Disorder-Averaged Free Energy and Introduction of Replicas 7

mechanical and structural properties. However, we are not interested in a specific cross-link configuration (and we cannot choose the cross-link configuration for 10²³ particles by hand anyway), but only intypical properties of the system. Therefore we have to calculate thedisorder-averaged free energy

[F_C] =−[lnZ_C]. (2.9)

In general it is very difficult to perform the disorder average, Eq. (2.7), of a logarithm. A successful way to perform the sum nevertheless is the replica trick:

Instead of averaging the logarithm itself, we take advantage of a representation of the logarithm:

−[F_C] = [lnZ_C] =

·

n→0lim

Z_Cⁿ−1 n

¸

= lim

n→0

[Z_Cⁿ]−1

n , (2.10)

and thus we try to calculate

[Z_Cⁿ] = 1 Z₁

X∞ M=0

XN i1,...,iM, j1,...,jM=1

1 M!

µ µ² 2N φ

¶_M

Z_C·Z_Cⁿ. (2.11)

At first, let’s have a look at the rightmost part of Eq. (2.11). At least for integer n, Z_C ·Z_Cⁿ can be written as a product Q_n

α=0Z_C. By doing that, we introduce n+1 copies of the system with identical cross-link configuration (one of

the copies, which we call the zeroth, stems from the Deam-Edwards distribution):

α=0H_ev({r^(α)_j }). Here we introduced the n+1 times replicated (and hence D(n+1)-dimensional) vectors

ˆ

r_j := (r⁽⁰⁾_j , ...,r⁽ⁿ⁾_j ). (2.13) This introduction ofn+1 replicas of the system¹is the main characteristic of replica theory and the reason for its name. One should keep in mind that the disorder is the same in all replicas; in particular M and i_e, j_e do not have replica indices (α). Assuming ergodicity, a possible way to imagine this replicated system is to see the particle conformations {r⁽⁰⁾_j },{r⁽¹⁾_j }, ...,{r⁽ⁿ⁾_j } as instances of the system taken at different times, with very long time differences in between. In this picture, the particle conformation in the zeroth replica, {r⁽⁰⁾_j }, corresponds to the particle conformation during cross-linking.

As we see in Appendix B.1.1, the sum over M in Eq. (2.11) can be performed and [Z_Cⁿ] can be written as: with the “replica free energy”

f˜_n+1{ˆr_j}=− µ² which incorporates the degrees of freedom of the system, the replicated particle po-sitions {ˆr₁, ...,rˆ_N}. That means, the disorder-averaged [Z_Cⁿ] for a D-dimensional system can be expressed as a partition function Z_n+1 that involves D(n+1)-dimensional vectors butdoes not involve a disorder average anymore. We will refer toZ_n+1 as the “replica partition function”.

Z₁ was originally introduced as a mere normalization constant in Eq. (2.8);

knowing that lim_n→0[Z_Cⁿ] = 1, we can confirm in Eq. (2.14a) that the notation for Z₁ is chosen consistently.

1or possibly onlynreplicas, if the cross-link distribution is not the Deam-Edwards distribution

2.4. REPLICA CALCULATION OF THE FREE ENERGY 9

Figure 2.2: Exemplary illustration of a snapshot of the model as in Fig.2.1, which only containslocalized particles. Additionally, the density plot behind the snapshot illustrates, how the thermally averaged density as defined in Eq. (2.15) might look like.

2.4.2 Introduction of the Replicated Density Fields

Later we want to introduce a field theory, so sooner or later we have to think about a suitable order parameter. A crucial requirement is that it can distinguish between the liquid and the amorphous solid state. A first idea would be the thermally averaged density

%^(α)_C (x) := 1 N

XN j=1

D

δ(x−r^(α)_j ) E

HC

. (2.15)

In the liquid phase, when no macroscopic cluster is present, any particle – even if it is part of a small cluster of connected particles – has equal probability to be found anywhere in the sample; hence this order parameter would simply be a constant.

In the gel phase, on the other hand, a macroscopic cluster is present, which is a cluster that contains an infinite number of particles asN → ∞. A particle of that cluster can not pass through the entire system, because cross-linking to the cluster will restrict its movement to fluctuations around a mean position (as illustrated in Fig.2.2). Hence we will refer to such a particle as a “localized particle”.

The emerging density fluctuations may be utilized to distinguish the gel from the sol phase for a given cross-link configuration C. However, quantities we are interested in, should not be restricted to a certain cross-link configuration, but should be disorder averaged. And when averaging over all cross-link configurations, the macroscopic cluster can be embedded anywhere in the sample and hence the probability to find a particle at any place in the sample should be the same – as in the liquid phase.

To overcome this problem, we take advantage of the fact that the cross-link configuration is the same for all replicas and hence the position of a localized particle should be correlated among the replicas: In the context of gelation, the crucial

(a) (b)

0 2 4 6 8 10

x^H1L 0

2 4 6 8 10

x^H2L

Ξ

0 2 4 6 8 10

x^H1L 0

2 4 6 8 10

x^H2L

Figure 2.3: For one spatial dimension, the replicated density O(x⁽¹⁾, x⁽²⁾), the prob-ability to find a particle at x⁽¹⁾ in replica 1 and at x⁽²⁾ in replica 2, is illustrated;

white means high values. The red line is the angle bisector x⁽¹⁾ =x⁽²⁾. (a) Exam-ple with two localized particles fluctuating around the mean positions R₁ = 2 and R₂ = 7. The blue arrow (↔) represents the localization length. (b) For the disor-der average, we have to integrate the mean positions over the system size. Thus, O(x⁽¹⁾, x⁽²⁾) becomes a rectilinear ridge along the axisx⁽¹⁾=x⁽²⁾.

question is, whether or not|r^(α)_j −r^(β)_j |, withα6=β, stays finite for a certain fraction of particles. With that in mind, we define the replica density:

O(ˆx) := 1 N

XN j=1

δ(ˆx−ˆr_j) = 1 N

XN j=1

δ(x⁽⁰⁾−r⁽⁰⁾_j )· · ·δ(x⁽ⁿ⁾−r⁽ⁿ⁾_j ), (2.16)

which is the joint probability to find a particle at x⁽⁰⁾ in replica 0, and the same particle atx⁽¹⁾in replica 1,... and atx⁽ⁿ⁾in replican. A thermally averaged example of O(ˆx), is illustrated in Fig. 2.3a for the case of one spatial dimension and two replicas: Alocalized particle which is found at positionxin one replica, will have a higher probability to be close to that position also in other replicas. The extent of the particle fluctuations, which we calllocalization length, is also constituted by the length scale of these correlations.

As before, after performing the disorder average, themean positions have equal probability to be anywhere in the sample; however, a particle of the macroscopic cluster still has a high probability to be found at close-by positions among the different replicas. This situation is illustrated in Fig. 2.3b, again for one dimension and two replicas: Due to translational invariance, the disorder-averaged replicated density does not depend on the mean position x_cm = (x⁽¹⁾+x⁽²⁾)/2, but only on the difference (x⁽¹⁾ −x⁽²⁾)/2 between the replicas. That latter dependence still represents the extent of the particle fluctuations ξ. In Sec.2.6 we are going to find

2.4. REPLICA CALCULATION OF THE FREE ENERGY 11 an Ansatz for the replicated density field and mathematically formulate the aspects introduced here.

For completeness, we also define the replicated density in Fourier space:

O(ˆq) := 1 N

XN j=1

exp(iˆqrˆ_j), (2.17a)

O^(α)(q) := 1 N

XN j=1

exp(iqr^(α)_j ) = O(0, ...,0,q

α↑

,0, ...,0). (2.17b) The first definition is the full Fourier transform of Eq. (2.16) and contains correla-tions between replicas. The latter definition is the ordinary Fourier density of the particle configuration of a given replicaα.

Im Dokument Aggregation and Gelation in Random Networks (Seite 17-21)