Kurt Kremer
(4) Addition and subtraction of these two equations yields
4 Polymer dynamics
To obtain information about dynamics, we have to follow the slow, physically realistic, simulation path. There are many tricks to vectorise or simplify the algorithms, but these only influence the prefactors of the power laws shown in Table 1. Faced with a naturally very slow process, we are forced to employ extremely simple models.
To test the applicability of any chosen method we need a basic model for dynamics, the Rouse model [23]. This is still the only model on a ‘molecular level’, which can be solved analytically. The Rouse model treats the dynamics of a Gaussian random walk in the overdamped (Brownian) limit. All the complicated inter-chain and intra-chain interactions are summarised in the viscous and random forces from the heat bath. Thus, we totally disregard excluded volume and topological constraints beyond the plain chain connectivity. This model, which represents the dynamics of molten but unentangled (short) chains can readily be solved analytically (see McLeish, this volume).
Here we recap the essentials. With ri the position of the i-th monomer, the Rouse
160 Kurt Kremer
I
10’’ 1 oo 4 1
Figure 7. Static structure function for N = 100 at the collapse transition point for the simple
PE0
model. The bigger spheres denote the hydrated monomers. For the expanded configurations these are equally distributed all over the chain, while for the collapsed ones only a few surface sites remain available to build up a solvation shell. The asymptotic slopes for the two cases of q-1/0.588 (swollen coil) and q-4 (collapsed globule) are indicated by straight lines; from[do].
equation of motion is
(15) dr .
C L = -
dt 4 2 r i - ri+l-
ri-l)+
f i ( t ).
Here
C
is the monomeric friction constant, K the bond spring constant and f i ( t ) the random (heat bath) force with ( f i ) = 0, and (via the fluctuation dissipation theorem)( f i ( t ) f j ( O ) ) = 2CT6(t)&. In the discrete version, the Rouse amplitudes X,, are given by
with relaxation times (for large N )
Eigenmodes decay exponentially, ( X p ( t ) X , , ( 0 ) ) / ( ~ X ~ ~ ) = e x p ( - t / r p ) , resulting in an over- all chain diffusion constant D = kBT/CN and a single chain dynamic structure function S(q, t ) scaling as In(S(q, t)/S(q, 0) )
c x
-q2&/6.The most natural quantity to measure from a simulation is the mean square displace- ment of the individual monomers, g l ( t ) = (Iri(t)
-
r,(O)l*). One finds (see Figure 8)Computer simulations in soft matter science 161
A
TIN: Tl N2 T,N?N,
In t
Figure 8. A sketch of the monomer mean square displacements for the standard Rouse model and for the reptation model. The time and length scales are indicated for the reptation case.
while the centre of gravity of the chain always follows r h ( t ) 0: t .
This model is a basis for the dynamic interpretation of stochastic algorithms. Besides the standard requirements for MC procedures (detailed balance etc.) we need the following property (for an extensive discussion of details see [3]): to simulate polymer dynamics, any algorithm with stochastic (or other artificial) dynamics must involve only local moves, and must yield Rouse dynamics for Gaussian random walks. Below we concentrate on lattice models; however, the generalisation to continuous space is obvious.
f f
a--.&*--. e---.
Figure 9. Typical kink jump moves for a dynamic Monte Carlo simulation.
The standard procedure is the kink jump method, Figure 9, as follows: (i) select monomer at random; (ii) select trial local move at random; (iii) check the move is topo- logically permitted; (iv) implement a Metropolis check; (v) decide on acceptance; (vi) start over again.
In a simulation we are mainly interested in SAWS, but to check the validity of a method for dynamics we need to recover Rouse behaviour for ideal, Gaussian chains. To
162 Kurt Kremer
check a given set of local move rules for the SAW, the test for Rouse behaviour is actually performed on non-reversing random walks (NRRWs). NRRWs are walks which do not allow direct back-folding but do not have any long range excluded volume interaction.
(Such walks are ideal Gaussian coils on large length scales, but have the same local conformations as SAWS and consequently the same set of local moves.)
To achieve Rouse dynamics, the set of local moves must contain moves which create new bond vectors {b,} inside the chain. In Figure 9, there are 2-bond moves and 3-bond moves. The 2-bond moves only exchange positions of bonds along the chain but do not change the set of bond vectors. For rings, an algorithm only containing such moves would never equilibrate, while for open chains new bond vectors could only diffuse in from the freely rotating chain ends. This would increase the relaxation time at least by a factor of N . Local rotations, such as the 3-bond move, are needed, so that {bi} is not conserved and Rouse behaviour recovered. The minimal number of bonds involved in d = 3 varies from 2 in continuum or on the fcc lattice to 4 on the diamond lattice. But in d = 2, for the 180" rotation of the 3-bond move we again only exchange bonds. To circumvent this difficulty, one can use off-lattice models, where both the bond lengths and angles vary. For such a model, the excluded volume constraints are usually time consuming to evaluate. Distances have to be calculated at every time step between several monomers.
This results in the loss of the main advantage of the MC methods compared with the MD.
A now frequently used alternative approach is the bond fluctuation (BF) method of Carmesin and Kremer [41], which combines advantages of lattice simulations and contin- uous space. Figure 10 illustrates the method for d = 2 and d = 3. Each monomer consists
Figure 10. An illustration of the bond fluctuation model f o r a 2-d branched polymer and for a 3-d linear polymer. Typical elementaw moves are indicated. Each monomer is represented by a square (cube) of four (eight) occupied sites on a 2-d (3-d) unit lattice.
of 2d lattice sites. In addition to the excluded volume interaction, the bond length 1 is restricted to a maximum extension to avoid bond crossing. On the square lattice, one has the constraint that 1 < 1 <
fl.
For d = 3, the situation is slightly more com- plicated. In this case a set of 108 different bonds are allowed [42, 43, 441. Since eachComputer simulations in soft matter science 163
monomer occupies 2d sites, but every jump only requires 2d-' empty sites, the method works effectively at high densities. (It also suffers less from non-ergodicity problems than the standard methods do.) For the 3-d study of the dynamics of polymer melts, densities as large as 0.5 were used, although higher densities are no problem. This corresponds to a very high physical chain density. Skolnick et al. [45] find for SAWS on a simple cubic lattice at
4
= 0.5 a static screening length for the excluded volume interaction of about 12 monomers, indicating that they are in a semi-dilute regime, whereas Paul et al. [46]find for BFM, and
Q
= 0.5 ( d = 3) a value of about 2. On a scale larger than a trimer the excluded volume interaction is already screened as it is in a polymer melt (see McLeish, this volume).A first serious application was the study of a 2-d melt of chains [47]. There one would expect the chains to segregate for entropic reasons [25]. In a Monte Carlo study using the bond fluctuation model chains of up to N = 100 monomers at a density of up to 80% occupied lattice sites were investigated. Figure 11 shows the segregation procedure clearly.
1 0 0
8 0
60
4 0
20
20 4 0 6 0 80 I [
Figure 11. A 2-d p o l y m e r m e l t with N = 100 and @ = 0.8 f o r a n almost completely equilibrated sample [47,l.
More interesting, however, is the dynamzcs of the d = 2 melt. The chains cannot cut through each other; on the other hand, they are compact objects with ( R 2 ( N ) ) 0: N cx Rd.
One finds a typical 2-d soft sphere liquid with an average of 6 chains surrounding a given chain. The Rouse model does not take into account any constraints on motion besides the connectivity. A first test is to measure the Rouse mode relaxation spectrum. The autocorrelation function for each mode shows an almost perfect single exponential decay.
On the other hand we can also measure the diffusion constant
Do
of the chains directly from the mean square displacements. Each test yields an independent estimate of the164 Kurt Kremer
monomeric friction constant
C,
which should be consistent, if the Rouse model describes the motion. For d = 2, however, one finds that >CO,
and that (mode is governed by the conformational relaxation of the chain, whileDo
is controlled by motion of the chain without really reorganising its internal conformational structure. Recently this picture was impressively supported by experiments on 2-d confined DNA molecules [48]. An extension, which puts these results into a much wider perspective, is 2-d polymer glasses [49]. It could be shown, for this case, that time temperature superposition (see McLeish, this volume) directly followed from the Rouse dynamics.More relevant to most actual experiments, of course, is the study of 3-d melts, where one wants to test the concept of chain reptation. Besides standard kink-jump and BF MC, another alternative is to perform a MD simulation. Simulations using the MD method usually employ the bead-spring model [50, 91, with, in addition, each monomer weakly coupled to a heat bath. (Technically, this is a hybrid method or ‘noisy MD’.) Each polymer chain consists of N monomers of mass m connected by an anharmonic spring.
The monomers interact through a repulsive Lennard-Jones potential given by Equation 1 with r, = 2If6a. For connected monomers we add an attractive interaction
For melts the parameters K = 30e/a2 and
&
= 1.50 usually are chosen, while in solution softer potentials are mostly used [9].In this algorithm, the equation of motion for monomer i is
Here
r
is the bead friction which acts to couple the monomers to the heat bath, and Wi(t) describes the random force actingon each bead, with (Wi(t)W,(t’)) = &,b(t-t’)!2k~T!?.Strict MD runs for polymers typically exceed the stability limits of a microcanonical simulation; these small damping and noise terms extend stability and define a canonical ensemble. We have used
r
= 0 . 5 ~ ~ ’ and T = 1.06 in most cases and m = 1. This value of I? is large enough to stabilise the run but small enough not to produce in itself Rouse-like behaviour on length and time scales of the order of a few bond lengths. Thusr
and Ware not be confused with
C
and f in the Rouse Equation 15 for a single chain; in this simulation, the Rouse friction is dominated instead by collisions with other chains. The program can be vectorised [51, 521 or parallelised [53]. Most older runs used a predictor- corrector scheme and Gaussian random numbers. One can however use equally distributed random numbers, so long as these have the correct mean value and second moment. The use of a Verlet algorithm then allows for time steps as large as 0.0127- yielding up to about 300000 particle timesteps per second for a typical vector processor or more than 100000 particle steps per processor or on a Cray T3E. With this method a huge variety of systems have been studied. Most recently a melt of chains of length N = 10000 was simulated [54, 551.Computer simulations in soft matter science 165
4.1 MC versus MD for melts
Which method is best to use for studying the dynamics of dense polymeric systems? I think that the choice should be between the BF Monte Carlo method and MD, or variants of these two approaches. For the athermal case (hard core repulsion only) we can compare the CPU time to reach the crossover time for entanglements -re (see Section 4.2 below) as a measure of the relative speed of the algorithms. By doing this we estimate that BF Monte Carlo is somewhat faster than MD on vector computers. However the inclusion of soft interactions has a much stronger effect on slowing down BF than it does for MD.
(The main speed advantage of the MC comes from the simple acceptance test for the moves compared to the more complicated force calculations of MD.) Which method to use therefore depends on the particular question under consideration, whether it is better to work in the continuum (for example, to study shear flow), and whether it is acceptable to have stochastic dynamics on all time scales: using MD, the Rouse behaviour for short chains (or early times) is a consequence of the interactions, while it is built in explicitly in MC. If one is interested in the behaviour of gels or polymer networks under swelling or elongation, or the forces between polymer brushes [56, 571, then a continuum simulation using MD is probably more appropriate; generally speaking, the MD method is more flexible. Note also that with increasing computer power, workstations are becoming more and more important. They typically do not take advantage of especially fast integer arithmetic, reducing even further the advantage of the BF MC approach compared to continuum methods.
4.2
The dynamics of polymer melts is observed experimentally to change from an apparent Rouse-like behaviour to a dramatically slower dynamics for chains exceeding the charac- teristic length N e . (For longer chains, one observes a much slower diffusion, D K N - 2 , and an increased viscosity, 77 0: N3.4.) There are several theoretical models which try to explain this behaviour. However, only the reptation concept of Edwards and de Gennes [23] and variants of this approach take the non-crossing of the chains explicitly into ac- count. This approach is the only one which, at least qualitatively, can account for a wide variety of different experimental results, such as neutron spin echo scattering, diffusion and viscosity. While it cannot explain all experimental data it does remarkably well, particularly considering its conceptual simplicity.
The idea of reptation is that the topological constraints imposed by the surrounding on each chain cause a motion along the polymers own coarse-grained contour. The diameter of the tube, to which the chain is constrained, is the diameter of a subchain of length Ne, namely dT 0: N:/?-. The chains follow the Rouse relaxation up to the time T~ 0: N,'. For longer times the constraints become dominant and the chain moves along its own contour.
To leave the tube, the chain has to diffuse along the tube a distance of the order of its own contour length, dTN/Ne. In order to leave this original tube the chain needs a time
Td cx N 2 ( N / N e ) , giving D K N - 2 and 17 K N 3 . The difference between the predicted and the measured exponent for 77 is still not completely understood. For the mean square displacements of the monomers, g ~ ( t ) , the model (Figure 8) predicts (a) the standard Rouse behaviour g 1 ( t ) 0: t1I2 for t