2.2. Rigid Bond Model and Empty Bond Model
Let
be the vertex position of the lattice (
Figure 3a),
be the bond length (
Figure 3b), and
be the unit normal vector of the triangle (
Figure 3c). These variables
are integrated into the partition function
Z for 2D models (here, we discuss 2D surface models), and
Z is given by
where the symbol
in
indicates that the model implicitly depends on a constant
, which will be used to define the crystalline bond below. The symbol
denotes the 3D integration for the internal vertices. The other
denotes 2D integration for the boundary vertices on the upper and lower boundaries with a periodic boundary condition such that
where
and
denote the
Z component of the vertex position
on the upper and lower boundaries (no confusion is expected for the symbol of the partition function and that of the coordinate symbol). The second equation in Equation (
3) denotes a constraint that the diameter of the boundary circle is given by
D (
Figure 2a). The reason why
becomes a 2D integration is that the boundary vertices are allowed to move not only along the boundary circle of fixed diameter
D, but also along the
Z direction due to the periodic boundary condition. For the 3D thick cylinder model,
becomes a one-dimensional integration because no periodic boundary condition is imposed.
The symbol
in Equation (
2), which has values in
, is another dynamical variable defined on the bond
i. The value of
represents whether the bond
i is rigid or amorphous:
The rigid bond is a rigid body, which is allowed to move by 3D rotation and translation. In other words, two vertices
i and
j, which are connected by a rigid bond, move with a constraint
, where
is described in the following paragraph. In contrast, this constraint on
is not imposed on any two vertices connected by an amorphous bond. Another model is the “empty bond” model, which is defined by replacing “rigid” with “empty” in Equation (
4) such that
The variable
in Equations (
4) and (
5) can be changed from
to
when the bond length
is
using the constant
in
according to the Monte Carlo update procedure described in a later section. If
is changed from
to
, then the bond length
is fixed to the length
, which is the length of bond
i just before the change in
in the rigid model. In the case of the empty bond model, the length
of the crystalline bond is changeable to a new length
if
, and this
is also changeable depending on the update of
, which is the same as in the case of the rigid bond model. The constraint
is always satisfied in both models. The empty bond is almost close to the rigid bond except that the length
is changeable to
. This fact implies that the empty bond length
is also prohibited from extending to
. The bond configurations for amorphous, rigid, and empty bonds are illustrated in
Figure 4a–c.
We should comment that the total number of degrees of freedom of variable is slightly reduced by crystallization. Indeed, in the rigid bond model, a crystalline bond is allowed to move by three-dimensional translation and two-dimensional rotation, where the rotational degrees of freedom are given as 2 because the bond is a one-dimensional object. Therefore, the 6 degrees of freedom, which is the sum of the degrees of freedom for translations of the terminal vertices of the bond, is reduced to 5 by crystallization. Thus, the total number of degrees of freedom is reduced by , which is the total number of crystalline bonds, in the rigid bond model. In the case of the empty bond model, the total number of reductions in the degrees of freedom of variable is effectively considered to be because the crystalline bond length is allowed to shrink even though the extension is not allowed, and, for this reason, the reduction per crystalline bond is roughly estimated to be .
The Hamiltonian for the rigid and empty bond models is given by
and
where the first two terms
and the third term
correspond to the tensile energy and the bending energy, respectively, and the final two terms
are constraint potentials. The symbol
in the definition of
is the variable
on the bond connecting the vertices
i and
j. No difference between the rigid and empty bond models is in the definition of
except that the length
of crystalline bond
is allowed to be
in the case of the empty bond model. In contrast, the length of bond
is prohibited from being
for
in both models. In this sense, this definition of
is specific to the models for SIC in this paper. In the bending energy
,
is defined on the bond shared by triangles 0 and
i (see
Figure 3c). This definition of the bending energy is also specific to the models for SIC in this paper. These specific definitions of
and
come from the fact that the crystalline state is implemented in the context of the Gaussian bond model for polymers [
13].
Due to the factor , the Gaussian bond potential is identical to , which is given by the sum of bond length squares . The reason why the sum is used for instead of is because the discrete form of the quadratic term is naturally given by , and the same summation convention is used for . The numerical factor in is determined such that it is identical to the factor in .
Note that
is the constraint potential for
in both models. From this constraint
, the length
of the empty bond is only allowed to be
, as mentioned above; however, this is strictly prohibited for the rigid bond because
should always satisfy
from the definition of
. The second constraint
prohibits the crystalline bond
i from being consecutively connected (
Figure 5a), or, in other words, every crystalline bond
i must be isolated or linked to the amorphous bond
j (
Figure 5b). This constraint
also plays a role in the constraint on the total number of crystalline bonds because the maximum number of crystalline bonds is limited by a lattice geometric (or topological) reason.
As a consequence of the constraint
, no constraint is imposed on the upper limit for the crystallization ratio
such that
, where
is defined by
where
is the total number of crystalline bonds and
is the total number of bonds. The value of
depends on the dimension of the lattice, that is, whether it is a 2D or 3D lattice; however, in both cases,
is reasonable in the sense that it is close to the experimental value, such as
[
1,
2,
3,
4], which will be shown in the presentation section.
The mean value
of physical quantity
(where the symbol
Q denotes an arbitrary quantity) is given by
We simply write the mean value Q by removing the symbol henceforth. To calculate these multiple integrations and the sum over all possible configurations of , we use Metropolis Monte Carlo (MC) simulations, which are described in the following section. Note that all physical quantities are calculated with the dynamical variables and , which are the vertex position and the spin variable at the vertices , respectively.
The 2D continuous forms of the Hamiltonian corresponding to the discrete
and
in Equation (
6) are given by
where
is the inverse of a metric tensor
given by the
matrix, and
g is the determinant. To obtain the discrete Hamiltonian, we simply assume that
, which is called the Euclidean metric. By the replacements
and
(see
Figure 3b), and by the symmetrization of the obtained expression of the Hamiltonian due to the three possible local coordinate origins in a triangle, we have the discrete
and
in Equation (
6). More detailed information on the discretization is given in Ref. [
20].
2.3. 3D Model
The three-dimensional (3D) continuous Hamiltonian is obtained from the 2D version in Equation (
10) simply by replacing the 2D integration
with the 3D integration
. In the 3D case, the metric tensor
is also replaced by a
matrix. Thus, these 3D continuous
and
can also be converted to discrete forms on the 3D lattice, which is a thick cylinder (
Figure 6a) discretized by tetrahedrons (
Figure 6b). On this thick cylinder, the vertices of tetrahedrons are distributed only on the inner and outer surfaces, and, therefore, the thickness is negligible compared with the diameter. Note also that in the 3D case, only the empty bond model is studied. The reason is that, as we will see in the presentation section, there is almost no difference between the results of the 2D empty and rigid models, or the 2D empty model is slightly better for the stress relaxation behavior, and, moreover, the empty model is more simple than the rigid model in its definition.
The 3D version of the constraint potentials
and
has the same expressions as those in Equation (
7), and, hence, their expressions are not written below.
where
in
and
denotes the sum over the tetrahedrons (
Figure 6a). The symbol
is the mean value of the total number of tetrahedrons around a bond and is given by
where
is the total number of tetrahedrons around bond
i, and
is the total number of bonds. Due to this factor
, the term
is identical to
, which is given by the sum of bond length squares
. The reason why the summation convention
is used instead of the simple form
for
is that the discrete
is naturally given by the sum over tetrahedrons as in the 2D case described above, and the same summation convention is also assumed for
. The symbol
in
is the internal angle of the triangle (
Figure 6c),
in
is the sum over all internal angles
i, and
and
in the definition of
are
at the bonds
in which the internal angle is
(
Figure 6c). The symbol
for the coefficient of
is the same as the bending rigidity
for
of the 2D model in Equation (
6); however, no confusion is expected. In the case of the 2D model,
only resists pure bending. In contrast,
of the 3D model in Equation (
11) resists all deformations of the tetrahedron except the homologous deformation.
2.4. Simulation Technique
The standard Metropolis technique is used to update the variables
and
[
21,
22]. For the update of
at vertex
i, a new position
is generated by three uniform random numbers
inside a sphere of fixed radius
such that
, where
is a positive number. The new
is accepted with the probability
, where
is the change in the Hamiltonian due to the update of
such that
under the constraints
and
on the bond length
and
, respectively. The constraint
is explicitly imposed on the update of
, and this
also implicitly imposes a constraint on the bond length
. Indeed, the constraint
on
in Equation (
6) or Equation (
11) for
together with
allows the bond length
to have only
for the rigid bond model and
for the empty bond model. The rate of acceptance for the update of
depends on the number
, which is fixed so that the acceptance rate is approximately in the range between
and
.
The update of to the new is performed independently of the old with the probability , which is the same expression for the update of . In this update of , the change from to , which is called crystallization, is allowed only when . This crystallization process, from to , causes a sudden decrease in for both rigid and empty models. The energy also discontinuously decreases in the empty model, while it remains unchanged in the rigid bond model. Therefore, the crystallization process is determined only by in both rigid and empty bond models. In contrast, the melting process, from to , is not constrained by ; therefore, this process is determined only by the energies in both rigid and empty bond models. Indeed, in the rigid bond model, a sudden increase is expected in for the melting process, and in the case of the empty bond model, a sudden increase is also expected in for this process. Therefore, the melting process is determined only by the change in energy in both rigid and empty bond models according to the probability .
One more point on the update of that should be emphasized is that this update is performed only once every 100 Monte Carlo sweeps (MCSs), where one MCS is composed of N consecutive updates of . The reason why the variables and are updated in such an asymmetrical manner is because a symmetrical or almost symmetrical update of causes a configuration with a nonuniform distribution of crystalline bonds, where “symmetrical” means that the total number of updates of and in one MCS is the same. As a consequence of this nonuniform distribution of crystalline bonds, for the large strain region becomes lower than that for the intermediate strain region. This effect will be described in further detail in the presentation section.
2.5. Frame Tension as Tensile Stress
Detailed information on how to calculate the tensile stress is given in Ref. [
20], and, here, we start with the formula. The formula for the stress in the 2D models is given by
where
and
T are the Boltzmann constant and the temperature, respectively,
corresponds to the true area of the cylinder, and
is the simulated frame tension. We should note that
on the left-hand side should be replaced by
for the 3D model because the integration for the boundary vertices in
Z of Equation (
2) is the two-dimensional (one-dimensional) integration for the 2D (3D) model. The total number of degrees of freedom
is given by
(rigid bond model) and
(empty bond model). As mentioned in
Section 2.2, the total number of degrees of freedom for the vertex move is slightly reduced by crystallization. For this reason, the number
in Equation (
13) is replaced by
.
We briefly describe how to calculate the tensile stress
in Equation (
13) and how to change the unit from
to
. The tensile stress is not directly calculated in our models because the materials represented by 2D or 3D lattices are very thin. Indeed, the 2D lattice is a cylindrical surface, and the 3D lattice is also regarded as a surface because its thickness is almost negligible compared to the diameter. In contrast, the frame tension
is calculable as a response to a mechanical constraint, which is imposed by fixing the height
H of the cylindrical surface (
Figure 7a,b). For these reasons,
is used to obtain
. The calculation formula of
is actually obtained by a property of the partition function
Z in Equation (
2) under the scale change such as
, which is a change of variables in the multiple-integration of
Z. This property in
Z is called scale invariance [
23], and is expressed by
for any
. From this, we have
. This derivative of
Z is directly calculated from the expression
, which is from the 2D model partition function in Equation (
2), where the periodic boundary condition is assumed on the boundary vertices. Hence, the factor of
is calculated by
from the fact that the boundary integration is two-dimensional. Note also that the total number of degrees of freedom
should be replaced by
, as mentioned above. Except for the total number of degrees of freedom and difference in the symbols for energies, the calculation of
is exactly the same as those written in Ref. [
20]. Thus, we obtain the surface tension
. By multiplying
to this
, we have the stress
in Equation (
13) in the unit of
. The reason for the multiplication of
is as follows: First, we should note that the simulation unit is determined by fixing
(Nm) and
(m), and, for this reason, all quantities with length units should be multiplied by the lattice spacing
a in the physical unit. For this reason, by replacing the area
A in the denominator of
by
, we obtain
with the unit of (
). Multiplying
to this
, we have
with the unit of (
). This the physical unit of the frame tension
. Finally, to obtain the quantity that has the unit of (
), we have to multiply
to this quantity once again, and this leads to the expression of
in Equation (
13).
By assuming room temperature for
T, we have
The symbol is the lattice spacing or the bond length, which corresponds to the coarse-grained distance between polymer segments, and, hence, it can be fixed to an arbitrary number larger than the van der Waals distance (∼). This implies that the magnitude of the calculated can be controlled arbitrarily such that is comparable to experimental data by using this parameter .
The diameter
D, which is equal to the initial height
, is fixed such that the height strain
is zero for
. For this purpose, MC simulations should be performed to obtain the correct value of
, which depends on the other parameters, such as
b and
, before the start of the production simulations.
Once
is obtained, the next task that should be done before the production simulations is to fix the constant
in
of Equation (
7). Using the value of
, we fix
to
for both the 2D lattice of size
10,230 and the 3D lattice of size
. This value of
is crucial to the shape of the crystallization ratio
vs. strain, which influences the final result of the stress–strain diagram. However, the results are almost independent of a small variation of this value of
. The factor
in
is expected to be dependent on the size
N in general because
depends on
as in Equation (
1), and this
determines
N such that
for the 2D lattice of size
10,230 and
for the 3D lattice of size
. We should note that the value of
or
depends on the experimental data of stress–strain curves to be fitted and on whether the model is 2D or 3D. In this paper, we assume Equation (
16) for
because the experimental data of stress–strain curves are well reproduced, as we see in the following section.