Attraction between Opposing Planar Dipolar Polymer Brushes

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Attraction in opposing planar dipolar polymer brushes Jyoti P. Mahalik, Bobby G. Sumpter, and Rajeev Kumar Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01566 • Publication Date (Web): 01 Aug 2017 Downloaded from http://pubs.acs.org on August 2, 2017

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Attraction in opposing planar dipolar polymer brushes J. P. Mahalik,†,‡ Bobby G. Sumpter,†,‡ and Rajeev Kumar⇤,†,‡ Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN-37831 E-mail: [email protected]

Abstract We use a field theory approach to study the effects of permanent dipoles on interpenetration and free energy changes as a function of distance between two identical planar polymer brushes. Melts (i.e., solvent-free) and solvated brushes made up of polymers grafted on non-adsorbing substrates are studied. In particular, the weak coupling limit of the dipolar interactions is considered, which leads to concentration dependent pairwise interactions and the effects of orientational order are neglected. It is predicted that a gradual increase in the dipole moment of the monomers can lead to attractive interactions at intermediate separation distances. Since classical theory of polymer brushes based on the strong stretching limit (SSL) and the standard self-consistent field theory (SCFT) simulations using the Flory’s

parameter always predicts repulsive interac-

tions at all separations, our work highlights the importance of dipolar interactions in To whom correspondence should be addressed Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN37831 ‡ Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN-37831 ⇤ †

1


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tailoring and accurately predicting forces between polar polymeric interfaces in contact with each other.

Introduction Polymer brushes (i.e., anchored polymers) are routinely used in order to tailor properties of surfaces such as mica or gold in order to achieve desired functionalities such as stimuli responsiveness, 1–7 and colloidal stabilization. 6,8–13 For example, it is well known that unmodified/bare nanoparticles tend to aggregate due to short-range attractive van der Waals and depletion interactions. Such an aggregation can be avoided by anchoring polymers on the nanoparticles leading to desirable well-dispersed formulations in solutions as well as in melt states. In the literature, term “colloidal stabilization” is used to describe such a modification of surface properties by the anchored polymers. The underlying origin of such a stabilization of colloidal dispersions is the entropic penalty of the polymer chains under compression leading to strong repulsive interactions, which can overwhelm short-range attractive van der Waals interactions. A number of experimental reports, 13–30 self consistent field theory (SCFT), 31–40 molecular dynamics (MD) simulations 26,41–49 and studies based on scaling theory 23,43,50,51 have focused on understanding the interactions between two surfaces bearing anchored or adsorbed polymers. One of the ways to characterize these interactions is by measuring the normal force versus separation distance e.g., using surface forces apparatus 52 (SFA) or atomic force microscopes (AFM). The effects of various parameters such as chain polydispersity, 19 presence or absence of solvent, 15,41 solvent quality, 23,44,46,48 temperature, 16,22,30 polymer molecular weight, 28,49 grafting density, 25,27,29,49 chain architecture (whether linear/cyclic or diblock/triblock copolymer), 13,35,53,54 surface roughness, 45 and its curvature (whether flat or cylindrical or spherical) 34 on the force-distance relations have been investigated. Experimentally, it is difficult to measure force as a function of distance for perfectly pla-

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nar polymer brushes. Instead, forces are measured for two crossed cylindrical brushes e.g., in a typical SFA experiment. A typical force-distance curve for two opposing cylindrical brushes shows distinct regimes: onset of interaction, weak repulsion and strong repulsion 16 as the distance between the two substrates is gradually reduced. The experimentally measured force-distance relation between any two curved polymer brushes can be related to the free energy changes of two opposing planar polymer brushes using the Derjaguin approximation. 52 The Derjaguin approximation and free energy estimation based on the analytical calculations possible by the use of the strong stretching limit (SSL) allows one to model the experimental force-distance curves 31–33,36,37 in a facile and transparent manner. The effects of grafting density, molecular weight, solvent quality, and the polydispersity on the force-distance relations can be readily studied using such a model. Quantitative modeling of experimental data with such a simple analytical model based on the SSL is truly remarkable. However, the SSL based model (originally developed by Semenov 55 and extended by Milner, Witten, Cates, 31–33 Zhulina and co-workers 36,37 ) and its extensions involving numerical treatments based on the standard saddle-point approximation 56 where the constraints of strong stretching are relaxed, can primarily be used for systems where the force is repulsive (or positive) irrespective of the separation distance. For many applications e.g., in colloidal dispersions and polymer nanocomposites, the attraction between polymer grafted surfaces is not desirable in order to attain well-dispersed suspensions. However, in other applications requiring interfacial reinforcements, the attraction is desirable. It has been shown 35,57,58 that chain conformations play an important role in inducing attraction between two opposing polymer brushes. For example, telechelic homopolymers that possess a strongly adsorbing group on each end 35,38,53 or linear tri-block copolymers 13 with ends that can form bridges between the two surfaces can induce weak attractive interactions between opposing surfaces. For strongly adsorbing groups, most of the polymers tend to form loops on each substrate and each loop can be considered equivalent to two grafted chains of half the length of the loop. As the formation of loops leads to 3


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purely repulsive interactions and only a small fraction of chain form bridges, 35,38 the bridging induced attraction is expected to be weak 35 (i.e., a fraction of the thermal energy). Such an attraction between the two opposing brushes formed by telechelic poly(styrene) brushes containing zwitterionic end-groups was measured using SFA. 59 However, it was argued 57 that in addition to the bridging interactions originating from the adsorption of the ends at the opposite substrate, attraction among the telechelic chain ends can cause the attraction in the force-distance curves. The idea of aggregates in the form of dimers and multi-mers containing chain ends was proposed 57 and it was argued that the dimer and multi-mer formation will always be preferred over the adsorption of the chain ends on the opposite substrate. Experimentally, it is difficult to verify such details of the multi-mer formation but the tendency for a polymer to form a bridge can be readily measured by studying interactions between a polymer brush and a bare substrate e.g., using an AFM tip. 27 In a polymer brush, chain conformations are determined self-consistently by the parameters such as grafting density, interactions with the grafting substrate and solvent. Due to the bridging mentioned above, adsorbing polymer chains can lead to attraction between two opposing polymer brushes. 59 However, it is not clear if the attraction can be sustained in the force-distance curves for two opposing neutral but polar polymer brushes in the absence of the adsorption on the grafting substrates. In particular, can the two opposing neutral polymer brushes ever exhibit attractive interaction? Based on the theoretical arguments, 57,58 polymer adsorption should play a minor role in inducing the attraction and if the monomers attract each other then attraction can originate solely due to interpenetration. Most of the polar polymers such as poly methyl methacrylate (PMMA), polyethylene oxide (PEO) 60,61 or polypeptides 62 are expected to exhibit such an attraction due to dipole-dipole and higher order multipolar interactions. Despite a large number of literature reports related to PMMA, PEO etc., we are not aware of any experimental study exhibiting attractive interactions between two opposing polymer brushes where the attraction originating from the adsorption of polymers to the opposing substrate can be neglected. This has led us to investigate the 4


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effects of dipolar interactions on the free energy versus normal separation distance between two identical planar brushes made up of polymers grafted on purely repulsive substrates. We have studied the effects of dipolar interactions on the free energy as well as interpenetration using a recently developed field theoretical approach. 63 In this work, it is shown that the opposing polymer brushes can exhibit attractive interaction and intermix with each other for intermediate separation distances in melts (i.e., solvent-free) as well as in the presence of solvents. This paper is organized as follows: in the Theory section the field theory and numerical scheme for an opposing dipolar brush are presented, in the Results and Discussion section, numerical and analytical results are presented, and we conclude with the Conclusions section.

Theory Field theory for opposing dipolar melt brushes We consider two opposing identical planar polymer brushes, where each brush is formed by n mono-disperse flexible Gaussian chains grafted uniformly on an uncharged and repulsive surface (see Fig. 1). Grafting density of each brush is taken to be

(defined as the number

of chains per unit area). For the field theoretical formulation 56,64 described in this article, every chain is represented as a continuous curve of length N b so that N is the number of Kuhn segments, each of length b. An arc variable t↵ is used to represent a segment along the backbone of ↵th chain so that t↵ 2 [0, N ]. t↵ = 0 and t↵ = N represent the grafted and the free end, respectively, of the ↵th chain. R↵ (t↵ ) represents the position vector for a particular segment, t↵ , along the ↵th chain and R↵ (t↵ = 0) = r↵ (0) represents the position vector for the grafted end of the ↵th chain. Every Kuhn segment along a chain is assigned a permanent dipole moment epp so that pp is the characteristic length of the dipole and e is the charge of an electron. The orientation of such a dipole assigned to a segment at t↵ along a chain contour is represented by a unit 5


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Figure 1: Schematic of the two opposing planar dipolar polymer brushes considered in this work. The dipolar chains are modeled as continuous curves (black colored) containing electric dipoles (represented by black arrows). vector u↵ (t↵ ). Consideration of “point” dipole-dipole interactions leads to divergences in the field theory, 65 which can be regularized by smearing charge distribution over a small volume. The smearing leads to introduction of additional length scales, which characterizes the size of the charge distribution in a dipole. A mathematically convenient and physically relevant choice is the Gaussian distribution model, 63 which introduces an additional length scale characterizing the size of the charge distribution of a dipole. In this work, we take the smearing approach based on the Gaussian distribution for modeling the dipole-dipole interactions. The field theory is constructed by considering two distinct brushes and the symmetry is

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imposed in the end to keep track of different contributions in the free energy. Subscripts 1 and 2 are used for indexing the two brushes. Partition function for the opposing brushes at a distance H apart can be written as

Z = ⇤

3

P

k=1,2

npk Nk

Y Z

k=1,2

exp [ Ho {R↵k }

npk Y

D[R↵k ]

r↵k (0) ↵ =1 k

Hw {R↵k }

npk Nk Z Y Y

du↵k (t↵k )

↵k =1 t↵k =0

Hdd {R↵k , u↵k (t↵k )}]

(1)

where ⇤ is the de Broglie’s wavelength, npk and Nk are the number of chains and number of the Kuhn segments in a chain in the brush k, respectively. Ho {R↵k } is the well-known Wiener measure for the flexible polymer chains 64 ✓ ◆2 npk Z N k X 3 X @R↵k (t↵k ) Ho {R↵k } = dt↵k 2 2b @t↵k 0 k ↵ =1 k=1,2

(2)

k

so that bk is the Kuhn segment length of a chain in the brush k. Hw {R↵k } is the energetic contributions originating from the short-range repulsive hard-core interactions and the attractive dispersive interactions between the segments (excluding the permanent dipole-dipole interactions). Hw can be expressed using the Edwards’s formulation 64 for local short-ranged interactions Z 1 X X Hw {R↵k } = wkk0 drˆ ⇢p,k (r)ˆ ⇢p,k0 (r) 2 k=1,2 k0 =1,2

(3)

where wkk0 is the excluded volume parameter describing the strength of short-range interactions between the segments of type k and k 0 . ⇢ˆp,k (r) represents the microscopic number density of the segments at a certain locations r belonging to brush k. ⇢ˆp,k (r) is defined as

⇢ˆp,k (r) =

npk Z X

↵k =1

Nk

ˆ p,k [r dt↵k h

R↵k (t↵k )]

0

7


(4)

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so that 1

ˆ p,k (r) = h

(2a2p,k )3/2

exp

"

⇡r2 2a2p,k

#

(5)

ˆ p,k characterizes the Here, we have used the notation r = |r|. The functional form of h Gaussian distribution of density and ap,k represents a measure of its spatial extent. 63 It should be noted that the Edwards’s formulation of introducing excluded volume parameters is limited to non-negative values of wkk0 i.e., for repulsive short-ranged interactions between the segments. Negative values of wkk0 may lead to extremely high local density of the polymer segments which ultimately leads to numerical instabilities. Generally, a three body virial term 56 is included in the Hamiltonian to stabilize the system against attractive interactions. In the current model, higher order viral terms arise from the dipolar interactions. In Eq.

1, Hdd is the electrostatic contribution to the Hamiltonian, which takes into

account the dipole-dipole interactions between the segments. Hdd can be written as 63,65,66 lBo Hdd {r, r } = 2 0

Z

dr

Z

dr0

(rr · Pâve (r))(rr0 · Pâve (r0 )) |r r0 |

(6)

where lBo = e2 /4⇡✏o kB T is the Bjerrum length in vacuum, e is the charge of an electron, ✏o is the permittivity of vacuum, kB is the Boltzmann constant and T is the absolute temperature. Pâve is given by 63,65,66 Pâve =

Z

"

#

X

pp,k ⇢¯k (r, u) u

ˆ p,k (r dt↵k h

R↵k (t↵k )) [u

du

k=1,2

(7)

so that

⇢¯k (r, u) =

npk Z X

↵k =1

Nk

u↵k (t↵k )]

(8)

0

Eq. 6 represents the dipolar interaction terms and the higher order multipolar terms are

8


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completely neglected. This approximation is valid for molecules bearing permanent dipoles such as those considered in this work. 66 Using the Hubbard-Stratonovich transformation 56 for exp[ Hdd /kB T ] and integrating out orientational degrees of freedom, 63 Eq. 1 can be expressed as,

Z = ⇤

3

P

Y Z

k=1,2 npk Nk

k=1,2



D[ ] 1 exp ⇣ 8⇡lBo

Z

npk Y

r↵k (0) ↵ =1 k

D[R↵k ] exp [ Ho {R↵k }

Hw {R↵k }]

dr (r)r2r (r)I1 { , R↵1 }I2 { , R↵2 }

(9)

where ⇣ is a normalizing factor,

Ik = (4⇡)

npk Nk

exp

"Z

dr0 ˆp,k (r0 ) ln

(

and we have defined ˆp,k (r) =

npk Z X

↵k =1

)# R ˆ p,k (r r0 )|] sin[pp,k | dr (r)rr h R ˆ p,k (r r0 )| pp,k | dr (r)rr h

Nk

dt↵k [r

(10)

(11)

R↵k (t↵k )]

0

so that

⇢ˆp,k (r) =

Z

ˆ p,k (r dr0 h

r0 ) ˆp,k (r0 )

(12)

Eq. 9 can be expressed as, 63 1 Z = ⇣

Z Y

k=1,2

{D[

p,k ]D[wp,k ]}

Z

D[ ] exp



Fo kB T

¯ H kB T

(13)

where we have defined " # P Fo ⇤3 k=1,2 npk Nk P = ln Q Qn kB T (4⇡) k=1,2 npk Nk k=1,2 ↵pk Qp,↵k {0} k =1 9


(14)

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and Z 1 X X = wkk0 dr⇢p,k (r)⇢p,k0 (r) 2 k=1,2 k0 =1,2

¯ H kB T

npk

X X

i

XZ

drwp,k (r)

p,k (r)

k=1,2

Z

1 dr (r)r2r (r) 8⇡l Bo k=1,2 ↵k =1 ( ) R 0 ˆ XZ sin[p | dr (r)r h (r r )|] p,k r p,k dr0 p,k (r0 ) ln R ˆ pp,k | dr (r)rr hp,k (r r0 )| k=1,2 ¯ p,↵ {wp,k }] ln[Q k

(15)

¯ p,↵ {wp,k } is the partition function of ↵th chain in brush k. It is given by where Q k ¯ p,↵ {wp,k } = Q k

1 Qp,↵k {0}

Z

D[R↵k ] exp r↵k (0)

"

3 2b2k

Z

Nk

dt↵k 0

✓

@R↵k @t↵k

◆2

i

Z

Nk

dt↵k wp,k (R↵k ) 0

# (16)

Qp,↵k {0} is the partition function of the same grafted chain in the absence of any interactions and is given by

Qp,↵k {0} =

Z

D[R↵k ] exp r↵k (0)

"

3 2b2k

Z

Nk

dt↵k 0

✓

@R↵k @t↵k

◆2 #

(17)

In Eq. 36, we have used the notation

⇢p,k (r) =

Z

ˆ p,k (r dr0 h

10

r0 )

p,k (r

0

)


(18)

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Weak Coupling Limit (WCL) for Dipolar Interactions 63,65 The electrostatic part of the free energy Eq. 13 can be expressed as, 

✓ Z D[ ] 1 ln[Zelec { p,k }] = ln exp dr (r)r2r (r) + ⇣ 8⇡lBo ( )◆ R 0 ˆ XZ sin[p | dr (r)r h (r r )|] p,k r p,k dr0 p,k (r0 ) ln R ˆ pp,k | dr (r)rr hp,k (r r0 )| k=1,2

(19)

Eq. 19 can be simplified to the following form in the WCL, 

D[ ] ln[Zelec { p,j }] ' ln exp ⇣

✓

1 2

Z

dr

Z

0

0

0

dr (r)EW CL (r, r ) (r )

◆

(20)

where 0

EW CL (r, r ) =

X p2p,k Z r)+ dr00 3 k=1,2

1 r2r (r 4⇡lBo

0

p,k (r

00

ˆ p,k (r )rr h

ˆ p,k (r0 r00 ) · rr0 h

r00 ) (21)

The functional integral appearing in Eq. 20 can be evaluated in an approximate manner 63 by ignoring the non-local contributions in Eq. 21. This leads to

ln[Zelec {

p,k }]

' ln[Zelec,W CL {

p,k }]

1 8⇡

'

Z

drf (r) ln[✏(r)]

(22)

where ✏(r) is the local dielectric function, given by

✏(r) = 1 +

4⇡lBo X 2 p ⇢p,k (r) 3 k=1,2 p,k

(23)

In Eq. 22, R

f (r) = R

dr0 dr

P

P 0

k=1,2

p2p,k

2 k=1,2 pp,k

11

p,k (r

0

)ˆ gp,k (r 0 ˆ p,k (r )hp,k (r


r0 ) r0 )

(24)

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so that 2

⇡ ˆ ˆ p,k (r) hp,k (r) 44a2p,k h a2p,k

gˆp,k (r) =

erf

⇣p

⇡ |r| 2 ap,k

|r|

⌘3 5

(25)

Using Eqs. 13, 19, 20 and 22, the partition function can be expressed as,

Z'

Z Y

D[

p,k ]D[wp,k ] exp

k=1,2



Fo kB T

¯ neu H kB T

¯ elec H kB T

(26)

where, Z ¯ neu H 1 X X = wkk0 dr⇢p,k (r)⇢p,k0 (r) kB T 2 k=1,2 k0 =1,2 np,k

X X

k=1,2 ↵k =1

i

XZ

drwp,k (r)

p,k (r)

k=1,2

¯ p,↵ {wp,k }] ln[Q k

(27)

and Z ¯ elec H 1 = drf (r) ln ✏(r) kB T 8⇡

(28)

Laterally homogeneous opposing planar brushes In this subsection, we apply the general theory presented in the previous subsection to the case of two identical and laterally homogeneous brushes. We investigated the two opposing polymer brushes in equilibrium, using the saddle-point approximation 56 for the field variables p,k

and wp,k in Eq. 26. The approximation evaluates the functional integrals over the field

variables by the value of the integrand at the saddle-point and leads to coupled non-linear ? equations. In the following, we have used the notations ⇢?p,k = ⇢p,k and wp,k = iwp,k to

represent the field variables at the saddle point. Optimization of the integrand in Eq. 26 12


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with respect to wp,k , we obtain (for the laterally homogeneous case so that z axis is along the normal to the grafting plane)

? p,k (z)

=

npk  X ¯ p,↵ {wp,k } ln Q k w p,k ↵ =1

= ? wp,k =wp,k

k

R Nk dt↵k qz↵k (z, t)¯ qk (z, Nk 0R A dzqz↵k (z, t↵k )¯ qk (z, Nk =1

np,k X

↵k

t ↵k ) t ↵k )

(29)

where A is the area of a grafting plane. qz↵k as well as q¯k satisfy the same modified diffusion equation of the form 

@qk (z, t) = @t

b2k @ 2 6 @z 2

but with different initial conditions qz↵k (z, 0) = (z ? p,k (z)

(30)

? wp,k (z) qk (z, t)

z↵k ) and q¯(z, 0) = 1. As

p,k (r)

⌘

for the laterally homogeneous case, Eq. 18 can be written in the form ⇢?p,k (z)

=

Z

H

¯ p,k (z dz 0 h

z0)

p,k (z

0

(31)

)

0

where the limits of integration arises due to the fact that

p,k (z)

= 0 for z outside the bound

ˆ p,k (cf. Eq. 5) over the of 0 and H. Also, we have used the result that the integration of h in-plane degrees of freedom gives unity. Optimization of the integrand in Eq. 26 with respect to ? wp,k (z)

=

X

wkk0 ⇢?p,k0 (z)

k0 =1,2

lBo p2pk + 6

Z

lBo p2p,k + 6

dz 0 [¯ gp,k (z

z0)

13

Z

? p

gives,

¯ p,k (z dz 0 h ¯ p,k (z f¯(z 0 )h


z0)

f¯(z 0 ) ✏¯(z 0 )

z 0 )]

ln[¯✏(z 0 )] [¯✏(z 0 ) 1]

(32)

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where 4⇡lBo X 2 ? p ⇢ (z) 3 k=1,2 pk p,k R 0P 2 dz ¯p,k (z z 0 ) k=1,2 ppk g f¯(z) = R 0 P 2 ¯ dz z0) k=1,2 ppk hp,k (z  ✓r ◆ ⇡2 ⇡ |z| 2 g¯p,k (z) = 1 erf 4a4p,k 2 ap,k " # 2 1 ⇡z ¯ p,k (z) = p h exp 2a2p,k 2ap,k

(33)

✏¯(z) = 1 +

p,k (z

0

p,k (z

)

0)

⇡ exp 2a4p,k

(34) "

⇡z 2 a2p,k

#

(35) (36)

Symmetric case and limit of point-like distribution: ap,k ⌘ ap ! 0 ¯ p,k (z) ! (z), which leads to ⇢? (z) = In this limit, h p,k

p,k (z)

(cf. Eq. 31). Also, in this limit,

f¯(z) ! ⇡/2a3p so that Eq. 28 can now be expressed as, 63 Z H ¯ elec H A = dz ln ✏¯(z) kB T 16a3p 0

(37)

which in turn, leads to ? wp,k (z) =

X

wkk0 ⇢?p,k0 (z) +

k0 =1,2

1 (✏p 1) 16a3p ⇢o ✏¯(z)

(38)

where, ✏p = 1 + 4⇡lBo p2p ⇢o /3, pp,k ⌘ pp and ⇢o is the reference number density. Eq. 38 is the limiting form of Eq. 32 for the point-like distibution model.

14


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Page 16 of 37

Strong stretching limit (SSL): free energy changes due to compression of a single brush It is worthwhile to consider the implications of the additional term in Eq. 38 on the free energy of compressing a single brush. Such a calculation can be readily done in the strong stretching limit. 31,36,55 Consider the compression of brush k = 1 by the second grafting plane designated by k = 2. In this case, we need to consider only wp,1 and wk2 = 0. In the strong stretching limit, the field must be parabolic so that ? ? wp,1 (z) = wp,1 (H) +

3⇡ 2 [H 2 2 2 8N1 b1

z 2 ] = w11 ⇢o cp,1 (z) +

1 (✏p 3 16ap ⇢o [1 + (✏p

1) 1)cp,1 (z)]

(39)

? where ⇢?p,1 (z)/⇢o = cp,1 (z). The unknown wp,1 (H) can be determined by using the mass balRH ance equation 0 dzcp,1 (z) = np,1 N1 /⇢o A = 1 N1 /⇢o . In the limit of ✏p ! 1, the calculations

can be done analytically, leading to ? wp,1 (H)

 ✏p 1 1 N1 = + w11 ⇢o 16⇢o a3p ⇢o H

cp,1 (z) =

1 N1

⇢o H

+

8N12 b21 (w11 ⇢o

(✏p 1)2 16⇢o a3p

⇡2H 2 4N12 b21 ✓

3⇡ 2 (✏p 1)2 /(16⇢o a3p ))

(40) H2 3

z

2

◆

(41)

These equations reveal that the dipolar interactions lead to renormalization of the excluded volume parameter so that wef f ⇢o = w11 ⇢o

(✏p

1)2 /(16⇢o a3p ). The free energy can be

expressed (cf. Eq. 26) in the form F Fo w11 = AkB T 2

Z

H 0

As ✏¯(z) = 1 + (✏p

dz⇢?2 p,1 (z)

Z

np,1

H 0

? dzwp,1 (z)⇢?p,1 (z)

1X ¯ 1 ? ln Qp,↵1 {wp,1 }+ A ↵=1 16a3p

Z

H

dz ln ✏¯(z) 0

1)⇢?p,1 (z)/⇢o , in the limit of ✏p ! 1, we can expand the last term in in

15


(42)

Page 17 of 37

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Eq. 42 in powers of ✏p 1 16a3p

Z

1, so that H

0

dz ln ✏¯(z) !

(✏p

(✏p 1)2 32a3p

1) 1 N1 16⇢o a3p

Z

H 0

(43)

dzc2p,1 (z)

The chain partition function in the strong stretching limit (SSL) can be expressed as np,1

1X ¯ ? ln Qp,↵1 {wp,1 }= A ↵=1

1



? N1 wp,1 (H) +

3⇡ 2 H 2 8N1 b21

(44)

Using Eqs. 39, 40, 41, 43, 44 and Eq. 42, we obtain an analytical form of free energy of compression of a single brush, written as (✏p 1) 1 N1 FSSL Fo = + AkB T 8⇢o a3p where H ? = (22/3

1/3

5/3 2/3 2/3 1 N1 wef f ⇡ 2/3 25/3 b1

"

H? + H

✓

H H?

◆2

1 5

✓

H H?

◆5 #

(45)

1/3

N wef f b2/3 )/⇡ 2/3 . Eq. 45 represents the free energy of a single com-

pressed brush and it is multiplied by 2 to obtain the free energy of two opposing brushes so that the interactions between the opposing brushes are not considered. Such a simplified analytical form of free energy can be compared to the full numerical SCFT results at very limited conditions. Those conditions are discussed in detail in the next section.

Numerical methods We have used a pseudo-spectral method based on the sine transform 56 for solving the modified diffusion equations after writing the equations in dimensionless form. All of the lengths p were made dimensionless by Rgo = N/6b. Due to the initial condition for one of the propagators (qz↵k ) involving a delta function, a fixed grid size of 0.1Rgo was used so that the changes in the free energy are unaffected by the numerical errors resulting from the numerical evaluation of the delta function.

s = 0.01 and ap /b = 0.5 were used in all the

calculations. The propagators in Eq. 30 were solved at

s/2 and

s followed by mixing

them using Richardson extrapolation 67 for fourth order numerical accuracy. For investigat16


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Page 18 of 37

ing non-local effects of the dielectric, Eq. 37 is mapped onto Eq. 51 of Ref. 65 By fixing p z = ap , qmax = ⇡/2/ap is obtained from Eq. 51 of Ref., 65 where qmax is the wave-vector introduced to regularize divergent integrals in Ref. 65 Including the non-local effects of the

local dielectric, contribution from the dipolar interactions can be expressed in an equation similar to Eq. 37, in the form

Z H Z H  ¯ elec H A A @ ln ✏¯(z) = dz ln ✏¯(z) + p dz 3 kB T 16ap 0 @z 32 2⇡ap 0

2

(46)

The free energy of the brushes separated by H/Rgo = 25 was taken as the reference for the computation of the changes in the free energy,

F . Chain end distribution is obtained by

evaluating one of the propagators at s = N , qz↵k (z, N ). 68 Opposing brushes immersed in a solvent bath were also considered in order to compare the qualitative behavior of such brushes with the melt brushes. Free energy of a single polymer brush immersed in a solvent bath is presented in Ref. 63 The theoretical formalism from this reference was generalized to the case of opposing brushes in a similar manner as we have shown in Appendix A and B for the melt brushes. In the modified formalism, q¯ and qz↵ described in Numerical Methods section of Ref. 63 are also solved for the opposing brush by grafting one end of the chain belonging to the opposing brush at the second to last grid of the simulation box. Similar to the melt brushes, H/Rgo = 25 (denoted as Hmax /Rgo ) was chosen as the reference state. For any two brushes with separation distance H/Rgo < Hmax /Rgo , solvent free energy equivalent to displaced volume for a translation by Hmax /Rgo was added. In particular,

FSolv /(AkB T ) = (H/Rgo

Hmax /Rgo )(1

H/Rgo

ln(✏s )/(16a3s )) was

added to the free energy of the two brushes at any separation distance H/Rgo < Hmax /Rgo . This free energy is obtained from equations 11, 18 and 21 of Ref., 63 which corresponds to pure solvent with dielectric ✏s and length scale of smearing, as . For simplicity as = ap = a was taken. The additional contribution to the free energy ensures that the total volume

17


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Langmuir

and number of the molecules stay the same while comparing the free energies at different separation distances.

Results and Discussion Numerical results for an opposing brush are compared against the analytical theory to determine the range of applicability of the analytical theory. Eq. 45 represents the free energy of a single compressed brush and it is multiplied by two to obtain the free energy of two opposing identical brushes separated by distance H. This, in turn, means that interactions between the opposing brushes are not considered explicitly in such a treatment. Such a simplified analytical form of free energy can be compared to the full numerical SCFT results at very limited conditions. Furthermore, Eq. 45 needs to be compared with the expression derived by Milner, Witten and Cates. 31 In Eq. 45, term on the right hand side, depending on H/H ? is identical with Eq. 31 in Ref. 31 if we substitute b2 = 3 due to differences in the prefactors for the Weiner measure of the entropic terms in our work and Ref. 31 Also, for the comparisons, it must be realized that Eq. 31 in Ref. 31 represents the free energy per chain, while Eq. 45 represents the free energy per unit area. In other words, Eq. 45 needs to be divided by

to directly compare with Eq. 31 in Ref. 31 Also, the H dependent part in Eq.

45 is identical to Eqs. 1-3 in Ref. 33 where a Wiener measure with a constant prefactor v was used (for comparisons with our work, v = 3). The additional term on the right hand side in Eq. 45 quantifies the self-energy of the dipoles and brings chemical specificity into the free energy of compression. However, this term is independent of H. Numerical results for the opposing brushes are compared with Eq. 45 for different cases in the limit ✏p ! 1 as shown in Figure 2. Reasonable agreements among the numerical results and Eq. 1.3. As Eq.

45 are observed but deviations greater than 10kB T are found for ✏p = 45 is derived based on the assumption that ✏p ! 1, such deviations are

expected. Figure 2 shows that the free energy - separation curves are positive for this

18


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εp 1 Numerical 1 Analytical 1.1 Numerical 1.1 Analytical 1.3 Numerical 1.3 Analytical

160

(F-F0)/AkBT

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 37

120

80

40 0.1

0.2

0.3

H/H*

0.4

0.5

0.6

Figure 2: Comparisons of the free energies for the opposing brushes estimated using numerical SCFT and the analytical calculations based on the SSL (cf. twice the free energy given by Eq. 45). Different curves corresponds to polymers of different dielectric constants. The dielectric constant is a function of the dipole moment of the polymer segments. Other parameters used in the calculations are: N = 150, b = 1, wpp = 0.2, and b2 = 0.5. The results agree well for ✏p ! 1 and strong deviations are observed for ✏p as high as 1.3, which highlights the breakdown of approximations involved in the analytical calculations. The numerical results obtained after including the non-local effects of the dielectric were almost identical to the curves obtained with purely local dielectric effects in the limit of ✏p ! 1. H ? is defined as per Eq. 45 for comparing the numerical and the analytical results. .

19


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Langmuir

highly compressed system. Similar curves have been reported in the literature. 16,17,31–33 It is well known for polymer brushes that the free energy changes exhibit repulsive interactions if polymer segments interact only with short range Flory type interactions. 69 The origin of the repulsions lies in the entropic penalty for the chains due to compression and domination of repulsive interactions parameterized via wpp in the limit of ✏p ! 1. We should point out that the numerical calculations converged only for highly compressed systems and the results are shown only for H/H ? < 0.6. This is a limitation of the numerical technique in the limit of ✏p ! 1. However, the calculations converged for higher values of ✏p , irrespective of the separation distance. Since the analytical results for ✏p = 1 agrees well with the corresponding numerical calculations for small separate distance, therefore we have plotted the analytical results for ✏p = 1 in Figure 3 for reference purposes, which clearly shows that the opposing polymer brushes are never attractive for ✏p = 1 irrespective of the separation distance. For larger values of ✏p (5  ✏p  15), numerical results for the free energy versus separation are shown in Figure 3. The free energy at large separation is taken to be the reference and is subtracted to compute the changes in the free energy due to interactions between the brushes. It should be clear from Figure 3 that the brushes don’t interact with each other at large separations (e.g.,

F = 0 for H/Rgo > 12 while considering ✏p = 5). At short

separation distances so that H/Rgo < 6,

F > 0 for all the cases presented in Figure 3

so that repulsions are predicted at such distances. In the intermediate separation distances, F < 0 highlighting attraction between the brushes. Strength of the free energy changes in the attractive wells seen in Figure 3 are of the order of kB T and increases monotonically with an increase in the parameter ✏p

1 (see the inset in Figure 3). The onset of the

attractive interactions is close to twice the equilibrium height of the brushes and the height decreases with an increase in ✏p

1 in qualitative agreement with the behavior of a dipolar

brush immersed in a solvent of dielectric constant lower than the polymer. 63 However, the attraction sets in at distances larger than twice the equilibrium height of a single brush. For example, the arrows in Figure 3, show the separation distances equal to twice the equilib20


Langmuir

6 4

∆Fmin/AkBT

8

∆F/AkBT

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 22 of 37

εP

-1

-1.5

-2

2 6

0

9 12 εp-1

15

1 analytical 5 non-local 5 local 7 non-local 10 non-local 10 local 12 non-local 15 non-local 15 local

-2 5

10

H/Rgo

15

20

Figure 3: separation distance (H/Rgo ), where p Changes in the free energy as a function of the 2 Rgo = N/6b, N = 150, b = 1, wpp = 0.2, and b = 0.5. For a given ✏p , free energy of the same two brushes at the longest distance is taken to be the reference free energy for computing the changes. Results from two sets of calculations are presented here. One includes a gradient term of the local dielectric (termed non-local) in the Hamiltonian and the other ignores this term (labeled local in this figure). The two arrows show the separation distance equal to twice the equilibrium height of a single brush for ✏p = 5 (black arrow) and ✏p = 15 (cyan arrow) , which included the non-local effects. It should be clear that the brushes don’t interact at distances much larger than twice the equilibrium height of the brushes, attract at intermediate distances and repel at short distances. The inset shows values of the free energy changes at the minima as a function of ✏p 1. The analytical result of free energy (Eq. 45) for ✏p = 1 with respect to a reference H/H ? = 1 is plotted for reference.

21


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Langmuir

rium heights of the brushes. This is a direct outcome of the long-range nature of the dipolar interactions. Furthermore, additional effects due to gradients of the local dielectric functions are included in the field theoretic Hamiltonian 65 and results from such calculations are also included in Figure 3. These non-local effects are found to be negligible in affecting the free energy-separation curves for ✏p ! 1 (see Figure 2) and at short distances for ✏p

5 (see

Figure 3). However, the non-local effects are found to be significant in affecting the strength and range of the attractive interactions between the brushes, which is about 3Rgo for all the cases considered here. In other words, the non-local effects are found to be negligible for the long and the short distances exhibiting no and repulsive interactions, respectively. The polymers possessing permanent dipoles are investigated in this study. The ✏p of PEO 70 and PNIPAM 71 are about 5, and 17, respectively, which are used as bounds on the values of ✏p in Figure 3. As interactions between the brushes stay repulsive for all separation distances in the absence of dipolar interactions, 31 we can attribute the attraction at intermediate separation distances to the attractive dipolar interactions. Density profiles of the segments for different separation distance are shown in Figure 4 for ✏p = 5 and ✏p = 15. Both of the panels in Figure 4 exhibit similar features. At a large separation distance, the brushes do not interact with each other and exhibit no interpenetration. At intermediate distances (e.g., see H/Rgo 2 (10.7, 10.9) in Figure 4(a)), the brushes interpenetrate, which can be seen as decrease in the maximum densities close to the grafting substrates and extension of the brushes. The maximum separation distance exhibiting interpenetration corresponds to the onset of attractive interactions between the brushes. The extent of the interpenetration and drop in the maximum density near a grafting substrate depends on ✏p

1. For example,

the drop is about 0.2 and 0.35 in the case of ✏p = 5 and ✏p = 15, respectively. Bringing the brushes closer leads to an increase in the overall density everywhere inside. We should point out that no interpenetration is expected for the case of ✏p = 1 i.e., in the absence of the dipolar interactions which leads to purely repulsive interactions between the brushes. This 22


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is in agreement with the extrapolation of

Page 24 of 37

Fmin /(AkB T ) in Figure 3 (inset) to ✏p = 1. The

attractive dipolar interactions lead to the interpenetration, which shows up as an attractive well in the free energy-distance plots. It has been argued that chain ends tend to aggregate in the case of brushes made up of telechelic polymers 57 and attraction between opposing telechelic brushes has been explained in terms of such aggregation. In our work, each segment on the brush has long-range attractive interactions with all the other segments and it is worthwhile to analyze the end distribution for the dipolar brushes exhibiting attraction at intermediate distances. In the field theory, distribution of the chain ends can be quantified by the chain propagators 68 (see SI for the details). Such results are shown in Figure 5. The results point out that the interpenetration takes place at the expense of pulling of the chain ends out of the polymer brushes. This can be seen by comparing the end distributions for H/Rgo = 8.4 and H/Rgo = 10.2 in Figure 5. The former case is near the onset of attraction and the latter shows no interactions between the brushes. It can be seen in Figure 5 that a single peak appears (instead of two in the case of non-interacting brushes) in the interpenetrating brushes exhibiting attraction. The peak in the distribution appears at distances larger than the equilibrium height of the same brush. Hence, we can infer that the interpenetration takes place by pulling of the chain ends out of the polymer brushes. This is in qualitative agreement with the idea of aggregation of chain ends leading to attraction between opposing telechelic brushes. 57 We have extended our investigations to opposing polymer brushes immersed in a dielectric solvent 63 (see the SI for the details). Presence of solvent leads to changes in the free energy landscape. In this case,

✏ = ✏p /✏s

1 becomes the relevant parameter so that ✏s is the

dielectric constant of the solvent. Changes in the free energy as a function of the separation distances are shown in Figure 6. It is shown that an increase in magnitude of to an attraction well and strength of the attractions can be tuned by

✏ can lead

✏, in qualitative

agreement with the melt/solvent-free brushes. Hence, irrespective of the presence or absence 23


Page 25 of 37

6

H/Rgo 14.2 10.9 10.7 10.2 8.4 5.2

(a)

ρp*(z/Rgo)

5 4 3

0.2

2 1 0 0

2

4

6

7

8

z/Rgo

(b)

6 5

ρp*(z/Rgo)

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

10

12

14

H/Rgo 10.2 9.2 8.4 7.9 6.7 5.2

4 3

0.35

2 1 0 0

2

4

z/Rgo

6

8

10

Figure 4: Density profiles of the segments as a function of the separation distance for (a) ✏p = 5 and (b) ✏p = 15. All the other parameters are the same as in previous figures. Both cases exhibit qualitatively similar features: two independent brushes at large separation, interpenetrated brushes at intermediate separation and strongly compressed brushes with higher densities near the grafting substrates. The interpenetration at intermediate distances takes place by extension of the brushes and resulting decrease in the density of the segments near the grafting substrates. The decrease in the density is more pronounced for ✏p = 15 compared to ✏p = 5. 24


Langmuir

H/Rgo

0.03

qzαk(z/Rgo, N)

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 37

10.2 9.2 8.4 7.9 6.7 5.2

0.02

0.01

0 0

2

4

z/Rgo

6

8

10

Figure 5: Distribution of chain ends quantified by the chain propagator (qz↵k defined in Eq. B-2 of SI) corresponding to the segment density profiles shown in Figure 4(b) (✏p = 15). The chain ends are distributed in the interior of each brush. At distances exhibiting attraction between the two brushes, the chain ends get pulled out to intermix with each other beyond the equilibrium height of the brush (e.g., compare results for H/Rgo = 8.4 with H/Rgo = 10.2).

25


Page 27 of 37

of solvent, a non-monotonic free energy - normal separation distance is predicted when the dipolar interactions are sufficiently strong.

3

∆ε 0 0.1 -0.1 0.2 -0.2 0.3 -0.25

2.5 2

∆F/AkBT

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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1.5 1 0.5 0 6

8

10

H/Rgo

12

14

Figure 6: Changes in the free energy as a function of the separation distance for two opposing brushes in the presence of solvent. The free energy of the same two brushes at the largest separation is taken as the reference state. The parameters used for generating these curves are: N = 150, b = 1, ↵ps = 0.55, b2 = 0.1, ✏s = 80 and a/b = 0.2 (ap = as = a). A fixed grid size, z = 0.1Rgo and a timestep s = 0.0001 are used for all the calculations. Just like the melt case, the brushes repel each other for low ✏ = ✏p /✏s 1 and exhibit attraction at intermediate separation distances for higher values of ✏.

Conclusions Changes in the free energy as a function of separation distances between two identical opposing dipolar brushes were studied using a field theory approach. Brushes made up of 26


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Page 28 of 37

polymers grafted on repulsive substrates were considered, which avoid polymer adsorption induced attraction between the brushes (termed as bridging interactions in the literature). Even in the absence of polymer adsorption, we showed that dipolar interactions can lead to attractive interactions between the brushes. The origin of the attractive interactions lies in the long-ranged and concentration dependent dipolar interactions, which causes interpenetration of the brushes at intermediate separation distances by the pull-out of the chain ends. The strength of the attraction and the distances at which such attraction appear depends on dielectric mismatch parameter, which is ✏p

1 and ✏p /✏s

1 for solvent-free and sol-

vated brushes, respectively. At short distances, the interactions stay repulsive independent of magnitude of the dielectric mismatch parameter. This work highlights the importance of dipolar interactions in opposing polymer brushes, which can cause colloidal destabilization by inducing attractions at intermediate separations.

ACKNOWLEDGMENTS This research was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. JPM acknowledges support from the Laboratory Directed Research and Development program at ORNL. RK acknowledges several discussions with Prof. P. Pincus regarding the attraction between opposing polymer brushes.

References (1) Stuart, M. A. C.; Huck, W. T. S.; Genzer, J.; Muller, M.; Ober, C.; Stamm, M.; Sukhorukov, G. B.; Szleifer, I.; Tsukruk, V. V.; Urban, M.; Winnik, F.; Zauscher, S.;

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Luzinov, I.; Minko, S. Emerging Applications of Stimuli-Responsive Polymer Materials. Nat. Mater. 2010, 9, 101–113. (2) Motornov, M.; Roiter, Y.; Tokarev, I.; Minko, S. Stimuli-Responsive Nanoparticles, Nanogels and Capsules for Integrated Multifunctional Intelligent Systems. Prog. Polym. Sci. 2010, 35, 174–211. (3) Mendes, P. M. Stimuli-Responsive Surfaces for Bio-Applications. Chem. Soc. Rev. 2008, 37, 2512–2529. (4) Lee, H. I.; Pietrasik, J.; Sheiko, S. S.; Matyjaszewski, K. Stimuli-Responsive Molecular Brushes. Prog. Polym. Sci. 2010, 35, 24–44. (5) Dai, S.; Ravi, P.; Tam, K. C. pH-Responsive Polymers: Synthesis, Properties and Applications. Soft Matter 2008, 4, 435–449. (6) Minko, S. Responsive Polymer Brushes. Polym. Rev. 2006, 46, 397–420. (7) Xin, B. W.; Hao, J. C. Reversibly Switchable Wettability. Chem. Soc. Rev. 2010, 39, 769–782. (8) Chen, K.; Ma, Y. Q. Interactions Between Colloidal Particles Induced by Polymer Brushes Grafted Onto the Substrate. J. Phys. Chem. B 2005, 109, 17617–17622. (9) Claesson, P. M.; Poptoshev, E.; Blomberg, E.; Dedinaite, A. Polyelectrolyte-Mediated Surface Interactions. Adv. Colloid Interface Sci. 2005, 114, 173–187. (10) Fritz, G.; Schadler, V.; Willenbacher, N.; Wagner, N. J. Electrosteric Stabilization of Colloidal Dispersions. Langmuir 2002, 18, 6381–6390. (11) Marla, K. T.; Meredith, J. C. Simulation of Interaction Forces Between Nanoparticles: End-Grafted Polymer Modifiers. J. Chem. Theory Comput. 2006, 2, 1624–1631.

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(12) Striolo, A.; Egorov, S. A. Steric Stabilization of Spherical Colloidal Particles: Implicit and Explicit Solvent. J. Chem. Phys. 2007, 126, 014902. (13) Toprakcioglu, C.; Dai, L.; Ansarifar, M. A. Conformations of Block Copolymers Terminally Adsorbed at the Solid-Liquid Interface. J. Macromol. Sci. A 1992, 29, 139–146. (14) Klein, J.; Kumacheva, E.; Mahalu, D.; Perahia, D.; Fetters, L. J. Reduction of Frictional Forces Between Solid-Surfaces Bearing Polymer Brushes. Nature 1994, 370, 634–636. (15) Cai, L. L. J.; Granick, S. Chains End-Grafted in a Theta-Solvent and Polymer Melts: Comparison of Force-Distance Profiles. Adv. Colloid Interfac. 2001, 94, 135–150. (16) Forster, A. M.; Mays, J. W.; Kilbey, S. M. Effect of Temperature on the Frictional Forces Between Polystyrene Brushes. J. Polym. Sci. Pol. Phys. 2006, 44, 649–655. (17) Klein, J. Shear of Polymer Brushes. Colloid. Surface. A 1994, 86, 63–76. (18) Klein, J.; Kamiyama, Y.; Yoshizawa, H.; Israelachvili, J. N.; Fredrickson, G. H.; Pincus, P.; Fetters, L. J. Lubrication Forces Between Surfaces Bearing Polymer Brushes. Macromolecules 1993, 26, 5552–5560. (19) Milner, S. T.; Witten, T. A.; Cates, M. E. Effects of Polydispersity in the End-Grafted Polymer Brush. Macromolecules 1989, 22, 853–861. (20) Taunton, H. J.; Toprakcioglu, C.; Fetters, L. J.; Klein, J. Interactions Between Surfaces Bearing End-Adsorbed Chains in a Good Solvent. Macromolecules 1990, 23, 571–580. (21) Hamilton, W. A.; Smith, G. S.; Alcantar, N. A.; Majewski, J.; Toomey, R. G.; Kuhl, T. L. Determining the Density Profile of Confined Polymer Brushes With Neutron Reflectivity. J. Polym. Sci. Pol. Phys. 2004, 42, 3290–3301. (22) Banquy, X.; Charrault, E.; Giasson, S. Normal and Lateral Interactions between Thermosensitive Nanoparticle Monolayers in Water. J. Phys. Chem. B 2010, 114, 9721– 9728. 29


Page 31 of 37

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

(23) Kilbey, S. M.; Watanabe, H.; Tirrell, M. Structure and Scaling of Polymer Brushes Near the Theta Condition. Macromolecules 2001, 34, 5249–5259. (24) Dhinojwala, A.; Granick, S. Surface Forces in the Tapping Mode: Solvent Permeability and Hydrodynamic Thickness of Adsorbed Polymer Brushes. Macromolecules 1997, 30, 1079–1085. (25) Liao, W. P.; Kuhl, T. L. Steric Forces of Tethered Polymer Chains as a Function of Grafting Density: Studies with a Single Diblock Molecular Weight. Macromolecules 2012, 45, 5766–5772. (26) de Beer, S.; Kutnyanszky, E.; Schon, P. M.; Vancso, G. J.; Muser, M. H. SolventInduced Immiscibility of Polymer Brushes Eliminates Dissipation Channels. Nat. Commun. 2014, 5, 3781. (27) Goodman, D.; Kizhakkedathu, J. N.; Brooks, D. E. Attractive Bridging Interactions in Dense Polymer Brushes in Good Solvent Measured by Atomic Force Microscopy. Langmuir 2004, 20, 2333–2340. (28) Goodman, D.; Kizhakkedathu, J. N.; Brooks, D. E. Evaluation of an Atomic Force Microscopy Pull-Off Method for Measuring Molecular Weight and Polydispersity of Polymer Brushes: Effect of Grafting Density. Langmuir 2004, 20, 6238–6245. (29) Goodman, D.; Kizhakkedathu, J. N.; Brooks, D. E. Molecular Weight and Polydispersity Estimation of Adsorbing Polymer Brushes by Atomic Force Microscopy. Langmuir 2004, 20, 3297–3303. (30) Yu, Y.; Kieviet, B. D.; Liu, F.; Siretanu, I.; Kutnyanszky, E.; Vancso, G. J.; de Beer, S. Stretching of Collapsed Polymers Causes an Enhanced Dissipative Response of PNIPAM Brushes Near Their LCST. Soft Matter 2015, 11, 8508–8516.

30


Langmuir

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

(31) Milner, S. T.; Witten, T. A.; Cates, M. E. Theory of The Grafted Polymer Brush. Macromolecules 1988, 21, 2610–2619. (32) Milner, S. T. Polymer Brushes. Science 1991, 251, 905–914. (33) Milner, S. T. Compressing Polymer Brushes - a Quantitative Comparison of Theory and Experiment. Europhys. Lett. 1988, 7, 695–699. (34) Ball, R. C.; Marko, J. F.; Milner, S. T.; Witten, T. A. Polymers Grafted to a Convex Surface. Macromolecules 1991, 24, 693–703. (35) Milner, S. T.; Witten, T. A. Bridging Attraction by Telechelic Polymers. Macromolecules 1992, 25, 5495–5503. (36) Zhulina, E. B.; Borisov, O. V.; Priamitsyn, V. A. Theory of Steric Stabilization of Colloid Dispersions by Grafted Polymers. J. Colloid. Interf. Sci. 1990, 137, 495–511. (37) Zhulina, E. B.; Borisov, O. V.; Pryamitsyn, V. A.; Birshtein, T. M. Coil Globule Type Transitions in Polymers .1. Collapse of Layers of Grafted Polymer-chains. Macromolecules 1991, 24, 140–149. (38) Tang, W. H.; Witten, T. A. Inducing and Measuring Bridging in Telechelic Polymer Brushes. Macromolecules 1996, 29, 4412–4416. (39) Matsen, M. W. Investigating the Dominant Corrections to the Strong-stretching Theory for Dry Polymeric Brushes. J. Chem. Phys. 2004, 121, 1938–1948. (40) Matsen, M. W.; Gardiner, J. M. Autophobic Dewetting of Homopolymer on a Brush and Entropic Attraction Between Opposing Brushes in a Homopolymer Matrix. J. Chem. Phys. 2001, 115, 2794–2804. (41) Traskelin, P.; Kuhl, T. L.; Faller, R. Molecular Dynamics Simulations of Polystyrene Brushes in Dry Conditions and in Toluene Solution. Phys. Chem. Chem. Phys. 2009, 11, 11324–11332. 31


Page 32 of 37

Page 33 of 37

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

(42) Elliott, I. G.; Kuhl, T. L.; Faller, R. Compression of High Grafting Density Opposing Polymer Brushes Using Molecular Dynamics Simulations in Explicit Solvent. J. Phys. Chem. B 2013, 117, 4134–4141. (43) Galuschko, A.; Spirin, L.; Kreer, T.; Johner, A.; Pastorino, C.; Wittmer, J.; Baschnagel, J. Frictional Forces between Strongly Compressed, Nonentangled Polymer Brushes: Molecular Dynamics Simulations and Scaling Theory. Langmuir 2010, 26, 6418–6429. (44) Yin, F.; Bedrov, D.; Smith, G. D. A Molecular Simulation Study of the Structure and Tribology of Polymer Brushes: Comparison of Behavior in Theta and Good Solvents. Eur. Polym. J. 2008, 44, 3670–3675. (45) de Beer, S.; Muser, M. H. Alternative Dissipation Mechanisms and the Effect of the Solvent in Friction Between Polymer Brushes on Rough Surfaces. Soft Matter 2013, 9, 7234–7241. (46) Cho, Y. K.; Dhinojwala, A.; Granick, S. Apparent Hydrodynamic Thickness of Densely Grafted Polymer Layers in a Theta Solvent. J. Polym. Sci. Pol. Phys. 1997, 35, 2961– 2968. (47) Elliott, I. G.; Kuhl, T. L.; Faller, R. A Molecular Dynamics Technique to Extract Forces in Soft Matter Systems Under Compression With Constant Solvent Chemical Potential. J. Chem. Theory Comput. 2012, 8, 1072–1077. (48) de Beer, S. Switchable Friction Using Contacts of Stimulus-Responsive and Nonresponding Swollen Polymer Brushes. Langmuir 2014, 30, 8085–8090. (49) Kreer, T. Polymer-brush Lubrication: A Review of Recent Theoretical Advances. Soft Matter 2016, 12, 3479–3501.

32


Langmuir

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

(50) Sirchabesan, M.; Giasson, S. Mesoscale Simulations of the Behavior of Charged Polymer Brushes under Normal Compression and Lateral Shear Forces. Langmuir 2007, 23, 9713–9721. (51) Abraham, T.; Giasson, S.; Gohy, J. F.; Jerome, R. Direct Measurements of Interactions Between Hydrophobically Anchored Strongly Charged Polyelectrolyte Brushes. Langmuir 2000, 16, 4286–4292. (52) Israelachvili, J. Intermolecular and Surface Forces; Academic Press Limited, San Diego, CA, 1992. (53) Bhatia, S. R.; Russel, W. B. End-capped Associative Polymer Chains Between Nanospheres: Attractions in Ideal Solutions. Macromolecules 2000, 33, 5713–5720. (54) Erbas, A.; Paturej, J. Friction Between Ring Polymer Brushes. Soft Matter 2015, 11, 3139–3148. (55) Semenov, A. N. Contribution to the Theory of Microphase Layering in Block-copolymer Melt. Sov. Phys. JETP 1985, 61, 733–742. (56) Fredrickson, G. H. The Equilibrium Theory of Inhomogeneous Polymers; Oxford University, New York, 2006. (57) Klein, J. Entropic Interactions: Neutral and End-functionalized Chains in Confined Geometries. J. Phys-Condens. Mat. 2000, 12, A19–A27. (58) Zilman, A. G.; Safran, S. A. Entropically Driven Attraction Between Telechelic Brushes. Eur. Phys. J. E 2001, 4, 467–473. (59) Eiser, E.; Klein, J.; Witten, T. A.; Fetters, L. J. Shear of Telechelic Brushes. Phys. Rev. Lett. 1999, 82, 5076–5079. (60) Lea, A. S.; Andrade, J. D.; Hlady, V. Measurement of Steric Exclusion Forces with the Atomic-force Microscope. Acs Sym. Ser. 1993, 532, 266–279. 33


Page 34 of 37

Page 35 of 37

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

(61) de Beer, S.; Kutnyanszky, E.; Muser, M. H.; Vancso, G. J. Preparation and Friction Force Microscopy Measurements of Immiscible, Opposing Polymer Brushes. Jove-J. Vis. Exp. 2014, 94, e52285. (62) Kornreich, M.; Malka-Gibor, E.; Zuker, B.; Laser-Azogui, A.; Beck, R. Neurofilaments Function as Shock Absorbers: Compression Response Arising from Disordered Proteins. Phys. Rev. Lett. 2016, 117 . (63) Mahalik, J. P.; Sumpter, B. G.; Kumar, R. Vertical Phase Segregation Induced by Dipolar Interactions in Planar Polymer Brushes. Macromolecules 2016, 49, 7096–7107. (64) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press, Oxford, 1986. (65) Kumar, R.; Sumpter, B. G.; Muthukumar, M. Enhanced Phase Segregation Induced by Dipolar Interactions in Polymer Blends. Macromolecules 2014, 47, 6491–6502. (66) Kumar, R.; Sumpter, B. G.; Kilbey, S. M. Charge regulation and local dielectric function in planar polyelectrolyte brushes. J. Chem. Phys. 2012, 136, 234901. (67) Ranjan, A.; Qin, J.; Morse, D. C. Linear Response and Stability of Ordered Phases of Block Copolymer Melts. Macromolecules 2008, 41, 942–954. (68) Holt, A. P. et al. Controlling Interfacial Dynamics: Covalent Bonding versus Physical Adsorption in Polymer Nanocomposites. Acs Nano 2016, 10, 6843–6852. (69) Kim, J. U.; Matsen, M. W. Interaction Between Polymer-grafted Particles. Macromolecules 2008, 41, 4435–4443. (70) Kumar, M.; Sekhon, S. S. Role of Plasticizer’s Dielectric Constant on Conductivity Modification of PEO NH4F Polymer Electrolytes. Eur. Polym. J. 2002, 38, 1297– 1304.

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(71) Ahmed, Z.; Gooding, E. A.; Pimenov, K. V.; Wang, L. L.; Asher, S. A. UV Resonance Raman Determination of Molecular Mechanism of Poly(N-isopropylacrylamide) Volume Phase Transition. J. Phys. Chem. B 2009, 113, 4248–4256.

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For Table of Contents Title: Attraction in opposing planar dipolar polymer brushes Authors: J. P. Mahalik, Bobby G. Sumpter, Rajeev Kumar

Change in Free Energy, ∆F/AkBT

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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4

εp 5 10 15

3 2 1 0 -1 -2

6

7

8

9

10

11

Distance between substrates, H/Rgo

12

Figure 7: Changes in the free energy as a function of the separation distance showing attraction at intermediate separations. Strength of the attraction is predicted to depend on the monomeric dipole moment, which characterizes the static dielectric constant of the polymer in the melt, ✏p .

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Attraction between Opposing Planar Dipolar Polymer Brushes

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