Supporting Information:

The influence of side-chain conformations on the

phase behavior of bottlebrush block polymers

Yuguo Chen1, Xinghua Zhang2,∗ and Ying Jiang1,3∗

1School of Chemistry & Key Laboratory of Bio-Inspired Smart Interfacial Science and

Technology of Ministry of Education & Center of Soft Matter Physics and Its Applications,

Beihang University, Beijing 100191, China

2School of Science, Beijing Jiaotong University, Beijing 100044, China,

3Beijing Advanced Innovation Center for Biomedical Engineering, Beihang University, Beijing

100191, China

E-mail: zhangxh@bjtu.edu.cn; yjiang@buaa.edu.cn

∗To whom correspondence should be addressed

1

Electronic Supplementary Material (ESI) for Soft Matter.This journal is © The Royal Society of Chemistry 2020

Theoretical formalism of SCFT

Figure S1: Schematic of the bottlebrush diblock polymer chain composed of side chains A (blue),B (red) and the backbone C (black). The grafting points along the backbone are labelled as τi,i = 1,2,3, · · · ,m and the free ends are labelled as τ0 and τm+1, respectively.

We consider a system of volume V , which is comprised of n monodisperse and indistinguish-

able bottlebrush polymer chains. Each of chain has m side chains including mA A-chains and

mB =m−mA B-chains, which uniformly divide the C backbone into m+1 equal segment, as illus-

trated in Fig.S1. We assume that each monomer on all blocks shares the same statistical length b

and then occupies the same volume ρ−10 . The degree of polymerization of side chains A and B is

NA and NB, while the degree of polymerization of backbone is NC. Thus, the summed total seg-

ment number of a single bottlebrush polymer is N = mANA+mBNB+NC. The persistence length

for the individual block is λα, α = A,B,C. The volume fraction for each component is calculated

by fA =mANA/N, fB =mBNB/N, and fC = NC/N. The ith grafting point on the backbone is located

at τi = i× fC/(m+ 1), where the grafting point index i ranges from 1 to m. Then, two remaining

free ends of backbone are designated as τ0 and τm+1, respectively.

The Helmholtz free energy per chain for multicomponent block copolymers in the mean field

approximation1 can be written as

FnkBT

= −lnQ+1V

∫dr

∑α,β

Nχαβϕα(r)ϕβ(r)−∑α

ωα(r)ϕα(r)− ξ(r)[1−∑α

ϕα(r)] (1)

2

The Flory-Huggins parameter χαβ describes the interaction energy between dissimilar species.

The average density for species ϕα(r) which is conjugated to the mean field ωα(r) is enforced by

the Lagrangian multiplier ξ(r) due to the incompressibility constraint to∑αϕα(r) = 1. kB is the

Boltzmann constant and T is the temperature of the system. For the wormlike chain model used

here, the single chain partition function for bottlebrush copolymers can be computed by

Q =1

4πV

∫dr∫

duqC(r,u, fC) (2)

The backbone propagator qC(r,u, t) represents the probability of finding the t terminal, starting

from the free end τ0, which is located at a spatial position specified by r and points to a direction

specified by the unit vector u.

In our model, the backbone is divided into m+1 blocks. The propagator within the ith block is

designated as

qC(r,u, t) = q(i)C (r,u, t) here, τi ≤ t < τi+1 i = 0,1,2, · · · ,m (3)

which satisfies the modified diffusion equation2

∂

∂tq(i)

C (r,u, t) =[

Nb2λC∇2

u−Nbu · ∇r−ωC(r)]q(i)

C (r,u, t) (4)

This equation is subject to the following initial conditions

q(0)C (r,u,0) = 1 (5)

and

q(i)C (r,u, τi) = q(i−1)

C (r,u, τi)×1

4π

∫duqα(r,u, t = fα/mα) i = 1,2, · · · ,m (6)

Herein, we assume that the side chain is grafted on the backbone in an orientation-independent

manner. It means that side chains are grafted to the backbone with the equal probability for all

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directions. The species index α =A,B is determined by the component of the side chain grafted at

the specific point along the backbone. The propagator for the side chain satisfies

∂

∂tqα(r,u, t) =

[Nb2λα∇2

u−Nbu · ∇r−ωα(r)]qα(r,u, t) (7)

which is subject to the initial condition qα(r,u,0) = 1 for the free end.

Because the side chains grafted at two sides of the backbone are distinct, a complementary seg-

ment distribution function for the backbone q̃(i)C (r,u, t) is also required. It represents the probability

of finding the t terminal belonging to the ith block along the backbone, starting from the other free

end τm+1, which is at the spatial position r and points in a direction specified by the unit vector −u.

The propagator within the ( j−1)th block is designated as

q̃C(r,u, t) = q̃( j−1)C (r,u, t) here, τ j−1 < t ≤ τ j j = m+1,m,m−1, · · · ,1 (8)

which satisfies the modified diffusion equation

∂

∂tq̃( j−1)

C (r,u, t) =[− Nb

2λC∇2

u−Nbu · ∇r+ωC(r)]q̃( j−1)

C (r,u, t) (9)

This equation is subject to the following initial conditions

q̃(m)C (r,u, t = fC) = 1 (10)

and

q̃( j−1)C (r,u, τ j) = q̃( j)

C (r,u, τ j)×1

4π

∫duqα(r,u, t = fα/mα) j = m,m−1,m−2, · · · ,1 (11)

where the species index α = A,B is determined by the component of the side chain grafted at the

junction point. Similarly, the other segment distribution function for the side chain propagating

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from the ith junction point satisfies

∂

∂tq̃(i)α (r,u, t) =

[− Nb

2λα∇2

u−Nbu · ∇r+ωα(r)]q̃(i)α (r,u, t) (12)

which is subject to the initial condition

q̃(i)α (r,u, t = fα/mα) =

14π

∫duq(i−1)

C (r,u, t = τi)q̃(i)C (r,u, t = τi)[

14π

∫duqα(r,u, t = fα/mα)

]2 (13)

Minimization of free energy functional Eq. (1) with respective of functions ϕα(r), ωα(r), ξ(r),

we directly arrive at a set of saddle-point approximation equations

ωA(r) = NχABϕB(r)+NχACϕC(r)+ ξ(r) (14)

ωB(r) = NχABϕA(r)+NχBCϕC(r)+ ξ(r) (15)

ωC(r) = NχACϕA(r)+NχBCϕB(r)+ ξ(r) (16)

ϕA(r)+ϕB(r)+ϕC(r) = 1 (17)

ϕA(r) =1

4πQ

mA∑i=1

∫du∫ fA/mA

0dtqA(r,u, t)q̃(i)

A (r,u, t) (18)

ϕB(r) =1

4πQ

m∑i=mA+1

∫du∫ fB/mB

0dtqB(r,u, t)q̃(i)

B (r,u, t) (19)

ϕC(r) =1

4πQ

m∑i=0

∫du∫ τi+1

τi

dtq(i)C (r,u, t)q̃(i)

C (r,u, t) (20)

In the present work, we are mainly interested on the lamellar phase self-assembled by the

symmetric bottlebrush copolymers. The system can be simplified further that only one spatial

variable z and one orientational variable θ are concerned. Then, Eqs.(4), (7), (9) and (12) can be

numerically solved by the pseudo-spectral methods.1,3 Once the propagators are calculated, we are

ready to revise the external fields ωα according to the newly obtained component density. In order

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to accelerate the convergence, we use the Anderson mixing scheme,4,5 which updates the original

field by incorporating the results from several preceding iterations. The free energy in Eq. (1) is

used to analyze the thermodynamical stability of a specific morphology.

The degree of microphase segregation for bottlebrush copolymers strongly depends on the

number of side chains m. For a fixed Flory-Huggins parameter χ, a larger m commonly results

in a stronger phase segregation. In order to estimate the degree of segregation varying with the

increase of m, we set NχAB = (m/2)(NA+NB)χ̃AB, where (NA+NB)χ̃AB = 15 corresponding to the

intermediate degree of segregation is fixed. On account of the radial stretching of side chains, we

assume that the backbone is neutral to side chains, i.e. χAC = χBC = 0. Then, the distribution of the

backbone is affected by the steric congestion of side chains and the incompressibility constraint,

as a consequence of A-B segregation. The present model facilitates us to understand the impact

of polymer conformations of the backbone and side chains on the self-assembly behaviors for

bottlebrush block polymers.

References

(1) Fredrickson, G. H. The Equilibrium Theory of Inhomogeneous Polymers; Oxford University

Press, 2006.

(2) Jiang, Y.; Li, S.; Chen, J. Z. The European Physical Journal E 2016, 39, 91.

(3) Jiang, Y.; Zhang, W.-Y.; Chen, J. Z. Y. Phys. Rev. E 2011, 84, 041803.

(4) Thompson, R. B.; Rasmussen, K. O.; Lookman, T. The Journal of Chemical Physics 2004,

120, 31–34.

(5) Matsen, M. W. The European Physical Journal E 2009, 30, 361.

6

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

(z)

z/D0

=0.60 =0.40 =0.27 =0.20

(a)

C(z

)

z/D0

(b)

Figure S2: The density profiles for (a) the side chain A and (b) the backbone C in the relativelength unit over one domain spacing of lamellar phase with the various grafting density σ. Thechain flexibility parameters LA/2λA = 40 and LC/2λC = 7.5 are used for the block A and the blockC, respectively.

7

-1.0 -0.5 0.0 0.5 1.00

2

4

6

0.25 0.50 0.750

5

10

15

203040

fC/4

P C(u

, tC)

u=cos

tC = 0 tC = fC/8 tC = fC/4 tC = 3fC/8 tC = fC/2

tC=0

fC/83fC/8

fC/2

(c)

(b)

P C(z

, tC)

z/D0

(a)

Interface

Figure S3: The probability distribution of the backbone segments in lamellae at the given graftingdensity σ = 0.53. (a) The schematic chain conformation of bottlebrush polymers which are specif-ically investigated in plots (b) and (c). (b) The orientational probability distribution PC(u, tC) ofthe backbone monomer labelled by the contour variable tC. (c) The positional probability distri-bution PC(z, tC) of backbone monomers labelled by the contour variable tC. The chain flexibilityparameters LA/2λA = 40 and LC/2λC = 7.5 are used for the block A and the block C, respectively.Note, we calculate the probability distribution for the bottlebrush molecule whose junction pointis located at one specifically selected interface by imposing a restriction of a the interface-locationdependence. Thus, herein, the orientation perpendicular to the interface is identified by u = −1,which is distinguishable from u = 1.

8

1 10 1001

2

3

4

W/b

LSC/2 SC

=0.35 =0.41 =0.47

Figure S4: The interfacial width W in the relative length unit over b as the function of the chainflexibility parameter LSC/2λSC of side chains for the given grafting density as labelled in the plot.

9

0.25 0.50 0.750

4

8

12

16

0.25 0.50 0.750

4

8

12

16

0.25 0.50 0.750

4

8

12

16

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

Interface

P C(z

, tC)

z/D0

(c1)

LSC/2 SC= 1.25

LSC/2 SC= 6.67

(b1)

P C(z

, tC)

z/D0

tC=0.06fC

tC=0.18fC

tC=0.30fC

tC=0.42fC

LSC/2 SC= 50

(a1)

P C(z

, tC)

z/D0

Interface

tC=0

(c2)

P C(u

, tC)

u=cos

(b2)

P C(u

, tC)

u=cos

(a2)

P C(u

, tC)

u=cos

Figure S5: Positional and orientational probability distributions of the backbone segments specif-ically for three different side-chain conformations parameterized by the flexibility parameter (a)LSC/2λSC = 50, (b) LSC/2λSC = 6.67, and (c) LSC/2λSC = 1.25 at the grafting density σ = 0.47.The probability functions follows the same definitions in Fig. S3. The chain flexibility parameterLC/2λC = 17 for the backbone is used here.

10

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

P A, g

rafti

ng p

oint(u

; tC) tC = 0.06fC

tC = 0.24fC tC = 0.48fC

(a1)

(a2)

P A, m

iddl

e(u ; t

C)

(a3)

P A, fr

ee e

nd(u

; tC)

u=cos

(b1)

(b2)

(b3)

u=cos

(c1)

(c2)

LSC/ SC=3.33LSC/ SC=5.0

(c3)

u=cos

LSC/ SC=20

Figure S6: The orientational probability distributions for different segments along the side-chain A with the various side-chain flexibility (a) LSC/2λSC = 20, (b) LSC/2λSC = 5.0, and (c)LSC/2λSC = 3.33 at a given grafting density σ = 0.47. The probability function PA,γ(u; tC) repre-sents the orientational probability of finding the specific segment γ along the side-chain A attachedto the backbone segment labelled by the contour variable tC. The definition on tC is same as theone in Fig. S3a. The side-chain segment index γ denotes the grafting point (1), middle segment(2), and free end (3) of the specified side chain. The orientational variable is defined as u = cosθ,where the angle θ is graphically defined in Fig. S3a. Note, herein, the orientation perpendicular tothe interface is identified by u = −1, which is distinguishable from u = 1, because the restriction ofone specifically selected interface is imposed on the computation of PA,γ(u; tC).

11

0.00 0.25 0.50 0.75 1.000

2

4

6

8

3040

0.00 0.25 0.50 0.75 1.000

2

4

6

820

30

0.00 0.25 0.50 0.75 1.000

2

4

6

8

3040

LSC/2 SC=5.0(b)

P A, (z

; tC)

z/D0

Interface

free grafting end pointtC=0.06fC: tC=0.24fC: tC=0.48fC:

(a)

P A, (z

; tC)

z/D0

LSC/2 SC=20

LSC/2 SC=3.33(c)

P A, (z

; tC)

z/D0

Figure S7: The positional probability distributions for two different ends of one side-chain A withthe various side-chain flexibility (a) LSC/2λSC = 20, (b) LSC/2λSC = 5.0, and (c) LSC/2λSC = 3.33at a given grafting density σ = 0.47. The probability function PA,γ(z; tC) represents the positionalprobability of finding the specific segment γ along the side-chain A attached to the backbonesegment labelled by the contour variable tC. The definition on tC is same as the one in Fig. S3a.The side-chain segment index γ denotes the free end (solid lines) and the grafting point (dottedlines) of the specified side chain. Note, we calculate PA,γ(z; tC) for the bottlebrush molecule whosejunction point is located at one specifically selected interface as indicated, by imposing a restrictionof the interface-location dependence.

12