Surfactant-Induced Patterns in Polymer Brushes - Langmuir (ACS

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Surfactant Induced Patterns in Polymer Brushes Daniil E. Larin, and Elena N Govorun Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01850 • Publication Date (Web): 31 Jul 2017 Downloaded from http://pubs.acs.org on August 2, 2017

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Surfactant Induced Patterns in Polymer Brushes Daniil E. Larin, Elena N. Govorun*

Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie gory, Moscow, 119991 Russia

The properties of surfaces with grafted macromolecules are determined by a fine structure of the macromolecular layer, whereas the mixtures of macromolecules with surfactants are very rich in structure types. Using the scaling mean-field theory, we consider the self-assembly in polymer brushes into various patterns induced by interactions with low-molecular surfactants. The interaction energies of the parts of a surfactant molecule with the polymer units are assumed to be greatly different. With increasing the grafting density, the formation of lamellae perpendicular to the grafting plane, continuous layer with oblong or round pores, or homogeneous brush is predicted. The driving force of the pattern formation is a gain in the interaction energy of surfactant molecules oriented at the lateral surfaces of lamellae or pores. The process of pore formation in a homogeneous brush caused by a temperature change at definite grafting densities is described as the first-order phase transition. It is accompanied by a step-wise extension of the brush and by orientational ordering of surfactant molecules. The transitions between the other patterns are of the second order. The thickness of lamellae and the distance between pores are approximately twice the surfactant molecule size except for the extremely high grafting densities. The diagrams of brush patterns are presented and discussed.

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INTRODUCTION The development of experimental techniques of the polymer brush synthesis and investigation increased interest to such systems over the last decade. Morphological, mechanical, tribological and adhesive properties of brushes are analyzed using reflection spectroscopy, force microscopy, and synchrotron radiation.1-9 For many practical applications, it is important to study the polymer brush interactions with other molecules such as proteins and surfactants and with living cells.10-18 In theory and computer simulations, the polymer brush characteristics are predicted depending on the solvent composition and quality and on the macromolecular architecture.19-28 Polyelectrolyte brushes can be laterally inhomogeneous because of the macromolecular selfassembly controlled by the interplay of hydrophobic, electrostatic, and steric interactions.25-28 The formation of bundles and maze-like structures was first predicted in theory and computer simulations.25 Further, the more detailed theoretical analysis of polyelectrolyte brushes under spatial confinements gave the richer variety of structures such as micelles, micelles coexisting with nonaggregated chains, stripes and layers with solvent-filled holes.27 The collapse of polyelectrolyte brushes in a poor solvent in the presence of multivalent ions can proceed with the formation of pinned micelles and cylindrical bundles.28 Investigation of the interactions of poly(methacrylic acid) brushes with oppositely charged surfactant led to the conclusion that the surfactant is not exclusively bound electrostatically, but also through hydrophobic interactions.16 The textures with liquid-crystal domains of micrometer scale were observed for the side-chain liquid-crystal polymer brushes.9 The poly(acrylic acid) brushes formed micellar islands or a continuous film with dimples, and a flat homogenous film was not observed even at the high grafting densities.4 The other type of polymer systems where the nanoscale segregation is common and actively studied nowadays is mixtures of polymers and surfactants.29-44 Such systems are widely used in different applications for the stabilization of dispersed systems, the removal of metal anions from aqueous solutions, health care, сontrolled sequestration of drugs, enhanced oil recovery and hydraulic fracturing, food science, and cosmetics.29,30,39-42 Polymer-surfactant complexes can be used for the catalysis and for the synthesis of polymer microparticles as well.43,44 In terms of the weak segregation theory, the formation of ordered micellar, cylindrical, and lamellar structures is predicted for the solutions of polymers with surfactants and for the bulk-phase of large polymer globules in the presence of a surfactant.35,36 In the diagrams, the widest range of the system parameters corresponds to the lamellar phase. For a single polymer globule in solution, the addition of a surfactant leads to a change in the globule volume and to a decrease in the surface tension and thickness, as analyzed in the mean-field theory approach.37 The strong interaction of surfactant molecules with monomer units can lead to the globule transformation into a system of ACS Paragon Plus Environment

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quite small aggregates as was observed in the molecular dynamics simulations.38 In those considerations,35-38 the surfactant molecules were represented as dimers with the different interaction energies of hydrophobic and polar parts with their surrounding. The dimer model of surfactant molecules was then used by us for the analysis of possible lateral aggregation in polymer brushes with the formation of strands (cylindrical bundles).14 It was predicted that the pattern formation can be favorable when compared to a homogeneous brush, because the appearance of the aggregate lateral surfaces permits polar groups of the surfactant molecules to orient outwards at the surfaces gaining the interaction energy. In the previous theoretical consideration of a polymer brush in the presence of a surfactant, the interactions between macromolecules and whole surfactant micelles were considered.17 It was assumed that such interactions effectively change the polymer excluded volume that led to a decrease in the monomer unit concentration with increasing the distance from the grafting plane, the surfactant concentration increased with this distance in all cases. The lateral aggregation of macromolecules was not addressed there. In the present paper, we generalize the approach of our previous work14 and analyze the lateral aggregation of macromolecules in a brush induced by dimer surfactant molecules. Using the scaling type mean-field theory and considering a variety of possible structures, we calculate the free energy accounting for the contribution of the oriented surfactant molecules into the lateral surface free energy. The polymer-containing regions in the brush are assumed to be homogeneous. The macromolecules and surfactants are nonionic. In the limit of a strong degree of the surfactant molecule amphiphilicity, we predict the formation of lamellae, a porous layer with oblong pores and with round (cylindrical) pores, and of a homogeneous brush depending on the grafting density. The characteristic sizes of polymercontaining regions and pores are determined by the surfactant molecule length. We calculate the lateral surface free energy contribution taking into account the surface curvature by direct integration over the surface region without expansion in powers of the curvature.

THE MODEL We consider a layer of m macromolecules grafted to a plane of area S. The macromolecules consist of N monomer units of volume p and size a along the chain, where each monomer unit represents a Kuhn segment of the polymer chain. The surface density of grafting is equal to ns=m/S, the surface area per macromolecule is correspondingly =1/ns. The grafting density is assumed to be large enough ( « Na2) for strong stretching of macromolecules in the brush. The layer is surrounded by amphiphilic (surfactant) molecules consisting of two parts (P and H) at a distance l>>a between their centers (Figure 1). The volumes of these parts and the volume of ACS Paragon Plus Environment

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the whole surfactant molecule are equal to P, H, and s=P+H, respectively. It is assumed that

Н >>P, and therefore sH, which permits to neglect an angular dependence of the steric interactions between surfactant molecules. The contacts of macromolecules with H groups are much more favorable than of those with P groups. Surfactant molecules are assumed to be homogeneously distributed outside the polymer brush. The average interaction energy of monomer units with the surfactant is characterized by the Flory-Huggins parameter . The surfactant amphiphilicity is characterized by the difference >0 in the energy of interactions of a P group with the pure polymer and with the pure surfactant (in units of kBT).

Figure 1. Grafted layer of macromolecules surrounded by surfactant molecules.

We assume that amphiphilicity of surfactant molecules causes the self-assembly of macromolecules with surfactant molecules in the brush. The possibility of the formation of strands and lamellae perpendicular to the grafting plane and of a porous layer is considered (Figure 2). Note that such self-assembly could be favorable in comparison with a homogeneous polymer brush only if the free energy of surfaces of polymer-containing regions is negative. For a polymer globule, it was shown that the surface thickness decreases in the presence of a surfactant.37 Accordingly, the surface thickness is assumed to be of the order of a monomer unit size a in the considered strong segregation limit ( » 1).

Homogeneous Polymer Brush At first let us write out the expressions for the free energy F0 of a homogeneous polymer brush with the polymer volume fraction p and with chain ends at a distance H from the grafting plane in terms of the Alexander-de Gennes theory:45,46


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F0 f ( h)  m el  N s ln(1   p )  mN (1   p ) , k BT k BT

(1)

where kB is the Boltzmann constant, T is the thermodynamic temperature, fel is the elastic energy of a stretched macromolecule, Ns=mNp(1p)/(sp) is the total number of surfactant molecules in the brush. The elastic free energy is taken in the form of a freely-jointed chain model47,48







f el (h)  k BTN hL1h   ln L1h  sinh L1h  , h 

1 H , L(h)  cothh  . Na h

(2)

The brush height and volume fraction are related by the expression

p 

N p H

.

(3)

The equilibrium value of the polymer volume fraction p corresponds to the free energy minimum ( F0  p  0 ) and satisfies the equation

p 1 1 2 ~ L (h)  (ln(1   p )   p )   p  0 ,



s

~ h  1 ( p  ) ,

(4)

~ where   a  p . The corresponding minimum value of the free energy of a homogeneous

polymer brush, F0, was calculated from Eq. (1) and further compared with the free energies of the patterned polymer brushes.

Figure 2. Basic structure types of a polymer brush: (a) homogeneous polymer layer, (b) strands (cylinders), (c) lamellae, and (d) porous layer (“inverse” cylinders). The cross-sections perpendicular to the grafting plane are shown. The polymer-containing regions are gray.


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Patterned Polymer Brush Now let us consider the self-assembly of macromolecules in the brush into strands, lamellae, or a porous layer (Figure 2). The polymer-containing regions are assumed to be homogeneous with chain ends at a distance H from the grafting plane. The brush free energy, F, is the sum of the free energy of the polymer-containing regions in the form (1) and of the lateral surface free energy, Fsurf: F f ( h) F  m el  Ns ln(1  p )  mN (1  p )  surf . kBT kBT kBT

(5)

The surface free energy should be calculated separately for the considered types of the brush structure. Besides, the relations between the polymer volume fraction p and brush height H are different. Let Mc be the total number of strands (cylinders) of radius R with the lateral surface area Sc and with the surface tension c (b), Ml be the total number of lamellae of thickness 2R at the distance d between them with the lateral surface area Sl and with the surface tension l (c), and Mp is the total number of pores of radius R with the lateral surface area Sp and with the surface tension p (d). The lateral surface area is equal to (a) S c  M c 2RH ; (b) Sl  M l 2HS (2R  d ) ; (c) S p  M p 2RH

(6)

The polymer volume fraction is given by the formulae:

~ N m p p   ; ; (a)  p  (b) (c) , (7)   p p ~  R 2 ) H 2 RH R 2 H (m p ~ ~ ~ mc  m M c , ml  m(2R  d ) (M l S ) , and m p  m M p are the numbers of macromolecules ~ N m c p

~ N m l p

per cylinder, lamella’s part of unit length, and pore, respectively. Note that S/(2R+d) is the net length of lamellae (b) along the grafting plane. We represent the surface tension i (i = c (a), l (b), or p (c)) as a sum of two contributions:

i =0+surf i, 0 is determined by the polymer conformational restrictions and polymer interactions with surfactant molecules on average and surf i is determined by the gain in the interaction energy of P groups with their surroundings originated from the surfactant molecule orientation at the surface. The first contribution is assumed to be proportional to the polymer volume fraction p: 0 = kBTs0p (s0 > 0). The second contribution is equal to

surf i = nsurf i kBTp,

(8)

where nsurf i is the number of oriented surfactant molecules per unit area (surface density), p is the interaction energy gain per one P group (p >>1). The surface density of surfactant molecules for the considered patterns can be found as


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(b)

(c)

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nsurf c 

nsurf l 

1  p

s 1  p

s

1  p  r l2    R dr    l  2R  s   R l R

l

(d)

nsurf p 

(9)

1  p

s

R l



R

1  p  r l2    dr  l R s  2 R 

From Eqs. (6)-(9), the surface free energy is equal to (b)

(c)

(d)

Fsurf k BT

Fsurf k BT

Fsurf k BT



 a p l l 2  S c  2mN  ~ s0   (1   p )  2    R  k BT s  R 2R   



p  a l S l  mN  ~ s0   (1   p )  k BT s R  R

c



(10)

l

p k BT

Sp 

2  ~ a    s0   p (1   p ) l  l    R 2R 2   s (R 2 )  1  R  

2mN ~ m p

~ (R 2 ) in the denominator (d) is equal to the ratio of the where ~ s0  s0 p a , the fraction m p grafting plane area to the area of pores in the cross-section parallel to the grafting plane. The area of the grafting plane per strand, lamella’s part of unit length, or pore relates to the pattern sizes R and d (Figure 2) as (b)

~ , 6R  d 22 tan 30 0  m c

(d)

~  6R  d 2 2 tan 30 0  m p

~ , (c) 2R  d  m l

(11)

The distance d between neighbor strands or lamellae is assigned to be greater than the surfactant size l to prevent P groups from joining alien aggregates. If P group is in an alien aggregate its surfactant molecule does not contribute to the decrease of the surface tension (8), whereas it is such decrease that can lead to the appearance of a pattern in the brush. That is why the sizes of polymerfree regions are limited in our model: d  l for strands and lamellae (b and c) and 2R > l for pores (d). For an each structure type, the equilibrium values of the polymer volume fraction (p), the ~ , m ~ , or m ~ ) and pattern sizes (R and d) should correspond to the free aggregation number ( m c l p

energy F (Eq. (5)) minimum with respect to these parameters. The minimum value among the free energies of the considered patterns corresponds to the favorable structure of the brush.

Transient Patterns Along with the basic patterns of the polymer brush (strands, lamellae, and cylindrical pores, Figure 2), we consider two transient patterns. One describes merged polymer strands, or the finite


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parts of lamellae (Figure 3e). Another one describes oblong pores and has the form of “inverse sausages” in a cross-section parallel to the grafting plane (Figure 3f). An additional geometrical parameter of those patterns is the length L. The merged strands (e) and pores (f) transform to single round strands (b) and pores (d), respectively, at L  0 and to the lamellar pattern (c) at L  .

Figure 3. Transient structure types of a polymer brush: (e) the merged polymer strands and (f) the oblong pores (“inverse sausages”). The cross-sections perpendicular to the grafting plane are shown. The polymer-containing regions are gray.

Let M1 and M2 be the total numbers of the merged strands (e) and oblong pores (f) respectively. The corresponding areas of the lateral surfaces, S1 and S2, are equal to

S1  M1 2( L  R) H ,

(e)

(f)

S 2  M 2 2( L  R) H

(12)

The polymer volume fraction is (e)  p 

~ N m 1 p (R 2  2 RL ) H

(f)  p 

;

~ N m 2 p

(13)

~  (R 2  2 RL)) H ( m 2

~  m M , i = 1 for the pattern of merged strands (e) and i = 2 for the pattern of merged where m i i pores (f). The free energy of the lateral surface consists of the free energies of the flat and curved parts: (e) Fsurf  2M1  c  RH   l  LH 



(f) Fsurf  2M 2  p  RH   l  LH



(14)

Using Eqs. (8), (9), (12-14), the lateral surface free energy (14) can be rewritten as (e)

(f)

Fsurf k BT Fsurf k BT





 ~ a  2mN L l  l  L   s0       p (1   p )   1   i    , (  2 L R)  R  R s R  2 R  R  

(15)

 (1   p ) l   ~ a  L l  L   s0       p   1       i  R s R  2 R  R   R 2  (  2 L R)  R  2mN

~ m 2

where 1 = 1 and 2 = 1. The area of the grafting plane per aggregate of strands (e) and per pore (“inverse sausage”, f) is equal to (Figure 3) ACS Paragon Plus Environment

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(e,f)

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6R  d 2

2



~, tan 30 0  L(2 R  d )  m i

i = 1,2

(16)

The sizes of polymer-free regions are assigned to be greater than the surfactant size l: d  l (e) and 2R  l (f). The free energy F (Eq. (5)) with the surface contribution Fsurf (15) has minimum with

~ ), and pattern sizes (R, L and respect to the polymer volume fraction (p), aggregation number ( m i d) at the equilibrium values of these parameters of the transient patterns.

RESULTS AND DISCUSSION In our model, the brush free energy per monomer unit F/(mN) given by Eqs. (5), (10), and (15) is independent of the polymer length N which is a consequence of homogeneity of polymercontaining regions in the case of long enough polymer chains. Therefore, the polymer length N has no effect on the brush pattern and its parameters. The free energies F for all considered patterns of the polymer brush with the embedded surfactant (Figures 2, 3) and the equilibrium pattern parameters were calculated for different values of the grafting density ns, the amphiphilicity parameter , the bond length l, and the Flory-Huggins parameter . The dependences of the free energy F of different patterns on the grafting density ns are plotted in Figure 4. The free energy minimum of the transient pattern of merged strands (e) corresponds to L, that is to the lamellar pattern (c) and the curves for these patterns coincide. The curve for the transient pattern of oblong pores (or “inverse sausages”, f) comprises the regions of lamellae (below point A), of finite values of L (between points A and B), and of round pores (above point B). The single strands are not favorable in comparison with lamellae, and the corresponding curve (b) is above the curve (c). The homogeneous brush (a) is favorable only at very high grafting densities. The equilibrium sizes of polymer-free regions are equal to the surfactant size l for all patterns that provide maximum of the side surface area and, correspondingly, minimum of the free energies Fsurf and F with respect to the size of those regions (d = l for strands, merged strands, and lamellae and 2R = l for pores). At given values of the parameters, an existing pattern should correspond to the free energy minimum among all of the considered patterns. The analysis of the theoretical model enables us to map out the diagrams of the brush patterns in the coordinates ns (the grafting density) and  (the amphiphilicity parameter) and ns and p/s (the ratio of the polymer unit volume and surfactant molecule volume), plotted in Figures 5 and 6, respectively.


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Figure 4. Free energies of the different patterns in a polymer brush with the surfactant vs the grafting density ns = 1/ : (a) a homogeneous brush (the marked curve), (b) round strands (dashed), (c) lamellae (blue), (d) round pores (red), (e) merged strands (coincides with the blue curve (c)), and (f) oblong pores (coincides with the blue curve (c) below point A and with the red one (d) above point B). The parameters are:  = 10, p/s = 1, l/a = 10,  = 0.5, s0 = 1, p/a3 = 1. Possible values of the volume ratio p/s are of the order of unity. For example, the volume per cetyltrimethylammonium bromide (CTAB) molecule CTAB estimated from its partial molar volume VM  230 cm3/mol for the very small CTAB concentration cCTAB = 0.0006 mol/l in water49 gives CTAB = VM/NA  0.38 nm3 (NA is the Avogadro number). The volume occupied by the Kuhn segment of a poly(acrylic acid) molecule can be found as PAA = lKM/(LNAPAA), where the values of the contour length L, molar mass M, and Kuhn segment length lK  1.65 nm were determined experimentally.50 The density, PAA, (mass to volume ratio) of pure poly(acrylic acid) can be estimated form the standard characteristics of the solution (the density =1.150 g/ml for 35 wt. % in water) available from MERCK. Assuming that  = 0.35PAA + 0.65w, w = 1.0 g/ml, the values   1.43 g/ml and PAA  0.72 nm3 can be found. Thus, the ratio of the polymer segment volume to the surfactant molecule volume is equal to PAA/CTAB  1.9. If the polymer-containing regions in the brush have convex lateral surfaces, then such patterns (strands and merged strands) have greater surface free energy because of the positive curvature dependent term (the last one in Eq. (10)) and therefore they are less favorable than the patterns with flat or concave lateral surfaces (lamellae or porous layer). This is conditioned by a difference corresponds to the difference in the surface density of surfactant molecules nsurf (Eq. (9))


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at the convex and concave lateral surfaces. As a result, the formation of only lamellae and porous layers of different types is predicted. With increasing the grafting density, lamellae perpendicular to the grafting plane, continuous layer with oblong or round pores, or homogeneous brush are formed. At high enough values of  and p/s, the region of the layer with round pores can be subdivided into a region of hexagonally packed pores at the distance 2l between them and a region of low porosity with sparse pores (the thin shaded regions near the homogeneous brush domain in Figures 5 and 6). An increase in temperature corresponds to a decrease in  that can lead to the transitions from lamellae to a porous layer with oblong pores and then with hexagonally packed round pores and from a layer of high porosity with round pores to a layer with sparse pores and then to a homogeneous brush. The transitions between the patterned states can be described as the secondorder phase transitions, whereas the transition between the patterned brush and homogeneous one is of the first order. It is accompanied by a step-wise extension of the brush and by orientational ordering of surfactant molecules. Those transitions can proceed at definite grafting densities only, otherwise, the brush swells or shrinks without changing its structure type. The patterned polymer brushes are considerably more extended compared to a homogeneous brush, in which the interaction energy of polymer and surfactant is determined by the parameter  only. For grafting densities corresponding to the region of lamellae, the difference in height between the patterned brush at  = 7-10 and the homogeneous brush at  = 0 is around 3-3.5 times. Both the temperature increase under the fixed p/s (and other parameters) and the increase in the surfactant volume, s, under the fixed  enlarge the region of a homogeneous brush and shift the transition values of the grafting density to lower numbers. Recall that the surfactant molecule volume is assumed to be dominated by its polymer-embedded part in our model. The change in the Flory-Huggins parameter  in the range 0 > a), whereas the pattern types and polymer volume fractions are virtually independent of it under the fixed values of  and p/s. Such peculiarity can be related to the form of the surface free energy for curved lateral surfaces (Eqs. (10) and (15)), where the contributions with  depend on the curvature l/R rather than on l.


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Figure 5. Diagram of brush patterns in the coordinates ns =1/ (the grafting density) and  (the amphiphilicity parameter) at p/s = 0.5 (thin black curves) and p/s =1 (thick blue curves); l/a = 10,  = 0.5, s0 = 1, p/a3 = 1. The regions of lamellae and homogeneous layer are denoted by “lam” and “hom” respectively, the regions of a layer with oblong pores (f) are shaded with dashes, and the region of a layer with sparse round pores is shaded with grey.

Figure 6. Diagram of brush patterns in the coordinates ns =1/ (the grafting density) and p/s (the ratio of the polymer unit volume and surfactant molecule volume) at  = 7 (thin black curves) and

 = 10 (thick blue curves); l/a = 10,  = 0.5, s0 = 1, p/a3 = 1. The other notations are as in Figure 5.


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Figure 7. The ratio of the area of polymer-containing regions, S*, in the cross-section parallel to the grafting plane to its total area, S, vs the grafting density ns = 1/;  = 10, p/s = 1, l/a = 10,  = 0.5, s0 = 1, p/a3 = 1.

The observed patterns can be characterized by the area fraction of the polymer-containing regions (in the cross-section parallel to the grafting plane) relatively to the total area of the grafting plane,  = S*/S. The lamellae correspond to c =2/3, the hexagonally arranged round pores at the distance 2l between them to d1 =1   (18 3 )  0.90; for the transient pattern of oblong pores, this ratio changes from c to d1 (Figures 7). The layer with sparse pores (near the transition to a homogeneous brush) is characterized by the ratio  somewhat greater than d1.


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Figure 8. The mean polymer volume fraction  p (the thick curves) in the brush and the polymer volume fraction p (the thin curve) in the polymer containing regions vs the grafting density ns =1/ at  = 10, p/s =1 (the blue curve),  = 7, p/s = 1 (the green curve), and =10, p/s =0.5 (the black curves); the other parameters l/a = 10,  = 0.5, s0 = 1, p/a3 = 1.

The dependences of the mean polymer volume fraction  p in the brush and of the volume fraction p in the polymer-containing regions on the grafting density are plotted in Figure 8 for the different values of the parameters  and p/s. The mean polymer volume fraction  p in the whole layer of height H near the grafting surface is equal to n~ mN p    p  s SH h

(17)

At a given grafting density, the mean volume fraction  p is inversely proportional to the brush height h = H/(Na). The greater value of the amphiphilicity parameter  under the fixed p/s corresponds to a stronger polymer chain stretching and lower value of  p (compare the thick green and blue curves). The greater surfactant molecule volume (the lower value of p/s) corresponds to the lower lateral surface density of surfactant molecules and lower polymer chain stretching and, therefore, the higher value of  p (compare the thick black and blue curves). The mean volume fraction  p increases monotonically with the grafting density. It is important to specify the applicability conditions of the present model. In addition to the condition for a brush regime of grafted macromolecules,  « Na2, it is assumed that the free energy contribution from the macromolecule parts near the grafting surface, which are not incorporated ACS Paragon Plus Environment

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into the microstructure, is negligible. These parts are strongly stretched and their characteristic length is equal to l, then, l/a « N. In our model, macromolecules self-assemble because of the high interaction energy gain produced by the surfactant molecule orientation, so that the product p should exceed several kBT. For moderate grafting densities and not extremely strong chain stretching, the polymer volume fraction p can be estimated considering the free energy of a lamellar structure consisting of only the elastic contribution in the gaussian form, fel/(kBT) = 1.5H2/(Na2), and the surface contribution of oriented surfactant molecules. That gives the additional relation between  and :  > 3(ps/a2)1/2. At lower grafting densities (larger ), the macromolecules possibly can self-assemble with surfactant molecules, however, the brush structure cannot be described in terms of the present model. Note that we consider the behavior of the two-component system of polymer with surfactant and do not take into account that they can be immersed in solvent. Although the dilution by solvent can be expected to weaken the effect of amphiphilicity of surfactant molecules, the pattern formation should be observed in the three-component solutions as well. Besides, the presence of solvent could provide high mobility of macromolecules and molecules of surfactant necessary for attaining a favorable pattern. In our model, surfactant molecules are considered as amphiphilic dimers with large polymer-compatible H parts and steric interactions between P parts are not taken into account. However, it can be predicted from the common sense that such steric repulsion could lead to the possibility of cylindrical strands (bundles) formation at moderate grafting densities and to the shift of all transitions between patterns to higher values of grafting densities.

CONCLUSIONS In this paper, the self-assembly in the polymer brush attached to a planar surface in the presence of surfactant is considered in the mean-field theory. The pattern formation in the direction parallel to the grafting plane is analyzed in the limiting case of very high degree of amphiphilicity of surfactant molecules with large polymer-compatible H parts. We present the pattern diagrams of polymer brushes comprising lamellae perpendicular to the grafting plane, polymer layers with pores of different types, and a homogeneous brush. The distance between lamellae and the pore diameter are determined by the length of a surfactant molecule, whereas the lamella thickness and the distance between pores are nearly twice that length. It was found that the process of pore formation in a homogeneous brush caused by a temperature change can be described as the first-order phase transition, it is accompanied by the step-wise extension of the brush. The transitions between the other patterns are of the second order.


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The transitions proceed at definite grafting densities, otherwise, the brush just changes its extension under fixed structure type. The pattern formation in nonionic polymer brushes is predicted for the first time. Such patterns could be, for example, a prototype of thin polymer films with nanochannels if additionally a chemical process of polymer crosslinking is carried out.

ACKNOWLEDGMENTS The research was financially supported by the Russian Science Foundation (Project No. 1413-00745).

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Surfactant-Induced Patterns in Polymer Brushes - Langmuir (ACS

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