Theory of brushes formed by Ψ-shaped
macromolecules at solid-liquid interfaces.
Ekaterina B. Zhulina1,2 , Frans A.M. Leermakers3 , Oleg V. Borisov1,2,4
1
Institute of Macromolecular Compounds
of the Russian Academy of Sciences,
St.Petersburg, Russia
2
St.Petersburg National University of
Informational Technologies, Mechanics and Optics,
3
Laboratory of Physical Chemistry and Colloid Science,
Dreijenplein 6, 6703 HB Wageningen, the Netherlands
4
CNRS, Université de Pau et des Pays de l’Adour UMR 5254,
Institut des Sciences Analytiques et de Physico-Chimie pour
l’Environnement et les Matériaux, 64053 Pau, France
May 9, 2015
Supporting Information
Appendix A. Stretching functions of stem and branches,
and force balance
When the stem and the branches are noticeably extended with respect to
their Gaussian sizes but are still far from their full extension, the conformations of macromolecules can be described by the so-called trajectories that
specify the most probable height z(j) above the surface of the monomer with
ranking number j. The stretching function Ei = dz(j)/dj of the free branch
with number 1 ≤ i ≤ q can be represented as
q
Ei (zi , z) = k zi2 − z 2 ,
z0 ≤ z ≤ zi
(1)
where zi is the position of the end segment of branch i, and z0 is the position
of the branching point. Eq 1 assures vanishing tension at the end of each
1
free branch, that is, at z = zi . By using normalization condition for the free
branches ,
Z zi
dz
= ni ,
i = 1, ...q
(2)
z0 Ei (zi, z)
one finds the relationship between position zi of the free end of i-th branch
and height z0 of the branching point,
zi =
z0
cos(kni )
The stretching function E0 of the stem yields
√
E0 (z0 , z) = k λ2 − z 2
(3)
(4)
where λ is the unknown constant. By using normalization condition for the
stem,
Z z0
dz
= n0
(5)
E0 (z0, z)
0
one finds constant λ in eq 4 for the stretching function E0 ,
λ=
z0
sin(kn0 )
(6)
In the linear elasticity regime, the balance of elastic normal forces at the
branching point of a dendron is equivalent to the balance of the stretching
functions,
q
X
E0 (z0, z0 ) =
Ei (zi , z0 )
(7)
i=1
Eq 7 together with eqs 1, 4, 6 and 3 provide the relationship to determine
k as a function of the degrees of polymerization of the stem, n0 , and of the
branches, ni ,
q
X
tan(kn0 ) ·
tan(kni ) = 1
(8)
i=1
In the case when the stem and all the arms have the same length, i.e.,
ni = n (i = 0, 1, 2, ..q), eq 8 provides the previously obtained expression
r
1
1
q
1
(9)
k = arctan √ ≡ arccos
n
q
n
q+1
2
Appendix B. Chemical potential of a Ψ-shaped macromolecule with polydisperse free branches
Consider a Ψ-shaped macromolecule with the stem composed of n0 monomers,
and free branches with ni monomers each, inserted in the brush with molecular potential U (z), eq T2 (here and below the numbers of equations with
the prefix ”T” refer to the main text of the paper)
3
U (z)
= 2 k 2 (H 2 − z 2 )
kB T
2a
The penalty µ(z0 ) for insertion of such a dendron with its branching point
located at height z0 above the grafting surface is given by
µelastic + µinter
µ(z0 )
=
=
kB T
kB T
z0
Z
3
= 2
2a
E0 (z0 , z)dz +
0
Z
0
z0
i=q Z
X
i=q
Z
Ei (zi , z)dz
+
z0
i=1
X
U (z)
dz +
E0 (z0 , z)
i=1
!
zi
zi
z0
U (z)
dz
Ei (zi , z)
!
(10)
The terms in the first brackets account for the elastic contributions of the
stretched stem and branches of the inserted dendron, respectively. The
terms in the second brackets are due to the interactions between the inserted
dendron and neighboring macromolecules, expressed through the molecular
potential U (z). The stretching functions of the stem, E0 , and of the free
branches, Ei , are specified by eqs 4, 1 and 6, 3 as
q
√
2
2
Ei (zi , z) = k zi2 − z 2
(11)
E0 (z0 , z) = k λ − z ,
with
λ=
z0
,
sin(kn0 )
zi =
z0
cos(kni )
(12)
By substituting E0 (z0 , z), Ei (zi , z), and U (z) in eq 10, we reduce µ(z0 ) to
the following expression
µ(z0 )
=
kB T
3
2a2
( Z
2
0
z0
E0 (z0 , z)dz + k 2
Z
0
z0
X
)
Z zi
i=q Z zi
2
2
(H 2 − λ2 )
(H
−
z
)
i
dz +
2
Ei (zi , z)dz + k 2
dz
E0 (z0 , z)
E
(z
,
z)
i
i
z0
z0
i=1
3
" Z q
#)
X
Z zi
i=q
zi
2
2
(H 2 − λ2 )
(H
−
z
)
i
√
p
dz +
2
zi2 − z 2 dz +
dz
=
2
2
2
2
λ −z
z
−
z
0
0
z
z
0
0
i
i=1
( Z
)
i=q
z0 √
X Z zi q
3k
2
=
λ2 − z 2 dz + (H 2 − λ2 )kn0 +
2
zi2 − z 2 dz + (H 2 − zi2 )kni
2a2
0
z
0
i=1
3k
2a2
( Z
2
z0
Z
√
2
2
λ − z dz +
z0
i=q
X 3k 2 H 2 n
+
ni +
0
2a2
i=1
3k
2a2
" Z
2
z0
√
λ2 − z 2 dz − λ2 kn0
i=q Z
X
+
2
0
zi
#
q
zi2 − z 2 dz − zi2 kni
z0
i=1
(13)
By using the expression for indefinite integral
"
#
Z √
z z r
2
2
b
z
b2 − z 2 dz =
arcsin
+
1− 2
2
b
b
b
with b = λ or b = zi , we calculate µ(z0 ) in eq 13 as
"
µ(z0 )
3k 2 H 2 N
3k
=
+ 2 λ2
kB T
2a2
2a
z0
λ
r
z2
1 − 02
λ
!
−
i=q
X
i=1
zi2
z0
zi
s
z2
1 − 02
zi
"
#
i=q
X
3kz02
3k 2 H 2 N
1
3k 2 H 2 N
+
−
=
tan(kn
)
=
i
2a2
2a2 tan(kn0 ) i=1
2a2
!#
=
(14)
Note that expression in square brackets is equal to zero due to force balance
in the branching point, eq T6. Eq 14 indicates that the insertion penalty
µ(z0 ) does not depend on the position of branching point, z0 , and is thereby
equal to the chemical potential µ of a Ψ-shaped macromolecule in the brush,
µ
3k 2 H 2 N
=
kB T
2a2
By substituting brush thickness H from eq T5,
H/a = (
one finds
(15)
2vσN 1/3
)
k2
µ
3k 2 N 2vσN 2/3 3k 2/3 N 5/3
=
( 2 ) =
(σv)2/3
kB T
2
k
21/3
4
(16)
where k is the minimal root of eq T9. By using the relationship between free
energy F (σ) per macromolecule in the brush with grafting density σ, and its
chemical potential µ,
∂[σF (σ)]
5
= F (σ)
∂σ
3
µ=
we find the free energy of the brush (per dendron)
F
9k 2 H 2 N
9k 2/3 N 5/3
=
(2vσ)2/3
=
kB T
10a2
10
(17)
The expressions for the free energy per macromolecule, F , and brush
thickness, H, reduce to the known expressions for linear chain brushes when
q = 1. For example, when q = 1, n1 = 2n0 , and N = n0 + n1 = 3n0 , solution
of eq T9 (see eq T10 with u = 2) yields k = π/(6n0 ), and
H/a = 3n0 (
8σv 1/3
8
) = ( 2 )1/3 N (σv)1/3
2
π
π
9π 2/3
F
=
N (σv)2/3
kB T
10
where N is the number of monomers in a linear chain.
Appendix C. Ψ-shaped macromolecules with weakly polydisperse free branches
For arbitrary degrees of polymerization of the branches ni (i = 1, 2, ..., q) eq
T6 must be solved numerically. However, at relatively low polydispersity of
the branches, eq T6 can be simplified. We introduce the average degree of
polymerization n of the free branches,
Pq
ni
n = i=1
q
and the differences ∆ni = (ni − n) as a measure of deviation of each branch
length from the average one. Then eq T6 can be rewritten as
tan(kn0 ) ·
q
X
tan[k(n + ∆ni )] = 1
i=1
5
Assuming ∆ni /n 1, and by using the Taylor’s expansion up to the second
order with respect to the small parameter k∆ni ,
tan[k(n + ∆ni )] = tan(kn) +
1
tan(kn)
(k∆n
)
+
(k∆ni )2 + ... (18)
i
2
2
cos (kn)
cos (kn)
one finds
q
(
tan(kn0 ) · tan(kn) ·
k 2 n2 X (∆ni )2
q+
cos2 (kn) i=1 n2
)
≈1
(19)
P
Note that summation of the second term in eq 18 gives zero because qi=1 ∆ni =
0. By averaging the expression in figure brackets in eq 19 (at fixed values of
n and n0 ), and introducing the polydispersity parameter δ as
q
1 X (∆ni )2
(∆n)2
δ=
=
1
q i=1 n2
n2
(20)
eq 19 is rewritten as
tan(kn0 ) · tan(kn) · 1 +
1
k 2 n2
δ ≈
2
cos (kn)
q
(21)
As it follows from eq 21, polydispersity of the free branches (δ ≥ 0) decreases the value of k compared to the case of monodisperse branches (δ = 0)
with degree of polymerization n = n, and makes the molecular potential
in eq T2 more flat. As a result, the thickness H of the brush composed
of Ψ−shaped macromolecules with average degree of polymerization n of
the free branches would increase compared to that for macromolecules with
monodisperse branches with n = n. This prediction is qualitatively valid for
both neutral and charged Ψ-brushes. However, the quantitative estimate for
polydispersity-induced increase in brush thickness H depends on the relationship between the molecular potential U (z) and polymer density profile
φ(z).
To specify how the value of k depends on the number q of free branches
with polydispersity δ each, we assume (for simplicity) that the stem has
degree of polymerization n0 = n, and expand the solution of eq 21 with
respect to small parameter δ up to linear in δ term. This leads to
δ
1
1 1
δ
k(δ) ≈ k(δ = 0) · 1 − √ arctan √
≈ √ · 1−
(22)
2 q
q
n q
2q
where the last equality in eq 22 is valid when q 1. Therefore for highly
branched macromolecules the effect of branch polydispersity becomes less
noticeable than for macromolecules with small number of branches.
6