Supporting information
0.42
0.40
0.38
h (1/dh)
0.36
0.34
0.32
eq.1
eq.2
eq.3
0.30
0.28
0.26
0.5
0.6
0.7
0.8
0.9
1.0
cA
Figure 1 Comparison of the dependence of exponent λh calculated by Rh~Nλ on cA under
different equations of Rh.
Rh  N
N
2
N
 
i 0 j 0( i  j )
N
N
N
 
Rh  N ( N  1)
Rh  N 2
1
rij
i 1 j 1( i  j )
N
 
i 1 j 1( i  j )
1
rij
eq.1
1
rij
eq.2
eq.3
According to reference 1, the equation 1 and 3 are given in the page of 243 and 21
respectively. However, both eq.1 and eq.3 should be more suitable for infinitely large
polymers. For small molecules, there may be large deviations because the calculated number
in numerator and denominator is not identical. In order to keep the consistency of numerator
and denominator, we proposed the eq.2 in this work. Since the degree of polymerization is
finite for all hyperbranched molecules obtained in our simulation, we think eq.2 is more
accurate. With the growing up of the polymers, the discrepancy caused by various equations
gradually decreases. This trend can be seen in Figure 1, with the increase of cA, the λhs
calculated by different equations gradually close to each other. For infinitely large polymers,
the results calculated by these three equations should be the same.
4
c0=0.1
4
c0=0.3
3
Rhz*
c0=0.5
3
c0=0.7
2
Rgz*
c0=0.9
1
0.2
2
0.4
0.6
0.8
1.0
cA
1
0.2
0.4
0.6
0.8
1.0
cA
Figure 2 Dependence of the dimensionless z-average radius of gyration, Rgz*, and z-average
hydrodynamic radius Rhz*, on the conversion of A groups, cA, under various monomer
concentrations, c0, for AB4 type polymerization.
5
4
k12=1/10
k12=1/3
3
Rhz*
4
k12=1
k12=3
2
k12=10
Rgz*
3
1
0.2
0.4
0.6
0.8
1.0
cA
2
1
0.2
0.4
0.6
0.8
1.0
cA
Figure 3 Dependence of the dimensionless z-average radius of gyration, Rgz*, and z-average
hydrodynamic radius Rhz*, on the conversion of A groups, cA, at various reactivity ratios, k12,
with monomer concentration c0 = 0.2 for AB2 type polymerization.
1.3
c0=0.1
c0=0.3
1.2
c0=0.5
1.1
c0=0.7
c0=0.9
(Rgz/Rhz)
1.0
0.9
0.8
0.7
0.6
0.5
0.2
0.4
0.6
0.8
1.0
cA
Figure 4 Dependence of the ratio of z-average radius of gyration, Rgz, to z-average
hydrodynamic radius, Rhz, γ = Rgz/Rhz on conversion of A groups, cA, at various monomer
concentrations, c0, with the reactivity ratio k12 =1 for AB2 type polymerization.
1.2
1.1
 (Rgz/Rhz)
1.0
0.9
c0=0.1
0.8
c0=0.3
c0=0.5
0.7
c0=0.7
c0=0.9
0.6
0.5
0.2
0.4
0.6
0.8
1.0
cA
Figure 5 Dependence of the ratio of z-average radius of gyration, Rgz, to z-average
hydrodynamic radius, Rhz, γ = Rgz/Rhz on conversion of A groups, cA, at various monomer
concentrations, c0, for AB4 type polymerization.
1.4
k12=1/10
1.3
k12=1/3
1.2
k12=1
k12=3
 (Rgz/Rhz)
1.1
k12=10
1.0
0.9
0.8
0.7
0.6
0.5
0.2
0.4
0.6
0.8
1.0
cA
(a)
1.3
1.2
1.1
(Rgz/Rhz)
1.0
0.9
k12=1/10
k12=1/3
0.8
k12=1
k12=3
0.7
k12=10
0.6
0.5
0.2
0.4
0.6
0.8
1.0
cA
(b)
Figure 6 Dependence of the ratio of z-average radius of gyration, Rgz, to z-average
hydrodynamic radius Rhz, γ = Rgz/Rhz on the conversion of A groups, cA, at various reactivity
ratios, k12, with monomer concentration (a) c0 = 0.2 and (b) c0 = 0.7 for AB2 type
polymerization.
(a)
(b)
(c)
Figure 7 (a) Schematic illustration of the 3d bond fluctuation lattice model of H. Deutsch and
R. Dickman.2 (b) Conformation of an AB2 type monomer. (c) Conformation of an AB4 type
monomer.
In this model each ABn (n=2, 4) unit takes up eight lattice sites (one cubic cell). In Figure 7
(b) and (c), each bead represents a unit which can be classified as linear (orange), branched
(gray) and terminal (blue) units. The difference between AB2 and AB4 type monomers can be
seen from the gray unit surrounded by a black cycle. AB2 type monomer can connect with 3
other units while AB4 type monomer can connect with 5 other units.
Reference
1. Grosberg, A. Y.; Khokhlov, A. R., Statistical Physics of Macromolecules. AIP Press:
New York, 1994.
2. Deutsch, H.; Dickman, R. J Chem Phys 1990, 93, 8983-8990.