Overlapping Distribution for Polymers

Appendix I

Overlapping Distribution for Polymers Let us first consider how the basic idea behind the overlapping distribution method can be applied to the Rosenbluth insertion scheme. The simplest approach would be to consider the histogram of the potential energy change on addition or removal of a chain molecule (see section 7.2.3). However, for chain molecules, this approach differs from the original Shing-Gubbins approach in that it has little, if any, diagnostic value. For instance, if we consider the chemical potential of hard-core chain molecules, the distributions of A/d will always overlap (namely, at A/d = 0), even in the regime where the method cannot be trusted. Here, we shall describe an overlapping distribution method based on histograms of Rosenbluth weights [305]. This method will prove to be a useful diagnostic tool. Consider again a model with internal potential energy U-int and external potential energy Uext. In what follows, we shall compare two systems. The first, denoted by 0, contains N chain molecules (N >__0). The second system, denoted by 1, contains N 4-1 chain molecules. In addition, both systems may contain a fixed number of other (solvent) molecules. Let us first consider system 1. Around every segment j of a particular chain molecule (say, ~), we can generate k - 1 trial directions according to an internal probability distribution given by equation (11.2.19). Note that the set does not include the actual orientation of segment j. We denote this set of trial orientations by k-1 {'Yrest(J)} -- I - I {'Y}J'~ j'=l where the subscript rest indicates that this set excludes the actual segment j. The probability of generating this set of trial directions is given by Prest(j ),

Appendix I. Overlapping Distribution for Polymers

574

given by equation (11.2.19). Having thus constructed an umbrella of trial directions around every segment 1 < j _< e, we can compute the Rosenbluth weight Wi of molecule L Clearly, W~ depends on all coordinates of the remaining N molecules (for convenience, we assume that we are dealing with a neat liquid), on the position ri and conformation F~ of molecule ~, and on the e sets of k - 1 trial directions:

{Crest}- U{~Y}rest(j). j=l We now define a quantity x through

X - lnW~ (QN+I ~{Frest}), where we use Q to denote the translational coordinates r and conformational coordinates F of a molecule. Next, consider the expression for the probability density of x, 191(x)"

dQN+ld{['rest} exp [-~U(QN+I)] I-Ij=l r Prest(j )6(x - In }/V~) V,(x) -

ZN+,

'

'

where ZN+I

I dQN+'d{Frest}exp [-13U(Q N+' )] I-I Prest(j) j=l

-

I... IdQ +, exp

=

The second line of this equation follows from the fact that all Pint(~) are normalized. We shall now try to relate pl (x) to an average in system 0 (i.e., the system containing only N chain molecules). To this end, we write U(Q N+I ) a s U ( Q TM) = U~x(QTM, Q~) + Uint(Qi). Second, we use the fact that

exp [- ffui~t(i)] - Zid x I-I Pint(j), j=l where

Kid -- I dFi I-[ exp [-- ~U.int(j )] . j=l Our expression for p l(X) now becomes p,(x)

=

Zid I dQN drid{rtrial} exp [-~U(QN)] ZN+I x r I Ptrial(J)exp [-ffU~x(j)] 6(x - In Wi).

j=l

Appendix I. Overlapping Distribution for Polymers

575

We use the symbol {rtrial} to denote the set of all trial segments, that is, the "umbrella" of trial directions around all segments of the chain molecule, plus the segments themselves. Next, every term exp(-f3ttex(j)) is multiplied and divided by Zj, defined as

k Zj - E

exp(-[3Uext(j')).

j'=l This allows us to write, for p l (x), p~(•

=

VZid ZN+I

x E [ dQNdsid{Ftrial} exp [-f3U(QN)] Psel(Qi)Wi6 ( x - lnWi), trials 4

where we have transformed from real coordinates ri to scaled coordinates st by factoring out V, the volume of the system. Here, Psel(Qt) denotes the probability of selecting the actual conformation of the molecule from the given set of trial segments according to the rule given in equation (11.2.20). Finally, we multiply and divide by ZN and employ the fact that the 6 function ensures that W~ = exp(x):

pl(x)--eX ( VZidZN)ZN+I x

Etrials ~ dQ TMdsid{Ftrial} exp(--~U(QN ))Psel(Qi )6(x - In ]/Vi) ZN

Finally, we obtain pl(X) = e •

VZidZN ZN+I

po(x)

or lnpl (x) = x + [3~x + lnpo(x). Hence, by constructing a histogram of In w both in system 0 (with N chains) and in system 1 (with N + 1 chains), we can derive the excess chemical potential of the chain molecules by studying In pl (x) - ln p0(x). As in the original Bennett/Shing-Gubbins scheme [182-184], the method works only if there is a range ofx values where we have good statistics on both pl (x) and p0(x). The advantage of this overlapping distribution scheme over the simple Rosenbluth particle insertion method is that, with the present method, sampling problems for long chains will manifest themselves as a breakdown of the overlap of p0 and pl. Figure 1.1 shows an example of an application of this overlapping distribution method to hard-sphere polymers.

Appendix I. Overlapping Distribution for Polymers

576

40

'

40

I

,

!

,

,]~ .................. i! P r . . . . . . . . . . . . . . .

g-f

g.f

~,...~i

20

!

,,,u ,~

....

"

20

0

++

.,~e'""~+ +

-20

-20

f

, 0 - ' ~ -+0

,

-40

,

-30

,

,

-20

In W

,

~

- 0

,

-+0

0

,

-40

,

-30

,

,

-20

,

~

- 0

,

0

In W

Figure 1.1: The functions f(ln W) - p0(ln W)+ 89 In W, g(ln W) - pl (In W ) !2 In W for fully flexible chains of hard sphere of length (left) e - 8 and (right) e = 14 in a hard-sphere fluid at density p(x3 = 0.4. Note that the overlap between the distributions decreases as the chains become longer. The difference g(ln W) - f(ln W) is the overlapping distribution estimated for ~ t ex. For the sake of comparison, we also show the value for [3~ex, obtained using the Rosenbluth test particle insertion method (dashed lines).