Effects of the macromolecular architecture
on polymer conformations and dynamics
Fabio Ganazzoli
Dip. Chimica, Materiali e Ing. Chimica “G. Natta”
Politecnico di Milano - Italy
MIPOL 2017
16th February 2017
Overview
§  Introduction: macromolecular topology
§  Equilibrium properties
§  Dynamical behavior
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
2
Overview
§  Introduction: macromolecular topology
§  Equilibrium properties
§  Dynamical behavior
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
3
Introduction: macromolecular topology - 1
§ Use of statistical mechanics to calculate average properties of
macromolecules.
§ Many properties are universal and only depend on the number
of N repeat units (chemical atoms for chemists, beads or
segments for physicists): thus we often obtain universal
power laws as a function of N for static properties (molecular
dimensions), or of time for dynamic properties.
§ Variables affecting the macromolecular behavior: molar mass,
temperature, solvent, external forces (e.g., shear rate), pH,
pressure (most often ignored), …
§ A further variable: polymer topology
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
4
Introduction: macromolecular topology - 2
§  Molecular topology is related with the presence of units with
an unlike functionality (e.g., chain ends, branch points)
§  All these topologies can be synthetized with an excellent
control of the molecular architectures
§  Dendrimers however stand apart, having some peculiar
properties, besides the large number of peripheric units
(which can be functionalized!)
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
5
Introduction: macromolecular topology - 3
§  Simplest model: freely jointed chain, or a random-walk
2
Rg
R 2g
=
lin
star
Nl2
=
/6
3f − 2
f2
(for N → ∞)
R 2g
lin
2
Rg
1 2
Rg
=
2
lin
ring
Ring and star are more compact,
but have the same N-dependence
(f − 1)G +1 − 1
Dendrimer: N increases exponentially with G: N = f
f −2
G = 2,
f=3
MIPOL 2017 - 16th February 2017
If N → ∞ i.e., G >> 1:
R 2g = (G + 1) l2 ∝ ln N
Fabio Ganazzoli
6
Introduction: macromolecular topology - 4
Rg ∝ N1/2
Rg ∝ (ln N)1/2
d = N / R g3
∝ N / N3/2 = N-1/2
d = N / R g3
∝ N / (ln N)3/2
Hence
d → 0 for N → ∞
Hence
d → ∞ for N → ∞
Dendrimers: the intramolecular density diverges with an
increasing molar mass ⇒ there is a limiting value for G
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
7
Overview
§  Introduction: macromolecular topology
§  Equilibrium properties
§  Dynamical behavior
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
8
Equilibrium properties – excluded volume 1
§  In a good solvent, the chain shows an
osmotic swelling compared to the
random walk
§  The result can be described as a Self-Avoiding Walk (SAW)
due to net repulsive interactions among repeat units
Two-body interactions
Three-body interactions
repulsive in a good solvent,
attractive in a poor solvent
(depending on T)
always repulsive
(red and blue bead cannot
freely approach the grey one)
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
9
Equilibrium properties – ideal Θ state – 1
§  The Θ state is achieved when A2 = 0 for a vanishing
interaction between two molecules.
Branched polymers: connectivity yields
more repulsive 3-body interactions
⇒ lower T to enhance (weakly) attractive
2-body interactions
Star polymers
Θ = Θ∞ – ϕ⋅χf (N/f)–1/2
1
0.8
Θ
2
2
8 10
15
g S = 〈R S 〉star/ 〈R S 〉lin
0.6
0.4
gS
Θ state, PS
Θ state, PE
Random Walk
PS
PIP
PE
Θ
0.2
2
4
6
20
f = no. of arms
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
10
Equilibrium properties – ideal Θ state – 2
Dendrimers have a larger multiplicity
of interactions
G=2
● ○ PE-like
▲r
PS-like
More generally: what is the ideal, or the Θ,
or the unperturbed state with branched polymers?
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
11
Overview
§  Introduction: macromolecular topology
§  Equilibrium properties
§  Dynamical behavior
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
12
Polymer dynamics – 1
§  Equilibrium thermal fluctuations at equilibrium
Coarse-grained Rouse Zimm model
(bead-and-spring)
§  masses and friction with the medium
are concentrated on the beads
§  Hookean (harmonic) springs allow
transmission of forces
§  Over-damped motion: inertia terms are neglected
§  Dynamic equations are decoupled through a transformation
to normal modes (eigenvectors) characterized by
characteristic relaxation times (eigenvalues)
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
13
Polymer dynamics – 2
§  The normal modes are "labelled" by an index p = 0, 1, 2, …
describing the coordinated (in phase) motion of ∼ N/p units
§  The normal modes are independent and relax exponentially
with time through a relaxation time τp ∝ (N/p)β
§  Any molecular symmetry leads to some degeneracy of
the relaxation times, dictating the dynamical behavior
(rheological properties)
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
14
Polymer dynamics – 3
§  In linear chains τp ∝
(N/p)β
10000
1000
100
τp
10
1
1
R 2g
10
100
p
∝ N 2ν
for linear, star and ring polymers
RT
τ p ∝ Nβ-1
Whence the intrinsic viscosity [η] =
∑
2Mηs p
§  Ring chains have only the even modes
(p = 2, 4, 6, …) with two-fold degeneracy
because of its topology
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
●
●
15
Polymer dynamics – 4
§  Star polymers: same relaxation times as in
the linear chain formed by two arms only, but
i) even (symmetrical) modes = concertated
motion of the arms, unit multiplicity;
ii) odd (antisymmetrical) modes = independent
motion of the arms, multiplicity f – 1.
Star with
f = 5 arms
§  Dendrimers:
increasing multiplicity
at more local scale
(larger symmetry)
G=2
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
16
Dynamical behavior – Complex modulus
§  Under an oscillating shear deformation, we get the complex
modulus G’ (elastic response) and G” (dissipative response)
2
(ωτ 2)
[G’] = ∑
1 + (ωτ 2)
p
log [G’], log [G”]
p
a)
log ωτ1
Star
f=5
MIPOL 2017 - 16th February 2017
∑ 1 + (ωτ 2)
2
p
log [G’], log [G”]
[G”] =
2
p
ωτ p 2
p
G=6
b)
log ωτ1
G=2
Fabio Ganazzoli
T = Θ ----good solvent
17
Dynamical behavior – Quasi-elastic scattering - 1
§  It provides space and time information
§  q = |q| = 4π sin(ϑ/2) / λ = modulus of the scattering vector,
(ϑ = scattering angle, λ = wavelength), q-1 ≈ probed distance
§  With neutrons, we can get the the incoherent dynamic
structure factor S(q, t) due to H atoms:
1
exp {− iq ⋅ [r j (t ) − r j (0 )] }
S(q, t) =
∑
N +1 j
(
≅ exp − q 2Dt
where
1
⎡ q 2 2 ⎤
exp⎢−
r jj (t ) ⎥
∑
N +1 j
⎣ 6
⎦
)
[r (t ) − r (0)]
2
j
MIPOL 2017 - 16th February 2017
j
2
r
= 6 D t + jj (t )
Fabio Ganazzoli
18
Dynamical behavior – Quasi-elastic scattering - 2
Intramolecular
(segmental) dynamics
1
Diffusion of c.o.m.
q
t
§  Characterization:
∂ ⎛ S (q, t ) ⎞
o  First cumulant Ω(q) = − ln⎜
⎟
∂t ⎝ S (q,0 ) ⎠ t →0
o  Fit of the scattering curves with stretched exponentials
Sinc(q,t) / Sinc(q,0) = exp[-(t/τ)β]
where:
0.5 < β < 1 (= diffusion)
τ = τeff q-α
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
19
Dynamical behavior – Quasi-elastic scattering - 3
§  Experimental data obtained from molten linear and cyclic
PDMS above Tg fitted with Sinc(q,t) / Sinc(q,0) = exp[-(t/τ)β]
o  Scattering curves due to methyl rotation and to
segmental dynamics with β ≅ 0.62 – 0.65
o  larger τeff for cyclic chains compared to linear chains
(τ = τeff q-α)
80
From incoherent
QENS data, V. Arrighi
et al., unpubl. results
t eff (ps)
60
40
20
linear PDMS
cyclic PDMS
0
0
2000
4000
6000
M w (g/mol)
MIPOL 2017 - 16th February 2017
8000
Fabio Ganazzoli
10000
20
Dynamical behavior – Quasi-elastic scattering - 4
§  Theoretical results were obtained for a FJ (freely jointed) chain
and realistic PDMS using the RIS method (Rotational Isomeric
States), fitting the calculated Sinc(q,t) / Sinc(q,0) with the
stretched exponential exp[-(t/τ)β]
o  For the segmental dynamics
•  FJ β ≅ 0.5; PDMS β ≅0.65 (slightly less for small rings)
o  larger τeff for cyclic chains compared to linear chains
- Constraint of ring closure ⇒ slower relaxation
- Further constraint of conformational rigidity
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
21
Conclusions
§  Branched polymers are quite common, and the
branching topology affects many important properties
§  Model branched chains (e.g. rings or regular stars) have
an aesthetic appeal, and can be synthetized with an
excellent control of the molecular architecture
§  These regular molecular architectures may have
practical uses, but basically they provide a stringent test
of theoretical approaches
§  Dendrimers have aroused great interest in the nanofield due to the possibility to incapsulate suitable hosts
end/or to functionalize the vast number of free ends
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
22
Acknowledgments
I would like to thank
§  prof. Giuseppe Allegra (Polimi)
§  prof. Giuseppina Raffaini (Polimi)
§  dr. Roberto La Ferla (Bracco)
§  prof. Valeria Arrighi (HW University, UK)
Thank you for your attention!
MIPOL 2017 - 16th February 2017
Fabio Ganazzoli
23