between that of a cylinder and a plane as is also the case for the 3-D junction. The elastic response of morphology ν ) 1/2 is similar to that of a weakly curved junction where c/b is between 10 and 50. All other contributions to the free energy of 3-D junction are calculated in a similar way, using eq A7 and performing volume integrals of the free energy density. The volume element for the toroid is calculated using eqs A3 and A4 in eq 20, where R ) b. Equations 13-18, 24, and 26 defining free energy density and eqs 28-30 expressing conditions of the brush swollen to equilibrium apply without change. Appendix B. Contributions to the Chemical Potential and to the Osmotic Pressure Below we give the expressions for different contributions needed in eqs 28-30. For the free energy of mixing, we have

(A7)

where Fjun, Ftor, and Fpl are the free energies per chain for the entire junction and for its toroidal and planar parts and ηtor and ηpl are the volume fractions of these parts in the junction, respectively. These volume fractions are calculated from eq A2. Hence, assuming that rpl ) b, the geometrical factors for the junction are also given by the corresponding linear combinations

ν)2 ν)1 branch, ν ) 1/2

∂ωFlory 1 ) [(1 - 2φA)χAS - ln(1 - φA) - 1] ∂φA VA

(B1)

and

(A8)

∂ωFlory 1 - ωFlory ) - (φA2χAS + ln(1 - φA) + φA) ∂φA VA (B2)

We now write eq 21 that describes unidimensional morphologies for different ν in the form of eq A6. Using eq 31 we obtain

For the electrostatic free energy, taking into account eq 13 and performing algebra, we obtain

φA tor Cjun g ) ηtorCg + ηpl10

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Φion A2 ∂ωion 1 )ln(1 + K1(√A2 + 1 - A)) ∂φA φA √ 2 VA A +1 K1 A2 -A VA √ 2 A +1 1 ) ln(1 - R) (B3) V 2 A 1 + K1(√A + 1 - A)

(

)

and

φA

∂ωion - ωion ) ∂φA φA

K1 VA

(

Φion

√A2 + 1

A2

- Φion -

)

-A

√A2 + 1 1 + K1(√A2 + 1 - A)

) Φion(√A2 + 1 - 1) (B4)

where A ≡ (R(r)cA(r))/Φion and K1 ) Rb/(1 - Rb). For the elasticity of the swollen brush, we use two different approximations, eq 24 or eq 26. We have, respectively

Figure 12. Optimal major radius of curvature vs salinity for the inner element of toroid and for the micellar junction as a whole. pH ) 7; the aggregate corona is described using the model of a parabolic brush; copolymer parameters are given in Figure 2b. Optimal core radii and corona thickness are shown in Figure 14.

3aKA 3aKA ∂ωstrA ∂ωstrA - ωstrA ) ), φ A 2 ∂φA ∂φ 2[φAσa(y)] φA[σa(y)]2 A (B5) in approximation (1), and

∂ωstrA R2 ) BA (1 - y)2, ∂φA VA

φA

∂ωstrA - ωstrA ) 0 ∂φA

(B6) in approximation (2), i.e., in this approximation, there is no contribution from chain elasticity to the osmotic pressure throughout the brush. Appendix C. Optimal Geometry of a Junction and Aggregates of Other Shapes For 3-D model of the junction, the free energy Fjun(b,c,rpl) was minimized with respect to the toroid radii b and c, and the thickness of the bilayer patch, rpl. The optimal geometry of the junction and the sequence of free energies for aggregates of different shapes depend on the solution salinity and pH. The contribution of the bare interfacial tension tends to contract the surface area and favors structures of smaller curvature. Elasticity opposes stretching of chains away from the surface and hence favors larger surface area per chain and curving of aggregates. Repulsion caused by the electrostatic forces and excluded volume promotes chain-stretching but also expands the surface. The resulting subtle balance of these opposite trends defines the optimal aggregate shape. For the inner part of toroid, the average area per molecule increases with c approaching its limiting value for cylinder. When electrostatic forces dominate, the free energy of the toroidal part of the junction, Ftor(b,c), is a monotonically decreasing function of c. Because at large c we have Ftor(b,cf∞) ) Fcyl(b), the cylinders are more stable than the aggregates of toroid geometry. When electrostatic interactions are weaker (owing to a weaker dissociation or to screening), a minimum of Ftor(b,c) appears at a finite c. This behavior is

Figure 13. The excess free energy (per chain) of an optimal aggregate of a specified shape over that of an optimal lamella ∆Ω ≡ Ω - Ωlam. Curves for micelle branch show results from the unidimensional model (ν ) 1/2) and from 3-D model of Figure 1. Copolymer parameters are given in Figure 2b. (a) pH ) 7, the aggregate corona is described using the model of a parabolic brush. (b) pH ) 13, the aggregate corona is described using the model of the Alexander-de Gennes brush.



Figure 14. Characteristics of optimal aggregates of different morphologies vs solution salinity. pH ) 7; the aggregate corona is described using the model of a parabolic brush; copolymer parameters are given in Figure 2b. (a) The area per chain at the aggregate core; (b) Equilibrium thickness of the swollen corona; (c) Equilibrium size of the hydrophobic core. For 3-D model of a junction, separate curves are shown for its toroidal and planar parts in (b) and (c).

illustrated in Figure 12 that shows optimal c versus salinity for the toroid and for the junction. We have different dependence of the optimal c on salinity for the toroid piece of the junction and for the junction as a whole, including its planar patch. In situations where lamellae have the lowest free energy the size of the planar element of the junction increases and starts to dominate in the junction free energy. The growth of junction’s planar element implies an increase of c. At large c the free energy of the junction reduces to that of a planar aggregate: Fjun(rpl,b,cf∞) ) Fpl(rpl) as we see from eqs A7 and A2. Hence Fjun(rpl,b,c) is a decreasing function of c and the junctions are unstable with respect to infinite planar aggregates. At intermediate salinity Fjun(rpl,b,c) has a minimum at finite c. Examples in Figures 13 and 14 show optimal characteristics of the 3-D junction and aggregates of other shapes. Results are also shown for the unidimensional model of the branch. Over the whole salinity interval the free energies from

the unidimensional model and from the 3-D model are close, both using the approximation of a parabolic brush (Figure 13a) and the Alexander-de Gennes brush (Figure 13b) for coronae. The unidimensional model and the 3-D model lead to similar optimal dimensions of the branch. Figure 14a shows that the optimal average areas per chain are very close. The thickness of corona and the size of the core of the unidimensional branch are intermediate between the corresponding linear dimensions of the toroidal and planar parts of 3-D junction, see Figure 14, panels b and c. Thus we may conclude that our effective unidimensional description of a branch is a viable approximation. Nevertheless, minor differences in the free energies in the transition zones (shown in the inserts of Figure 13) lead to qualitatively different results for different models of the branch. Zones of stable branches are only predicted from the unidimensional model. For the 3-D model, calculations

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over large intervals of pH, salinity and copolymer parameters (chain length and composition, bare interfacial tension and excluded volume corresponding to good and theta solvents) show that the inner piece of toroid is never more stable than both the lamellae and cylinders, in contrast to the results obtained from a similar model for classical ionic surfactants.18 Thus within our specific 3-D model too large free energy penalty is caused by curving swollen polyelectrolyte corona into a saddle-like pattern. Possibly this is one of model’s deficiencies: toroidal aggregates are observed experimentally in polyelectrolyte systems.1,40 References and Notes (1) Fo¨rster, S.; Abetz, V.; Mu¨ller, A. H. E. AdV. Polym. Sci. 2004, 166, 173. (2) Moffit, M.; Khougaz, K.; Eisenberg, A. Acc. Chem. Res. 1996, 29, 95. (3) Zhulina, E. B.; Borisov, O. V. Macromolecules 2005, 38, 6726. (4) Zhulina, E. B.; Birshtein, T. M.; Borisov, O. V. Eur. Phys. J. E 2006, 20, 243. (5) Daoud, M.; Cotton, J.-P. J. Phys. (Paris) 1982, 43, 531. (6) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (7) Ahrens, H.; Fo¨rster, S.; Helm, C. A.; Kumar, N. A.; Naji, A.; Netz, R. R.; Seidel, C. J. Phys. Chem. B 2004, 108, 16870. (8) Marcus, R. A. J. Chem. Phys. 1955, 23, 1057. (9) Lyatskaya, Y. V.; Leermakers, F. A. M.; Fleer, G. J.; Zhulina, E. B.; Birshtein, T. M. Macromolecules 1995, 28, 3562. (10) Hamley, I. W. The physics of block copolymers; Oxford University Press: New York, 1998. (11) Helfand, E.; Tagami, Y. J. Chem. Phys. 1972, 56, 3592. (12) Helfand, E.; Sapse, A. J. Chem. Phys. 1975, 62, 1327. (13) Ermoshkin, A. V.; Semenov, A. N. Macromolecules 1996, 29, 6294. (14) Olmsted, P. D.; Milner, S. T. Phys. ReV. Lett. 1994, 72, 936. (15) Likhtman, A. E.; Semenov, A. N. Macromolecules 1994, 27, 3103. (16) Victorov, A. I.; Radke, C. J.; Prausnitz, J. M. Mol. Phys. 2005, 103, 1431. (17) Victorov, A. I.; Radke, C. J.; Prausnitz, J. M. Phys. Chem. Chem. Phys. 2006, 8, 264.

Victorov et al. (18) Andreev, V. A.; Victorov, A. I. Langmuir 2006, 22, 8298. (19) Plotnikov, N.; Victorov, A. I. Fluid Phase Equilib. 2007, 261, 26. (20) Semenov, A. N. SoV. Phys. JETP 1985, 61, 733. (21) Safran, S. A. Statistical thermodynamics of surfaces, interfaces, and membranes; Addison-Wesley Pub.: Reading, MA, 1994. (22) Li, H.; Witten, T. A. Macromolecules 1994, 27, 449. (23) Lauw, Y.; Leermakers, F. A. M.; Cohen Stuart, M. A.; Borisov, O. V.; Zhulina, E. B. Macromolecules 2006, 39, 3628. (24) Zilman, A. G.; Tlusty, T.; Safran, S. A. J. Phys.: Condensed Matter 2003, 15, S57. (25) Fo¨rster, S.; Hermsdorf, N.; Leube, W.; Schnablegger, H.; Regenbrecht, M.; Akari, S.; Lindner, P.; Buttcher, C. J. Phys. Chem. B 1999, 103, 6657. (26) Zhulina, E. B.; Borisov, O. V. Macromolecules 2002, 35, 9191. (27) Xu, L.; Zhu, Z.; Borisov, O. V.; Zhulina, E. B.; Sukhishvili, S. A. Phys. ReV. Lett. 2009, 103, 118301(1). (28) Ball, R. C.; Marko, J. F.; Milner, S. T.; Witten, T. A. Macromolecules 1991, 24, 693. (29) Milner, S. T. Macromolecules 1994, 27, 2333. (30) Lee, A. S.; Butun, V.; Vamvakaki, M.; Armes, S. P.; Pople, J. A.; Gast, A. P. Macromolecules 2002, 35, 8540. (31) Groenewegen, W.; Egelhaaf, S. U.; Lapp, A.; van der Maarel, J. R. C. Macromolecules 2000, 33, 3283. (32) Khougaz, K.; Astafieva, I.; Eisenberg, A. Macromolecules 1995, 28, 7135. (33) Zhang, L.; Barlow, R. J.; Eisenberg, A. Macromolecules 1995, 28, 6055. (34) Kriz, J.; Masar, B.; Pospisil, H.; Plestil, J.; Tuzar, Z.; Kiselev, M. A. Macromolecules 1996, 29, 7853. (35) Wesley, R. D.; Cosgrove, T.; Thompson, L.; Armes, S. P.; Billingham, N. C.; Baines, F. L. Langmuir 2000, 16, 4467. (36) Hariharan, R.; Biver, C.; Mays, J.; Russel, W. Macromolecules 1998, 31, 7506. (37) Amiel, C.; Sikka, M.; Schneider, J. W., Jr.; Tsao, Y.-H.; Tirrell, M.; Mays, J. W. Macromolecules 1995, 28, 3125. (38) Fo¨rster, S.; Hermsdorf, N.; Buttcher, C.; Lindner, P. Macromolecules 2002, 35, 4096. (39) Muller, F.; Guenoun, P.; Delsanti, M.; Dem’e, B.; Auvray, L.; Yang, J.; Mays, J. W. Eur. Phys. J. E 2004, 15, 465. (40) Cui, H. G.; Chen, Z. Y.; Wooley, K. L.; Pochan, D. J. Soft Matter 2009, 5, 1269.

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