Review Article

A Review on Water Vapor Pressure Model for Moisture Permeable Materials Subjected to Rapid Heating PUBLIC ACCESS

[+] Author and Article Information
Liangbiao Chen

Department of Mechanical Engineering,
Lamar University,
Beaumont, TX 77710
e-mail: goodbill2008@gmail.com

Jiang Zhou

Department of Mechanical Engineering,
Lamar University,
Beaumont, TX 77710
e-mail: zhoujx@lamar.edu

Hsing-Wei Chu

Department of Mechanical Engineering,
Lamar University,
Beaumont, TX 77710
e-mail: chuhw@lamar.edu

Guoqi Zhang

Department of Microelectronics,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands
e-mail: G.Q.Zhang@tudelft.nl

Xuejun Fan

Department of Mechanical Engineering,
Lamar University,
Beaumont, TX 77710
e-mail: xuejun.fan@lamar.edu

1Corresponding authors.

Manuscript received September 6, 2017; final manuscript received March 7, 2018; published online April 23, 2018. Assoc. Editor: Rui Huang.

Appl. Mech. Rev 70(2), 020803 (Apr 23, 2018) (16 pages) Paper No: AMR-17-1062; doi: 10.1115/1.4039557 History: Received September 06, 2017; Revised March 07, 2018

This paper presents a comprehensive review and comparison of different theories and models for water vapor pressure under rapid heating in moisture permeable materials, such as polymers or polymer composites. Numerous studies have been conducted, predominately in microelectronics packaging community, to obtain the understanding of vapor pressure evolution during soldering reflow for encapsulated moisture. Henry's law-based models are introduced first. We have shown that various models can be unified to a general form of solution. Two key parameters are identified for determining vapor pressure: the initial relative humidity and the net heat of solution. For materials with nonlinear sorption isotherm, the analytical solutions for maximum vapor pressure are presented. The predicted vapor pressure, using either linear sorption isotherm (Henry's law) or nonlinear sorption isotherm, can be greater than the saturated water vapor pressure. Such an “unphysical” pressure solution needs to be further studied. The predicted maximum vapor pressure is proportional to the initial relative humidity, implying the history dependence. Furthermore, a micromechanics-based vapor pressure model is introduced, in which the vapor pressure depends on the state of moisture in voids. It is found that the maximum vapor pressure stays at the saturated vapor pressure provided that the moisture is in the mixed liquid/vapor phase in voids. And, the vapor pressure depends only on the current state of moisture condition. These results are contradictory to the model predictions with sorption isotherm theories. The capillary effects are taken into consideration for the vapor pressure model using micromechanics approach.

Polymers or polymer composites play a significant role in the advancement of technology in microelectronics, deformable electronics, and microelectromechanical system [1,2]. However, most polymers and their composites are inevitable to moisture absorption, which will alter physical properties and greatly compromise material performance [36], such as diminished glass transition temperature Tg [7], volumetric swelling [811], loss in modulus and strength [1215], and interfacial adhesion degradation [1621]. Moreover, the “encapsulated” moisture in material may cause popcorn blistering failure subjected to the increasingly hostile environment, for example, to a higher reflow temperature with the new lead-free solders and nanoparticle sintering process [2228]. Water vapor pressure grows exponentially with temperature and thus becomes substantially high at a temperature of 300 °C. Extremely high steam pressure is generated due to the phase change of the encapsulated moisture at elevated temperature. As an extreme case, the softened material is further damaged by water vapor when a macroscopic crack develops and propagates to the exterior. An audible sound will be produced with a sudden release of water vapor. This phenomenon is the so-called popcorn failures and well observed in plastic integrated circuit packages [16,17,2326,28].

Figure 1 shows an example of polymer adhesive film rupture in a stacked-chip microelectronic package during reflow soldering process after moisture absorption, which followed an industrial standardized moisture sensitivity test [28]. Massive cohesive film ruptures were observed at the bottom layer of polymeric thin films that are used for chip bonding to the substrate (Fig. 1(a)). Figure 1(b) reveals the details of the damage of film from a top-down microscopic view of the film.

Vapor pressure-induced failures were also reported for graphite/polyimide laminates used in aerospace applications [29,30]. As shown in Fig. 2, both dry and moisture-saturated specimens were tested under thermal spikes from room temperature to 310 °C. The recorded strain-temperature data (Fig. 2(a)) indicate that moisture-saturated specimens experienced much larger deformation during heating in comparison to the dry ones, with an abrupt increase of strain occurring at 300 °C. The microscopy in Fig. 2(b) shows that many spherical microvoids were generated by high water vapor pressure, which grew and filled the entire midplane layers of the laminate at 310 °C.

Encapsulated moisture is a unique form of mechanical load, which evolves and is coupled with moisture diffusion, water vapor flow, phase change, heat transfer, and more significantly, material's microscopic degradation [31,32]. Over the last two decades, numerous studies have been made, predominately in microelectronics packaging community, to gain the fundamental understanding associated with moisture absorption and vapor pressure evolution [17,3340]. One of the pioneering works in Ref. [25] treated the moisture diffusion by heat conduction equations. Then, stresses induced by moisture and temperature are modeled in the combined form for a packaging structure subjected to encapsulation and reflow soldering. Interface gaps are explicitly modeled by rigorous contact mechanics and interfacial mechanics. Time-dependent vapor pressure is also modeled. Yet, vapor pressure-induced failures still persist with today's new product development in three-dimensional integrated circuit technology. Many new materials, such as super-hydrophobic coating, are effective in protecting device/component from water attack, but they have shown negligible resistance to ambient humidity environment, and thus no resistance to blistering failure during rapid heating [5,41].

Vapor pressure evolution during rapid heating is presumed to be a dominant driving force for failure. Different theories and models have been developed to characterize vapor pressure in relation with moisture diffusion and microscopic feature. However, a variety of formulations for vapor pressure yield different results, and no consensus has been reached on the assumption and results obtained in each model. The aim of this paper is to provide a comprehensive review and comparison of different theories and models for water vapor pressure in moisture permeable materials. We recognize that the existing theories can be grouped into two categories: Henry's law-based and micromechanics-based models. For the former, by assuming an existence of a “macrocavity” in material, or introducing porosity, several theories and models using different assumptions and approaches are present in the literature [3436,42,43], which give different results. This paper re-examines the existing theoretical solutions based on a generalized Henry's law approach. The models that consider the effect of the dynamics of vapor flow are also discussed [37,44,45]. Furthermore, the generalized frameworks to consider non-Henry sorption isotherms for vapor pressure prediction are developed in this paper.

Different from Henry' law-based vapor pressure models, micromechanics-based model assumes that moisture resides in micro/nanosized pores or voids, which are responsible for vapor pressure development. This paper will review the two-phase micromechanics-based vapor pressure model [3840] and the semi-infinite media pressure model [46], and the results are compared to Henry's law-based models. The paper further develops a modified micromechanics-based model to consider the capillary effect that could significantly lower the calculated water vapor pressure.

The paper is organized as follows: In Sec. 2, various analytical solutions based on Henry's law, using one-dimensional (1D) problem, are presented and compared. A unified formulation is developed, with which the history effect of initial relative humidity and the presence of unphysical pressure are discussed. Section 3 reviews two porous media models that consider water vapor flow in water vapor pressure analysis. Section 4 proposes two generalized frameworks to consider non-Henry moisture sorption isotherms. Section 5 reviews and compares micromechanics-based models against the isotherm-based models. A modified model is developed in Sec. 6 to consider the capillary effect on water vapor pressure in microvoids. Concluding remarks are drawn at the end.

Shirley's Solution.

Consider a moisture permeable material in equilibrium with a humid ambient with ambient vapor pressure, Pamb. Henry's law establishes a relationship between the ambient vapor pressure and the moisture concentration in solid matrix, as follows [4749]: Display Formula


where Csat is the saturated moisture content in the solid, S is Henry solubility that is dependent on temperature T, S is a prefactor, ΔHs is the heat of solution, and R is universal gas constant.

Henry's law in Eq. (1) can also be written as Display Formula


where Psat is saturated water vapor pressure, kH=(SPsat)1 is called Henry constant, and RHamb is ambient relative humidity.

To apply Henry's law for water vapor pressure analysis, a cavity is typically assumed at the place of interest in material, thus Display Formula


where Pcav is cavity pressure and Ccav is the moisture concentration at cavity surface.

Shirley [35] conducted an analytical study on a 1D cavity pressure problem as depicted in Fig. 3. The 1D problem represents a typical and simplified microelectronic assembly of an epoxy molding compound and lead frame. The compound can absorb a significant amount of moisture, while the lead frame (made of copper in most of the applications) is water-impermeable. Between the compound and the lead frame, there is a cavity to represent a crack or void along the interface. The system is initially in equilibrium with the preconditioning ambient with RH0 and T0, and then subjected to a reflow soldering process simulated by a step change from T0 to the reflow temperature T1. The ambient humidity is also changed to RH1 (<RH0) to represent a drying process. To analyze this problem, two main assumptions are made in Ref. [35]: (1) the material and cavity are rigid and (2) the heating is uniform across the whole model.

To describe moisture diffusion, the Fick's second law was used Display Formula


where C is moisture concentration at location x and time t, D1 = D(T1) is diffusion coefficient at reflow temperature T1, and w is thickness. Water diffusivity is often described by Arrhenius equation as D(T)=Dexp(ED/RT), where D is a prefactor and ED is activity energy [35,36].

The initial condition is written as Display Formula


where S0 = S(T0) and Psat0 is Psat at T0. The boundary condition at material surface x = 0 is Display Formula


where S1 = S(T1) and Psat1 is Psat at T1.

At cavity surface x = w, the step change of temperature at t = 0 leads to a jump in cavity pressure according to Boyle's law, as Pcav (t = 0+) = P0T1/T0 (P0 is the initial cavity pressure). Thus, there is sudden change in moisture concentration at the cavity interface x = wDisplay Formula

which serves as one boundary condition. In addition, the mass of moisture must be balanced at the cavity interface, as Display Formula

where l is the cavity width, Rw is the gas constant for water molecules, and water vapor density ρcav=Pcav/(RwT)=Ccav/(RwTS1) based on the ideal gas law and Henry's law.

Equations (4)(8) can be solved by the superposition of the existing solutions in VanSant's report [50], resulting in the Shirley's solution for moisture concentration, as [35] Display Formula




where x¯=x/w, t¯=D(T1)t/w2, h=RwT1S1w/l, and γn is the positive roots of γntan(γn)=h.

Cavity pressure is then evaluated by substituting CShirley(1.0,t¯,h) into Eq. (3) or Henry's law, as Display Formula

which is the Shirley's solution to cavity pressure.

For demonstration, the calculated cavity pressure is plotted in Fig. 4 for different values of h with the material properties given in Ref. [35]. In general, cavity pressure becomes higher as h increases (a larger h means a greater ratio between the material thickness and cavity size). There exists a maximum pressure Pmax (<Psat1) for the case of h →∞, which occurs when the cavity size approaches zero. The “zero-size” cavity solution was found by taking the limit of Eq. (9) with h →∞, as [35] Display Formula


where γn = (2n + 1)π/2 and Pcav0Shirley=Ccav0Shirley/S1. Shirley [35] showed that the zero-cavity solution in Eq. (11) resembles the widely used solution of 1D diffusion models where no moisture transfer at cavity surface was considered.

The maximum vapor pressure Pmax can be obtained by using Eq. (11) at short times (t¯ ≪ 1), where the summation of the series becomes 1/2 and Ccav0ShirleyC0, yielding [35] Display Formula


Equation (12) can be rewritten by approximating Psat using the Clausius–Clapeyron (CC) equation [35,36,51] Display Formula

with P = 3.8208 × 1010 Pa and ΔHvap40.2×103 J/mol by fitting the steam table in Ref. [52]. Substituting Eq. (13) into Eq. (12) yields Display Formula

Equation (14) indicates that Pmax is proportional to RH0, and weakly dependent on T0 if the difference between ΔHs and ΔHvap is small. For a typical epoxy molding compound [35], ΔHs=38.7 kJ/mol, so ΔHsΔHvap1.5 kJ/mol. The results in Fig. 4 showed that the resulting Pmax is significantly lower than Psat1, and more accurately lower than the product of Psat1 and RH0. This, however, could be directly explained from Eq. (14). More discussions will be provided in Sec. 2.6.

Hui's Solution.

Hui et al. [36] conducted another analytical study on a similar problem illustrated in Fig. 5. An isolated crack-like cavity was assumed in the middle of a moisture permeable material. Due to the symmetry in geometry, the 1D problem can be stated exactly the same as the problem studied in Ref. [36]. Similar assumptions are also adopted, including the neglect of solid deformation and uncoupling with thermal diffusion. However, the methods used in the two works are different.

The same diffusion equation as stated in Eq. (4) was normalized and solved in Ref. [36]. However, an integral form of the boundary condition at x = w is used Display Formula


where the normalized variables are: g = 1 − C/C0, X=dω1x/l, α=dω1w/l, τ=d2ω12D1t/l2, d=RwC0T0/Psat0, and ω1=T1Psat0/(T0Psat1). Equation (15), in fact, is an integration form of Eqs. (7) and (8).

Applying Laplace transformation and residual theorem, Hui et al. [36] obtained the exact solution to normalized concentration g at cavity surface, as Display Formula


where g0=1C1/C0 and un is the roots of utan(u)=α.

Then the cavity pressure was calculated by utilizing Eq. (3) based on Henry's law, as Display Formula


Equation (17), however, is only valid for kH(T0)=kH(T1) and RH0 = 100% [36,42]. To gain an insight into the assumption, we rewrite kH by replacing Psat with the CC equation in Eq. (13), obtaining Display Formula


where ΔHs*=ΔHsΔHvap called the net isosteric heat of sorption. Equation (18) indicates that the assumption kH(T0)=kH(T1) is used in Hui's solution corresponding to the case of ΔHs*=0 kJ/mol.

The form of Hui's solution is apparently different from Shirley's solution given in Eq. (10). In addition, parameters in Ref. [36] were set as RH0 = 100% and ΔHs* = 0 kJ/mol, whereas in Ref. [35], RH0 = 85% and only ΔHs* = −1.5 kJ/mol was applied to represent the property of typical epoxy molding compound. In the following, we provide a generalized and unified solution, which is applicable to any choice of material parameters.

Chen's Solution.

Chen et al. [42] presented a general nonlinear equation with temperature-dependent Henry's law for moisture sorption. For temperature-dependent kH given by Eq. (18) and varying RH0, Chen's solution can be obtained as follows: Display Formula


where g is obtained from Eq. (16) and CcavChen(τ)=C0(1g(X=α,τ)).

In the case of the infinitely large geometry (i.e., w → ∞), the solution of Eq. (19) can be reduced to [36] Display Formula


For another special case of the zero-size cavity, when a different set of normalized variables are used: x¯=x/w and t¯=D(T1)t/w2, the solution to g in Eq. (20) becomes Display Formula


where γn = (2n + 1)π/2. Equation (21) indeed resembles with Eq. (11), yielding Ccav0Chen=Ccav0Shirley. The corresponding pressure solution for zero-size cavity is Display Formula


It is worth noting that the Chen's solution in Eq. (19) and its special solutions in Eqs. (20) and (22) have the same mathematical form except for the difference in the concentration term.

As a comparative study, Fig. 6 plots the numerical results of Shirley's solution, Hui's solution, and Chen's solution with different material properties. With RH0 = 100% and ΔHs*=1.5 kJ/mol, Chen's solution matches Shirley's solution exactly. With RH0 = 100% and ΔHs=40.2 or ΔHs* = 0.0 kJ/mol, Chen's solution matches exactly with Hui's solution that assumes temperature-independent kH. Therefore, Chen's solution provides a unified and general solution for vapor pressure calculation under the assumption of linear and temperature-dependent Henry's law. Discussions of additional important results obtained from Chen's solution will be described in Sec. 2.6.

Wong's Equation.

Wong et al. [34] reported another formula to calculate the water vapor pressure in a cavity or delamination, as Display Formula


where Wd=Ccav/Csat, where Csat is saturated moisture concentration for a given RH. Equation (23) assumes that there is no moisture diffusion into the nonexpendable cavity [34]. Therefore, it applies that Ccav=Ccav0Shirley=Ccav0Chen and Csat=C0 for the 1D problem. Dividing Eq. (23) with Psat1 yields Display Formula


Equation (24) indicates that Wong's equation is a special case of Chen's solution by assuming zero-size cavity.

Bhattacharyya's Equation.

Bhattacharyya et al. [43] proposed an empirical equation based on thermodynamics and the assumption of no moisture diffusion during the heating, as Display Formula


where ρg(T)=Psat(T)/RwT. Notice that RwT0RH0ρg(T0)=Psat0RH0 based on ideal gas law and Henry's law, so Eq. (25) can be rewritten as Display Formula


Therefore, Bhattacharyya's equation becomes the solution of PmaxChen if ΔHs=2740R=22.78 kJ/mol or ΔHs* ≈ −17.4 kJ/mol.


Chen's solutions in Eqs. (19), (20), and (22) have the following unified form: Display Formula


where Ccav is moisture concentration in the cavity (e.g., for a zero-size cavity, Ccav=Ccav0). Equation (27) can be considered as a unified equation of Shirley's solution, Hui's solution, Wong's equation, and Bhattacharyya's equation (after certain transformations). The unified solution for the maximum pressure can be obtained by letting Ccav=C0 in Eq. (27), as [42] Display Formula


Equation (28) can be derived by dividing PmaxShirley in Eq. (10) with Psat1, indicating that they represent the same solutions. Unlike Shirley's equation in Eq. (10), Chen's or unified solution in Eq. (28) gives an explicit relation between Pmax and Psat1. Some important implications of Eq. (28) are discussed as follows.

Effect of RH0.

To demonstrate the effect of RH0, Fig. 7 plots the normalized cavity pressure, Pcav/Psat1, versus normalized time, τ, under different various values of parameter, α (see definitions of τ and α in Eq. (15)). Generally, the pressure experiences a rapid increase at the early stage of heating, but gradually decreases due to moisture loss. The maximum pressure increases as α increases (or thickness increases), reaching the solution for w = ∞ given in Eq. (20) when α > 100. However, Fig. 7 also shows that the normalized pressure is significantly lower than RH0 (a dashed horizontal line in the figure) even for α = . This is considered as the capping effect of RH0, meaning that the maximum vapor pressure could not surpass the product of saturated vapor pressure and RH0 for the given parameter settings (i.e., ΔHs*=1.5 kJ/mol). The effect of different ΔHs* is further discussed in the following.

Effect of ΔHs*.

Figure 8 plots the normalized cavity pressure for negative, zero, and positive values of ΔHs*. As ΔHs* increases from a negative value to zero, the normalized pressure could approach to but never exceed RH0. However, when ΔHs* is a positive value (i.e., ΔHs*= 3.0 kJ/mol), the normalized pressure surpasses RH0 and may become even greater than 1.0 after τ > 5.0. This means that the cavity pressure is higher than Psat1. Although the saturated vapor pressure represents the maximum water vapor pressure at a particular temperature, with a positive value of ΔHs*, the predicted vapor pressure can be greater than the saturated water vapor pressure.

Mathematically, the following three different scenarios for a heating process (T1 > T0) can be obtained from Eq. (28), as: Display Formula


Literature shows that ΔHs* could range from −5.0 kJ/mol for microcrystalline [49,53] and 6.0 kJ/mol for certain epoxy network [54]. Therefore, all the three scenarios in Eq. (29) can occur. Bhattacharyya's equation even adopted an extreme case with ΔHs* ≈ −17.4 kJ/mol.

Figure 9 plots Pmax/Psat1 versus a wide range of ΔHs* from −5.0 to 6.0 kJ/mol. Two reflow temperatures (T1) are compared, which shows that the dependence or sensitivity of cavity pressure on ΔHs* increases as T1 increases. Generally, when ΔHs*0, cavity pressure is restricted by RH0 (=85%), or PmaxPsat1·RH0. The results that were given in Shirley's solution, Hui's solution, and Bhattacharyya's equation fall into this category. However, when ΔHs*>0, it becomes possible to exceed the saturated vapor pressure. For example, if ΔHs* = 2.0 kJ/mol, the maximum cavity pressure reaches saturated vapor pressure Psat1 at T1 = 215 °C even RH0 < 100%. Above ΔHs* (=2.0 kJ/mol), there exists unphysical pressure greater than Psat1. The findings of capping effect of RH0, as well as unphysical pressure, pose a great challenge for using the water vapor pressure models based on the linear sorption isotherm or Henry's law.

Convection-Only Model.

Water vapor pressure models reviewed in Sec. 2 only consider moisture transport through diffusion during rapid heating. At elevated temperature, however, convective vapor flow may become dominant over diffusion for a porous media. Consider a special case where vapor flow is the only mechanism for moisture transport at higher temperature, a “convection-only” model is then developed to describe the vapor flow during rapid heating [37] Display Formula


where Pv is the vapor pressure (in pores), ϕ is the porosity, KH is the Henry solubility constant (=1/kH when ΔHs* = 0 kJ/mol), κ is the Darcy's vapor permeability, μ is the vapor viscosity, and T˙ is the heating rate. Since the porosity is introduced, vapor pressure at any location in media may be calculated. The above formulation, however, does not consider material deformation, which is consistent with the model assumptions in Sec. 2. Equation (30) may be solved numerically using finite difference method.

The convection-only model can be applied to the 1D problem depicted in Fig. 5. The equivalent porosity is calculated as ϕeq = l/(w + l), and vapor permeability is set to be κ = 1.1 × 10−20 m2 based on Ref. [44]. The Henry solubility constant is calculated as KH = 1/kH = 20.85 kg/m3. Since KH or kH is temperature-independent, it is valid that ΔHs*=0 kJ/mol. The stepping temperature change is approximated with a high heating rate (i.e., T˙ = 10,000 K/s). Figure 10 compares Chen's analytical solution and numerical solution of the convection-only model with l = 0.01 cm and w = 0.2 cm (i.e., ϕeq = 0.5%). Different results are obtained from the two models, mainly due to the difference of transport mechanism between moisture diffusion and vapor convection. The maximum pressure based on the convection-only model is slightly lower than Chen's solution. The similarity between the two different models is due to the use of the same Henry's law.

Convection-Diffusion Model.

A more general porous media model has been developed by Chen et al. [44] to consider both moisture diffusion in the bulk material and vapor flow in the pores, as schematically shown in Fig. 11. In this model, water is partitioned into two different states: one is in vapor form that flows through pore network, and the other is in liquid form that is dissolved into a solid matrix. Considering both the vapor convection and moisture diffusion mechanism, the final governing equation in solving pore vapor pressure is [44] Display Formula


where Pv is the vapor pressure (in pores), ϕ is the porosity, KH is the Henry solubility constant (=1/kH when ΔHs* = 0 kJ/mol), κ is the Darcy's vapor permeability, μ is the vapor viscosity, D is the diffusivity, and T˙ is the heating rate.

The convection-diffusion (CD) model in Eq. (31) was applied to study water vapor pressure within the sample of epoxy molding compound in Ref. [44]. The problem is similar to the 1D problem in Fig. 5 (without considering the “cavity” in the middle). The material is initially saturated at RH = 85% at T = 300 K and is heated to 600 K at a rate of 1 K/s. Figure 12 compares water vapor pressure results for three different cases: convection-diffusion, convection only, and diffusion only, respectively. The heating process stops at t = 300 s, so there exists a turning point for all the curves of vapor pressure results. The diffusion-only model gives a pressure up to two times higher than that given by the CD model, while the convection-only model predicts a pore pressure lower than the CD model. This is mainly because the diffusion-only model ignores the vapor flow at high temperatures and results in a slower desorption for the buildup of pore pressure, whereas the convection-only model overestimates the vapor flow by using a higher vapor permeability κ(=1.1 × 10−20 m2) than CD model (κ = 6.9 × 10−21 m2) and results in faster desorption and lower vapor pressure [44]. Generally, the CD model or convection-only model predicts a lower pressure because the interconnected pore network within porous media provides another fast path for moisture transport in the form of vapor flow.

To consider temperature-dependent KH in CD model, Eq. (31) has been further extended to as [45] Display Formula


where KH = KH(T). With this model, the moisture “overshooting” at the material interface, as reported in Refs. [28] and [40], has been simulated in Ref. [45]. The numerical studies in Ref. [45] also showed that the vapor pressure predicted from the model in Eq. (32) correlated well with experimental observations of cohesive failures of polymer thin films in a stacked-chip microelectronic package.

Nonlinear Moisture Sorption Models.

Vapor pressure models in previous Secs. 2 and 3 are based on Henry's law, which assumes a linear and temperature-dependent relationship between saturated moisture concentration Csat and ambient relative humidity RHamb. However, moisture permeable material may exhibit nonlinear moisture sorption behavior. As shown in Fig. 13, there are five typical types of moisture absorption curves to describe the relationship between Csat and ambient water activity (which is denoted as aw,eq and is equal to RHamb). These sorption isotherms include Henry sorption, nonlinear sorption for hydrophobic materials [55,56], Langmuir sorption [57,58], nonlinear sorption for high water interaction [59], and sigmoidal sorption curve [60].

Table 1 lists some existing mathematical models to describe the moisture sorption isotherms (the temperature effect is not included in those models). In the following, two generalized frameworks are developed to evaluate vapor pressure with the consideration of these nonlinear functions.

A Concentration-Based Framework.

Let the moisture diffusion within solid matrix be considered by Fick's second law as given in Eq. (4). A general function fC is applied at the material interface and cavity surface, as Csat = fC (aw,eq, T) to describe the nonlinear isotherm. The cavity pressure then can be evaluated by Display Formula


where aw,cav=fa(Ccav,T) and fa=fC1. Accordingly, the initial and boundary conditions are changed to Display Formula


where the values of aw,eq for initial and reflow conditions are equal to RH0 and RH1, respectively.

It is interesting to notice that Eq. (34) is independent of the function fC for the special case of zero-size cavity, as long as the same C0 and C1 are used. In other words, the same solution to Ccav0 as given in Eq. (11) would apply to all types of sorption models. However, the cavity pressure Pcav0 will still depend on the sorption function fC.

The corresponding maximum pressure can be obtained with Ccav0=C0, as Display Formula


To compare with the solutions based on Henry's law, one can transform Eq. (35) through the CC relation in the below equation [6467]: Display Formula


where aw0=fa(C0,T0)=RH0, aw1=fa(C0,T1), Qst*=QstΔHvap, and Qst is the isosteric heat of sorption. Note that Qst*=ΔHsol* for Henry's law. Subsisting Eq. (36) into Eq. (35) yields Pmaxconc, the maximum pressure based on the concentration-based framework Display Formula


Equation (37), in fact, results in the same maximum pressure as Eq. (28) based on Henry's law. This indicates that the maximum pressure would be independent of the sorption models. As a result, the effects of RH0 dependence and unphysical pressure discussed in Sec. 2.6 are applicable to both Henry's law and non-Henry sorption models.

As an example, Fig. 14 plots three different sorption isotherms: one is based on Henry's law and the other two are based on Ferro Fontan (FF) model (see Table 1). They cross at one point (aw,eq = 0.85, Csat = 9.781) for T = 85 °C, and another point (aw,eq = 0.71, Csat = 9.781) for T = 215 °C. Model parameters for low-temperature are given, and the high-temperature isotherms are derived using the CC relation in Eq. (36) with Qst= 38.7 kJ/mol or Qst*=1.5 kJ/mol. The three different isotherms can be applied to calculate water vapor pressure by solving Eqs. (4), (33), and (34). The results are compared in Fig. 15 where both the moisture concentration and cavity pressure are plotted for a zero-size cavity problem according to Ref. [35]. It can be seen that moisture concentration deviates greatly due to the difference in the sorption isotherm, but the cavity pressure is the same at the early stage and only deviates slightly at a later time. The maximum pressure is lower than Psat1 at 215 °C due to the use of a negative Qst*, which is consistent with Eq. (37).

An Activity-Based Framework.

The vapor pressure models previously discussed utilize concentration-based diffusion theory described by Fick's second law. A more fundamental equation for moisture diffusion is based on the gradient of chemical potential, as [68,69] Display Formula


where J is the water flux, B is the mobility of water molecules, and μw is the water chemical potential that can be written as [68] Display Formula


where μw0 is reference chemical potential.

Introducing the generalized solubility K as [36,68,70] Display Formula


and assuming a uniform temperature (or T = 0), one can obtain a general, activity-based diffusion theory, as [70] Display Formula

in which moisture diffusivity D = BRT. Because water activity is a state variable and always continuous, the activity-based model in Eq. (41) does not require normalization at the material interface. This is convenient for studying multimaterial systems where the concentration-based model must be normalized due to the concentration discontinuity. It was also found that Eq. (41) returns to Fick's second law under the assumption of Henry's law [70].

The generalized solubility K in Eq. (41) is a material property that can be evaluated from the sorption function fc, as Display Formula


where function fc can be given by Table 1 and the CC relation in Eq. (36). With K determined, the initial and boundary conditions for Eq. (41) are changed to Display Formula

which may be solved by numerical approaches. Once the water activity is solved, cavity pressure is readily obtained by using the relationship Pcav = Psat·aw,cav.

To derive the maximum vapor pressure, consider a sufficiently short period of time when the moisture concentration at the cavity remains to be C0. In this case, the following relationship holds at cavity surface: Display Formula


where aw0 and aw1 are the water activity corresponding to C0 at T0 and T1, respectively. Applying Eq. (36) to Eq. (44) leads to Display Formula


The maximum pressure is corresponding to aw1 and can be written as Display Formula


Equation (46) shows that the two frameworks proposed in this paper give a consistent solution and Pmax is independent of moisture sorption isotherms.

Micromechanics-based vapor pressure models are reviewed as follows, including the two-phase micromechanics-based model, the one-phase semi-infinite medium model, and other empirical models. They are also compared with Henry's law-based models reviewed in Sec. 3.

Two-Phase Micromechanics-Based Model.

Fan et al. [39] developed a micromechanics-based vapor pressure model, or Fan's model, to calculate water vapor pressure in moisture permeable materials. A representative elementary volume (REV) is taken from the material, as shown in Fig. 16. It assumes that the total volume of the studied REV can be divided into free volume or microvoids, where moisture resides, and the occupied volume for the solid only. The porosity ϕ in REV defines the collective free volume or microvoids. In the micromechanics-based model, moisture only resides in microvoids with two possible states or phases: a pure vapor phase or a mixed phase with both liquid and vapor, as shown in Fig. 16.

Introduce the apparent moisture density ρa, as Display Formula


Phase transition of water vapor occurs when ρa reaches the saturated vapor density, or ρg=Psat/(RwT). If the apparent density ρa is smaller than ρg, only the vapor phase exists; otherwise, the moisture is in a mixed phase. Therefore, water vapor pressure in a microvoid can be evaluated by Display Formula


To use this model, C(x,t) is first solved based on applicable diffusion theory (i.e., Fickian or non-Fickian models reviewed in Ref. [71]). Then water vapor pressure can be evaluated with Eq. (48). Applications and more discussions on the micromechanics-based models can be found in Refs. [3840].

To compare this model with Henry's law-based models, a cavity problem defined in Fig. 3 is revisited. The equivalent porosity is computed as ϕeq=l/(w+l) similar to the use of the convection-only model in Sec. 3. The results are given in Fig. 17, where Chen's unified solution in Eq. (27) is also plotted for comparison. It can be seen that Fan's model predicts a pressure staying at the saturated vapor pressure for the most time of the heating process. However, Chen's analytical solution, which is based on temperature-dependent Henry's law with a negative ΔHs*(=−1.5 kJ/mol), gives much lower vapor prediction.

Alpern's One-Phase Model.

Alpern et al. [46] proposed a model for calculating the water vapor pressure in a semi-infinite body with a cavity. It assumes that all the absorbed moisture is driven into the cavity during the diffusion process. An analytical equation was derived to calculate the amount of moisture entering into the cavity [46] Display Formula


where A is the body surface. It also assumed that moisture accumulated in the cavity is always in vapor form and never condenses. Applying the ideal gas law yields Display Formula


where V is the volume of the cavity. Using the elastic plate theory for both rectangular and cylindrical semi-infinite bodies, the relationship between the cavity pressure and V can be obtained [46]. Alpern's model is basically a one-phase model without considering the phase change of water vapor. For the 1D problem, Alpern's equation yields the following equation for cavity pressure: Display Formula


Figure 18 compares Alpern's solution with Fan's model and Henry's law-based solution. It can be seen that Alpern's one-phase model yields much higher pressure than the other two solutions when the cavity size is small, far beyond the saturated vapor pressure.

Sawada's Model.

Sawada et al. [72] proposed the following empirical equation for calculating water vapor pressure in cavity/delamination: Display Formula


where G1 is an empirical constant. Similar empirical equations were also used in other works [73,74]. The use of Sawada's and other similar equations, however, is very limited because G1 is usually unknown and is difficult to determine through experiments.

Consideration of Capillary Effect in Microvoids.

According to Kelvin's equation, the saturation point may be altered by capillary effect for small-size voids [7577] Display Formula


where Psat is modified vapor pressure at phase change, σ is the surface tension of water, Vm is the water molar volume (0.018 L/mol), θ is the surface wetting angle for the pore surface, and r is pore radius. Equation (53) assumes a cylindrical shape of the pore. For a perfect wetting condition, θ = 0 deg.

Based on Eq. (53), the corresponding saturated vapor density for a microvoid with a size of r should be modified as Display Formula


A modified Fan's model thus can be obtained Display Formula


Case Study.

To investigate capillary effect, various void sizes and water surface tension coefficients at different temperatures are considered. The pore radius ranges from 0.1 to 100 nm based on typical values for polymeric materials [78]. Both constant surface tension (σrm = 0.072 N/m [7981]) and temperature-dependent surface tension are considered. For the latter, the temperature-dependent σ can be evaluated by the following linear equation: Display Formula

which simulates about 85% decrease of surface tension at 600 K according to the atomic studies in Ref. [82]. Figure 19 shows the ratio of Psat/Psat under various temperatures. Perfect wetting is considered with θ = 0 deg. When a constant surface tension is used, the capillary effect on the ratio of Psat/Psat is significant as shown in Fig. 19(a). On the other hand, for temperature-dependent σ described by Eq. (56), the capillary effect becomes less significant at high temperatures as seen from Fig. 19(b).

Figure 20 presents the vapor pressure results of modified Fan's model for the 1D problem of a step change from 85 °C to 215 °C. The solution of moisture concentration is obtained on an equivalent porosity of 5% using the diffusion parameters given in Ref. [35]. Various void sizes are compared, with r = 0.5 nm, 1.0 nm, 10 nm, and ∞. It can be proved that the original Fan's solution is obtained at r = ∞ when capillary effect vanishes. However, when a very small void size is considered (e.g., 0.5 nm), the maximum vapor pressure drops by 75% for a constant surface tension and 50% for temperature-dependent surface tension. Like original Fan's model, the unphysical pressure will not occur in the modified Fan's model with consideration of phase change and capillary effects.

High vapor pressure is induced by the encapsulated moisture in moisture permeable material. The vapor pressure evolves and is coupled with moisture transport, water vapor flow, phase change, heat transfer, material's deformation, and material's microscopic change. In this paper, we present a comprehensive review of various theories and models to evaluate water vapor pressure subjected to rapid heating. In Secs. 2 and 3, Henry's law, a linear and temperature-dependent equation, is applied. Such a treatment establishes a relationship between the pressure in cavity or pores (or pore network) and the local moisture concentration in solid matrix. Even though various models provide various forms of analytical or numerical solutions, which give different results, those solutions can be unified to a general form of solution. With specific assumptions and parameter settings, the generalized solution can return to each specific solution in each model. Two key parameters appear in the unified solution for determining the magnitude of vapor pressure in cavity: the initial relative humidity RH0, in which the material is preconditioned and saturated, and ΔHs*, the net solution heat, a material's property, respectively. Although the saturated vapor pressure represents the maximum water vapor pressure at a particular temperature, with a positive value of ΔHs*, the predicted vapor pressure can be greater than the saturated water vapor pressure. Such an unphysical pressure solution needs to be further studied in the future. Moreover, with a nonpositive value of ΔHs*, the predicted maximum vapor pressure is equal to or less than the product of the saturated vapor pressure and RH0 (≤ 1), which means that the vapor pressure is less than the saturated vapor pressure, and is directly affected by the initial relative humidity. This implies that the vapor pressure in cavity will depend on the history of moisture absorption—what initial relative humidity is applied before rapid heating, not only the present state of moisture condition.

In porous media with interconnected pore network, vapor flow provides another path for moisture transport. In this paper, two models considering vapor flow are presented, which are the so-called convection-only model and “convection-diffusion” model. For both models, water is partitioned into two different states: one is in vapor form that flows through pore network, and the other is liquid form that is dissolved into solid matrix. Henry's law is applied to both external boundary and the interface between pores and solid matrix. However, the convection-only model considers vapor flow as dominant in moisture transport; thus, diffusion is neglected. As a result of vapor flow, the predicted vapor pressure in pores or cavity by both models is generally lower than that by the models given in Sec. 2.

For materials with nonlinear sorption isotherms, two generalized frameworks: namely, concentration-based and activity-based approaches, respectively, are presented in this paper. The analytical solutions for the maximum vapor pressure are derived under both frameworks. It is found that the maximum pressure is mainly governed by the net isosteric heat of sorption Qst* and does not depend on the types of sorption isotherms (Qst* = ΔHs* for Henry's law).

For vapor pressure models using either Henry's law or nonlinear sorption isotherm, the predicted vapor pressure can potentially be greater than the saturated water vapor pressure. Moreover, the predicted maximum vapor pressure is always proportional to the initial relative humidity, implying that the vapor pressure will depend on the history of moisture absorption.

For micromechanics-based vapor pressure model, moisture is assumed to reside in microvoids or free volumes in material. The magnitude of the vapor pressure depends on the state or phase of moisture in voids. When moisture is in vapor phase, the vapor pressure follows the ideal gas law. When moisture is in the mixed liquid/vapor phase, vapor pressure stays at the saturated vapor pressure (Sec. 5). Furthermore, this model has been extended to take the capillary effects into consideration. It is found that considering the capillary effects will lower the predicted vapor pressure. Two completely different results are derived from the micromechanics-based model: (1) the maximum vapor pressure stays at the saturated vapor pressure provided that the moisture is in the mixed liquid/vapor phase in voids and (2) the vapor pressure depends only on the current state of moisture condition, not a direct relationship with the initial relative humidity condition.

Material's deformation will alter the volume or size of voids or cavity during rapid heating. Therefore, to accurately evaluate vapor pressure buildup, solid or structure deformation and heat diffusion should also be considered. However, to fundamentally understand the vapor pressure theory, in particular, to obtain the analytical solutions, this paper focuses on the problems and solutions that do not consider the deformation. Also for simplicity, one-dimensional problems have been used to develop most of the analytical solutions or governing equations for determining vapor pressure. When the solid deformation is considered, numerical simulation such as coupled-field finite element analysis can be applied, but the fundamental theory of the vapor pressure presented in this paper remains same. Poroelasticity, poroplasticity, and the fully coupled constitutive relation at a multiscale level can be applied and integrated with vapor pressure models to fully understand the material's behavior with moisture under elevated temperature [31,32,8389].

  • A =

    cavity surface, m2

  • aw, aw,eq =

    water activity and equilibrium ambient water activity, respectively

  • B =

    mobility of water molecules, mol × m2/s/J

  • C =

    moisture concentration, kg/m3

  • Ccav, ρcav =

    moisture concentration and density at cavity surface, respectively, kg/m3

  • Csat =

    saturated moisture concentration, kg/m3

  • C0, C1 =

    initial and boundary moisture concentration for preconditioning and reflow, respectively, kg/m3

  • D =

    moisture diffusivity coefficient, m2/s

  • D1 =

    diffusion coefficient at T1, m2/s

  • D, ED =

    prefactor (m2/s) and activity energy (J/mol), respectively, for calculating D

  • fC, fa =

    water sorption isotherm functions, fC = C(aw, T); fa = aw (C, T) = fC1

  • G1 =

    empirical constant for Sawada's model, Pa m3/kg/K

  • g, X, τ =

    normalized concentration, position, and time for Hui's solution

  • h =

    normalized parameter in Shirley's solution

  • J =

    water flux, kg/m2/s

  • K =

    generalized solubility which is a function of T and aw, kg/m3

  • kH =

    Henry's constant, kH = 1/(SPsat), m3/kg

  • k =

    prefactor for calculating kH, k=1/(pS), m3/kg

  • KH =

    Henry solubility constant (=1/kH when ΔHs* = 0 kJ/mol)), kg/m3

  • M =

    mass of water vapor in cavity, kg

  • Pamb, Psat =

    ambient vapor pressure and saturated water vapor pressure, Pa

  • Pcav, Pmax =

    cavity pressure and its maximum pressure during heating, Pa

  • Psat0, Psat1 =

    saturated water vapor pressure at T0 and T1, respectively, Pa

  • Pv =

    water vapor pressure in free volumes or pores, Pa

  • P0 =

    ambient water vapor pressure for preconditioning, Pa

  • P =

    prefactor in Clausius–Clapeyron relation for Psat, Pa

  • Psat =

    modified saturated vapor pressure in microvoids, Pa

  • Qst =

    isosteric heat of water sorption, kJ/mol

  • Qst* =

    net isosteric heat of water sorption, Qst* = QstΔHvap , kJ/mol

  • r =

    radius of microvoids, nm

  • R =

    universal gas constant, 8.314 J/mol/K

  • Rw =

    water gas constant, 461.6 J//kg/K

  • RHamb =

    ambient relative humidity

  • RH0, RH1 =

    initial and reflow relative humidity, respectively, RH = Pamb/Psat

  • S =

    Henry solution coefficient, kg/m3/Pa

  • S =

    prefactor to calculate S, kg/m3/Pa

  • T =

    temperature, K

  • T˙ =

    heating rate, K/s

  • T0, T1 =

    preconditioning and reflow temperature, respectively, K

  • V =

    cavity volume, m3

  • Vm =

    molar volume of water, =1.8 × 105 m3/mol

  • Wd =

    dynamic wetness at cavity surface, WdCcav/Csat

  • w, l =

    thickness and cavity size, respectively, m

  • x¯, t¯ =

    normalized position and time for Shirley's solution

  • α =

    normalized parameter in Hui's and Chen's solution

  • ΔHs =

    heat of solution to calculate S, J/mol

  • ΔHvap =

    heat of vaporization for water, J/mol

  • ΔHs* =

    net heat of solution, ΔHs*=ΔHsΔHvap, J/mol

  • θ =

    water contact angle of microvoids, deg

  • κ, μ =

    vapor permeability (m2) and viscosity (Pa·s), respectively

  • μw, μw0 =

    water chemical potential and its reference value, J/mol

  • ρa, ρg =

    apparent and saturated moisture density, kg/m3

  • ρg =

    modified saturated vapor density, kg/m3

  • σ(T), σrm =

    water surface tension at T and room temperature, respectively, N/m

  • ϕ, ϕeq =

    porosity and equivalent porosity, respectively

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