Modelling the diffusive behavior of polymer foam immersed in water

Abstract

This study introduces a refined methodology for modeling water diffusion in polymeric foams. The approach is grounded in Darcy’s law, which governs diffusion in porous media. A modified diffusion law is proposed to describe the complex interplay between the foam’s mechanical state and the dynamics of water transport. The resulting diffusion kinetics follows a three step mechanism controlled by a parameter that captures the interaction between Darcy-governed diffusion and diffusion influenced by mechanical properties. Notably, at low values of this parameter, the kinetics closely approximate those predicted by Fick’s laws. The framework is further extended to sandwich structures composed of composite skins and a polymer foam core. In this configuration, the diffusion process proceeds in two distinct phases: an initial pseudo-step corresponding to skin saturation, followed by diffusion within the foam core. A series of experimental studies corroborate this intricate diffusive behavior, underscoring the robustness and applicability of our model.

Keywords

polymer foam sandwich structure diffusive behavior Darcy’s law water immersion porous media

General introduction

Polymeric foams are used in a variety of applications, including the construction, automotive, household and packaging industries. These materials are characterized by advantageous physical properties, including low specific mass, effective thermal and acoustic insulation, and superior mechanical shock absorption capabilities.¹

Polymeric foams are extensively employed in various industrial contexts as critical components of sandwich structures,² where a high stiffness-to-weight ratio is of paramount importance.^1–3 This is particularly relevant in the aeronautical and naval transport sectors, as well as in high-tech applications such as aerospace. The incorporation of polymeric foams into these structures offers a number of benefits, including enhanced mechanical rigidity and a notable reduction in overall mass compared to traditional materials. The foam market is primarily dominated by amorphous polymers, such as polystyrene (PS), polyurethane (PU), and polyvinyl chloride (PVC). However, these sandwich structures are often subjected to severe environmental conditions, including high humidity and temperature fluctuations, which can lead to significant structural degradation, especially when hydrophilic materials are used in the foam core.^4–10

In these polymer-based structures, moisture diffusion can give rise to several phenomena, including a decline in physical and mechanical properties due to plasticization. Additionally, differential swelling can result in the generation of a stress field within the thickness of the material, induced by the non-uniform distribution of the water content during the transient phase of the water uptake process. Consequently, the occurrence of low mechanical stress in conjunction with hygro-thermal stress can result in material damage. This assertion is supported by the findings of previous studies, as reported in Refs. 7,11–13.

In general, the diffusion of moisture in polymeric foams is predicted by Fick’s law in a manner analogous to that observed in conventional polymers with a dense structure (see Refs. 14–19). However, this diffusion model does not accurately represent the diffusion process in polymeric foams. The porous structure of these materials differs completely from that of typically dense polymers, where free volume and micro-porosities play a key role in the diffusion process. The use of conventional diffusion models, such as Fick’s law, raises significant concerns regarding the accuracy of durability predictions for these structures. To address this issue, a more refined modeling approach for moisture diffusion in polymeric foams subjected to hygroscopic loading is proposed in this work. The methodology is based on the principles of liquid diffusion in porous media, specifically employing Darcy’s law.^20–23

The application of Darcy’s law can be justified by considering that water absorption by the polymeric matrix induces hygroscopic swelling, which generates internal hydromechanical potential gradients responsible for fluid transport through the connected porous network; under the assumptions of slow flow (Re <<<1), sufficient local saturation ensuring the continuity of the fluid phase, porous connectivity allowing water percolation, small deformations of the polymeric skeleton, homogenized behavior at the macroscopic scale, a constant temperature (isothermal), and a local thermodynamic equilibrium between the fluid and the solid, the flow can be described by Darcy’s law, where the pressure (p) is interpreted as an effective pressure related to hygroscopic swelling rather than a simple conventional hydraulic pressure. In this approach, swelling gradients induce water potential gradients that can be regarded as an equivalent driving pressure for flow in the porous medium, which is consistent with Maurice Anthony Biot’s work on poroelasticity, the hydro-mechanical formulations developed in Poromechanics, the models of cellular materials presented in Cellular Solids: Structure and Properties, as well as the theories of polymer swelling proposed by Paul J. Flory through the Flory–Huggins solution theory.^24–27

This framework allows the derivation of a novel differential equation governing diffusion within the foam. As a result, this study offers a more consistent analysis of foam durability compared to traditional decoupled models, which often rely on classical Fickian principles.

Problem statement

In this study, we examine the diffusion of fluid through foam in accordance with Darcy’s law. In this context, the mass flux, J [kg.m⁻²s⁻¹] of a fluid passing through the porous medium is defined as a function of the mass concentration C [kg.m⁻³], as expressed in equation (1)^19,20:

J_{Darcy} = \bar{v} C

(1)

\bar{v}

is the average mass flow velocity according to Darcy’s law, given by Refs. 19–23:

\bar{v} = - \frac{k}{μ} (\nabla P - ρ_{f} φ eg)

(2)

In expression (2) g represents the acceleration due to gravity, expressed in meters per second squared (m.s⁻²). Therefore, g corresponds to gravity, which influences the movement of the fluid under the effect of its weight in the porous medium.

The intrinsic permeability k [m²] of foam is a property that is independent of the diffusing fluid and depends solely on the structure and connectivity of the foam cells. It is therefore assumed to be independent of the diffusing fluid content. µ represents the dynamic viscosity µ [kg.m⁻¹. s⁻¹] of the fluid at a given temperature. P [Pa] is the pressure between two points. ρ_f represents the density of the diffusing fluid, which in this case is water. The linear fraction of the void along the thickness e of the foam, which we will refer to as the diffusion path, is represented by ϕ.

From the mass conservation equation, one obtains:

\frac{\partial C}{\partial t} + \nabla J_{Darcy} = 0

(3)

\frac{\partial C}{\partial t} = \nabla ⌈ (\frac{k}{μ} (\nabla P - ρ_{f} φ eg)) C ⌉

(4)

This implies:

\frac{\partial C}{\partial t} = \frac{k}{μ} (Δ P - φ eg \nabla ρ_{f} - ρ_{f} φ e \nabla g) C + (\frac{k}{μ} (\nabla P - ρ_{f} φ eg)) \nabla C

(5)

Since the density of the penetrating fluid (in this case, water), as well as gravity g, are constants, ∇ρ_f = 0 and ∇g = 0, so equation (5) simplifies as follows:

\frac{\partial C}{\partial t} = \frac{k}{μ} [C Δ P + \nabla P \nabla C - ρ_{f} φ eg \nabla C]

(6)

Substituting the water content w for the concentration C ( $C = ρ_{m} w$ ), where $ρ_{m}$ is the dry density of the foam, the equation (6) yields the following expression;

\frac{\partial w}{\partial t} = \frac{k}{μ} [w Δ P + \nabla P \nabla w - ρ_{f} φ eg \nabla w]

(7)

In equation (7), the evolution of the water content is a function of the pressure within the foam. In this case, at any given moment in time, it is necessary to determine the pressure, which is referred to as the hydrostatic pressure. This represents the sum of two distinct contributions: the hygroscopic pressure Pi, which is due to hygroscopic loading, and the external pressure P_ext, which is due to external mechanical loading applied at the foam boundary. Equation (7) indicates that diffusion is coupled to the mechanical state. This is due to the introduction of Darcy’s law (3) into the diffusion equation (4), which then becomes a function of the pressure gradient. In the transient state, the pressure gradient incorporates the contribution of differential swelling in the foam thickness. As the foam approaches saturation (steady state), this contribution tends to disappear.

Determination of pressure P

As previously stated, the pressure P to be determined is simply the hydrostatic pressure, which represents the sum of the hygroscopic pressure Pi, due to the hygroscopic loading and the external pressure P_ext, due to the external mechanical loading applied to the foam boundary. These two loads induce internal stresses. Under the assumption that the foam under study is isotropic and obeys an elastic behavior, its hygro-elastic behavior law can be expressed as follows:

σ_{ij} = \frac{E}{(1 + ν) (1 - ν)} ε_{kk} δ_{ij} + \frac{E}{(1 + ν)} ε_{ij}^{tot} + \frac{E β Δ w}{1 - 2 ν} δ_{ij}

(8)

In equation (8), the variable βrepresents the hygroscopic expansion coefficient of the foam. The symbols $E, ν$ represent, respectively, the Young’s modulus and the Poisson’s ratio of the foam. On the other hand, $ε_{i j,} ε_{i j}^{t o t}$ are respectively the trace of the strain tensor and the total strain, which are expressed by the relations below?

{\begin{cases} ε_{ij}^{tot} = ε_{ij}^{m} + ε_{ij}^{hyg} \\ ε_{ij}^{m} {= S}_{ijkl} σ_{kl} \end{cases}

(9)

The first part $ε_{ij}^{m}$ represents the mechanical strain, due to mechanical loading. $S_{i j k l}$ denotes the relaxation tensor. The second part of the right-hand side, $ε_{ij}^{hyg}$ , represents the strain due to hygroscopic loading. Let $Δ w$ denote the difference between the water content in the foam at time t and w the initial water content: $w_{0}$ : $Δ w = w - w_{0}$ . The expression for total strain can be rewritten as follows:

ε_{ij}^{tot} = \frac{- ν}{E} σ_{kk} δ_{ij} + \frac{1 + ν}{E} σ_{ij} + β Δ w δ_{ij}

(10)

In (10), $δ_{i j}$ is the Kronecker delta:

δ_{ij} = {\begin{cases} δ = 1 if i = j \\ δ = 0 if i \neq j \end{cases}

(11)

Conversely, the hydrostatic pressure within the foam, resulting from external mechanical and hygroscopic loading, is defined as follows:

P = - \frac{1}{3} σ_{kk}

(12)

In order to express the problem in terms of stresses, it is necessary to use the compatibility equations (also known as the Mitchell-Beltrami equations):

(1 + ν) σ_{i j k l} + σ_{k l i j} + E β (\frac{1 + ν}{1 - v} δ_{i j} L_{, k l} + L_{, i j}) = 0

(13)

Assuming i = j and that a quasi-static equilibrium is maintained at all times,²⁸ one obtains:

σ_{i j, j} = 0

(14)

This yields the following equation:

\nabla^{2} (σ_{kk} + 2 β \frac{E}{(1 - ν)} w) = 0

(15)

In light of the aforementioned considerations (given by equation (12)), one may conclude that:

Δ P = {Δ P}_{i} + {Δ P}_{ex} = β (\frac{2 E}{3 (1 - ν)}) Δ w + {Δ P}_{ex}

(16)

In the absence of external loading, the following simplification occurs, so that equation (16) reduces to:

{Δ P}_{i} = β (\frac{2 E}{3 (1 - ν)}) Δ w

(17)

Finally, by introducing the following substitution rules:

Y_{c} = \frac{2 E}{3 (1 - ν)}

(18)

{Δ P}_{i} = β Y_{C} Δ w

(19)

The solution of equation (17) is then given by:

P_{i} (z, t) = β Y_{c} w (z, t) + K_{1} (t) z + K_{2} (t)

(20)

The constants K₁ and K₂ are determined from the equilibrium conditions. The final form of the solution to equation (18) is expressed in accordance with the methodology proposed by Ref. 16:

P_{i} (z, t) = β Y_{c} (w (z, t) - \bar{w} (t))

(21)

where

\bar{w} (t)

is the average moisture content within the foam at a given instant t, expressed by:

\bar{w} (t) = \frac{1}{e} \int_{0}^{e} m (z, t) dz

(22)

From equation (21), one obtains:

Δ P = \nabla P_{i} = β Y_{c} \nabla w

(23)

\nabla \bar{w} = 0

Since

\bar{w} (t)

only depends on time.

Replacing (19) and (23) in (7) yields:

\frac{\partial w}{\partial t} = \frac{k}{μ} [β Y_{c} (w Δ w + {(\nabla w)}^{2}) - φ e ρ_{f} g \nabla w]

(24)

Assuming:

η = β Y_{c}

(25)

Then

ζ_{m} = \frac{k}{μ} β Y_{c}

(26)

And:

ζ_{g} = e \frac{k}{μ} ρ_{f} g

(27)

η

ζ_{m}

and

ζ_{g}

are respectively a term related to the foam’s mechanical properties, while the other two represent diffusion velocities (whose dimensions correspond to the diffusion coefficient [L²T⁻¹]). The first is linked to diffusion, which depends on the foam’s permeability and mechanical properties. The second is linked to diffusion, which is attributable to gravity.

e represents the path of the diffusing species (in this case, the thickness of the foam in the direction of diffusion) $φ$ and is the linear porosity along this thickness. In this case, equation (24) can be expressed as follows:

{\frac{\partial w}{\partial t} = ζ}_{m} [w Δ w + {(\nabla w)}^{2} - φ \frac{ζ_{g}}{ζ_{m}} \nabla w]

(28)

The following ratio is introduced:

χ = \frac{ζ_{g}}{ζ_{m}} = \frac{e ρ_{f} g}{η}

(29)

χ

is the ratio between diffusion due to gravitational force (Darcy) and diffusion driven by the stress induced by the swelling of the diffusing species. It is notable that the values of

χ

are very low, due to the fact that

η ≫ e ρ_{f} g

, which means that the diffusion process due to gravity (according to Darcy’s law) is always very slow compared to the diffusion process governed by the mechanical state.

Substituting by $χ$ in (28), one obtains:

{\frac{\partial w}{\partial t} = ζ}_{m} [w Δ w + {(\nabla w)}^{2} - φ χ \nabla w]

(30)

The case $χ \to 0$ i.e., when $η ≫ e {φ ρ}_{f} g$ , or when $φ \approx 0$ corresponds to scenarios where the porosity is either very low or completely absent, characteristic of dense materials. In such cases, the influence of gravity on diffusion becomes negligible in comparison with that driven by the mechanical state. Consequently, equation (30) simplifies to the following expression:

{\frac{\partial w}{\partial t} = ζ}_{m} [w Δ w + {(\nabla w)}^{2}]

(31)

Equation (31) describes diffusion in a highly dense material. In this context, the value in question represents a diffusion velocity, analogous to the diffusion coefficient. Referring to equation (26), which defines this coefficient, it can be concluded that the diffusion coefficient results from several parameters, including intrinsic permeability, dynamic viscosity, hygroscopic expansion coefficient, and mechanical properties. Therefore, the diffusion coefficient can be defined as follows:

D_{m} = \frac{k}{μ} β Y_{c}

(32)

In a similar manner, a diffusion coefficient attributed to gravity, as given by equation (27), can be defined as follows:

D_{g} = φ e \frac{k}{μ} ρ_{f} g

(33)

The diffusion coefficient is significantly influenced by the type of foam cells (open or closed) and their porosity, as demonstrated by expression (32). In the case when the foam exhibits strong porosity, i.e., $φ \approx 1$ , the coefficient D_g can be expressed as follows:

D_{g} = e \frac{k}{μ} ρ_{f} g

(34)

The relationship between the two diffusion coefficients is defined as follows:

D_{g} = φ χ D_{m}

(35)

Thus, the diffusion equation (30) can be expressed as:

{\frac{\partial w}{\partial t} = D}_{m} [w Δ w + {(\nabla w)}^{2} - φ χ \nabla w]

(36)

It is crucial to highlight that the diffusion coefficient of the foam D_m can be as curtained by fitting the theoretical curve derived from equation (36) to the gravimetric data obtained during hygroscopic aging. Subsequently, the gravity-related diffusion coefficient D_g can be calculated using equation (35). The intrinsic permeability k of the foam can then be obtained from equation (34).

Determination of the moisture content within the foam

The average content at any given time throughout the thickness of the foam is obtained using equation (22). To achieve this, it is necessary to solve the diffusion equation (36) to obtain the moisture field w (z, t) across the entire thickness e of the foam at each instant. To facilitate the resolution of equation (36), we resort to a dimensionless approach. This involves introducing a dimensionless time $τ = \frac{D_{m} t}{e^{2}}$ and a dimensionless thickness $Z = \frac{z}{e}$ , which allows equation (36) to be rewritten as follows:

\frac{\partial w}{\partial τ} = w Δ w + {(\nabla w)}^{2} - φ χ \nabla w

(37)

In this case, the diffusion equation depends on two parameters: the linear fraction of porosity in the foam and the coefficients of the diffusion ratio $χ$ . In the absence of porosity or with very low porosity such as in the case of a dense polymer $χ$ depends on the hygroscopic expansion coefficient $β$ , the Young’s modulus E, and the Poisson’s ratio $ν$ of the foam, which can be obtained from the literature. The nonlinear coupled equation (37) is solved in Mathematica software, by using a finite difference method with an implicit time stepping scheme, which is unconditionally stable and, combined with a consistent spatial discretization, ensures convergence of the numerical solution under the considered conditions.

Figure 1 presents the evolution of the general diffusion kinetics of foam, derived from equation (36). The moisture content w (t) is normalized by the moisture content value at saturation Ws and plotted as a function of the coefficient $χ$ . This figure was generated assuming a linear porosity fraction $φ$ in the foam of 95%.

Figure 1.

Evolution of the diffusion kinetics in a polymeric foam as a function of the diffusion ratio $χ$ .

Parametric tests performed for porosities within the realistic range 0.9–0.99 showed similar trends in the resulting curves, confirming that the qualitative behavior of the model is not significantly affected by small variations in porosity.

It can be observed that, for the specific value $= 1$ , which corresponds to $D_{g} = D_{m}$ , the observed diffusion kinetics closely resemble Fickian diffusion. This behavior is characteristic of foams with open cells that do not exhibit swelling tendencies, or of closed cells with a low hygroscopic expansion coefficient.

In this context, the rate of diffusion is not significantly impeded. As the values of $χ$ decreases, indicative of foams with closed cells that exhibit a pronounced tendency to swell (thus presenting high hygroscopic expansion coefficients a significant deviation of the foam’s diffusion kinetics from Fickian behavior is observed. Specifically, the diffusion kinetics manifest as a dual-phase process comprising an initial pseudo-plateau followed by a definitive saturation plateau. The visibility of the pseudo-plateau intensifies with decreasing $χ,$ reaching its maximum when $χ$ approaches zero. This condition signifies an almost complete absence of the diffusion phenomenon due to gravitational effects, as represented in equation (31).

It is important to note that in this particular case, the time required to achieve saturation is considerably longer than in other scenarios. When $χ$ is between 0 and 1, the evolution of the diffusion kinetics can be interpreted as follows: during the initial diffusion phase, water infiltrates the foam cells. As the cells become saturated, a pseudo-saturation plateau appears. However, the swelling of the foam cells inhibits the maintenance of this pseudo-saturation, thereby enhancing the cells’ capacity to accommodate additional water. This phenomenon initiates a secondary diffusion process, which is illustrated by the subsequent diffusion curve. Notably, in the second stage of diffusion, the time required to reach saturation is approximately half that of the first stage.

The curve profile is consistent with the experimental data on water uptake in polymeric foams during immersion, as reported in several studies, including Ref. 18. These studies reveal a diffusion process that is characterized by distinct stages, which is particularly evident in the water uptake curve of closed-cell polyurethane (PU) foam. In the initial stage, the water diffusion exhibits a linear trend, followed by a slower absorption phase over an extended duration. Subsequently, a new quasi-linear absorption phase emerges, marked by a significantly reduced absorption rate compared to the initial stage. The authors attribute the high diffusivity observed in the first stage to the rapid initial penetration of water into the first open cells at the cut surface of the foam during sample preparation. In studies on the water uptake of closed-cell semi crystalline polyethylene terephthalate foam, although the pseudo-plateau was less pronounced in the water uptake curves, the authors were able to detect water droplets in the closed cells.^29,30

In the second stage, the process is dominated by moisture diffusion along the cell walls, driven by the moisture concentration gradient. The progressively decreasing moisture absorption rate can be attributed to the internal stress of the material, which is induced by water sorption and manifests as elastic swelling. Similar profiles to these experimental curves have been reported in Ref. 17, which examined the water uptake of closed-cell PVC foam (H100) under both immersion and exposure to a humid air environment. Additionally, in Ref. 31, the authors characterized the absorption behavior of closed-cell PVC foams over a period of two and a half years during immersion.

The moisture absorption curve observed during the immersion phase unambiguously presents non-Fickian behavior. This multi-step absorption process can be attributed to the gradual diffusion within the polymeric cell wall structure, in conjunction with the ingress of water into the cellular cavities. Each stage is distinguished by a linear increase in mass, which then reaches a plateau. These apparent saturation plateaus are associated with localized saturation within the layers of the cell walls, which is influenced by internal stresses. Similar findings have been documented in multiple studies that explicitly highlight the non-Fickian diffusivity of polymeric foams, which typically follows a mechanism comprising three distinct stages.^17,18,29,32 Similar multistep behavior in foam cores has been observed by Earl and Shenoi³¹ and has been attributed to the foam structure. Earl and Shenoi state that the cellular structure of the foam allows water uptake by the thin cell walls of the first cell layers at the core surface which quickly saturate and trigger moisture ingress into the cell network by the movement of water within the foam cavities and cell walls. They also suggest that water penetrates only a couple of millimeters from the foam surface.

Modeling of a sandwich structure with composite skins and a polymer foam core

Contextual framework of the problem

This section presents a comprehensive analysis of a sandwich structure comprising facings or skins fabricated from a unidirectional fiber-reinforced polymer-matrix composite, coupled with a core constructed from polymer foam, as illustrated in Figure 2. The sandwich structure, assumed to be of infinite length, is subjected to hygroscopic loading applied to both facings, which initiates a diffusion process whereby water initially penetrates the composite facings before subsequently diffusing into the polymer foam core. It is proposed that the diffusion of water within the composite material is consistent with Fick’s law, which suggests that the diffusion process is independent of external mechanical stresses. Accordingly, the interaction between mechanical stresses and diffusion is deemed to be insignificant for this material system. In contrast, the diffusion mechanisms of water within the polymer foam are governed by a diffusion law that is coupled with mechanical effects, quantitatively represented by equation (36). The primary objective of this modeling study is to systematically evaluate the influence of the foam’s diffusive characteristics on the overall effective diffusive behavior of the sandwich structure. By elucidating these interrelationships, it is aimed to deepen the understanding of how material properties and environmental conditions synergistically impact the structural integrity and performance.

Figure 2.

Schematic of a sandwich structure subjected to hygroscopic loading applied to both facings.

For the composite skin, the hygroscopic problem is expressed by:

\frac{\partial w}{\partial t} = D_{c} \frac{\partial^{2} w_{c}}{\partial z^{2}}

(38)

with

D_{c}

, the diffusion coefficient of the composite.

For the composite, the following initial and boundary conditions are considered:

{\begin{cases} w_{c} (\frac{- b}{2} - a, t) = W_{c} \\ w_{c} (\frac{b}{2} + a, t) = W_{c} \\ w_{c} (z, 0) = 0 \end{cases}

(39)

For the polymer foam core, the hygroscopic problem is expressed by:

{\begin{array}{l} \frac{\partial w}{\partial t} = D_{m} [w_{m} Δ w_{m} + {({\nabla w}_{m})}^{2} - φ χ {\nabla w}_{m}] \\ w_{m} (\frac{- b}{2}, \frac{- b}{2}, t) = α w_{c} (\frac{- b}{2}, \frac{- b}{2}, t) \\ α = \frac{W_{m}}{W_{c}} \end{array} at the foam / composite intreface

(40)

w_m is the moisture content in the foam core, while w_c stands for the moisture content in the composite skin.

α

is the jump parameter, which represents the ratio between the maximum moisture content capacities of the foam and the composite skin.

Wm and Wc are respectively the maximum moisture content capacities of the polymer foam core and the composite skin.

To quantify the influence of the water transport behavior of the foam on the overall water transport characteristics of the sandwich structure, a dimensionless resolution of the diffusion problem within the sandwich structure is performed. In this context, the diffusion equation within the facings can be expressed as follows:

\frac{{\partial w}_{c}}{\partial τ} = ψ \frac{{\partial^{2} w}_{c}}{\partial Z^{2}}

(41)

with:

ψ = \frac{D_{c}}{D_{m}}

(42)

In the case of the foam, the diffusion equation is given by equation (37). It should be noted that the boundary conditions remain the same in both cases.

In this instance, the diffusion problem within the sandwich structure will be expressed as follows:

With respect to the composite facings:

{\begin{cases} \frac{{\partial w}_{c}}{\partial τ} = ψ \frac{{\partial^{2} w}_{c}}{\partial Z^{2}} \\ W_{c} (\frac{- b}{2} - a, t) = 1 \\ W_{c} (\frac{b}{2} + a, t) = 1 \\ w_{c} (Z, 0) = 0 \end{cases}

(43)

whereas, in the foam, the system of equations to be solved satisfies the following conditions:

{\begin{array}{l} \frac{\partial w}{\partial τ} = w_{m} Δ w_{m} + {({\nabla w}_{m})}^{2} - φ χ {\nabla w}_{m} \\ w_{m} (\frac{- b}{2}, \frac{- b}{2}, t) = α w_{c} (\frac{- b}{2}, \frac{- b}{2}, t) \\ α = \frac{W_{m}}{W_{c}} \end{array}

(44)

Figure 3 illustrates the evolution of water diffusion within a sandwich panel comprising a polymer foam core and composite facings. Initially a more rapid diffusion rate is assumed within the foam compared to the composite facings, for a specific value of the diffusion coefficient $ψ = 0.1$ . This assumption implies that diffusion occurs 10 times faster in the foam than in the composite skin, which is consistent with the behavior observed for hydrophobic foams.

Figure 3.

Evolution of the diffusion kinetics in a sandwich structure comprising a polymer foam core and composite facings.

In examining multiple values for the parameter $χ$ in Figure 2, it was found that the profile of the diffusion kinetics remained uniform irrespective of the specific value of $χ$ . In the early stages of the diffusion process, the kinetics curve exhibits a linear behavior, subsequently transitioning to a pseudo-equilibrium phase characterized by a modest saturation level associated with moisture uptake by the facings.

Subsequently, the curve rises sharply, indicating the onset of a second diffusion phase that is more pronounced than the initial phase. This second phase corresponds to diffusion within the polymer foam. This phenomenon has been previously documented in Ref. 33 and in 29 for a sandwich structure consisting of glass fiber composite facings with a vinyl ester matrix and a core made of H200 PVC foam. A comparable profile of water uptake has been observed in Ref. 32 for a sandwich structure with GFRP facings and a PET core.

It can be observed that the moisture content attained in the transient regime is markedly elevated when the values of $χ$ approach zero, suggesting that diffusion is predominantly influenced by mechanical conditions. Furthermore, the ageing time required to reach saturation times is considerably longer, nearly doubling that observed in the foam alone, clearly illustrating the protective effect of the facings on the overall structure.

Conclusion

The objective of this study is to develop a mathematical model for the diffusion of water in polymer foams. The proposed model for these materials is founded upon Darcy’s law. A novel diffusion equation has been formulated, which demonstrates coupling with the mechanical state. By introducing a parameter that represents the ratio between diffusion governed by Darcy’s law and that influenced by the stress associated with the swelling of the diffusing species, it is shown that the diffusion kinetics curve exhibits three phases, one of which corresponds to a plateau between the initial and final stages. The existence of this plateau is contingent upon the values of the diffusion ratio $χ$ . In instances where this parameter has low values, the plateau diminishes thereby causing the diffusion kinetics to converge towards that described by Fick’s law. This phenomenon has been observed innumerous studies examining water absorption by polymer foams. Moreover, the modeling of water diffusion in a sandwich structure comprising polymer matrix composite facings, where diffusion follows Fick’s law, and a polymer foam core, where water diffusion is governed by the novel diffusion law based on Darcy’s equation, reveals a two-step diffusion curve. The initial step of the process involves diffusion within the composite, resulting in a pseudo-plateau, followed by a second step representing the diffusion process within the foam. This phase culminates in a saturation plateau for the sandwich structure. It is noteworthy that the pseudo-plateau, which indicates the conclusion of the diffusion process in the composite facings, is less pronounced at lower values of the parameter.

In future work, classical poroelastic theory will be considered in order to develop a more realistic and more general modeling framework that fully couples fluid flow and skeleton deformation. Specifically, the model presented in this paper may be extended to account for a variable intrinsic permeability dependent on the water content, including possible pore clogging or microstructural changes due to swelling. Sensitivity analyses will then be conducted to assess the influence of such variations on the predicted moisture uptake kinetics and pseudo-plateau behavior.

Footnotes

ORCID iD

Gueribiz Djelloul

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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