The scattering intensity

$I\left(\mathit{q}\right)$ is a function of the density field

$\rho \left(\mathit{R}\left(i,j,k\right)\right)$ in the observed system, where

$\mathit{R}\left(i,j,k\right)$ is the position of the grid point

$\left(i,j,k\right)$, which is the index of a three-dimensional (3D) regular mesh grid. We can obtain

$\rho \left(\mathit{R}\left(i,j,k\right)\right)$ from the position vector,

$\mathit{r}\left(n\right)$ (where

$n=1,\cdots ,MN$), which denotes the position of the

nth segment of the polymer. Using

$\rho \left(\mathit{R}\left(i,j,k\right)\right)$,

$I\left(\mathit{q}\right)$ can be calculated as follows:

For efficient computing of the numerator in Equation (7), we often use 3D fast Fourier transformation (3D-FFT), where:

When the scattering elements are points,

$I\left(0\right)$ is given by the product of squares of the scattering factors of the elements and the number of elements. In the present work, the scattering factors were set to unity so that

$I\left(0\right)=MN=\left|{\displaystyle \sum}_{i,j,k}\rho \left(\mathit{R}\left(i,j,k\right)\right)\right|$. For the insertion of additional midpoints, we must keep the total scattering factors the same; therefore, we scaled the scattering factor of each scattering site, which consists of segments of DPD chains and the additional midpoints. The intensity of each point on a 2DSP,

$I\left({q}_{x},{q}_{y}\right)$, is defined on the plane of

$I\left(q\right)$, with

${q}_{z}=0$. To calculate the X-ray and neutron scattering patterns corresponding to the all-atomistic molecular dynamics (AAMD) model, the density field

$\rho \left(\mathit{R}\left(i,j,k\right)\right)$ can be estimated from the atomic position and atomic weight. For the X-ray and neutron scattering patterns, the weight for

$\rho \left(\mathit{R}\left(i,j,k\right)\right)$ corresponds to the atomic number and the scattering length density, respectively, of the atoms. In general, the chain picture is required to calculate the wide-angle scattering intensity, which is related to the orientation distribution of bonds. To overcome this problem, we propose using TA to calculate the 2D WAXS patterns of dense polymer melts with KG chains under a rapid shear flow.

Figure 1a shows an illustration of the TA process based on the insertion of particles in the middle of bonds. In our previous work [

44], the total volume is assumed to be conserved from the viewpoint of the scattering length density. For the fine-graining of dense KG chains with extra monomers placed in the middle of bonds, the effective volume of each monomer shrinks by 1/2. Based on Equation (7),

$I\left(\mathit{q}\right)$ is independent of the scaling factor,

α, for

ρ. Here,

α is proportional to the volume of one monomer and corresponds to the scattering length density for X-ray scattering. Owing to the conservation of the total volume,

α = 1/2 because the number of monomers becomes 2

N after the TA. Thus, we can assume that the diameter of the polymer chains decreases to 2

^{−1/3}. For the DPD chains, the concept of TA is effective. Owing to the large distribution of bond lengths when compared to those of KG chains, we assumed multiple (a fixed number and an adaptive number) midpoints,

${n}_{\mathrm{mid}}$. Here, the TA for KG chains [

44] corresponds to the cases with

${n}_{\mathrm{mid}}$ = 1, and the scaling factor

α of the effective volume of each scattering site used to calculate

$\rho \left(\mathit{R}\left(i,j,k\right)\right)$ is given by

$\alpha =1/\left({n}_{\mathrm{mid}}+1\right)=1/2$. Thus, we used

$\alpha =1/\left({n}_{\mathrm{mid}}+1\right)$ for multipoint TA. For adaptive TA, the scaling factor,

$\alpha =1/\left(n+1\right)$, for each bond was set based on the number of divisions,

$n,$ obtained for each bond based on the dividing distance,

${l}_{\mathrm{ATA}}$. When multiple midpoints (i.e., multiple scattering sites) were placed on a bond, we used a fine mesh for

$\rho \left(\mathit{R}\left(i,j,k\right)\right)$ in the 3D-FFT.

As shown in our previous investigation [

44] on KG chains, the observed anisotropy of the 2DSP on the

y axis is as small as the random noise. Thus, the circular average with rotational symmetry on the

${q}_{y}-{q}_{z}$ plane seems to be reasonable even though it ignores the anisotropy on the

y axis. Therefore, we assumed rotational symmetry of the system to improve the statistical accuracy of the 2DSP. Note that the number of grids,

n, is given by

$\text{}{L}_{\mathrm{pbc}}/\Delta x$ in the present work. Here, the length of each side of the box with periodic boundary conditions (PBCs) is

${L}_{\mathrm{pbc},x}=\text{}{L}_{\mathrm{pbc},y}=\text{}{L}_{\mathrm{pbc},z}=\text{}{L}_{\mathrm{pbc}}$ and the mesh size in real space is

$\Delta x=\Delta y=\Delta z$. In this case, the mesh size for the 3D-FFT is given by

$\Delta {q}_{x}=\Delta {q}_{y}=\Delta {q}_{z}=\text{}2\mathsf{\pi}/{L}_{\mathrm{pbc}}$. For the circular averaging, the mesh size for the 2D WAXS pattern is given by

$\Delta {q}_{\parallel}=\Delta {q}_{x}$ and

$\Delta {q}_{\perp}=\Delta {q}_{y}=\Delta {q}_{z}$.