Thermodynamics, Microstructures, and Solubilization of Block

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Thermodynamics, Microstructures, and Solubilization of Block Copolymer Micelles by Density Functional Theory Shun Xi, Le Wang, Jinlu Liu, and Walter G Chapman Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b04336 • Publication Date (Web): 11 Mar 2019 Downloaded from http://pubs.acs.org on March 11, 2019

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Thermodynamics, Microstructures, and Solubilization of Block Copolymer Micelles by Density Functional Theory Shun Xi, Le Wang, Jinlu Liu, and Walter Chapman∗ Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, United States E-mail: [email protected]

Abstract Block copolymer micelle is one of the most versatile self-assembled structures with applications in drug delivery, cosmetic products, and micellar-enhanced ultrafiltration. The key to design an effective block copolymer to form micelles is to understand how molecular architecture affects critical micelle concentrations, micellar dimensions, and partitioning of solute into the micelle. In this work we studied micelles from nonionic block copolymers using interfacial statistical associating fluid theory(iSAFT) a density functional theory, which explicitly includes the hydrogen bonding between surfactants and water, and water/water. We are able to predict and explain how micellar thermodynamic properties depend on polymer chain architecture. Dimension and aggregation of micelles are investigated for block copolymers with different hyrophobes and hydrophiles. The effects of temperature and pressure on micelle stability are also captured by the theory. The enhanced solubility of hydrophobic substance in water by micelle loading is demonstrated, and predicted solute distribution answers the question about the locus of benzene in micelles from a theoretical perspective.

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Introduction Block copolymers consist of blocks that are chemically different but covalently bonded. 1 When block copolymers are dissolved in a selective solvent which has stronger affinity with one block, the block copolymers can spontaneously aggregate into micelles. 2,3 For instance, Pluronic surfactants in aqueous solution form spherical micelles with a hydrophobic core of poly(propylene oxide) (PPO) blocks and a hydrophilic corona of poly(ethylene oxide) (PEO) blocks. This consequent aggregate structure is a result of general thermodynamic balance between entropy and enthalpy. Hydrophobic PPO blocks collapse to create a hydrophobic environment and minimize the exposure to unlike water molecules with a cost of entropy. The surrounding hydrophilic PEO blocks become an interfacial layer to further reduce unfavorable hydrophilic and hydrophobic interactions. This meso-scale micellar structure is where the system free energy reaches a minimum. The chemically opposite environments of micellar core and bulk open the door to many applications. The hydrophobic core of micelles is attractive to hydrocarbons and non-watersoluble aromatic compounds. Pluronic for instance has been successfully engineered as a micellar drug carrier for its enhanced solubility, metabolic stability, and circulation time. 4 A combined formula of sodium lauryl sulfate with poly(ethylene oxide) is known to effectively solubilize hydrophobic sebum and exogenous contaminants on skin by forming spherical micelles. Derivatives of the formula have been commercialized as micellar cleansing products in the cosmetic industry. 5 Poly(ethylene glycol) alkylether surfactants have been investigated for applications in micellar-enhanced ultrafiltration(MEUF). MEUF process can efficiently encapsulate toxic organic compounds from aqueous stream and then be rejected by molecular sieves in a membrane. The solubilization of micelles reduces the energy cost of the separation process. 6,7 Large complexity in surfactant molecules makes it a challenging task to select highperformance surfactant for the above applications. For instance, the design of a MEUF 2


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process requires surfactants to form micelles with a suitable range of dimension that the separation can achieve both high flow rate and high selectivity. 7 Critical micelle concentration(CMC) is an important physical property in designing surfactants. It is the concentration, above which surfactant molecules self-assemble into micelles. It is known that temperature and surfactant chain architecture can significantly affect CMCs over several orders of magnitude. To experimentally determine CMCs, light scattering and small-angle X-ray scattering require a long relaxation time to reach a micellar thermodynamic equilibrium before measurement. It is desirable to have a physical model that can be exploited to predict CMCs of surfactants at different conditions. The earliest theoretical approach uses scaling models of block copolymers under the assumption that core and corona are uniformly stretched chains. The Alexander-de Gennes model correlates the dimension of micellar core with hydrophobe chain length. 8 Daoud and Cotton 9 extend previous scaling theory and classify micellar regions into uniform core, unswollen brush and swollen brush. Volume profile in this theory is correlated with aggregation number by power law relation. A mean-field version of scaling theory for micelle in polymeric solvent was later developed by Leibler et al. 10 An interfacial free energy function by Helfand and Tagami is included in this formalism. 11,12 Modern scaling theory using a mean-field approach developed by Ekaterina, Zhulina, and Rubinstein has been applied to model the hydrodynamic radius, aggregation number, polymophysims of diblock copolymers micelles 13 and polypeptide micelles 14 with experimental measurement. Free energy of transfer method by Nagarajan and Ganesh, 15 and Puvvada and Blankschtein 16 treats the diffusive interface of micellar system as a step-function-like interface between two pseudo phases. Chemical potentials of surfactants in micelle phase and solution phase are derived by including additional surfactant parameters. This method provides some predictive power in modeling CMCs but cannot explain the difference between triblock and diblock copolymers. Lattice self-consistent field theory(LSCF) by Scheutjens and Fleer, 17 and by Borisov 3


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et al. 18 is an inhomogeneous physical theory in studying micellar self-assembly based on the analysis of grand potential. The incorporation of full Flory-Huggins interaction parameters between segments, and segment volumes assumes that each segment takes a constant lattice space. The theory predicts a continuous volume fraction of different segments as a function of lattice distance. It has been applied to model CMCs of PEO/PPO and Pluronic block copolymer micelles. 19 LSCF predicts a diffusive interface between micelle core and corona. Polyelectrolyte block micelles have also been modeled using LSCF to understand the response of polyelectrolyte block micelles to pH. 20 Classical density functional theory(DFT) applied in this work provides more detailed molecular physics by explicitly including the solvent and the effect of association on molecular distribution. The theory is based on the general thermodynamic principle that the grand potential is minimized at the equilibrium density distribution of molecules. The key advantage of DFT is that thermodynamic consistency is required. 21 Once the bulk surfactant concentration is known, the theory is able to predict equilibrium density as micelle or solution without additional parameters. There have been several successful DFTs in modeling hard sphere mixtures, 22 interfacial tensions between liquid-vapor interfaces 23,24 and liquid-liquid interface, 25 adsorptions in nanopores, 26,27 confined associating fluids. 28 One particularly successful DFT is inhomogeneous statistical associating fluid theory(iSAFT). It is a natural extension of statistical associating fluid theory (SAFT) developed by Chapman et al. 29 iSAFT is a perturbation theory with the reference fluid described by Rosenfeld’s fundamental measure theory for hard sphere mixture fluid. Wertheim’s first order thermodynamic perturbation theory TPT1 30 is then extended for inhomogeneous fluids by Tripathi, 31 and Jain 32 to take into account tangential bonding between segments. Long-range attractions are treated in the spirit of Weeks-Chandler-Andersen(WCA) theory as a mean-field approximation(MFA). 33 Previous success of iSAFT has demonstrated its accuracy and thermodynamic consistency in modeling macromolecular systems against molecular simulation. For block copolymers, iSAFT have been applied to model lipid bilayers, 34 block copoly4


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mer self-assembly under confinement, 32 tethered block copolymers, 35 dendrimer molecules in different solvents, 36 associating block copolymers, 37 block copolymer and nanoparticle, 38 formation of block copolymer micelles, 39 and hybrid density-gradient DFT. 40 In contrast to other theories, iSAFT is a better candidate to model non-ionic micellar systems as it can accurately include the association contribution to total free energy functional for inhomogeneous fluid. 41 Both the hydrogen bonding between poly(ethylene oxide) and water molecules, and water/water association are crucial to understand hydrophobicity and micelle formation. The explicit description of water association expanding the head group is key to this process. We will address how the difference in chain structures between diblock copolymers and triblock copolymers causes variations in thermodynamic properties of spherical micelles. Spherical micelles are formed in a dilute block copolymer solution. However, hexagonal, lamellar and cubic mesophase micelles are formed when block copolymers are significantly concentrated. 42 In this work we focus on the study of spherical micelles. We demonstrate that strong degree of hydrogen bonding at a micellar interface is predicted by theory when a micelle is thermodynamically stable. The aggregation and dimension of micelles as a function of hydrophobe and hydrophile are investigated. We show the temperature and pressure effects in CMCs as a result of including compressibility. Finally we study the solubilization capacity of block copolymer micelles and provide a microscopic description of micellar swelling process. For comparison purpose, we consider the diblock copolymer micelles of poly(ethylene oxide)-b-poly(1,2-butylene oxide) (PEO-b-PBO), and triblock copolymer micelles of poly(ethylene oxide)-b-poly(1,2-butylene oxide) (PEO-b-PBO-b-PEO).

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Method Intermolecular Potential The potential models for block copolymers, solvents, and solutes are described by fully flexible chains of hard spherical segments bonded at hard sphere contact that can attract each other by Lennard-Jones(LJ) attraction in the spirit of perturbation theory. Off centered sites with short-ranged squared-well attraction are used to model hydrogen bonding between water molecules, and association between ethylene oxide groups and water molecules. The pair potential between spherical segments is given by:

u(rij ) = uhs (rij ) + uatt (rij ) +

XX Ai

uass Ai Bj (ri , rj )

(1)

Bj

uhs is hard sphere potential, uatt is LJ attractive potential, and uAi Bj is directional and short-ranged associating potential. Hard sphere potential is given by:    ∞, rij < σij hs u (rij ) =   0, otherwise

(2)

By taking the hard sphere fluid as reference fluid, the WCA type of LJ attractive potential is given by:

   −LJ − uLJ (rcut ), 0 < rij ≤ σij    uatt (rij ) = uLJ (rij ) − uLJ (rcut ), rmin < rij < rcut      0, rij ≥ rcut

and uLJ (rij ) = 4LJ

"

σij rij

6

12

−

σij rij


6 #

(3)

(4)

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Association potential is given by:

uass Ai Bj (rij ) =

   −ass , rij < rc , θA < θc , θB < θc i j Ai Bj otherwise

  0,

(5)

θc and rc are set to 27° and 1.05σ.

iSAFT Free Energy Functional Modified iSAFT is used to determine equilibrium structure of a micelle by minimizing the system’s grand potential. The total Helmholtz free energy functional of iSAFT can be decomposed into an ideal contribution, hard sphere contribution, hard-chain contribution, LJ attractive contribution and association contribution similar to perturbation theory for a homogeneous fluid. 21,43

A[ρ] = Aid [ρ] + Ahs [ρ] + Aatt [ρ] + Achain [ρ] + Aassoc [ρ]

(6)

The ideal contribution has an exact form given by neglecting thermal de Broglie wavelength:

id

Z

βA [ρ] =

dr1

M X

ρi (r1 )[ln ρi (r1 ) − 1)]

(7)

i=1

where M stands for the total number of segment types over all chains and ρi (r) is the density of spherical segments at position r. The chain free energy functional provides correction to the exact ideal chain functional. 32 Hard sphere contribution is calculated from Rosenfeld’s Fundamental Measure Theory. 22

hs

βA [ρ] =

Z dr1 Φ [nα (r1 )]

7


(8)

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The hard sphere free energy density Φ[nα (r)] is given by:

Φ [nα (r)] = −n0 ln(1 − n3 ) +

n1 n2 n32 nv1 · nv2 n2 (nv1 · nv2 ) + − − 1 − n3 24π(1 − n3 )2 1 − n3 8π(1 − n3)2

(9)

It constructs an excess free energy density by using four scalar weighted densities and two weighted vector densities. Z nα [ρ(r)] =

dr1

M X

(10)

ρi (r1 )ωαi (r − r1 )

i=1

These weighted densities nα [ρ(r)] includes surface averaged, volume averaged, mean curvature averaged and Gaussian curvature averaged densities. Rosenfeld’s free energy functional agrees with the Percus−Yevick compressibility equation of state in the bulk limit. A mean-field version of dispersion contribution is used in this work as it has been demonstrated to provide a good approximation for polymeric systems by comparing against molecular simulation results. 32,38 1X βA [ρ] = 2 i,j att

Z

dr1 dr2 βuatt ij (|r2 − r1 |)ρi (r1 )ρj (r2 )

(11)

|r2 −r1 |>σij

The association free energy functional was first proposed by Chapman 44 based on extension of Wertheim’s bulk TPT1 and further developed by Chapman and Segura, 28,45 and extended to associating polymer by Bymaster and Chapman. 41

βA

assoc

Z [ρ] =

dr1

M X i=1

ρi (r1 )

X Ai

χA (r1 ) 1 + ln χAi (r1 ) − i 2 2

(12)

where χAi (r1 ) is the fraction of segment i at position r1 not bonded at site A. The inhomogeneous unbonded fraction involves solving integral equations in real space in order

8


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to provide accurate degree of association: "

Z

χAi (r1 ) = 1 +

dr2

M X

#−1 ρj (r2 )

j=1

X

χBj (r2 )∆Ai Bj (r1 , r2 )

(13)

Bj

The association strength is similar to its original form in SAFT except that it contains an inhomogeneous cavity function:

ij ∆Ai Bj (r1 , r2 ) = κ[exp(βass Ai Bj ) − 1]y (r1 , r2 )

(14)

where κ is a constant that inlcudes the bonding volume and orientation constraints, and ass Ai Bj is the association energy between site A and site B from species i and j, respectively. We approximate the inhomogeneous correlation function y ij (r1 , r2 ) for the reference hard sphere fluid between segment i and j as the geometric average of bulk radial distribution functions at contact evaluated at average density ρ¯i (r): 31

y ij (r1 , r1 ) = [gij (¯ ρ(r1 ), σij )]1/2 [gij (¯ ρ(r2 ), σij )]1/2

(15)

The average density is given by: 3 ρ¯i (r1 ) = 4π(σi )3

Z dr2 ρi (r2 )

(16)

|r2 −r1 |>σi

where σi is segment diameter for species i. The chain free energy functional is derived from the association free energy functional at the limit of complete association. 32 The detailed free energy functional derivatives are given in Supporting Information. The total grand potential which is crucial to determine the equilibrium state point is therefore: Z βΩ[ρ] =

dr1

M X i=1

n(Ai ) ρi (r1 ) Di (r1 ) + − 1 + βAhs + βAatt + βAassoc 2

9


(17)

n(Ai ) is the total number of association sites on segment i, and Di (r1 ) is functional derivative:

Di (r) =

βδAchain βδAhs βδAatt βδAass − − − δρi (r) δρi (r) δρi (r) δρi (r)

(18)

The equilibrium density distribution can be calculated by solving Euler-Lagrange equation: δΩ[ρ] δA[ρ] = − (µi (r) − Viext (r)) = 0 δρi (r) δρi (r)

(19)

Corona Core

Figure 1: Coarse-graining scheme of a spherical micelle formed by block copolymer E30 B10 . Hydrophilic and hydrophobic chains are in red and green, respectively. Each circle represents a coarse-graining segment in theory.

Figure 2: Hydrogen bonding between water and water, and water and coarse-grained PEO. Lone pairs of electrons are in red. Protons are in yellow.

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Micelle Model and Parameters Fig. 1 shows the coarse-grained molecular model of a spherical micelle from diblock copolymers. The model takes into account hydrogen bonding between PEO and water by assuming that the lone pairs of electrons on oxygen in PEO can only associate with hydrogen sites on water. Water molecules are modeled with 4 association sites, 2 hydrogen sites to represent 2 protons, and 2 oxygen sites to represent 2 lone pairs of electrons. Water molecules can self-associate or cross associate with block copolymers. Because of the extended alkyl side chain in poly(1,2-butylene oxide), the electron lone pairs on poly(1,2-butylene oxide) are shielded from association with water molecules and the hydrophobicity of poly(1,2-butylene oxide) is much higher than poly(propylene oxide). 46 The coarse-grained poly(1,2-butylene oxide) is therefore modeled as hydrophobic chains without association sites. Chain connectivity of diblock and triblock copolymers is treated as tangentially bonded segment model as in SAFT model. 29 The consideration of SAFT chain stiffness by Walker et al. 47 will be necessary to accurately model micelles formed by branched block copolymers. A schematic figure to demonstrate the association in the system is in Fig. 2. Density distribution of segments in the micelle is assumed spherically symmetric. To compare results based on our micelle model with experimental data, we scale the iSAFT chain segment number mi to the number of block copolymer chain unit Ni . A summary of estimated parameters and scaling relations for pure components is are presented in Table 1. Six pairs of binary interaction parameters and binary association parameters are listed in Table 2. Parameters are obtained from different sources by determining the nature of binary interaction from independent data. This approach is pragmatic but it has been used to model block copolymer micelles and capture the impact of molecular architecture by LSCF. 19,48 A more systematic approach to determine the binary interaction parameters for diblock copolymers will be the subject of future work dependent on the availability of data for the vapor pressure of PBO in water and the interfacial tension of PBO and PEO polymer melt

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mixture. Some of the parameters are estimated following the route described by Wang et al. 39 The small LJ interaction energy between PBO segments and water, and between segments of PBO and PEO is chosen to mimic the fact that PBO is six times more hydrophobic than PPO. 49,50 The coarse-grained approach to model the PBO unit requires a small LJ interaction energy between PBO and other hydrophilic components. Benzene is modeled as a linear chain molecule 51 and the parameters are estimated from vapor pressure and saturated liquid density data. The binary interaction parameter kij = 0.0009T − 0.2269 between benzene and water is estimated using aqueous solubility data from Arnold. 52 The predicted solubility between benzene and water is given in Supporting Information. Pair interaction parameters between benzene and other components are calculated by Lorentz-Berthelot rules with tuning parameters. Ni is the number of real repeated chain units of copolymers in experiment, and mi is the coarse-grained iSAFT chain length. The scaling constant Ni /mi is estimated based on CMC data for PEO-b-PBO diblock copolymers of various lengths and the same scaling constant is used for triblock copolymers. The scaling relations for poly(ethylene oxide)(E) and poly(1,2-butylene oxide)(B) coarse-grained segments are NB = 5(mB − 2) and NE = 6mE . The additional intercept used in the scaling from coarse-grained chainlength to degree of polymerization of PBO group is due to the fact that a surfactant with too few number of hydrophobic segments cannot form a micelle. Table 1: Summary of Parameters ˚ σ(A) Water 3.0 Benzene 3.2 PEO 3.0 PBO 3.0

LJ /kB (K) ass /kB (K) 328.60 1747.30 260.05 328.60 1747.30 294.05

κ m 4.43E−3 1 2 4.43E−3 6 5

Numerical Procedure The equilibrium density distribution is determined by minimizing the grand potential given in Eq 19. Details of calculating functional derivatives are given in Supporting Information. 12


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Table 2: Values and Sources of Binary Interaction Parameters LJ /kB (K) ass /kB (K) source Water-PEO 328.60 1747.30 averaged Water-PBO 100.00 estimated kij p Water-Benzene i j (1 − kij ) solubility PEO-PBO 100.00 estimated kij Benzene-PEO 292.32 averaged Benzene-PBO 276.53 averaged

ref 39 39 52 39

Grand potential is calculated by Eq 17 after a density distribution is obtained. Mathematical operation to reduce the multi-dimensional convolution to a one-dimensional integral in spherical coordinate is outlined in our previous work. 39 A grid spacing of 0.05σ in radial distance is used in this work to balance time complexity and accuracy. The fixed-point iteration is stopped once the total residual of the density profiles from last iteration is less than 1E−5.

Results and Discussion Spherical Micelles Formed by Triblock and Diblock Copolymers There has been great interest in understanding the behavior of spherical micelles from a theoretical point of view. Diblock PBO-b-PEO(Em Bn ) and triblock PEO-b-PBO-b-PEO(Em Bn Em ) micelles have been experimentally studied 53–56 and summarized. 57 Here we investigate the chain architecture effect by comparing theoretical calculations with experimental results of micelles formed by both Em Bn and Em Bn Em copolymers. Bulk density of micellar solution is assumed to be equivalent to water density within the dilute regime to allow conversion of data from volumetric concentration to mole fraction. The results in this section and next section are predicted at a condition of 1 atm and 30°C. CMCs are located by an extensive search for the cross over of inhomogeneous grand potential and bulk aqueous phase grand potential. 39 A comparison between the CMCs of triblock and diblock copolymer micelles at varying 13


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0

log10 [cmc/ (mol dm-3)]

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-2 triblcok -4

diblock EₘBₙEₘ iSAFT EₘBₙ iSAFT EₘBₙEₘ EXP EₘBₙ EXP

-6

-8 4

6

8

10

12

14

16

Hydrophobe Length(NB) Figure 3: Critical micelle concentrations of diblock copolymer micelles(diamond) and triblock copolymer micelles(circles) at 30°C. Solid line represents theory prediction for diblock copolymer micelle. Dashed line represents theory prediction for triblock copolymer micelle. Markers represent experimental results. 53–56 hydrophobe length is shown in Fig. 3. The predicted CMCs of both triblock and diblock copolymer micelles have log-linear behaviors with respect to the hydrophobe chain length in our model. Theory quantitatively agrees with CMC data of diblock copolymer micelles. Both theoretical and experimental CMCs of triblock copolymer micelles are higher than diblock copolymer micelles by orders of magnitude. As shown in Fig. 3 theory predicts a shift in CMC between diblock micelle and triblock micelle. We attribute the increase of CMC to the formation of smaller micelles by triblock copolymers as illustrated in Fig. 4. The shift in CMC is related to a microscopic change in density distribution of micelles. This is reflected in Fig. 5 for overall micellar segment density profiles of E36 B10 and E18 B10 E18 at their equilibrium states. Both triblock copolymers and diblock copolymers have equivalent number of hydrophobes and hydrophiles. Therefore they are energetically identical but their chain structures are quite different. This entropic difference leads to macroscopically different CMCs. The triblock copolymers are entropically disfavoured in the micellar state because 14


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Figure 4: Schematic figures of micelles formed by different block copolymers with same molecular weight. Left: spherical micelles from tiblock copolymers with bended hydrophobe; right: micelles from diblock copolymers with un-bended hydrophobe. of the restriction of the two blocks at the interface. 58,59 The chain free energy of modified iSAFT predicts this entropic effect. The predicted total micellar radius and core radius of triblock copolymer micelles by iSAFT are smaller than the diblock micelles. This agrees with experimental observation that triblock copolymers have to bend to form spherical micelles. A detailed segment density profile in Fig. 6 shows this microscopic structure. Each peak represents a coarse-grained segment in iSAFT. The hydrophobe segment density profile of diblock copolymer micelle has four distinct peaks that means each segment has a different average location from the micellar core. Triblock copolymer micelle has two distinct density peaks for hydrophobe means two hydrophobe segments are spherically overlapped due to chain conformation as shown in Fig. 4. Experimental measurements also show that the maximum radius of spherical micelles formed from a linear diblock copolymer will be up to two times that from surfactant of triblock architectures. 60 Theoretical study of Pluronic and Tetronic surfactant micelles by Hurter et al. 19 using LSCF theory showed that at low molecular weight a branched block copolymer architectures with higher entropy formed a smaller micelles with less concentrated hydrophobic core due to a higher degree of mixing of hydrophobe and hydrophile segments in this region. Equilibrium density distribution in this work also shows that the hydrophobic core of micelles from triblock copolymers is less concentrated. The predicted degree of hydrogen bonding when forming block copolymer micelles are

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1.0 0.8

E₁₈B₁₀E₁₈ PBO E₁₈B₁₀E₁₈ PEO E₁₈B₁₀E₁₈ Water E₃₆B₁₀ PBO E₃₆B₁₀ PEO E₃₆B₁₀ Water

0.6

𝜌𝜎3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4 0.2 0.0 0

3

6

r/𝜎

9

12

15

Figure 5: Density distribution of spherical micelles formed by diblock copolymers E36 B10 (solid line) and triblock copolymers E18 B10 E18 (dashed line) at 30°C and 1 atm. They are both predicted at equilibrium. shown in Fig. 7. The degree of hydrogen bonding of a segment not bonded at an association site is given in Eq 13. Each proton of water can self-associate with the lone pairs of electrons of water and cross associate with lone pairs of ether group of PEO. The proton of water has the highest degree of association at micellar interface when a spherical micelle is formed. The degree of hydrogen bonding of water at the micellar interface is higher than water in aqueous phase as shown in Fig. 7. This enhanced hydrogen bonding at micellar interface is crucial to micelle formation.

Dimension and Aggregation One of the appealing aspects of using inhomogeneous fluid theory to study micellar selfassembly is that segment density distribution of spherical block copolymer micelles reveals microscopic structure at the molecular level. 31 Here we investigate how hydrophobe and hydrophile have different contributions to the dimension and aggregation of a spherical micelle.

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0.8 E₃₆B₁₀ PBO

𝜌𝜎3

0.6

E₃₆B₁₀ PEO

0.4 0.2 0 0

3

6

r/𝜎

9

12

15

0.8 E₁₈B₁₀E₁₈ PBO E₁₈B₁₀E₁₈ PEO

0.6

𝜌𝜎3

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0.4 0.2 0 0

3

6

r/𝜎

9

12

15

Figure 6: Detailed coarse-grained segment density of micelles from diblock and triblock copolymers as in Fig. 5: Top: diblock copolymer micelle; Bottom: triblock copolymer micelle. Green and red are for coarse-grained PBO and PEO segments. Both diblock copolymers and triblock copolymers have four coarse-grained hydrophobes. Two of the coarsegrained hydrophobes of triblock copolymers are overlapped with another two coarse-grained hydrophobes as suggested by Fig. 1. 17


Langmuir

1 H₂O E₁₈B₁₀E₁₈ PEO E₁₈B₁₀E₁₈ H₂O E₃₆B₁₀ PEO E₃₆B₁₀

0.8 0.6

𝜒A

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 39

0.4 0.2 0 0

5

10

15

r/𝜎 Figure 7: Degree of association for spherical micelles formed by diblock copolymers(solid line) and triblock copolymers(dashed line). Blue lines are are the fraction of water not bonded at proton association site. Red lines are the fraction of PEO segments not bonded at electron lone pair association site. Micelles are formed at the same conditions as Fig. 5. We first examine the hydrophobe effect onto micellar conformations. Fig. 8 shows the density profiles of hydrophile and hydrophobe at varying hydrophobe length. Radii of micelles formed at critical micelle concentrations increase by approximately one coarse-grained hydrophobe dimension for each additional hydrophobe coarse-grained segment. This implies that the hydrophobic cores of micelles are densely packed. Fig. 9 shows the change of dimension of micelles formed by block copolymers at varying hydrophile length. Here the number of hydrophobes is fixed but the number of hydrophile varies. Compared to the effect of hydrophobe length in Fig. 8, the dimension of a micelle is found to be highly insensitive to hydrophile length. At a fixed hydrophobe B10 , an increase in PEO groups shrinks micellar core but results in a more spread density distribution of PEO block groups. This is due to the stronger repulsion against micelle core from longer hydrophile. The dimension of an overall micelle, however, increases if we treat the vanishing boundary of hydrophile as the radius of overall micelle. Borisov et al. 18 using scaling theory also show that the radius of a 18


Page 19 of 39

starlike micelle is insensitive to hydrophile length. 1 E₃₀B₁₀ PBO E₃₀B₁₅ PBO E₃₀B₂₀ PBO E₃₀B₂₅ PBO E₃₀B₁₀ PEO E₃₀B₁₅ PEO E₃₀B₂₀ PEO E₃₀B₂₅ PEO

0.8 0.6

𝜌𝜎3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4 0.2 0 0

3

6

9

12

15

r/𝜎 Figure 8: Density distribution of diblock copolymer micelle at varying hydrophobe length at 30°C and 1 atm. Density distribution for water is omitted for clarity. Micelles are formed at the same conditions as Fig. 5. Both increase of hydrophile length and hydrophobe length lead to an increase of micellar radius in experiments. In the experimental study of micelles formed from Em Bn Em copolymers with different Em block and constant Bn block, 1nm increment in radius is observed by every 33 additional hydrophilic ethylene oxide chain unit. 61 In contrast to the study of hydrophile block, 1nm increment in micellar hydrodynamic radius is observed for every additional hydrophobic butylene oxide chain unit. 58 The experimentally observed change in micellar dimension due to varying hydrophobe and hydrophile qualitatively agrees with our theoretical calculations. Aggregation number of a spherical micelle can be determined from the relation: p = R

ρB (r)4πr2 dr/mB . ρB and mB are inhomogeneous hydrophobe segment density, and coarse-

grained iSAFT segment number. The aggregation numbers for the micelles shown in Fig. 8 are 38, 53, 69, 83 for diblock copolymers E30 B10 , E30 B15 , E30 B20 , E30 B25 , respectively. This agrees with the experimental findings that the aggregation number increases for block 19


Langmuir

1 E₃₀B₁₀ PEO E₃₆B₁₀ PEO E₄₂B₁₀ PEO E₄₈B₁₀ PEO E₃₀B₁₀ PBO E₃₆B₁₀ PBO E₄₂B₁₀ PBO E₄₈B₁₀ PBO

0.8 0.6

𝜌𝜎3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 39

0.4 0.2 0 0

3

6

9

12

15

r/𝜎 Figure 9: Density distribution of diblock copolymer micelle at varying hydrophile length at 30°C and 1 atm. Density distribution for water is omitted for clarity. Micelles are formed at the same conditions as Fig. 5. copolymer with longer hydrophobe. 62 The aggregation numbers for the micelles with increasing hydrophile in Fig. 9 are 38, 31, 26, 23 for diblock copolymers E30 B10 , E36 B10 , E42 B10 , E48 B10 , respectively. Chaibundit et al. 61 found that the aggregation numbers for micelles formed by PEO-PBO and PEO-PBO-PEO copolymers decreased with longer hydrophile by using static light scattering. A typical aggregation number 20-100 for micelles formed by PEO-PPO and PEO-PBO copolymers is reported by Alexandridis 63 based on a series of experimental study.

Pressure and Temperature Effects Pressure variation of the CMC is interesting though usually ignored because of the challenge to accurately measure a small variation of CMC over an extreme range of pressure. Experimental evidence indicates that a change of several hundreds of mega-pascals are required to observe the effect of pressure on self-assembly. 64 For micellar solutions, pressure

20


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leads to three phases in general: dilute surfactant solutions, micelles, and crystal phases. 65 Understanding the pressure effects onto micellar self-assembly from a theoretical perspective is a starting point for further study of mesophase self-assembly of surfactants. Besides the physico-chemical interest, high pressure is also of great interest for application of food processing under high pressure. 66 In this work the critical micelle concentrations at different pressures from 0.1 MPa to 200 MPa are determined. We first start with diblock copolymer E30 B10 at 30°C. The predicted CMC at 1 atm is 1.95E−7 in mole fraction. The CMC then increases under a compression of bulk solution and reaches a maximum of 2.05E−7 at P=40 MPa. The CMC then monotonically decreases under further compression as shown in Fig. 10(a) and Fig. 11. An experimental measurement of CMC of n-alkyl polyethylene ether surfactant in water reaches a maximum at 150 MPa 67 shown in Fig. 11. The compression effect is studied for diblock copolymers with different hydrophobe lengths. Fig. 10(b) shows the predicted CMC of E30 B15 . CMC for E30 B15 at the same condition is two orders of magnitude lower than E30 B10 as a result of having a longer hydrophobe. The window of equal CMCs in pressure is narrowed for surfactant with longer hydrophobe. The pressure maximum is almost diminished for E30 B15 . A much less pronounced pressure maximum is found around P=1 MPa for E30 B15 . Further extending the hydrophobe leads to a disappearance of pressure maximum CMC for micelles formed by E30 B20 . We attribute the disappearance of a pressure maximum to the imbalance between hydrophobic force and hydrophilic force. As the hydrophobe gets extended, the pressure at which CMC reaches a maximum decreases. For a block copolymer molecule whose hydrophobic chain is dominant, the partial molar volume of a pseudo micellar phase is always less than the solution phase as seen for E30 B20 . Therefore a very hydrophobic block copolymer micelle does not have a CMC-pressure maximum. It is interesting to note that this pressure maximum behavior is different from the folding and unfolding of protein under compression. 21


Langmuir

2.10E-07 2.00E-07

XCMC

1.90E-07 1.80E-07 1.70E-07 1.60E-07 1.50E-07 0

50

100

150

200

Pressure(MPa)

(a) 1.29E-10 1.29E-10

XCMC

1.29E-10 1.29E-10 1.29E-10 1.28E-10 1.28E-10 0

1

2

3

4

5

6

Pressure(MPa)

7

8

9

10

(b) 6.62E-14 6.60E-14 6.58E-14

XCMC

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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6.56E-14 6.54E-14 6.52E-14 6.50E-14 6.48E-14 0

2

4

6

8

10

Pressure(MPa)

(c)

Figure 10: Predicted absolute CMCs at various pressure conditions and 30°C for micelles formed by: (a)E30 B10 (b)E30 B15 (c)E30 B20 . X stands for mole fraction.

22


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10% E₃₀B₁₀ E₃₀B₁₅ E₃₀B₂₀

0% -10% -20%

2.1 1.9 XCMC 104

Relative XCMC to XCMC at 1 atm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

-30% -40%

1.7 1.5 1.3 0

100 200 300 Pressure (MPa)

-50% 0

50

100

Pressure(MPa)

150

200

Figure 11: Relative CMCs of block copolymers at different pressures at 30°C. The inset is digitized data of measured critical micelle concentrations of pentaethylene glycol monooctyl ether (C8 E5 ) ether under different high pressure conditions at 30°C. 67 Under compression unfolded protein first become folded at pressure close to ambient pressure then become unfolded by compression. 68 Temperature dependence of CMCs is shown in Fig. 12. Two diblock copolymers with different hydrophobes are examined. Logarithm of CMCs are plotted against reciprocal temperatures. The slope is related to the heat of micellization. Similar temperature dependence is seen for block copolymers with different hydrophobes. CMCs for non-ionic surfactants in water are reported to slightly decrease with temperature from experiments. However, increased CMCs with temperature for n-dodecyl polyethylene glycol monoether at high temperatures are reported by Chen et al.. The increase of CMCs at higher temperature is due to the breakdown of water structures around the hydrophobe. 69 Temperature dependence of CMCs from lattice Grand Canonical Monte Carlo simulation studies of block copolymer micelles by Kim and Jo, 70 and by Panagiotopoulos et al 71 show a negative reciprocal temperature dependence of CMCs which are similar to this work.

23


Langmuir

-3

E₃₀B₁₅ E₃₀B₁₀

-4


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-5 -6 -7 -8 -9 -10 0.0031

0.0032

0.0033

0.0034

0.0035

0.0036

0.0037

1/T(K-1) Figure 12: Temperature of critical micelle concentration of two block copolymers E30 B15 and E30 B10 at 1 atm. Temperature varies in the range between 273K to 312K.

Micellar Solubilization and Locus of Solute Solubilization of benzene is studied in this work to shed light on the design of micellar carrier for effective container of less soluble pharmaceuticals as well as catalytic activity of micellar systems. 72 Since most pharmaceuticals are aromatic derivatives, we show results of solubilization of benzene in the micelles formed by E30 Bn . Solubilization of benzene is shown in Fig. 13. Molar solubilization ratio 73 is plotted against percentage of bulk aqueous phase concentrations of benzene solute. The ratio is defined by Eq 20

M SR =

Nsolute Nsurf actant

(20)

Bulk concentrations of block copolymers are fixed at CMCs and solute concentration varies. The molar solubilization ratio is the total number of benzene molecules over the total number of surfactants within a single spherical micelle. As the concentration of benzene solute 24


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1.6

Molar Solubilization Ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

E₃₀B₁₀ E₃₀B₅

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

% of Bulk Benzene Solubility Figure 13: Predicted molar solubilization ratio of benzene to E30 Bn in aqueous solution at the condition of 1 atm and 30°C by theory. increases in continuous aqueous phase, more benzene is solubilized by each surfactant. At higher concentration, micelles formed by more hydrophobic block copolymers have higher loading capacities of benzene solutes. When the bulk concentration of benzene is close to the solubility limit in the aqueous phase, molar solubilization ratio of E30 B10 is almost twice that of E30 B5 . The enhancement of solubilization by having a more hydrophobic block copolymer is due to formation of more non-polar micro-environment in the micellar core. This phenomenon has been confirmed by several studies of micellar solubilization of Pluronic with different hydrophobes. Kozlov et al 74 showed that the partition coefficient of solute between micellar phase and aqueous phase has a log-linear dependence on block copolymer hydrophobicity. The partition coefficients of Naphthalene in Pluronic micelles measured by Hurter 48 is linear with weight fractions of PPO groups at constant molecular weight. Beyond the solubilization capacity of micelles, understanding the micellar micro-environment is important for broader applications of block copolymer micelles. Density distribution of

25


Langmuir

1

1

0.8

0.8 Water Benzene PBO PEO

0.4

Water Benzene PBO PEO

0.6

𝜌𝜎3

𝜌𝜎3

0.6

0.4 0.2

0.2

0

0 0

5

r/𝜎

10

0

15

(a) 1% Solubility

5

r/𝜎

10

15

(b) 10% Solubility

1

1

0.8

0.8 Water Benzene PBO PEO

0.4

Water Benzene PBO PEO

0.6

𝜌𝜎3

0.6

𝜌𝜎3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 39

0.4

0.2

0.2

0

0 0

5

r/𝜎

10

15

0

(c) 30% Solubility

5

r/𝜎

10

15

(d) 70% Solubility

Figure 14: Density profile during the formation of a swollen micelle. From (a)-(d) refers to the equilibrium prediction of distribution of benzene, block copolymers and water molecules from a low to high bulk benzene concentration.

26


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benzene, PBO, PEO, and water predicted at different bulk aqueous phase benzene concentrations are shown in Fig. 14. The calculations at varying solute concentrations is to mimic the solute loading process by adding more solute into micellar solution. At 1% of bulk solubility, benzene is predominantly located at the corona region at very low concentration. More benzene penetrates into the micellar core at 10% of bulk solubility but the amount of benzene is not enough to create a swollen interior. At 30% a benzene interior is formed and it continues to swell with more benzene from bulk region. Figures to demonstrate the swelling process during solubilization are given in Fig. 15. Eriksson and Gillberg 75 used 1

H NMR to determine the location of benzene solute in cetyltrimethylammonium bromide

(CTAB) micelles and they concluded that benzene is located at corona at low concentration but it moves to micellar core at higher concentrations. Mukerjee and Cardinal, 76 however, conclude that benzene locates at micellar interior even at low concentrations. Based on the study in this work, theory reconciles the conflict. Benzene solute always exists at the interfacial region of micelles as shown in Fig. 14. However, no swollen interior can be detected at low benzene concentration as shown in Fig. 14 (c). As the solubilization ratio approaches 0.8, a new benzene micro-environment is formed to replace the hydrophobic core of micelles without solute. Therefore a clear transition from swollen micelle to microemulsion 77 is explained by theory. The conclusion by Eriksson and Gillberg 75 is a situation in Fig. 14 (c) and in Fig. 15 (b). The consistent interfacial locus of benzene solute has been observed from molecular simulation of oil-water interface with toluenes, and it is confirmed by Nagarajan 78 using the transfer of free energy approach. LCSF 19 underestimates the amount of benzene derivatives in micellar core and concludes no transition from swollen to microemulsion. The detailed solute distribution from iSAFT shows promise as a powerful tool to design more effective drug delivery carrier. For instance, a drug delivery carrier with drug loaded in the interfacial region has faster release rate and is named a burst-release carrier. 75

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(a)

(b)

Page 28 of 39

(c)

Figure 15: Swollen process implied by iSAFT: (a)infinite dilute solute in aqueous phase; (b)intermediate solute in aqueous phase; (c)close to bulk solubility. Hydrophobe is in green, hydrophile is in red and benzene is in black.

Conclusion iSAFT has been applied to study thermodynamics, microstructures, and solubilization of block copolymer micelles of poly(ethylene oxide)-b-poly(1,2-butylene oxide). The incorporation of detailed chain physics and an associating free energy functional in the iSAFT formalism explains micellar dimension and critical micelle concentration between the aggregation of triblock and diblock copolymers. Theory showed the reduction in micellar radius is from the bending of triblock copolymer hydrophobe. CMC predictions are in quantitative agreement with experimental data. The phenomenon of pressure maximum CMC is captured by theory and it predicts that this maximum disappear for surfactants with highly unbalanced structures. Solubilization of micelles is investigated by taking benzene as an example solute. The theory predicts the enhancement of solubility by forming micelles with different hydrophobicity, and the results qualitatively match experimental data. Additional insight is provided from the theory regarding the locus of aromatic solute in a micelle. The calculated density profile during the formation of a swollen micelle reconciles the conflict arguments about the locus of benzene in micelles at different solute concentrations.

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Acknowledgement The authors are grateful to Dr. Irina Smirnova who shares key insight based on the experimental study of micelles in her group. Leilei Zhang is gratefully acknowledged for his inspiring discussion about swollen micelles. The authors thank the Robert A. Welch Foundation(Grant No. C-1241) for financial support.

Supporting Information Available • Free Energy Functional Derivatives • Predicted Solubility of Benzene in Water

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Graphical TOC Entry 0

iSAFT model


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-2 triblcok diblock

-4

EₘBₙEₘ iSAFT EₘBₙ iSAFT EₘBₙEₘ EXP EₘBₙ EXP

-6 -8 4

39

6 8 10 12 14 Hydrophobe Length(NB)


16

Thermodynamics, Microstructures, and Solubilization of Block

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