An Investigation on Characteristics of Linear Homopolymer Molecules
Using Monte Carlo Simulation with
Random Walk and Self-Avoiding Walk Models
Idtisak Paopo, Pajaree Rittiyong, Siripon Anantawaraskul*
Department of Chemical Engineering, Faculty of Engineering,
Kasetsart University, 50 Phaholyothin Rd., Jatujak, Bangkok, Thailand, 10900
*
Tel (662) 942-8555 ext 1231 Fax (662) 561-4621
E-mail fengsia@ku.ac.th
ABSTRACT
In this work, polymer characteristics (e.g., radius of gyration and end-to-end distance) of linear,
monodisperse homopolymer molecules were investigated using Monte Carlo simulation with
random walk (RW) and self-avoiding walk (SAW) models in two-dimension square lattice and
three-dimension cubic lattice.
The results show quantitative relationships between the number of steps (i.e., molecular
weight) and the chain characteristics. The results from our simulation were validated with results
from theory and data from literature for the limiting cases and a good agreement was obtained.
The results from the present work will serve as a basis for further investigation of the
characteristics of polymer molecules having various complex architectures of interests.
KEYWORDS: Monte Carlo simulation, random walk (RW) model, self-avoiding walk (SAW)
model, Size and shape of polymer molecule
INTRODUCTION
A molecular size is the basic characteristic of a polymer molecule. Difference in molecular sizes
is the basis of the fractionation process in gel permeation chromatography (GPC), an important,
widely used polymer characterization technique. In this technique, polymer molecules are
fractionated by the hydrodynamic volume (reflected through the radius of gyration) to provide the
information of molecular weight distribution (MWD). The knowledge of the relationship between
molecular size and molecular weight is crucial and could help improve our understanding of this
important characterization technique, especially in the case of polymers with complex chain
architectures (e.g., branched, crosslinked, star, and comb polymers) [1-9]. These polymers with
complex chain architectures have been found to be very useful and increasingly focused in both
academia and industry. Therefore, understanding the characteristics of these polymers would be
of interests.
Two of the parameters that are commonly used for representing size and shape of
polymer molecules are the mean square radius of gyration (<Rg2>) and mean square end-to-end
distance (<Re2>). The radius of gyration is the distance of the segments from the center of gravity
of the polymer coil and the end-to-end distance is simply the distance between both ends of the
molecule. These parameters of each polymer molecule can be defined as follows [10]:
 Re  (ωN  ωo ) 2
2
 R g 
2
1
2(N step  1)2
(1)
N
 (ω
i
 ω j )2
(2)
i,j  0
where Nstep is the number of steps (or segments) and ωi is a coordinate vector of structural unit i
([x, y] for 2D square lattice and [x, y, z] for 3D cubic lattice).
In this work, we take a first step by investigating the relationship between molecular size
(e.g., radius of gyration and end-to-end distance) and molecular weight in the case of linear,
monodisperse homopolymers using Monte Carlo simulation technique together with random
walk (RW) and self-avoiding walk (SAW) models in both 2 and 3 dimensions. The results from
the present work will serve as a basis for further investigation of the characteristics of polymer
molecules having various complex architectures.
MONTE CARLO SIMULATION WITH RANDOM WALK (RW) AND SELF-AVOIDING
WALK (SAW) MODELS
For generating each polymer chain, Monte Carlo simulation technique was used with random
walk (RW) or self-avoiding walk (SAW) models. Monte Carlo simulation is a widely used
computational technique, especially for describing the statistical nature of polymer chains [1-4].
Details of this simulation approach with random sampling technique have been described in the
literature [10]. This simulation approach is chosen in this study because of its main advantage
over other methods (e.g., renormalization, exact-enumeration), as it allows us to probe
information directly in the regime of fairly long polymer chain [10]. Especially with the advances
in the computing technology, researchers now can easily perform some computation works,
which are once considered too time-consuming, with a good resolution.
Models of single polymer molecule
Both RW and SAW models used in this study represent a polymer chain in different
situations. In the RW model, a polymer chain is generated with the assumption that allows the
chain to pass through the regions of space that are already occupied by other segments of the
same chain. A polymer chain simulated using this model can represent the situation where a
polymer molecule is surrounded by other chains of the same type. Because various interactions
cancel out due to a collection of chains of the same type, a polymer chain in this situation
behaves as if it could “pass through itself” [11].
In the SAW model, each polymer chain is generated with a condition that no segment is
allowed to be in the same position. This model is well known for describing a polymer chain in a
good solvent (excluded volume condition). In this work, we used SAW model with an algorithm
called dimerization (algorithm based on the principle of “divide and conquer”) together with one
called non-reversal random walks (NRRWs or memory-2 walks) [10].
Both RW and SAW models were investigated in 2D square lattice and 3D cubic lattice.
Figure 1 shows examples of polymer chains with 100 steps obtained from different models and
different lattices. In each model, linear polymers with the number of steps (Nstep) of 15-200 and
the number of chains (Nchain) of 200,000 were studied to establish the quantitative relationships
between Nstep and chain characteristics.
2D RW model
2D SAW model
6
4
0
2
0
-10
Y
Y
-2
-4
-20
-6
-8
-30
-10
-5
0
5
10
15
20
25
30
-6
-4
-2
0
2
X
4
6
8
10
X
3D SAW model
3D RW model
8
4
6
2
4
2
0
Z
Z
0
-2
-2
-4
16
14
12
10
-6
-8
8
6
-6
4
6
8
6
4
4
2
2
0
Y
-2
-4
X
8
-10
2
-8
6
4
0
2
-6
-8
-2
-2
0
0
-2
Y
X
-4
-4
-4
Figure 1 Examples of polymer molecules with Nstep= 100 generated from different models and
different lattices.
Figure 1 Examples of polymer molecules with Nstep = 100 generated
from different models and different lattices.
Model validations
To validate our model, we compared results from our 2D-SAW model with the data from
2D-SAW model in the literature [12] in the case of Nstep = 15 and 20 (see Table 1) and a good
agreement was obtained. This indicates that our algorithm is reliable can be used in a
comprehensive investigation of long chain molecules. The comparisons between the results from
other models (2D-RW, 3D-RW, and 3D-SAW) and the results from theory and literature [11-13]
give a similar impressive agreement.
Table 1 Comparison between the characteristic data obtained
from the present work with Nchain = 200,000 and literature [12]
Nstep
15
20
Present
work
6.7842
10.2470
Radius of gyration
Literature % Difference
[12]
6.7843
0.001%
10.2477
0.01%
Present
work
47.2140
72.1034
End-to-end distance
Literature % Difference
[12]
47.2177
0.01%
72.0765
0.04%
RESULTS AND DISCUSSION
Effect of the number of chains (Nchain)
It is well known that when using Monte Carlo simulation, a large amount of polymer chains has
to be generated in order to obtain a good representation of all molecules. A small number of
chains typically can not adequately represent the characteristics of the overall population and
often lead to unreliable results.
In order to determine the appropriate number of chains, a number of simulations was
carried out with different number of chains in the case of Nstep = 15 and plotted in Figure 2. From
Figure 2, it is obvious that the value representing characteristic of polymer molecule (mean
square radius of gyration in this case) converges to a specific value, as Nchain increases. From
these results, we chose the appropriate Nchain to be 200,000 because at this Nchain the standard
deviation is already small and further increase in Nchain does not give a significant improvement
of results. This Nchain will be used through out this work.
2
Mean square radius of gyration, <Rg >
7.6
7.4
7.2
7.0
6.8
6.6
6.4
6.2
6.0
1e+1
1e+2
1e+3
1e+4
1e+5
1e+6
1e+7
1e+8
Number of chains (Nchain)
Figure 2 Effect of the number of chains on the mean square radius of gyration
obtained using 2D SAW model with 15 steps.
Effect of the number of steps (Nstep)
The relationship between the molecular weight and polymer size is of interests in this study.
Figure 3 illustrates relative sizes of polymer molecules with different numbers of steps obtained
from the 2D SAW model. The quantitative effect of chain length (represented by Nstep) on
polymer size (represented by mean square radius of gyration and mean square end-to-end
distance) in case of 2D SAW model is shown in Table 2 and Figure 4. The results show that as
the number of steps increases, both mean square radius of gyration and mean square end-to-end
distance exponentially increase and the ratio between the mean square radius of gyration and
mean square end-to-end distance converges to ~0.1403. This is in a very good agreement with the
results (0.14026 ± 0.00007) reported earlier by Li et al. [13].
30
30
Nstep= 20
20
Nstep= 50
20
10
10
0
0
-10
-10
-20
-20
-30
-30
-20
-10
0
10
20
30
-20
30
0
10
20
30
0
Nstep= 100
20
-20
0
-30
-10
-40
-20
-50
-20
-10
0
10
20
Nstep= 200
-10
10
-30
-30
-10
30
-60
-30
-20
-10
0
10
20
30
Figure 3 Relative sizes of polymer chains having different numbers of steps
obtained using 2D SAW model
Table 2 Effect of the number of steps on the characteristics of polymer chains
(Nchain = 200,000) for 2D-SAW model
<Rg2>
6.78427
10.24708
28.11626
51.17896
78.2744
109.3031
143.4247
Nstep
15
20
40
60
80
100
120
<Re2>
47.21401
72.10342
199.6772
364.7749
557.1227
779.1172
1,023.5282
<Rg2>/<Re2>
0.143692
0.142116
0.140809
0.140303
0.140498
0.140291
0.140128
1200
Radius of gyration
End-to-end distance
1000
Distance
800
600
400
200
0
0
20
40
60
80
100
120
140
Number of steps (Nstep)
Figure 4 Effect of the number of steps on the sizes of polymer chains
A careful look at the results indicates that root of both mean square radius of gyration and
mean square end-to-end distance is proportional to Nstep0.75 as showed by the linear relationship in
Figure 5. We obtained the following relationships from our simulation results.
 Re  1 / 2  0.8839N step
3/ 4
 Rg  1 / 2  0.3312N step
3/ 4
2
2
(3)
(4)
This again is in a good agreement with the results from theory [11], which indicates that
 Re  1 / 2  N step

 Rg 

2
2
1/ 2
 N step
(5)
(6)
where  
3
and d is the dimensionality of space (2 for square lattice and 3 for cubic lattice).
d2
The results we obtained in Eq. (3) and (4) will be useful as a basis for comparison with polymer
molecules with complex chain architectures in the further investigation.
35
Radius of gyration
End-to-end distance
30
Distance
1/2
25
<Re2>1/2 = 0.8839 Nstep0.75
r2 = 0.9999
20
15
10
<Rg2>1/2 = 0.3312 Nstep0.75
5
r2 = 0.9998
0
0
10
20
30
40
Nstep0.75
Figure 5 Relationship between Nstep0.75 and square root of characteristic distances
As mentioned before that RW and SAW models describe polymer molecules in different
situations, the information about quantitative relationships between the number of steps and
polymer characteristics for all the models is of important. In this article, we mainly show the
results from 2D-SAW model to convey the key concept of our approach. The results of other
models (2D-RW, 3D-RW, and 3D-SAW models) have also been validated and investigated in the
same manner. Table 3 shows the summary of the final quantitative relationships we obtained for
all investigated models.
Table 3 Summary of quantitative relationships between the number of steps and polymer
characteristics (<Rg2> or <Re2>) of all investigated models
Model
2D-RW
Relationship for radius of gyration
Relationship for end-to-end distance
 Rg  0.1667N step
 R g  0.9832N step
3D-RW
 Rg  0.1683N step
 Rg  1.007N step
2
2
2
2
2D-SAW
 Rg  1 / 2  0.3312N step
3D-SAW
 Rg 1 / 2  0.4102N step
2
2
3/ 4
 Re  1 / 2  0.8839N step
3/5
 Re 1 / 2  1.0301N step
2
2
3/ 4
3/5
The results from Table 3 indicate that for RW models, the dimensionality (2D or 3D) does
not affect the relationship between Nstep and polymer characteristics (Rg and Re). However, for
SAW models, the dimensionality has a significant effect due to the self-avoiding or excluded
volume effect.
CONCLUSION
In this study, we investigated polymer characteristics (e.g., radius of gyration and end-to-end
distance) of linear polymer molecules using Monte Carlo simulation with random walk (RW) and
self-avoiding walk (SAW) models in two and three dimensions. For the limiting cases, the
present results from simulation were validated with results from theory and data from literature
and a good agreement was obtained.
From this investigation, we obtained quantitative relationships between the number of
steps (i.e., molecular weight) and polymer characteristics (Rg or Re) for all the models. These
relationships developed in this work can provide the information of the characteristics of long
chain molecules, cases mainly disregarded in the past due to the long computational time
required. This information will be useful as a basis for further research works on the
characteristics of polymer molecules having various complex architectures, which are under
investigation in our laboratory.
ACKNOWLEDGEMENT
The authors would like to thank the financial support for establishing computing facilities in
polymer computation laboratory by petroleum and petrochemical technology consortium (ADB
project).
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