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40
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69149098564
-
-
note
-
The dimensionless compressibility g and the effective bond length b depend in general on both the overlap strength x=ε/T and the chain length N. We refer to the limiting values g (x) = limN→∞ g (x,N) and b (x) = limN→∞ b (x,N) of asymptotically long chains if not stated otherwise. The chain length dependence of the compressibility is due to the translational entropy of the chains [Eq.]. The much stronger N -effects observed for the effective bond length are due to swelling of the chains as described by Eq. and discussed in Sec..
-
-
-
-
41
-
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1142305344
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1292-8941,. 10.1140/epje/e2004-00019-2
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Eur. Phys. J. e
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-
-
Semenov, A.N.1
Johner, A.2
-
45
-
-
69149100218
-
-
For the computational definition of G (q) see Sec.. Note that our normalization of the structure factor is consistent with Eq. (5.39) of Ref., i.e., G (q) is dimensionless. Quite generally, we present in this paper the properties normalized by the total number of monomers nmon, i.e., essentially by the degree of freedom of the system rather than by its volume V.
-
For the computational definition of G (q) see Sec.. Note that our normalization of the structure factor is consistent with Eq. (5.39) of Ref., i.e., G (q) is dimensionless. Quite generally, we present in this paper the properties normalized by the total number of monomers nmon, i.e., essentially by the degree of freedom of the system rather than by its volume V.
-
-
-
-
46
-
-
69149099910
-
-
Following Refs. the "bulk modulus" ṽ 1/gρ is sometimes called "effective excluded volume" which refers to the fact that convergence and self-consistency of the perturbation theory (Ref.) suggest to substitute (renormalize) the bare excluded volume v of the monomers by the explicitly measured ṽ summing also over higher virial contributions, not only pair interactions. However, since higher virial contributions become negligible for weak excluded volume interactions one expects that at leading order v (x) ≈ ṽ (x) for x1. This will be cross-checked in Sec. (Fig.). Since the perturbation calculation summarized by Eq. is anyway restricted to the same x -regime [Eq.] we do not use ṽ in the main text.
-
Following Refs. the "bulk modulus" ṽ 1/gρ is sometimes called "effective excluded volume" which refers to the fact that convergence and self-consistency of the perturbation theory (Ref.) suggest to substitute (renormalize) the bare excluded volume v of the monomers by the explicitly measured ṽ summing also over higher virial contributions, not only pair interactions. However, since higher virial contributions become negligible for weak excluded volume interactions one expects that at leading order v (x) ≈ ṽ (x) for x1. This will be cross-checked in Sec. (Fig.). Since the perturbation calculation summarized by Eq. is anyway restricted to the same x -regime [Eq.] we do not use ṽ in the main text.
-
-
-
-
47
-
-
69149095889
-
-
Since the overlapenalty is the only energy scale in this study one may either vary the overlaparameter ε or the temperature T. Since the presentation of thermodynamic properties (especially Figs.) becomes, however, slightly simpler if T is the control parameter we fix arbitrarily ε=1. The inverse temperature Β=1/T and the dimensionless overlastrength x=ε/T are thus numerically equal. We keeboth notations for dimensional reasons and for future generalization to models with more than one energy scale.
-
Since the overlap penalty is the only energy scale in this study one may either vary the overlap parameter ε or the temperature T. Since the presentation of thermodynamic properties (especially Figs.) becomes, however, slightly simpler if T is the control parameter we fix arbitrarily ε=1. The inverse temperature Β=1/T and the dimensionless overlap strength x=ε/T are thus numerically equal. We keep both notations for dimensional reasons and for future generalization to models with more than one energy scale.
-
-
-
-
48
-
-
69149095034
-
-
A similar intrachain energy contribution to the free energy arises also from Eq. (5.43) of Ref. if an upper cutoff qmax is introduced for the wave vectors q to avoid the ultraviolet divergence. Such an upper cutoff is justified by the discreteness of the monomers of real polymers. This leads necessarily to a nonuniversal free energy contribution.
-
A similar intrachain energy contribution to the free energy arises also from Eq. (5.43) of Ref. if an upper cutoff qmax is introduced for the wave vectors q to avoid the ultraviolet divergence. Such an upper cutoff is justified by the discreteness of the monomers of real polymers. This leads necessarily to a nonuniversal free energy contribution.
-
-
-
-
49
-
-
69149109469
-
-
Interestingly, according to Ref. the chain connectivity, i.e., the presence of attractive forces between bonded monomers, does not change the excluded volume v (x) -as one would expect naively-but rather gives rise to an additional term scaling differently with density.
-
Interestingly, according to Ref. the chain connectivity, i.e., the presence of attractive forces between bonded monomers, does not change the excluded volume v (x) -as one would expect naively-but rather gives rise to an additional term scaling differently with density.
-
-
-
-
50
-
-
69149111300
-
-
The Ginzburg parameter is defined here in terms of the measured compressibility g (x) and the measured effective bond length b (x). Equation is consistent with Eq. (5.46) of Ref..
-
The Ginzburg parameter is defined here in terms of the measured compressibility g (x) and the measured effective bond length b (x). Equation is consistent with Eq. (5.46) of Ref..
-
-
-
-
51
-
-
69149102086
-
-
Since the intrachain form factor is essentially the Fourier transform of the segmental size distribution (Refs.), R (s) is related to an experimentally relevant quantity.
-
Since the intrachain form factor is essentially the Fourier transform of the segmental size distribution (Refs.), R (s) is related to an experimentally relevant quantity.
-
-
-
-
54
-
-
0000109109
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s /g) 4 3 u we have calculated. This is due to the fact that we consider the size of inner chain segments which involves the two additional graphs in the perturbation calculation illustrated on the right of Fig. 14 given in Ref.. Only the first diagram shown in this figure is taken into account for the total chain size.
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s /g) 4 3 u we have calculated. This is due to the fact that we consider the size of inner chain segments which involves the two additional graphs in the perturbation calculation illustrated on the right of Fig. 14 given in Ref.. Only the first diagram shown in this figure is taken into account for the total chain size.
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Attractive interactions similar to the ones used in Refs. may be easily added to the Potts spin formulation of the soft BFM. The simulation of polymer blends requires additional Potts spin lattices interacting as indicated in Eq. for the two lattices used to obtain the chemical potential in Sec..
-
Attractive interactions similar to the ones used in Refs. may be easily added to the Potts spin formulation of the soft BFM. The simulation of polymer blends requires additional Potts spin lattices interacting as indicated in Eq. for the two lattices used to obtain the chemical potential in Sec..
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0 and μex ∼N may arise due to the long-range correlations which lead to the swelling described by Eq.. This effect is, however, too weak to be visible in our data. It can be demonstrated from the number distribution of equilibrium polymers (Refs.) which differs slightly from the exponential behavior generally admitted (Refs.).
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0 and μex ∼N may arise due to the long-range correlations which lead to the swelling described by Eq.. This effect is, however, too weak to be visible in our data. It can be demonstrated from the number distribution of equilibrium polymers (Refs.) which differs slightly from the exponential behavior generally admitted (Refs.).
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e (N) [rather than from R (s) as suggested in Sec.], could be refined.
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Effects of a tunable local chain rigidity will be discussed using this off-lattice scheme since lattice artifacts are known to arise for stiff BFM chains (Ref.).
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Effects of a tunable local chain rigidity will be discussed using this off-lattice scheme since lattice artifacts are known to arise for stiff BFM chains (Ref.).
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