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Full PaPer

The Presence of a Wall Enhances the Probability 
for Ring-Closing Metathesis: Insights from Classical 
Polymer Theory and Atomistic Simulations

Ingo Tischler, Alexander Schlaich, and Christian Holm*

I. Tischler, Dr. A. Schlaich, Prof. C. Holm
Institute for Computational Physics
University of Stuttgart
70569 Stuttgart, Germany
E-mail: holm@icp.uni-stuttgart.de

The ORCID identification number(s) for the author(s) of this article 
can be found under https://doi.org/10.1002/mats.202000076.

DOI: 10.1002/mats.202000076

In this work, we investigate the impact 
of geometric confinement on the pro-
cess of ring-closing of a single chain. To 
do so we assume that the probability of a 
ring-closing event is linearly related to the 
probability that the two ends of a chain are 
located within a center-to-center distance 
between zero and some reaction length λ. 
We start out by analyzing the end-to-end 
distance distribution for ideal Gaussian 
chains, and for molecular dynamic simula-
tions of an atomistic representation of 18- 
and 28-monomer long alkane oligomers, 
where one of the end monomers is fixed in 
space, and located a distance d away from a 

reflecting, that is, inert and impenetrable, surface. This set-up is 
motivated by the idea that a possible catalyst can be attached to 
a confining wall via molecular linker groups of various lengths 
at a certain distance. The comparison between the ideal chain 
and the atomistic oligomers is performed by mapping the oli-
gomer conformational properties to an equivalent freely-jointed 
chain, whose statistical properties can be calculated analytically 
via classic polymer theory.[9] Previous analytic work on RCM for 
bulk systems has been done, for example, in refs.  [10, 11]. Our 
analysis of this model suggests that the ring-closing probability 
of a tethered ideal chain is always enhanced compared to a free 
ideal chain, and that the two investigated united atom oligomers 
can show both an enhancement, and a diminishing, of the ring-
closing probability, which depends on the tether distance d.

This article is organized as follows: First we describe the 
polymer theory and the notation for our set-up. In Section 3, we 
describe the investigated atomistic oligomer model, followed by 
the results in Section 4. We finish our article with the conclu-
sions and outlook for further studies.

2. Theory

The theory of ideal Gaussian polymer chains (or classical 
random walks) is well understood,[9] and is summarized in 
the following section. Starting the random walk (RW) at the 
origin, a displacement of fixed length b, in one of the Cartesian 
directions is chosen randomly. Starting again from this point 
in space, the procedure is repeated N times. If the probability 
to take a step in any direction is equally likely, the distribution 
of walks of a certain length (i.e., the end-to-end distance of the 
ideal polymer) is given by the binomial distribution. For long 
RWs, N ≫ 1, the central limit theorem can be applied, and the 

The probability distribution of chain ends meeting when one end of the 
polymer is fixed to a certain distance to a reflecting wall is investigated. For 
an ideal polymer chain the probability distribution can be evaluated analyti-
cally via classic polymer theory. These analytical predictions are compared to 
atomistic MD simulations of one tethered alkane chain close to the wall. The 
results demonstrate that a confining wall can lead to a significant increase in 
the return probability for the chain ends, and thus, can increase the occur-
rence of ring-closing reactions. It is further demonstrated that the excess 
return probability shows a maximum at a certain distance, thereby yielding an 
optimal catalyst position in the ring-closing reaction.

1. Introduction

Macrocyclization reactions[1–3] are a common tool for drug dis-
covery and production in industry. Due to the bioactivity of 
macrocyclic molecules, they can be used as an antitumoral, anti-
biotics, or an antifungal. On the commercial side, they are often 
used as perfume components. One particular reaction is the ring-
closing metathesis (RCM) of dienes.[4–6] As depicted in Figure 1, a 
competing pathway for this reaction is the oligomerization via the 
acyclic diene metathesis (ADMET), resulting in a so-called a ring-
chain-equilibrium[7] that will diminish the efficiency of the RCM.

To increase the selectivity of ring-closing over polymerization, 
it is possible to decrease the concentration of the dienes, how-
ever, this is not feasible when trying to upscale the catalysis. In 
ref.  [8] a novel biomimetic approach was suggested where the 
catalyst was brought into a cylindrical confining space, such that 
only very few (ideally one) substrates were able to enter the pore, 
resulting in preferential RCM and suppressed ADMET. Experi-
ments of this metathesis in SBA-15 nano-tubes by Ziegler et al.[8] 
showed an increase in the selectivity due to confinement effects.

© 2020 The Authors. Macromolecular Theory and Simulations published 
by Wiley-VCH GmbH. This is an open access article under the terms 
of the Creative Commons Attribution License, which permits use, 
distribution and reproduction in any medium, provided the original 
work is properly cited.

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probability of finding RWs with an end-to-end vector x, can be 
approximated by a Gaussian of the form

( )
3

2 2

3
2

3

2

2

2



π
= 





−

P x
R

e
x

R  (1)

where we have introduced the length of the random walk, 
R2 = Nb2.

Considering all RWs which have the same end-to-end dis-
tance | |e

=R x  from the origin we obtain the probability distri-
bution Pe(Re) by integrating Equation  (1) over the surface of a 
sphere of radius Re.

( ) 4
3

2
e e e

2
2

3
2

3

2
e
2

2π
π

= 





−

P R R
R

e
R

R  (2)

To investigate interfacial and confinement effects on a 
random walk we introduce a reflecting wall at zw  =  −d (see 
Figure  2). A reflection occurs if in step i, a displacement is 
selected that would end in zi − zw < 0. In this case, the z-coor-
dinate is mirrored at the wall such that zi = |zi − zw|. While this 
approach does not conserve the step length b upon a collision 
with the wall, it has the advantage of a well-defined inert, and 

impenetrable wall, without any additional degrees of freedom. 
The corresponding RW now starts a distance d away from the 
surface, which corresponds to the distance of a stiff linker fixing 
polymer in space. The probability distribution function to find 
the other end of the polymer at position 



x, is then given by[12]

( , ) ( ) (2 )
0

� � ��
= + − ≥ −

< −






P x d P x P d e x z d

z d
z  (3)

Integration over the surface of a sphere of size Re to obtain 
the end-to-end distance yields

( , )
( )

2
sin( ) 1 d

( )

2
1 cos( )

6

with
arccos

e e
e e

0

e

6 cos( )

e

e e
e

2

e

6 cos( ) 6 ( )

e

e

e
e

e e e

2

e e

2
e

2

∫ θ θ

θ

θ
π

π

= +






















= − + −






















=
≤

− 





>









θ θ

θ

( )

( )

− +

− + − +

P R d
P R

e

P R R

dR
e e

R d

d

R
R d

d d R

R

d d R

R

d d R

R

 (4)

which describes the end-to-end distribution of a random walk 
confined by a flat wall at distance d.

We now investigate the influence of the end-to-end prob-
ability distribution Equation (4) on the ring-closing probability. 
Whilst the latter depends on many variables including the local 
environment, diffusion, and transport of the substrate etc., here 
we assume that this scales linearly with polymer return prob-
ability Ppr(d, λ). This assumption holds to leading order, as the 
ring-closing can only happen if the two polymer ends meet at 
least a characteristic distance λ away from catalytic center, as 
indicated in Figure  2. The reaction radius λ thus describes a 
sphere around the origin encapsulating the details of the cata-
lytic reaction. The polymer return probability is correspondingly 
obtained by integrating the end-to-end distribution up to λ by

( , ) ( , ) dpr

0

e e∫λ =
λ

P d P R d Re  (5)

Figure 1. Scheme of different pathways for a diene metathesis reaction. The catalyst is shown in red, the green particles denote those carbon atoms 
that share a double bond. Only these can attach to the catalyst. The other backbone carbon atoms are depicted in blue. a) The different pathways of 
an acyclic diene metathesis is seen. A ring-closing metathesis is observed. In the center top, and center bottom stages, the catalyst can accept a bond 
from the green particles. In the center of the picture the bonds have been formed. The reaction can either continue forward, (clockwise) or backward 
(counter clockwise). This depends on the order with which the bonds will break. The right path for both reactions is the same up to the point where 
either (a) a carbon chain is attached to the catalyst, or (b) the two ends of the chain close onto themselves to form a ring.

Figure 2. Illustration of the polymer model. One end of the chain (red) 
is held in place at a fixed linker length d. Re marks the end-to-end dis-
tance between the beginning and the end (green) of the chain. The 
dashed circle indicates the reaction radius λ of the sphere over which 
Equation (5) is integrated.

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It is instructive to compare the polymer return probability 
close to an interface to the case where the polymer is in bulk 
(“free” case, i.e., the polymer does not interact with the surface 
in the limit d  →  ∞). To this end, we define the excess return 
probability due to the wall as

( , )
( , )

( , )
1ex

pr

pr

λ
λ
λ

=
∞

−P d
P d

P
 (6)

Equation  (6) directly yields the increase in return 
probability— which we assume to be proportional to the 
selectivity increase — for an ideal polymer attached to a surface 
at a linker distance d.

To connect the ideal polymers that underlie a RW to chemi-
cally realistic alkane chains below, we rescale all lengths in our 
model by the Kuhn length defined as:[9]

( 1) cos
2

e
2

max

e
2

θ= =
− 





b
R

R

R

n l
 (7)

Here, Rmax is the maximal end-to-end distance of the pol-
ymer in equilibrium, n is the number of monomers, l is the 
bond length, and θ is the bond angle of the chemically realistic 
chains. This allows the mapping of the end-to-end distance to 
an equivalent freely jointed chain of segment length b, with the 
corresponding number of Kuhn segments

max=N
R

b
 (8)

Using such a mapping, any polymer will display the 
same average and-to-end distance as a RW, in the limit of 
long polymers.

3. Polymer Simulation

We model n-alkane chains using a chemically realistic united 
atom force field, with potential energy functions summarized 
in Table  1. The solvent was treated implicitly via Langevin 
dynamics with the Verlet integration scheme. To match the dif-
fusion coefficient of methane, we chose a friction coefficient 
of γ = 2.566/kBT. We set the temperature to T = 300K, and the 
timestep to Δt  = 8 fs. The total time over which each simula-
tion was sampled was 160=t  µs. In analogy to the experiments 
performed by Ziegler et al.,[8] simulations for two sets of chain 
lengths n ∈ {18, 28} were performed using the ESPResSo soft-
ware package.[14]

In line with the theoretical considerations above we get 
rid of additional simulation parameters by employing a 
purely reflective wall. Particle positions which, after a posi-
tion update are within the wall, are reflected according to 
zi =  |zi − zw|, and their velocities in z-direction are reversed. 
The position of the n  = 1 monomer was fixed at a distance 
d from the reflecting wall and the end-to-end distance distri-
bution was sampled for d  ∈ [0Å, 19Å]. The sampling reso-
lution was Δd  = 0.25Å for 4Å <d  < 14Å, and Δd  = 1Å in all 
other cases.

4. Results

The resulting end-to-end distance probability distribution from 
Equation  (4), and the united atom MD simulations, are found 
in Figure 3. For the random walk theory one can easily proof 
that the random walk case for d/b = 0, that is, the random walk 
starts on the wall, is equal to the end-to-end distance probability 
distribution of an unconfined (free) random walk. This is due 
to symmetry reasons.

Moving the starting location slightly away from the wall 
results in a shift toward smaller end-to-end distances. If one 
increases the distance d to values of about the average end-to-
end distance of the free random walk, the end-to-end distance 
starts to approach the free case again. This time, however, the 
peak of the distribution gets larger, while the longer end-to-end 
distances appear less frequently. For even larger values of d, the 
peak shifts back to larger end-to-end distances, where again Pe 
lies above the free RW. For even larger values of d, the distribu-
tion will converge to that of a free random walk which is clear 
as the effects of the wall are rarely noticed by the monomers. 

Table 1. United atom force field parameters taken from ref.  [13]. The 
Lennard–Jones interactions have only been accounted for if between the 
two particles of the chain, there are at least three other particles (1–4 
exclusion).

Pair bond potential

( )
2

( )0
2= −E r

k
r rr

Pair r0 [Å] k [kcal mol−1/Å2]

CH2 CH2 1.53 899

CH2 CH3 1.53 899

Angle bond potential

( )
2

( )0
2θ θ θ= −θE

k

Pair θ0 [°] kθ [kcal mol−1/rad2]

CH2 CH2 CH2 112 120

CH2 CH2 CH3 112 120

Dihedral bond potential

( )
1
2

[ (1 cos( )) (1 cos(2 )) (1 cos(3 ))]1 2 3φ φ φ φ= − + − + −φ φ φE k k k

Pair k 1
φ  [kcal mol−1] k 2

φ  [kcal mol−1] k 3
φ  [kcal mol−1]

CH2 CH2 CH2 
CH2

1.6 −0.8 3.24

CH2 CH2 CH2 
CH3

1.6 −0.8 3.24

LJ-parameters

( ) 4
12 6

ε σ σ= 



 − 













E r

r r

Pair σ [Å] ε [kcal mol−1]

CH2 ⋅⋅⋅ CH2 4.009 0.09344

CH2 ⋅⋅⋅ CH3 4.009 0.14546

CH3 ⋅⋅⋅ CH3 4.009 0.22644

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This behavior can be observed for both chain lengths. The 
main difference is the observation that the longer chain prefers 
longer end-to-end distances.

From our simulations using the united atom model, we 
observe that the n = 18 chain for d/b = 0 varies strongly from 
the n  = 28 chain. In both cases, it is important to note that 
the particle is practically embedded in the wall, and the chain 
starting from that point can only extend in the direction away 
from the boundary. Therefore, excluded volume effects con-
tribute significantly to the end-to-end distribution, and corre-
spondingly, Pe(Re) is shifted to larger end-to-end distances. For 
longer chain lengths, and large distances, the excluded volume 
effects become less important, and the distribution functions 
approach the analytical solution for the RW (see Figure 4b,d). 
Notably, at a certain distance d, the distribution functions for 
realistic polymer chains prefer to favor shorter end-to-end dis-
tances when compared to the RW. The simulation of chain 
length n = 28 is long enough to behave as predicted from a RW 
model, with the exception of the excluded volume at short end-
to-end distances. The shorter chain, however, does not fit the 
model, and tends toward larger end-to-end distances.

Investigating the excess return probability (Figure  4), we 
observe a maximum in all 4 cases considered. For the analyt-
ical solution, we observe a dependency on the reaction radius 
λ, which is not present in the simulation data. This reaction 
radius, however, is not of experimental significance, as it cor-
responds to details of the reaction. The maximum itself on the 
other hand, tells us that we can have a significant increase in 
the polymer return probability, which further translates into 

a boost of the ring-closing probability. For our simulation we 
therefore would have expanded the ring-closing probability by 
18.5%, given the optimal distance to the wall.

As can be observed from our simulation results in 
Figure  4c,d, the simulation show a range where the excess 
return probability is negative. This is both due to the excluded 
volume of the LJ-particles, which repel each other, and the stiff-
ness of the polymer, and thus is expected to also correspond to 
real polymers anchored close to a wall.

5. Conclusion

We have investigated how the probability distribution of end-to-
end distances for a single polymer chain changes when one end 
of the polymer is fixed at a certain distance from a reflecting 
wall. For an ideal Gaussian chain, this can be done analytically 
via classical polymer theory. By assuming that the ring-closing 
probability is linearly related to both polymer ends finding each 
other within a certain reaction radius λ, we have investigated 
the excess return probability, which reveals the corresponding 
change in the ring-closing probability due to the wall constraint. 
The theory of classical random walks yields a non-monotonic 
excess return probability that is always positive, and that displays 
a chain length dependent maximum, that varies monotonically 
with d. Since the Gaussian chain lacks any molecular interac-
tions, we also performed the same investigations by performing 
molecular dynamics simulations of a united atom alkane model. 
The comparison with the random walk model was performed 

Figure 3. End-to-end probability distributions obtained via random walk theory (top) using Equation (4) and simulations (bottom) for the polymer 
chain lengths 18 (left) and 28 (right). The theoretical model used the Kuhn length b, and corresponding Kuhn segments N, calculated from the 
simulation data (Equations (7) and (8)). For the shorter chain these parameters are b18 = 8.5Å, and N18 = 1.71. For the longer chain, they are b28 = 
6.8Å and N28 = 3.49.

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by mapping the conformational properties of the united atom 
model, onto that of an equivalent freely jointed chain. For both, 
theory and simulation, we found an optimum in the excess 
return probability, which demonstrates a linker-length which 
maximizes the ring-closing probability. For the simulation of 
a polymer with n  = 18 monomer units, we could measure an 
increase of up to 18.5% in comparison to a free polymer chain, 
whereas for the n = 28 chain, the increase was reduced to about 
14.5%, albeit with a broader maximum. The optimal distances 
for the spacer length were for both cases about 7 Å, although 
the distribution is much broader for the longer chain. Interest-
ingly, the ring-closing probability depended only weakly on λ 
for the interacting chain simulations.

The advantage of single chain simulations is that they run 
very fast and allow for an easy change of the confining geom-
etry and chain parameters. In the future we plan to perform 
multi-chain particle based simulations, in order to further study 
the influence of more complicated confining geometries on the 
selectivity of the RCM of dienes.

Acknowledgements
This work was funded by the German Research Foundation (DFG) via 
Project-ID 358283783 – SFB 1333.

Open access funding enabled and organized by Projekt DEAL.

Conflict of Interest
The authors declare no conflict of interest.

Keywords
confinement, Gaussian chain statistics, MD simulations, polymer 
theory, random walk, ring-closing metathesis

Received: October 12, 2020
Published online: 

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Figure 4. Excess return probability as defined in Equation (6). In the top row plots, a,b) we plot Pe of the analytical RW solution, for three values of 
λ/b, calculated via the corresponding parameters of the equivalent freely jointed chain. The dotted line indicates the position of maxima for other 
values of λ/b. In the bottom plots, c,d) we display the corresponding Pe of the simulation results for the united atom alkane chains with n monomers.

Macromol. Theory Simul. 2020, 2000076