Coarse graining

We describe a systematic approach to renormalise the intra-chain interactions towards a coarser level for three different polycarbonates [82]. The three modifications of the basic polycarbonate structure are BPA-PC, BPZ-PC, and TMC-PC (Figure 16). Although the backbone sequence is the same these have remarkably different physical properties. For the first two (BPA and BPZ) the glass transition temperature TG is roughly the same (TG 420K), while the third one has a glass transition temperature which is about 80 to 100 K higher, around 500 K. On the other hand, BPA-PC is ductile while BPZ-PC is brittle; TMC-PC is less ductile than BPA but much less brittle than BPZ. This is also reflected in the difference of the generalised activation energy within a Vogel-Fulcher fit (see Kob, this volume, and Equation 24 below): BPA and TMC have roughly the same activation energy while BPZ has a significantly higher one. Not only are the glass transition properties different, but also the entanglement chain lengths Ne in the melt are significantly different. For BPA-PC an extremely short entanglement length of Ne ⁼ 7 monomers is reported. This length increases through Ne = 9 monomers (BPZ) to Ne ⁼ 14 monomers for TMC-PC. In particular, the extremely short entanglement length for BPA- PC is not understood. Considering other well-studied polymers like polyethylene or PDMS (polydimethylsiloxane), one would expect it to be larger by a factor of at least 5 to 10.

Whether this is the result of a special local chain structure (banana shaped repeat units joined by almost pivot-like junctions) is beyond the scope of the present discussion, but is a matter of current studies.

172 Kurt Kremer

Figure 16. Three modifications of BPA repeat units tested for the coarse graining proce- dures. From top to bottom: BPA-PC, BPZ-PC, TMC-PC.

The coarse graining procedure is explained in detail here for BPA-PC. Ideally, the method is parameter free and as simple as possible. In addition we would like to stay as close as possible to the chemical structure in order to be able to reintroduce the chemical details later without too many problems. The coarse-grained monomers have to be designed to be easily be identified with specific chemical groups of the polymer itself. Considering the chemical structure of the three different polycarbonates a ^{2 : l} mapping onto spherical beads seems to be a first reasonable choice, as illustrated in Figure 17. The resulting coarse-grained structure then only has four relevant internal degrees of freedom: the bond length C between carbonate and isopropylidene group, a, the carbonate-isopropylidene-carbonate bond angle,

p,

the isopropylidene-carbonate- isopropylidene bond angle and 19, the torsion angle.

To arrive at the coarse-grained interactions from the microscopic model one can imag- ine a number of empirical fitting procedures. Here, we follow a different route. The coarse- grained potentials not only have to include energetic aspects of the microscopic model but also entropic terms from the different possibilities of local conformations. We first use intra-chain distribution functions to construct the bonded potentials in the coarse-grained model. Knowing the potential functions of the detailed chemical system, it is rather straightforward to perform an MC simulation to a very high accuracy at a given temper- ature of the conformations of individual free random walks. The probability distribution functions of conformations of such a model system are only dependent on temperature and originate from the bonded interactions along the backbone of a chain. The potentials for the microscopic models are derived from ab initio quantum chemistry calculations. Using the microscopic model to generate configurations, we sample the probability distribution function P(C, CY,

p,

²⁹⁾ for the coarse-grained model in the limit of single isolated random walks.

The coarse-grained distribution function is temperature dependent via the Boltzmann weights of the different states of the microscopic model. The most crucial assumption now is, that the distribution function of the set of variables factorises into independent

Computer simulations in soft matter science 173

Figure 17. Illustration of the mapping procedure for a 2:l mapping where the repeat unit of a BPA-PC chain is replaced b y two monomers of a generalised bead spring chain. The geometrical centres of the carbonate group and the geometrical centre of the isopropylidene group respectively are mapped onto the centres of the new spherical bead [82].

distribution functions of the individual variables:

P(4 ^{Q l}

P ,

8 ) = P ( ~ ) P ( ~ ) P ( B ) P ( ~ ) ‘

The distribution function P is determined at each temperature separately, and may be written

U _is a generalised potential function at that temperature, already expressed in units of k B T . (In effect, this allows one to keep the simulation temperature at k B T = 1, which is of technical advantage for molecular dynamics simulations.) From Equation 22 we get the forces

d Fe = --lnP(l) de F,, = -- d l n P ( 0 ) . . .

da (23)

as they originate from the conformations of the coarse-grained model. This approach avoids the fitting of a functional form of the coarse-grained potential functions to the microscopic parameters. The only fitting procedure is a smoothing for potentials in order to get rid of the scatter in the sampling and to stabilise the resulting force. There is no need to determine the partition function explicitly, since it only shows up as a constant in the potential and thus does not alter the resulting forces.

Using this model we can now simulate dense polymer systems. The volume of the effective hard-sphere beads of the coarse-grained model is adjusted to give the same Van- der-Waals volumes as in the experimental case (normalised to the simulation density). In

174 Kurt Kremer

the case of BPA-PC the carbonate group and the isopropylidene group are represented by spheres with a radius of 3.02A and 3.11A respectively. No further specific excluded volume interactions, nor any directional interaction is taken into account. The simulation density is adjusted to the experimental mass density in every case, and then the volume held constant; after this, there is no freedom left to adjust parameters.

If this procedure is to reproduce the essential aspects of the different chemical species, then not only should the static structure which comes out of this simulation compare well to the experimental systems, but also the dynamics. A detailed discussion of the dynamic properties is given elsewhere, but note that for the range of temperatures which we are investigating here, it seems reasonable to assume that the simulation time scales linearly with the physical time. (Possible deviations originating from the different shape of the potentials instead of taking different temperatures are neglected at this stage.)

MD melt simulations were performed as described before (Equation 19). The excluded volume interactions of the monomers are taken into account through a repulsive Lennard- Jones interaction. For the present system the background friction

r

is about 100 times weaker than the monomer-monomer friction. For static properties, the mapping between simulation and physical units requires a length, fixed by equating the the mass density in simulation and experiment. Starting from a mass density of ppc ⁼ 1.05g/cm3 (BPA-PC at 500 K) and the simulation number density of p i l . i ~ = 0 . 8 5 ~ - ~ , we arrive at a length scaling of ^{F =} 5.56A for the present case.

To compare the dynamic properties, a time scale is also required. This is found by using the Rouse model to calculate the melt viscosity in terms of the centre-of-mass diffusion constant of the chains, and equating this to the observed viscosity at some reference temperature. This is possible as we have one case where the highest experimental temperature and the lowest simulation temperature coincide. For the present example one gets ^{T =} 2.21 x lo-'' seconds, where ^T is the simulation (Lennard-Jones) time unit. The simulation time step is typically 6t = 0.017. However, the absolute comparison of dynamic quantities should only be taken as indicative since the experimental systems and the simulations systems comprise different chain lengths, and also the effect of polydispersity might alter this absolute scale by some prefactor.

Compared to other molecular dynamics simulations of microscopic models, the sim- ulation time step is roughly three orders of magnitude larger than usual. Taking the simplicity of the potentials and the short range nature of the interactions into account the resulting speed-up is of order lo4. The simulated systems typically comprised be- tween 1000 and 10000 model monomers on chains of 20 or 60 model monomers. For our cubic simulation box this means that one can easily simulate systems of up to 125A3. As it turns out, the inter-chain interactions strongly modify the angular distribution func- tions compared to the isolated chain. These and other static properties are discussed in Section 7.2 below, where chemical detail is reintroduced.

For the coarse-grained model we first check the dynamical properties as a function of temperature, especially approaching the glass transition temperature. The properties of many materials when approaching the glass transition temperature are well described by the so-called Vogel-Fulcher behaviour

D = Doexp

(--)

^A0

T - TVF

Computer simulations in soft matter science 175

Simulation (N=20) Experiment

A0

Simulation (N=20) Experiment

given here for the tracer diffusion constant D of the chains: ^A0 is a generalised activation energy and T ~ F , _the so called Vogel-Fulcher temperature, typically is about 80 degrees below the calorimetric glass transition temperature. The prefactor

Do

_{is a} hypothetical high temperature diffusion constant. For the present situation

D O

is easy to determine, because it simply corresponds to the freely jointed polymer melt with athermal excluded volume, and all the chemistry dependent intra-molecular interactions set to zero. (Exten- sive computer simulations are available for that case.)

Figure 18 gives a Vogel-Fulcher plot of the three polycarbonate modifications. The

322 407 292

387 477 392

BPA-PC TMC-PC BPZ-PC

1305 1363 1443

1012 1073 1534

0.6

0.4

0.2

o'oO.O 200.0 400.0 600.0 600.0 10W.O 1200.0 T/K

Figure 18. Vogel-Fulcher plot of the chain diffusion constants D for the three diflerent polycarbonate modifications, as indicated in the figure, for N = 20 model monomers [82].

results qualitatively match the experimental situation, namely that the Vogel-Fulcher temperature for TMC-PC is about 80-100 degrees above the Vogel-Fulcher temperature of BPZ-PC and BPA-PC while the generalised activation energy, which in Figure 18 is the slope of the lines, is roughly the same for BPA and TMC but is different for BPZ-PC.

Even quantitatively the results are not that different from the typical experimental value as Table 2 shows.

I

TVF

I

BPA-PC

I

TMC-PC

I

BPZ-PC

1

Table 2. Activation energies A0 (below) and Vogel-Fulcher temperatures T ~ F (above) for experiment and simulation. While the shift for TVF (simulated) is consistent with expectations, the deviations (about 30%) for the experimental determination of A0 are probably due to the large polydispersity of the typical commercial samples.

176

20.0

-

15.0

-

n D

z

10.0

-

5.0

-

Kurt Kremer

25.0

+ ,

- Simulation

0

ExDeriment 4

I

+AmorPous Cell 1

0.0 0.5 1 .o 1.5 2.0 2.5

Figure 19. Coherent structure function S(q) in absolute units in comparison to amor- phous cell simulation and neutron scattering data [82, 841.

In document - - - SOFT AND FRAGILE MATTER (pagina 176-181)