# A finite excluded volume bond-fluctuation model:
Static properties of dense polymer melts revisited

J.P. Wittmer<EMAIL_ADDRESS>Institut Charles Sadron, 23 rue du
Loess, 67037 Strasbourg Cédex, France  A. Cavallo  Institut Charles Sadron, 23
rue du Loess, 67037 Strasbourg Cédex, France  T. Kreer  Institut Charles
Sadron, 23 rue du Loess, 67037 Strasbourg Cédex, France  J. Baschnagel
Institut Charles Sadron, 23 rue du Loess, 67037 Strasbourg Cédex, France  A.
Johner http://www.ics-u.strasbg.fr/~etsp/welcome.php Institut Charles Sadron,
23 rue du Loess, 67037 Strasbourg Cédex, France

###### Abstract

The classical bond-fluctuation model (BFM) is an efficient lattice Monte Carlo
algorithm for coarse-grained polymer chains where each monomer occupies
exclusively a certain number of lattice sites. In this paper we propose a
generalization of the BFM where we relax this constraint and allow the overlap
of monomers subject to a finite energy penalty $\varepsilon$. This is done to
vary systematically the dimensionless compressibility $g$ of the solution in
order to investigate the influence of density fluctuations in dense polymer
melts on various static properties at constant overall monomer density. The
compressibility is obtained directly from the low-wavevector limit of the
static structure factor. We consider, e.g., the intrachain bond-bond
correlation function, $P(s)$, of two bonds separated by $s$ monomers along the
chain. It is shown that the excluded volume interactions are never fully
screened for very long chains. If distances smaller than the thermal blob size
are probed ($s\ll g$) the chains are swollen according to the classical Fixman
expansion where, e.g., $P(s)\sim g^{-1}s^{-1/2}$. More importantly, the
polymers behave on larger distances ($s\gg g$) like swollen chains of
incompressible blobs with $P(s)\sim g^{0}s^{-3/2}$.

###### pacs:

05.10.Ln, 61.25.hk, 61.25.hp

##  I Introduction

#### The bond-fluctuation model.

The classical bond-fluctuation model (BFM) is an efficient lattice Monte Carlo
(MC) algorithm for coarse-grained polymer chains where each monomer occupies
exclusively a certain number of lattice sites on a simple cubic lattice
Carmesin and Kremer (1988); Deutsch and Dickman (1990); Deutsch and Binder
(1991). It was proposed in 1988 by Carmesin and Kremer Carmesin and Kremer
(1988) as an alternative to single-site self-avoiding walk models, which
retains the computational efficiency of the lattice without being plagued by
severe ergodicity problems. The key idea is to increase the size of a monomer
which now occupies a whole unit cell of the lattice, as illustrated in Fig. 1.
The multitude of possible bond lengths and angles allows a better
representation of the continuous-space behavior of real polymer solutions and
melts. The BFM algorithm has been used for a huge range of problems addressing
the generic behavior of long polymer chains of very different molecular
architectures and geometries: statics Deutsch and Dickman (1990); Deutsch and
Binder (1991); Paul et al. (1991a); Müller and Paul (1994); Wilding and Müller
(1994); Müller et al. (2000a); Stukan et al. (2002); Wittmer et al. (2004);
Wittmer et al. (2007a); Beckrich et al. (2007); Wittmer et al. (2007b); Meyer
et al. (2008) and dynamics Paul et al. (1991b); Wittmer et al. (1992); Kreer
et al. (2001); Mattioni et al. (2003); Azuma and Takayama (1999) of linear
chains and rings Müller et al. (1996); Müller et al. (2000b), polymer blends
and interfaces Müller (1995, 1999); Cavallo et al. (2003), gels and networks
Sommer and Saalwächter (2005), glass transition Binder et al. (2003),
(co-)polymers at surfaces Werner et al. (1999), brushes Wittmer et al. (1994);
Kopf et al. (1996); Wittmer et al. (1996), thin films Khalatur et al. (2000);
Mischler et al. (2001); Cavallo et al. (2005), equilibrium polymers Wittmer et
al. (1998); Wittmer et al. (2007c); Beckrich et al. (2007) and general self-
assembly Cavallo et al. (2006, 2008), …. For recent reviews see Refs.
Baschnagel et al. (2004); Müller (2005).

#### A BFM version allowing a systematic compressibility variation.

As sketched in Fig. 1, we propose here a generalization of the BFM where we
relax the no-overlap constraint and allow the overlap of monomers subject to a
finite energy penalty $\varepsilon$. This is done to vary systematically the
strength of density fluctuations in dense solutions and melts to study their
influence on static and dynamical properties. More specifically, we want to
test the standard perturbation theory of weakly interacting three-dimensional
polymer melts Doi and Edwards (1986) and to verify whether certain long-range
correlations Wittmer et al. (2004); Wittmer et al. (2007a); Beckrich et al.
(2007); Wittmer et al. (2007b); Meyer et al. (2008), which have been found
recently for incompressible melts, are also present in melts with finite
compressibility. We will see that this is indeed the case if one considers
properties on scales larger than the screening length $\xi\approx bg^{1/2}$ of
the density fluctuations where $b$ indicates the effective bond length of the
chain and $g$ the number of monomers spanning the “thermal blob” de Gennes
(1979); Doi and Edwards (1986); foo (a). Interestingly, $g$ is related to the
isothermal compressibility of the solution $\kappa_{\text{T}}$ Semenov and
Johner (2003); Semenov and Obukhov (2005); Beckrich et al. (2007), i.e. the
thermodynamic property which measures the strength of the density
fluctuations. It can thus be determined directly experimentally or in a
computer simulation from the low-wavevector limit of the total monomer
structure factor McQuarrie (2000); Frenkel and Smit (2002); foo (b)

$G(q)\equiv\frac{1}{n_{\text{mon}}}\sum_{n,m=1}^{n_{\text{mon}}}\exp\left(-i\bm{q}\cdot(\bm{r}_{n}-\bm{r}_{m})\right)\stackrel{{\scriptstyle
q\to 0}}{{\Longrightarrow}}g\equiv T\ \kappa_{\text{T}}\rho,$ (1)

with $\bm{q}$ being the wavevector, $\bm{r}_{i}$ the position of monomer $i$,
$n_{\text{mon}}=\rho V$ the total monomer number, $\rho$ the monomer number
density, $V=L^{3}$ the volume of the system and $T\equiv 1/\beta$ the
temperature. (Boltzmann’s constant $k_{\text{B}}$ is set to unity throughout
this paper.) Due to the definition given in Eq. (1), $g$ is also called
“dimensionless compressibility” foo (c). We will show that the variation of
the operational parameter $x=\varepsilon/T$ foo (d), characterizing the
strength of the overlap penalty, allows us to scan $g(x)$ over four orders of
magnitude foo (a). This puts us in the position to test various theoretical
predictions which are sketched below.

#### Thermodynamic properties for $x\ll 1$.

To characterize the proposed soft BFM model we will first investigate
thermodynamic properties such as the mean overlap energy per monomer $e$, the
excess chemical potential $\mu_{\text{ex}}$ or the (already mentioned)
dimensionless compressibility $g$ as functions of $x$. In the limit of weak
overlap penalties these properties have been calculated long ago by Edwards
theory Doi and Edwards (1986) and we will compare our numerical results with
his predictions. For later reference we postulate here the contributions to
the free energy per monomer $f(\beta)$ relevant for this comparison

$\displaystyle\beta f(\beta)$ $\displaystyle=$ $\displaystyle\beta
e_{\text{self}}$ (2) $\displaystyle+$
$\displaystyle\frac{1}{N}\left(\log(\rho/N)-1\right)$ $\displaystyle+$
$\displaystyle\frac{1}{2}v(x)\rho\
\underline{-\frac{1}{12\pi}\frac{1}{\xi(x)^{3}\rho}}\ +\ldots.$

The first term is due to the constant intrachain self-energy discussed in Sec.
III.1. It is due to the reference energy chosen in this study and it is not
accessible experimentally foo (e). The second and third line of Eq. (2) can be
readily obtained from the literature, e.g., by integrating the osmotic
pressure given by Eq. (5.45) or Eq. (5.II.5) of Doi and Edwards (1986) with
respect to the density $\rho$. (The first line can be considered as the
integration constant with respect to this integration.) The second line
represents the translational invariance of monodisperse chains of length $N$
(van’t Hoff’s law). Due to this contribution the compressibilities depend in
general on $N$ as will be discussed in Sec. III.4. The (bare) excluded volume
interaction between the monomers is accounted for by the first term in the
third line with $v(x)$ being the second virial coefficient of a solution of
unconnected monomers. The underlined term represents the leading correction to
the previous term due to the fact that the monomers are connected by bonds
summing over the density fluctuations to quadratic order. As one expects de
Gennes (1979), the corresponding correlations of the density fluctuations
reduce the free energy by about one $k_{\text{B}}T$ per thermal blob of volume
$\xi^{3}$. Consistently with Refs. Doi and Edwards (1986); Semenov and Obukhov
(2005); Meyer et al. (2008) the correlation length $\xi$ of the density
fluctuations has been defined here as

$\xi^{2}(x)\equiv b^{2}(x)g(x)/12\approx b^{2}(x)/(12v(x)\rho).$ (3)

Note that $g(x)$ and $b(x)$ refer respectively to the dimensionless
compressibility and the effective bond length of asymptotically long chains
foo (a, f). The density fluctuation contribution to the free energy will be
demonstrated numerically from the scaling of the specific heat $c_{\text{V}}$
with respect to overlap strength $x$ and density $\rho$ (Sec. III.2). We note
finally that Eqs. (2) and (3) are supposed to apply as long as

$G_{\text{z}}(x)\equiv\frac{1}{\sqrt{g(x)}\ b^{3}(x)\rho}\ll 1,$ (4)

with the Ginzburg parameter $G_{\text{z}}$ being the small parameter of the
perturbation theory foo (g). This restricts — as we shall see — the related
predictions to overlap penalties with $x\ll 1$.

#### Conformational properties of asymptotically long chains for all $x$.

This paper aims ultimately to characterize an important intrachain
conformational property, the size of chain segments of arc-length $s$ in very
long chains where finite-$N$ effects may be neglected foo (h). Specifically,
we will investigate the mean-square end-to-end distance
$R^{2}(s)=\left<\left(\bm{r}_{m=n+s}-\bm{r}_{n}\right)^{2}\right>$ where the
average is performed over all possible pairs of monomers $(n,m=n+s)$. The
total chain end-to-end distance is $R_{\text{e}}(N)\equiv R(s=N-1)$. If
appropriately plotted Wittmer et al. (2007b), the segment size $R(s)$ will
allow us to obtain an estimation of the effective bond length $b(x)$ by
extrapolation foo (a). Our numerical results will be compared with an
analytical prediction obtained by a standard one-loop perturbation calculation
very similar to the analytic results already presented elsewhere Wittmer et
al. (2004); Wittmer et al. (2007a); Beckrich et al. (2007); Wittmer et al.
(2007b); Meyer et al. (2008). Details of the calculation for general soft
melts will be presented in a future paper Wittmer et al. (2009). Focusing in
this paper mainly on computational issues, we only quote here the “key
relation” put to the test

$1-\frac{R^{2}(s)}{b^{2}s}=\frac{c_{\text{s}}}{g^{1/2}}\left(\frac{1}{\sqrt{u}}-\frac{\sqrt{\pi/8}}{u}\left(1-e^{2u}\mbox{erfc}(\sqrt{2u})\right)\right),$
(5)

where we have used $u=s/g(x)$ for the reduced curvilinear arc-length and
$\mbox{erfc}(x)$ for the complementary error function Abramowitz and Stegun
(1964). The correction to Gaussianity expressed by the r.h.s. terms of the key
prediction is always positive and corresponds to a weak swelling of the chain.
The “swelling coefficient” $c_{\text{s}}=\sqrt{24/\pi^{3}}/\rho b^{3}$ Wittmer
et al. (2007b) measures the strength of the effect. While Eq. (2) fails for
large $x$ where the Ginzburg parameter $G_{\text{z}}$ becomes of order unity,
Eq. (5) is supposed to hold for all $x$ provided that the considered segment
length $s$ is large enough. As explained in Sec. II of Ref. Wittmer et al.
(2007b), the relevant small parameter of the perturbation theory is here the
so-called “correlation hole potential” of the chains

$u(s)\approx G_{\text{z}}\sqrt{g/s}\approx 1/\sqrt{s}\ll 1\mbox{ for }s\gg 1$
(6)

rather than $G_{\text{z}}$. For a discussion of chain end effects in
incompressible melts see Ref. Wittmer et al. (2007b).

#### Check of limiting behavior.

Although no formal derivation of Eq. (5) is given, we verify briefly whether
the suggested key prediction is reasonable by discussing the limits of large
and small reduced arc-length. We remind first that according to Flory’s
“ideality hypothesis” Flory (1949); de Gennes (1979), polymer chains in the
melt are thought to display Gaussian statistics for segment sizes somewhat
larger than the persistence length de Gennes (1979); Doi and Edwards (1986);
Flory (1949) which implies that the r.h.s. of Eq. (5) is traditionally assumed
to vanish (exponentially) for finite $s$. At variance to this, the expansion
of the complementary error function Abramowitz and Stegun (1964) for $u=s/g\gg
1$ yields

$1-\frac{R^{2}(s)}{b^{2}s}\approx\frac{c_{\text{s}}}{\sqrt{s}},$ (7)

suggesting in fact that corrections to Gaussianity must be taken into account
for all finite $s$. As one expects, however, Gaussianity is still recovered in
three dimensions if $s\to\infty$. (Incidentally, this does not hold for
effectively two-dimensional melts as may be seen from the discussion of
ultrathin polymer films in Refs. Semenov and Johner (2003); Cavallo et al.
(2005); Meyer et al. (2007).) Most remarkably, the explicit compressibility
dependence drops out for large $u$ for all $g(x)$ provided that the chains are
long enough, such that $g\ll s\ll N$. Eq. (7) is precisely the relation which
has been discussed in detail both theoretically and numerically Wittmer et al.
(2004); Wittmer et al. (2007a); Beckrich et al. (2007); Wittmer et al. (2007b)
for highly incompressible melts with full excluded volume interactions
($x=\infty$). Hence, this is the expected limiting behavior if the polymer
chains are renormalized in terms of an incompressible packing of thermal blobs
with $g$ monomers. In the opposite limit of small reduced arc-lengths,
$u=s/g\ll 1$, the expansion of Eq. (5) yields

$1-\frac{R^{2}(s)}{b^{2}s}\approx\frac{c_{\text{s}}}{\sqrt{g}}\left(\sqrt{\frac{\pi}{2}}-\frac{4}{3}\sqrt{u}\right).$
(8)

This is consistent with the classical expansion result of the chain size in
terms of the “Fixman parameter” $z\approx
v\sqrt{s}/b^{3}\approx\sqrt{u}/(\sqrt{g}b^{3}\rho)$ Doi and Edwards (1986);
foo (c) since the deviations from ideality expressed by the last term of Eq.
(8) become then proportional to $-c_{\text{s}}\sqrt{u/g}\approx-z$, in
agreement with the leading correction term for the total chain size
$R_{\text{e}}(N)$ presented in textbooks foo (i).

#### Outline.

In Section II the algorithm is introduced and some technical details are
discussed. First we summarize the classical BFM without monomer overlap (Sec.
II.1), which has been used in previous work Wittmer et al. (2007b) providing
the start configuration for the present study, and introduce then its
generalization with finite overlap penalty (Sec. II.2). The central Section
III presents our numerical results starting with the thermodynamic properties
(Secs. III.1-III.4), in particular the dimensionless compressibility $g(x)$.
We characterize then (Secs. III.5-III.7) various intrachain properties, as for
instance the effective bond length $b(x)$ or the bond-bond correlation
function $P(s)$, as functions of $g(x)$. A synopsis of our results is
presented in Section IV.

##  II Algorithm and technical details

###  II.1 Bond fluctuation model without monomer overlap

#### A lattice Monte Carlo scheme for topology conserving polymers.

We have used the three-dimensional version of the bond fluctuation model
Wittmann and Kremer (1990); Deutsch and Dickman (1990); Deutsch and Binder
(1991); Paul et al. (1991a); Wittmer et al. (1992) where each effective
coarse-grained monomer is represented by a cube of eight adjacent sites on a
simple cubic lattice, as illustrated in Fig. 1. Even the partial overlap of
monomers is forbidden. The lattice constant $a$ is naturally chosen as the
unit length. Polymers of length $N$ consist of $N$ cubes connected by $N-1$
bonds, as shown in the sketch for $N=3$. These bonds are taken from the set

$\left\\{P(2,0,0),P(2,1,0),P(2,1,1),P(2,2,1),P(3,0,0),P(3,1,0)\right\\}$ (9)

of allowed bond vectors, where $P$ stands for all permutations and sign
combinations of coordinates. This corresponds to 108 different bond vectors of
5 possible bond lengths ($2$, $\sqrt{5}$, $\sqrt{6}$, $3$, $\sqrt{10}$) and
$100$ possible angles between consecutive bonds. The smallest $13$ angles do
not appear for the classical BFM without monomer overlap, because excluded
volume forbids sharp backfolding of bonds. If only local Monte Carlo (MC)
moves of the monomers to the six nearest neighbor sites are performed — called
“L06” moves Wittmer et al. (2007b) — this set of vectors ensures automatically
that polymer chains cannot cross. Topological constraints, e.g. in ring
polymers Müller et al. (1996), hence are conserved and the polymer dynamics
may be expected to be of reptation type Paul et al. (1991a); Müller et al.
(2000b); Kreer et al. (2001); Wittmer et al. (2007b). It is this fact which
has originally motivated the choice of allowed bonds. We keep it for
consistency with previous work although the non-crossing constraint is
irrelevant for the present study. Note that the classical BFM algorithm with
L06 moves is strictly speaking not ergodic, since some (thermodynamically
irrelevant) configurations may be easily constructed which are not accessible
starting from an initial configuration of stretched linear chains Baschnagel
et al. (2004).

#### Obtaining athermal start configurations with topology violating moves.

The algorithm is up to now athermal and the only control parameters are the
monomer density $\rho$ and the chain size $N$. Melt conditions are realized
for $\rho=0.5/8$, where half of the lattice sites are occupied Paul et al.
(1991a). We use (if not stated otherwise) periodic simulation boxes of linear
dimension $L=256$ which contain $n_{\text{mon}}=\rho L^{3}=2^{20}\approx
10^{6}$ monomers. This large system size allows to eliminate finite-size
effects even for the longest chain lengths studied. Our simulations have been
carried out by a mixture of local, slithering snake Kron (1965); Wall and
Mandel (1975); Mattioni et al. (2003), and double-bridging MC moves
Karayiannis et al. (2002); Banaszak and de Pablo (2003); Auhl et al. (2003);
Baschnagel et al. (2004) which allow us to equilibrate polymer melts with
chain lengths up to $N=8192$. Instead of the more realistic but very slow L06
dynamical scheme we make so-called “L26” jump attempts to the 26 sites of the
cube surrounding the current monomer position. This permits the crossing of
chains which dramatically speeds up the dynamics, expecially for long chains
$(N>512)$. Details of the equilibration procedure and possible caveats are
discussed in Ref. Wittmer et al. (2007b). We stress that if these topology
violating MC moves are included all possible configurations become accessible,
i.e. the BFM becomes fully ergodic. Concerning the static properties ergodic
and non-ergodic BFM versions are, however, practically equivalent. This has
been confirmed by comparing various static properties and by counting the
number of monomers which become “blocked” once one returns to the original
local L06 moves.

###  II.2 Bond fluctuation model with finite excluded volume penalty

#### The model Hamiltonian.

Fig. 1 shows how finite energy penalties are introduced. The overlap of two
cube corners on one lattice site ($N_{\text{ov}}=1$) corresponds to an energy
cost of $\varepsilon/8$, the full overlap of two monomers ($N_{\text{ov}}=8$)
to an energy $\varepsilon$. More generally, with $N_{\text{ov}}$ being the
total number of interacting cube corners the total interaction energy of a
configuration is

$E=\frac{\varepsilon}{8}N_{\text{ov}}.$ (10)

With the energies of the final ($E_{\text{f}}$) and the initial configurations
($E_{\text{i}}$) we accept the MC move according to the Metropolis criterion
Baschnagel et al. (2004); Landau and Binder (2000) with probability
$\mbox{min}(1,\exp[-(E_{\text{f}}-E_{\text{i}})/T])$. We set arbitrarily
$\varepsilon=1$ and vary the ratio $x=\varepsilon/T$ foo (d) starting from
$x=\infty$ corresponding to the athermal classical BFM and systematically
increase the temperature $T$ as shown in Table 1 for unconnected monomers
($N=1$).

#### Second virial coefficient.

To illustrate this interaction we indicate already here the second virial of
an imperfect gas of unconnected monomers, $v=\int
d{\bm{\delta}}(1-e^{-E({\bm{\delta}})/T})$, which is shown below to be useful
for roughly characterizing the effective strength of the potential.
$\bm{\delta}$ stands for a possible lattice vector between the centers of two
interacting cubes. It is easy to see that there are 8 vectors corresponding to
$N_{\text{ov}}=1$ (as in Fig. 1), 12 to $N_{\text{ov}}=2$ (overlap of two cube
corners), 6 to $N_{\text{ov}}=4$ (overlap of two faces), and 1 to
$N_{\text{ov}}=8$ (full overlap). This leads to a second virial

$v=8\times(1-\exp(-x/8))+12\times(1-\exp(-x/4))+6\times(1-\exp(-x/2))+1\times(1-\exp(-x))$
(11)

given in units of the lattice cube volume $a^{3}$. We note that the second
virial becomes constant, $v=27$, in the low temperature limit ($x\gg 1$)
corresponding to the classical athermal BFM result Deutsch and Dickman (1990).
In the opposite high temperature limit ($x\ll 1$) it decays as

$v\approx 8x-\frac{27}{16}x^{2}.$ (12)

#### Implementation of the algorithm.

We briefly explain the implementation of BFM chains with the soft overlap
Hamiltonian, Eq. (10). Following Paul et al. (1991a, b); Wittmer et al.
(1992); Kreer et al. (2001); Mattioni et al. (2003) it is convenient to keep
one list of the monomer positions in absolute space (since we are also
interested in dynamical properties) and one for the corresponding lattice
positions. We identify each of the 108 allowed bonds of the set [Eq. (9)] with
a unique bond index and keep a list of these indices. This allows to verify
rapidly whether an attempted bond vector is acceptable. Since the bond index
(being less than 128) can be encoded as a byte, this compressed list is stored
when we write down the configuration ensemble for further analysis. Additional
lists allow to handle efficiently the periodic boundary conditions, the change
of a bond index for a given monomer move and the local interactions relevant
for local L06 or L26 moves.

Since the soft overlap Hamiltonian allows the occupation of a lattice site by
more than one monomer, it is not possible to use a compact boolean occupation
lattice (corresponding to a spin $\sigma=0$ and $\sigma=1$ for an empty or
filled lattice site) or an integer lattice filled with the monomer indices as
in previous implementations. Instead we have mapped Eq. (10) onto a Potts spin
model Landau and Binder (2000)

$E=\frac{1}{2}\sum_{\bm{r}}\sigma(\bm{r})\sum_{\bm{\delta}}J(\bm{\delta})\sigma(\bm{r}+\bm{\delta})-\frac{1}{2}\varepsilon
n_{\text{mon}}$ (13)

with constant monomer number
$n_{\text{mon}}=\sum_{\bm{r}}\sigma(\bm{r})\stackrel{{\scriptstyle!}}{{=}}L^{3}\rho$
using a Wigner-Seitz representation of the cubic lattice following Müller
Müller (1995, 2005). In this representation an integer spin variable
$\sigma(\bm{r})$ counts the number of BFM monomers ($\sigma=0,1,2,\ldots$)
with cubes centered at a Wigner-Seitz lattice position $\bm{r}$. In other
words, each cube is not represented by 8 lattice entries for the cube corners,
but just by one for its center. Since we have now to compute the interaction
between cube centers instead of cube corners, the coupling constant $J$
characterizing the interaction between two spins depends only on the relative
distance $\bm{\delta}$:

$J(\bm{\delta})=\varepsilon\left\\{\begin{array}[]{ll}1/8&\mbox{if
$\bm{\delta}=P(1,1,1)$ (cube corners),}\\\ 1/4&\mbox{if $\bm{\delta}=P(1,1,0)$
(cube edges),}\\\ 1/2&\mbox{if $\bm{\delta}=P(1,0,0)$ (cube faces),}\\\
1&\mbox{if $\bm{\delta}=P(0,0,0)$ (full overlap),}\\\
0&\mbox{otherwise.}\end{array}\right.$ (14)

Since the interaction is still short-ranged and the values of $J$ are readily
tabulated, this remains an efficient rendering of the monomer interactions.
Note that the first term on the r.h.s. of Eq. (13) contains a constant self-
interaction contribution of the $n_{\text{mon}}$ monomers with themselves for
$\bm{\delta}=\bm{0}$, which is substracted by the second term foo (j).

#### Equilibration and system properties of high-molecular melts.

As already stated we have used as start configurations the equilibrated BFM
configurations without monomer overlap from our computationally much more
expensive previous studies obtained with topology violating moves Wittmer et
al. (2007b). Since the soft BFM simulations are also ergodic, these are the
relevant reference configurations. Starting with these configurations we
increase the temperature. As one may expect, the configurational properties
essentially are found unchanged for low temperatures ($x\gg 5$). Local L26
moves need to be added to global slithering-snakes moves for $x\geq 1$.
Otherwise the pure snake motion will become ineffective as it is well known
from a previous study of the snake dynamics without overlap Mattioni et al.
(2003). Simple slithering-snake moves without additional local moves are
sufficient for smaller $x$. We have crosschecked our results in this regime
for $N=2048$ and $N=8192$ using boxes of linear size $L=512$ by starting our
simulations with Gaussian chains at $x=0$ and increasing then the penalty.
Tables 2 and 3 present some system properties obtained for our reference
density $\rho=0.5/8$. Averages are performed over all chains and at least 100
configurations. Table 2 summarizes the properties extrapolated for
asymptotically long chains. Similar information is given in Table 3 for a
constant penalty $x=1$ as a function of chain length $N$. Density effects have
been studied only briefly for chains of length $N=8192$ and weak overlap
penalties ($x\ll 1$). This has been done to investigate the intrachain
contributions to the mean energy. We begin the discussion of our numerical
results by addressing this issue.

##  III Computational results

###  III.1 The mean overlap energy

#### Qualitative behavior.

From the numerical point of view the simplest thermodynamic property to be
investigated here is the mean interaction energy per monomer,
$e=\left<E\right>/n_{\text{mon}}$, due to the Hamiltonian, Eq. (10). Fig. 2
presents the dimensionless energy $y=e/\varepsilon$ as a function of the
reduced overlap penalty $x=\varepsilon/T=\varepsilon\beta$ foo (d) for melts
at monomer number density $\rho=0.5/8$ and various chain lengths $N$ as
indicated. We increase the temperature $T$ from the right to the left starting
with configurations obtained recently Wittmer et al. (2007b) for the classical
athermal BFM. As one expects, the interaction energy increases exponentially
for small $T$ and levels off for large $T$ where chains and their monomers
freely overlap ($x\ll 1$). The data for unconnected beads ($N=1$) represented
by the filled spheres and polymer chains ($N\gg 1$) are broadly speaking
similar, especially for large overlap penalties, $x>1$. Interestingly, the
mean energy of polymer melts increases more strongly for $x\ll 1$ as can be
seen better from the log-linear data representation chosen in the inset of
Fig. 2. Also shown in the inset is the mean intrachain self-energy per chain
monomer $e_{\text{self}}$ (filled triangles) obtained for the largest chain
length $N$ available for a given $x$. As can be seen also from Table 2, about
half of the energy of polymer melts for all $x$ is due to these intrachain
interactions. For $x\ll 1$ the self-energy becomes
$e_{\text{self}}/\varepsilon\approx 0.18$ which is exactly the observed energy
difference between polymer and bead systems.

#### Second virial contribution.

Before addressing this point let us first consider the energy of unconnected
soft BFM beads for which the second virial coefficient $v(x)$ has been given
in Eq. (11). Since $e=\partial_{\beta}(\beta f(\beta))$ the mean energy per
bead becomes to leading order McQuarrie (2000)

$y\approx\frac{1}{2}\rho\frac{\partial v(x)}{\partial
x}=\frac{\rho}{2}\left(\exp(-x/8)+3\exp(-x/4)+3\exp(-x/2)+\exp(-x)\right)$
(15)

corresponding to the first term in the third line of Eq. (2). Eq. (15) is
represented by the dashed line in Fig. 2. It corresponds to a Arrhenius
behavior with $y\approx\rho\exp(-x/8)/2$ (dash-dotted line) in the low
temperature region and, as expected, to $y\rightarrow\frac{1}{2}8\rho=4\rho$
for large temperatures. This simple formula predicts well the bead data over
the entire range of $x$ (underestimating slightly the mean energy at $x\approx
10$) and yields also a remarkable fit for polymer chains with larger overlap
penalties.

#### Self-energy in the high temperature limit.

The above-mentioned energy difference between polymer chains and beads for
$x\ll 1$ has been accounted for by the first free energy contribution
indicated in Eq. (2). This contribution is further investigated in Fig. 3
presenting data for such a high temperature ($x=0.001$) that the entropy
dominates essentially all conformational properties. The self-energy of a
chain is thus given by the probability $p(s,\bm{\delta})$ that a random walk
of $s$ BFM bonds returns to a relative position $\bm{\delta}$ with respect to
a reference monomer at $\bm{r}$. The self-energy per monomer is then

$e_{\text{self}}=\frac{2}{N}\sum_{\bm{\delta}}\sum_{s=2}^{N-1}(N-s)J(\bm{\delta})p(s,\bm{\delta})$
(16)

where the first sum runs over all positions with non-vanishing coupling
constant $J(\bm{\delta})$ as defined in Eq. (14). The probability
$p(s,\bm{\delta})$ and the weights $J(\bm{\delta})p(s,\bm{\delta})$ can be
tabulated in principle for small $s$. Since the return probability decreases
strongly with $s$, these model-specific small-$s$ values dominate the
integral, Eq. (16). As can be seen from the inset of Fig. 3 for single chains
(corresponding to an overall density $\rho=0$) the self-energy per monomer
becomes $e_{\text{self}}\approx 0.18\varepsilon$ for large $N$. The weak chain
length dependence visible in the panel stems from the upper integration
boundary over the Gaussian return probability which leads to a chain length
correction linear in $t\equiv 1/\sqrt{N-1}$. This is indicated by the bold
line presented in the panel. Also shown in the panel are energies for our
reference density $\rho=0.5/8$. They are shifted vertically by the mean field
energy $4\rho$ assuming that densities fluctuations of different chains do not
couple. The main panel presents the mean energy $e$ and the mean self-energy
$e_{\text{self}}$ as functions of the density $\rho$ for chains of length
$N=8192$. The self-energy (triangles) stays essentially density-independent.
The total interaction energy sums over the self-energy and mean-field energy
contributions as shown by the dashed line. The self-energy contribution can
only be neglected for very large densities corresponding to volume fractions
larger than unity.

#### Temperature dependence in the high temperature limit.

Summarizing Eqs. (2), (3) and (12) the energy per bead should scale to leading
order in $x$ as

$y\approx 0.18+4\rho-\
\underline{\frac{24^{3/2}}{\pi}\frac{\sqrt{x\rho}}{l^{3}\rho}}+\ldots\mbox{
for }x\ll 1$ (17)

where the two $x$-independent contributions have already been discussed above.
The underlined term stems from the density fluctuation contribution for long
polymer chains predicted by Edwards Doi and Edwards (1986) indicated in Eq.
(2). Here we have approximated the effective bond length $b(x)$ by the mean-
squared bond length $l\sim x^{0}$. As can be seen from Table 2 this
approximation (further discussed in Sec. III.6) is justified for $x\ll 1$. Eq.
(17) is indicated by the bold line in the inset of Fig. 2. It yields a
reasonable description of the temperature dependence of the mean energy for
small $x$. Since the energy is dominated by the two constant contributions to
Eq. (17) for $x\leq 0.001$ and since higher expansion terms become relevant
for $x>0.1$, the predicted $\sqrt{x}$-decay corresponds unfortunately only to
a small $x-$regime. In order to show that it is indeed the density fluctuation
term which dominates the temperature dependence for $x\ll 1$ we will consider
in the next paragraph the specific heat $c_{\text{V}}$, i.e. the second
derivative of the free energy with respect to $\beta$.

###  III.2 Energy fluctuations

#### Specific heat.

The fluctuations of the interaction energy are addressed in Fig. 4 displaying
the enthalpic contribution to the specific heat per monomer,
$c_{\text{V}}=-\beta^{2}\partial^{2}_{\beta}(\beta
f(\beta))=\frac{1}{T^{2}}\left(\left<E^{2}\right>-\left<E\right>^{2}\right)/n_{\text{mon}}$
McQuarrie (2000). Using again the second virial of soft BFM beads, Eq. (11),
one obtains the simple estimate for the specific heat

$c_{\text{V}}=\frac{\rho}{2}x^{2}\left(\frac{1}{8}\exp(-x/8)+\frac{3}{4}\exp(-x/4)+\frac{3}{2}\exp(-x/2)+\exp(-x)\right)$
(18)

represented by the dashed line. In the large-$x$ limit, this corresponds to
the exponential decay, $c_{\text{V}}\approx\rho x^{2}\exp(-x/8)/16$, indicated
by the dash-dotted line. For barely interacting beads ($x\ll 1$), Eq. (18)
yields a power-law limiting behavior, $c_{\text{V}}\approx\frac{27}{16}\rho
x^{2}\sim 1/T^{2}$. As can be seen from the plot, Eq. (18) predicts the energy
fluctuations of BFM beads for essentially all $x$, slightly underestimating
again the maximum of $c_{\text{V}}$ at $x\approx 10$. Since the specific heat
for beads and polymer chains is similar for $x>1$, the virial formula is also
good for polymer chains in this limit.

#### High temperature limit for polymer melts.

Strong chain length effects are, however, visible for high temperatures ($x\ll
1$) where the specific heat is found to increase monotonously with $N$. This
can better be seen from the inset where the specific heat is plotted as a
function of the reduced chain length $u=N/g$ with $g$ being the dimensionless
compressibility determined below in Sec. III.4. (Since $e$ and $c_{\text{V}}$
correspond to different derivatives of the free energy $f$ with respect to
$\beta$, there is obviously no thermodynamic inconsistency in the finding that
$c_{\text{V}}$ reveals much larger chain length effects than $e$.) For large
chains with $u\gg 1$ this increase levels off at a chain length independent
envelope

$c_{\text{V}}\approx\frac{24\sqrt{6}}{\pi}\frac{\rho^{1/2}}{l^{3}}x^{3/2}N^{0}+\ldots$
(19)

as anticipated by the density fluctuation contribution predicted in Eq. (2).
In contrast to Eq. (17) for the mean energy the density fluctuation term does
now correspond to the leading contribution to the numerically measured
property. This increases the range where the density fluctuation contribution
can be demonstrated to over three decades in $x$. Eq. (19) is indicated by the
bold lines in the main panel and the inset of Fig. 4.

#### Scaling with chain length $N$.

We have still to clarify the scaling observed for $u=N/g(x)\ll 1$ in the inset
of Fig. 4. Chains which are smaller than the thermal blob ($u\ll 1$) behave
obviously as random walks and the density fluctuations decouple from the
interaction strength $x$. Due to this factorization the specific heat for
these short chains must scale as $x^{2}$, just as for beads. This is shown in
the main figure for $N=16$ (thin solid line). Consistency with Eq. (19)
implies the scaling $c_{\text{V}}(x,N)\approx x^{3/2}\rho^{1/2}h(u)$ with
$h(u)$ being a universal function scaling as $h(u)\sim u^{0}$ in the large-$u$
limit. Since $c_{\text{V}}\sim x^{2}$ and $g(x)\sim 1/(x\rho)$ (as shown in
Sec. III.4) for $u\ll 1$, it follows that $h(u)\approx u^{1/2}$ as confirmed
by the dashed line indicated in the inset. Hence, $c_{\text{V}}\sim\rho
x^{2}N^{1/2}$ for $u\ll 1$.

###  III.3 Chemical potential

#### Scaling of the chemical potential.

Fig. 5 presents the excess chemical potential per monomer,
$y\equiv\mu_{\text{ex}}/TN$, obtained using thermodynamic integration (as
explained below) for three chain lengths $N=1$, $64$, and $2048$ as functions
of the overlap penalty $x=\varepsilon/T$. As one expects, the chemical
potential increases first linearly with $x$ and then levels off. Chain length
effects are again small on the logarithmic scale chosen in the plot foo (k).
For large $x$ the chemical potential becomes slightly larger for beads
($y\approx 2.64$) than for long chains where $y\approx 2.1$ (dash-dotted
line). That the chemical potential of polymer chains is reduced compared to
melts of unconnected beads is of course expected for all $x$ due to the
(effectively) attractive bond potential. For weak interactions this reduction
should be described by the density fluctuation contribution to the free energy
[Eq. (2)] which corresponds to an excess chemical potential

$y=\frac{\partial(\beta f(\beta)\rho)}{\partial\rho}\approx v(x)\rho\left(1-\
\underline{\frac{3\sqrt{3}}{\pi}\frac{(v(x)\rho)^{1/2}}{b(x)^{3}\rho}}+\
\ldots\right)\mbox{ for }x\ll 1$ (20)

with $v(x)$ being the second virial of unconnected beads. The dashed line in
Fig. 5 presents the leading contribution $v(x)\rho$ for unconnected beads, the
bold line in addition the underlined connectivity contribution given in Eq.
(20). It turns out that the simple second virial approximation provides a much
better fit of the data over the entire $x$-range than the full prediction.
(The weak underestimation of the chemical potential for $x>10$ must be
attributed to higher order correlations relevant in this limit.) That the
density fluctuation contribution overestimates the reduction of the chemical
potential for $x>1$ is in agreement with Eq. (4) and the Ginzburg parameters
indicated in Table 2. Hence, Eq. (20) in principle can be tested only for
$x\ll 1$. Unfortunately, in this limit the relative correction, scaling as
$\sqrt{x/\rho}$, becomes too small to allow a fair test of the theory. A
systematic variation of the density and an improved numerical accuracy of the
chemical potentials measured are warranted to achieve this goal.

#### Thermodynamic integration.

We now explain how the data of Fig. 5 have been obtained numerically. The
simple insertion method due to Widom Frenkel and Smit (2002) obviously becomes
rapidly inefficient with increasing $x$, especially for longer chains.
Slightly generalizing the method suggested in Müller and Paul (1994); Wilding
and Müller (1994) we therefore have performed a thermodynamic integration
Frenkel and Smit (2002)

$\beta\mu_{\text{ex}}=\int_{\lambda(\varepsilon)}^{1}d\lambda\frac{\left<N_{\text{sg}}\right>}{\lambda}$
(21)

over discrete values of the interaction affinity
$\lambda=\exp(-\varepsilon_{\text{sg}}\beta/8)$ characterizing the excluded
volume interaction of a ghost (g) chain that is gradually inserted into an
equilibrated system (s). $\left<N_{\text{sg}}\right>$ refers to the mean
number of lattice sites where system and ghost monomer cube corners overlap at
a given interaction $\lambda$. Generalizing the Potts spin mapping, Eq. (13),
of the excluded volume interactions for homopolymers presented above, we use
now two spin lattices, $\sigma_{\text{s}}(\bm{r})$ describing (as before) the
interaction of the system monomers and $\sigma_{\text{g}}(\bm{r})$ the ghost
chain. The spin lattices are kept at the same temperature and are both
characterized by the same (arbitrary) energy $\varepsilon=1$ which has to be
paid for a complete overlap of two system monomers or two ghost monomers. The
interaction of both spins is described by

$\Delta
E_{sg}=\sum_{\bm{r}}\sigma_{\text{s}}(\bm{r})\sum_{\bm{\delta}}J_{\text{sg}}(\bm{\delta})\sigma_{\text{g}}(\bm{r}+\bm{\delta})$
(22)

with coupling constants
$J_{\text{sg}}(\bm{\delta})\sim\varepsilon_{\text{sg}}$ defined as in Eq. (14)
taken apart the energy parameter $\varepsilon$ which is replaced by the
tuneable interaction energy $\varepsilon_{\text{sg}}$. Starting with decoupled
system and ghost configurations at $\varepsilon_{\text{sg}}=0$, i.e.
$\lambda=1$, we gradually increase the interaction parameter up to
$\varepsilon_{\text{sg}}=\varepsilon$, i.e.
$\lambda(\varepsilon)=\exp(-\varepsilon\beta/8)$, always keeping the coupled
system at equilibrium. Monitoring the distribution of the number
$N_{\text{sg}}$ of overlaps between system and ghost cube corners we use
multihistogram methods as described in Müller and Paul (1994); Wilding and
Müller (1994) to improve the precision of the integral.

The mean overlap number $\left<N_{\text{sg}}\right>$ (devided by $8N$) is
shown in the inset of Fig. 5 as a function of $\lambda$ for $N=2048$ and two
inverse temperatures $x=3$ and $x=100$. Starting from $\lambda=1$ the overlap
number decreases monotonously with increasing coupling between system and
ghost monomers, i.e. decreasing $\lambda$. Interestingly, a power law behavior
$\left<N_{\text{sg}}\right>/N\approx\lambda^{1/4}$ is found empirically for
large $x\gg 10$ (dashed line). Fitting this power law and integrating then
analytically Eq. (21) provides a useful crosscheck of the numerical
integration using the multihistogram analysis. This is a technically important
finding, since the multihistogram analysis requires overlapping distributions
of $N_{\text{sg}}$ and hence much more equilibrated intermediate values
$\lambda$ as indicated for $x=100$. A detailed explanation for the observed
power law still is missing, but it is presumably due to the systematic
screening of the long range correlations of the ghost chain which is swollen
at $\lambda=1$ becoming more and more Gaussian as it feels the compression due
to the surrounding host chains de Gennes (1979).

###  III.4 The compressibility

#### Compressibility $g(x,N)$ and excess compressibility $g_{\text{ex}}(x)$.

To test the key relation Eq. (5) announced in the Introduction we need
accurate values for the dimensionless compressibilities
$g(x)\equiv\lim_{N\to\infty}g(x,N)$ of asymptotically long chains. As
suggested by Eq. (1), we compute first the dimensionless compressibility
$g(x,N)=\lim_{q\to 0}G(q)$ from the low-$q$ limit of the total monomer
structure factor for different overlap penalties $x$ and chain lengths $N$
(see below for details). These raw data are presented in Fig. 6 as a function
of $x$. As one expects, $g(x,N)$ decreases monotonously with overlap strength
$x$. In contrast to the thermodynamic integration performed for the chemical
potential [Eq. (21)], the structure factor measures the complete
compressibility, not just the excess contribution. The strong $N$-dependence
visible in the plot thus is expected from the translational entropy of the
chains. As can be seen, e.g., from Eq. (2) or from the virial expansion of
polymer solutions de Gennes (1979), the compressibility can be written in
general as

$\frac{1}{g(x,N)}=\rho\ \frac{\partial^{2}(\beta
f(\beta)\rho)}{\partial\rho^{2}}=\frac{1}{N}+\frac{1}{g_{\text{ex}}(x,N)}$
(23)

for all $x$ with $g_{\text{ex}}(x,N)$ being the excess contribution to the
compressibility which may, at least in principle, depend on $N$ foo (k). As
can be seen from the inset of Fig. 6, all compressibilities collapse, however,
on one $N$-independent master curve if one plots $1/g(x,N)-1/N$ as a function
of $x$, even the compressibilities obtained for unconnected beads ($N=1$).
Within numerical accuracy the $N$-dependence observed for $g(x,N)$ can
therefore be attributed to the trivial osmotic contribution and the excess
compressibility $g_{\text{ex}}\sim N^{0}$ is thus identical to the
compressibility $g(x)$ of asymptotically long chains. The bold line indicated
in the inset presents the best values of $g(x)$ summarized in Table 2. These
values have been obtained from the excess compressibilities for the largest
chain length available for $x\geq 0.001$. A precise numerical determination of
$g_{\text{ex}}(x)$ becomes impossible for even smaller overlap penalties. We
thus have used for the smallest $x$-values sampled the theoretical prediction

$\frac{1}{g(x)}\approx v(x)\rho\left(1-\
\underline{\frac{3\sqrt{3}}{2\pi}\frac{(v(x)\rho)^{1/2}}{b(x)^{3}\rho}}\ldots\right)\mbox{
for }x\ll 1$ (24)

due to the free energy [Eq. (2)] postulated in the Introduction. The prefactor
$v(x)\rho$ representing the bare monomer interaction is indicated by the
dashed line in the main panel of Fig. 6. Hence, $g(x)\approx 1/(8x\rho)$ for
weak interactions, i.e. the compressibility increases linearly with
temperature. The underlined term is the leading correction due to the density
fluctuation contribution to the free energy. It implies that the excess
compressibilities for polymer melts and unconnected beads cannot be completely
identical. However, as before for the chemical potential, the difference is
far too small to be measurable in the limit where Eq. (24) applies. Although
this result is unfortunate from the theoretical point of view, the data
collapse observed in the inset suggests that it is acceptable to numerically
estimate the long chain compressibility $g(x)$ by computing the structure
factors of rather short chains.

#### Total monomer structure factor.

We now turn to the total structure factor $G(q)$ shown in Fig. 7 to explain
how the compressibilites $g(x,N)$ have been obtained. Only chains of length
$N=2048$ are presented for clarity. The total monomer structure factor is
obtained by computing
$G(q)=\frac{1}{n_{\text{mon}}}\left<[\sum_{n}\cos(\bm{q}\cdot\bm{r}_{n})]^{2}+[\sum_{n}\sin(\bm{q}\cdot\bm{r}_{n})]^{2}\right>$
where the sums run over all the $n_{\text{mon}}$ monomers of the box and the
wavevectors are commensurate with the cubic box of linear length $L$. Since
the smallest possible wavevector is $2\pi/L$, it thus is important to have
large box sizes to scan over a sufficiently important $q$-range allowing a
reasonable determination of $g(x,N)$. Note that around and above $q\approx 2$
monomer structure and lattice effects become important. Since only smaller
wavevectors are of interest if one is interested in universal physical
behavior, we will focus below on wavevectors $q<1$. For comparison, we have
also included the single chain form factor
$F(q)=\frac{1}{N}\left<[\sum_{n=1}^{N}\cos(\bm{q}\cdot\bm{r}_{n})]^{2}+[\sum_{n=1}^{N}\sin(\bm{q}\cdot\bm{r}_{n})]^{2}\right>$
Doi and Edwards (1986) for $x=0.001$ (bold line). Note that the qualitative
shape of $F(q)$ — decaying monotonously with $q$ from its maximum value
$F(q=0)=N$ — depends very little on the temperature (not shown). We remind
that the “random phase approximation” (RPA) formula de Gennes (1979); Doi and
Edwards (1986)

$\frac{1}{G(q)}=\frac{1}{F(q)}+\frac{1}{g(x)}$ (25)

relates the total structure factor to the measured form factor. Eq. (25) is of
course consistent with Eq. (23) in the $q\to 0$ limit. It allows to directly
fit for the excess compressibility $g_{\text{ex}}(x)$ using the measured
structure factor $G(q)$ and form factor $F(q)$, at least in the $x$-range
where the RPA approximation applies. As may be seen from the figure, $G(q)$
indeed decreases systematically with $x$, i.e. with decreasing $g(x)$. For
large temperatures ($x\leq 3$) it also decays monotonously with $q$, again in
agreement with Eq. (25). Interestingly, this becomes qualitatively different
for larger excluded volume interactions ($x>3$) where the total structure
factor is essentially constant (in double-logarithmic coordinates), very
weakly increasing monotonously with $q$. The RPA formula apparently does not
apply in this limit in agreement with Eq. (4). Fortunately, this is of no
concern for our main purpose — to compute $g(x)$ — since in precisely this
limit the compressibility is readily obtained from a broad plateau (even for
much smaller boxes) which in addition becomes chain length independent, as we
have already seen from the inset of Fiq. (6). Using boxes with $L=256$ it is
possible to directly measure the plateau values for $x\leq 0.3$. For smaller
$x$ we have simulated boxes with $L=512$ containing $n_{\text{mon}}\approx
8.4\cdot 10^{6}$ monomers and corresponding to a smallest wavevector $q\approx
0.01$. This box size becomes again insufficient for the largest temperatures
we have simulated, as shown in Fig. 7 for $x=0.001$ (dashed line). It is for
these values where the RPA formula, Eq. (25), allowing to fit the deviation
from the (barely visible) plateau, has been particulary useful.

#### Approximated RPA formula.

We finally note that in the intermediate wavevector regime (where $q$
corresponds to distances much smaller than the radius of gyration and much
larger than the monomer size) the general RPA Eq. (25) may be written as

$\frac{1}{G(q)}=\frac{1}{g(x)}+\frac{1}{12}b^{2}(x)q^{2}=\frac{1}{g(x)}\left(1+(q\xi)^{2}\right)$
(26)

which justifies the definition given in Eq. (3) for the correlation length of
the density fluctuations $\xi$. Here we have used that the form factor becomes
$F(q)\approx 12/b^{2}q^{2}$ Doi and Edwards (1986). This assumes that
corrections to Gaussian chain statistics may be ignored Wittmer et al.
(2007a); Beckrich et al. (2007) and that finite chain size effects are
negligible. From the numerical point of view the approximated RPA Eq. (26) has
the disadvantage that the effective bond length $b(x)$ needs to be determined
first. As shown in Fig. 8, it has the advantage that it allows for an
additional test of the values of $g(x)$ and $b(x)$ indicated in Table 2. The
main panel presents the rescaled structure factor $G(q)/g(x)$ for chains of
length $N=8192$ as a function of $Q\equiv q\xi$ with $\xi$ being obtained from
$g(x)$ using Eq. (3). All data collapse on the master curve $1/(1+Q^{2})$
indicated by the bold line provided that the wavevector $q$ remains
sufficiently small and no local physics is probed. That the used
compressibilities are accurate is emphasized further in the inset where
$g(x)/G(q)-1$ is plotted as a function of $Q^{2}$ using only sufficiently
small wavevectors $q$. According to Eq. (26) all data should collapse on the
bisection line in double-logarithmic coordinates if the correct
compressibilities are used. This is indeed the case. Unfortunately, even this
rather precise method still does not allow to demonstrate the density
fluctuation contribution in Eq. (24) for $x\ll 1$ since the same scaling
collapse is obtained for the simple choice $1/g(x)\equiv v(x)\rho$. Please
note the weak deviations visible for $x=1$ which is due to the breakdown of
the RPA formula for large $x$ mentioned above.

###  III.5 Bond properties

Up to now we focused on some thermodynamic features of the soft BFM model,
i.e. on large-scale properties. Turning to configurational issues we begin by
characterizing local-scale features of the algorithm. (Readers only interested
in universal properties may wish to skip this paragraph.)

#### Mean bond length.

By definition of our version of the BFM algorithm the bond length is allowed
to fluctuate strongly between $2$ and $\sqrt{10}$. One expects that switching
on the excluded volume interaction, i.e. decreasing the temperature, will
suppress large bonds due to the increasing pressure. The mean bond length
commonly is characterized by the root-mean-square length
$l=\left<\bm{l}^{2}\right>^{1/2}$. (Other moments would yield similar
results.) The mean bond length rapidly becomes ($N>20$) chain length
independent Paul et al. (1991a). As a function of overlap penalty $x$ the bond
length shows a monotonous decay between $x\approx 3$ and $x\approx 20$ as can
be seen from Fig. 9. As other local properties, $l$ becomes constant in the
small-$x$ and large-$x$ limits (dashed horizontal lines). The value
$l(x=0)=2.718$ gives the lower bound for the effective bond length $b(x)$ of
asymptotically long chains (stars) obtained below.

#### Mean bond angle and local chain rigidity.

Defining the bond angle $\theta$ between two subsequent bonds by the scalar
product $\cos(\theta)=\bm{e}_{n}\cdot\bm{e}_{n+1}$ of the normalized bond
vectors $\bm{e}_{i}=\bm{l}_{i}/|\bm{l}_{i}|$, the local chain rigidity may be
characterized by $\left<\theta\right>$ and $\left<\cos(\theta)\right>$. Note
that $\left<\theta\right>$ and $\left<\cos(\theta)\right>$ can be regarded as
chain length independent, just as the mean bond length. As can be seen from
Table 2, the local rigidity is negligible for $x\ll 1$, i.e.
$\left<\theta\right>\approx 90^{\circ}$ and $\left<\cos(\theta))\right>\approx
0$ due to the symmetry of the distribution $p(\theta)$ with respect to
$90^{\circ}$. The rigidity then increases around $x\approx 1$ and becomes
constant again for large $x$ where $\left<\theta\right>\approx 82.2^{\circ}$
and $\left<\cos(\theta)\right>\approx 0.106$. The increase of the local
rigidity for larger excluded volume interactions is of course expected due to
the suppression of immediate backfoldings corresponding to bond angles
$\theta>143^{\circ}$ Wittmer et al. (1992). The distribution $p(\theta)$
therefore becomes lopsided towards smaller $\theta$ (not shown). It is well
known Doi and Edwards (1986) that for chains characterized by the “freely
rotating” (FR) chain model such a local rigidity would lead to an effective
bond length $b(x)=l(x)\sqrt{c_{\text{FR}}}$ with
$c_{\text{FR}}=(1+\left<\cos(\theta)\right>)/(1-\left<\cos(\theta)\right>)$.
This simple model, indicated by the crosses in Fig. 9, yields a qualitatively
reasonable trend (monotonous increase of the effective bond length at
$x\approx 1$) but fails to fit the directly measured effective bond lengths
quantitatively.

###  III.6 Chain and segment size

#### Total chain size $R_{\text{e}}(N)$.

One way to characterize the total chain size is to measure the second moment
of the chain end-to-end distance
$R_{\text{e}}^{2}(x,N)=\left<(\bm{r}_{N}-\bm{r}_{1})^{2}\right>$. (Other
moments yield similar behavior Wittmer et al. (2007b).) We consider the
effective bond length $b(x,N)\equiv R_{\text{e}}(x,N)/\sqrt{N-1}$ to compare
the measured chain size with the ideal chain behavior which is commonly taken
as granted de Gennes (1979); Doi and Edwards (1986); Flory (1949) and which is
the basis of our perturbation calculation. We use the notation
$b(x)\equiv\lim_{N\to\infty}b(x,N)$ for the effective bond length of
asymptotically long chains foo (a). The effective bond lengths $b(x,N)$ for
$N=64$ and $N=2048$ and the asymptotic limit $b(x)$ — obtained by
extrapolation as described below — are presented in Fig. 9 as functions of
$x$. Obviously, $b(x,N)\to l(x=0)$ for all $N$ in the small-$x$ limit.
$b(x,N)$ increases then in the intermediate $x$-window before it levels off at
$x\approx 10$. The swelling due to the excluded volume interaction is the
stronger the larger the chain length, i.e. $b(x,N)$ increases monotonously
with $N$. This swelling therefore cannot be attributed to a local persistence
length as described, e.g., by the freely-rotating chain model.

The chain length effects can be seen better in Fig. 10 where we have plotted
$b(x,N)$ for several penalties $x$ as a function of $t=1/\sqrt{N-1}$. The
choice of the horizontal axis is motivated by Eq. (7) suggesting the linear
relation

$b^{2}(x,N)\approx b^{2}(x)\ (1-c(x)c_{\text{s}}(x)t)$ (27)

for $u=N/g\gg 1$ with $c_{\text{s}}(x)\equiv\sqrt{24/\pi^{3}}/\rho b(x)^{3}$
being the swelling coefficient defined in Sec. I. $c(x)$ is an additional
numerical prefactor of order unity which has been introduced in agreement with
Eq. (19) of Ref. Wittmer et al. (2007b). The reason for this coefficient is
that the corrections to Gaussian behavior differ slightly for internal chain
segments [as described by Eq. (7)] and the total chain size which is
characterized in Fig. 10. It has been shown that $c\to 1.59$ for large $N$
Wittmer et al. (2007b). However, since this value corresponds to the limit of
a very slowly converging integral Wittmer et al. (2007b) it is better to use
Eq. (27) as a two-parameter fit for $b(x)$ and $c(x)$ and to crosscheck then
whether the fitted $c$ is of order unity. As shown in the figure for three
overlap penalties, this method can be used reasonably for overlap penalties as
low as $x\approx 0.1$, albeit with decreasing $x$ systematically
underestimating the “true” $b$-values indicated in Table 2. Please note that
$N/g\approx 400$ for $x=0.1$ and $N=8192$. Chains with $N\gg 8192$ would be
required to use this method for even smaller $x$. In this limit it is better
to use as a simple first step the value $b(x,N=8192)$ of the largest chain
length simulated as a (rather reasonable) lower bound for $b(x)$.

#### Segment size $R(s)$.

As we have already stressed in Ref. Wittmer et al. (2007b), it is technically
better to extrapolate for the effective bond length $b$ using the distribution
$R(s)$ of the mean-squared size of segments of curvilinear arc-length $s$
defined in Sec. I. It follows from $c>1$ that the total chain ratio
$b^{2}(x,N)$ converges less rapidly to the asymptotic Gaussian behavior as
$R(s)^{2}/s$. (See Fig. 4 of Ref. Wittmer et al. (2007b).) The ratio of
$R^{2}(s)/s$ as a function of $s$ is plotted in the inset of Fig. 11 for
$N=2048$ and for several $x$. As can be seen, it increases systematically with
segment length $s$. The swelling levels off for large $s$, but rather
gradually. Therefore it would not be appropriate to identify the maximum
around $s\approx N$ as the asymptotic plateau. This again would yield an
underestimation of $b(x)$. A more precise method to obtain $b(x)$ uses the
predicted correction, Eq. (7), to the Gaussian limit. We recommend to plot,
using double-logarithmic coordinates, $1-R(s)^{2}/b^{2}s$ as a function of
$c_{\text{s}}/\sqrt{s}$ and to tune $b(x)$ until the data for intermediate
chain segments with $g\ll s\ll N$ collapses on the bisection line. This one-
parameter fit yields good estimates down to $x=0.01$ where $N/g\approx 40$.
Since the corresponding plot is very similar to Fig. 5 of Ref. Wittmer et al.
(2007b), it is not reproduced here. We rather show in Fig. 11 a scaling plot
motivated by the key relation, Eq. (5), which uses our best values of $g(x)$
and $b(x)$ for asymptotically long chains (Table 2). The data collapse on the
theoretical prediction (bold line) is remarkable, especially considering that
$u$ covers seven orders in magnitude. The “Fixman limit”, Eq. (8), for $u\ll
1$ (dashed line) fits the data with the smallest overlap penalty $x\approx
0.001$ confirming the chosen value of $b$. The limiting behavior for $u\gg 1$
[Eq. (7)], characterizing an incompressible melt of thermal blobs of length
$g$, is indicated by the dash-dotted line.

#### Predicting the effective bond length.

Up to now, we have used theory to improve the fit of $b(x)$, rather than to
predict it from the thermodynamic properties and local model features such as
the bond length $l(x)$. The increase of the effective bond length for weakly
interacting and asymptotically long polymer melts has been calculated long ago
by Edwards [see Eq. (5.55) of Ref. Doi and Edwards (1986) or Eq. (11) of Ref.
Wittmer et al. (2007b)]. Reformulated using our notations and substituting the
bare excluded volume parameter $v(x)$ by $1/g\rho$ Semenov and Obukhov (2005);
Wittmer et al. (2007b); foo (c) his result reads

$b^{2}=l^{2}\left(1+\frac{\sqrt{12}}{\pi}G_{\text{z}}\right)\mbox{ with
}G_{\text{z}}=\frac{1}{\sqrt{g}b_{\text{r}}^{3}\rho}$ (28)

where $b_{\text{r}}$ is the bond length of the unperturbed reference chain of
the calculation and $G_{\text{z}}(b_{\text{r}},g)$ the relevant Ginzburg
parameter quantifying the strength of the interaction acting on a chain
segment of length $s=g$ (see Eq. (6) of Ref. Wittmer et al. (2007b)). Since
$G_{\text{z}}$ becomes small for large compressibilities $g(x)$, one expects
good agreement with our data for small $x$. The question is now what actually
might be the best reference bond length to allow a prediction over the
broadest possible $x$-range. The simplest choice to associate $b_{\text{r}}$
with the bond length $l(x)$ yields the dash-dotted line indicated in Fig. 9.
As can be seen, this choice of $b_{\text{r}}$ allows a reasonable prediction
only up to $x\approx 0.01$. The predictive power of Eq. (28) can be
considerably improved over nearly two decades up to $x\approx 1$ if one
applies the formula iteratively starting with $b_{\text{r}}=l$ and using the
effective bond length obtained as input for the Ginzburg parameter ($b\to
b_{\text{r}}$) in the next step. This recursion relation converges rapidly as
shown by the bold line indicated in Fig. 9 obtained after 20 iterations. This
iterative renormalization of the bond length of the reference chain and the
associated Ginzburg parameter has been suggested by Muthukumar and Edwards
Muthukumar and Edwards (1982). Essentially the same result is obtained up to
$x\approx 1$ if one sets directly $b_{\text{r}}=b$ using the measured
effective bond length (not shown), i.e. these values correspond to the fix-
point solution of Eq. (28). The Ginzburg parameters $G_{\text{z}}$ obtained
using the measured $b(x)$ are listed in Table 2. Note that $G_{\text{z}}<0.34$
for $x<1$ where Eq. (28) fits our data nicely. The fix-point solution of Eq.
(28) does not capture correctly the leveling off of $b(x)$ setting in above
$x\approx 1$. Since the Ginzburg parameter becomes there of order one (Table
2), this is to be expected. In summary, we have shown that the iteration of
Eq. (28) allows a good prediction for $b(x)$ for $x<1$ such that
$G_{\text{z}}\ll 1$. If reliable values for the compressibility $g(x)$ are
available [by means of the extrapolation method implied by Eq. (23)], this is
the method of choice if one cannot afford to simulate very long chains.

###  III.7 Bond-bond correlation function

#### Motivation and theoretical prediction.

We return now to the deviations from Flory’s ideality hypothesis predicted in
Eq. (5) for the mean-square segment size $R^{2}(s)$. As we have seen above
(Fig. 11), this property requires to substract a large Gaussian contribution
$b^{2}s$ from the measured $R^{2}(s)$ to demonstrate the existence and the
scaling of the deviations. Unfortunately, this requires as a first step the
precise determination of the effective bond length $b(x)$ for asymptotically
long chains which might not always be available. Indeed we have used in the
preceeding Sec. III.6 the fact that the scaling of Eq. (5) critically depends
on this accurate value to improve the estimation of the effective bond length
$b(x)$ for asymptotically long chains. Hence, it would be nice to demonstrate
directly the scaling implied by our key prediction without any tuneable
parameter. The trick to achieve this is similar to our demonstration of the
density fluctuation contributions to the free energy, Eq. (2), presented in
Sec. III.2: We consider the curvature of $R^{2}(s)$, i.e. its second
derivative with respect to $s$, to eliminate the large Gaussian contribution.
In principle this can be achieved by fitting $R^{2}(s)$ by a sufficiently high
polynomial whose second derivative with respect to $s$ then is compared to the
theory. Following Wittmer et al. (2004); Wittmer et al. (2007b) we use a more
direct numerical route where we compute the well-known bond-bond correlation
function $P(s)\equiv\left<\bm{l}_{m=n+s}\cdot\bm{l}_{n}\right>/l^{2}$ with
$\bm{l}_{i}=\bm{r}_{i+1}-\bm{r}_{i}$ denoting the bond vector between two
adjacent monomers $i$ and $i+1$ and $l^{2}$ the mean-square bond length (Sec.
III.5). The average is performed as before over all chains and all pairs of
monomers $(n,m+s)$ possible in a chain of length $N$. We use this definition
rather than the more common first Legendre polynomial
$\left<\bm{e}_{n}\cdot\bm{e}_{m}\right>$ since it allows to relate the bond-
bond correlation function to the segment size by

$P(s)=\frac{1}{2l^{2}}\frac{d^{2}}{ds^{2}}R^{2}(s).$ (29)

This formula is obtained from
$\left<\bm{l}_{n}\cdot\bm{l}_{m}\right>\approx\left<\partial_{n}\bm{r}_{n}\cdot\partial_{m}\bm{r}_{m}\right>=-\partial_{n}\partial_{m}\left<(\bm{r}_{n}-\bm{r}_{m})^{2}\right>/2$.
Using Eq. (29) the key prediction, Eq. (5), implies for the bond-bond
correlation function

$P(s)=\frac{c_{\text{P}}}{g(x)^{3/2}}\left(\frac{4}{\sqrt{u}}-4\sqrt{2\pi}e^{2u}\mbox{erfc}(\sqrt{2u})\right)$
(30)

where we have introduced the coefficient
$c_{\text{P}}=c_{\text{s}}(b/l)^{2}/8$. Eq. (30) corresponds to the limiting
behavior

$P(s)\approx\frac{c_{\text{P}}}{g^{3/2}}\frac{4}{\sqrt{u}}$ (31)

for small reduced arc-lengths $u\ll 1$. The explicit compressibility
dependence drops out in the opposite limit ($u\gg 1$) where the bond-bond
correlation function becomes

$P(s)\approx c_{\text{P}}/s^{3/2},$ (32)

in agreement with Eq. (7). Please note that $c_{\text{P}}$ depends implicitly
on the compressibility. (Obviously, both asymptotic behaviors could have been
obtained directly from the corresponding limits for $R^{2}(s)$, Eqs. (8) and
(7).)

#### Numerical confirmation.

The bond-bond correlation function $P(s)$ for different overlap penalties $x$
is presented in Fig. 12 for chains of length $N=2048$. As can be seen from the
unscaled data shown in the inset, $P(s)$ approaches a power law with exponent
$\omega=1/2$ (dashed line) in the limit of weak overlap penalties in agreement
with Eq. (31). For $x\geq 1$ our data is compatible with an exponent
$\omega=3/2$ (dash-dotted line) as suggested by Eq. (32). Hence, we have
demonstrated without any tunable parameter that Flory’s ideality hypothesis is
systematically violated for all segment lengths $s$ and all overlap penalties
$x$.

We consider now the prefactors and the scaling with $x$. As suggested by Eq.
(30), the main figure presents $P(s)/(c_{\text{P}}/g(x)^{3/2})$ as a function
of the reduced arc-length $u=s/g(x)$ using the dimensionless compressibilities
$g(x)$ and effective bond lengths $b(x)$ from Table 2. The data collapse is
remarkable as long as $1\ll s\ll N$. The relation Eq. (30) is indicated by the
bold line; it is in perfect agreement with the simulation data foo (l). The
asymptotic power law behavior with exponents $\omega=1/2$ for $u\ll 1$ and
$\omega=3/2$ for $u\gg 1$ is shown by the dashed and dash-dotted lines,
respectively. As predicted by Eq. (32), one recovers the power law
$P(s)=c_{\text{P}}/s^{3/2}$ — already observed for incompressible melts
Wittmer et al. (2004); Wittmer et al. (2007b) — for scales larger than the
thermal blob irrespective of the blob size $g$. This demonstrates that the
exponent $\omega=3/2$ is not due to local physics on the monomer scale, since
for $s\gg g\gg 1$ distances much larger than the monomer or even the thermal
blob are probed.

##  IV Conclusion

#### Thermodynamic properties of a BFM version with finite overlap penalty.

In this paper we have discussed a generalization of the standard bond-
fluctuation model (BFM) where the monomers may overlap subject to a finite
energy penalty $\varepsilon$ (Fig. 1). This allows us to switch on
systematically the excluded volume interaction between the monomers as
suggested by perturbation theory Doi and Edwards (1986) and to tune the
density fluctuations of the solution at constant monomer density. In this
study we have focused on dense polymer melts containing flexible linear chains
which are athermal apart from the finite overlap penalty. The central
thermodynamic parameter characterizing these systems is the excess part of the
dimensionless compressibility $g=T\kappa_{\text{T}}\rho$ of the solution which
has been obtained directly from the low-wavevector limit of the static
structure factor (Figs. 7 and 8). Scanning the overlap penalty (or,
equivalently, the temperature $T$ foo (d)) from $x=\varepsilon/T=\infty$ (no
overlap) down to $x=0.0001$ leads to a variation of $g(x)$ over four orders of
magnitude (Fig. 6). This allows for a systematic study of the thermodynamic
properties (Figs. 2-6) and the intrachain configurational statistics (Figs.
9-12). Particular attention has been paid to the thermodynamic properties of
weakly interacting melts ($x\ll 1$). We have verified that our results are
consistent with the free energy, Eq. (2), postulated in agreement with Edwards
Doi and Edwards (1986). The main result of this part of our study is that we
have been able to demonstrate the density fluctuation contribution to the free
energy induced by the chain connectivity from the scaling of the specific heat
$c_{\text{V}}$ with respect to overlap penalty $x$ (Fig. 4).

#### Intrachain conformational properties: Violation of Flory’s ideality
hypothesis.

The broad variation of $g(x)$ puts us into a position to test the recently
proposed Eq. (5) predicted by perturbation theory Meyer et al. (2008); Wittmer
et al. (2009) describing the systematic swelling of chain segments as function
of the segment size $s$ and the compressibility $g(x)$. As outlined in the
Introduction, this key relation suggests that the repulsive interactions
between chain segments in the same chain are not fully screened at variance to
Flory’s ideality hypothesis for polymers in dense melts de Gennes (1979). The
violation of the ideality hypothesis is demonstrated in Fig. 11 for the mean-
square segment size $R^{2}(s)$ and in Fig. 12 for the bond-bond-correlation
function $P(s)$. We show that data obtained for systems with very different
compressibilities $g(x)$ can be superimposed on the predicted master curves if
plotted as a function of the reduced arc-length $u=s/g$. The scaling of $R(s)$
allows a precise determination of an important intrachain property, the
effective bond length $b(g)$ for asymptotically long chains (Fig. 9). These
values compare well for $x<1$ with the fix-point solution of the recursion
relation, Eq. (28) Doi and Edwards (1986); Muthukumar and Edwards (1982);
Wittmer et al. (2007b). The bond-bond correlation function $P(s)$ being the
second derivative of $R^{2}(s)$ with respect to $s$ allows an even more direct
test of the predicted deviations. The reason is that the large Gaussian
contribution $b^{2}s$, which must be subtracted from $R^{2}(s)$ (see the
vertical axis of Fig. 11), drops out due to the differentiation. In contrast
to Flory’s hypothesis, $P(s)$ does not vanish rapidly on scales corresponding
to the local persistence length (Fig. 12). In perfect agreement with theory
foo (l), the scaling plot shows two power law regimes characterized by
exponents $\omega=1/2$ for small ($u\ll 1$) and $\omega=3/2$ for large ($u\gg
1$) reduced arc-length.

The central result of this study is that even for polymer melts with finite
overlap penalty excluded volume interactions are not fully screened. If
distances smaller than the thermal blob size are probed the chains are swollen
according to the standard Fixman parameter expansion. More importantly, even
on distances larger than the thermal blob size ($s/g\gg 1$) deviations from
ideal chain behavior are found. Interestingly, in this limit the explicit
compressibility dependence drops out and the relations established for
incompressible melts Wittmer et al. (2004); Wittmer et al. (2007b) are
recovered. This shows that soft melts behave on large scales as incompressible
packings of blobs.

#### Outlook.

Since the presented soft BFM is fully ergodic (in contrast to the classical
BFM) and very efficient, it may be an interesting alternative to various
popular coarse-grained simulation approaches using soft effective interaction
parameters Likos (2001); Eurich and Maass (2001); Yatsenko et al. (2004);
Müller and Smith (2005); Daoulas and Müller (2006). The presented model is
part of a broader attempt to describe systematically the effects of correlated
density fluctuations in dense polymer systems, both for static Semenov (1996);
Semenov and Johner (2003); Semenov and Obukhov (2005); Obukhov and Semenov
(2005) and dynamical Semenov (1997); Semenov and Rubinstein (1998); Mattioni
et al. (2003) properties. This also involves the comparison with (off-lattice)
molecular dynamics simulation using a standard bead-spring model which is
discussed elsewhere Meyer and Müller-Plathe (2001); Wittmer et al. (2007b);
Meyer et al. (2008); foo (m). An important unresolved question is for instance
whether recently predicted long-range repulsive forces of van der Waals type
(“Anti-Casimir effect”) Semenov and Obukhov (2005); Obukhov and Semenov (2005)
can be demonstrated numerically from specific non-analytic deviations from the
RPA formula, Eq. (25), at intermediate overlap strengths ($x\approx 1)$. In
order to do this, we are currently improving the statistics of our data.

In this paper we have discussed only static properties of the soft BFM.
Similar scaling behavior also has been obtained for the static Rouse mode
correlation function which displays systematic deviations from the scaling
expected for ideal chains Meyer et al. (2008). We currently are working out
how these deviations may influence the dynamics for polymer chains without
topological constraints. (These constraints can be switched off even for
$x=\infty$ by using the “L26” local moves described in Sec. II.) Conceptually
important issues can be addressed if the (artifical) slithering-snake dynamics
is analysed and compared to predictions of the “activated-reptation dynamics”
hypothesis suggested by Semenov Semenov (1997); Semenov and Rubinstein (1998)
for real, although extremely long polymer chains. If no overlap is allowed,
the slithering-snake dynamics is known to show anomalous curvilinear diffusion
and correlated motion of neighboring snakes Mattioni et al. (2003). Since for
$x=\infty$ the lattice might influence the results, it is important to verify
if qualitative similar behavior is also found for soft BFM melts with thermal
blobs much larger than the local monomer scale and how the anomalous
curvilinear diffusion changes with compressibility.

###### Acknowledgements.

We thank H. Meyer and A.N. Semenov (both ICS, Strasbourg, France), S.P.
Obukhov (Gainesville, Florida) and M. Müller (Göttingen) for helpful
discussions. A generous grant of computer time by the IDRIS (Orsay) is also
gratefully acknowledged. We are indebted to the Université de Strasbourg and
the ESF STIPOMAT programme for financial support. J.B. acknowledges financial
support by the IUF.

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  * foo (c) Following Semenov and Obukhov (2005); Beckrich et al. (2007); Wittmer et al. (2007b) the “bulk modulus” $\tilde{v}\equiv 1/g\rho$ is sometimes called “effective excluded volume” which refers to the fact that convergence and self-consistency of the perturbation theory Doi and Edwards (1986) suggests to substitute (renormalize) the bare excluded volume $v$ of the monomers by the explicitly measured $\tilde{v}$ 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 to leading order $v(x)\approx\tilde{v}(x)$ for $x\ll 1$. This will be crosschecked in Sec. III.4 (Fig. 6). Since the perturbation calculation summarized by Eq. (2) is anyway restricted to the same $x$-regime [Eq. (4)] we do not use $\tilde{v}$ in the main text.
  * foo (d) Since the overlap penalty is the only energy scale in this study one may either vary the overlap parameter $\varepsilon$ or the temperature $T$. Since the presentation of thermodynamic properties (especially Figs. 2-5) becomes, however, slightly simpler if $T$ is the control parameter we fix arbitrarily $\varepsilon=1$. The inverse temperature $\beta=1/T$ and the dimensionless overlap strength $x=\varepsilon/T$ are thus numerically equal. We keep both notations for dimensional reasons and for future generalization to models with more than one energy scale.
  * foo (e) A similar intrachain energy contribution to the free energy arises also from Eq. (5.43) of Doi and Edwards (1986) if an upper cut-off $q_{\text{max}}$ is introduced for the wavevectors $q$ to avoid the ultra-violet divergence. Such an upper cut-off is justified by the discreteness of the monomers of real polymers. This leads necessarily to a non-universal free energy contribution.
  * foo (f) Interestingly, according to Edwards Doi and Edwards (1986) 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.
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  * foo (h) Since the intrachain form factor is essentially the Fourier transform of the segmental size distribution Wittmer et al. (2007b); Beckrich et al. (2007), $R(s)$ is related to an experimentally relevant quantity.
  * Wittmer et al. (2009) J. P. Wittmer, A. Johner, J. Baschnagel, and S. P. Obukhov (2009), in preparation.
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  * Meyer et al. (2007) H. Meyer, T. Kreer, A. Cavallo, J. P. Wittmer, and J. Baschnagel, Eur. Phys. J. Special Topics 141, 167 (2007).
  * foo (i) See, e.g., Eq. (2.100) of Ref. Doi and Edwards (1986), which presents the excluded volume effect for the total chain size $R_{\text{e}}(N)$ as obtained by expansion with respect to the bare excluded volume coefficient $v$. As already mentioned Semenov and Obukhov (2005); Wittmer et al. (2007b); foo (c), $v$ must be replaced by the effective modulus $\tilde{v}=1/g\rho$. Note that the leading correction term $\frac{4}{3}z\sim\sqrt{u}$ differs by a numerical coefficent of order one from the corresponding deviation $\frac{c_{\text{s}}}{\sqrt{g}}\frac{4}{3}\sqrt{u}$ 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. Wittmer et al. (2007b). Only the first diagram shown in this figure is taken into account for the total chain size.
  * Wittmann and Kremer (1990) H. P. Wittmann and K. Kremer, Comp. Phys. Comm. 61, 309 (1990).
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  * Landau and Binder (2000) D. P. Landau and K. Binder, _A Guide to Monte Carlo Simulations in Statistical Physics_ (Cambridge University Press, Cambridge, 2000).
  * foo (j) Attractive interactions similar to the ones used in Müller (1995, 1999); Cavallo et al. (2003) 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. (22) for the two lattices used to obtain the chemical potential in Sec. III.3.
  * foo (k) Small corrections to $g(x)\sim N^{0}$ and $\mu_{\text{ex}}\sim N$ may arise due to the long-range correlations which lead to the swelling described by Eq. (5). This effect is, however, too weak to be visible in our data. It can be demonstrated from the number distribution of equilibrium polymers Wittmer et al. (1998); Wittmer et al. (2007c) which differs slightly from the exponential behavior generally admitted Semenov and Johner (2003); Beckrich et al. (2007).
  * Muthukumar and Edwards (1982) M. Muthukumar and S. Edwards, J. Chem. Phys. 76, 2720 (1982).
  * foo (l) Our theoretical result, Eq. (30), has already been applied successfully to the bond-bond correlation function obtained from single-chain-in-mean-field simulation Daoulas and Müller (2006). Over the range of arc-lengths $s$ studied in Ref. Daoulas and Müller (2006) ($0.04\leq u=2/g<4$) Eq. (30) accounts for the shape of $P(s)$, but quantitatively lies slightly above the simulation data, perhaps because the effective bond length $b$, which is determined from $R_{\text{e}}(N)$ [rather than from $R(s)$ as suggested in Sec. III.6], could be refined.
  * Likos (2001) C. Likos, Physics Reports 348, 267 (2001).
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  * Müller and Smith (2005) M. Müller and G. D. Smith, J. Polym. Sci. B: Polymer Physics 43, 934 (2005).
  * Daoulas and Müller (2006) K. C. Daoulas and M. Müller, J. Chem. Phys. 125, 184904 (2006).
  * Semenov (1996) A. N. Semenov, Journal de Physique II France 6, 1759 (1996).
  * Obukhov and Semenov (2005) S. P. Obukhov and A. N. Semenov, Phys. Rev. Lett. 95, 038305 (2005).
  * Semenov (1997) A. N. Semenov, in _Theoretical Challenges in the Dynamics of Complex Fluids_ , edited by T. McLeish (Kluwer, Dordrecht, 1997), pp. 63–86.
  * Semenov and Rubinstein (1998) A. N. Semenov and M. Rubinstein, Eur. Phys. J. B 1, 87 (1998).
  * Meyer and Müller-Plathe (2001) H. Meyer and F. Müller-Plathe, J. Chem. Phys. 115, 7807 (2001).
  * foo (m) Effects of a tunable local chain rigidity will be discussed using this off-lattice scheme since lattice artefacts are known to arise for stiff BFM chains Wittmer et al. (1992).

$x$ | $e$ | $c_{\text{V}}$ | $\mu_{\text{ex}}/T$ | $g$
---|---|---|---|---
0.0001 | 0.2499 | 1.14E-09 | 5.0E-05 | $\approx 1$
0.001 | 0.2499 | 1.08E-07 | 5.0E-04 | $\approx 1$
0.01 | 0.2489 | 1.05E-05 | 0.0049 | $\approx 1$
0.1 | 0.2400 | 0.00093 | 0.049 | 0.95
0.3 | 0.2228 | 0.00709 | 0.14 | 0.88
1.0 | 0.1799 | 0.04854 | 0.43 | 0.69
3.0 | 0.1147 | 0.21538 | 1.04 | 0.47
10 | 0.0321 | 0.57795 | 2.15 | 0.25
20 | 0.0223 | 0.36671 | 2.55 | 0.20
30 | 0.0012 | 0.16318 | 2.62 | 0.20
50 | 8.9E-05 | 0.02874 | 2.63 | 0.20
100 | 2.2E-07 | 0.00039 | 2.635 | 0.20
$\infty$ | 0 | 0 | 2.635 | 0.20
Table 1: Various properties for soft BFM beads ($N=1$) at monomer number
density $\rho=0.5/8$ (corresponding to a volume fraction $0.5$) and linear box
size $L=256$ as a function of the reduced overlap strength $x=\varepsilon/T$.
The limit $x=\infty$ corresponds to the classical BFM without monomer overlap,
the limit $x=0$ to non-interacting monomers. Indicated are the mean energy per
bead $e$, the specific heat $c_{\text{V}}$ per bead, the excess part of the
chemical potential $\mu_{\text{ex}}/T$, and the dimensionless compressibility
$g(x,N=1)$. Within statistical accuracy we obtain below $x\approx 0.1$ the
ideal gas compressibility, $g\approx 1$, and above $x\approx 10$ the
compressibility for a melt without monomer overlap.

$x$ | $e$ | $e_{\text{self}}$ | $c_{\text{V}}$ | $\mu_{\text{ex}}/(TN)$ | $g$ | $l$ | $b$ | $\left<\theta\right>$ | $\left<\cos(\theta)\right>$ | $c_{\text{s}}$ | $G_{\text{z}}$
---|---|---|---|---|---|---|---|---|---|---|---
0 | 0 | 0 | 0 | 0 | $\infty$ | 2.718 | 2.72 | $90^{\circ}$ | 0 | - | 0
0.0001 | 0.42 | 0.18 | 2.5E-07 | 4.9E-05 | 20094 | 2.718 | 2.75 | $90^{\circ}$ | 0 | 0.68 | 0
0.001 | 0.42 | 0.18 | 8.8E-06 | 4.9E-04 | 2029 | 2.718 | 2.75 | $89.99^{\circ}$ | 1.7E-04 | 0.68 | 0.017
0.01 | 0.39 | 0.17 | 2.2E-04 | 4.5E-03 | 209 | 2.718 | 2.80 | $89.9^{\circ}$ | 1.1E-03 | 0.65 | 0.052
0.1 | 0.32 | 0.15 | 4.5E-03 | 0.05 | 22 | 2.719 | 2.92 | $89.4^{\circ}$ | 9.2E-03 | 0.57 | 0.14
0.3 | 0.26 | 0.12 | 0.015 | 0.1 | 7.1 | 2.720 | 3.01 | $88.5^{\circ}$ | 0.021 | 0.52 | 0.22
1 | 0.18 | 0.08 | 0.06 | 0.4 | 2.4 | 2.721 | 3.13 | $86.9^{\circ}$ | 0.043 | 0.46 | 0.34
3 | 0.11 | 0.05 | 0.3 | 0.9 | 0.85 | 2.721 | 3.21 | $84.9^{\circ}$ | 0.069 | 0.42 | 0.52
10 | 0.03 | 0.01 | 0.5 | 1.8 | 0.32 | 2.670 | 3.24 | $82.9^{\circ}$ | 0.096 | 0.41 | 0.83
20 | 0.004 | 0.002 | 0.26 | 2.0 | 0.25 | 2.643 | 3.24 | $82.4^{\circ}$ | 0.104 | 0.41 | 0.94
30 | 9.7E-04 | 4.0E-04 | 0.11 | 2.0 | 0.25 | 2.638 | 3.24 | $82.3^{\circ}$ | 0.105 | 0.41 | 0.94
50 | 7.1E-05 | 2.9E-05 | 0.019 | 2.1 | 0.25 | 2.636 | 3.24 | $82.2^{\circ}$ | 0.106 | 0.41 | 0.94
100 | 2.2E-07 | - | 4.5E-04 | 2.1 | 0.25 | 2.636 | 3.24 | $82.2^{\circ}$ | 0.106 | 0.41 | 0.94
$\infty$ | 0 | 0 | 0 | 2.1 | 0.25 | 2.636 | 3.24 | $82.2^{\circ}$ | 0.106 | 0.41 | 0.94
Table 2: Various properties for asymptotically long BFM chains at number
density $\rho=0.5/8$ as a function of $x=\varepsilon/T$. Apart from the
properties already presented in Table 1 for beads we indicate here the
intrachain self-energy $e_{\text{self}}$, the root-mean-square bond length
$l=\left<\bm{l}_{n}^{2}\right>^{1/2}$, the effective bond length $b$, the mean
angle $\left<\theta\right>$ and the mean cosine
$\left<\cos(\theta)\right>=\left<\bm{e}_{n}\cdot\bm{e}_{n+1}\right>$ of two
subsequent bonds with $\bm{e}_{n}=\bm{l}_{n}/|\bm{l}_{n}|$ being the
normalized bond vector, the swelling coefficient
$c_{\text{s}}\equiv\sqrt{24/\pi^{3}}/\rho b^{3}$, and the Ginzburg parameter
$G_{\text{z}}=1/\sqrt{g}b^{3}\rho$. The excess part of the chemical potential
of a chain is given in units of the chain length $N$ (column 4). The effective
bond length $b(x)$ has been obtained using an extrapolation scheme implied by
Eq. (7) and discussed in Sec. III.6.

$N$ | $e$ | $c_{\text{V}}$ | $\mu_{\text{ex}}/(TN)$ | $g$ | $l$ | $b$
---|---|---|---|---|---|---
1 | 0.1799 | 0.0485 | 0.43 | 0.69 | - | -
4 | 0.1767 | 0.0482 | 0.40 | 1.5 | 2.717 | 2.77
16 | 0.1811 | 0.0691 | 0.37 | 2.0 | 2.720 | 2.89
64 | 0.1819 | 0.0562 | 0.35 | 2.3 | 2.721 | 2.99
256 | 0.1820 | 0.0877 | 0.35 | 2.4 | 2.721 | 3.05
1024 | 0.1820 | 0.0871 | 0.34 | 2.4 | 2.721 | 3.08
2048 | 0.1820 | 0.0639 | 0.34 | 2.4 | 2.721 | 3.09
4096 | 0.1820 | 0.0697 | 0.34 | 2.4 | 2.721 | 3.10
8192 | 0.1820 | 0.0795 | 0.34 | 2.4 | 2.721 | 3.11
Table 3: Various properties for BFM melts of number density $\rho=0.5/8$ at
overlap strength $x=\varepsilon/T=1$ as a function of chain length $N$. For
small chains the overlap energy $e$ and its fluctuation $c_{\text{V}}$
increase slightly while the chemical potential per bead decreases. The
compressibility $g(x=1,N)$ becomes chain length independent for $N>64$. The
chain length dependence visible for small $N$ is described by Eq. (23), i.e.
the data is consistent with an excess compressibility $g_{\text{ex}}(x)\approx
2.4N^{0}$ for all $N$. The last column indicates the rescaled end-to-end
distance $b(x=1,N)\equiv R_{\text{e}}(N)/(N-1)^{1/2}$ which approaches the
effective bond length $b(x)\approx 3.13$ of asymptotically long chains
monotonously from below, just as for classical BFM melts Wittmer et al.
(2007b). Interestingly, $b(x=1,N)$ has not yet reached the asymptotic limit
$b(x)$ even for $N=8192$ albeit all other quantities indicated can be regarded
(within statistical accuracy) as independent of chain length above $N\approx
256$.

Figure 1: Sketch of the bond-fluctuation model (BFM) with finite excluded
volume penalty. The BFM algorithm represents monomers by cubes of length $a$
on a simple cubic lattice (of lattice constant $a$) which are connected by a
set of allowed bond vectors given by Eq. (9). Two short chains of length $N=3$
are shown. The classical BFM model Carmesin and Kremer (1988); Deutsch and
Binder (1991); Baschnagel et al. (2004) assumes that all lattice sites are at
most occupied once. We relax this constraint and penalize double occupation by
a finite interaction energy $\varepsilon$ which has to be paid if two cubes
totally overlap. A corresponding fraction of the energy penality is associated
with a partial monomer overlap, as sketched in the figure for two cube corners
occupying the same lattice site. Varying systematically the ratio
$x=\varepsilon/T$ allows us to put to a test various theoretical results
obtained by perturbation calculation Wittmer et al. (2004); Beckrich et al.
(2007); Wittmer et al. (2007b); Wittmer et al. (2009) for flexible polymer
chains in the melt.

Figure 2: Reduced mean overlap energy per monomer $y=e/\varepsilon$ as a
function of the overlap penalty $x=\varepsilon/T$ for several chain lengths
$N$ as indicated. The energy decreases monotonously with increasing $x$. The
decay becomes Arrhenius-like for $x\gg 10$ (dash-dotted line). The dashed line
indicates the energy predicted from the second virial of soft BFM beads, Eq.
(15). The main figure demonstrates the weak chain length dependence on
logarithmic scales, especially for strong excluded volume interactions
($x>1$). Inset: Same data plotted with linear vertical axis emphasizing the
higher mean energy for long polymers ($N>64$) for $x\ll 1$ caused by a self-
energy contribution $e_{\text{self}}/\varepsilon\approx 0.18$. The self-
energies are indicated by the triangles. The bold line shows the temperature
dependence predicted by Eq. (17).

Figure 3: Reduced mean energy $e/\varepsilon$ (spheres) and self-energy
$e_{\text{self}}/\varepsilon$ (triangles) as functions of the number density
$\rho$ for $N=8192$, $L=512$ and $x=0.001$. As shown by the dashed line,
$e(\rho)$ is a superposition of the mean field energy $4\rho$ and the
(essentially) constant self-energy $e_{\text{self}}/\varepsilon\approx
0.18N^{0}x^{0}\rho^{0}$. Inset: $e/\varepsilon-4\rho$ as a function of chain
length $1/\sqrt{N-1}$ for our reference density $\rho=0.5/8$ and for a single
chain ($\rho=0$). The linear slope (bold line) is expected from the return
probability of Gaussian chains.

Figure 4: Specific heat per bead $c_{\text{V}}$ vs. $x$ for chain length $N$
as indicated. The dashed line indicates the energy fluctuations predicted from
the second virial, Eq. (18), which fits nicely the data of soft BFM beads
($N=1)$ over six decades. While the chain length appears not to matter for
strong excluded volume interactions, the energy fluctuations are found to
increase strongly with $N$ for $x\ll 1$. For short chains we observe
$c_{\text{V}}\sim\rho N^{1/2}x^{2}$ as can be seen for $N=16$ (thin solid
line). The chain length effect saturates for long chains where
$c_{\text{V}}\approx\rho^{1/2}x^{3/2}N^{0}$ (bold line) in agreement with Eq.
(19). Inset: $c_{\text{V}}/(\rho^{1/2}x^{3/2})$ as a function of the reduced
chain length $u=N/g(x)$ with $g(x)$ being the dimensionless compressibility
(Table 2).

Figure 5: Excess chemical potential $y=\mu_{\text{ex}}/TN$ as a function of
the inverse temperature $x=\varepsilon/T$. Increasing linearly (dashed line)
for small $x$ it levels off for large $x\gg 1$ (dash-dotted line). The dashed
line indicates the simple second virial approximation $y\approx v(x)\rho$ for
unconnected beads, fitting successfully the data below $x\approx 1$. The bold
line corresponds to the high temperature prediction Eq. (20) taking into
account the density fluctuation contribution induced by the chain
connectivity. Inset: The chemical potential has been obtained by thermodynamic
integration over the excluded volume interaction of an inserted ghost chain
generalizing the method suggested in Ref. Müller and Paul (1994). The mean
number of lattice sites where monomers and ghost monomers overlap,
$\left<N_{\text{sg}}\right>$, is presented for $N=2048$ as a function of
$\lambda=\exp(-\varepsilon_{\text{sg}}/8T)$ for $x=3$ and $x=100$. A power law
increase of $\left<N_{\text{sg}}\right>$ is found for large $x$ (dashed line).

Figure 6: Dimensionless compressibility $g(x,N)$ as a function of $x$ for
different chain lengths $N$ using the same symbols as in Fig. 4. Main panel:
Raw data as obtained from the low-wavevector limit of the structure factor.
Chain length effects become irrelevant for $x\geq 0.1$ if $N\geq 64$ and for
$x>0.001$ if $N\geq 2048$. The data are compared to the simple second virial
approximation $1/v(x)\rho$ (dashed line) which reduces to $1/(8x\rho)$ for
$x\ll 1$. As one expects, the compressibility levels off for large $x$ and
becomes identical to the value $g\approx 0.25$, known for the classical BFM
Wittmer et al. (2007b) (dash-dotted line). Inset: As suggested by Eq. (23) the
excess part of the inverse compressibility $1/g(x,N)-1/N$ becomes chain length
independent, i.e. the data points for all $N$ collapse. The master curve
indicated by the bold line corresponds to the long chain limit
$g(x)=\lim_{N\to\infty}g(x,N)$ indicated in Table 2.

Figure 7: Total structure factor $G(q)$ as a function of wavevector $q$ for
$N=2048$ for different overlap penalties $x=\varepsilon/T$ as indicated. For
comparison, we have also included the single chain form factor $F(q)$ for
$x=0.001$. The low-wavevector limit of the structure factor is used to
determine the dimensionless compressibility $g(x,N)$ [Eq. (1)]. Only for
$x\leq 3$ does the structure factor decay monotoneously with $q$ as suggested
by the RPA formula, Eq. (25). $G(q)$ becomes essentially constant for smaller
temperatures except for wavevectors corresponding to the first sharp
diffraction peak (called here “Bragg peak”). The box size $L=256$ allows only
a direct and fair determination of $g(x,N)$ for $x>0.1$. We have been forced
to increase the box size to $L=512$ for smaller $x$ as may be seen for an
example with $x=0.1$ (dash-dotted line). As shown by the bold dashed line, the
RPA formula is used to improve the estimation of $g(x,N)$ for small $x$.

Figure 8: Rescaled total structure factor $G(q)/g(x)$ as a function of the
reduced wavevector $Q\equiv q\xi$ for chain length $N=8192$ and several $x\leq
1$ as indicated. The screening length $\xi$ of the thermal blob is obtained
according to Eq. (3) using the dimensionless compressibility $g(x)$ and the
effective bond length $b(x)$ from Table 2. The bold line compares the data
with the approximated RPA, Eq. (26). If replotted as indicated in the inset
the data collapse on the bisection line. Deviations from the RPA formula
become visible for larger $x$ as shown for $x=1$ (crosses).

Figure 9: The (effective) bond length as a function of the reduced overlap
penalty $x=\varepsilon/T$. The data for the root-mean-square bond length
$l(x)$ and the effective bond length $b(x)$ for asymptotically long chains are
listed in Tab. 2. The dash-dotted line indicates the effective bond length as
predicted by Eq. (28) assuming $b_{\text{r}}=l(x)$ for the bond length of the
reference chain. The bold line shows the fix points obtained by iteration of
Eq. (28) using as an input for the Ginzburg parameter the effective bond
length of the previous iteration step: $b\rightarrow b_{\text{r}}$. See the
main text for details.

Figure 10: Rescaled mean-square chain end-to-end distance $b(x,N)^{2}\equiv
R_{\text{e}}^{2}(N)/(N-1)$ as a function of $t=1/\sqrt{N-1}$ for different $x$
as indicated. The chains only remain Gaussian on all scales and all $N$ for
extremely small $x$. For $x\geq 0.1$ one observes $b(x,N)^{2}$ to decay
linearly in agreement with Eq. (27). This can be used for a simple two-
parameter fit for $b(x)$ as indicated for $x=0.1$, $1.0$ and $\infty$. Note
that the coefficient $c$ is slightly above unity as expected from Eq. (19) of
Ref. Wittmer et al. (2007b).

Figure 11: Segment size $R(s)$ for overlap penalty $x$ as indicated for chain
length $N=2048$. Inset: $R(s)^{2}/s$ as a function of segment length $s$
increases monotonously approaching from below the asymptotic limit for large
$s$, i.e. the chains are swollen. Main figure: As suggested by Eq. (5), the
rescaled data
$\left(1-R^{2}(s)/b^{2}(x)s\right)/\left(c_{\text{s}}(x)/g(x)^{1/2}\right)$ is
plotted as a function of the reduced arc-length $u=s/g$. The data collapse is
successful for $1\ll s\ll N$ which confirms the values $g(x)$ and $b(x)$ for
asymtotically long chains (Table 2). The bold line shows the full prediction
from Eq. (5). We indicate the limiting behavior for small and large $u$ by the
dashed and dash-dotted lines, representing respectively Eq. (8) and Eq. (7).

Figure 12: Bond-bond correlation function $P(s)$ for different overlap
penalties $x$ as indicated in the figure. Inset: $P(s)$ as a function of
segment length $s$ in log-log coordinates. The data approaches a power law
behavior, $P(s)\sim 1/s^{\omega}$, with exponent $\omega=1/2$ for small $x$
(dashed line) and $\omega=3/2$ for $x\geq 1$ (dash-dotted line). Main panel:
Rescaled bond-bond correlation function
$P(s)/\left[c_{\text{P}}(g)/g^{3/2}\right]$ plotted as a function of $u=s/g$
as suggested by Eq. (30). For large $u$, where an incompressible packing of
thermal blobs is probed, all data collapse onto the dash-dotted line as
predicted by Eq. (32), i.e. $P(s)$ becomes independent of the compressibility
$g$. That this holds not only for the classical BFM with $x=\infty$ (stars)
but also for finite $x$ is the central result of this study.