Phase Transitions and Relaxation Processes in Macromolecular Systems: The Case of Bottle-brush Polymers

Abstract

As an example for the interplay of structure, dynamics, and phase behavior of macromolecular systems, this article focuses on the problem of bottle-brush polymers with either rigid or flexible backbones. On a polymer with chain length $N_b$, side-chains with chain length $N$ are endgrafted with grafting density $\sigma$. Due to the multitude of characteristic length scales and the size of these polymers (typically these cylindrical macromolecules contain of the order of 10000 effective monomeric units) understanding of the structure is a challenge for experiment. But due to excessively large relaxation times (particularly under poor solvent conditions) such macromolecules also are a challenge for simulation studies. Simulation strategies to deal with this challenge, both using Monte Carlo and Molecular Dynamics Methods, will be briefly discussed, and typical results will be used to illustrate the insight that can be gained. Comment: 8 pages, 7 figures, to be published in the proceedings of "NIC Symposium 2010", eds. G. Muenster et al. (NIC, Juelich, 2010)

Full text

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arXiv:0910.4309v1 [cond-mat.soft] 22 Oct 2009
Phase Transitions and Relaxation Processes in
Macromolecular Systems: The Case of Bottle-brush
Polymers
Hsiao-Ping Hsu1, Wolfgang Paul1,2, Panagiotis E. Theodorakis1, and Kurt Binder1
1Institute for Physics,
Johannes Gutenberg University Mainz, 55099 Mainz, Germany
E-mail: {hsu, wolfgang.paul, theodora, kurt.binder}@uni-mainz.de
2Institute for Physics,
Martin-Luther University Halle-Wittenberg, 06120 Halle, Germany
E-mail: wolfgang.paul@physik.uni-halle.de
As an example for the interplay of structure, dynamics, and phase behavior of macromolecu-
lar systems, this article focuses on the problem of bottle-brush polymers with either rigid or
flexible backbones. On a polymer with chain length Nb, side-chains with chain length N are
endgrafted with grafting density σ. Due to the multitude of characteristic length scales and
the size of these polymers (typically these cylindrical macromolecules contain of the order of
10000 effective monomeric units) understanding of the structure is a challenge for experiment.
But due to excessively large relaxation times (particularly under poor solvent conditions) such
macromolecules also are a challenge for simulation studies. Simulation strategies to deal with
this challenge, both using Monte Carlo and Molecular Dynamics Methods, will be briefly dis-
cussed, and typical results will be used to illustrate the insight that can be gained.
1 Introduction
Theso-called“bottle-brushpolymers”consist of a longmacromoleculeservingas a “back-
bone” on which many flexible side chains are densely grafted1,2. Since the chemical syn-
thesis of such complex polymeric structures has become possible, these molecular bottle-
brushes have found much interest for various possible applications: the structure reacts
very sensitively to changes of solvent quality (due to change of temperature of the solu-
tion, pH value, etc.), enabling the use of these molecules as sensors or actuators on the
nanoscale3,4. For some conditions these bottle-brush molecules behave like stiff cylindri-
cal rods, and hence they can serve as building blocks of supramolecular aggregates, or
show orientational order in solution as nematic liquid crystals do. On the other hand, these
molecular bottle-brushes are also very “soft”, i.e, they show only very small resistance to
shear deformation; bottle-brush molecules of biological origin such as aggrecan which oc-
curs in the cartilage of mammalian (including human!) joints are indeed held responsible
for the excellent lubrication properties (reducing frictional forces) in such joints5.
However, for being able to control the function of these complex macromolecules one
must be able to control their structure, i.e., one must understand how the structure depends
on various parameters of the problem: chain length Nbof the “backbone”, chain length N
of the side chains, graftingdensity σ of the side chains along the backbone,solvent quality,
just to name the most importantones of these parameters. This is the reason why computer
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(a) (b)
(c)(d)
Figure 1. Snapshot pictures of simulated bottle-brush polymers with a flexible backbone containing Nb= 131
effective monomers. At each backbone monomer one side chain is grafted (grafting density σ = 1) and side
chain lengths are N = 6 (a), N = 12 (b), N = 24 (c) and N = 48 (d). The chains are described by the bond
fluctuation model on the simple cubic lattice (see Sec. 2) and the only interaction considered is the excluded
volume interaction between effective monomers.
simulations are needed in order to understand such bottle-brushes: even in the case of
good solvent, the structure of these objects is very complicated as the snapshot pictures6,7
(Fig. 1) of the simulated models show, and there occur a multitude of characteristic length
scales (Fig. 2)6,7that are needed to describe the structure; it is very difficult to extract this
information from experiment, and hence simulations are valuable here as they can yield a
far more detailed picture.
Of course, the large size of these bottle-brush polymers (even the coarse-grained mod-
els described in the following sections contain of the order of several thousand or even
more than 10000 effective monomers per polymer) is a serious obstacle for simulation,
too; in fact, developing efficient models and methods for the simulation of macromolecu-
lar systems is a longstanding and important problem8,9. We hence shall address this issue
of proper choice of both suitable models and efficient algorithms in the next sections.
2 Bottle-brush polymers with rigid backbones in good solvents
If one assumes the backbone of the bottle-brush to be completely stiff, the problem is re-
duced to grafting side chains of length N to a straight line. Although this limiting case
seems somewhat artificial, from the point of view that one wishes to model real systems, it
is a very useful test case: there is no reason to assume that approximationsthat already fail
in this rather simple limit become accurate for the more complicated case of flexible back-
bones; moreover, this case is rather simple to simulate, and the analysis of the simulation
data is relatively straightforward.
For the study of this case, polymers were simply represented as standard self-avoiding
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Re
lp
ls
L
R
Rcs
e,bb
cc
lb
Figure 2. Multiple length scales are needed for the structural characterization of a bottle-brush polymer: a coarse-
grained view describes this object as a flexible spherocylinder of length Lcc and cross-sectional radius Rcs,
which is locally straight over a length ℓp (“persistence length”). A further length scale of interest is the end-
to-end distance?Re,bbof the backbone chain. This coarse-grained view is experimentally obtained by atomic
force microscopy imaging of bottle-brush polymer adsorbed on substrates, for instance. A “microscope” with
resolution on the scale of atoms would see the monomers of the backbone chains (connected by backbone bond
vectors?ℓb) and monomers of the side chains (connected by bond vectors?ℓs). A mesoscopic length of interest
then is the end-to-end distance?Reof the side chains.
walks on the simple-cubic lattice (i.e., beads of the chains are occupied lattice sites, con-
nected by nearest-neighbor links on the lattice, and multiple occupancy of sites is forbid-
den8,9).
As a simulation method, pruned-enriched Rosenbluth methods (PERM)10,11were
used12. In the Rosenbluth method, all side chains are grown simultaneously step by step,
choosing only from sites which are not yet occupied, and the statistical weight of the poly-
mer configuration is computed recursively. In the PERM algorithm, one does not grow a
single polymer at a time, but a large “population” of equivalent chains is grown simultane-
ously but from time to time configurations with very low statistical weight are killed, and
configurations with large statistical weight are “cloned” (“go with the winners”strategy10).
The advantage is that (unlike dynamic Monte Carlo algorithms8,9) this method does not
suffer from “critical slowing down” when N gets large, and data for all N (up to the max-
imum value studied) are obtained in a single simulation (see11for more details).
Now onehypothesispopularin mostexperimentalstudies (e.g.13,14) is the factorization
approximation15for the structure factor S(q) (that describes the small angle scattering
intensity of neutrons, X-rays or light from dilute solutions of bottle-brushespolymers)into
a contribution due to the backbone (S(q)) and due to the side chains (Ss(q))
S(q) ≈ Sb(q)Ss(q) ≈ Sb(q)Scs(q)
(1)
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(a)
0.01
0.1
1
0.001 0.01 0.1 1
Sb(q) / Lb
q
Lb = 32
Lb = 64
Lb = 128
(b)
1
10
100
1000
10000
0.01 0.1 1
Ss(q)
q
q-1/0.588
σ=1, nc=128
nc= 64
nc= 32
σ=1/2, nc=128
nc= 64
nc= 32
(c)
0.001
0.01
0.1
1
10
100
1000
0.01 0.1 1
Sbs(q)
q
σ=1/2, Lb=256
Lb=128
Lb= 64
σ=1, Lb=128
Lb= 64
Lb= 32
(d)
0.01
0.1
0.01 0.1
q
1
q Scs(q)
Lb = 32
ρ(r) -> Scs(q)
Figure 3. a) Log-log plot of the normalized scattering function of the backbone Sb(q)/Lb, Lbbeing the back-
bone length, Lb= Nba where a(= 1) is the lattice spacing. Three choices of Lbare shown. b) Log-log plot of
the scattering from all the side chain monomers, for N = 50, and two choices of the grafting density c) Log-log
plot of the scattering due to interference contributions from monomers in side chains and the backbone (for the
same parameters as in b). d) Log-log plot of qScs(q) vs. q in the range 0.01 ≤ q ≤ 1, comparing the actual
scattering due to the side chains with the prediction resulting from Eq. (2), using ρ(r) as actually recorded in the
simulation12.
where q is the absolute value of the scattering wave vector. In the last step the side chain
scattering was approximated by Scs(q). This cross sectional scattering is in turn approx-
imated related to the radial density ρcs(r) perpendicular to the cylinder axis at which the
side chains are grafted as (the constant c ensures proper normalization12)
Scs(q) = c−1?|
?
d2? rρcs(r)exp(i? q ·? r)|2?.
(2)
Obviously, Eq. (1) neglects interference effects due to correlations between the monomer
positions in the side chains and in the backbone.
monomers in the side chains as the average of a square {Eq. (2)} ignores correlations
in the occupation probability in the z-direction along the bottle-brush axis. All such corre-
lations do contribute to the actual structure factor, of course, when it is computed from its
definition,
Writing the scattering due to the
S(q) =
1
Ntot
Ntot
?
i=1
Ntot
?
j=1
?c(? ri)c(? rj)?sin(q|? ri−? rj|)/(q|? ri−? rj|).
(3)
Here c(? ri) is an occupation variable, c(? ri) = 1 if the site i is occupied by a bead, and
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1e-04
0.001
0.01
0.1
1
0.1 1 10 100
S(q)
q Rg
experiment
simulation
Figure 4. Structure factor S(q) of bottle-brush polymer obtained13from scattering experiment mapped to the
simulated model (cf. text) by requiring that the total gyration radius Rg is matched, to fix the translation factor
for the length scale7.
0.001
0.01
10 8 6 4 2 0.5
ρcs(r)
r [nm]
Gaussian
simulation
Figure 5. Log-log plot of the density profile ρcs(r) vs. r, for the systems shown in Fig. 4. For fitting the
experimental data, a Gaussian form for ρcs(r) was assumed13.
zero otherwise, and the sums extend over all Ntotmonomers (in the side chains and in the
backbone).
In the simulation, one can go beyond experiment by not only recording the total scat-
tering S(q), but also the contributions Sb(q), Ss(q) and Scs(q) individually (Fig. 3). One
can see that the approximation S(q) ≈ Sb(q)Ss(q) leads to a relative error of the order
of a few % only (cf. the difference in ordinate scales between parts b) and c), while the
assumption Ss(q) ≈ Scs(q) [with Eq. (2)] leads to appreciable errors at large q (see part
d)). Thereforethe use ofEqs. (1),(2)to analyzeexperimentscanlead toappreciableerrors,
when one wishes to predict12the radial density distribution ρ(r).
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1/2
N
1/2
N
Figure 6. Schematic diagram of states of a bottle-brush polymer with a rigid backbone under poor solvent condi-
tions in the plane of variables x = N1/2σ and y = N1/2(1 − T/θ), as proposed by Scheiko et al.19. The lines
indicate (smooth) crossovers between the different states19, which we have characterized by snapshots from our
simulations20. For further explanations see text.
3 Bottle-brushes with flexible backbones in good solvents
In the case of flexible backbones, the PERM algorithm no longer provides an efficient
sampling of configuration space; there “dynamic” Monte Carlo algorithms are used, but
with “unphysicalmoves” allowing chain intersections and large configurationalchanges in
a single step, as achieved by the Pivot algorithm8,9. Using the bond fluctuation model8,9,
the “L26 algorithm”16was used17for local moves allowing bond intersection and never-
theless respecting excludedvolume. In this way well-equilibrated data for systems with up
to Nb= 259 monomers at the backbone and up to N = 48 monomers in the side chains
could be studied (for σ = 1). Examples of snapshot pictures have already been given in
Fig. 1.
Adjusting the physical meaning of the lattice spacing to correspond to 0.263 nm, the
structure factor of the simulated model matches almost perfectly an experimental result13
(for slightly different numbers of chemical monomers, Nexp
ference is irrelevant, since there is no one-to-one correspondence between covalent bonds
and the “effective bonds” of the model) see7Fig. 4. From the simulation one can directly
extract the cross-sectional density profile ρcs(r) and compare it7to the approximateexper-
imental result (Fig. 5), which was obtained from fitting S(q) to Eqs. (1), (2)13. One sees
that the analysis of the experiment could predict roughly correctly the distance on which
the profile ρcs(r) decays to zero, but does not account for its precise functional form. A
further interesting finding7,17is the result that the persistence length ℓp(cf. Fig2) depends
strongly7on Nb(at least if one uses the textbook definitions18), and hence is not a useful
measure of local chain stiffness.
b
= 400, Nexp= 62; this dif-
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1
10
0.01 0.1 1
σλ
σ Rg
0
0.2
0.4
0.6
0 0.1 0.2 0.3 0.4
S(q)
q
σ=0.38
σ=0.57
σ=0.76
σ=1.14
σ=1.51
Figure 7. Log-log plot of wavelength λ (dimensionless by normalization with σ) versus grafting density σ (di-
mensionless by normalization with the radius of gyration of the side chains, Rg). The wavelength was extracted
either from the z-dependence of a normalized correlation function at radial distance r ≈ 3 Lennard-Jones units
σ (stars) or extracted from the peak position of its Fourier transform S(q) (squares) [inset]. All data refer to
temperature T = 1.5 (in Lennard Jones units). Blue and red symbols refer to N = 35, orange and green ones to
N = 50. Inset shows S(q) vs. q for N = 35 and grafting densities σ ≥ 0.38.
4 Bottle-brushes in poor solvents
While isolated single chains in dilute solution collapse to dense globules18when the tem-
perature T is lowered below the Theta temperature T = θ, for a bottle-brushthe constraint
of the chains being grafted to a backbone leads to an interesting diagram of states, consid-
ering 1 − T/θ and σ as variables (Fig. 6)19. Near T = θ and σN1/2→ 0 the side chains
collapse to dense globules when N1/2(1 − T/θ) ≫ 1. For T ≈ θ and N1/2σ ≫ 1 one
has a cylindrical structure (as in the previous sections) which collapses to a dense cylinder
when N1/2(1 − T/θ) ≫ 1. However, interestingly at intermediate grafting densities and
N1/2(1−T/θ) > 1 a laterally inhomogeneous“pearl necklace-structure”was predicted19.
This problem can neither be studied efficiently by the PERM method nor by the bond
fluctuation algorithm of Sec. 3 - both methods get very inefficient for dense polymer con-
figurations. Thus, instead Molecular Dynamics simulations of the standard Grest-Kremer-
type bead-spring model8,9were carried out20. Fig. 7 shows20characteristic wavelength λ
plotted vs. grafting density σ. One finds that for small σ a trivial periodicity given by 1/σ
occurs, while for σRg≥ 0.2 (Rg= ?R2
σ, given by about that λ ≈ 8Rg. Of course, mapping out the phase behavior precisely for
a wide range of N, σ and T is a formidable problem, due to very long relaxation times of
the collapsed states in Fig. 6.
g?1/2) the periodicity is essentially independent of
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5Concluding Remarks
The above examples have illustrated that polymers with complex architecture pose chal-
lengingproblemswith respect totheir structureand thermodynamics. Investigationoftheir
relaxation behavior is another interesting problem, to be tackled in the future. The above
exampleshavealso illustratedthe needto carefullyadjustboththemodelandthe algorithm
to the problems one wants to deal with.
Acknowledgments
One of us (H.-P. Hsu) received support from the DFG (SFB625/A3), another (P. E.
Theodorakis) from the MPG via a Max-Planck Fellowship of the MPI-P. We are grate-
ful to the NIC J¨ ulich for computer time at the JUMP and SoftComp cluster.
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