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Density modeling of polyurethane box foam.


Polyurethanes (PU) play a significant role in global industry with over three-quarters of the consumption in the form of foams [1], They are extensively used in the field such as rigid insulations in the walls of refrigerators and buildings, high-resilience and flexible foam cushions, and elastomeric wheels and tires.

Foams can expand from during polymerization if gases or vapors are produced in concert with the polymerization. A common mechanism for gas generation is the chemical reaction between water and isocyanate forming carbon dioxide. In addition, exothermic polymerization reactions can provide heat to evaporate volatile blowing agents such as methyl formate (MF) and pentane. For rigid foams, the physical properties of most interest are density, compressive strength, and thermal conductivity.

Both cell size and cell number contribute to change of density and compressive strength [2], For PU foam, to a first approximation, there exists a linear relationship between density and compressive strength [3]. Density impacts the thermal conductivity due to the lower thermal conductivity of the gas phase relative to the resin phase [4], Therefore, accurately predicting density during polyurethane foaming is both important in its own right and as a critical step to ultimately predict other physical properties of foams [5-7].

Baser and Khakhar [8, 9] carried out a detailed experimental study of chemical and physical blowing agents with measurement of both temperature and density changes during the foaming process. The theoretical model was based on the hypothesis that the foam was under a single pseudohomogeneous phase, and the variation of density with time was obtained by applying energy and mass balances along with the kinetic and thermodynamic relations. Later, Tesser et al. [10] developed a mathematical model to simulate the temperature and density of the foams with the evaluation of kinetics of the reaction between polyol and isocyanate during the foaming process with an emphasis on Flory-Huggins modeling of blowing agent activity.

Gibson and Ashby [11] came up with an elastic modulus based on the relative density for closed-cell (cc) foam taking into account of three components: the strut flexion, the stretching of the cell walls, and the gas pressure inside the cc. Saint Michel et al. [12] characterized the influence of the density and compared the experimental results with Gibson and Ashby [11] and Christensen et al. [13] in the linear domain and then extended to the nonlinear domain.

Avalle et al. [14] tested several types of foams and performed a study based on the Gibson model and the Rusch (of the phenomenological type) model [15-17] together with two other models namely a modified Gibson model and a new phenomenological model to characterize the mechanical properties. Additionally, available experimental data were used to obtain the relationship between material density and model parameters.

These previous works illustrate the importance of quantifying foam density in foam formulation. The work of this article differs in two ways from this other work: (a) the current work is based on over a dozen fundamental ordinary differential models for the reactions and physical processes and (b) this work evaluates sources of blowing agent loss in addition to solubility of blowing agents in the resin phase.

This work builds on the work of Zhao et al. [18] on modeling the foaming process using a series of ordinary differential equations with Arrhenius constants and enthalpy parameters. Although the emphasis of Zhao's work was the prediction of polyol mixture performance in formulations, the emphasis of this work is the modeling and simulation of foam height (density).


Blowing Agents

As a chemical blowing agent, water reacts with isocyanate (RNCO) and produces carbamic acid (RNHCOOH) which further decomposes into carbon dioxide and thus generates gas bubbles in the resin according to reaction Schemes 1 and 2 forming an amine (RN[H.sub.2]).

RNCO + [H.sub.2]O [right arrow] RNHCOOH (1)

RNHCOOH [right arrow] RN[H.sub.2] + C[O.sub.2] + HEAT (2)

MF serves as a physical blowing agent that does not rely on any chemical reactions. Table 1 summarizes physical properties of MF. The MF is initially soluble in the mixture of isocyanate and polyol at ambient temperature.

Equation 3 is used to estimate the vapor pressure of MF as a function of time [19].

ln ([P.sup.sat.sub.i]/[P.sub.C]) = [(1 - x).sup.-1] [([A.sub.i])x + ([B.sub.i])[x.sup.1.5] + ([C.sub.i])[x.sup.3] + ([D.sub.i])[x.sup.6]] (3)

x = 1--T/[T.sub.c] where [T.sub.c] is critical temperature in Kelvins, [P.sup.sat.sub.i] is vapor pressure, in bars, [P.sub.c] is critical pressure, in bars, Eqs. 3 and 4 are valid at the temperature range of 220-487.2 K.

The vapor pressure is used in the modified Raoult's Law (Eq. 4) equation which is used in combination with heat and energy balances to estimate the amount of MF that evaporates from the resin phase and proceeds to form bubbles/cells. It should be noted that because the resin phase is comprised of macromolecules, mass fraction was used instead of mole fraction.

[x.sub.i][gamma][P.sup.sat.sub.i] = [y.sub.i]P (4)

It is assumed that the carbon dioxide and vapor-phased MF are ideal with fugacity equaling to 1. Deviation from ideality is accommodated by modifying the activity coefficient. Mass balance equations are solved under the constraint of Raoult's law to determine the extent to which MF evaporates. During analysis of the data, performance at an activity coefficient of 1.0 is compared to 15 to evaluate the extent that actual foam density varies from the ideal model.


The isocyanate and petroleum-based polyols used in this study were RUBINATE M isocyanate, Poly G76-635, Voranol 360, and Jeffol R315x from Huntsman Company and Dow Chemical Co. and their specifications are shown in Tables 2. AUV-dimethylcyclohexylamine and N, N, N', N"', N"-pentamethyldiethylenetriamine were used as gelling catalyst and blowing catalyst, respectively. Momentive L6900 was used as the surfactant for rigid foams, Tris (1-chloro-2-propyl) Phosphate (TCPP) was used as fire retardant, and distilled water was use as chemical blowing agent.

Experimental Procedure

Gelling and Foaming Experiment. Experiments were performed to create polyurethane gels as well as rigid polyurethane foams. The gel reactions were used as a control to assist in interpreting the data for the foam reactions. Table 3 provides the control formulation used in these studies. The amount of water and MF in this formulation was used as parameters to better understand the foaming process.

The following steps were used in both the gelling and foam experiments.

1. Polyols, water, blowing catalyst, gelling catalysts, and surfactant (B-side components) were added into a plastic cup successively.

2. These B-side components were mixed for 10-15 s.

3. The mixture was allowed to degas for 2 min.

4. Thereafter, preweighed isocyanate (A-side material) was added and mixed at the same speed for 7-10 s.

5. The reacting mixture was then quickly poured into a box with aluminum foil lining, and the foam was allowed to rise and sit at ambient conditions (25[degrees]C) during curing.

All the B-side chemicals were added in the foam reaction, whereas water and blowing catalyst were not added in the gel reaction.

A high-speed mixer blade attached to a floor-model drill press was used to mix the chemicals. LabVIEW software was used to monitor the temperature of the gel or foam reactions for the first 15 min with a type-k thermocouple attached through a National Instruments SCB-68 box to a National Instruments PCI 6024E data acquisition card.

Mass Loss Test. Experiments were also performed to evaluate the amount of mass loss during box foaming. A cardboard box containing the reacting foam mixture was placed on a digital balance accurate to 0.01 g. The following sequence describes the manner in which mass was balanced and measured:

1. Polyols, water, blowing catalyst, gelling catalysts, and surfactant (B-side components) were added and stirred in a plastic cup. The B-side component was weighed and set aside in a second plastic cup. About 2 min were allowed for degassing.

2. The B-side components were mixed and stirred with A-side for 15 s.

3. The mixtures were quickly poured into the cardboard box (designated at t = 0 s) located on the balance. The foam was allowed to rise and sit

at ambient conditions during curing.

4. Residual masses in the mixing cup and stirrer were measured (by mass differences) and recorded for subsequent mass balance calculations.


Foaming Height Studies

Figure 1 demonstrates the effect of water content on the height of PU foams. Increasing water content leads to increasing height of the box foam and decreasing density. A doubling of the water content resulted in an approximate doubling of the height.

As a physical blowing agent, MF does not react with isocyanate; it provides an additional degree of freedom to control foam volume. Because of a lower boiling point and volatility, MF volatilizes and dissolves in polymer phase and consequently bubbles inside the foams come into being. Figure 2 illustrates how more MF produces more vapor which contributes to the volume increase. Based on inspection, water appears to be more effective in forming cells; the subsequent discussion pursues a quantification and insight into the differences.

Increased amounts of both water and MF provide lower foam densities. These trends are qualitatively consistent with the mechanisms of Eqs. 2 and 4. Figure 3 compares performances to what the ordinary deferential equation model predicts for extreme cases of the activity coefficient equal to 1 and 15. Even in the extreme case of high activity coefficients, the actual foam height is much less than what is projected by the model. Temperatures in the foam exceed 100[degrees]C during curing, and as such, little MF remains in the liquid even if high activity coefficients impacted performance. The model trends show that the activity coefficient predominantly has an impact during foam rise when the temperature ranged from MF's boiling point to about 40[degrees]C greater than MF's boiling point.

Actual foam height only attained about 63% of what was projected with full use of the blowing agents and ideal gas law calculation of the gas volume. Possible factors leading to the inconsistency are low-efficiency water-isocyanate reaction, the rupture of carbon dioxide and MF gas cells, and higher internal pressure. Further studies were conducted that evaluated the mass of the foam during the foaming process.

Mass Loss Studies

The box used to contain the rigid foam during the foaming process was placed on a digital balance accurate to 0.01 g to monitor weight loss during the foaming reaction. Possible sources for weight loss are (1) air's buoyant force applied by the increasing volume of the foam, (2) rupture of foam cells (bubbles) with collapse of the cell or replacement of the vapors by new vapors, and (3) evaporation/escape of MF, water, or carbon dioxide along the upper surface of the foam.

Figure 4 illustrates the mass loss for a sample that used only MF blowing agent and distinguishes between buoyant force and true weight loss. The volume caused by buoyancy was calculated using Eq. 5.

[V.sub.b] = [DELTA]V * [[rho].sub.a] = ([V.sub.0] - [V.sub.exp]) x [[rho].sub.a] (5)

[V.sub.b] is volume of the buoyancy, [[rho].sub.a] is density of the air, [V.sub.o] is initial volume before the growth of the foam, which corresponds to the total volume of isocyanate, polyols, water, and MF, [V.sub.exp] is experimental volume of the foam.

For the foam of Fig. 4, [V.sub.o] = 92.5 [cm.sup.3] and [V.sub.exp] = 572.7 [cm.sup.3] resulting in an equivalency of 0.57 g of buoyancy. It is presented from Fig. 4 that the total amount of loss is 2.13 g, which indicates 1.38 g of mass loss attributed to the inefficiency of blowing agent (much of which is attributed to the box-foaming technique rather than the blowing agent).

Table 4 summarizes the buoyancy forces and mass loss of several foams. [DELTA]m and [DELTA]h are total mass loss from experiment and height decrease associated with the release of the blowing agents. The foam in Fig. 4 stopped expanding at 75 s because an increase of viscosity facilitated the rigidity of the foam that did not allow further expansion. During the first 75 s, the mass loss was resulted from the buoyancy applied by the increasing volume of displaced air and the rupture of blowing agent characterized as [DELTA][m.sub.2] and [DELTA][m.sub.3] in Fig. 4. Afterwards, no expansion occurred in the system, and all the mass loss was attributed to the escape of gases in the cells.

Figures 5 and 6 compare actual to model heights where the model heights are for no loss of blowing agent versus losses as summarized by Table 4. The results substantiate that a primary mechanism for the inefficiency of blowing agent is the loss of blowing agent through the upper surface of the foam through cell rupture (convection) or through the surface evaporation. Loss from the top surface could be through diffusion straight from the resin to the air or it could be from cells/bubbles that rupture through the top surface.

Modeling Loss of Blowing Agent Efficacy

Possible sources of inefficacy of blowing agents include: (1) higher than atmospheric pressure in cells, (2) solubility of blowing agent in the polymer phase, and (3) release of blowing agent through the upper surface. The impact of these is summarized by Eq. 5 in terms of gas volume.

V = [V.sub.i] - [V.sub.p] - [V.sub.s] - [R.sub.b] (6)

[V.sub.i] is ideal volume of the foam without the gas rupture, [V.sub.p] is volume loss due to the pressure, Vs is volume loss due to the solubility of the MF, [R.sub.b] is release of the blowing agent.

This approach does not distinguish between the susceptibility of MF versus carbon dioxide to these mechanisms. Under the assumption that the pressure buildup has a minor contribution, Fp is cancelled out in Eq. 6.

Closed cell content is a property commonly measured in rigid foam. A reasonable hypothesis is: if the final foam has a high cc content, the foam would have maintained a high cc content during the entire foaming process; and cc's correlate this low release of blowing agent. The Eq. 7 is an example correlation that can be tested with data where cc is close cell content (%).

[R.sub.b] = [k.sub.2]{1 - cc) (7)

Two assumptions to simplify this model are assuming ideal gas behavior and assuming that only MF partitions in the resin phase. Substituting Eq. 6 to 8, where [n.sub.1] and [n.sub.2] are moles of MF and carbon dioxide, and [S.sub.1] is the fraction of MF that is in the resin (soluble).

V = [[([n.sub.1] + [n.sub.2])/RT/P] - [[n.sub.1]RT]/P] [S.sub.1] - [k.sub.2](1 - cc) (8)

By setting [V.sub.o] equals to the first two terms of the three on the right-hand side of Eq. 8 and expressing volume in term of density, Eq. 8 can be represented as a linear equation where the extent to which a varies from [k.sub.2] provides a point of discussion. Equation 9 displays a relationship between density and close cell content used to interpret the data.

[1/[[rho].sup.o]] - [1/[rho]] = [k.sub.2]cc + a (9)

Figure 7 summarizes this trend for density versus close cell content data collected as summarized by Table 5. The figure illustrates that while [k.sub.2] is close to a, the best fit is not with them equal. Furthermore, the data illustrate that the primary cause for them not being equal is the concentration of data at a nonzero limit as cc approaches 1.

For foam cells to expand, they must have a pressure greater than the surroundings. For foams with nearly 100% cc content, the strength of the resin to preserve cell integrity is at its greatest relative to foam with an open cell nature, weak cell walls, and respective low cc content. The conclusion is that the internal pressure is significant, is a function of cc, and is likely near a value of 2.02 atm at 100% cc content.

The lower dashed line of Fig. 7 represents a linear relation for internal cell pressure. Within the standard deviation of the data, the linear model (solid line) as represented by a combination of the two dashed lines is substantiated.

Future work on this area would include several topics of interest, including: (a) development of fundamentally based (not simple linear equations) for the relations of cell pressure and release to cc., (b) acquiring more accurate data, and (c) developing fundamentally based models that can predict cc based on the properties of the reagents used to make the foam and the foaming formulation.

Figure 8 displays the comparison between modeling and experimental density, which presents slight deviation caused by higher internal bubble pressure and potential experimental error. It is predictable to estimate the density with the aid of MATLAB Simulation when we have the foaming formulation.


The measuring of mass and height in combination with computer-based simulation of the foaming process provided a closure of the mass balance of blowing agents used to make rigid box foams. The results indicated that evaporation losses during the mixing and degassing steps leads to a loss of efficacy of the blowing agent. This conclusion would broadly apply to blowing agents that rely on low boiling points which form gas cells in urethane foams at temperatures between 30 and 80[degrees]C.

The primary sources of inefficient use of blowing agent are loss of the physical blowing during open-air mixing and degassing. This loss of blowing agent would not apply to in-line mixers used for commercial production and should be taken into account with scaling up box or cup foams commercial processes.

Although previous modeling work that was able to correlate inefficiency with Flory-Huggins parameters appeared to exaggerate the impact of activity coefficients, the activity coefficients considered in this were reasonably close to 1.0. A correlation relates to lack of efficacy of blowing agents to both closed cell (cc) content and the buildup of pressure in the cells. The implication is that if cc content is able to be simulated from a foam recipe, increasingly accurate estimates/simulations of density can also be attained. This was a major finding of this study as it identifies that the modeling of cell formation and rupture is critical to accurately model foam density.


PU              polyurethanes
cc              closed-cell
MF              methyl formate
ODE             ordinary deferential equation
MW              molecular weight
Fn              functionality
Eq. Wt          equivalent weight
[gamma]         solubility
R               ideal gas constant (L atm/K mol)
T               temperature (K)
P               pressure (atm)
t               time (s)
V               volume (ml)
[rho]           density (g/ml)
m               mass weight (g)
h               height (cm)
[T.sub.c]       critical temperature (K)
[P.sub.c]       critical pressure (bar)
[p.sub.sat]     vapor pressure (bars)
[V.sub.b]       volume of the buoyancy
[[rho].sub.a]   density of the air
[V.sub.i]       initial volume before the growth of the foam
[V.sub.exp]     experimental volume of the foam
[V.sub.i]       ideal volume of the foam without the gas rupture
[V.sub.p]       volume loss due to the pressure
[V.sub.s]       volume loss due to the solubility of the MF
[R.sub.b]       release of the blowing agent


The authors thank the United Soybean Board for financial support of the experimental studies used to validate the modeling work. Thanks to FSI Company providing foam formulas and technology support.


[1.] G. Avar, Polyurethanes (PU), Vol. 10, Kunststoffe International, 123 (2008).

[2.] H. Fan, A. Tekeei, G.J. Suppes, and F.-H. Hsieh, Physical Properties of Soy-Polyol Based Polyurethane Foams Reinforced with Microspheres and Nanoclay, American Society of Agricultural and Biological Engineers Annual International Meeting, 5, 3520 (2011).

[3.] H. Fan, A. Tekeei, GJ. Suppes, and F-H. Hsieh, J. Appl. Polym. Sci., 127(3), 1623 (2013).

[4.] V. Yakushin, L. Bel'kova, and I. Sevastyanova, Properties of Rigid Polyurethane Foams Filled with Glass Microspheres, Mech. Compos. Mater., 48(5), 579 (2012).

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[7.] L.T. Yang, C.S. Zhao, C.L. Dai, L.Y. Fu, and S.Q. Lin, J. Polym. Environ., 20(1) 230 (2012).

[8.] S.A. Baser and D.V. Khakhar, Polym. Eng. Sci., 34(8) 642 (1994).

[9.] S.A. Baser and D.V. Khakhar, Polym. Eng. Sci., 34(8) 632 (1994).

[10.] R. Tesser, M. DiSerio, A. Sclafani, and E. Santacesaria, J. Appl. Polym. Sci., 92(3) 1875 (2004).

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[13.] R.M. Christensen, J. Meehan. Pliys. Solids, 34(6), 563 (1986).

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[18.] Y. Zhao, M.J. Gordon, A. Tekeei, F.-H. Hsieh, and G.J. Suppes, J. Appl. Polym. Sci. (in press).

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Lu Shen, (1) Yusheng Zhao, (1) Ali Tekeei, (1) Fu-Hung Hsieh, (2) Galen J. Suppes (1)

(1) Department of Chemical Engineering, University of Missouri-Columbia, Columbia, Missouri 65211

(2) Department of Biological Engineering, University of Missouri-Columbia, Columbia, Missouri 65211

Correspondence to: Lu Shen; e-mail: Contract grant sponsor: United Soybean Board.

DOI 10.1002/pen.23694

Published online in Wiley Online Library (

TABLE 1. Specifications of methyl formate (MF).

MW                              60
Physical state                Liquid
Boiling point ([degrees]C)     31.5
[P.sub.c] (bar)               60.075
[T.sub.c] (K)                  487.2
[A.sub.i]                    -6.99601
[B.sub.i]                     0.89328
[C.sub.i]                    -2.52294
[D.sub.i]                    -3.16636

TABLE 2. Specifications of RUBINATE M isocyanate and three dif-
ferent polyols.

  Fn                                2.70
  Sp. gravity @25[degrees]C         1.23
  % NCO                             31.2
  Eq. wt.                           135
  Viscosity cps @25[degrees]C       190
Product                          OH number
  Poly G76-635                      635
  Voranol 360                       360
  Jeffol R315x                      315

TABLE 3. Foaming formulation of rigid polyurethane foam.

                                       Weight/g       Equivalents
B-side materials
  Poly G76-635                           13.84           0.1569
  Voranol 360                            15.68           0.1008
  Jeffol R315x                             4             0.0225
  Dimethylcyclohexylamine                0.12
    (gelling catalyst)
  Pentamethyldiethylenetriamine       0.32 (foam
    (blowing catalyst)              reaction only)
  Momentive L6900                         0.6
  TCPP                                     2
  Distilled water (blowing agent)     1.04 (foam          0.08
                                    reaction only)

A-side material
  RUBINATE M                            61.548           0.4559

TABLE 4. Results of mass loss of different samples.

                            [DELTA]m       [DELTA]h    [DELTA]m from
Foam type                (g equivalent)      (cm)         buoyancy

Foam only with MF 1           2.13           8.83           0.57
Foam only with MF 2           2.12           8.48           0.52
Foam only with water 1        2.22           12.47          1.72
Foam only with water 2        2.26           11.61          1.78
Gel 1                         0.38           2.02           0.10
Gel 2                         0.29           1.54           0.11

                         [DELTA]m from    Efficiency      Efficiency
                            blowing       of blowing    calculated by
Foam type                    agent         agent (%)      height (%)

Foam only with MF 1           1.38           42.4            40.0
Foam only with MF 2           1.42           41.0            33.2
Foam only with water 1        0.50           80.2            80.1
Foam only with water 2        0.48           81.1            82.7
Gel 1                         0.28            --              --
Gel 2                         0.18            --              --

TABLE 5. Experimental and modeling density from samples displayed in
Fig. 7 with comparison of efficiency, close cell content and
quantities of blowing agent.

Close cell    Efficiency (%)      Density        Density
content (%)                     exp (g/ml)    model (g/ml)

87.55              54.29           33.00          21.43
90.30              53.64           33.90          21.76
91.66              69.16           33.36          26.55
94.06              69.12           28.50          24.75
94.63              58.05           42.22          26.76
91.96              57.27           52.50          34.24
95.46              50.80           57.20          33.12
94.40              66.21           44.85          35.17
94.31              71.16           33.31          26.83
98.20              58.72           42.09          28.81
99.66              67.48           39.80          27.31
93.99              62.89           44.79          29.96
89.10              56.32           43.63          27.01
98.80              63.79           40.03          27.12
94.84              62.00           48.60          30.14
97.76              71.20           39.59          30.58
83.22              54.48           27.05          18.32
73.16              50.21           24.59          16.26
93.39              59.68           32.32          21.83
71.80              50.67           31.5           17.19
56.83              44.32           25.72          12.63
94.96              55.33           34.76          18.60
90.27              60.88           41.32          26.67
74.62              50.00           36.87          28.42
78.70              50.00           32.76          27.01
91.36              55.33           36.32          21.14
97.63              63.79           36.10          23.59
97.67              67.26           32.91          24.58
98.53              64.16           33.37          22.39
98.53              64.37           30.99          21.10
96.78              68.61           33.36          24.92

Close cell    Quantities of    Quantities of
content (%)       MF (g)         Water (g)

87.55              2.40             1.04
90.30              2.41             1.02
91.66              2.42             1.06
94.06              2.45             1.01
94.63              2.39             0.76
91.96              2.40             0.50
95.46              2.48             0.50
94.40              2.47             0.51
94.31              1.51             1.01
98.20              1.53             1.05
99.66              1.50             1.04
93.99              0.52             1.01
89.10              0.50             1.07
98.80              0.50             1.04
94.84              0.00             1.05
97.76              0.00             1.04
83.22              2.35             1.53
73.16              2.29             1.97
93.39              2.20             1.03
71.80              2.40             1.50
56.83              2.30             2.00
94.96              2.32             1.02
90.27              2.33             1.03
74.62              2.25             1.00
78.70              2.20             0.98
91.36              2.42             1.05
97.63              2.40             1.06
97.67              2.42             1.05
98.53              2.41             1.05
98.53              2.41             1.06
96.78              2.41             1.05
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Article Details
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Author:Shen, Lu; Zhao, Yusheng; Tekeei, Ali; Hsieh, Fu-Hung; Suppes, Galen J.
Publication:Polymer Engineering and Science
Article Type:Report
Geographic Code:1USA
Date:Jul 1, 2014
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