# Multiscale Modeling and Simulation of Polymer Blends in Injection Molding: A Review

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Framework of Multiscale Modeling

**Macroscale (~10**The solid plastic pellets melt at a certain temperature above the melting point and are injected into the mold at a certain speed under the pressure of the injection machine. The polymer melt expels the air from the cavity until it fills the entire mold cavity and finally cools and solidifies to get the product as the designed mold cavity. The polymer blend melt is considered to be a continuous fluid, with the inside microstructure neglected. Injection molding of polymer melt is a non-Newtonian, non-isothermal, and unsteady process of mass, momentum, and heat transfer with the moving polymer–air front.

^{−3}m, ~1 s):**Mesoscale (~10**Mesoscale morphology is formed in the compounding and granulation stages prior to injection molding. In the equipment, such as roll mills, mixing machines or screw extruders, a sea-island-like multiphase structure emerges [3,4] when the micron-sized droplets are dispersed throughout the matrix. During the injection molding stage, the dispersed droplets inside the blend melt undergo complex morphology evolution under the combined action of shear, pressure and heat, and interfacial tension [5]. Finally, the blend morphology is frozen inside the product as the polymers cool and solidify after the flow ceases.

^{−6}m, ~10^{−6}–10^{−3}s):## 3. Mesoscopic Modeling of Droplet Morphology Evolution

#### 3.1. Ellipsoid Droplet Models

**q**to characterize the microstructure.

**n**, and

**I**is the unit tensor.

#### 3.1.1. Basic Quantities

^{3}= ab

^{2}. It is apparent that a larger Df indicates a more deformed droplet. θ is the angle between the principal axis of the droplet and the reference direction, usually the flow direction surrounding the droplet.

_{d}/η

_{m}and the capillary number Ca = η

_{m}γR/Γ, where η

_{d}and η

_{m}are the viscosities of the droplet and the matrix, γ is the shear rate, and Γ is the interfacial tension between the phases of the droplet and matrix. The capillary number characterizes the competition of the viscous stress that drives the droplet deformation against the interfacial tension that maintains the original shape. Droplets exhibit different orientations when different forces dominate, as described in detail below.

#### 3.1.2. Deformation

**G**can be expressed by the following equation [8]:

_{r}= η

_{m}R/Γ is the surface-tension relaxation time, e

_{ij}and w

_{ij}are the deformation rate tensor and the vorticity tensor, respectively, both of which can be obtained from the velocity gradient tensor L

_{ij}: e

_{ij}= (L

_{ij}+ L

_{ji})/2, w

_{ij}= (L

_{ij}− L

_{ji})/2. e

_{ij}with superscript A denotes the externally applied deformation rate, while e

_{ij}without superscript is related to the deformation of the droplet itself.

_{1}and f

_{2}are phenomenological model parameters for which certain choices have been proposed:

_{2}is simplified to ${f}_{2}=5/(2p+3)$ the model is referred to as the MM1 model.

_{ij}is the transformation matrix that rotates the droplet axis in alignment with the coordinate system.

#### 3.1.3. Breakup

_{crit}, the pattern of droplet breakup depends on the viscosity ratio [2]: when p is much smaller than 1, the droplet is extremely stretched into an S-shape and small droplets are released at both ends; when p ≈ 1, the droplet gradually necks out from the middle part until it breaks up into two sub-droplets, with some smaller satellite droplets in between. When Ca far exceeds Ca

_{crit}, the droplet stretches into an elongated fiber.

_{crit}(i.e., the ratio of the local capillary number to the critical value) can be defined. Depending on the cases, most researchers [14,17,18] have adopted the following general rule to describe droplet behavior.

- (1)
- k* < 0.1, droplets do not deform;
- (2)
- 0.1 < k* < 1, droplets deform, but do not break up;
- (3)
- 1 < k* < 4, droplets deform and split into two major sub-droplets;
- (4)
- k* > 4, droplets form fibers with the affine deformation of the medium.

_{crit}decides whether the deformed droplets reach equilibrium shapes or breakup into sub-droplets. Ca

_{crit}of a specific droplet depends on the viscosity ratio and the ambient flow field type. The following empirical de Bruijn formula [19] is commonly used to write Ca

_{crit}as a function of p for simple shear fields [20].

_{0}splits completely into two sub-droplets of the same diameter without considering the effect of surrounding droplets, the diameter of the split sub-droplet, d ≈ 0.794D

_{0}, can be obtained according to the principle of volume conservation.

_{b}, a statistically significant average of the breakup process, can also be calculated. The rate of change of the total number of droplets N

_{d}is first obtained as:

_{0}is the droplet diameter. This leads to the rate equation for droplet breakup [18].

_{0}is the fiber diameter before the breakup, which can be approximated as the critical diameter d* at the moment of fiber breakup, and X

_{m}represents the main wave number. When p = 1, X

_{m}≈ 0.56, the diameter of the droplet after breakup d ≈ 2d* is obtained using the above equation.

#### 3.1.4. Coalescence

_{coll}can be expressed as (see [14]):

_{loc}is the local residence time.

_{exp}depends on the activity of the interface, which depends on the viscosity ratio p: the larger the value of p, the less active the interface is. For inactive interfaces with p much larger than 1, the liquid film discharge probability can be calculated using the following equation [14]:

_{c}is the critical thickness of the liquid film at the time of the breakup, which can be obtained by experiment, and k* is defined as before.

_{coal}can be expressed as the product of p

_{coll}and p

_{exp}.

_{eq}is the diameter at equilibrium and ${D}_{eq}^{0}$ is the diameter at zero component (i.e., without any dispersed phase), obtained by extrapolation. Using the above equation, the coalescence constant C of a certain blend can be obtained by preparing a series of materials with different components, i.e., calculated by the slope of the curve of D

_{eq}versus ϕ

^{4/3}. Using the coalescence probability formula obtained earlier, an approximate formula for the radius of the new particle after coalescence can also be obtained. Assuming the interaction of two droplets of the same size, the volume conservation principle yields (see [22]):

_{coal}= 0; R = 2

^{1/3}R* when coalescence is complete, i.e., p

_{coal}= 1.

#### 3.1.5. Size Distribution

_{1}), the following equation describes the change in the number of droplets of volume kV

_{1}, n

_{k}with time t [23]:

_{f}(i) is the number of fragments formed at the breakup of a droplet of volume iV

_{1}, and ω(i, j) is the probability that a fragment formed by the breakup of a droplet of volume jV

_{1}will have volume iV

_{1}.

_{αp}is the apparent viscosity of the blend and E

_{DK}is the volume energy.

_{c}:

^{0}is the droplet radius for ϕ = 0, and t

_{B}* is the dimensionless breakup time, which is a function of p and is independent of Ca.

_{c}, the approach led to the following equation:

#### 3.2. Phase Field Models

**u**is the velocity of the flow field, D is a diffusion coefficient of the m

^{2}/s unit, $\mu ={c}^{3}-c-\gamma {\nabla}^{2}c$ is the chemical potential at that location, and $\sqrt{\gamma}$ is the thickness of the two-phase transition region (interface).

**j**

_{i}of species i can be written as

_{N}for the last component is inferred from a material balance equation $\sum _{i}{\varphi}_{i}}=1$, where ϕ

_{i}is the volume fraction of species i. For a binary system with components 1 and 2, the transport equation for species i can then be written as:

_{ij}, can be obtained by considering the generalized N-component Landau–Ginzburg free-energy functional equation for inhomogeneous systems enclosed within a dimensionless volume.

#### 3.3. Lattice Boltzmann Method

_{σ}

_{,α}(

**x**,t) is the probability of finding a fluid particle of component σ at position

**x**and time t with the discrete velocity

**e**

_{α}.

**x**from a particle of component $\overline{\sigma}$ at spatial position ${\mathit{x}}^{\prime}$ as

**x**is the sum of the forces acting on it by all neighboring particles.

**x**and let G(|

**e**

_{α}|) be just a function of |

**e**

_{α}|, i.e., the interaction between the particles is isotropic, then the interaction force

**F**(

**x**) can be further expressed as:

**e**

_{α}|

^{2}) is the weight that is used to calculate the isotropic interaction force.

_{σ}and momentum ρ

_{σ}

**u**

_{σ}are defined as the zeroth and first-order moment of the distribution function:

## 4. Macroscopic Mold-Filling Flow Simulation

**u**, P, and T are the velocity, pressure, and temperature of the mold-filling flow;

**f**is the external force field; ρ, η, c

_{p}, and λ are the density, viscosity, specific heat, and heat conductivity of the polymer melt; Φ is the heat dissipation.

## 5. Scale-Bridging Strategies

#### 5.1. Parameter-Based Methods

**A**is the area tensor, converted from the ellipsoid morphology tensor

**G**.

#### 5.2. Particle-Based Methods

## 6. Outlook and Summary

- Polymer melts are mostly viscoelastic fluids. Although the generalized Newtonian fluid model has been able to accurately simulate the filling flow process in most cases, for some products with high requirements on mechanical properties, optical properties, or geometric accuracy, the residual stress caused by viscoelasticity during the filling and packing process is often not negligible. Therefore, how to establish a stable and efficient method for solving the viscoelastic flow solution for the actual product forming process is also an important topic worth studying.
- The energy equation of the filling flow process is significantly convection-dominant, and the boundedness of the discrete scheme of the convection term in the equation has an important impact on the accuracy and stability of the whole filling flow simulation. Therefore, it is necessary to study the discrete scheme of the convection diffusion equation with high accuracy under an unstructured grid to satisfy the boundedness.
- In the simulation of the mold-filling flow process, the solution of the algebraic equation system occupies most of the computational time, among which the solution of the velocity-pressure coupled algebraic equation system takes the most time. Therefore, for the research and development of efficient solution methods for the velocity-pressure-coupled algebraic equation system, shortening the process is also an important part of the next work.

- When the fraction of the blends exceeds a certain range (greater than about 40%), the dispersed phase no longer exists in the form of isolated droplets, and the simulation algorithm based on the ellipsoidal assumption becomes invalid and other morphology models could be considered, such as the interfacial tensor model.
- Most current models of droplet morphology evolution are limited to Newtonian fluids due to the non-uniformity and strong nonlinearity of the viscoelastic constitutive equations. However, the elasticity of the polymer melt has a significant effect on the evolution of droplet morphology, so the role of component elasticity on phase morphology should be considered, for example, by introducing empirical parameters into the models.
- The evolution equation of droplet size distribution is an important way to parameterize the microstructure of the blend. However, the current evolution models are still based on the ellipsoidal droplet assumption, which cannot characterize the complex morphology of the dispersed phase, so there is a need to establish the evolution equations of the dispersed phase distribution based on the tensor form or the component concentration form in the future.

- Compared with traditional Eulerian methods, such as the finite volume method and the finite element method, the SPH method, based on the Lagrangian description, has the natural advantage of automatically recording polymer history in simulating polymer melt flow; however, poor numerical stability, high computational cost, and difficulty in boundary handling confine its further application in simulating polymer processing, which needs to be addressed in the future.
- The rheological constitutive relationship of polymers is the key to realizing the coupling between macroscopic and mesoscopic scales; however, it is not easy to establish the constitutive relationship of polymer blends in traditional equation form. It is a promising alternative to use the current data-driven modeling method based on deep learning to propose the constitutive relationship of polymer blends.
- Since the size of the dispersed phase droplets in the blend is very small and their number is very large, simulating the morphological evolution of the entire dispersed phase during the mold-filling flow is still unaffordable under current computing power, so it is necessary to investigate the use of parallel computing and GPU accelerometers to increase the efficiency of the simulation and the use of multidimensional fractal theory for the parametric description of dispersed phase morphology.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

η_{m} | polymer viscosity of matrix |

η_{d} | polymer viscosity of droplet |

γ | shear rate |

Γ | interfacial tension |

R | droplet radius |

D_{0} | droplet diameter |

Ca | capillary number |

Ca_{crit} | critical capillary number |

p | viscosity radio |

G | ellipsoid droplet tensor |

L | velocity gradient tensor |

Df | droplet deformability |

L | major axis of droplet |

B | minor axis of droplet |

W | width of droplet |

θ | orientation angle of droplet |

τ_{r} | surface-tension relaxation time |

e_{ij} | deformation rate tensor |

w_{ij} | vorticity tensor |

f_{1} | MM model parameter |

f_{2} | MM model parameter |

k* | simplified capillary number |

d | diameter of the split sub-droplet |

d* | critical diameter of fiber breakup |

t_{b} | time required for droplet breakup |

N_{d} | total number of droplets of volume V |

X_{m} | main wave number |

p_{coll} | collision probability |

t_{loc} | local residence time |

p_{exp} | liquid film discharge probability |

h_{c} | critical thickness of the liquid film for breakup |

D_{eq} | diameter at equilibrium |

${D}_{eq}^{0}$ | diameter at zero component |

R* | radii of droplets after coalescence |

nk | number of droplets of volume kV_{1} |

C(i, j) | coalescence coagulation kernel |

F(i) | overall breakup frequency |

n_{f}(i) | number of fragments formed at breakup of a droplet of volume iV1 |

ω(i, j) | probability that a fragment formed by the breakup of a droplet of volume jV1 will have volume iV1 |

η_{αp} | apparent viscosity of the blend |

EDK | volume energy |

R_{0} | droplet radius for ϕ = 0 |

t_{B}* | dimensionless breakup time |

R_{c} | critical value for breakup |

R* | ratio of R and Rc |

D | diffusion coefficient |

μ | chemical potential |

c | concentration of the fluid |

ξ | thermal noise |

f_{σ,α}(x,t) | particle probability distribution function |

eα | discrete particle velocity |

G(|eα|) | interaction function of pseudo-potential lattice Boltzmann method |

${g}_{\sigma \overline{\sigma}}$ | interaction constant |

w(|e_{α}|2) | weight function of LBM |

ρ | density |

u | velocity |

P | pressure |

T | temperature |

c_{p} | specific heat |

λ | heat conductivity |

η | viscosity |

A | area tensor |

dim | dimension |

v_{σ} | kinematic viscosity of component σ |

ρ_{σ} | density of component σ |

τ_{σ} | relaxation time of component σ |

j_{σ} | momentum flux of component σ |

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**Figure 2.**Morphology of the dispersed phase in PDMS/PB blends. (

**a**) Before shearing; (

**b**) during shearing; (

**c**) 25 s after shearing stop; (

**d**) 89 s after shearing stop (experimental results from the literature [6]).

**Figure 3.**Schematic diagram of droplet geometry model parameters in the (

**a**) flow plane and (

**b**) rotational plane.

**Figure 5.**The principle of the pseudo-potential model: (

**a**) the schematic of the drop–interface–matrix structure; (

**b**) the pseudo-potential force between particles at the interface.

**Figure 6.**Procedure of the parameter-based scale-bridging methods: (

**a**) macroscopic mold-filling simulation; (

**b**) polymer chains conformation.

**Figure 7.**Particle-based scale-bridging method: (

**a**) a fluid particle trace in the flow around circular cylinder; (

**b**) a fluid particle containing droplets.

Ellipsoid Models | Phase Field Models | LBM | |
---|---|---|---|

Morphology | ellipsoid | arbitrary | arbitrary |

Interface tracking | √ | × | × |

Interface type | sharp | diffuse | diffuse |

Flow filed inside drops | × | √ | √ |

Physical domain size | large | medium | small |

Source of model parameters | physical properties | first-principles calculations | physical properties |

Solving method | implicit | implicit | explicit |

Computation cost | low | medium | high |

External field incorporation | × | √ | √ |

Phase transition incorporation | × | √ | √ |

Mid-Plane Model | Surface Model | Solid Model | |
---|---|---|---|

Thin-wall laminar flow assumptions | √ | √ | × |

Incompressibility assumption | √ | √ | √ |

Inertia and volume forces | × | × | × |

Heat transfer in direction of flow | √ | √ | × |

Internal heat source items | × | × | × |

Constant physical parameters | √ | √ | √ |

Planar-shaped flow front | √ | √ | × |

Grid size | small | medium | large |

Algorithm complexity | simple | complex | more complex |

Calculation time | short | ordinary | long |

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**MDPI and ACS Style**

Deng, L.; Fan, S.; Zhang, Y.; Huang, Z.; Zhou, H.; Jiang, S.; Li, J.
Multiscale Modeling and Simulation of Polymer Blends in Injection Molding: A Review. *Polymers* **2021**, *13*, 3783.
https://doi.org/10.3390/polym13213783

**AMA Style**

Deng L, Fan S, Zhang Y, Huang Z, Zhou H, Jiang S, Li J.
Multiscale Modeling and Simulation of Polymer Blends in Injection Molding: A Review. *Polymers*. 2021; 13(21):3783.
https://doi.org/10.3390/polym13213783

**Chicago/Turabian Style**

Deng, Lin, Suo Fan, Yun Zhang, Zhigao Huang, Huamin Zhou, Shaofei Jiang, and Jiquan Li.
2021. "Multiscale Modeling and Simulation of Polymer Blends in Injection Molding: A Review" *Polymers* 13, no. 21: 3783.
https://doi.org/10.3390/polym13213783