Foam injection moulding (FIM) enables lightweight thermoplastic parts, but current process simulations do not resolve microstructure formation. This work presents a micro-scale CFD framework for FIM that captures gas–melt interaction and bubble morphology. A two-phase, compressible volume-of-fluid solver (OpenFOAM) with surface tension and viscoelastic Phan–Thien–Tanner rheology is coupled to a nucleation pre-processor based on classical nucleation theory, which places bubbles stochastically using macro-scale pressure and temperature histories. The approach was demonstrated on a plate geometry using a 2D through-thickness section to investigate bubble nucleation, deformation, coalescence, and interaction under realistic process conditions. The simulations reproduced characteristic morphology trends across the thickness. In particular, the predicted aspect ratio and orientation show the expected skin–core behaviour and agree qualitatively with experimental observations. These results demonstrate that the framework can describe morphology development beyond simplified spherical-cell assumptions and provides a proof of concept for multiscale coupling between macro-scale process conditions and micro-scale foam structure evolution. A simplified surrogate growth representation was used to enable bubble expansion; however, a physically based mass-transfer model is required for quantitatively accurate growth kinetics. 1. Introduction The thermoplastic foam injection moulding (FIM) process is widely used for the production of parts in the automotive, packaging and leisure industries. Physically foamed thermoplastic injection moulded components have many advantages over compact components. The foam structure allows for significant weight reduction while maintaining the same volume. Furthermore, the pressure of the blowing agents in the bubbles reduces shrinkage and warping. In the context of sustainability and ever-increasing technical requirements, the demand for foamed plastic components is therefore rising continuously [ 1, 2 Most commercial injection moulding simulation software still represents the foam microstructure only in highly simplified ways, with spherical/isotropic cells, fixed diffusion and without shear-induced deformation, coalescence, or collapse. The only results are the average cell radius and cell density. So, unlike compact moulding where simulation is accurate enough to display lab experiments, reliable prediction for foamed parts remains limited. The deficit becomes more severe in complex geometries because local pressure–temperature–shear histories differ strongly across the cavity, which affects the local microstructure [ 3]. However, the full potential of foamed components can only be exploited if the spatially inhomogeneous anisotropic foam structure and thus the mechanical and thermal properties of the components can be accurately predicted and designed to suit the specific application [ 4, 5 Modelling and simulation of the foam formation process have therefore been the focus of research for many years. For example, Chu and Xu investigated nucleation and cell growth in a CO 2-loaded melt of polymethyl methacrylate using molecular dynamics simulations. They concluded that the bubble size distribution becomes increasingly homogeneous as the process pressure rises [ 6]. Complementary macro-to-micro-evidence shows that the final cell size distribution and morphology also depend on viscoelastic growth and depletion effects. Direct measurements combined with 3D simulations demonstrate how process pressure histories control elastic energy storage and, hence, the final cellular structure [ 7 Park et al. investigated the solubility of physical blowing agents, such as CO 2, in polymers. Solubility was determined using a modified pvT measuring device for linear and branched polypropylene. It was found that the solubility of physical blowing agents in polymer melts can be increased with increasing process pressure and decreasing temperature. The results were also compared with semi-empirical equations of state for predicting solubility. The potential for improvement in the equations of state for prediction was pointed out [ 8 The research focuses in particular on cell nucleation and cell growth. Both macro-scale and micro-scale analyses have been carried out with regard to nucleation. Leung et al. investigated the influence of nucleating agents (e.g., talc) on the heterogeneous nucleation of CO 2-loaded polystyrene melts. Bubble growth was visualised in situ using a special experimental setup. It was shown that the use of nucleating agents is accompanied by local pressure fluctuations, which promotes the nucleation of new cells. A comparison with the nucleation simulation by Wang et al. showed good agreement with the experiment [ 9, 10, 11 On the micro-scale, several numerical simulation models have been developed and investigated that describe the nucleation, growth and solidification of cells, taking into account different boundary conditions. Arefmanesh and Advani developed a numerical approach called the ‘Cell Model’ to describe cell growth under non-isothermal boundary conditions and low process pressures. The numerical model was compared with experiments in terms of the bubble density (bubbles/cm 3) and gas concentration. There were significant deviations between the experiment and the model, which can be attributed to the many simplifications [ 12 Taki used a numerical approach to describe the nucleation and growth of spherical cells and compared this with experimental investigations. For constant pressure release rates, good agreement with the experimental results was achieved [ 13 Ataei et al. used a modified Lattice Boltzmann Method (LBM) to describe cell growth at the micro-scale level. Here, the two-phase (liquid–gas) problem was simplified to one phase (liquid) and a comparatively low process pressure was used (15 MPa). The cell growth model is based on the ‘cell model’, which involves several simplifications: no interaction between neighbouring cells, incompressible melt, isothermal bubbles, constant diffusion rates, etc. However, the model is capable of performing a 2D or 3D simulation of the foaming process for specified boundary conditions. The model was then compared with the ‘Cell Model’ by Arefmanesh and Advani. A comparison of the simulative and experimental results with regard to the cell structure was not carried out [ 14, 15 Within the scope of commercially available simulation tools, modelling based on a homogeneous nucleation theory and the simplified modelling of cell growth is the state of the art [ 3, 16]. The bubble geometry is assumed to be spherical and is not dependent on the flow behaviour of the plastic melt [ 17]. Shear effects, coalescence, bubble collapse, etc., are not taken into account. Due to locally inhomogeneous process variables resulting from high pressure variations, thermal processes and complex flow processes, such simplifications (symmetrical bubble geometry, constant diffusion rates) are not permissible for precise modelling. The widespread adoption of FIM is hindered by the insufficient ability to predict the development of the cell structure for a specific geometry with sufficient accuracy, taking into account the local flow and solidification behaviour. Wang et al. calculated the weight reduction of complex foamed moulded parts using commercial simulation tools and found good agreement with experimental results [ 17]. However, since local cell orientation or coalescence are not taken into account, the calculated effective properties are insufficient for accurate component design. The result of the structural simulation is highly dependent on the predictive accuracy of the injection moulding process simulation [ 16, 17 Therefore, a multi-scale simulation approach is described to calculate the pressure, temperature and gas concentration on a macro-scale and simulate the gas cell geometry on a micro-scale [ 18]. The present work focuses on the development of a micro-scale simulation model in OpenFOAM that resolves bubble nucleation, deformation, coalescence, and gas–melt interactions based on macro-scale process conditions. To enable bubble expansion within the current solver framework, a simplified surrogate growth representation is employed, in which diffusion-driven mass transfer is not explicitly resolved. This approximation was used to study the qualitative behaviour of bubble–flow interaction and to establish a proof of concept for multiscale coupling. The model was subsequently used to analyse the phenomenological deformation of cells under characteristic injection moulding flow conditions. 2. CFD Model for Micro-Scale FIM 2.1. Software, Solver and Numerics All simulations were performed with OpenFOAM v11 (OpenFOAM Foundation, London UK). The two-phase, compressible, immiscible flow was modelled with the solver compressibleVoF, which employs a volume-of-fluid (VoF) phase-fraction formulation with bounded-phase transport (MULES) and an interface compression term to maintain a sharp interface. Pressure–velocity coupling followed the PIMPLE algorithm (outer and inner correctors), and surface tension was represented by the Continuum Surface Force (CSF) model. Because quasi-static effects (buoyancy, hydrostatics) are negligible in the present micro-scale model, gravity was disabled (g = (0,0,0)) [ 19]. This assumption can be justified by considering the Bond number, which compares buoyancy and surface tension forces. Using conservative estimates with a characteristic bubble size of up to 300 µm, a maximum density difference of Δρ ≈ 1200 kg/m 3, and a minimum surface tension of σ = 0.0178 N/m, the Bond number is approximated as Bo ≈ 0.06. For more typical conditions (L~100 µm), the Bo is on the order of 10 −3. Since Bo ≪ 1, surface tension forces clearly dominate over buoyancy, and gravitational effects can be neglected. 2.2. Governing Equations 2.3. Geometry and Mesh Therefore, the thickness and flow length are created as a 2D infinite-width model, as the velocity, temperature and pressure in the width direction are assumed to be negligible for a uniform flow front. We adopt this rationale at the micro-model scale to focus computational effort on the evolution that governs gas–melt segregation and the local pressure/temperature histories. The computational domain is a straight channel with a length (L) of 4.0 mm and a thickness (Th) of 3.0 mm. A structured mesh was generated with blockMesh using uniform spacings, ΔT = 0.015 mm and ΔL = 0.02 mm, yielding 200 × 200 × 1 cells. Although strong temperature gradients occur near the walls, the chosen mesh resolution must also be sufficient to resolve the smallest bubbles and their interfaces throughout the domain. A graded mesh with coarser resolution in the bulk would reduce the ability to capture bubble deformation and coalescence, while refining both wall and bulk regions would exceed the available computational budget. Therefore, a uniformly fine mesh was selected as a compromise to adequately resolve both wall gradients and bubble-scale phenomena. The planar 2D treatment with empty patches is standard in OpenFOAM when no through-width physics is modelled. Boundary patches are named left (inlet), right (outlet), top/bottom (walls), and front/back (empty). 2.4. Phases and Thermophysical Models Of the two phases considered, the gas phase is treated as a perfect gas, and the ideal-gas equation of state (EOS) is used to model the frequently used blowing agents nitrogen and carbon dioxide. The gas viscosity and gas Prandtl number are chosen as constant parameters because the temperature dependency is negligible in the temperature and pressure range of the foam injection moulding process. The relevant domain ranges from the melt temperature to cell stabilisation. The plastic melt’s density (ρ m) is given by a reciprocal polynomial EOS, where C i are the polynomial coefficients of any order (N): 1 ρ m = C 0 + C 1 T + C 2 T 2 − C 3 p − C 4 p T (2) This compact form captures the dominant thermal expansion with a quadratic temperature dependence and allows for a modest pressure coupling. As the focus was on the filling and foaming regime rather than precise packing or warpage prediction, this available function in OpenFOAM was used instead of implementing a full Tait law [ 27 The viscoelastic behaviour of the polymer melt is described using the built-in Phan–Thien–Tanner (PTT) model available in OpenFOAM v11. In this implementation, the polymeric extra-stress tensor is solved as an additional transport equation and incorporated directly into the momentum equation, replacing the turbulent stress contribution. Thus, the solver operates as a viscoelastic compressible VoF formulation without the use of external rheology libraries. The melt behaves as a shear-thinning, viscoelastic fluid. We therefore model μ(T, γ ˙ ) and the elastic stresses with a multi-mode Phan–Thien–Tanner (PTT) constitutive law, so that the simulation captures both the viscous and history-dependent (elastic) contributions to stress. PTT is a network-based model that reproduces shear-thinning and normal stress differences and provides a finite, rate-dependent extensional viscosity in uniaxial/planar/biaxial extension; depending on the parameters, it can also exhibit pronounced strain-/rate-dependent extensional thinning, which is relevant when the material is strongly stretched. In Equation (3), τ denotes the polymeric extra-stress tensor, U the velocity, and ∇U the velocity gradient tensor. The operator symm(.) extracts the symmetric part of a tensor, and tr(τ) denotes the trace. The material parameters are the polymeric viscosity scale ( υ M ), the relaxation time (λ) and the nonlinearity parameter (ε). The left-hand side ∂τ/∂t + ∇ (Uτ) represents the unsteady transport of polymer stress with the flow, while the term 2 symm(τ ∇U) provides the frame-invariant convected kinematics typical of differential viscoelastic models. The term (2νM/λ) symm(∇U) acts as the deformation-driven stress source, and the exponential factor exp[−(ελ/νM)tr(τ)] introduces the characteristic PTT network damping, which regularises stress growth and yields a finite, rate-dependent extensional response. PTT has also been used in polymer foaming cell growth simulations, supporting its relevance for the micro-mechanics of foaming [ 30 The heat capacity and thermal conductivity of the melt were set as temperature-independent for this work. However, tabulated temperature-dependent values can be implemented in this model if required. The melt viscosity parameter ( υ M ) is defined using temperature-dependent tabulated values. The initial condition is a fully liquid channel (φ = 0 everywhere). Gas nuclei and their distribution arise from the nucleation model described in Section 3. The VoF transport herein simply carries the gas fraction once present. However, the pressure field inside the bubbles is not driven by diffusion from the solved gas in the melt towards the nucleated bubbles, as it is not part of the compressibleVoF solver. The gas concentration cannot be used as a state variable, and the cell growth kinetics is thus described with a substitution model (see Section 3.5). 2.5. Boundary and Initial Conditions Both the inlet and outlet are prescribed with time-dependent pressure boundary conditions derived from macro-scale simulation results. No velocity or mass flux is imposed; instead, the flow is driven entirely by the resulting pressure gradients. The velocity field therefore develops self-consistently from the governing equations under physically consistent boundary conditions. Temperatures at the inlet are set with zero gradient (convective inflow). The top and bottom walls are no-slip for velocity, and the walls’ thermal conditions are imposed from the macro-model as the time-dependent wall temperature. 2.6. Temporal and Algorithmic Settings The time integration uses an adaptive time step controlled by both the flow Courant number and the VoF Courant number: maxCo = 0.25 and maxAlphaCo = 0.25. These bounds are typical for VoF with explicit interface compression and help avoid aliasing of capillary waves without explicitly computing a capillary time-step limit. The PIMPLE settings were nOuterCorrectors = 3, nCorrectors = 3, and nNonOrthogonalCorrectors = 0. Interface compression employed the OpenFOAM interface compression scheme with a compression coefficient of 1 and MULES bounding. Discretisation used second-order schemes in space and a first-order Euler in the time scheme. 2.7. Surface Tension and Capillary Treatment Surface tension was included through the CSF model of Brackbill et al., where curvature is obtained from the phase fraction field. This formulation is the default in OpenFOAM’s VoF solvers [ 32]. While various geometric VoF curvature options exist, we retained the standard algebraic CSF/VoF for consistency with the intended multiscale workflow. To promote stability with explicit surface tension and interface compression, we limited maxAlphaCo, as stated above; no additional capillary time-step constraint was enforced. While geometric VoF approaches can provide improved interface curvature estimation and reduce spurious currents, their implementation for compressible, viscoelastic two-phase flows is currently limited. Therefore, the standard CSF formulation available in OpenFOAM v11 was employed to enable a consistent multiscale framework. The focus of this study is on the coupling of bubble nucleation, growth representation, and flow interaction rather than on highly accurate curvature reconstruction. 3. Bubble Nucleation In the present work, the initial bubble population for the micro-scale foam injection moulding simulations was generated in a pre-processing step from a physically based nucleation model ( Figure 1). It was programmed with Python 3.11.9 and imported the pressure and temperature results of the OpenFOAM simulation as input parameters. The goal was to obtain a spatially inhomogeneous distribution of bubble centres, radii and internal pressures that was consistent with the local thermodynamic state of the polymer melt and could be used as the initial condition for the two-phase flow solver. 3.1. Local Nucleation Rate Nucleation is treated as a stochastic process described by classical nucleation theory (CNT) [ 33, 34, 35]. The local homogeneous nucleation rate (N hom) describing the number of critical nuclei formed per unit volume and time is taken as: N hom = c hom · f 1 · e x p − ∆ G h o m ∗ k B · T (4) where c hom is the local gas concentration during the homogeneous nucleation, k B is the Boltzmann constant, and T is the local melt temperature. The factor f 1 is a frequency factor representing the attempt frequency for molecular attachment to a critical nucleus. For gas bubble nucleation in a liquid, the critical free energy ( ∆ G h o m ∗ ) barrier is the maximum of the excess Gibbs free energy. The maximum is reached at the critical radius (r*), which is defined as follows [ 34, 35]: r ∗ = 2 σ Δ p (5) where σ is the polymer–gas surface tension and Δp is the effective pressure difference between the gas in the nucleus and the surrounding polymer melt. The corresponding critical Gibbs free energy barrier for a homogeneous nucleation is: ∆ G h o m ∗ = 16 π · σ 3 3 · Δ p 2 (6) For heterogeneous nucleation on a substrate, the critical nucleus has the shape of a spherical cap rather than a full sphere. The corresponding free energy barrier is reduced by a geometric factor (f(θ)) that depends on the contact angle (θ) between the nucleus and substrate [ 34, 35]. Therefore, the critical Gibbs energy for heterogeneous nucleation is reduced by this factor: ∆ G h e t ∗ = f ( θ ) · ∆ G h o m ∗ , f ( θ ) = 1 4 ( 2 + c o s θ ) ( 1 − c o s θ ) 2 (7) The ∆ G h o m ∗ is the homogeneous barrier defined above (Equation (4)). For 0 r b u b b l e : ω i , o l d · F · 1 − r − r b u b b l e G · r b u b b l e − r b u b b l e (16) 3.5. Initial Cell Properties and Emulation of Diffusion-Controlled Growth The following approach is introduced as a temporary surrogate to enable bubble expansion within the current modelling framework ( Figure 3). It is not intended to provide a physically accurate description of diffusion-controlled growth but rather to allow for a qualitative investigation of bubble–flow interaction in the absence of an explicit mass-transfer model. For the initialisation in OpenFOAM, the initial radius, pressure and temperature for each cell need to be defined. The bubble temperature (T bub,i) is assumed to be the same as the local melt temperature (T m,i). The radius of the bubble is then determined as the critical radius (r*). However, in the real foaming process, bubbles nucleate at nanometre-to-sub-micrometre critical radii and grow by diffusion of dissolved gas into the bubble, while the polymer is still sufficiently mobile [ 44]. Therefore, such small bubbles cannot be resolved by the computational mesh, and a larger initial radius must be prescribed. Furthermore, cell growth arrests locally once the polymer cools and solidifies either before or after reaching a depictable size. Consequently, one criterion for whether a bubble can be initialised is that the smallest depictable radius (r min) is reached until bubble stabilisation. Therefore, a temperature limit (T limit) 20 K above the no-flow temperature is used as a nucleation criterion. The value of 20 K is currently a fitting parameter that reflects the cooling that occurs during the cell’s growth phase, which is necessary for it to reach a significant size. If the temperature at the position chosen by the probability distribution is below the limit, the current iteration of the loop is skipped. The next criteria of the initial radius results from the substitute model for cell growth kinetics. As explained in Section 2.4, the CFD model is currently lacking a physical cell growth model due to the mass in a bubble being constant and no description of the gas diffusion. Therefore, a surrogate cell growth model assigns higher initial pressure instead of increasing the pressure over time. This forces the bubbles to grow during the simulation. The term ‘surrogate’ is used here to emphasise that the model does not calculate physical mass transfer but mimics its macroscopic effect on bubble expansion. But as a consequence, the cell growth is faster than in reality, and the initial pressure difference between the bubble and the melt can lead to divergence in the CFD simulation. Thus, the initial bubble radius and pressure are determined by the model described as follows. The aim is to choose an effective initial gas pressure ( p g , 0 e f f ) for an initial radius (r ini) in a way that, under the process pressure and temperature history, the bubble volume evolution approximates the desired change from r 0 to the final bubble radius r f. For diffusion-controlled cell growth kinetics in a supersaturated liquid, classic analyses by Epstein and Plesset and subsequent extensions to polymer melts show that, over a wide range of times, the bubble radius satisfies: r t 2 − r 0 2 ≈ 2 K t g r o w , (17) where r 0 is the initial radius, t g r o w is the growth time, and K is a growth constant that depends on the gas diffusivity, solubility and supersaturation [ 45, 46]. Neglecting r 0 compared to the final radius implies: r t ∝ t g r o w (18) In a non-isothermal injection moulding process, the available growth time is limited by the local freezing of the polymer. So, the growth time is the difference between the start of the nucleation (t nuc) and the time (t freeze) at which the plastic melt drops below the above-described limit temperature (T limit). This growth time distribution is derived from the macro-scale. Except for areas with T(t nuc) p m , 0 · F p . F p is the factor of overpressure, which is usable by the CFD simulation and is defined by tests. The new initial radius ( r 0 , new ) with the same gas amount is calculated with the ideal-gas law: r 0 , new = r 0 , old p g , 0 e f f p g , 0 , n e u e f f 1 / 3 with p g , 0 , n e u e f f = p m , 0 · F p (22) In other words, all gas that would have diffused into the bubble during microscopic growth from r 0 to r f is effectively assigned at the beginning of the CFD simulation at the numerically resolvable radius (r 0). The resulting larger gas mass causes the bubble to expand under the subsequent pressure drop. The expansion is slowed down by the inertia and resistance of the melt viscosity in a way that approximates the desired effective growth. While this approach does not reproduce the detailed kinetics of diffusion-controlled growth, it provides a physically interpretable way to initialise bubble sizes and internal pressures in a two-phase flow simulation without explicit mass transfer, consistent with classical bubble growth theory and with observed variations in the final cell size across the moulded part [ 44, 45 An optional distance check to avoid overlapping bubbles can also be activated. However, the gas depletion (Equation (16)) already reduces this probability depending on the chosen depletion parameters. 4. Demonstration Case Simulation Setup To demonstrate the two-phase CFD simulation, an injection moulded part with a simple plate geometry was used ( Figure 5). The mould of the part has a cold runner that distributes the melt over the whole plate width and ensures a uniform flow front. The modelled materials were the PC LED 5102, Covestro AG, Leverkusen, Germany, and nitrogen gas. Their material and model parameters are listed in Table 1 and Table 2. The zero-shear viscosity is defined as: η 0 = D 1 · e x p − A 1 ( T − T r e f ) A 2 ( T − T r e f ) (23) with the reference temperature (T ref) and the fitted coefficients A 1, A 2 and D 1. The parameters of the van’t Hoff equation are approximated from literature values [ 4, 47 The boundary conditions of the micro-scale simulation, like the wall temperature and pressure development, were derived from a macro-scale simulation. It was conducted with the injection moulding simulation software Moldex3D 2024, CoreTech System Co., Ltd., Zhubei, Taiwan, and its foam injection moulding module ( Figure 6). The process parameters are displayed in Table 3. The cooling channels were considered, and a transient cooling simulation was calculated. Due to the different available material models, the macro-simulation was conducted with a Tait model and modified cross-WLF model instead. The graphs showing the results of the nucleation algorithm were created with a Python script written by the generative artificial intelligence Claude Haiku 4.5. It used the data of the nucleation algorithm and OpenFOAM simulation as input. This work presents a micro-scale simulation framework for foam injection moulding that combines a two-phase compressible VoF solver with viscoelastic rheology and a CNT-based nucleation model driven by macro-scale process conditions. The framework enables the simulation of bubble nucleation, deformation, coalescence, and interaction with the surrounding flow field. In a plate geometry case study (polycarbonate with N 2), the model qualitatively captured these mechanisms in a 2D through-thickness section. The simulated deformation reproduced the characteristic aspect ratio and orientation trends observed in the experiments on a qualitative level. A simplified surrogate growth approach was used to enable bubble expansion within the current solver framework. The results show that this approximation introduces non-physical oscillations and cannot reproduce realistic growth kinetics. These are interpreted mainly as artifacts of the growth substitution strategy, although their amplitude is amplified by the stiffness of the coupled two-phase viscoelastic CFD problem. The resulting accelerated early growth can exaggerate deformation near the injection stop. Sensitivity to the initial radius and overpressure underscores the need for a physically consistent growth mechanism and careful parameter calibration. Nevertheless, the presented framework provides a proof of concept for multiscale coupling, and its ability to capture essential micro-scale mechanisms governing foam morphology has been demonstrated. Future work will therefore introduce diffusion-controlled mass transfer to the gas phase to investigate cell growth kinetics without artefacts from the surrogate growth model. The improved model can then be validated against measured through-thickness cell size gradients. Incorporating temperature-dependent thermophysical properties, exploring 3D domains where needed, and tightening the macro–micro-coupling loop are expected to improve quantitative predictiveness for microstructure-aware component design. Author Contributions Conceptualisation, D.C.F. and M.S.; methodology, D.C.F.; software, M.S.; validation, D.C.F.; formal analysis, D.C.F.; investigation, D.C.F.; resources, C.H.; data curation, D.C.F.; writing—original draft preparation, D.C.F.; writing—review and editing, M.S. and C.H.; visualisation, D.C.F.; supervision, C.H.; project administration, C.H.; funding acquisition, C.H. All authors have read and agreed to the published version of the manuscript. Funding The presented research was funded by the Deutsche Forschungsgemeinschaft (DFG). We would like to extend our thanks to the DFG. Project number 493892533. Institutional Review Board Statement Not applicable. Data Availability Statement The data and materials for this publication are available upon request at the following link: http://hdl.handle.net/21.11102/e6bc97b7-a112-480b-8545-8870534b2f03 (accessed at 7 July 2026). Acknowledgments During the preparation of this manuscript, the author used Claude Haiku 4.5 for the purposes of writing Python scripts to generate images from the simulated and calculated data. The authors have reviewed and edited the output and take full responsibility for the content of this publication. Conflicts of Interest Author Malte Schön was employed by the company 3M Deutschland GmbH. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Abbreviations The following abbreviations are used in this manuscript: CNT Classical nucleation theory CSF Continuum surface force DFG Deutsche Forschungsgemeinschaft EOS Equation of state FIM Foam injection moulding LBM Lattice Boltzmann method N 2Nitrogen PC Polycarbonate VoF Volume of fluid References The process of determining the local nucleation rate with the pressure, temperature and gas concentration distribution as input before progressing to the next steps in Figure 2. The process of determining the local nucleation rate with the pressure, temperature and gas concentration distribution as input before progressing to the next steps in Figure 2. The process of determining the bubble position with the local nucleation rate, temperature and melt pressure distribution as input. The process of determining the bubble position with the local nucleation rate, temperature and melt pressure distribution as input. The process of determining the bubble properties with the bubble location, local melt pressure and temperature over time as input. The process of determining the bubble properties with the bubble location, local melt pressure and temperature over time as input. An exemplary microscopy image of a section cut from a N 2 foamed PC part and its average cell radius distribution together with the fit (Equation (19)). An exemplary microscopy image of a section cut from a N 2 foamed PC part and its average cell radius distribution together with the fit (Equation (19)). The part geometry, position and size of the micro-scale simulation. The part geometry, position and size of the micro-scale simulation. The process setup of the macro-simulation with cooling channels and the unused heating ceramics of the fixed mould half. The process setup of the macro-simulation with cooling channels and the unused heating ceramics of the fixed mould half. The pressure and pressure gradient along the flow direction development of the macro-scale simulation at the position of the micro-scale simulation domain ( a) and the development of the gas volume and volume fraction ( b). The pressure and pressure gradient along the flow direction development of the macro-scale simulation at the position of the micro-scale simulation domain ( a) and the development of the gas volume and volume fraction ( b). The nucleated bubble distributions (red circles) with the overpressure factors 1.0, 1.5 and 2.0, including the resulting initial bubble pressures, as well as the common initial distributions of the nucleation probability ( a), temperature ( b) and pressure ( c). The nucleated bubble distributions (red circles) with the overpressure factors 1.0, 1.5 and 2.0, including the resulting initial bubble pressures, as well as the common initial distributions of the nucleation probability ( a), temperature ( b) and pressure ( c). Simulated bubble shape (black) and velocity field with different nucleation times. Simulated bubble shape (black) and velocity field with different nucleation times. Qualitative comparison of experimental and simulated (nucleation at 2.4 s) cell aspect ratios and orientations with blue gas cells and green matrix. Qualitative comparison of experimental and simulated (nucleation at 2.4 s) cell aspect ratios and orientations with blue gas cells and green matrix. The simulated bubble shape (black) over time with overpressure factors of 1, 1.5, and 2.0, including the resulting velocity distribution. The simulated bubble shape (black) over time with overpressure factors of 1, 1.5, and 2.0, including the resulting velocity distribution. The pressure development in the centre of the micro-scale simulation for different overpressure factors and the pressure field with black gas bubbles shortly after bubble initialisation. The pressure development in the centre of the micro-scale simulation for different overpressure factors and the pressure field with black gas bubbles shortly after bubble initialisation. Table 1. Material parameters of PC and its PTT as well as cross-WLF model parameters for micro-scale simulation. Table 1. Material parameters of PC and its PTT as well as cross-WLF model parameters for micro-scale simulation. Material Parameter Value Unit Thermal conductivity 0.23 [W/(m∙K)] Heat capacity 2000 [J/(kg∙K)] PTT υ M 0.1 [Pa∙s] PTT λ 0.004 [s] PTT ε 0.01 [-] EOS C 0୬.୨୫୭ ୍ଠ ୧୦ −4[m 3/kg] EOS C 1୫.୮୩୭ ୍ଠ ୧୦ −7[m 3/(kg∙K)] EOS C 2−୩.୩୨୭ ୍ଠ ୧୦ −12[m 3/(kg∙K 2)] EOS C 3−୮.୧୧୫ ୍ଠ ୧୦ −13[m 3/(kg∙Pa)] EOS C 4୨.୯୮୭ ୍ଠ ୧୦ −15[m 3/(kg∙Pa∙K)] Cross-WLF A 17.9837 [-] Cross-WLF A 2162 [K] Cross-WLF D 11207 [Pa∙s] Cross-WLF T ref518 [K] Table 2. Material parameters of nitrogen and its solubility parameters in polymer melt. Table 2. Material parameters of nitrogen and its solubility parameters in polymer melt. Material Parameter Value Unit Prandtl number 0.7 [-] Heat capacity 1045 [J/(kg∙K)] Molecular weight 28.9 [g/mol] Surface tension 0.0178 [N/m] Contact angle [ 48] 30 [°] Reference Henry coefficient, k H (298 K) [ 4, 47] 6.5 [mol/(m 3∙MPa)] Enthalpy of solution, ∆ H S 10,500 [J/mol] Viscosity ୨.୮୪ ୍ଠ ୧୦ −5[Pa∙s] Table 3. Process parameters of macro-scale foam injection moulding simulation. Table 3. Process parameters of macro-scale foam injection moulding simulation. Process Parameter Value Unit Melt temperature 300 [°C] Mould temperature 110 [°C] Volume rate 100 [cm 3/s] Gas concentration 0.6 [weight-%] Switch-over point 95 [vol-%] Holding pressure time 0 [s] Cooling time 25 [s] Mould-opening time 5 [s] Table 4. Model parameters of nucleation algorithm. Table 4. Model parameters of nucleation algorithm. Model Parameter Value Unit Maximum bubble radius, r f150 [µm] Nucleation time, Δ t n u c 0.1 [s] Gas depletion factor 0.9999 [-] Depletion range multiplier 4.0 [-] Minimum bubble radius, r min69 [µm] Critical growth temperature, T l i m i t 450 [K] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license.
