Simulation of the percolation of water into rigid polyurethane foams at applied hydraulic pressures.
Rigid polyurethane foams are used primarily for thermal insulation, construction, and packaging [1, 2]. Rigid polyurethane foams, which are resistant to penetration by water at high hydraulic pressures, can be used for underwater buoyancy applications. For such applications, when the foam is placed deep inside water, cell windows of the foam may rupture and allow water to penetrate inside the foam, causing a reduction in the buoyancy of the foam. Thus a high hydraulic resistance is required for foams used in underwater buoyancy applications.
Foam structure typically consists of large number of tiny cells, in the shape of irregular polyhedra. The lamella of foam material that separates two adjacent cells is called a cell window. A strut is generated where three windows of three different cells meet. Typical cell windows and struts of a rigid polyurethane foam are indicated in a scanning electron micrograph in Figure 1. A foam is called open cell or closed cell, depending on the nature of these windows. If most of the windows are partially or fully ruptured then the foam is called open cell; on the other hand, if nearly all windows are intact it is called a closed cell foam. Foams for buoyancy applications must be closed cell and must have cell windows resistant to rupture under pressure.
Previous experimental studies [3-7] have shown that at low applied hydraulic pressure, very little water penetrates into the foam and thus buoyancy losses are very small. However, at higher pressures the foam loses its buoyancy very rapidly beyond a threshold value. The results indicate typical percolation behavior. Experimental results have also shown that strength of the windows increases as they become smaller and thicker [6, 7], increasing the hydraulic resistance of the foam. A detailed understanding of how foam cell structure affects its hydraulic resistance is of practical importance.
The problem discussed earlier is analogous to the well studied bond percolation problem, [8, 9] with each cell window acting as a bond. As the pressure is increased, the fraction of ruptured cell windows increases (bonds become conductors) and the volume of connected cells filled with water (conducting clusters) increases. There are, however, differences between a typical bond percolation problem and the present system. Polyurethane foams have a distribution of cell sizes and the strength of cell windows (resistance to rupture under pressure) is also distributed over a range of values. In contrast, in the percolation problem a regular matrix with bonds of equal conductivity is generally used. The number of windows a particular cell has, which corresponds to the coordination number in the percolation problem, also varies about an average value of 12.
We note that there are several experimental works on percolation phenomena such as increase in dielectric constant of a random metal-insulator composite  and decrease in resistivity of the conductive adhesives [11-14] and conductor filled polymers  near the percolation threshold. However, there is no previous theoretical work related to water percolation in foam.
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We consider here the percolation of water into model foams by means of simulations. The foam is constructed by randomly distributing points in space, which are the centers of the cells. No two centers are closer than a specified distance and all centers closer than another specified distance are taken to be neighbors. The strength of the windows separating two adjacent cells is taken to be a function of the distance between the cell centers in one set of computations. In another set of computations, the strength of the windows is assigned randomly from a specified distribution. The objective of this work is to understand the percolation of fluid into rigid closed cell foams in terms of the cell-level properties of the foam. Such an understanding would help to devise formulations to obtain the desired foam microstructure. In the following sections the simulation method and the experimental details are described first followed by results and discussion. Conclusions of the study are given in the final section.
The simulation is done in three steps. In the first step the coordinates of the centers of the cells are generated. The centers are assumed to be uniformly distributed and the coordinates of each cell center are three random numbers generated in the range (0, L). The generated cell is accepted if its center is not closer than the minimum distance ([d.sub.min]) to any other cell center. In this way, a foam with a desired number of cells ([N.sub.c]) in the cube of volume [L.sup.3] is generated. The generation of a model foam structure is illustrated in Figure 2. All the variables used in the simulation are dimensionless. All lengths are made dimensionless by [n.sup.1/3], where n is the number density of cells. All pressures are made dimensionless with a characteristic pressure such that the prefactor in the dimensionless strength functions is unity.
In the second step, the neighboring cells for a particular cell are determined. The centers of the cells, which are within a distance, [d.sub.max], from the center of a particular cell are called neighbors of that cell (see Fig. 2). For simplicity, in the rest of this paper the distance between two centers of the cells is referred to as the distance between two cells. Clearly, the average number of neighbors (nb[.sub.avg]) of a cell must increase with increase in [d.sub.max]. The value of [d.sub.max] is adjusted until the average number of neighbors of the cells achieves the desired value. For each cell, the neighboring cells, the number of neighbors, and distances from its neighbors are then determined (see Fig. 2).
In the third step the fraction of gas volume filled with water ([phi]) at different applied pressures ([P.sub.app]) is calculated. Each cell is assigned a volume of a sphere, whose radius is half of the average distance of that cell from its neighbors. The total gas volume of the foam is calculated as the sum of the volume of individual cells. A hypothetical window is assumed between two neighboring cells with its strength specified as detailed later. A dry cell is assumed to get filled with water if any of its neighbors is already filled with water and the applied pressure is sufficient to rupture the window between them. At the start of the calculation, cells at a certain small distance from the surface of the foam are assumed to be filled with water. A small pressure is applied to the foam boundaries and every cell window between a wet cell and a dry cell is checked to evaluate if the applied pressure is greater than its rupture strength ([P.sub.rup]). If this criterion is satisfied the cell window is assumed to have ruptured and the dry cell under consideration gets filled with water (becomes a wet cell). This process is repeated until no more cell windows rupture at that applied pressure. The volume of the cells filled with water is used to calculate the volume fraction [phi], at that applied pressure. The applied pressure is increased in small steps, each time achieving an equilibrium state, to obtain the variation of [phi] as a function of applied pressure.
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The percolation behavior for the penetration of water into the foam depends on the rupture strength distribution of the cell windows. We define rupture strength here as the critical pressure on the foam lamella at which it ruptures. Experimental results show that strength of the windows of the foams increases with decrease in cell window area [4, 7]. Therefore it is reasonable to assume that the strength of a window is inversely related to the distance between the two concerned cells (d). Several different expressions for the rupture strength are considered here, the simplest being an inverse relation [P.sub.rup] = 1/d. The strength is scaled by the prefactor in each case considered, as mentioned earlier. In the case of the model foams the relation [P.sub.rup] = 1/d indicates that the maximum value of the rupture strength ([P.sub.rup]) is 1/[d.sub.min] and the minimum is 1/[d.sub.max], However, in real foams the minimum value of [P.sub.rup] may be zero because some windows are partially or fully ruptured during foam formation. To take into account this aspect, the expression for the rupture strength is modified to [P.sub.rup] = 1/d - 1/[d.sub.max], so that the minimum strength of the windows is zero when d = [d.sub.max]. The other functional forms considered for the rupture strength are [P.sub.rup] = (1/d)[.sup.1/2] - (1/[d.sub.max])[.sup.1/2], [P.sub.rup] = exp(1/d) - exp(1/[d.sub.max]), [P.sub.rup] = 1/ln(d) - 1/ln([d.sub.max]), and [P.sub.rup] = 1/[d.sup.2] - 1/[d.sub.max.sup.2]. Some variants of these cases are considered as specified next.
In one set it is assumed that the strength of a window between two cells (i and j) is an inverse function of the radius of the smaller cell ([r.sub.ij,min]) between the two cells. This implies that for two dissimilar cells the bigger one does not have any contribution to the strength of the window between them. In the second set we assume that strength of a window is an inverse function of the sum of the radii ([r.sub.i] + [r.sub.j]) of the concerned cells. This assumption implies that the average sizes of a pair of cells contribute to the strength of the window between them. The two different expressions for the strength of the windows are [P.sub.rup] = 1/[r.sub.ij,min] - 1/[r.sub.max] and [P.sub.rup] = 1/([r.sub.i] + [r.sub.j]) - 1/(2[r.sub.max]), where [r.sub.max] is the radius of the largest cell. In these two expressions the radius of the ith cell, [r.sub.i] is the half of the average distance of the cell from its neighbors.
We also consider that the rupture strength is not directly dependent on cell size but assumed to be random with a given distribution. Such an assumption may be justified if the strengths of the weakest windows are determined by defects, which are randomly distributed. The following expression is used to generate different distributions for the fraction of windows, f([P.sub.rup]) between ([P.sub.rup], [P.sub.rup] + d[P.sub.rup]).
f([P.sub.rup]) = [B[exp(B[P.sub.rup]) - 1]]/[exp(B[P.sub.rup,max]) - 1 - B[P.sub.rup,max]]. (1)
In this expression, B is a model parameter and [P.sub.rup,max] = 1/[d.sub.min] - 1/[d.sub.max] is the maximum rupture strength for the rupture strength function [P.sub.rup] = 1/d - 1/[d.sub.max]. By changing the value of the constant B, different distributions are generated.
The raw materials required to make water blown rigid polyurethane foam are polyol, isocyanate, catalysts, and surfactant. The polyol used was a sucrose-based polyether polyol (DC 9911, Huntsman International, India). The polyol had a hydroxyl number of 440 mg of KOH per gram of the polyol and an equivalent weight of 128 g/mol. The isocyanate was a polymeric diphenyl methane diisocyanate (MDI) (SUPRASEC 5005, Huntsman International). The isocyanate had an equivalent weight of 132 g/mol. The catalysts used were dibutyltin dilaurate (DBTDL) (Lancaster, England), a polymerization catalyst and triethanolamine (TEA) (Spectrochem, India) which catalyses both the polymerization and blowing reaction between isocyanate and water, which produces carbon dioxide. In the foam formulations, Tegostab B8404 and Cresmer B246M were used as surfactant and were donated by Goldschmidt AG Germany and ICI (Mumbai, India) respectively. All materials were used as received without further purification.
Initially polyol mixtures were prepared by stirring 250 g of the polyol in a one-liter plastic container with required amounts of water, catalysts, and surfactant for half an hour. The details of the formulations used are given in Table 1. The polyol mixture was mixed thoroughly with a predetermined amount of isocyanate for 15 s using a high-speed stirrer at 2800 rpm. The same speed and mixing time was maintained for all formulations. The mixing time was the maximum time possible that still left adequate time for pouring into the mold before the foam started rising. The mixture was immediately poured into a stainless steel mold. The mold was closed and kept at room temperature for 15 min. The foam was then removed from the mold. The mold had the dimensions of 127 x 127 x 317 [mm.sup.3] and was coated with a mold-release agent (wax). All characterizations were done after curing the foams at room temperature for at least one day. All foams were uniform in color according to visual observation, indicating uniform mixing of the reactants.
Hydraulic Resistance. Four samples of cubical shape with a size of 45-48 mm were cut from each foam. The weight of the samples was measured and then the samples were immersed in a sealed container completely filled with water. The container was connected to a pressure gauge and a hydraulic hand operated pump. The pressure inside the container was raised to a specific value using the hydraulic pump. After one hour the foam samples were taken out and water from the surface of the samples was removed by a piece of cloth and again weighed. During the one-hour equilibration period the pressure was found to decrease due to water absorption by the foam samples. Thus the pressure was checked at intervals of 5-10 min, and if there was any decrease, the pressure was raised to the specified value using the hand pump. The buoyancy loss was calculated by using the following formula
percentage buoyancy loss = [[[[rho]'.sub.f] - [[rho].sub.f]]/[[[rho].sub.w] - [[rho].sub.f]]] x 100, (2)
where [[rho].sub.f], [[rho]'.sub.f] and [[rho].sub.w] are the initial foam density, density of the foam after water absorption, and the density of water, respectively. The buoyancy loss was measured for different hydraulic pressures in the range 0-3 MPa. It was assumed that volume of the foam samples remained constant in the experiment.
Scanning Electron Microscopy. Scanning electron micrographs of selected samples were taken on an environmental scanning electron microscope (model Quanta 200; FEI; Brno, Czech Republic).
RESULTS AND DISCUSSION
In the following subsections the water-filling pattern for model 2D foam is described first. Subsequently, the effects of sample size, different rupture strength functions, average number of neighbors, minimum distance between the neighbors, distribution of cell volumes, different assumptions for the rupture strength, and randomly assigning rupture strength of the windows on the hydraulic resistance of the model foams (in 3D) are discussed. Finally the simulation results are compared with experimental results.
Water-Filling Pattern in 2D
One of the foams is constructed in 2D to qualitatively show the water-filling pattern. In an area ([L.sup.2]) of [1000.sup.2], [100.sup.2] cells ([N.sub.c]) are generated. The minimum distance ([d.sub.min]) between any two cells is 8.50. The average number of neighbors (nb[.sub.avg]) is fixed at 5.00 by setting a maximum distance ([d.sub.max]) of 13.67 between any two neighbors. The strength of the windows ([P.sub.rup]) is assumed to be an inverse function ([P.sub.rup] = 1/d - 1/[d.sub.max]) of the distance between the two concerned cells. The water-filling pattern for the model 2D foam is shown in Figure 3 in terms of a number of equilibrium snapshots at different values of applied pressure ([P.sub.app]). The wet cells are colored black and the dry cells gray. The values of the applied pressure ([P.sub.app]) and fraction filled with water ([phi]) at that pressure are given for each snapshot. The snapshots shown in Figure 3 are all concentrated in the region of rapid increase of the water fraction ([phi]). The figure shows that water progresses along paths of least resistance forming fractal like structures. Structures join up leaving a few dry islands and eventually the islands become smaller and disappear. The variation of the fraction of volume filled with water with applied pressure for this case is shown in Figure 4. The position of each snapshot on the hydraulic resistance curve (i.e., the [phi] vs. [P.sub.app] profile) is also shown in Figure 4. Figure 4 shows typical percolation behavior. At low pressures there is little ingress of water with increasing pressure but beyond a threshold value there is a sharp increase. This is followed by a slower increase in [phi] with pressure until the foam is completely filled. A consequence of having a fraction of cell windows of nearly zero strength is that there is a gradual increase in [phi] followed by a sharp increase. This is in contrast to the classical percolation problem in which the conductivity is zero until the percolation threshold.
Effect of Sample Size
In rigid polyurethane foams there are nearly [10.sup.6] cells/[cm.sup.3] and a typical sample used in the buoyancy loss experiments [4-7] was about [10.sup.2] cells/[cm.sup.3]. Simulations with [10.sup.8] cells are not feasible. Hence we consider model foams of different sizes to determine how large the foam must be for the results to be nearly independent of sample size.
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Foams of cubical shape of different sizes are constructed keeping the cell density fixed. The different parameters of the foams are given in Table 2 (Sets 1 and 2). Foams in Set 1 are generated independent of each other (the centers of the cells for different foams are generated giving different seeds for the generation of random numbers), whereas, foams in Set 2 are generated taking a part from the biggest foam of that set (i.e., from the foam with L = 1250, [N.sub.c] = [125.sup.3], and [N*.sub.c] = [N.sub.c.sup.1/3] = 125 corresponding to [10.sup.-3] cells per unit volume). All foams have the same minimum distance ([d.sub.min]) between any two cells and the same maximum distance ([d.sub.max]) between any two neighbors. The average number of neighbors (nb[.sub.avg]), average distance ([d.sub.avg]) between any two neighbors, and average volume of the cells ([v.sub.avg]) are calculated and given in Table 2 (for Sets 1 and 2). Table 2 shows that [d.sub.avg] and [v.sub.avg] are almost constant for all the foams in Set 1 and 2. Thus morphologies of the foams are identical; however, they differ in size. As the sample size decreases, the ratio of the cells at the surface to the cells in the bulk increases, and the numbers of neighbors of the cells at the surface are lower than that of the cells in the bulk. The average number of neighbors (nb[.sub.avg]) decreases with decrease in sample size because of this.
Figure 5 shows the effect of sample size on the hydraulic resistance of the two sets of foams. The graphs (5a and 5b) show that hydraulic resistance of the foam does depend on the chosen sample size; however, beyond a certain size (L = 750 and [N.sub.c] = [75.sup.3] there is no change in hydraulic resistance with sample size. The same trend is observed for both the Set 1 and Set 2 foams. Therefore, this sample size is considered suitable for further study. In the remaining part of this paper all foams have [75.sup.3] cells in a dimensionless volume of [750.sup.3].
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Effect of Different Rupture Strength Functions
The effect of different rupture strength functions on the hydraulic resistance and on the distribution of the strength of the windows ([P.sub.rup]) is shown in Figure 6 for the foam 1D. The effect of the rupture strength functions on the distribution of the strength of the windows ([P.sub.rup]) is shown in Figure 6a. Figure 6b shows the effect of rupture strength functions on the hydraulic resistance. The curves in Figure 6b are almost identical and only the scales along pressure axis are different due to the nature of those functions. At lower pressures, [phi] remains almost constant but at higher pressures, [phi] increases very steeply. The curves reach a plateau for [phi] greater than about 0.90. The same kind of behavior was observed experimentally [4-7]. Figure 7 shows window strength distribution and volume fraction of water absorbed versus applied pressure for the quadratic strength function. The curves are qualitatively similar to those in Figure 6 but of different magnitude. A threshold pressure ([P.sub.th]) is defined as the point of intersection of the two straight lines, representing the best straight lines for the data at low pressures and at high pressures. A typical calculation of threshold pressure is shown in Figure 8. The threshold pressure and the applied pressure corresponding to [phi] = 0.90 are marked in the hydraulic resistance curve in Figure 7b.
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It is interesting to note that only 9% windows have strength less than the threshold pressure ([P.sub.th]). These windows are distributed throughout the foam and not all these windows are ruptured at the applied pressure, [P.sub.app] = [P.sub.th], since at low pressures only the cells, which are near the surface, are ruptured. This is evident from the water-filling pattern shown in Figure 3. Thus at the threshold of failure of the foam much less than 9% of the cell windows are ruptured. Furthermore, only 19% of the windows have strength less than the applied pressure at which nearly 90% of the gas volume is filled with water. Thus less than 19% of the windows of the foam must be ruptured to fill 90% of its volume with water. This indicates that the weaker windows control the hydraulic resistance of the foam. Furthermore, only a small fraction of weak windows is sufficient to permit nearly complete filling of the foam with water. The percentages of cell windows with strength less than the threshold pressure and less than applied pressure corresponding to [phi] = 0.90 are given in Table 3 for the different model foams and the different strength functions used. Although the values vary a little for the different foams, they are nearly independent of the strength functions. In Figure 7a, the percentage of cell windows with strength less than the threshold pressure and the percentage less than applied pressure corresponding to [phi] = 0.90 are marked by a solid and a dotted line, respectively.
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For comparison of different profiles a relative pressure, P*, is defined as P* = [P.sub.app]/[P.sub.th]. The percolation curves shown in Figures 6b and 7b are plotted in terms of the relative pressure in Figure 7c. The figure shows that [phi] vs. P* curves for the five different rupture strength functions are nearly same. This is consistent with the results given in Table 3. The results are independent of the strength functions because only the weakest windows ([P.sub.rup] [approximately equal to] 0) determine the percolation curve and all the distributions are quite similar in this region.
Effect of Average Number of Neighbors (nb[.sub.avg])
The effect of average number of neighbors (nb[.sub.avg]) on the hydraulic resistance of the foam is shown in Figure 9a. The figure shows that hydraulic resistance decreases with increase in average number of neighbors from 11 to 13. This is due to two reasons. To increase the nb[.sub.avg] from 11 to 13 (foams 1C-1E) the maximum distance between any two neighbors ([d.sub.max]) is increased from 14.24 to 15.10 (Table 2). Therefore some windows become weaker. At the same time with increase in average number of neighbors the probability of getting at least one weak cell window increases. The percolation curves are rescaled with the threshold pressure and plotted in Figure 9b and the figure shows that the [phi] vs. P* profiles for three foams are nearly same.
Effect of Minimum Distance ([d.sub.min]) between the cells
The effect of minimum distance between any two cells, [d.sub.min], on foam properties is shown in Figure 10. The distributions of the strength of the windows of the foams (1A, 1B, and 1D) are given in Figure 10a. The hydraulic resistances of the foams are compared in Figure 10b. The hydraulic resistance of the foam decreases slightly with decrease in [d.sub.min]. Figure 10a shows that as [d.sub.min] decreases, the maximum strength of the windows increases. However, a consequence of this is that the fraction of the weakest windows increases slightly (Fig. 10a). Since it is the weaker windows that control the hydraulic resistance of the foam, the hydraulic resistance of the foam thus decreases with decrease in [d.sub.min]. However, the rescaled percolation curves show no effect of [d.sub.min] (Fig. 10c).
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Effect of Distribution of Cell Volumes
Instead of calculating the volume of a cell from the mean distance from neighbors as described earlier, simulations are carried out with arbitrary volumes assigned to each cell picked randomly from a specified distribution. The different distributions of the cell volumes used are shown respectively in Figure 11a-11c. In another case it was considered that volumes of all cells are same. The percolation curves of the foam (1D) calculated for the different volume distributions are shown in Figure 11d. There is no noticeable effect of cell volume distribution on hydraulic resistance curve. This is because a large number of cells are ruptured at any applied pressure. For example, even at low pressure ([P.sub.app] = 0.0004) a large number of cells (7582) are ruptured even though [phi] is low ((0.018). Since the probability of rupture of a cell is independent of its volume, the distribution of volumes of cells is uniformly sampled. Thus for the different distributions, the average volumes of the wet cells are nearly same as the average volume of all cells (Table 4) and hence there is no effect of the distribution of volumes of the cells on the percolation curve.
Effect of Different Assumptions for the Rupture Strength
The effect of different assumptions for the rupture strength ([p.sub.rup] = 1/d - 1/[d.sub.max]; [P.sub.rup] = 1/[r.sub.ij, min] - 1/[r.sub.max] and [P.sub.rup] = 1/([r.sub.i] + [r.sub.j]) - 1/2[r.sub.max]) of the windows of the foam 1D on the hydraulic resistance is shown in Figure 12. Figure 12a shows that the nature of the distribution of the strength of the windows is different for different assumptions. When [P.sub.rup] = 1/d - 1/[d.sub.max] the distribution is nearly uniform with windows of lower strength being more numerous. However, for the other two assumptions the distributions are close to Gaussian. The distribution for [P.sub.rup] = 1/([r.sub.i] + [r.sub.j]) - 1/2[r.sub.max] is narrower than the distribution for [P.sub.rup] = 1/[r.sub.ij, min] - 1/[r.sub.max]. The hydraulic resistance curves of the foam with three different distributions of the window strength are shown in Figure 12b. The curves are significantly different from each other. Figure 12c shows the rescaled percolation curves and are different for different assumptions made for the strength of the windows. This is because of the significantly different distribution of the rupture strengths in the three cases. The threshold pressure ([P.sub.th]) and the applied pressure [P.sub.app|[phi] = 0.90] along with the corresponding percentage of windows with strength less than these pressures are given in Table 5 for these three assumptions. For the assumption [P.sub.rup] = 1/d - 1/[d.sub.max] when only 19% windows of the foam are ruptured, 90% of its volume gets filled with water. However, when the strength of a window depends on the radius of the smaller cell ([P.sub.rup] = 1/[r.sub.ij, min] - 1/[r.sub.max]) 90% of the foam's volume gets filled with water only after 80% of its windows are ruptured. Rupture of nearly 49% windows must occur to fill 90% of the foam's volume if strength of a window depends on the average size of a pair of cells ([P.sub.rup] = 1/([r.sub.i] + [r.sub.j]) - 1/2[r.sub.max]). This is because there is a sharp reduction in the proportion of the weakest windows in both the modified distributions.
Effect of Randomly Assigning Rupture Strength of the Windows
The effect of randomly assigning rupture strength of the windows from the distribution given in Eq. 1 is shown in Figure 13 for different values of the model parameter B. Figure 13a shows the distributions of the strength of the windows. The figure shows that for B [less than or equal to] 1.0 the distribution is linear and as the value of constant B increases the distribution becomes narrower and the fraction of weak windows decreases. Note that the window strength distributions for B = 0.01 and B = 1.0 are the same and so only the results for B = 0.01 are shown. The fraction of gas volume filled of the foam 1D at different applied pressure with the earlier-mentioned distributions of strength of the windows is shown in Figure 13b. The figure shows that the threshold pressure increases and the curve becomes steeper with increase in the value of B. The percentage of weaker windows decreases with increase in B and for this reason threshold pressure increases with B. The distribution becomes narrower with increasing B and thus the percolation curve becomes steeper as value of B is increased. The curves in Figure 13b are rescaled and plotted in Figure 13c. The figure shows that nature of the rescaled percolation curve changes as the distribution of the window strength is changed. The percentages of windows having strength less than [P.sub.th] and [P.sub.app|[phi] = 0.90] are given in Table 5 for different values of B. The table shows that both at the threshold pressure and at the applied pressure [P.sub.app|[phi] = 0.90] nearly the same numbers of windows are ruptured for different values of B. About 28% of the windows must rupture to fill 90% of the volume of the foam with water.
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Comparison of Experimental and Simulation Results
Figure 14 shows experimental results for water absorption by polyurethane foam with increasing applied pressure. The error bars represent standard deviation for four samples. The foams in Figure 14a were made with various amounts of the catalysts, dibutyltin dilaurate (DBTDL) and triethanol amine (TEA). The hydraulic strength increases with increase in DBTDL concentration. DBTDL catalyses the polymerization reaction, but not the blowing reaction. Thus as the proportion of the DBTDL is increased there is a faster viscosity increase. This reduces the rate of capillary force driven drainage from the cell windows into the struts, resulting in thicker cell windows, which are stronger . The experimental results for another two foams made with two commercially available surfactants Cresmer B246M and Tegostab B8404 are shown in Figure 14b. Details of the properties of these two foams are reported in a previous work .
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The simulation results of the foam 1D for different assumptions for the rupture strength ([P.sub.rup] = 1/d - 1/[d.sub.max]; [P.sub.rup] = 1/[r.sub.ij, min] - 1/[r.sub.max] and [P.sub.rup] = 1/([r.sub.i] + [r.sub.j]) - 1/2[r.sub.max]) are compared with the experimental results in Figures 15a and 15b. The rescaled experimental curves show a linear increase followed by sharp increase beyond the threshold pressure, which is also roughly linear. B246M and B8404 data collapse to a single curve but P_50, P_75, and P_100 show different rates of increase beyond threshold pressure. The figures show that there is good agreement between the experiments and the predictions up to threshold pressure for all cases. The theoretical prediction for the rupture strength function [P.sub.rup] = 1/d - 1/[d.sub.max] is in qualitative agreement with the experimental results for the 100% DBTDL foam (P_100), but not the others. The prediction for the function, [P.sub.rup] = 1/[r.sub.ij, min] - 1/[r.sub.max], is qualitatively similar to the experimental results. The prediction of this function agrees with the experimental results of B246M and B8404. However, it increases faster than P_50 and P_75 but slower than P_100. For [P.sub.rup] = 1/([r.sub.i] + [r.sub.j]) - 1/2[r.sub.max], the prediction is in agreement with the experimental results up to threshold pressure; however, beyond that the model prediction increases faster than the experiemental results.
The rescaled percolation curves for different values of the parameter B (shown in Fig. 13c) for the randomly assigned rupture strength of the windows are compared with the experimental results in Figures 16a and 16b. The theoretical predictions are in agreement with the experimental results up to the threshold pressure. However, beyond the threshold pressure the theoretical curves are qualitatively dissimilar to the experimental results.
Hydraulic resistance behavior of rigid polyurethane foam was studied in this work by means of simulations. Foams of cubical shape were constructed by generating the centers of the cells randomly, maintaining a minimum distance between any two cells. The effects of sample size, different rupture strength functions for the strength of the windows, minimum distance between the centers of the cells and distribution of cell volume on the hydraulic resistance were studied. It was found that the measured hydraulic resistance of the foam depends slightly on the sample size but beyond a certain size there is no change in hydraulic resistance. It was also found that the hydraulic resistance of the foam decreases as the minimum distance between the cells decreases. However, it is independent of the cell volume. Simulations indicate that it is mainly the weaker windows that control the hydraulic resistance of the foam.
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Different assumptions were made regarding the rupture strength of a window in the simulations. Initially, the strength of a window was assumed to be an inverse function of the distance between the centers of the two concerned cells. Different functional forms were considered. However, the rescaled percolation curves were found to be independent of the functional form of the rupture strength. Further assumptions were made to describe the hydraulic resistance of the polyurethane foam more quantitatively. In one case it was considered that the strength of a window between two cells is an inverse function of the radius of the smaller cell. In another case it was assumed that strength of a window is an inverse function of the sum of the radii of the two concerned cells. The rescaled percolation curves are different in these cases when compared with the case when the rupture strength depended only on the distance between the cells. Finally it was assumed that rupture strength of the windows to be randomly distributed and independent of cell size. An exponential distribution with one parameter was considered to generate the strength of the windows. Different rescaled curves were obtained for different parameter values.
Comparison between theoretical predictions and experiments indicate that [P.sub.rup] = 1/[r.sub.ij, min] - 1/[r.sub.max] gives the best prediction of the rescaled percolation curve. There is good qualitative agreement with experimental results for some cases and quantitative agreement with the other experimental results. The results presented indicate that the model gives a reasonably good description of the percolation process. Detailed modeling at the cell level to obtain strength functions based on the local geometry of the cells is required for obtaining more accurate results.
NOMENCLATURE (ALL THE PARAMETERS ARE DIMENSIONLESS)
L Length of the cube [N.sub.c] Total number of the cells [N.sub.c]* [N.sub.c.sup.1/3] [d.sub.ij] (= d) Distance between the centers of the two neighboring cells i and j [d.sub.min] Minimum distance between the centers of any two neighboring cells [d.sub.max] Maximum distance between the centers of any two neighboring cells [d.sub.avg] Average of all [d.sub.ij] nb[.sub.avg] Average number of neighbors of the cells [v.sub.avg] Average volume of the cells (cells are assumed to be spherical) [r.sub.i] Radius of the ith cell. [r.sub.ij,min] Radius of the smaller cell between two neighboring cells [r.sub.max] Radius of the largest cell [[rho].sub.c] = Cell density [N.sub.c]/L [P.sub.rup] Strength of the window, i.e. pressure required to rupture a window [P.sub.app] Applied pressure [P.sub.th] Threshold pressure P* = [P.sub.app]/ relative pressure [P.sub.th] [phi] Fraction of total volume of the cells filled with water
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Pravakar Mondal, D.V. Khakhar
Department of Chemical Engineering, Indian Institute of Technology--Bombay, Powai, Mumbai 400076, India
Correspondence to: D.V. Khakhar; e-mail: email@example.com
TABLE 1. Formulation used for making different foams. Formulations (pphp (a)) Ingredients B8404 B246M P_50 P_75 P_100 polyol 100.0 100.0 100.0 100.0 100.0 water 1.0 1.0 1.0 1.0 1.0 surfactant 3.0 3.0 3.0 (b) 3.0 (b) 3.0 (b) DBTDL (c) 0.5 0.5 1.26 1.89 2.53 TEA (d) 0.5 0.5 0.30 0.15 0.0 isocyanate index 105 (a) Parts per hundred g of polyol. (b) Tegostab B8404 used as surfactant. (c) Molecular weight 631.11. (d) Molecular weight 149. TABLE 2. Parameters for the different foams generated. Foam L [N.sub.c] [[rho].sub.c] [d.sub.min] [d.sub.max] Set 1 250 15625 0.001 8.50 14.60 500 125000 0.001 8.50 14.60 750 421875 0.001 8.50 14.60 1000 1000000 0.001 8.50 14.60 1250 1953125 0.001 8.50 14.60 Set 2 250 15873 0.001 8.50 14.60 500 125195 0.001 8.50 14.60 750 420616 0.001 8.50 14.60 1000 995564 0.001 8.50 14.60 1250 1953125 0.001 8.50 14.60 IA 750 753 0.001 7.50 14.68 IB 750 753 0.001 8.00 14.68 IC 750 753 0.001 8.50 14.24 ID 750 753 0.001 8.50 14.68 IE 750 753 0.001 8.50 15.10 Foam L [d.sub.avg] [v.sub.avg] Set 1 250 11.58 [+ or -] 1.81 811.9 [+ or -] 104.2 500 11.58 [+ or -] 1.81 810.4 [+ or -] 99.5 750 11.50 [+ or -] 1.81 810.4 [+ or -] 98.1 1000 11.58 [+ or -] 1.80 810.1 [+ or -] 97.6 1250 11.45 [+ or -] 1.79 809.9 [+ or -] 97.1 Set 2 250 11.55 [+ or -] 1.82 805.4 [+ or -] 102.0 500 11.57 [+ or -] 1.81 808.1 [+ or -] 99.0 750 11.49 [+ or -] 1.81 809.4 [+ or -] 98.1 1000 11.56 [+ or -] 1.80 809.9 [+ or -] 97.6 1250 11.45 [+ or -] 1.79 809.9 [+ or -] 97.1 IA 750 11.51 [+ or -] 2.06 806.8 [+ or -] 106.2 IB 750 11.54 [+ or -] 1.95 814.6 [+ or -] 102.4 IC 750 11.27 [+ or -] 1.71 767.6 [+ or -] 94.3 ID 750 ll.56 [+ or -] 1.84 820.3 [+ or -] 99.0 IE 750 11.84 [+ or -] 1.97 874.6 [+ or -] 102.8 No. of neighbors (nb[.sub.avg]) Foam L Min Max Avg (nb[.sub.avg]) [N*.sub.c] Set 1 250 2 18 10.95 [+ or -] 2.07 25 500 2 19 11.59 [+ or -] 1.84 50 750 3 20 11.82 [+ or -] 1.74 75 1000 2 19 11.93 [+ or -] 1.68 100 1250 2 19 11.99 [+ or -] 1.65 125 Set 2 250 2 18 11.36 [+ or -] 2.18 25 500 2 19 11.73 [+ or -] 1.89 50 750 1 19 11.87 [+ or -] 1.78 75 1000 2 19 11.94 [+ or -] 1.71 100 1250 2 19 11.99 [+ or -] 1.65 125 IA 750 1 20 11.99 [+ or -] 2.03 75 IB 750 1 20 11.97 [+ or -] 1.87 75 IC 750 2 18 11.01 [+ or -] 1.68 75 ID 750 3 20 12.00 [+ or -] 1.76 75 IE 750 3 21 13.00 [+ or -] 1.84 75 TABLE 3. Fraction of windows (%) having strength less than the threshold pressure ([P.sub.th]) and the applied pressure at which 90% of the foam volume is filled with water ([P.sub.app|[phi] = 0.90]). % of windows with strength less than Foam Function [P.sub.th] 1A [P.sub.rup] = 1/d - 1/[d.sub.max] 8.93 1B [P.sub.rup] = 1/d - 1/[d.sub.max] 9.01 1C [P.sub.rup] = 1/d - 1/[d.sub.max] 9.77 1D [P.sub.rup] = 1/d - 1/[d.sub.max] 8.62 [P.sub.rup] = (1/d)[.sup.1/2] - 8.66 (1/[d.sub.max])[.sup.1/2] [P.sub.rup] = exp(1/d) - 8.43 exp(1/[d.sub.max]) [P.sub.rup] = 1/ln(d) - 8.70 1/ln([d.sub.max]) [P.sub.rup] = 1/[d.sup.2] - 8.76 1/[d.sub.max.sup.2] 1E [P.sub.rup] = 1/d - 1/[d.sub.max] 8.12 % of windows with strength less than Foam Function ([P.sub.app|[phi] = 0.90]) 1A [P.sub.rup] = 1/d - 1/[d.sub.max] 19.57 1B [P.sub.rup] = 1/d - 1/[d.sub.max] 19.66 1C [P.sub.rup] = 1/d - 1/[d.sub.max] 21.49 1D [P.sub.rup] = 1/d - 1/[d.sub.max] 19.45 [P.sub.rup] = (1/d)[.sup.1/2] - 19.89 (1/[d.sub.max])[.sup.1/2] [P.sub.rup] = exp(1/d) - 19.46 exp(1/[d.sub.max]) [P.sub.rup] = 1/ln(d) - 19.33 1/ln([d.sub.max] [P.sub.rup] = 1/[d.sup.2] - 19.13 1/[d.sub.max.sup.2] 1E [P.sub.rup] = 1/d - 1/[d.sub.max] 18.02 TABLE 4. The numbers of wet cells, the fraction of the foam filled with water ([phi]) and the average volume (avg. vol.) of the wet cells at different applied pressures ([P.sub.app]) for different assumed distributions for the volume of each cell of the foam 1D. No. of Unimodal (a) [P.sub.app] wet cells [phi] Avg. vol. of the wet cells 0.0004 7582 0.0175 800.8 [+ or -] 122.5 0.0026 58,684 0.1439 848.5 [+ or -] 99.3 0.0032 22,0535 0.5402 847.7 [+ or -] 93.9 0.0038 29,3831 0.7152 842.3 [+ or -] 93.4 0.0050 36,0974 0.8704 834.5 [+ or -] 93.5 0.0110 41,7879 0.9928 822.2 [+ or -] 97.3 Bimodal (b) Avg. vol. of Trimodal (c) [P.sub.app] [phi] the wet cells [phi] Avg. vol. of the wet cells 0.0004 0.0180 12.541 [+ or -] 0.0180 27.700 [+ or -] 13.967 5.190 0.0026 0.1393 12.511 [+ or -] 0.1394 27.596 [+ or -] 13.971 5.201 0.0032 0.5229 12.499 [+ or -] 0.5230 27.555 [+ or -] 13.954 5.206 0.0038 0.6966 12.496 [+ or -] 0.6965 27.546 [+ or -] 13.942 5.207 0.0050 0.8557 12.497 [+ or -] 0.8558 27.550 [+ or -] 13.937 5.205 0.0110 0.9906 12.496 [+ or -] 0.9906 27.546 [+ or -] 13.935 5.205 (a) Avg. vol. of all cells, 820.309 [+ or -] 98.990. (b) Avg. vol. of all cells, 12.495 [+ or -] 5.205. (c) Avg. vol. of all cells, 27.545 [+ or -] 13.935. TABLE 5. Fraction of windows (%) having strength less than the threshold pressure ([P.sub.th]) and the applied pressure at which 90% of the foam volume is filled with water ([P.sub.app|[phi] = 0.90]) for different rupture strength function for the foam 1D. % of windows with strength Rupture strength less than function [P.sub.th] [P.sub.th] ([P.sub.app|[phi] = 0.90]) [P.sub.rup] = 0.0022 8.62 0.0056 1/d - 1/ [d.sub.max] [P.sub.rup] = 1/ 0.0160 0.95 0.0336 [r.sub.ij, min] - 1/[r.sub.max] [P.sub.rup] = 0.0080 3.02 0.0122 1/([r.sub.i] + [r.sub.j]) - 1/(2[r.sub.max]) Equation 1 B = 0.01 0.0118 5.64 0.0264 B = 50.0 0.0169 5.63 0.0324 B = 0.0245 5.74 0.0380 100.0 B = 0.0311 5.58 0.0416 150.0 Rupture strength % of windows with strength less than function ([P.sub.app|[phi] = 0.90]) [P.sub.rup] = 19.45 1/d - 1/ [d.sub.max] [P.sub.rup] = 1/ 79.56 [r.sub.ij, min] - 1/[r.sub.max] [P.sub.rup] = 48.77 1/([r.sub.i] + [r.sub.j]) - 1/(2[r.sub.max]) Equation 1 B = 0.01 27.97 B = 50.0 28.07 B = 28.03 100.0 B = 28.10 150.0
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|Author:||Mondal, Pravakar; Khakhar, D.V.|
|Publication:||Polymer Engineering and Science|
|Date:||Jul 1, 2006|
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