|  Marginally compact hyperbranched polymer trees
---|---
|  M. Dolgushev,a,b J.P. Wittmer,∗b A. Johner,b O. Benzerara,b H. Meyer,b and
J. Baschnagelb
|  Assuming Gaussian chain statistics along the chain contour, we generate by
means of a proper fractal generator hyperbranched polymer trees which are
marginally compact. Static and dynamical properties, such as the radial
intrachain pair density distribution $\rho_{\mathrm{pair}}(r)$ or the shear-
stress relaxation modulus $G(t)$, are investigated theoretically and by means
of computer simulations. We emphasize that albeit the self-contact density
$\rho_{\mathrm{c}}=\rho_{\mathrm{pair}}(r\approx 0)\sim\log(N/S)/\sqrt{S}$
diverges logarithmically with the total mass $N$, this effect becomes rapidly
irrelevant with increasing spacer length $S$. In addition to this it is seen
that the standard Rouse analysis must necessarily become inappropriate for
compact objects for which the relaxation time $\tau_{p}$ of mode $p$ must
scale as $\tau_{p}\sim(N/p)^{5/3}$ rather than the usual square power law for
linear chains.

##

††footnotetext: a Institute of Physics, University of Freiburg, Hermann-
Herder-Str. 3, D-79104 Freiburg, Germany††footnotetext: b Institut Charles
Sadron, Université de Strasbourg & CNRS, 23 rue du Loess, 67034 Strasbourg
Cedex, France.††footnotetext: ∗ E-mail<EMAIL_ADDRESS>
##  1 Introduction

#### Marginal compactness.

Natural selection quite generally has to strike a compromise between two
requirements.1, 2 On the one hand, biological structures have to be as compact
(volume-filling) as possible due to packing constraints and to reduce the
typical spatial distances $R$ over which materials are transported within
organisms and hence the time required for transport. On the other hand,
natural selection also tends to maximize the metabolic capacity of organs by
increasing the average surface $A$ where resources are exchanged with the
environment.3, 1, 2 This leads to the extensive surface areas of, e.g., gills,
lungs, guts, kidneys, sponges and diverse respiratory and circulatory systems.
As a consequence of both tendencies a broad range of structures in biology
form fractal networks 3, 1, 2, 4 which are, moreover, as marginally compact as
possible.5, 4 This notion implies that the fractal bulk dimension $d_{\rm f}$
and the fractal surface dimension $d_{\rm A}$ become similar approaching (from
below) the dimension $d$ of the embedding space.3 (It is assumed below that
$d=3$.) The average linear size $R(N)$ — characterized, e.g., by the radius of
gyration $R_{\mathrm{g}}(N)$ — and the surface $A(N)$ — obtained, e.g., from
the number of subunits interacting physiologically with the environment — thus
increase in the large-$N$ limit as

$\displaystyle R(N)$ $\displaystyle\sim$ $\displaystyle N^{1/d_{\rm f}}\approx
N^{1/d}\mbox{ and }$ (1) $\displaystyle A(N)$ $\displaystyle\sim$
$\displaystyle R^{d_{\rm A}}\sim N^{d_{\rm A}/d_{\rm f}}\approx N,$ (2)

i.e. every subunit has thus to leading order the same $N$-independent finite
probability to interact with the environment.222All subunits of open objects
($d_{\rm f}<d$) are at or close to the surface, i.e. $A(N)\sim N$. If one
insists on using the first relation of eqn (2) as the operational definition
of $d_{\rm A}$, this implies $d_{\rm A}=d_{\rm f}$ for open objects. The more
common (and perhaps mathematically more rigorous) definition of the surface
fractal dimension $d_{\rm A}$ assumes that the considered object is compact
($d_{\rm f}\equiv d$).3 Importantly, marginal compactness is observable
experimentally from the scaling of the structure (form) factor $F(q)$, i.e.
the Fourier transformed pair-correlation function of the relevant subunits of
the network under consideration ($q$ being the wavevector).3, 6, 7 As reminded
in Section 3.6, it can be shown5 that to leading order we have

$F(q)\approx N/[R(N)q]^{d}\sim N^{0}\mbox{ for }1/R(N)\ll q\ll 1/b$ (3)

with $b$ being a local length scale (lower cutoff) which is often set by the
size of the subunits of the network.

#### A controversial example.

Following a first brief comment by some of us,5 there appears to be a growing
(albeit not general) consensus 5, 8, 9, 10, 11, 12 that unknotted and
unconcatenated polymer rings in dense solutions and melts may reveal a similar
marginally compact behavior. That such rings should adopt increasingly compact
configurations has been expected theoretically due to the mutual repulsion
caused by the topological constraints.13, 14, 15, 10 Various numerical 16, 17,
18, 19, 20, 5, 8, 9, 21, 22 and experimental studies 23, 24, 25, 26 suggest
that the apparent fractal dimension $d_{\rm f}(N)$ approaches $d=3$ with
increasing mass $N$. Naturally, this begs the question of how to characterize
the surface of these assumed ultimately compact objects. Since there is no
obvious reason for a finite surface tension, a non-Euclidean irregular surface
is expected to be characterized by an apriori unknown fractal surface
dimension with $2<d_{\rm A}\leq d$. The limit $d_{\rm A}\to d^{-}$ is an
attractive scenario since all monomers are evenly exposed to the topological
constraints imposed by other chains, i.e. all subchains have the same self-
similar and isotropic statistics (no screening of topological interactions).
Interestingly, motivated by the behavior of melts of strictly two-dimensional
linear chains,27 a different hypothesis has been suggested in the recent
molecular dynamics (MD) simulations and numerical analysis of ring melts.18,
19, 20, 28 Various properties are fitted with power-law exponents
corresponding to (in our language) fractal surface dimensions $d_{\rm A}$
similar to $d$. The reported differences are, however, too small — considering
the limited number of decades available at present and that error bars must be
interpreted with care — to rule out marginal compactness merely on numerical
grounds. Much more important is the clever theoretical argument18 that
marginal compactness implies that the return probability $p(n)$ between two
tagged monomers must decay inversely with the mass $n$ of the chain
segment.333While for rings $n$ is equivalent to the arc-length $s$ along the
chain contour, this notion must be generalized for branched structures as
discussed in Section 3.3. Please note that our definition of $n$ is slightly
different from the one used in recent work on rings and branched polymers.29,
30, 31, 32, 33, 11 Since this yields in turn a logarithmically diverging self-
contact density

$\rho_{\mathrm{c}}\equiv\sum_{n=1}^{N}p(n)\sim\log(N),$ (4)

it is argued that, quite generally, a “mathematically rigorous fractal
structure" can not be marginally compact.18

Fig. 1: Iterative generation of marginally compact hyperbranched trees in
$d=3$ dimensions by means of a proper fractal generator: (a) A given segment
of curvilinear length $S_{I-1}$ is replaced by $M_{I}=8M_{I-1}=2^{3I}$
segments of length $S_{I}=S_{I-1}/4=S_{0}/2^{2I}$ with $I$ denoting the number
of iterations. (b) While the connectivity matrix is generated in a
deterministic manner, the particle positions are chosen randomly assuming
Gaussian chain statistics along the chain contours as shown for the iteration
$I=0\to I=1$.

#### Central goal of the current study.

Not focusing specifically on melts of rings, but addressing this argument for
general marginally compact structures, we want to show in the present work
that there is a simple, generic way out of this difficulty. As sketched in
Fig. 1, we investigate as a counter example a marginally compact hyperbranched
tree generated using one proper fractal generator. The present study
elaborates on one of the models introduced in a recent work on dendrimers and
hyperbranched stars with Gaussian chain statistics along the chain contour.34
As shown by our toy model, it is straightforward to generate mathematically
rigorous marginally compact objects. They simply do exist and this
irrespectively of whether some moments of $p(n)$ diverge with increasing mass
$N$. It is another issue, however, whether for the real system, one attempts
to describe using the mathematical model, there exist additional physical or
biological requirements which set constraints on moments of $p(n)$. Quite
generally, such a constraint may lead to a restriction of the parameter space
within which the fractal model may be applied and may set an upper bound
$N_{\star}$ for the system size $N$. The existence of an upper bound does not
invalidate per se a fractal description, at least not if a broad $N$-window
can be identified where the constraints are irrelevant. The physical (not
mathematical) restriction we need to address is the fact that the number of
close contacts of any reference monomer is generally limited due to excluded
volume constraints.444Hyperbranched trees with fractal dimension $d_{\rm f}=4$
have been discussed, e.g., in the context of randomly branched polymers (often
called “lattice animals”),35 dilute rings in a gel of topological obstacles 14
and as an early (mathematically perfectly legitimate) model describing the
topological interactions of unconcatenated molten rings.14, 23, 24 Obviously,
such a modeling approach must break down in $d=3$ dimension above an upper
mass limit $N_{\star}$. A sufficiently broad mass window exists, however,
where the average self-density remains below unity and the fractal model is
thus applicable.14 Please note that the limit set by the local excluded volume
constraint on the randomly branched lattice animals is much more severe as the
one, eqn (5), set on our marginally compact trees. We show below that albeit
the monomer connectivity in a marginal compact tree leads indeed to a
logarithmically diverging self-contact density, in agreement with eqn (4),
there exist an extremely broad $N$-window

$N\ll N_{\star}\approx Se^{\sqrt{S}},$ (5)

strongly increasing with the spacer length $S$ between the branching points of
network, where the excluded volume constraints can be neglected. (To make the
above exponential relation meaningful, prefactors must be specified. This will
be done in Section 3.4.) We thus argue that one cannot reject apriori a
marginally compact model such as the one proposed by Obukhov et al.9

#### Outline.

The fractal generator and the construction of the hyperbranched trees is
described in Section 2 where we also give some algorithmic details concerning
the MD and Monte Carlo (MC) simulations 36 used in the present work. Static
properties are then presented in Section 3 where we confirm eqn (3) and eqn
(5). We turn then in Section 4 to dynamical properties. The work is summarized
in Section 5. Appendix A gives some details concerning the prefactors expected
for the self-contact density $\rho_{\mathrm{c}}$. A comparison of the
structure factor of our model and (already published) simulated ring data5 is
given in Appendix B. Some properties of the “generalized Rouse model" (GRM)
37, 38, 39, 40, 41, 42, 43, 44 are recalled in Appendix C.

##  2 Model

Fig. 2: Sketch of some notations using an example with iteration $I=2$, number
of generations $G=8$, spacer length $S=2$, total monomer number $N=129$ and
total number of dangling ends $m_{\mathrm{e+E}}=29$. Some spacer chains of
lengths $S=2$ are marked by dashed circles, branching points by “B", dangling
ends at largest curvilinear distance $SG$ from the root monomer by “E", all
other dangling ends by “e". $R_{\mathrm{E}}$ denotes the root-mean-squared
distance between the root monomer and the ends “E". The $n(s)$ monomers with a
curvilinear distance $\leq s=4$ from the monomer $k$ are marked by filled
disks.

#### Gaussian chain statistics.

Treelike networks with Gaussian chain statistics have been considered
theoretically early in the literature 35, 45, 14, 46, 47 and have continued to
attract attention up to the recent past.37, 38, 39, 40, 41, 42, 43, 44, 48
Assuming translational invariance, these models have in common that the root-
mean-square distance $R_{\mathrm{s}}$ between two monomers $i$ and $j$ is
given by
$R_{\mathrm{s}}^{2}\equiv\langle(\underline{r}_{i}-\underline{r}_{j})^{2}\rangle=b^{2}s$
with $\underline{r}_{i}$ being the position of a bead $i$, $s$ the shortest
chemical distance (curvilinear distance) connecting both monomers and $b$ the
statistical segment size.49 Other moments are obtained from the distribution
$P(\underline{r},s)$ of the distance vector
$\underline{r}=\underline{r}_{i}-\underline{r}_{j}$ which, irrespective of the
specific topology, is given by

$P(\underline{r},s)=\left(\frac{d}{2\pi
R_{\mathrm{s}}^{2}}\right)^{d/2}\exp\left(-\frac{d}{2}\left(\frac{r}{R_{\mathrm{s}}}\right)^{2}\right)$
(6)

with $d=3$ being the spatial dimension. Such Gaussian trees may be modeled
theoretically as well as in MD and MC simulations using $n_{\mathrm{sp}}$
ideal harmonic springs connecting specified pairs of beads $i$ and $j$. The
potential energy $V$ may be written as

$V=\frac{K}{2}\sum_{l=1}^{n_{\mathrm{sp}}}r_{l}^{2}=\frac{K}{2}\sum_{i,j=1}^{N}A_{ij}\
\underline{r}_{i}\cdot\underline{r}_{j}$ (7)

with $K=d\mbox{$k_{\rm B}T$}/b^{2}$ being the spring constant, $l$ a label
standing for the spring between the monomers $i$ and $j$,
$r_{l}=|\underline{r}_{i}-\underline{r}_{j}|$ the corresponding monomer
distance and $\mathbf{A}=(A_{ij})$ the so-called connectivity or Laplacian
matrix.50, 51, 52 This is a $N\times N$ square matrix with diagonal elements
$A_{ii}$ denoting the number of bonds connected to monomer $i$ and off-
diagonal elements $A_{ij}=-1$ if the beads $i$ and $j$ are connected or
$A_{ij}=0$ elsewise. Due to their theoretical simplicity such Gaussian chain
trees (including systems with short-range interactions along the network)
allow to investigate non-trivial conceptual and technical issues, both for
static 38, 41, 34 and dynamical 47, 53, 54, 50, 39, 40, 41, 42, 51, 52
properties.

#### Construction of connectivity matrix.

It is assumed in the present study that the hyperbranched network is not
annealed, but irreversibly imposed by the fractal generator sketched in Fig.
1. Note that fractal self-similar hyperbranched trees generated in this way
are called “$\beta$-stars" in Polińska _et al._ 34 where a more detailed
description can be found. Basically, we start at iteration step $I=0$ with a
linear Gaussian chain of $1+S_{0}$ monomers created using the distribution eqn
(6). The “$1$” is required due to the “root monomer" sitting for all
iterations $I$ topologically in the middle of the network. This was done for
technical convenience to have a fixed reference monomer for the various
pointer lists used. (The iteration step $I=0\to I=1$ is thus slightly
different from all others.) At each iteration step $I-1\to I$ each spacer
subchain of length $S_{I-1}$ is divided into 4 subchains of equal length
$S_{I}=S_{I-1}/4=S_{0}/2^{2I}$ and 4 subchains of same length are added
laterally to the subchain as sketched in Fig. 1. The total number of spacer
chains $M_{I}$ thus increases as $M_{I}=8M_{I-1}=2^{3I}$ and the total mass
$N_{I}$ as $N_{I}-1=2(N_{I-1}-1)=2^{I}S_{0}=M_{I}S_{I}$. Note that the 4 new
spacer subchains are again created according to eqn (6). Importantly, $S_{0}$
may be chosen such that for the final iteration $I$ considered we have the
same monodisperse final spacer length $S\equiv S_{I}$. Below we shall often
use $N\equiv N_{I}$ for the total mass after $I$ iterations,
$s_{\text{max}}\equiv S_{0}=2^{2I}S$ for the largest possible curvilinear
distance and $G=s_{\text{max}}/2S=2^{2I-1}$ for the number of “generations" of
spacer chains of length $S$. We thus have by construction

$N-1=2^{dI}S=S(s_{\text{max}}/S)^{d/2}=S(2G)^{d/2}$ (8)

with $d=3$.555Note that the exponent $d/2$ corresponds to the exponent
$1/\rho$ used in various recent publications on melts of ring polymers and
annealed branched polymers.29, 30, 31, 32, 33, 11 At difference to these
studies $1/\rho$ is imposed here, not fitted or predicted. As a consequence
the typical distance $R_{\mathrm{E}}$ between the root monomer and the
monomers “E" (Fig. 2), as one observable measuring the tree size, must scale
as $R_{\mathrm{E}}=b\sqrt{SG}\sim N^{1/d}$.

$I$ | $G$ | $N$ | $m_{\mathrm{e+E}}$ | $m_{\mathrm{E}}$ | $s_{\text{max}}$ | $W_{\mathrm{1}}/N^{2}$ | $R_{\mathrm{E}}$ | $R_{\mathrm{g}}$ | $\rho_{\mathrm{c}}$
---|---|---|---|---|---|---|---|---|---
1 | 2 | 9 | 5 | 5 | 4 | 1.2 | 1.41 | 1.09 | 1.01
2 | 8 | 65 | 29 | 15 | 16 | 4.2 | 2.82 | 2.06 | 2.06
3 | 32 | 513 | 221 | 45 | 64 | 16 | 5.66 | 4.00 | 3.18
4 | 128 | 4097 | 1757 | 135 | 256 | 63 | 11.3 | 7.95 | 4.34
5 | 512 | 32769 | 14045 | 405 | 1024 | 252 | 22.6 | 15.9 | 5.52
6 | 2048 | 262145 | 112349 | 1215 | 4096 | 1008 | 45.3 | 31.4 | 6.71
Table 1: Some properties for spacer length $S=1$: iteration number $I$, generation number $G$, total mass $N$, total number of dangling ends $m_{\mathrm{e+E}}$, number of ends $m_{\mathrm{E}}=5\times 3^{I-1}$ with maximum distances $SG$ from root monomer, maximum possible curvilinear arc-length $s_{\text{max}}$, rescaled Wiener index $W_{\mathrm{1}}/N^{2}$ (Section 3.2), root-mean-squared distance $R_{\mathrm{E}}$ between root monomer and dangling end “E", radius of gyration $R_{\mathrm{g}}$ (Section 3.1) and self-contact density $\rho_{\mathrm{c}}$ (Section 3.4).  $S$ | $N-1$ | $s_{\text{max}}$ | $W_{\mathrm{1}}/N^{2}$ | $R_{\mathrm{E}}$ | $R_{\mathrm{g}}$ | $N/R_{\mathrm{g}}^{3}$ | $\rho_{\mathrm{c}}$
---|---|---|---|---|---|---|---
1 | $2^{15}$ | 1024 | 252.2 | 22.6 | 15.9 | 8.19 | 5.52
2 | $2^{16}$ | 2048 | 503.9 | 32 | 22.5 | 5.79 | 4.59
4 | $2^{17}$ | 4096 | 1007.5 | 45.3 | 31.7 | 4.09 | 3.85
8 | $2^{18}$ | 8192 | 2014.5 | 64 | 44.5 | 2.89 | 3.29
16 | $2^{19}$ | $2^{14}$ | 4028 | 90.5 | 71.5 | 2.05 | 2.85
32 | $2^{20}$ | $2^{15}$ | 8056 | 128 | 89.8 | 1.45 | 2.54
64 | $2^{21}$ | $2^{16}$ | 16112 | 181 | 126.9 | 1.03 | 2.31
128 | $2^{22}$ | $2^{17}$ | 32224 | 256 | 179.5 | 0.73 | 2.14
256 | $2^{23}$ | $2^{18}$ | 64448 | 362 | 253.9 | 0.51 | 2.02
512 | $2^{24}$ | $2^{19}$ | 123274 | 512 | 359 | 0.36 | 1.94
Table 2: Some properties for iteration $I=5$: spacer length $S$, total mass
$N-1$, maximum arc-length $s_{\text{max}}$, Wiener index
$W_{\mathrm{1}}/N^{2}$, root-mean- squared distance $R_{\mathrm{E}}$, radius
of gyration $R_{\mathrm{g}}$, $N/R_{\mathrm{g}}^{3}$ as a measure of the
typical density and self-contact density $\rho_{\mathrm{c}}$. Note that
$N/R_{\mathrm{g}}^{3}\sim 1/\sqrt{S}$ and that $\rho_{\mathrm{c}}$ converges
similarly down to $\rho_{\mathrm{c,lin}}=1.724/b^{3}$ for large $S$.

#### Technical comments.

Due to their Gaussian chain statistics many conformational properties can be
readily obtained using Gaussian propagator techniques or equivalent linear
algebra relations.38, 42, 43, 55, 44 However, some interesting properties,
such as the dynamical structure factor $F(q,t)$ discussed in Section 4.5, can
be more easily computed by direct simulation which are in any case necessary
if long-range interactions between the monomers are switched on. Following
Polińska _et al._ 34 we have used for static properties MC simulations mixing
pivot and local jump moves. To obtain dynamical properties we have in addition
sampled time series using local MC moves with maximum jump distance $\delta
r=0.5$ and velocity-verlet MD using a Langevin thermostat 36, 56 with an
extremely strong friction constant $\zeta=10$ suppressing all inertia effects.
(This is of course not the most efficient parameter to sample the
configuration space via MD.) The temperature $T$, Boltzmann’s constant $k_{\rm
B}$ and the statistical bond length $b$ are set to unity. We sample in
parallel $m=100$ uncorrelated chains. These are contained in a periodic cubic
simulation box of total density $\rho=Nm/V=1$ with $V$ being the volume. We
focus often on trees with a spacer length $S=1$ as summarized in Table 1. As
may be seen from Table 2 for $I=5$, a broad range of spacer lengths $S$ has
also been sampled. This is especially important for the scaling of the self-
contact density $\rho_{\mathrm{c}}$ presented in Section 3.4.

##  3 Static properties

Fig. 3: Inset: Rescaled root-to-end distance $R_{\mathrm{E}}/bN^{1/3}$ and
radius of gyration $R_{\mathrm{g}}/bN^{1/3}$ for different $N(I)$. Main panel:
Self-density $\rho_{\mathrm{cm}}(r)$ around chain center of mass for different
$I$. The data collapse if rescaled as indicated using $N(I)$ and
$R_{\mathrm{g}}(I)$.

###  3.1 Compactness

#### Typical chain size.

We demonstrate first that our hyperbranched trees become indeed compact, i.e.
eqn (1) holds, and this rapidly after one or two iterations. The data
presented in Fig. 3 have been obtained by MC simulations. The typical chain
size $R$ of the trees shown in the inset has been characterized using the
root-mean-squared distance $R_{\mathrm{E}}$, as sketched in Fig. 2, and the
standard radius of gyration $R_{\mathrm{g}}$ of the chain.6 The ratio
$R/bN^{1/3}$ is seen to be perfectly constant for both observables. Compared
to the sluggish crossover of ring melts, our simple model thus becomes rapidly
compact. We remind 34 that the Gaussian chain statistics along all chain
contours implies $R_{\mathrm{E}}^{2}=b^{2}SG$ and

$R_{\mathrm{g}}^{2}=\frac{1}{2}\sum_{s=0}^{s_{\text{max}}}w(s)R_{\mathrm{s}}^{2}\mbox{
with }R_{\mathrm{s}}^{2}=b^{2}s$ (9)

and $w(s)$ being the histogram of curvilinear distances discussed in Section
3.2. Both lengths $R_{\mathrm{E}}$ and $R_{\mathrm{g}}$ can thus be obtained
without explicit simulations.

#### Center of mass self-density $\rho_{\mathrm{cm}}(r)$.

The self-density $\rho_{\mathrm{cm}}(r)$ at a distance $r$ from the chain
center of mass is presented in the main panel. Plotting the rescaled density
$y=\rho_{\mathrm{cm}}(r)/(N/R_{\mathrm{g}}^{d})$ as a function of the
dimensionless distance $x=r/R_{\mathrm{g}}$ allows to collapse the data for
all iterations $I$. As already stated $N/R_{\mathrm{g}}^{d}\sim N^{0}$, i.e.
the plateau $\rho_{\mathrm{cm}}(r\approx 0)\approx 4$ for $x\ll 1$ does not at
all depend on $N$.666The deviation from this scaling observed for melts of
rings16, 18 is due to insufficient chain lengths and the sluggish crossover to
the compact limit in these systems.9 We note finally that the self-density
decreases as $\rho_{\mathrm{cm}}(r)\sim N/R_{\mathrm{g}}^{d}\sim 1/\sqrt{S}$
with the spacer length (Table 2).

###  3.2 Distribution $w(s)$

#### Definitions.

Central properties characterizing the monomer connectivity are the
distributions

$w_{k}(s)\equiv\frac{1}{N}\sum_{l=1}^{N}\delta(s-s_{kl})\mbox{ and
}w(s)\equiv\frac{1}{N}\sum_{k=1}^{N}w_{k}(s)$ (10)

for $0\leq s\leq s_{\text{max}}$ with $s_{kl}$ denoting the arc-length between
the monomers $k$ and $l$. $Nw_{k}(s)$ counts the number of beads with
curvilinear distance $s$ from a specific monomer $k$ and $Nw(s)$ corresponds
to the mean number of beads averaged over all reference monomers $k$. Both
distributions are normalized, i.e.
$\sum_{s=0}^{s_{\text{max}}}w_{k}(s)=\sum_{s=0}^{s_{\text{max}}}w(s)=1$. For
consistency with earlier work 34 we have included here the reference monomer
$k$ at $s=0$ into the distributions. This is accounted for by the
normalization. Trivially, $w(s=0)=1/N$. (For some properties considered below
it is more useful to exclude the reference monomer and to only consider
distances $s>0$. This leads to a renormalization factor $1/(1-1/N)$ if moments
with $s>0$ are considered using the histogram $w(s)$ defined for $s\geq 0$.)
The first moment

$\frac{1}{2}\sum_{s}sw(s)\equiv W_{\mathrm{1}}/N^{2}$ (11)

of the distribution, listed in Table 1 and Table 2, defines the Wiener index
$W_{\mathrm{1}}$.57, 58 According to eqn (9) this implies
$W_{\mathrm{1}}/N^{2}=(R_{\mathrm{g}}/b)^{2}$ for Gaussian trees. Associated
to the distributions $w_{k}(s)$ and $w(s)$ are the sums

$n_{k}(s)\equiv N\sum_{s^{\prime}=0}^{s}w_{k}(s^{\prime})\mbox{ and
}n(s)\equiv\frac{1}{N}\sum_{k=1}^{N}n_{k}(s)$ (12)

measuring, respectively, the total mass attached within an arc-length $s$ to a
specific monomer $k$ (as sketched in Fig. 2) and the corresponding
$k$-average. Due to the normalization of $w_{k}(s)$ and $w(s)$ we have
$n_{k}(s=s_{\text{max}})=n(s=s_{\text{max}})=N.$ A power-law ansatz $w(s)\sim
s^{\alpha-1}$ implies $n(s)\sim s^{\alpha}$. Since

$n(s)\approx\rho R(s)^{d}\approx S(s/S)^{d/2}$ (13)

must hold for a self-similar compact network with Gaussian chain statistics,
this implies $\alpha=d/2$.

Fig. 4: Distribution $w(s)$ measuring the number of monomer pairs with
curvilinear distance $s$. Main panel: Double-logarithmic representation of
$v=w(s)N$ vs. $u=s/S$ demonstrating eqn (15) for large $I$ (bold solid line).
The bold vertical arrow marks the crossover $s_{1}=S(2/c_{\star})^{2}$ between
the linear spacer limit (horizontal dashed line) and the intermediate power
regime (bold solid line), all other vertical arrows mark the breakdown of the
power law at $s_{2}=f_{2}s_{\text{max}}$. Inset: Linear representation of
$y=w(s)s_{\text{max}}$ vs. $x=s/s_{\text{max}}$ for $I=5$.

#### Three regimes of $w(s)$.

Various distributions $w(s)$ are presented in Fig. 4. There are basically
three regimes: (i) the linear spacer limit with

$w(s)=2/N\mbox{ for }s\ll s_{1}\equiv f_{1}S\mbox{ with }f_{1}\approx 0.64,$
(14)

(ii) the power-law regime expected according to eqn (13) for marginal compact
networks where

$w(s)=\frac{c_{\star}}{N}(s/S)^{3/2-1}=\frac{c_{\star}}{s_{\text{max}}}(s/s_{\text{max}})^{3/2-1}$
(15)

holds (with eqn (8) being used in the second step) for $s_{1}\ll s\ll
s_{2}\equiv f_{2}s_{\text{max}}$ with $c_{\star}\approx 5/2$ and $f_{2}\approx
0.2$ and (iii) the final cutoff beyond $s_{2}$ where non-universal finite
chain size effects become dominant. We comment now on the scaling of these
regimes.

#### Scaling with $s/S$.

Using a double-logarithmic representation the main panel presents a broad
range of $I$ and $S$ tracing $v=w(s)N$ as a function of $u=s/S$. This allows
to collapse the data for small and intermediate $s$. Deviations from this
scaling are seen for small $S<8$ and $u\approx 1$ due to too small spacer
lengths and for $s>s_{2}$ as marked by thin vertical arrows. The dashed
horizontal line indicates the limit eqn (14) for long spacer chains expressing
the fact that each monomer has essentially two neighbors at curvilinear
distance $s$. As expected from the scaling argument eqn (13), it is seen that
eqn (15) holds over more than two decades for the largest $I$ presented (bold
solid line).777Interestingly, essentially the same power-law exponent has been
fitted in recent simulations of three-dimensional melts of self-avoiding
branched polymers.31, 32 This suggests that marginal compactness remains
relevant if excluded volume is switched on provided that the connectivity is
annealed. The bold vertical arrow at $s_{1}/S\equiv
f_{1}=(2/c_{\star})^{2}\approx 0.64$ marks the crossover of the first two
regimes. Using eqn (9) and eqn (8) one verifies that eqn (15) is consistent
with $R_{\mathrm{g}}\approx b\sqrt{S}\ (N/S)^{1/d}$.

#### Scaling with $s/s_{\text{max}}$.

Using linear coordinates $y=w(s)s_{\text{max}}$ is plotted in the inset as a
function of $x=s/s_{\text{max}}$. Data for $I=5$ and two spacer lengths $S$
are shown. The rescaling with $s_{\text{max}}$ allows quite generally to
collapse data for large arc-lengths $s\gg S$. It is seen that $w(s)$ is a non-
monotonous distribution, having a maximum at $x\approx 1/2$ and vanishing for
$x\to 0$ and $x\to 1$. The vertical arrow marks the arc-length
$s_{2}/s_{\text{max}}=f_{2}\approx 0.2$ above which the power law eqn (15)
becomes inaccurate. The latter value depends somewhat on the criterion used.

Fig. 5: Return probability $p(n)$ for $I=4$, $5$ and $6$. Open symbols
indicate $S=1$. (a) Rescaled mass $y=n(s)/N$ vs. $x=s/s_{\text{max}}$ with
$n(s)$ being obtained using eqn (12). (b) Double-logarithmic representation of
the rescaled return probability $v=p(n(s))/p(s=S)$ vs. $u=n/S$ allowing the
scaling of all data for $s\ll s_{2}$. The linear chain power law $(2/u)^{3/2}$
is seen for large $S$ and small $u$ (dashed line). The bold line indicates the
power law $(5/3)/u$ for the self-similar marginal compact regime.

###  3.3 Return probability

We turn now to the return probability $p(n)$ mentioned in the Introduction.
Let us consider two monomers $k$ and $l$ being an arc-length $s$ apart as
shown in Fig. 2 for $s=4$. Using the distribution $P(\underline{r},s)$ given
by eqn (6) the return probability is simply

$p(s)\equiv P(\underline{r}=\underline{0},s)=\left(d/2\pi
b^{2}s\right)^{d/2}\mbox{ for all }s.$ (16)

It is now customary and useful (see Section 3.4) to express the return
probability in terms of the typical mass $n(s)$. According to eqn (12), $n(s)$
must be a monotonously increasing function and its inverse $s(n)$ may thus be
obtained from the distribution $w(s)$. This is confirmed in panel (a) of Fig.
5 where $y=n/N$ is plotted as a function of $x=s/s_{\text{max}}$. Following
eqn (15) we have

$n(s)=(2c_{\star}/3)\ S(s/S)^{3/2}\mbox{ for }s_{1}\ll s\ll s_{2}.$ (17)

This corresponds to $y=(5/3)x^{3/2}$ as indicated by the bold solid line. A
continuous crossover to unity is observed for larger $x$ (finite chain size).
Panel (b) of Fig. 5 presents the distribution $p(s(n))$ obtained using eqn
(16) and $s(n)$. The data are brought to collapse by plotting
$v=p(s(n))/p(s=S)$ as a function of $u=n/S$, i.e. we take the return
probability of the spacer chain as reference. Due to eqn (14) we have
$n(s)=2s$ which implies $v=(2/u)^{3/2}$ in the small-$u$ limit (dashed line).
Using again eqn (17) this leads to $v=(5/3)/u$ for large $u$ (bold line). The
return probability thus decreases as $p(n)\sim 1/(\sqrt{S}n)$ within the
$n$-range

$n_{1}=2s_{1}=2f_{1}\ S\ll n\ll n_{2}=(2c_{\star}/3)f_{2}^{3/2}\ N$ (18)

where we have used eqn (8) for the upper limit. The limits $n_{1}$ and $n_{2}$
are indicated by arrows in Fig. 5. Note that the representation of panel (b)
masks somewhat that the scaling fails above $n_{2}$. We are now ready to
embark on the key issue of this paper.

###  3.4 Self-contact density

#### Observable.

The self-contact density $\rho_{\mathrm{c}}$ presented in Fig. 6 has been
obtained using the sum

$\rho_{\mathrm{c}}=\frac{1}{1-1/N}\sum_{s=1}^{s_{\text{max}}}Nw(s)\times p(s)$
(19)

taking advantage of the distribution $w(s)$ characterizing the tree
connectivity. The prefactor $1/(1-1/N)$ appears for normalization reasons
(Section 3.2) since the sum runs from $s=1$ excluding the reference monomer at
$s=0$. This factor becomes rapidly irrelevant and will be ignored below. Since
according to eqn (12) $Nw(s)=\text{d}n(s)/\text{d}s$, the above sum (integral)
over $s$ becomes equivalent in the continuum limit to the integral over $n$

$\rho_{\mathrm{c}}\approx\int_{n=1}^{N}\text{d}n\
p(n)\approx\sum_{n=1}^{N}p(n),$ (20)

i.e. the weight $w(s)$ needed in eqn (19) naturally drops out if $n$ is used
as variable. We emphasize that being applicable for arbitrary tree networks,
eqn (20) is more general than the (formally identical) well-known expression
for unbranched (linear or closed loop) polymer chains.18 The self-contact
density $\rho_{\mathrm{c}}$ is thus equivalent to the surface below the data
in Fig. 5.

Fig. 6: Self-contact density $\rho_{\mathrm{c}}$: (a)
$y=\rho_{\mathrm{c}}/\rho_{\mathrm{c,lin}}-1$ vs. $N$ for a broad range of
spacer lengths $S$ using $\rho_{\mathrm{c,lin}}=1.724/b^{3}$ as reference. In
agreement with eqn (21) the data diverge logarithmically with mass $N$ at
constant spacer length $S$ (thin dashed lines) and decays as $y\sim
1/\sqrt{S}$ with spacer length $S$ at constant iteration $I$ (thin solid
lines). (b) Upper limit $N_{\star}(S)$ according to eqn (22) for $a=10\%$ and
$a=1\%$.

#### Scaling of data.

The main panel (a) of Fig. 6 presents the rescaled self-contact density
$y=\rho_{\mathrm{c}}/\rho_{\mathrm{c,lin}}-1$ as a function of $N$ for a broad
range of spacer lengths $S$. The self-contact density
$\rho_{\mathrm{c,lin}}\approx 1.724/b^{3}$ for arbitrarily long linear
Gaussian chains is used as reference. See Appendix A for details. As shown by
the dashed lines for a constant spacer length $S$ and by the thin solid lines
for a constant iteration $I$ our data are well described using

$y\equiv\frac{\rho_{\mathrm{c}}}{\rho_{\mathrm{c,lin}}}-1=\frac{c_{\mathrm{asym}}}{\sqrt{S}}\left[\log(N/S)-c_{\mathrm{corr}}\right]$
(21)

with $c_{\mathrm{asym}}\approx 0.319$ and $c_{\mathrm{corr}}\approx 3.4$. As
shown in Appendix A, the first coefficient $c_{\mathrm{asym}}$ is known
exactly. It stems from the contribution of the intermediate marginal compact
regime of $w(s)$ which must dominate the integral eqn (19) for asymptotically
large trees. For large $I$ and constant $S$ the self-contact density thus
diverges logarithmically as $\rho_{\mathrm{c}}/\rho_{\mathrm{c,lin}}\approx
c_{\mathrm{asym}}\log(N)/\sqrt{S}$ with $N$, while for constant $I$ and
increasing $S$ the data approach rapidly the linear chain reference (bold
horizontal solid line). The coefficient $c_{\mathrm{corr}}$, summarizing all
subdominant corrections to the asymptotic behavior, is more difficult to
predict due to the various crossovers and has been fitted. More details on
this minor technical point can be found at the end of Appendix A.

#### Upper mass bound.

The central consequence of eqn (21) is that if a local relative density
fluctuation $y=a$ is physically just acceptable, an allowed chain mass $N$
must satisfy the inequality

$N\ll N_{\star}(S)\equiv
S\exp\left(\frac{\sqrt{S}a}{c_{\mathrm{asym}}}+c_{\mathrm{corr}}\right).$ (22)

As may be seen from panel (b) of Fig. 6 for $a=10\%$ and $a=1\%$, the upper
limit $N_{\star}$ increases dramatically with $S$, i.e. the marginal compact
model provides a physically acceptable model over several orders of magnitude.
Please note that the indicated parameters $a$ are extremely conservative
considering that in normal excluded volume polymer fluids the local density
around a reference monomer may fluctuate by a factor of order unity.

###  3.5 Radial intrachain pair density distribution

Fig. 7: Radial intrachain pair density distribution $\rho_{\mathrm{pair}}(r)$:
(a) different $I$ at constant spacer length $S=1$, (b) different $S$ at
constant $I=5$. The bold horizontal lines indicate the self-contact density
$\rho_{\mathrm{c,lin}}$ for long linear chains. As shown by the dashed
horizontal lines $\rho_{\mathrm{c}}\approx\rho_{\mathrm{pair}}(r\approx 0)$.

The self-contact density $\rho_{\mathrm{c}}$ has been determined in Section
3.4 using the histogram $w(s)$ without explicit computer simulation. This has
allowed us to scan over a huge range of $N$. In addition to this we have
computed $\rho_{\mathrm{c}}$ from the $r\to 0$ limit of the directly simulated
radial pair density distribution $\rho_{\mathrm{pair}}(r)$. We remind that
$\rho_{\mathrm{pair}}(r)$ may be obtained by radially averaging over

$\rho_{\mathrm{pair}}(r)=\frac{1}{N}\sum_{k=1}^{N}\sum_{l\neq
k}\langle\delta(\underline{r}-(\underline{r}_{l}-\underline{r}_{k}))\rangle.$
(23)

The data presented in Fig. 7 have been computed by means of MC simulations
using pivot moves.34 As shown by the dashed horizontal lines,
$\rho_{\mathrm{c}}\approx\rho_{\mathrm{pair}}(r\approx 0)$ holds as expected.
We use a half-logarithmic representation with a rescaled horizontal axis
$x=r/R_{\mathrm{g}}$. As shown in the main panel (a) for a spacer length
$S=1$, $\rho_{\mathrm{pair}}(x)$ differs quite strongly for small $S$ from the
average density $\rho_{\mathrm{cm}}(x)$ around the chain center of mass
considered in Fig. 3: The data do not scale and increase logarithmically with
decreasing $x$. Note that $\rho_{\mathrm{pair}}(x)$ increases strongly above
the self-contact density $\rho_{\mathrm{c,lin}}$ for long linear Gaussian
chains indicated by the solid horizontal line. Spacer length effects are
considered in the inset of Fig. 7 for $I=5$. It is seen that
$\rho_{\mathrm{pair}}(x)$ decreases with $S$ approaching from above an
$S$-independent asymptote. We shall further analyse these pair correlations in
the next subsection where we discuss the related, but experimentally more
relevant intramolecular structure factor $F(q)$.

Fig. 8: Kratky representation $y=(F(q)/N)Q^{2}$ vs. $Q=qR_{\mathrm{g}}$ of the
static intrachain form factor $F(q)$ for different iterations $I$ as indicated
in the figure with $q$ being the wavevector and $R_{\mathrm{g}}(N)$ the radius
of gyration. The trees become marginally compact, i.e. $y\sim 1/Q$ for
sufficiently large $I$ (bold solid line). The dashed line indicates the
standard Porod scaling for a compact object with a well-defined non-fractal
surface. The dash-dotted line has been obtained according to eqn (44) of
Appendix B.

###  3.6 Static structure factor

Conformational properties of polymer chains can be determined experimentally
by means of light, small angle X-ray or neutron scattering experiments.7, 6
This allows to extract the coherent intramolecular structure (form) factor
$F(q)=\frac{1}{N}\langle||\sum_{k=1}^{N}\exp\left(\text{i}\underline{q}\cdot\underline{r}_{k}\right)||^{2}\rangle$
with $\underline{r}_{k}$ the monomer position and $\underline{q}$ the
wavevector. For polymer networks with Gaussian chain statistics the form
factor may be computed directly using 34

$F(q)=\sum_{s=0}^{s_{\text{max}}}w(s)P(q,s)$ (24)

with $P(q,s)$ being the Fourier transform of the segment size distribution
$P(\underline{r},s)$. Since for Gaussian chains $P(q,s)=\exp(-(aq)^{2}s)$ with
$a\equiv b/\sqrt{2d}$, the form factor is readily computed yielding, as one
expects, within numerical accuracy exactly the same results as obtained using
the configuration ensembles computed by means of MD and MC simulations. Data
for $S=1$ are presented in Fig. 8 where a Kratky representation is used.7 We
remind that for large $N$ and small $q\equiv||\underline{q}||$ the radius of
gyration $R_{\mathrm{g}}$, as one measure of the tree size, must become the
only relevant length scale. The form factor thus scales as $F(q)/N=f(Q)$ with
$Q=qR_{\mathrm{g}}$ being the reduced wavevector and $f(Q)$ a universal
scaling function with $f(Q)=1-Q^{2}/d$ in the “Guinier regime" for $Q\ll 1$ as
seen on the left side of the figure. The opposite large-$q$ limit probes the
density fluctuations within the spacer chains and on the scale of the
monomers. The data thus do not scale with $Q$ just as the pair correlation
density $\rho_{\mathrm{pair}}(r)$ in Fig. 7 did not scale for small
$r/R_{\mathrm{g}}$. Details on this limit are given in Appendix B. While one
expects $F(q)/N\sim 1/Q^{d_{\rm f}}$ in the intermediate wavevector regime of
“open" fractal objects ($d_{\rm f}<d$),49 compact fractals are described by
the “generalized Porod law"7, 5

$F(q)/N\sim 1/Q^{2d-d_{\rm A}}\ \mbox{ for }d_{\rm f}=d,d_{\rm A}<d.$ (25)

By matching these well-known limits, this demonstrates the key relation eqn
(3) for marginally compact objects. Alternatively, eqn (3) can be confirmed
using eqn (24) and eqn (15). As seen in Fig. 8, the data approach as expected
with increasing $I$ the slope $y\approx 4/Q$ (bold line). This confirms the
claimed marginal compactness of our trees. The standard Porod scattering 7 of
compact objects with a smooth surface ($d_{\rm A}=2$) corresponds instead to a
much steeper power-law envelope (dashed line).

##  4 Dynamical properties

Fig. 9: Scaling of mean-square displacements (MSD). Filled triangles indicate
the center-of-mass MSD $g_{\text{cm}}(t)$ for $I=3$, open symbols the monomer
MSD $g_{\text{mon}}(t)$, the bold solid lines the exponent $\delta=2/5$ from
eqn (28) with $\nu=1/3$. Inset: $g_{\text{mon}}(t)$ and $g_{\text{cm}}(t)$ vs.
time $t$ in MC steps for $I=3$ computed using local MC jumps with $\delta
r=0.5$. Also indicated is the GRM prediction for $g_{\text{mon}}(t)$ (pluses).
Main panel: Scaling collapse for $g_{\text{mon}}(t)/R_{\star}^{2}$ vs.
$x=t/\tau_{\star}$ for MC with different $I$ as indicated (open symbols) and
MD data for $I=4$ (crosses).

###  4.1 Introduction

We describe now some dynamical properties of our marginally compact trees
focusing on a spacer length $S=1$. We remind that all excluded-volume,
topological or hydrodynamic interactions between the $m=100$ chains of the
simulation box at $\rho=1$ are switched off. The Gaussian chain connectivity
is the only remaining potential. For clarity of the presentation we focus on
numerical data obtained using local MC jumps. Occasionally, we include the
corresponding results obtained by means of MD simulations or using the
generalized Rouse model (GRM). The principal goal is to show that the dynamics
is of a generalized Rouse-type characterized by an inverse fractal dimension
$\nu\equiv 1/d_{\rm f}=1/d$.

###  4.2 Mean-square displacements

Since the effective random forces acting on the monomers are uncorrelated, a
pure Fickian diffusion is expected for the center-of-mass of the chains.49 The
corresponding mean-square displacement (MSD) $g_{\text{cm}}(t)$ should thus
scale as

$g_{\text{cm}}(t)=2dDt\ \mbox{ with }D=\mbox{$k_{\rm B}T$}/\zeta N$ (26)

being the diffusion coefficient and $\zeta$ the friction coefficient. As shown
in Fig. 9, this is consistent with our simulations. We find that $\zeta\approx
44$ for MC and $\zeta=10$ for MD using appropriate units. (The latter value
for the MD simulations is imposed by the friction coefficient of the Langevin
thermostat.) Having thus determined the central parameter $\zeta$, this allows
us to compare the predictions of the GRM with our MC and MD simulations. Since
$R\sim N^{\nu}$, one expects a characteristic chain relaxation time 49

$\tau_{N}\sim R^{2}/D\sim N^{\beta}\mbox{ with }\beta\equiv 1+2\nu$ (27)

in agreement with eqn (49). The same scaling argument holds for the relaxation
time $\tau_{n}\sim n^{\beta}$ of a subchain of mass $n$ around an arbitrary
tagged monomer $k$ (Fig. 2). Compact (sub)chains ($\nu=1/d$) thus relax faster
in $d>2$ than Gaussian (sub)chains ($\nu=1/2$) of same mass.

The scaling of the relaxation time may be verified using the monomer MSD
$g_{\text{mon}}(t)$ as shown in Fig. 9 (open symbols). For large times
$t\gg\tau_{N}$ the monomers must follow the chain center-of-mass, i.e.
$g_{\text{mon}}(t)$ is given by eqn (26). A free diffusion
$g_{\text{mon}}(t)\sim tN^{0}$ is also observed for our MC simulations in the
opposite limit of very small times $t\ll\tau_{\text{mon}}\sim N^{0}$.
(Depending on the friction constant $\zeta$ a ballistic regime with
$g_{\text{mon}}(t)\sim N^{0}t^{2}$ appears for our MD simulations in this time
regime.) The chain connectivity matters, however, in the intermediate time
window for $\tau_{\text{mon}}\ll t\ll\tau_{N}$. The anomalous diffusion in
this regime can be understood using the standard scaling argument 49

$g_{\text{mon}}(t)\approx R^{2}(t/\tau_{N})^{\delta}\sim N^{0}\ \mbox{ for
}\tau_{\text{mon}}\ll t\ll\tau_{N}$ (28)

with $\delta=2\nu/\beta=1/(1+1/2\nu)$. Alternatively, one may obtain eqn (28)
from the center-of-mass mean-square displacement

$g_{\text{cm,k}}(t)\sim\frac{1}{n(t)}\times t\sim n(t)^{2\nu}\sim t^{\delta}$
(29)

of the $n(t)\sim t^{1/\beta}$ monomers dragged along by a reference monomer
$k$ within a time window $t$. As seen in Fig. 9 the power law $t^{2/5}$ (bold
solid lines) corresponding to $\nu=1/d$ perfectly fits the data. As shown in
the main panel of Fig. 9, all MC and MD data can be brought to collapse
($t\gg\tau_{\text{mon}}$) using as dimensionless coordinates the rescaled time
$x=t/\tau_{\star}$ and the rescaled MSD $y=MSD/R_{\star}^{2}$. The crossover
size $R_{\star}=2.2R_{\mathrm{g}}\sim N^{\nu}$ and the crossover time
$\tau_{\star}=R_{\star}^{2}/6D\sim N^{\beta}$ have been chosen to match both
asymptotic slopes at $x=1$.

###  4.3 Relaxation time of generalized Rouse modes

#### Monomer MSD revisited.

Due to the bilinearity of our model potential, eqn (7), the monomer MSD
$g_{\text{mon}}(t)$ presented in Fig. 9 can be directly predicted using the
GRM outlined in Appendix C. As shown elsewhere,49, 59, 60 the MSD can be
written as

$g_{\text{mon}}(t)=2dDt\left(1+\sum_{p=1}^{N-1}\frac{1-\exp(-t/\tau_{p})}{t/\tau_{p}}\right)$
(30)

with $\tau_{p}$ being the relaxation time of the mode $p>0$. As shown in the
inset of Fig. 9, eqn (30) yields the same results (crosses) as the MC
simulations (open triangles). The relaxation times $\tau_{p}$ have been
determined using $\tau_{p}=\tau_{\text{mon}}/\lambda_{p}$ with
$\tau_{\text{mon}}=\zeta/K$ being a convenient constant and $\lambda_{p}$ the
eigenvalues of the connectivity matrix $\mathbf{A}$. While the local time
scale $\tau_{\text{mon}}\sim N^{0}$ depends due to $\zeta$ on the simulation
method, the same eigenvalues $\lambda_{p}$ characterize MC and MD simulations.

#### Scaling of relaxation times $\tau_{p}$.

The reduced relaxation times $\tau_{p}/\tau_{\text{mon}}$, i.e. the inverse
eigenvalues $\lambda_{p}$, are shown in Fig. 10 for four different $I$.

Fig. 10: Generalized Rouse relaxation times
$\tau_{p}/\tau_{\text{mon}}=1/\lambda_{p}$ for four chain lengths $N(I)$. The
smallest eigenmode $p=1$ scales as $\tau_{p=1}/\tau_{\text{mon}}\approx
0.083N^{5/3}$ (bold solid lines). Inset: Half-logarithmic representation of
$1/\lambda_{p}N^{5/3}$ vs. $p$ for the first 23 eigenmodes. Degeneracies are
marked by horizontal lines. Main part: Double-logarithmic representation of
$1/\lambda_{p}$ vs. $p/N$.

We emphasize first that, as shown by the bold solid lines, the relaxation time
of the eigenmode $p=1$ increases as $\tau_{p=1}/\tau_{\text{mon}}\approx
0.083N^{5/3}$ as one expects from eqn (27) and $\tau_{N}\approx\tau_{p=1}$.
(Already the value for the tiny trees with $I=1$ comes close to this limit.)
The inset presents the first 23 eigenmodes using a half-logarithmic
representation where $\tau_{p}/\tau_{\text{max}}N^{5/3}$ is plotted as a
function of $p$. All relaxation times $\tau_{p}$ are given in the main panel
using a double-logarithmic representation with the rescaled mode number $p/N$
as horizontal axis. The coordinates have been chosen to verify the expected
data collapse with respect to the chain length $N$ in the regime each panel
focuses on. As seen from the inset, the relaxation times $\tau_{p}$ for small
$p$ do not decrease continuously but with clearly separated steps which become
more marked with increasing $N$. (The data become $N$-independent in this
limit.) These degeneracies reflect the symmetries of the tree generated by the
generator (Fig. 1). Despite these degeneracies the relaxation times $\tau_{p}$
collapse if plotted vs. $p/N$ as shown in the main panel. While this collapse
is only approximative for small $p/N$, as emphasized by the arrows pointing
downwards, it becomes clearly better with increasing $N$ and $p/N$.
Importantly, the relaxation times for all $p$ are within a narrow band between
$1/\lambda_{p=1}p^{5/3}$ (bold solid line) and $2/\lambda_{p=1}p^{5/3}$
(dashed line). It is seen that the relaxation times approach the upper bound
with increasing $p/N$. One may thus describe the relaxation times by an
effective power law

$\tau_{p}/\tau_{\text{mon}}\approx c(N/p)^{\beta}\mbox{ with
}\beta=1+2\nu=5/3$ (31)

as before and $c$ a numerical constant of order unity. Importantly, if we
replace the directly measured $\tau_{p}$ by eqn (31) with $c\approx 2$, one
obtains the same MSD $g_{\text{mon}}(t)$ as before (not shown). We note
finally that using the power law $\tau_{p}\sim(N/p)^{\beta}$ in eqn (30) for
times $t\ll\tau_{N}\approx\tau_{p=1}$ one confirms by replacing the sum by an
integral that $g_{\text{mon}}(t)\sim t^{2\nu/\beta}=t^{2/5}$ in agreement with
eqn (28).

Fig. 11: Shear-stress relaxation modulus $G(t)$. Inset: $G(t)$ for $I=3$
obtained with local jump MC and GRM prediction using eqn (32). Main panel:
$y=G(t)/(\mbox{$k_{\rm B}T$}\rho/N)$ vs. reduced time $x=2t/\tau_{p=1}$. The
predicted exponent $\gamma=3/5$ (bold lines) can be seen over several decades.
The dash-dotted line represents the GRM prediction $y\approx x^{-3/5}\exp(-x)$
for large $I$. The cutoff for $x\gg 1$ is visible for the smaller trees.

###  4.4 Shear-stress relaxation modulus

A central rheological property characterizing the linear shear-stress response
in fluids as well as in solids is the shear relaxation modulus
$G(t)\equiv\delta\sigma(t)/\delta\gamma$ defined as the ratio of observed
shear-stress increment $\delta\sigma(t)$ and applied shear-strain
$\delta\gamma$.6 Due to the simplicity of our Hamiltonian one may directly
compute $G(t)$ using the (slightly rewritten) relation eqn (4.158) from ref.
49

$G(t)=\mbox{$k_{\rm B}T$}n_{\mathrm{sp}}/V\
\sum_{p=1}^{N-1}\exp(-2t/\tau_{p})$ (32)

with $n_{\mathrm{sp}}$ being the number of springs and $\tau_{p}$ the mode
relaxation time described above, Fig. (10). An example for $I=3$ is given in
the inset of Fig. 11 (pluses). Using the shear-stress autocorrelation
function36 we have additionally obtained $G(t)$ from our equilibrium MC and MD
simulations as also shown in Fig. 11. This is, of course, the more general
method being not restricted to our specific Hamiltonian. The initial value
$G(t=0^{+})$ is given quite generally by the “affine shear elasticity"
$\mu_{\mathrm{A}}$, i.e. the ensemble-averaged second functional derivative of
the system Hamiltonian with respect to an affine shear strain.61 Using eqn
(B8) of Wittmer _et al._ 61 this yields $\mu_{\mathrm{A}}=\mbox{$k_{\rm
B}T$}n_{\mathrm{sp}}/V$ for a system of ideal springs. Since
$n_{\mathrm{sp}}/V\approx mN/V=\rho=1$ and $\mbox{$k_{\rm B}T$}=1$, this
implies $G(t=0^{+})=\mu_{\mathrm{A}}\approx 1$ in agreement with the MC and
GRM data presented in the inset of Fig. 11. Using general scaling arguments
(not restricted to Gaussian chain statistics)6 it is seen that the shear-
stress relaxation function should decay as

$G(t)\approx\mbox{$k_{\rm B}T$}\rho/N\ \left(\tau_{N}/t\right)^{\gamma}\
\exp(-t/\tau_{N})$ (33)

with $\gamma\equiv 1/\beta=1/(1+2\nu)$. The first factor $\mbox{$k_{\rm
B}T$}\rho/N$ stands for the ideal pressure of the chains and the last one for
the final exponential cutoff. If considered at a short local time
$t\approx\tau_{\text{mon}}\sim N^{0}$, eqn (33) becomes $G(t\approx
0)\approx\mbox{$k_{\rm B}T$}\rho$ in agreement with
$G(t=0^{+})=\mu_{\mathrm{A}}$. As seen from eqn (32), the final cutoff is
dominated by the relaxation of the smallest mode $p=1$ which suggests to set
$\tau_{N}=\tau_{p=1}/2$ for the longest shear-stress relaxation time. The main
panel presents again a dimensionless scaling plot. We trace the rescaled
relaxation modulus $y=G(t)/(\mbox{$k_{\rm B}T$}\rho/N)$ as a function of
$x=t/\tau_{N}$, i.e. we impose a final cutoff $y\approx\exp(-x)$. The power-
law exponent $\gamma$, characterizing the short time behavior, is again a
consequence of the scaling requirement that $G(t)$ cannot depend on $N$ in
this time regime. As may be seen from Fig. 11, the exponent $\gamma=3/5$ for
compact objects is confirmed over several orders of magnitude from our data
while the standard Rouse exponent $\gamma=1/2$ for $\nu=1/2$ (dash-dotted
line) is clearly ruled out. That $\gamma=1/\beta$ holds can be also
demonstrated by integration of eqn (32) using eqn (31). Paying attention to
the $p=1$ mode this leads to $y(x)\approx x^{-\gamma}\exp(-x)$ for large $I$
as indicated by the dashed line in the main panel. The main point we want to
make here is merely that it is in our view inconsistent to claim a value
$\gamma\neq 1/2$ for the shear-stress relaxation modulus and to attempt then
to fit $G(t)$ and other dynamical properties using a mode expansion in terms
of relaxation times $\tau_{p}$ characterized by an exponent $\beta=2$.888We
remind that an exponent $\gamma\approx 0.4$ has been fitted for dense
rings.23, 24 This compares nicely with the exponent $\gamma=3/7$ predicted
recently assuming compact rings.10 That our trees are characterized by a
significantly larger exponent $\gamma=3/5$ is caused, of course, by the faster
dynamics due to the missing topological constraints.

###  4.5 Dynamical structure factor

The vibrational or diffusive motion of folded proteins 62 or more general
natural or synthetic polymer-like structures 25, 26 can be studied
experimentally by means of dynamic light scattering or neutron spin echo
scattering.49, 6, 7 This allows to extract the single (intrachain) chain
dynamical structure factor

$F(q,t)=\frac{1}{N}\sum_{k,l=1}^{N}\langle\exp\left[\text{i}\underline{q}\cdot(\underline{r}_{k}(t)-\underline{r}_{l}(0))\right]\rangle$
(34)

which generalizes the static structure factor $F(q)=F(q,t=0)$ into the time
domain.

Fig. 12: Dynamical structure factor $F(q,t)$: (a) Rescaled relaxation time
$y=\tau_{q}q^{2}D$ vs. reduced wavevector $Q$ for MC ($I=3$ and $4$) and MD
($I=4$). (b) Reduced dynamical structure factor $y=-\log(F(q,t)/F(q))$ vs.
$x=t/\tau_{q}$. The dash-dotted line indicates the linear power-law slope
$\delta=1$ for small $x$, the dashed line the slope $\delta=2/5$ for $x\gg 1$.

In the low-wavevector limit the dynamical structure factor allows to probe
quite generally the overall translational motion of the chain using49

$F(q,t)\approx N\exp\left(-\frac{q^{2}}{2d}\ g_{\text{cm}}(t)\right)\ \mbox{
for }Q\equiv qR_{\mathrm{g}}\ll 1.$ (35)

We have verified that this relation holds for both our simulation methods in
the low-$Q$ limit (not shown).

An operationally simple way to quantify the decay of $F(q,t)$ for general
wavevectors is to define a relaxation time $\tau_{q}$ by the time needed for
$F(q,t)$ to reach some fraction of its initial value. We have chosen the
standard definition $F(q,t=\tau_{q})/F(q)=1/\mathrm{e}$. Following eqn (35)
this implies $\tau_{q}=1/q^{2}D$ for $Q\ll 1$. This suggests the scaling of
the reduced relaxation time $y=\tau_{q}q^{2}D$ as a function of the reduced
wavevector $Q$ shown in panel (a) of Fig. 12. Data for $I=3$ and $I=4$ for MC
and $I=4$ for MD are thus successfully brought to collapse. By construction,
$y\to 1$ for $Q\ll 1$ (thin horizontal line). The power law for the self-
similar $Q$-regime follows from

$\tau_{q}\approx\frac{1}{q^{2}D}Q^{-d_{\rm f}}\sim N^{0}/q^{\beta/\nu}\mbox{
for }Q\gg 1$ (36)

with $\beta=1+2\nu$. As shown by the bold solid line, $y\sim 1/Q^{3}$ holds as
expected for a marginally compact object. This implies $\tau_{q}\sim 1/q^{5}$
in the self-similar regime at variance to the standard $1/q^{4}$-scaling.49 By
setting $1/q\approx b(N/p)^{\nu}$ one confirms using eqn (36) that
$\tau_{p}\sim(N/p)^{\beta}$ with $\beta=5/3$ (Fig. 10).

Having characterized the relaxation time $\tau_{q}$, we attempt now to
describe the time-dependent decay of $F(q,t)$. We focus on data corresponding
to intermediate times $\tau_{\text{mon}}\ll t\ll\tau_{N}$ and wavevectors
$1/R_{\mathrm{g}}\ll q\ll 1/b$ allowing to probe the internal chain motion. As
shown in panel (b) of Fig. 12, we plot the reduced dynamical structure factor
$y=-\log(F(q,t)/F(q))$ as a function of the dimensionless scaling variable
$x=t/\tau_{q}$. Large $y$ corresponds to small density correlations, low $y$
to large ones. Using a double-logarithmic representation MC and MD data for
two chain lengths and several wavevectors $q$ are successfully brought to
collapse. (The MC data for $I=3$ correspond to slightly too small chains for
$q=0.6$.) Note that we have used the measured relaxation times $\tau_{q}$ for
the scaling, i.e. by construction $y(x=1)=1$ for all data. We could have
replaced $x=t/\tau_{q}$ by the more common scaling variable $x=t\Gamma_{q}$
using the rate 49

$\Gamma_{q}=(D/R_{\mathrm{g}}^{2})\ Q^{2+d_{\rm f}}\approx 1/\tau_{q}\sim
N^{0}q^{5}$ (37)

in agreement with eqn (36) and $d_{\rm f}=3$. One verifies that the standard
Rouse scaling with $\Gamma_{q}\sim q^{4}$ is not appropriate (not shown).

We have thus verified the scaling for the intermediate wavevector regime. The
observed universal function is qualitatively similar to the one for linear
Rouse chains for which a steepest decent argument shows that $y$ scales as
$g_{\text{mon}}(t)$. (See the discussion after eqn (4.III.10) in ref. 49.)
This leads for very small $x$, where the chain connectivity is less important
and the free monomer diffusion is probed, $g_{\text{mon}}(t)\sim tN^{0}$, to
the linear power-law slope ($\delta=1$) indicated by the dash-dotted line. The
steepest decent argument is also consistent with the fact that we observe for
$x\gg 1$ a power law $y\sim x^{\delta}$ with the same exponent
$\delta=2\nu/\beta=2/5$ (dashed line) as for $g_{\text{mon}}(t)$ in Fig. 9.
The data is thus more strongly curved ($\delta=1\to 2/5$) as for linear chains
($\delta=1\to 1/2$).

##  5 Conclusion

#### Summary.

We have investigated theoretically and by means of MC and MD simulations
several static (Section 3) and dynamical (Section 4) properties of marginally
compact hyperbranched polymer trees generated by means of a proper fractal
generator (Fig. 1). We have shown that our idealized hyperbranched trees are
after at most two iterations of the generator self-similar on all scales
(Figs. 4-8), marginally compact (Figs. 6 and 8) and much faster as unbranched
objects of same mass (Figs. 9-12). Two points have been emphasized in the
present work. First, since the return probability decays as $p(n)\sim
1/\sqrt{S}n$ for $n_{1}\approx S\ll n\ll n_{2}\approx N$ (Fig. 5), the average
density of self-contacts $\rho_{\mathrm{c}}$ must diverge logarithmically, eqn
(21), with the total mass $N$, however, with a prefactor strongly decaying
with the spacer length $S$ (Fig. 6). Second, the standard linear-chain Rouse
mode analysis commonly made in experimental studies 63, 25, 26 of the shear-
stress relaxation modulus $G(t)$ or the intrachain dynamical structure factor
$F(q,t)$ must necessarily become inappropriate for self-similar compact
objects (Figs. 9-12) for which the relaxation time of a subchain of mass
$n\approx N/p$ must scale as $\tau_{p}\sim(N/p)^{5/3}$ (Fig. 10).

#### Discussion.

The first emphasized point is the most crucial one of general importance
beyond the polymer-like systems we have focused on. As correctly stressed by
Halverson _et al._ 18, any description of a real system in terms of a
marginal-compact model must thus in principle break down for $N\to\infty$
since for excluded volume constraints the local volume density cannot exceed
unity. However, due to the much stronger dependence on the spacer size (Fig.
6), the logarithmic divergence becomes in practice rapidly irrelevant for a
broad mass window. Please note that the marginal compact model for melts of
rings advanced by some of us 5, 9 corresponds to spacer lengths $S$ set by the
entanglement length $N_{\mathrm{e}}$ of the corresponding linear chain
systems. Since $N_{\mathrm{e}}$ is rather large and considering the ring
masses $N$ available at present, this makes the (certainly simplified)
marginal-compact modeling approach at least conceivable. Coming back to the
living matter network systems mentioned in the Introduction, it should be a
fairly simple task for evolution to tune the length and other structural
properties of the spacers (twigs, blood vessels, bronchiae, …) connecting the
metabolically active subunits to increase the allowed mass window $N\ll
N_{\star}$. Marginal compact models are thus not only mathematically well-
defined (assuming one or several proper generators) but also of relevance for
the description of real physical and biological systems. In a nutshell, a
marginally compact Christmas tree cannot be made only using needles. One needs
twigs as well.

#### Outlook.

Excluded volume and all other effectively long-range interactions have been
assumed to be switched off in the present work. This is in line with Flory’s
ideal chain hypothesis for dense polymer melts 49 and is a necessary condition
for taking advantage of the GRM (Appendix C). However, considering that
Flory’s hypothesis only holds to leading order for linear chains,64 it may not
be justified for more compact fractal objects even for a large spacer length
$S$. Excluded volume effects in melts of hyperbranched chains will thus be
considered as a function of $S$ in future work. This should also address the
possibility to relax the quenched connectivity matrix $\mathbf{A}$, eqn (7),
by means of local MC moves allowing small dangling ends to hop along the
network following broadly the recent work by Rosa and Everaers.29, 30, 31

##  Appendix A Self-contact density $\rho_{\mathrm{c}}$

#### Linear chain reference.

The self-contact density $\rho_{\mathrm{c}}$ presented in Fig. 6 has been
rescaled using the self-contact density $\rho_{\mathrm{c,lin}}$ for
arbitrarily long linear Gaussian chains. Using eqn (19) with $Nw(s)=2$ this
reference is

$\rho_{\mathrm{c,lin}}=2\times(d/2\pi b^{2})^{d/2}\times\zeta(d/2)\approx
1.724/b^{3}$ (38)

with $\zeta(x)=\sum_{s=1}^{\infty}1/s^{x}$ being Riemann’s $\zeta$-function.65
Note that $\zeta(3/2)\approx 2.612$. The surprisingly large value of
$\rho_{\mathrm{c,lin}}$ stems mainly (by about $50\%$) from the two next and
the two next-nearest neighbors along the chain, i.e. this is a local-range
artifact of the extremely simplified Gaussian chain model.

#### Contributions to $\rho_{\mathrm{c}}$.

Let us trace back eqn (21) to the contributions $\rho_{\mathrm{c,1}}$,
$\rho_{\mathrm{c,2}}$ and $\rho_{\mathrm{c,3}}$ due to the three regimes of
the distribution $w(s)$ discussed in Sec. 3.2. For simplicity, we assume large
spacer chains and a large iteration number $I$, i.e. $1\ll s_{1}\approx S\ll
s_{2}\approx s_{\text{max}}\ll N$. The dominant contribution to
$\rho_{\mathrm{c}}$ thus stems from the second regime ($s_{1}\ll s\ll s_{2}$)
where $w(s)$ is given by eqn (15). Integration of eqn (19) from $s_{1}$ to
$s_{2}$ yields

$\frac{\rho_{\mathrm{c,2}}}{\rho_{\mathrm{c,lin}}}\approx\frac{c_{\mathrm{asym}}}{\sqrt{S}}\times\frac{d}{2}\times\log(s_{2}/s_{1})$
(39)

where we have introduced for convenience the constant

$c_{\mathrm{asym}}\equiv\frac{c_{\star}}{d\ \zeta(d/2)}\approx 0.319$ (40)

using that $c_{\star}\approx 5/2$ (Fig. 4). Since $s_{1}=Sf_{1}$ and
$s_{2}=s_{\text{max}}f_{2}$ and using eqn (8) this can be rewritten as

$\frac{\rho_{\mathrm{c,2}}}{\rho_{\mathrm{c,lin}}}\approx\frac{c_{\mathrm{asym}}}{\sqrt{S}}\left(\log(N/S)-\underline{\frac{d}{2}\log(f_{1}/f_{2})}\right)$
(41)

where the underlined term in eqn (41) gives one contribution to the
subdominant correction $c_{\mathrm{corr}}$ fitted in eqn (41). Other
subdominant corrections arise from the linear spacer regime for $1\leq s\ll
s_{1}$ where to leading order

$\frac{\rho_{\mathrm{c,1}}}{\rho_{\mathrm{c,lin}}}-1\sim-1/\sqrt{S}.$ (42)

The contribution $\rho_{\mathrm{c,3}}$ of the third regime beyond $s_{2}$ is
difficult to describe by an analytic formula. It is expected to be negligible,
$\rho_{\mathrm{c,3}}\ll\rho_{\mathrm{c,2}}$, due to the cutoff of $w(s)$ seen
in Fig. 4. In any case neither $\rho_{\mathrm{c,1}}$ nor $\rho_{\mathrm{c,3}}$
do matter in the large-$I$ limit. Summarizing all three terms this leads to
eqn (21) where we have lumped all subdominant contributions into the
coefficient $c_{\mathrm{corr}}$.

##  Appendix B Additional points concerning $F(q)$

#### Spherical preaveraging.

As reminded at the beginning of Section. 3.6, $F(q)$ is the ensemble average
of the squared Fourier transform $\hat{\rho}(\underline{q})$ of the
fluctuating instantaneous density
$\sum_{k}\delta(\underline{r}-\underline{r}_{k})$. Following Harreis _et al._
66, Götze and Likos 67 this begs the question of whether in the limit of large
marginally compact chains density fluctuations become sufficiently small
allowing to replace $\hat{\rho}(\underline{q})$ by the Fourier transform of
the averaged density profile $\rho_{k=1}(r)$ around the root monomer. (Per
symmetry the root monomer stands at the center of the preaveraged density
profile.) The (not shown) density $\rho_{k=1}(r)$ is very similar to the pair
correlation density $\rho_{\mathrm{pair}}(r)$ averaged over all monomers $k$
(Fig. 7). It may be obtained from the simulations or by means of

$\rho_{k=1}(\underline{r})=N\sum_{s=0}^{s_{\text{max}}}w_{k=1}(s)P(\underline{r},s)$
(43)

with $P(\underline{r},s)$ being given by eq. (6). Due to the spherical
symmetry this suggests using eqn (6.54) of ref. 7 that

$F(q)\approx\frac{1}{N}\ \left(\int\text{d}\underline{r}\ \rho_{k=1}(r)\
\frac{\sin(\underline{q}\cdot\underline{r})}{\underline{q}\cdot\underline{r}}\right)^{2}.$
(44)

Note that this is equivalent to the factorization suggested on page S1791 by
Götze and Likos 67. As revealed by the dash-dotted line in Fig. 8, eqn (44) is
not a useful approximation. This counter example shows that it is not possible
in general to determine from the measured form factor $F(q)$ the average
density profile and thus an effective interaction potential. While this
approach may work for colloids, collapsed polymer chains below the
$\theta$-point or densely packed dendrimers 66, 67 having a clearly defined
surface, it should be used with care for general compact objects. Compactness
does not necessarily imply negligible density fluctuations.

Fig. 13: Kratky representation $y=(F(q)/N)Q^{2}$ vs. $Q=qR_{\mathrm{g}}$
comparing our simple tree model for $I=3$ and different $S$ as indicated with
published data for ring melts obtained by MD simulations of a standard bead-
spring model.5 Both data sets are qualitatively similar, albeit with slightly
different prefactors for the $1/Q$-slopes. The thin solid line indicates the
Debye formula for linear Gaussian chains.49

#### Spacer length effects.

The data presented in Fig. 8 have been obtained for spacer chains of length
$S=1$. The observed strong increase of the data for large $q$ is simply due to
the discrete monomeric units used in our simulations. As shown in Fig. 13 for
$I=3$, a horizontal plateau gradually appears with increasing $S$
corresponding to the density fluctuations associated to the Gaussian spacer
chain. This plateau is roughly located at the minimum $y_{min}\sim
R_{\mathrm{g}}^{2}/N\sim 1/N^{1/3}$ of the data for $S=1$. It is thus not
possible by tuning the local physics, i.e. the spacer length $S$ and (as may
be shown) the statistical segment size $b$, to increase the range of the
intermediate wavevector regime. Since $y_{min}\sim 1/N^{1/3}$ decreases
extremely weakly with mass, the determination of the $1/Q$-scaling becomes
very difficult.

#### Comparison with ring data.

This sluggish convergence is by no means unique to our simple model, but is
generic for more-or-less compact polymers such as linear polymer melts in
strictly two dimensions 68, 69, 5 or melts of polymer rings.5, 8, 25, 26 This
may be seen from the ring data5 also included in Fig. 13. As expected from the
sluggish convergence of the radius of gyration, the reduced ring data approach
from above the slope $y=3/Q$ (bold dashed line) indicated to guide the eye.
The ring simulation data are thus not compatible with a generalized Porod law,
eqn (25), with a fractal surface dimension $d_{\rm A}<d$. Similar behavior is
also seen in other numerical studies 16, 17, 18, 8 and in recent experimental
work.25, 26 As emphasized by Wittmer _et al._ 8, it is important that the data
is correctly scaled used the measured radius of gyration as done in Fig. 13.
Interestingly, albeit our tree model and the ring data are qualitatively
similar, the respective scaling functions $f(Q)$ of both systems differ
slightly: The trees reveal stronger density fluctuations with a broader hump
and a slightly larger prefactor for the $1/Q$ power law. While for large
wavevectors the fit of the tree model onto the ring data can be improved by
tuning $S$ and $b$, this is impossible in the $Q$-scaling regime. It is
currently not clear whether it is possible to generalize the tree generator,
e.g., using a multi-fractal approach,9 to get a better match of the scaling
functions $f(Q)$.

##  Appendix C Generalized Rouse Model (GRM)

The dynamics of our ideal spring networks, eqn (7), is described by the
Langevin equation 49

$\zeta\underline{\dot{r}}_{i}+\partial_{\underline{r}_{i}}V(\\{\underline{r}_{i}\\})=\underline{f}_{i}(t),$
(45)

where the stochastic forces $\underline{f}_{i}(t)$ are represented by a white
noise and $\zeta$ is the friction constant. Due to the bilinear form of
$V(\\{\underline{r}_{i}\\})$, eqn (7), the set of Langevin equations (eqn 45)
is a linear set. It is solved by diagonalization of $\mathbf{A}$. The non-
vanishing ($p\geq 1$) eigenvalues $\lambda_{p}$ yield the relaxation times
$\tau_{p}\equiv\tau_{\text{mon}}/\lambda_{p}$ with
$\tau_{\text{mon}}=\zeta/K$. Using merely the eigenvalues $\lambda_{p}$ of the
connectivity matrix $\mathbf{A}$ (and not its eigenvectors), many dynamic
properties can be readily calculated 59, 60 as shown by the relations given in
the main text for the monomer MSD, eqn (30), and the shear-stress relaxation
modulus $G(t)$, eqn (32). Unfortunately, the computation of the dynamical
structure factor $F(q,t)$ requires in addition the determination of all
eigenvectors.

In the reminder we corroborate the scaling relations presented in Section 4
using a more general context stemming from the analysis of the eigenmode
spectrum.70 For fractals, the density of states $h(\lambda)$ usually possesses
a power law behavior 70

$h(\lambda)\equiv\frac{1}{N-1}\sum_{p=1}^{N-1}\delta(\lambda-\lambda_{p})\sim\lambda^{\frac{d_{s}}{2}-1},$
(46)

where $d_{s}$ is the so-called spectral dimension.70 Equation (46) stems from
solid state physics, where for regular lattices one gets a similar expression
having instead of $d_{s}$ the dimension of the lattice.71 However, in case of
fractals, the spectral dimension $d_{s}$ differs generally from the dimension
of the fractal lattice.70 We note also that the density of states $h(\lambda)$
is directly connected to the dependency of the eigenvalues $\lambda_{p}$ on
the mode number $p$. Indeed, rewriting $p=p(\lambda)$ and bearing in mind that
$h(\lambda)\sim\partial_{\lambda}p(\lambda)$, one gets
$\lambda_{p}\sim(p/N)^{2/d_{s}}$ and thus

$\tau_{p}\approx\tau_{\text{mon}}(N/p)^{\beta}\mbox{ with }\beta=2/d_{s}.$
(47)

Hence, the smallest non-vanishing eigenvalue $\lambda_{\min}$ and the
corresponding largest relaxation time $\tau_{N}$ are

$\lambda_{\min}\sim N^{-2/d_{s}}\mbox{ and
}\tau_{N}\approx\tau_{\text{mon}}N^{2/d_{s}}.$ (48)

Moreover, in many cases the spectral dimension $d_{s}$ is related to the
fractal dimension $d_{\rm f}$ of a Gaussian macromolecule by 70, 72

$d_{s}=\frac{2d_{\rm f}}{d_{\rm f}+2}\mbox{ hence, }\beta=1+2/d_{\rm f}.$ (49)

To see this relation, one considers the continuous limit
$NR_{\mathrm{g}}^{2}\sim\
\sum(1/\lambda)\rightarrow\int\mathrm{d}\lambda(h(\lambda)/\lambda)$.73 This
leads readily to $R_{\mathrm{g}}^{2}\sim N^{(2-d_{s})/d_{s}}$, if one takes
eqn (48) into account.74 It is important to note, that there are examples,
where the relation of eqn (49) is invalid, partly due to violation of the
$\lambda_{\min}\sim N^{-2/d_{s}}$ behavior, see, e.g., Sokolov 75. Now, the
scaling of eqn (46) determines the behavior of the dynamic variables. One
finds59, 76 for intermediate times $\tau_{\text{mon}}\ll t\ll\tau_{N}$

$g_{\text{mon}}(t)\sim t^{1-\frac{d_{s}}{2}}\textrm{ and }G(t)\sim
t^{-\frac{d_{s}}{2}}.$ (50)

Since $d_{\rm f}=3$, we have $d_{s}=6/5$. Hence, from the above relations we
get $\tau_{p}\sim(N/p)^{5/3}$, $g_{\text{mon}}(t)\sim t^{2/5}$ and $G(t)\sim
t^{-3/5}$ in agreement with the corresponding statements made in Section 4.

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