Polymer foaming with chemical blowing agents: experiment and modeling.
The foaming process to produce microcellular thermoplastics has been widely analysed in the last decades. Microcellular plastics are generally formed by cell nucleation and growth of bubbles in the polymer matrix. Chemical blowing agents (CBA) or physical blowing agents (PBA) are used to introduce the gas that creates the cellular structure. This work is focused on CBA foaming process.
A typical polymer foaming process involves several steps: first, the dissolution under an elevated pressure of a gas created by a chemical reaction from (CBA), or gas dissolved in supersaturated state in the molten polymer (PBA). Second, the nucleation of a population of gas clusters in the supersaturated solution upon the release of pressure to the ambient pressure and finally the growth of nucleated bubbles in the polymer to their ultimate equilibrium size. The final foam density depends on the original gas loading, the gas fraction which remains dissolved in the polymer matrix when it solidifies, the gas losses to the environment, and the depressurization rate. The cell size and cell size distribution depend on the kinetics of nucleation, the bubble growth process following nucleation, and the coalescence during expansion.
Different steps can be considered when processing such materials. First the polymer/gas solution formation, second the microcellular nucleation, and finally the cell growth and the resulting density reduction. In the first stage, the polymer/gas solution formation is accomplished by saturating a polymer under a high gas pressure, forming a single-phase supersaturated solution governed by the gas dissolution in the polymer matrix which is a function of pressure and, at a lesser extent, of temperature. Numerous studies have been carried out analysing the dependence of the solubility of gas in several polymers, mainly C[O.sub.2] in polypropylene [1-3].
In the second stage, it is necessary to submit the polymer/gas solution to a thermodynamic instability to nucleate microcells. This nucleation can be achieved by lowering the solubility of the solution through the temperature and the pressure of the system. Usually, a rapid pressure drop produces a high nucleation rate in the polymer matrix and in the ideal case this nucleation occurs instantaneously.
In plastic foaming, nucleation refers to the process of generating gas bubbles in a polymer melt through a reversible thermodynamic process. In the classical nucleation theory, there is a critical nucleus, which defines the minimum radius for a bubble to growth. Nucleated bubbles which size is larger than the critical nucleus radius will survive, whereas those smaller will collapse. Cell nucleation can occur homogenously or heterogeneously. The heterogeneous nucleation is usually 100 to 1000 times more favourable than homogeneous nucleation. Several additives or organic charges can be employed as nucleation sites in polymer foaming processes [4, 5],
Once the cells are nucleated, they continue to expand by diffusion of the dissolved gas from the polymer matrix into the bubbles. In this stage of the process, a deep knowledge of the physical parameters that govern the diffusion properties of gaspolymer systems is necessary [6-9]. The cells grow reducing the polymer density as the gas molecules diffuse into the nucleated cells. The rate at which the cells grow is limited by the diffusion rate and the rheology of the polymer/gas solution. The cell growth process is controlled also by the time allowed for the cells to grow before solidification, the temperature and pressure of the system, the presence of other bubbles, and so forth .
One of the main foaming processes involves CBA, which liberate gases under certain processing conditions either due to chemical reaction or thermal decomposition. Most CBAs produce nitrogen ([N.sub.2]) or carbon dioxide (C[O.sub.2]) after decomposition [11-13]. CBAs reactions can be endothermic or exothermic. Azodicarbonamide is the most representative exothermic CBA, commonly having a high gas yield, with decomposition temperatures between 170 and 200[degrees]C , Sodium bicarbonate and zinc bicarbonate are the most common endothermic blowing agents .
In this article, a simple experiment has been designed to analyse the foaming expansion as a function of time of a polypropylene containing three types of CBA, in static conditions (no flow). The expansion ratio has been measured by direct observation and from optical measurements and image analysis. A single bubble simulation based on differential scanning calorimetry (DSC) and thermo-gravimetrical analysis (TGA) experiments, assuming each CBA particle as a nucleation site and accounting for gas diffusion in the surrounding polymer matrix has been built. The sensitivity of the model to physical and processing parameters has been tested and the results are compared to the experiments.
The polypropylene compound was a 12% mineral filled (5% wt. talc et 7% wt. fibers), elastomer modified polypropylene (SUMIKA PP) with a melt flow index of 65 g/10 min (ISO R1133), a Newtonian plateau viscosity of 500 Pa s at 210[degrees]C, determined from rotational rheological measurements, and a density of 0.91 g/[cm.sup.3]. Three different endothermic CBA referred as CBA-1, CBA-2, and CBA-3 were analysed. These foaming agents are PE-based compounds with reactive elements (typically citric acid, sodium bicarbonate, or a mix of both components). The CBA-1 was composed of 35 wt% of citric acid and 35 wt% of sodium bicarbonate. The CBA-2 was composed of 70 wt% of citric acid and finally the CBA-3 was based in 70 wt% of sodium bicarbonate. In all the cases, the percentages represent the weight concentration respect to PE matrix. In the following, CBA refers to the compound and not to the reactive elements only [16-20],
Chemical foaming agents have been extensively employed in the last few years [16-20]. These products, with decomposition temperatures between 160 and 210[degrees]C, can be added directly into the hopper of an injection moulding machine in the form of pellets in proportions from 1% to 4 wt%.
Thermal characterization was carried out to determine the polypropylene fusion temperature and the decomposition temperatures of each reaction of the reactive elements included in the CBA pellets. Results are presented in Fig. 1. A heating rate of 6[degrees]C/min was chosen to assure the same heating rate as in the experimental foaming expansions.
The curve for PP shows the fusion between 160 and 170[degrees]C. For the three CBAs, the small peak around 90[degrees]C corresponds to the PE melting temperature. For the CBA-1 and CBA-3, the peaks between 150 and 175[degrees]C correspond to sodium bicarbonate decomposition reaction. A peak between 190 and 220[degrees]C caused by the decomposition of citric acid is visible for CBA-2 and CBA-1. Finally, a small peak about 240[degrees]C takes place in the CBA-1, caused by the coupling reaction of the remaining sodium bicarbonate and citric acid. This third reaction is only important at higher heating rates (above 20[degrees]C/min). For low heating rates, the importance of this reaction is negligible, as it can be seen in Fig. 1, All the decomposition reactions of the CBAs start after the polypropylene fusion, which assures that the gas obtained from the CBA can be diluted in the melted matrix.
TGA measurements were carried out to determine the quantity of gas released by the reactive elements in the CBA particles (Fig. 2). The relative weight loss refers to the original weight of the granule containing 30 wt% of polyethylene and 70 wt% of reactive elements. This loss is associated to the gas escaping the sample, assuming that the pressure conditions do not allow any gas dissolution in the polyethylene.
The TGA curve of CBA-3 shows that the decomposition reaction of sodium bicarbonate begins at 150[degrees]C, reaching a weight percentage of created gas around 24% at the end of the reaction, at about 210[degrees]C. In the case of CBA-2, the decomposition reaction of citric acid begins at 215 and ends around 300[degrees]C, with a weight percentage of gas created about 35%. Finally, the CBA-1 presents a first decomposition reaction which begins at 150[degrees]C. The citric acid decomposition reaction is probably coupled with the sodium bicarbonate reaction, as it can be observed in the small change in the slope about 200[degrees]C. The maximum quantity of gas generated at the end of the decomposition reactions is about 28 wt% at 300[degrees]C. In all the cases, the PE decomposition begins at 450[degrees]C.
It is possible to obtain the evolution of the total quantity of moles of created gas from the stoichiometry of the decomposition reactions. The sodium bicarbonate decomposition is (Reaction 1):
2NaHC[O.sub.3] [right arrow] [Na.sub.2]C[O.sub.3] + [H.sub.2]O+C[O.sub.2] (D
The citric acid decomposition is (Reactions 2 and 3):
[C.sub.6][H.sub.8][O.sub.7] [right arrow] [C.sub.6][H.sub.6][O.sub.6]+[H.sub.2]O (2)
[C.sub.6][H.sub.6][O.sub.6] [right arrow] [C.sub.5][H.sub.6][O.sub.4]+C[O.sub.2] (3)
Finally, a coupling reaction occurs when the sodium bicarbonate and citric acid react together to produce C[O.sub.2] and [H.sub.2]O (Reaction 4):
[C.sub.6][H.sub.8][O.sub.7] + 3 x NaHC[O.sub.3] [right arrow] [Na.sub.3][C.sub.6][H.sub.5][O.sub.7] + 3 x [H.sub.2]O+3 x C[O.sub.2] (4)
In CBA-3 and CBA-2 only Reactions 1 and 2 occur, respectively. As our experiments were carried out at [??] < 20[degrees]C/min, the coupling Reaction (4) for CBA-1 is neglected.
One gram of CBA contains 0.7 g of reactive element. The gas escaping the sample is composed of C[O.sub.2] (molar mass of 48 g) and H20 (molar mass of 18 g). From the stoichiometry of the reactions it is then possible to obtain the evolution of the number of moles of gas per gram of CBA generated during the decomposition reaction (Fig. 3). It is expected that after foaming and cooling of the samples, the [H.sub.2]O remains in the samples as condensed water vapor.
The objective is to analyze the foaming behavior of different CBA/PP samples, obtained by mixing the three CBAs with PP granules in the same proportions as in injected samples, usually between 1 and 4% wt. These materials are foamed in a steel mold and the expansion ratio and cellular structure will be analyzed.
Sample Preparation and Characterization. The challenge is to mix the components without activating the chemical reactions. A solid sample was fabricated starting from PP and CBA powders. Powders were mixed at room temperature in a fixed pro portion of 98 wt% of polypropylene and 2 wt% of CBA. Then, cylindrical samples of 9 mm height ([h.sub.i]) and 20 mm diameter ([phi]) were fabricated in a steel mold by compression under a pressure of 20 MPa at 60[degrees]C for 30 min. A total number of six samples, three groups of two samples with the same CBA, were fabricated, with a volume [V.sub.i] of 2.83 [cm.sup.3]. The density of all the samples ([[rho].sub.sample]) was about 0.905 g/[cm.sup.3], with a densification value up to 99.9%, calculated as:
Densification = 100 x ([[rho].sub.sample]/[[rho].sub.PP]) (5)
taking [[rho].sub.PP] = 0.91 g x [cm.sup.-3] as the density of the solid PP.
Figure 4 presents a typical SEM micrograph of the sample surface showing the distribution of the foaming agent (white particles) in the polypropylene matrix. The average particle size can be calculated by image analysis, using the ImageJ[R] software. The minimal observable size is 1 [micro]m. After binarization of SEM images (Fig. 5a), the apparent particle radius distribution is obtained and represented as a histogram (Fig. 5b).
The number average radius [bar.R] and the number average volume [bar.V] can be obtained assuming a spherical geometry of the CBA particles, which is far from reality (Eqs. 6 and 7):
[bar.R] = [[summation].sup.N.sub.i=1] [n.sub.i][R.sub.i]/[[summation].sup.N.sub.i=1][n.sub.i] (6)
[bar.V] = [[[summation].sup.N.sub.i=1] [n.sub.i][R.sup.3.sub.i]/[[summation].sup.N.sub.i=1][n.sub.i]] [[4[pi]/3] (7)
where N represents the total number of particles (N = 1800).
The average value of the reactive agent particle radius is R = 4.46 [micro]m and the average volume is [bar.V] 2.80 x [10.sup.-9] [cm.sup.3]. This calculation was performed in three different SEM micrographs, with a dispersion value of [+ or -] 10%.
Knowing that each CBA particle contains 70 wt% of reactive agent and 30 wt% of low-density PE, and using the mixing law (with [[rho].sub.reactive agent [approximately equal to] 2 g/[cm.sup.3] and [[rho].sub.LDPE] = 0.9 g/[cm.sup.3]), the density of a CBA particle is 1.67 g x [cm.sup.-3], which leads to an average mass of a reactive agent particle of 4.70 x [10.sup.-9] g. The total mass of reactive agents in the sample (mass of 2.5 g) is obtained knowing that each sample contains 2 wt% of CBA, in which the reactive agents represent 70 wt% (0.035 g). The total number of reactive agent particles ([N.sub.p]) in the solid sample is obtained simply by the ratio between the total mass of reactive agent in the sample and the average mass of a reactive agent particle: 0.035/4.70 x [10.sup.-9] = 7.45 x [10.sup.6].
It is also possible to obtain the distance between each particle (1), from the previous binarized image (Fig. 5a). The resulting histogram is presented in Fig. 6, together with the expression to calculate the number average value l'.
l' = [[summation].sup.N.sub.i=1][n.sub.i][l.sub.i]/[[summation].sup.N.sub.i=1][n.sub.i] (8)
The determination of l' from Eq. 8 gives a result of 74 [micro]m. Another way for calculating l' is to assume a uniform distribution of the reactive agent particles in the solid sample using crystallographic considerations. Considering for example a primitive cubic (PC) Bravais lattice, the number of reactive agent particles in the cubic cell ([N.sup.cell.sub.P]) is 1. The determination of the distance l' is related to the volume of the solid sample [V.sub.i], the number of reactive agent particles in each cubic cell [N.sup.cell.sub.P] and the total number of reactive agent particles [N.sub.P] through the expression:
[l'.sup.3][N.sub.P]/[N.sup.cell.sub.P] = [V.sub.i] (9)
Similar calculations can be carried out for the body-centered cubic and face-centered cubic bravais lattices, in which the number of reactive particles in each cubic cell is 2 and 4, respectively. Taking [N.sub.P] = 7.45 x [10.sup.6] [cm.sup.-3] and [V.sub.i] = 2.83 [cm.sup.3], the average initial distance between reactive agent particles calculated ranges from 76 to 87 [micro]m, which is similar to the value obtained from the binarized image. This distance l' will increase during the expansion process.
Foaming Experiment Under Pressure. Figure 7 presents the scheme of the foaming expansion experiment designed to analyze the behavior of the polypropylene/gas systems. The solid sample, with a number of reactive agent particles [N.sub.P], has an initial height [h.sub.i] and an initial volume [V.sub.i]. It is placed in a steel reservoir (height 200 mm) under an external pressure [P.sub.ext]. The inner diameter is 20 mm. The external pressure is applied with a weight deposited on a circular steel cap, with a diameter slightly lower than the inner cylinder diameter. The device is introduced in an oven at a temperature T during a time t. A type K thermocouple is introduced in the sample to monitor the evolution of the temperature. Another one is located inside the oven. At the end of the heating step, the device is removed from the oven and cooled down at ambient air. When cooling is finished the weight is removed. The foamed sample has a final height [h.sub.f], a final volume [V.sub.f], and a cellular structure with a cell density [N.sub.c] and an average cell radius R. The height increase during the foaming process is measured optically with a recording camera, with a precision of [+ or -] 0.5 mm. It is important to mention that the temperature will vary differently as a function of time at different points of the sample, due to heat conduction from the oven. This means that the foaming develops heterogeneously inside the sample, starting in the sample edges and propagating towards the core. The evolution of the temperature with time is also represented. The sample and oven temperatures are not equal until the last part of the experiment. The average heating ramp of the sample temperature is about 6[degrees]C/min, considering a total heating time of 32 min from ambient temperature to 210[degrees]C.
Two external pressures were tested to evaluate their influence on the expansion rate, gas diffusion and final pore radius: 0.25 MPa (Samples 1-1, 2-1, and 3-1) and 0.5 MPa (Samples 1-2, 2-2, and 3-2).
Results and Discussion. The measurement of the height variation started when the sample temperature reached 180[degrees]C, slightly above the melting point of the PP, which is located between 160 and 170[degrees]C according to DSC data (see Fig. 1). Figure 8 presents the samples height evolution from the beginning of the expansion measurements (about 23 min after starting the experience, see Fig. 7).
All samples show a remarkable volume expansion due to the gas creation and the foaming process. The expansion process takes about 4 min for CBA-1 and CBA-3, and 6 min for CBA2, leading to expansion ratios around 200% with [P.sub.ext] = 0.25 MPa and 140% with [P.sub.ext] = 0.5 MPa. After that time, the sample height reduces because of cooling and thermal shrinkage.
The samples foamed with the CBA-2 showed a higher expansion than samples foamed with the other two CBAs. This may be explained by the larger number of moles of gas generated by the chemical reactions, as shown on Fig. 3.
At the end of the process, shrinkage is clearly seen, especially in samples foamed with CBA-2. No shrinkage was observed in the lateral direction. For CBA-2 at 0.25 MPa, height reduces from 22.1 mm at 200 s to 20.1 mm at 260 s after (both times measured after starting the expansion at 23 min), which corresponds to a volume reduction of 10%. For the same CBA at 0.5 MPa, the height reduction begins at 170 s with a maximum value of 14.8 mm and a final value of 13.3 mm at the end of the experience, with a similar volume reduction value.
It is important to estimate the volume of gas retained in the sample to analyze possible gas diffusion outside the polymer. Assuming that the decomposition reaction is complete, the theoretical total volume of gas [V.sup.generated.sub.gas] can be calculated from perfect gas equation.
[V.sup.generated.sub.gas] = n[(t).sup.total] x R x T/[P.sub.ext] (10)
[P.sub.ext] is the external pressure (MPa), T is the foaming temperature, which is taken as a constant value of 210[degrees]C (483 K) in a first approximation, and SR is the gas constant (8.31 J x [mol.sup.-1] x [K.sup.-1]). The value of n[(t).sup.total] corresponds to the total number of moles of gas created in the solid sample for the different CBAs at the end of the reaction.
According to Fig. 3, the total number of moles of gas created per gram of CBA is 0.0092 mol/g in the case of the CBA-2 and CBA-3, and 0.0135 mol/g in the case of the CBA-1. The total number of moles of gas generated in the sample n[(t).sup.total] can be calculated by multiplying the previous values and the mass of CBA particles in each sample, in our case 2 wt% of 2.5 g, and assuming that only the 70% of each CBA particle contains reactive elements.
The gas volume generated at the end of the expansion, before any cooling and thermal shrinkage of the polymer is deduced from Eq. 10, and represented on Table 1, together with the final volume [V.sub.f], the expansion ratio after cooling and the foam density [rho].
The fraction of gas in the foamed polymer can be also determined from image analysis. Several optical micrographs of the fracture surface of the expanded samples are presented in Fig. 9. The structure is homogenous throughout the whole sample volume, which indicates that the thermal gradients do not influence the final foamed structure, even when the heating ramp may be locally different in the sample.
Figure 9 shows that CBA-2, containing citric acid as reactive element, induces a finer bubble size and higher expansion rate than the other CBAs (see Fig. 9b). The addition of sodium bicarbonate (CBA-1) or CBA based only on sodium bicarbonate (CBA-3), leads to larger bubble size but lower expansion rate.
Using the ImageJ[R] software previously presented, it is possible to quantify the average bubble radius [R.sub.bubble], the average distance between bubbles, (namely [l.sub.bubble]). and the total volume of gas in each sample [V.sub.gas]. As an example, Fig. 10 presents the histogram of both bubble radius, assuming that each bubble can be considered as a sphere (Fig. 10a), and the distance between bubbles (Fig. 10b). This parameter is calculated for each bubble, taking the minimum distance between the edge of this bubble and the surroundings ones. Results presented have been calculated from Test 1-1, but similar calculations have been performed for the other samples.
The average results are obtained from 10 micrographs of each sample with a maximum error of [+ or -] 5%. The software accounts for the number of bubble in each image and the average radius. The total cell number [N.sub.c] in the sample was calculated using Eq. 11 , which accounts for the 3D extrapolation starting from a 2D image:
[N.sub.c] = 6(1 - [[rho].sub.F]/[[rho].sub.P])/[pi][R.sup.3] [V.sub.f] (11)
[[rho].sub.p] represents the foam density, and [[rho].sub.P] the solid polymer density (0.91 g/[cm.sup.3]). [V.sub.f] represents the final volume of the sample after shrinkage. The experimental gas volume can be also easily calculated from expansion measurements, following the next equation:
[V.sup.direct.sub.gas] = [V.sub.f] - [V.sub.i] (12)
Both results (indirect method [V.sup.indirect.sub.gas], from ImageJ[R] analysis and direct method [V.sup.direct.sub.gas] from expansion measurements), together with the morphological determinations, are presented in Table 2. They are in the same range, and the slight differences between both measurements can be due to specific errors associated to the software employed to analyze the optical micrographs. The last column of Table 2, [V.sup.remainining.sub.CO2] represents the total volume of C[O.sub.2] remaining. This value is calculated assuming than after cooling and shrinkage [H.sub.2]O condensates into liquid water, and extracting the volume of water vapor generated from the total volume of gas generated calculated in Table 1. From the chemical Reactions (1), (2), and (3) both citric acid and sodium bicarbonate decompose in one mole of [H.sub.2]O and one mole of C[O.sub.2]. The ratio between the molar masses of both components is 18/(18 + 44) = 0.29, indicating that the 29% of the gas generated transforms into liquid water by condensation.
The proportion between the measured gas volume and the theoretical gas volume varies between 50 and 75%.
The order of magnitude of the thermal shrinkage between 210[degrees]C and room temperature for the polypropylene is around 20%. This cannot explain the measured difference. An incomplete chemical reaction is unlikely according to the DSC and TGA measurements. It could be speculated that a part of the gas generated has not been nucleated and does not produce any expansion. It could remain dissolved in the polymer matrix or have diffused outside the sample during the expansion process. To test these hypotheses, the foamed samples were re-heated up to 210[degrees]C to analyze a possible second expansion produced by the remaining dissolved gas, with negative results. This indicates that a small proportion of the gas has probably diffused out of the samples.
Another interesting parameter that can be analyzed from the foaming experiment is the coalescence. It is possible to define the ratio k between the final number of cells Nc and the initial number of reactive agent particles NP. It lies between 0.2 and 12%, indicating that coalescence is a very important phenomenon that will be discussed lately.
Many studies have been devoted to the development of numerical models for the bubble nucleation and growth in polymeric foaming process. In the classical work presented by Amon and Denson  a complete mathematical analysis of a bubble growth in a Newtonian matrix is presented. Bikard et al.  and Bruchon  solved the same problem with a 3D finite element method which allows accounting for the simultaneous growing of multiple bubbles. Koopmans et al.  introduced a viscoelastic multimode Maxwell behavior for the polymer matrix in a "bubble influence volume" surrounding the growing bubble. They also account for non-isothermal phenomena occurring at die exit in an extrusion process. Otsuki and Kanai  introduced a more realistic Phan-Thien Tanner viscoelastic constitutive equation which limits the dramatic increase of the Maxwell model elongation viscosity. Shafi et al. [27, 28] and Joshi et al.  developed a homogeneous nucleation model that they couple to the Newtonian Amon and Denson bubble growing model. Taki [30, 31] compares these calculation results to experiments performed under several pressure release rates. Feng and Bertelo  investigated bubble nucleation in a viscoelastic polymer melt (Oldroyd B constitutive equation) containing nucleating agents.
All these works assume that the polymer is saturated with gas that diffuses from the matrix to the bubble. In our case, the diffusion process is in the opposite direction from the bubble, which is supposed to be nucleated around the CBA particle, towards the polymer matrix. Recently Emami et al.  analyzed the bubble nucleation in non-pressurized foaming CBA systems starting from solid materials, composed of PP and CBA powders as in our experiments. It was observed that the nucleation process proceeded in two distinct stages, namely primary and secondary nucleation. Primary nucleation occurred in the interstitial regions of the sintered plastic powder and the agglomerated blowing agent particles acted as nucleation sites, and secondary nucleation occurs in the polymer melt. The visual observations indicated that most of the first generation of bubbles endured the entire foaming process, whereas most of the bubbles generated during secondary nucleation disappeared over time. These results support the previous assumption which considers each reactive agent particle as a nucleation site, with no further nucleation phenomenon.
In the following, a kinetic model for a single bubble expansion in a Newtonian fluid coupled with the gas diffusion in the surrounding polymer matrix is proposed. The nucleation phenomenon will be simplified, assuming that each reactive agent particle can be considered as a nucleation site. The model will be applied to the experimental conditions presented in the previous section. The sensitivity of the model to several unknown parameters has been tested. Comparison between calculation and experiments will be discussed.
Single Bubble Growth Model
A schematic of the bubble growth model is shown in Fig. 11. A reactive agent particle creates n[(t).sub.created] moles of gas by the chemical decomposition reaction deduced from the TGA curve. The bubble growth is governed by the competition between the gas which remains within the bubble and induces the growing mechanism (number of moles n(t)) and the moles of gas n[(t).sub.diffused] which diffuses in the surrounding polymer matrix at the external pressure [P.sub.ext]. The gas concentration C(r,t) propagates concentrically in the surrounding polymer melt. The average concentration of gas at the bubble surface, C[(r,t).sub./r=R], is related to the gas pressure inside the bubble [P.sub.gas] through the solubility factor K.
The following assumptions are made:
1. The bubble is spherically symmetric when it nucleates and remains so for the entire period of growth.
2. The polymer matrix is Newtonian.
3. The growth process is considered isothermal. Latent heat of reaction is neglected.
4. Inertia effects are neglected and the fluid is assumed to be incompressible, which is reasonable in the pressure range studied.
5. The gas released by the chemical reactions follows the ideal gas law.
6. The matrix is considered as an infinite medium and one single bubble is considered. This condition will be commented later, in terms of the gas diffusion distances, bubble radius and sample dimensions.
The first step consists in the determination of the number of moles of created gas n(r)created. Three different CBA kinetics are shown in Fig. 3. Considering the experimental foaming results, two different CBAs, CBA-1, and CBA-3, present similar results in terms of expansion rate and morphology. This indicates that the coupling reaction between citric acid and sodium bicarbonate has not occurred at the foaming temperature of the experience (210[degrees]C), as indicated in the DSC curves Fig. 1. For this reason, two different simulations will be presented, first for the PP + CBA-2 samples (2-1 and 2-2) with the decomposition of citric acid, and then for the PP + CBA-3 samples (3-1 and 3-2) with the sodium bicarbonate decomposition reaction.
Figure 12 presents the theoretical fitting of the experimental TGA data extrapolated to one single particle of reactive agent for both decomposition reactions, at a temperature of 210[degrees]C. The low heating rate (about 6[degrees]C/min according to the temperature measurements during the foaming experiment), permits, as a first approximation, the extrapolation of the non-isothermal results derived from the TGA to isothermal kinetics at 210[degrees]C. The total reaction time was between 100 and 200 s, much lower than the foaming time employed during the expansion experiment, which assures that the chemical reactions are complete. Different kinetic models can be found in the literature, such as the Kamal and Sourour model . In our case, the best fitting correlation was found using Boltzmann's exponential functions:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
Fitting parameters, as well as the quality of the correlation are presented in Table 3.
The differential equation for the bubble radius growth as a function of time writes :
dR/dt = R/4[eta] (n(t)R[T.sub.g]/4/3 [pi][R.sup.3] - [P.sub.ext] - 2[gamma]/R) (14)
R is the bubble radius, [eta] is the polymer viscosity and [gamma] is the surface tension. As explained before, the gas temperature [T.sub.g] is assumed to be constant (210[degrees]C) during the expansion process.
The variation of the number of moles of gas n(t) inside the bubble is derived from Eq. 15:
dn/dt = dn[(t).sub.created]/dt - dt[(t).sub.diffused]/dt (15)
The number of moles of gas which diffuse outside the bubble in the surrounding polymer matrix is obtained from the mass transfer of CBA at the gas-polymer interface and it can be expressed as follows:
dn[(t).sub.diffused]/dt = -4[pi][R.sup.2]D [partial derivative]C/[partial derivative]r|[sub.r=R] (16)
D is the diffusion coefficient, and C is the gas concentration of the diffused gas. The concentration profile is given by a diffusion equation around the bubble which writes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
u is the velocity field related to the bubble expansion, in spherical coordinates. The ratio between the convection term and the diffusion term is a Peclet number ([P.sub.c]) given by (see Joshi et al. ):
[P.sub.c] = convection/diffusion [approximately equal to] [u/R]/[D/[r.sup.2]] (18)
The diffusion coefficient value can be taken from the literature , with a typical value, for PP and PE/C[O.sub.2] systems, of D = [10.sup.-8] [m.sup.2]/s. The calculation without diffusion will show that the bubble radius goes from 5.[10.sup.-6] to 2.[10.sup.-4] m and the growth speed of the bubble radius is around [10.sup.-6] m/s, leading to a Peclet number of [10.sup.-4]. Therefore, convection can be neglected.
The numerical implementation requires defining boundary and initial conditions. Three physical parameters must be defined at time t = 0. First, the average size of the reactive agent particles can be considered as the initial bubble radius [R.sub.0], as a first approximation. This assumption is only an approximation, and the sensitivity of the calculations when changing the initial radius will be analyzed. The initial number of moles of gas n0 can be directly calculated from the gas perfect law using the initial radius [R.sub.0] = [bar.R] = 4.46 [micro]m, at the two different external pressures 0.25 MPa and 0.5 MPa. The boundary condition for the gas concentration at the bubble surface C(R, t) is described by Henry's law:
C(R, t) = [3/4] k n(t)R[T.sub.g]/[pi][R.sup.3] (19)
The numerical implementation of the previous equations is carried out by means of an incremental time marching approach during the decomposition reaction time (about 100 s, see Fig. 12). This approximation implies that the expansion process is limited to the decomposition reaction time, and that no further expansion occurs later. The variation of the number of moles of gas created is obtained by differentiating Eq. 13:
Two hundred and fifty iterations were used to solve the equations. We adjust the time step At in order to obtain an incremental radius variation less than 1%:
RESULTS AND DISCUSSION
Table 4 presents the physical parameters and the initial conditions employed in the foaming simulation.
Figure 13 presents the gas concentration profiles outside the bubble for five different reaction times and CBA-2, for an external pressure of 0.25 MPa. Results for the concentration profiles taking an external pressure of 0.5 MPa were almost equivalent, which indicates that the slight variation in the external pressure does not affect significantly the gas diffusion.
The quantity of gas which diffuses outside the bubble increases as the reaction develops. For reaction times close to 40 s, the gas diffusion penetration thickness is around 150 [micro]m, whereas for reaction times of 2 s, this distance is only 50 [micro]m. The gas concentration at the gas-polymer interphase decreases strongly with time, even if the quantity of gas generated inside the bubble n(t) increases according to the TGA kinetics. The reason for this decrease is related to the dependence of C(R,t) with bubble radius in a proportion C [varies] [R.sup.-3] (see Eq. 19). At high reaction times, the gas concentration at the interphase is negligible.
Figure 14 compares the number of moles of gas created by the decomposition reaction, diffused and retained in the bubble during the expansion process for an external pressure of 0.5 MPa.
These results show that, in the foaming conditions of Fig. 14, the diffusion process becomes noticeable from 20 s. From that time, the quantity of diffused gas increases, reaching a final value about 1.35 x [10.sup.-12] moles. On the other hand, the number of moles of gas created by the decomposition reaction reaches a value around 4 x [10.sup.-12] moles.
At the end of the reaction, the ratio between diffused and created gas is about 33%. This value is in reasonable agreement with the experimental values found previously in Table 2 (between 25 and 50% of diffused gas).
Figure 15 presents the predicted radius evolution for both CBA-2 and CBA-3.
The final bubble radius depends strongly on the external pressure and is obviously limited by the gas diffusion. For the CBA-2, when the external pressure is 0.25 MPa, the bubble radius reaches 530 [micro]m when gas diffusion is neglected, and about 130 [micro]m when the diffusion process is considered. When the external pressure is increased up to 0.5 MPa, these values are 330 and 90 [micro]m, respectively. The theoretical values obtained (considering the diffusion process), are in qualitative agreement with the foaming experiments values (120 and 85 [micro]m, respectively, see Table 2). A similar analysis can be performed for the CBA-3. In this case, the theoretical values obtained considering the diffusion process are about 300 and 180 [micro]m, for both external pressures of 0.25 and 0.5 MPa. These results are also in qualitative agreement with the foaming experiments (350 and 190 [micro]m, respectively, see Table 2).
Using the R(t) value allows to compute the velocity u at the bubble/polymer interface. The evolution of u for the CBA-2 and CBA-3 is similar, with values are around [10.sup.-6] m/s which justifies neglecting the convection term in the diffusion Eq. 17.
The sensitivity of the model to the initial radius R0, and the diffusion coefficient D and the viscosity [eta] to the bubble growth rate has been investigated (Fig. 16). The results are shown for CBA-3 for an external pressure of 0.25 MPa.
In Fig. 16a, three different values of the initial radius Ro were chosen (5 nm, 5 [micro]m, and 100 [micro]m). The lowest value corresponds to the typical critical radius value that can be found in the literature for homogeneous nucleation , The final predicted bubble radius is only slightly influenced by the initial radius value (between 290 and 320 [micro]m) but differences are obviously very important at intermediate time steps (till 30 s).
Varying the diffusion coefficient from [10.sup.-8] [m.sup.2]/s to 10 [m.sup.2]/s leads to a final bubble radius around 100 and 600 [micro]m, respectively (Fig. 16b). Typical diffusion values for a gaspolymer system are in the range between [10.sup.-8] and [10.sup.-10] [m.sup.2]/s. The chosen diffusivity value was taken directly from the literature , and corresponds to the diffusivity value of C[O.sub.2] into a PP matrix at 483 K. A more detailed analysis should include the diffusion process of water vapor into a PP matrix. It is possible to assume a value of the diffusion coefficient of water vapor slightly higher than the value for C[O.sub.2] due to the lower molar mass. However, the range of diffusion coefficients analyzed in Fig. 16b is expected to cover both C[O.sub.2] and water vapor diffusion process.
It can be observed that increasing viscosity 1/ from 200 Pa s to 1000 Pa s induces a decrease of the final bubble radius from 600 [micro]m to less than 100 [micro]m (Fig. 16c). The viscosity was determined experimentally from shear rheological measurements and presents a Newtonian plateau around 500 Pa.s at the strain rates encountered during the foaming process. During the foaming process, the flow around a single bubble is purely elongational, which means that the chosen viscosity value determined by shear measurements may be significantly underestimated. This justifies testing higher viscosity values in the single bubble growth model, as seen in Fig. 16c.
The influence of the initial number of moles [n.sub.0] and the influence of the initial radius [R.sub.0] are correlated from the gas perfect law ([n.sub.0] [varies] [R.sub.0.sup.3]).
Finally, as the number of bubbles is less important than the number of CBA particles and as the final radius of the bubbles is more important than the mean initial distance between the particles, it is clear that bubble coalescence appears quite early in the process. Due to the large important CBA particle size distribution (Fig. 5), one can believe that the biggest initial particles will induce the biggest bubbles in the first stage of their development (see Fig. 16a), which will coalesce with the small surrounding bubbles initiated by the smallest initial particles. Considering, for example, one bubble with a diameter of 100 [micro]m surrounded by eight bubbles with a diameter of 25 [micro]m (see Fig. 16a at t = 5 s), the final diameter of the central bubble will be only 108 [micro]m after coalescence. As a consequence, if 88% of the bubbles disappear during the foaming process, (which corresponds to the scenario of one bubble surrounded by eight smaller bubbles), the final bubble diameter will be only enhanced by 8%. This may explain the good agreement between the calculation which does not account for coalescence and the experimental observation where coalescence takes place.
A simple polymer expansion experiment has been designed to test the foaming behavior of different CBA, submitted to different pressure conditions. The bubble size and bubble size distribution depend on the blowing agent and on the applied pressure and careful bubble size measurements obtained by Image Analysis correlate well with the global macroscopic expansion of the foamed sample.
A single bubble expansion model assuming nucleation on each CBA particle, accounting for the different chemical reactions and for the gas diffusion from the bubble to the surrounding polymer matrix agrees fairly well with the experimental results, despite the strong hypothesis. This allows us to build a simple method to estimate the capability of a CBA to develop a foamed structure and the resulting mechanical properties. In this work, the expansion ratio varied between 1.4 at 0.5 MPa and 2 at 0.25 MPa. The cell size obtained when using CBA based in citric acid was much smaller than the obtained using sodium bicarbonate CBA, due to the number of moles of gas released. Also, a simple calculation using the single bubble growth model, with no dissolution or nucleation phenomena, represents well the experiments.
Further studies analyzing injection molding experiments with the same CBAs are in progress.
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Jose Antonio Reglero Ruiz, (1) Michel Vincent, (1) Jean-Franqois Agassant, (1) Tarik Sadik, (2, 3) Caroline Pillon, (2, 3) Christian Carrot (2,3)
(1) MINES ParisTech--Centre de Mise en Forme des Materiaux (CEMEF), UMR CNRS 7635, 1, rue Claude Daunesse, CS 10207, 06904--Sophia Antipolis Cedex, France
(2) Ingenierie des Materiaux Polymeres (IMP), UMR CNRS 5223, Universite de Saint-Etienne, Jean Monnet, F-42023, Saint-Etienne, France
(3) Universite de Lyon, F-42023, Saint-Etienne, France
Correspondence to: J. A. Reglero Ruiz; e-mail: jose-antonio.reglero_ruiz@ mines-partistech.fr
TABLE 1. Physical parameters and theoretical total gas volume generated during the expansion process for each sample. Sample CBA [P.sub.ext] n[(t).sub.total] (MPa) (mol) 1-1 1 0.25 3.22 x [10.sup.4] 1-2 0.5 2-1 2 0.25 4.72 x [10.sup.4] 2-2 0.5 3-1 3 0.25 3.22 x [10.sup.4] 3-2 0.5 Sample [V.sup.generated.sub.gas] [V.sub.t] Expansion ([cm.sup.3]) ([cm.sup.3]) ratio 1-1 5.17 5.62 1.94 1-2 2.58 3.98 1.38 2-1 7.58 6.31 2.18 2-2 3.79 4.17 1.44 3-1 5.17 5.84 2.02 3-2 2.58 3.92 1.36 Sample [rho] (g/[cm.sup.3]) 1-1 0.47 1-2 0.65 2-1 0.41 2-2 0.63 3-1 0.45 3-2 0.66 TABLE 2. Gas volume and morphological parameters in the expanded samples. Test CBA Bubble [R.sub.bubble] [V.sub.bubble] [L.sub.bubble] count ([micro]m) ([mm.sup.3]) ([micro]m) 1-1 1 64 360 0.195 50 1-2 1 108 170 0.021 42 2-1 2 241 120 0.007 25 2-2 2 292 85 0.003 23 3-1 3 79 350 0.180 56 3-2 3 121 190 0.029 38 Test [[rho].sub.F] [N.sub.c] [V.sup.indirect.sub.gas] (g/[cm.sup.3]) ([cm.sup.-3]) ([cm.sup.3]) 1-1 0.47 1.43-104 2.89 1-2 0.65 8.97-104 1.19 2-1 0.41 4.30-105 6 3.11 2-2 0.63 7.58- 10 s 1.35 3-1 0.45 1.65-104 2.90 3-2 0.66 6.24-104 1.19 Test [V.sup.direct.sub.gas] [V.sup.remaining.sub.CO2] ([cm.sup.3]) ([cm.sup.3]) 1-1 2.73 3.77 1-2 1.09 1.88 2-1 3.42 5.53 2-2 1.28 2.76 3-1 2.95 3.77 3-2 1.03 1.88 TABLE 3. Fitting parameters of the number of moles of gas created during the decomposition reaction of the CBAs. CBA [A.sub.1] (mol) [A.sub.2] (mol) [t.sub.0] (s) 2 4.17 x [10.sup.12] 2.95 x [10.sup.-14] 63.12 3 3.92 x [10.sup.-12] 2.18 x [10.sup.-14] 57.87 CBA [lambda](s) [R.sup.2] 2 8.59 0.9991 3 9.20 0.9994 TABLE 4. Physical parameters and initial conditions employed for the foaming simulation. Parameter CBA 2 Initial number of moles [n.sub.0] (3.56 or 7.13) x [10.sup.-14] Initial radius [R.sub.0] 4.46 x [10.sup.-6] Viscosity [eta] 500 Surface tension [gamma] (a) 0.020 Diffusion coefficient D (a) 8 x [10.sup.-9] Gas temperature [T.sub.g] 483 External pressure [P.sub.ext] (2.5 or 5) x [10.sup.5] Solubility parameter K (a) 1.15 x [10.sup.-4] Parameter CBA 3 Initial number of moles [n.sub.0] (3.56 or 7.13) x [10.sup.-14] Initial radius [R.sub.0] 4.46 x [10.sup.-6] Viscosity [eta] 500 Surface tension [gamma] (a) 0.020 Diffusion coefficient D (a) 8 x [10.sup.-9] Gas temperature [T.sub.g] 483 External pressure [P.sub.ext] (2.5 or 5) x [10.sup.5] Solubility parameter K (a) 1.15 x [10.sup.-4] Parameter Units Initial number of moles [n.sub.0] mol Initial radius [R.sub.0] m Viscosity [eta] Pa x s Surface tension [gamma] (a) J x [m.sup.-2] Diffusion coefficient D (a) [m.sup.2] x [s.sup.-1] Gas temperature [T.sub.g] K External pressure [P.sub.ext] Pa Solubility parameter K (a) mol x [m.sup.-3] [Pa.sup.-1] (a) Values taken from literature 
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|Author:||Ruiz, Jose Antonio Reglero; Vincent, Michel; Agassant, Jean-Francois; Sadik, Tarik; Pillon, Caroline|
|Publication:||Polymer Engineering and Science|
|Date:||Sep 1, 2015|
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