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Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL MODEL
  5. EXPERIMENT
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. REFERENCES

Operation of a flat slot die with rollers for the extrusion foaming of polymers, which has been originally designed by Benkreira et al. (Int. Polym. Proc., 19, 111 (2004)), is considered. The rotation of rollers makes it possible to independently control shear rate inside the die, in the bubble nucleation zone, thus influencing the cell density. In experimental studies of low-density polyethylene foaming, the influence of the roller rotation speed on the cell density at variable isobutane and talc concentrations and die outlet area has been determined. Based on the fluctuational nucleation theory, a simple model is proposed for estimating the number density of supercritical nuclei formed at the nucleation stage and evaluating the cell density in the foam with allowance for bubble coalescence effects. POLYM. ENG. SCI., 54:96–109, 2014. © 2013 Society of Plastics Engineers


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL MODEL
  5. EXPERIMENT
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. REFERENCES

At present, the extrusion foaming of polymers has become one of the main industrial methods of obtaining foamed plastics [1-3]. The essence of this method is as follows. A gas is supplied at a high pressure to the extruder, where it is dissolved in a polymer. After the outflow of polymer/gas solution through the die into the atmosphere, it is supersaturated and, as a result, transforms from homogenous into a two-phase state. According to the classical nucleation theory, a metastable phase transforms into the stable one by the fluctuational formation of small clusters of a new phase (nuclei) in the uniform medium. If the nucleus radius exceeds the critical one, it starts macroscopic growth. Generally speaking, the nuclei can not only be formed due to fluctuations but also be present in the solution because of incomplete dissolution of the foaming gas in the polymer. It becomes especially important to take this possibility into account when the time of mixing gas and polymer in the extruder is insufficient for the complete dissolution [4]. Microvoids in the polymer can also act as nuclei. The results of direct experimental observations [5, 6] and calculations [7] show that the nucleation is initiated inside the die, but the main foam growth occurs outside. In the course of gas diffusion from the solution into bubbles, the solution viscosity and strength increase [8] and the polymer melting or glass-transition temperature increases. All the above factors favor stabilization of the foam, thus preventing it from contraction. Along with the effective density of the foam, the cell density also significantly influences the final mechanical properties of the foam. For example, at equal densities of foamed polymer samples, the sample with a higher cell density has better strength characteristics. The cell density of the foam depends on the nucleation and bubble coalescence processes. The nucleation rate is mainly determined by the degree of solution supersaturation and the availability of nucleation centers. The latter lead to the increase in the nucleation rate owing to a decrease in the critical nucleus formation. In the case of polymer extrusion foaming, the pressure in solution after its outflow through the die into the atmosphere drops within a finite time interval rather than instantaneously. For this reason, some gas from the solution goes into the growth of previously formed bubbles, which results in reduction of the degree of solution supersaturation at later nucleation stages and, as a consequence, leads to a decrease in the cell density of the foam [9-11]. Along with the pressure in the die and the rate of its drop, the cell density is affected by shear stresses, whose growth leads to intensification of the nucleation process [5, 12-15]. The maximum cell densities (above109 cells/cm3) have been achieved by using the microcellular plastic technology [16, 17]. However, the technological “window” for producing microcellular plastics is rather narrow and, in addition, it is necessary to strive for obtaining higher pressure drop rates in the solution during efflux through the nozzle. High pressures in the die and large pressure drop rates can be achieved with a small nozzle outlet diameter, while its increase leads to catastrophic reduction in the cell density [18].

In the production of usual structural foams, whereby the die outlet diameters are large that is determined by the need for obtaining products with large cross-sections, the pressure drop rate in the die is low. Owing to the nucleation process extension in time, the cell density sharply decreases. In this connection, when the die with a large outlet size is used, the problem is to increase the cell density. To solve this problem, we propose to use a flat slot die with rotating rollers, which are placed as close as possible to die lips (Fig. 1). The impossibility of using rollers instead of the die lips is related to their bending under the disjoining pressure forces, resulting in a nonuniform thickness of the final product. A die with rotating rollers for the polymer foaming technology was originally proposed by Benkreira et al. [19]. In that pioneering work, the main attention was devoted to generating high pressures in the die for the purpose of obtaining low-density foams.

image

Figure 1. Scheme of the flat slot die with rollers.

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The rollers may be rotated either in the direction of polymer flow or in the opposite direction. In what follows, when rollers rotate in the polymer flow direction, their rotation speed is referred to as “positive”, while the term “negative” rotation speed is used for the opposite direction. By rotating rollers, it is possible to independently control shear stress in the region between them. When the shear stress increases, the bubble nucleation rate increases, resulting in an increase in the cell density. However, in this case, the coalescence of bubbles can also be intensified. A competition of these processes accounts for the appearance of a maximum in the dependence of the cell density on the roller rotation speed.

MATHEMATICAL MODEL

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL MODEL
  5. EXPERIMENT
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. REFERENCES

Nucleation Theory

The influence of rotating rollers on the cell density in the foam is related to the dependence of the bubble nucleation rate on the shear stress. The mechanism of this influence is multifactor and ambiguous. Thus, it has been ascertained [5] that the shear stress plays an important role in the process of nucleation near die walls. A nonuniformity of shear stresses across the channel can lead to the forced motion of gas molecules and their clusters, which also affects the nucleation conditions. Another interesting conclusion drawn in Ref.[5] was that, in the presence of shear stresses, the nucleation can take place in under-saturated solutions. According to the nucleation mechanism analyzed in Ref.[14], normal stresses in the flow of a viscoelastic liquid favor the growth of subcritical bubbles and, hence, intensify the nucleation.

It is necessary to perform more delicate experimental studies to estimate the influence of shear stresses on the nucleation rate according to one or another mechanism. In this context, one possible mechanism is considered in this work, although, several factors can probably act simultaneously in real practice. Thus, in a metastable liquid, the subcritical bubble growth rate can be limited by viscous forces, which determine the influence of viscosity on the nucleation rate [20, 21]. As the shear rate increases, the shear viscosity decreases, leading to an increase in the subcritical bubble growth rate and, as a result, in the nucleation rate. In the framework of the reptation theory [22], the viscosity of a polymer melt is determined by the formation of quasi-links between polymer macromolecules. The contacts of macromolecules, which are called quasi-links, limit their motion. As a result of the reptation motion of macromolecules, entanglements between them periodically disappear and they acquire mobility. The faster one macromolecule creeps relative to another, the lower the viscosity is. Shear deformations favor acceleration of the motion of one macromolecule relative to another. The dependence of the shear viscosity on the shear rate is based on this process. Generally speaking, the shear deformations not only affect the melt viscosity but also influence the coefficient of gas diffusion in the solution. This mechanism can be substantiated as follows. The transfer of a gas particle from the initial equilibrium position to the adjacent one is not active but passive, that is, determined by a random separation of surrounding particles with the formation of a microcavity (hole) in the immediate vicinity of this particle rather than by a random increase in the kinetic energy of this particle at invariable positions of the surrounding particles. The creeping of one macromolecule with respect to another is accelerated under the action of shear, thus contributing to new microcavity formation and, hence, increasing the diffusion coefficient.

According to the classical nucleation theory, the nucleus formation rate (per unit time) in the unit volume can be written as follows:

  • display math(1)

where Ct is the concentration of heterogeneous nucleation centers (per unit volume of the polymer); f is the pre-exponential (frequency) factor, which depends on the frequency at which molecules join to the nucleus; ΔGhet is the Gibbs free energy change for the critical nucleus formation; kb is the Boltzmann constant; and T is the absolute temperature. Relation (1) is valid for the heterogeneous nucleation, when bubbles are formed, for example, in cavities of talc particles.

At the initial growth stage, the bubble growth rate in a viscous liquid is limited by viscosity forces, while at the final stage, it is limited by gas diffusion from the solution into bubbles. This is related to the fact that, at the initial bubble growth stage, the thickness of the concentration boundary layer is small and the viscosity forces are responsible for the main resistance to its growth. As the bubble grows, the thickness of the concentration boundary layer increases with the time t as inline image which leads to an increase in the diffusion resistance and results in the passage to conditions where the control over the bubble growth rate belongs to diffusion.

In the case of the melt viscosity being a factor limiting the subcritical bubble growth rate, Kagan [20] obtained the following expression for the pre-exponential factor:

  • display math(2)

where f1 is a parameter that does not depend on viscosity and μ is the shear viscosity.

The Gibbs free energy change in Eq. (1) can be expressed as

  • display math(3)

where σ is the surface tension coefficient, S(θ) is a function that depends on the wetting angle θ at the polymer–gas interface, Pg is the gas pressure in the bubble, and P is the pressure in the polymer melt.

Let us consider, in a one-dimensional setting, the motion of a selected element of the solution in the die (Fig. 1) in the Lagrange coordinates, that is, in the moving frame related to this element. Note that, in this formulation of the problem, the polymer adhesion to walls is ignored. Generally speaking, residence times of various solution portions located near the surface and in the central part of the flow are different; however, in the context of the one-dimensional model, this factor is disregarded. In what follows, by any parameter value at a given point, we imply its value averaged over the cross-section of the channel in the die.

As the selected element moves to the exit of the die, the pressure drops and, at a certain moment of time, the polymer–gas solution proves to be supersaturated and nuclei start to form in the metastable solution. As the supercritical bubbles grow, the thickness of a diffusion boundary layer around them increases. If the criterion inline image is met (where Nn is the number of supercritical bubbles per unit volume of the polymer and t is the time), the concentration boundary layers around the bubbles merge together. On the assumption that Nn ≈ 106 1/cm3, t ≈ 1 s, and D ≈ 27 · 10−5 cm2/s, this criterion proves to be fulfilled. In what follows, it is assumed that the condition of merging concentration boundary layers around the bubbles is met, and by the gas concentration in solution, we imply its average value in the space between bubbles.

Supercritical bubbles start to grow directly in the die (closer to its exit), thus reducing the solution supersaturation degree and, as a result, the fluctuational nucleation rate. The current gas concentration in the solution can be estimated from the approximate balance equation for gas, the meaning of which is as follows: the gas mass concentrated in bubbles is equal to its loss from the solution:

  • display math(4)

where R is the average radius of supercritical bubbles, C0 is the initial gas concentration in the solution, C is the current gas concentration in the selected element of the solution, ρg is the gas density in bubbles, and ρp is the polymer melt density. In writing Eq. (4), we ignore differences in the bubble radii, which is possible due to small gas absorption from the solution at the nucleation stage. In addition, differences in densities of the polymer/gas solution and polymer were also disregarded. Assuming that the Henry law is valid and the supercritical bubble growth rate is limited by gas diffusion, the average bubble radius as the function of time is expressed as [23]:

  • display math(5)

where Kw is the Henry constant. Adopting the characteristic values of 1/Kw = 27 · 106 Pa, P = 106 Pa and C0 = 0.1%, one may readily check that the inequality C0 > KwP is true.

By combining Eqs. (4) and (5) and using the assumption that KwP is small compared to C0, we obtain the following expression for the gas concentration in the selected element of the solution as a function of the time:

  • display math(6)

The pressure variation in a selected element of the solution, as it moves to the die exit, can be written in a linearized form as follows:

  • display math(7)

where P0 is the pressure corresponding to the beginning of the nucleation and a1 is a parameter that defines the pressure drop rate. The latter parameter depends on the pressure gradient at the die exit and the throughput of the extruder. The pressure difference between the bubble and surrounding liquid is

  • display math(8)

Substituting relations (6) and (7) into Eq. (8), we obtain the following expression:

  • display math(9)

Pressure difference (9), as a function of the time, reaches a maximum at the following moment:

  • display math(10)

The maximum of function (9) coincides with the minimum of function (3) and, hence, corresponds to the maximum nucleation rate (1). As the selected element moves closer to the die exit, the pressure exhibits a drop that leads, on the one hand, to an increase in the degree of solution supersaturation and, on the other hand, to a decrease in the gas concentration in solution as a result of the absorption of previously formed bubbles. Thus, the nucleation rate goes through a maximum that depends on the time and, in addition, is sufficiently sharp due to the exponential behavior of function (1).

Using relation (7), it is possible to estimate the pressure drop time in the selected element from P0 to the atmospheric pressure (which is taken to be zero) as follows:

  • display math(11)

If the inequality t* < ta is valid, the fluctuational nucleation rate maximum is reached inside the die, while the nucleation intensity in the main foam growth zone is low. Taking into account relations (10) and (11), the criterion of t* < ta can be rewritten as follows:

  • display math(12)

From this relation, it is seen that the implementation of this regime is characteristic of high gas concentrations and low pressure drop rates in the die.

If the inequality t* < ta is true, the nucleation takes place mainly outside the die, in the main foam growth zone. After estimating the supercritical bubble number density, it is necessary to redefine criterion (12).

Let us first consider the case, whereby criterion (12) is fulfilled. The total number of bubbles formed as a result of the nucleation can be estimated by integrating Eq. (1) with respect to the time over the whole nucleation period:

  • display math(13)

The region near the maximum of the integrand, that is, in the vicinity of t = t*, mainly contributes to the integral, while the temporal dependence of the frequency factor f can be generally disregarded because the exponential changes are most substantial. To simplify the final result, let us expand (9) into Taylor's series near the maximum t = t* and retain terms up to the quadratic one inclusive:

  • display math(14)
  • display math

In deriving Eq. (14), it was assumed that the inequalities C0P0Kw and inline image are valid.

After substituting relation (14) into formula (3) and expanding this into series up to the quadratic term, we arrive at the following formula:

  • display math(15)

The critical bubble formation work passes through a minimum (Fig. 2) in the die. The minimum of function (15) corresponds to the maximum of nucleation rate (1).

image

Figure 2. Change of the Gibbs free energy along the die.

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For the approximate estimation of integral (13), it is possible to use the following: if some function y(t) has a maximum in the integration interval at t = t*, the integral ∫y(t)dt can be approximately estimated as [2πy3(t*)/|y(t*)|]1/2. Then, by factoring the pre-exponential multiplier out of the integral (13) and taking into account the fact that the region near t = t* mainly contributes to the integral, the estimation of (13) with allowance for (15) can be written as follows:

  • display math(16)

From relation (16), it follows that Nna11/4, that is, a decrease in the pressure drop rate in the die favors decrease in the number of supercritical bubbles in solution.

An increase in the shear rate in solution leads to a decrease in the viscosity and, in accordance with Eq. (16), to an increase in the bubble number density owing to the accelerated growth of subcritical nuclei. Function (16) exhibits a maximum depending on the gas concentration. At a small gas concentration, its growth leads to an increase in the degree of solution supersaturation and, hence, in the nucleation rate. However, this is accompanied by growth in the intensity of gas absorption from solution by the bubbles formed at the earlier stage, which leads to a decrease in the degree of solution supersaturation and to the nucleation slowing down at the later stage. In fact, a certain fraction of gas is spent for the growth of previously formed bubbles rather than for the formation of new bubbles. This circumstance accounts for the appearance of the extremum in function (16). By substituting the estimated bubble number density (16) into relation (12), it is possible to obtain a criterion of the regime corresponding to the applicability of relation (16). However, since the resulting formula is rather cumbersome, it is not given here.

If condition (12) is not valid, the nucleation mainly occurs outside the die at the atmospheric pressure (i.e., at P = 0. Using the same assumptions as those involved in deriving (15), substituting expression (6) into Eq. (3), taking into account that P = 0, and expanding into series, we eventually obtain the following formula:

  • display math(17)

According to Eq. (17), a minimum work of critical bubble formation corresponds to the time when the selected element goes out of the die, that is, t = ta and it is possible to ignore the nucleation in the die. One should note that the case considered here, as well as the previous one, are the limiting for the situation that apparently most often takes place in practice.

By substituting formula (17) into Eq. (13) and integrating over the entire period of time within which the nucleation takes place in the course of foam growth outside the die, it is possible to estimate the supercritical bubble number density. In order to simplify the final result, let us use the following approximate estimate of the integral,

  • display math

whereby the integrand assumes a maximum value on the integration interval boundary at t = ta. Upon substituting formula (17) into Eq. (13) and using the above estimation of the integral, we obtain an implicit equation for Nn. Solving this equation in the first approximation under the assumption that NnCt under the exponent sign, we eventually obtain the following expression for the number of bubbles:

  • display math(18)

Note that estimation (18) is valid in cases where the pressure in the die drops very rapidly, that is, when the criterion (12) is not met. According to relation (18), a decrease in the pressure drop rate leads to a catastrophic reduction in the cell density, and, in addition, this dependence is exponential. This conclusion is supported by numerous experimental results (see, e.g., [24]).

Depending on the talc concentration, function (18) goes through a maximum. The maximum density of nuclei in the foam corresponds to the following talc concentration:

  • display math(19)

It is seen from Eq. (19) that the lower the pressure drop rate in the die, the smaller the talc concentration at which the extremum in the cell density is reached. Therefore, on the passage to a die with large size of the outlet, it is necessary to use smaller talc concentrations to reach the highest cell density in the foam.

Coalescence of Bubbles

The cell density in the foam depends not only on the nucleation of bubbles but also on their coalescence (i.e., absorption of some bubbles by others). The coalescence of bubbles may be induced by various mechanisms, for example, by destruction of bubble walls. One possible mechanism of bubble coalescence is based on the idea that was for the first time considered by Lifshitz and Slezov (see, e.g., [25]) with respect to the problem of precipitation of a dissolved substance from a supersaturated solution. The essence of this mechanism as applied to the gas bubble coalescence in the supersaturated solution is as follows. At the late stage of foam growth, when the solution supersaturation becomes very small, the fluctuational formation of new nuclei is virtually excluded, since the critical bubble radius is too large. (Indeed, an increase in the critical radius, accompanying the progressive drop in the degree of solution supersaturation, results in that smaller the existing bubbles become subcritical and dissolve.) Thus, the absorption of small bubbles by large ones, that is, the growth of larger bubbles at the expense of dissolving smaller ones (coalescence process) plays a determining role at this stage, which is considered in detail below.

Based on the notions presented earlier, it is assumed that the coalescence process runs outside the die in the main foam growth zone, where the supersaturation degree is significantly reduced. The actual bubble size distribution curve is approximated by the uniform distribution function (Fig. 3). In other words, the probability of finding bubbles in an interval from R to R + dR does not depend on the bubble radius. All bubble sizes are in the interval of 0 – Rmax, where Rmax is the maximum radius of bubbles in the foam. The normalized bubble radius distribution curve can be written as 1/Rmax. This means that NdR/Rmax fraction of their total number (equal to N) is concentrated in an interval from R to R + dR. In this case, the current gas concentration in the solution can be written as follows:

  • display math(20)

The integral in Eq. (20) is equal to the total volume of all bubbles.

image

Figure 3. Pore size distribution curves.

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The critical bubble radius is given by the following formula:

  • display math(21)

All bubbles with radii smaller than the critical value (R < Rcr will disappear, and only bubbles from the interval Rcr < R < Rmax will be retained. Formally speaking, when the gas concentration in the solution approaches zero, the critical radius according to formula (21) becomes infinite, which corresponds to the coalescence of all bubbles. However, it is necessary to take into account that, when the gas is released from the solution, the viscosity and elasticity of the polymer melt grow up, that is, these forces “freeze” bubbles in the solution. Moreover, reduction of the diffusion coefficient slows down the dissolution of smaller bubbles. As a result, a certain amount of the foaming gas remains in the polymer matrix, and this retained gas fraction determines the maximum critical bubble radius. It can be assumed that this retained fraction is equal to the minimum residual gas content Cmin in the solution. In the given theory, this parameter is phenomenological and should be determined from experiment. Taking into account that the coalescence runs outside the die, where P = 0, it is possible to express the maximum critical bubble radius from formula (21) as

  • display math(22)

For σ = 30 · 10−3 N/m; 1/Kw = 27 · 106 Pa; Rcr = 10−4 m, it follows from formula (22) that Cmin = 2.2 · 10−6%, which is significantly smaller than C0 ≈ 0.0%. Using the approximation of CminC0 in Eq. (20), it is possible to estimate the maximum radius of foam bubbles as

  • display math(23)

For C0 ≈ 0.1%, ρp = 920 kg/m3, ρg = 1.86 kg/m3, and N = 105 cell/cm3, formula (23) yields Rmax ≈ 10−1 cm, which corresponds to experimental results for the above parameters.

At the nucleation stage, Nn bubbles are formed and, in the subsequent foam-growing process, only some bubbles but not all are retained, since NnRcr/Rmax fraction of bubbles are subject to coalescence. As a result, the residual number of bubbles is

  • display math(24)

Using the estimation of Rcr/Rmax < 1 and adopting the approximation of 1 − xex for x < 1, relation (24) can be rewritten as follows:

  • display math(25)

Substituting expressions (22) and (23) into relation (25), we obtain an implicit equation for the total number of bubbles in the foam. In order to further simplify this, it is possible to assume in the first approximation that NNn, and then the estimation for the cell density can be written as follows:

  • display math(26)

The final cell density is not equal to the number density of supercritical nuclei, since it is reduced due to the coalescence of bubbles.

EXPERIMENT

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL MODEL
  5. EXPERIMENT
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. REFERENCES

Extrusion Setup

A single-screw extruder with a screw diameter of 120 mm and a length-to-diameter ratio of 63 was used for the foam extrusion (Fig. 4). A helical-groove barrel was used in the feeding zone of the extruder to ensure a high coefficient of the polymer friction on walls of the feeding zone. The grooved barrel was cooled by a heat-transfer fluid with a temperature of 7°C. The screw rotation speed was controlled using an inverter provided with a motor feedback sensor. The above-mentioned measures ensured stable operation of the extruder. A foaming gas was fed from a vessel to the booster pump, which raised its pressure to 20 bar and then was supplied by a positive displacement pump directly to the extruder. The melt pressure and temperature prior to entering the die were measured by a Dynisco TDA 463-1/2-3.5C-15/46 sensor. The die temperature was measured by a thermocouple sensor.

image

Figure 4. Experimental setup for polymer foam extrusion: (1) pump; (2) throttle; (3) gas vessel; (4) pressure sensor; (5) positive displacement pump; (6) variable speed motor; (7) gear box; (8) hopper with polymer pellets; (9) extruder; (10) die; (11) melt temperature and pressure sensors; (12) cooler; (13) feedback sensor.

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Figure 5 shows a scheme of the experimental die. The die was heated using oil-filled channels. The die rollers with a diameter of 45 mm could rotate in both directions and the motor could smoothly control their rotation speed. A gap between the die lips could also be controlled and, in the experiments, it was equal to 1.45 and 2 mm. In order to avoid polymer losses via rollers, these were provided with labyrinth packing.

image

Figure 5. Experimental prototype of the extrusion die with rollers.

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Materials

Low-density polyethylene LDPE 15803-020 (State Standard 16337-77, Ufaorgsyntez company; melting index, 2.1 g/10 min; density, 921 kg/m3) was used as a polymer for foaming. Commercial isobutane (>98% pure was used as the foaming gas. Glycerol monostearate (1.2 wt% solution) was used to stabilize the foam. The nucleating agent was talc with a 10−5 average particle size.

Procedure

A stationary state was reached within 2 h after startup of the extruder. The temperature distribution over extruder zones was 7, 190, 160, 125, and 87°C. The melt temperature at the die entrance was maintained constant at 88°C, and the die temperature was 90°C. After setting a specified rotation speed of die rollers, the foamed polymer samples were taken in 20 min. After changing the gas and talc concentrations in the polymer, the samples were taken in 1 h (transient process time). The polymer mass flow rate through the extruder was 82 kg/h and remained unchanged during the experiments. The mass output rate was determined by weighing a foamed polymer sample extruded for 3 min. When the gas was supplied, the pressure behind the booster pump, which was placed immediately behind the tank (Fig. 4), was kept at a 25 bar level that ensured operation without cavitation in the positive displacement pump. The gas output rate was monitored by a mass flow meter.

Foam Structure Analysis

The samples of a foamed polymer were characterized by the bulk foam density, cell size, and cell density. Prior to analysis, the foam samples were aged for 1 month. In order to exclude unconditioned measurements and reduce accidental errors, the measurements were triply repeated in each regime, and the experimental value was determined as the arithmetic mean. The relative errors of experimental values were determined by conventional methods [26]. In order to eliminate the edge effects in a foamed polymer plate, a sample for analysis was cut from its central part. The foam density was determined by weighing a sample and measuring its volume as that of water displaced in a graduated cylinder. Since water in some samples could penetrate into pores, this absorption was minimized by forming a smooth impermeable film on the sample surface. The uncertainty of determining the foam density was within ±4%. Morphological features of the foamed polymer were determined by making microphotographs of a cut section of the sample stained with a dye and processing the image using a standard program package. The pore diameter was defined as the minimum diameter of a circumscribed circle in the pore cross section. The cross-sectional area was chosen so as to ensure that the total number of pores would be about 100. All pores were distributed over groups with respect to their diameter at an interval of 0.2 mm, the number of pores in each interval was counted, and the corresponding pore size distribution curves were constructed. The error of determination of the average pore diameter was about ±3%. The cell density was calculated per unit volume of the polymer. The total volume of the foam was taken equal to a sum of the volumes occupied by the polymer and pores (bubbles):

  • display math(27)

where Vp is the polymer volume, n is the number of bubbles in volume V, and d is the average pore diameter.

Neglecting the gas mass compared to that of the polymer, the foam density ρf can be expressed as follows:

  • display math(28)

According to definition, the cell density is

  • display math(29)

Combining Eqs. (27), (28), and (29), we eventually obtain the following expression for the cell density:

  • display math(30)

Using this formula, the cell density was evaluated with an error of about ±8%.

RESULTS AND DISCUSSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL MODEL
  5. EXPERIMENT
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. REFERENCES

We have studied the influence of the roller rotation speed on the cell density in the foam for variable gas content, talc concentration, and gap width between the die lips. The comparison of theoretical and experimental results had mainly a qualitative character, which was related to ambiguity of the mechanisms responsible for the influence of the shear on the bubble nucleation rate and coalescence.

Effect of Gas Concentration

Figure 6 shows the dependence of the cell number density on the roller rotation speed at a minimum isobutane concentration of 8.2 wt% used in these experiments. The average foam density for this gas content amounted to 24.3 kg/m3. The dependence of pressure in the die on the roller rotation speed is plotted in the same figure. An analysis of Fig. 6 shows that, at the given foaming gas concentration, the roller rotation rather insignificantly influences the cell density. This is related to the fact that the nucleation zone is located downstream the rollers and their rotation weakly influences the nucleation rate. In this case, the pore structure remained virtually unchanged when the rollers rotated in comparison to the case of immobile rollers (Fig. 7). An insignificant decrease in the cell density at a negative value of the roller rotation frequency (Fig. 6) is apparently related to dissipative heating of the polymer and the related contraction of the foam. The regime of die operation, whereby the nucleation of bubbles mostly proceeds outside the die, is realized in cases where criterion (12) is not valid. This is characteristic of low gas contents in solution and a high rate of pressure drop in the die (narrow gap width between die lips). In this case, the density of supercritical nuclei must be estimated using formula (18). A joint analysis of Eqs. (12) and (18) shows that the probability of this regime is highly sensitive to the gas concentration in solution. Indeed, even a very small increase in the gas concentration can lead to a shift of the main nucleation zone inward the die. This is confirmed by the results of experiments with a change in the character of N = N(ω) dependence, where the gas concentration in solution was increased from 8.2 to 10.8 wt% (Fig. 8). In this case, a change in the shear rate due to the rotation of rollers will influence the nucleation rate and, as a result, the bubble coalescence process.

image

Figure 6. Dependence of the cell density and pressure drop on the roller rotation speed. The positive speed corresponds to the roller rotation in the direction of the polymer flow, while the negative speed corresponds to the roller rotation in the opposite direction. The talc concentration is 1.95 wt%, the isobutane concentration is 8.2 wt%, and the slot width between the die lips is d = 1.45 mm.

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image

Figure 7. Foam morphology obtained for a talc concentration of 1.95 wt%, an isobutane concentration of 8.2 wt%, a slot width of d = 1.45 mm, and a roller rotation speed of (a) ω = 0 rpm and (b) ω = 40 rpm.

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image

Figure 8. Dependence of the cell density and pressure drop on the roller rotation speed for a talc concentration of 1.95 wt%, an isobutane concentration of 10.8 wt%, and a slot width of d = 1.45 mm.

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Figure 9 shows similar results of experiments in which the foaming gas content is higher than that in Fig. 8. The data in Fig. 9 show degeneracy of the extremal behavior of N = N (ω), which is caused by the intensification of bubble coalescence and the leveling of nucleation processes as factors influencing the resulting cell density. It should be noted that the average foam density at a gas content of 10.8 wt% was 16.7 kg/m3, while that at 12.1 wt% was 14.2 kg/m3. In the latter case, the surface of a foam exhibited characteristic incrustations, which showed that the macroscopic growth of bubbles proceeded inside the die. In this respect, data presented in Fig. 9 are not typical of the effect of rotating rollers on the cell density, rather offering an example with a limiting case of the gas content in solution prior to foam destruction.

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Figure 9. Dependence of the cell density and pressure drop on the roller rotation speed for a talc concentration of 1.95 wt%, an isobutane concentration of 12.1 wt%, and a slow width of d = 1.45 mm.

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Figure 8 shows that extrema in the N = N(ω) curve are present in the regions with both positive and negative values of the roller rotation speed ω. It should be noted that, when the extruder throughput was 82 kg/h, the average polymer flow velocity in the narrowest region between rollers was 0.11 m/s, while the linear roller speed, for example, at 8 rpm was 0.84 m/s. Thus, in the region of extremum in the N = N(ω) curve at ω > 0 the rollers rotated with a speed exceeding the average polymer flow velocity. Figure 10 shows the flow velocity profiles in the region between rollers; in particular, Fig. 10a refers to a case where the linear velocity of rollers exceeds the average velocity of polymer flow. In the corresponding range of roller rotation speeds, an increase in the speed leads to growth in the average shear rate.

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Figure 10. Effect of the roller rotation direction on the field of flow velocities between the rollers: (a) rollers rotate in the direction of the polymer flow; (b) rollers rotate against the polymer flow.

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Assuming that the power law equation inline image (where inline image is the shear rate, Kμ is the consistency index, and n is the power index), and taking into account Eqs. (26) and (16), we obtain the following relation:

  • display math(31)

It should be noted that this relation was derived using estimation (16) for the number of supercritical nuclei, in which case the nucleation proceeds inside the die. According to Eq. (31), the dependence of the cell density on the shear rate goes through a maximum. As the shear rate grows, the bubble nucleation rate in the supersaturated solution increases, but the increase in the number of supercritical bubbles is accompanied by their stronger coalescence. The existence of an extremum in the N = N(ω) curve is caused by a competition between the two processes.

These theoretical conclusions are confirmed on a qualitative level by the results of experiments presented in Fig. 8. As the roller rotation frequency increases, a growth in the shear rate leads to a decrease in the polymer melt viscosity. This, in turn, favors a growth of supercritical nuclei and eventually results in their number density increasing as Nn ∼ μ−1/2. However, this growth is limited by the coalescence of bubbles, which leads to a decrease in the cell density with a further increase in the roller rotation frequency. A less pronounced maximum observed in the region of negative values of the rotation frequency (Fig. 8) is probably caused by dissipative heating of the polymer melt and the related foam contraction.

Figure 11 shows the dependence of the pore size distribution curve on the roller rotation speed. Note that, at the time when the function N = N(ω) reaches its maximum for ω = 40 rpm, the pore size distribution curve is narrower than that observed for the rollers at rest and, in addition, the minimum pore radius remains virtually unchanged when the rollers rotate, while the maximum pore radius is changed. According to Eq. (22), upon completion of the coalescence stage, the minimum bubble radius depends only on the residual gas content in the polymer and the maximum pore radius (23) decreases with increasing number of bubbles formed.

image

Figure 11. Pore size distribution curves for a talc concentration of 1.95 wt%, an isobutane concentration of 10.8 wt%, a slot width of d = 1.45 mm and a roller rotation speed of (a) ω = 0 rpm; (b) 8 rpm; (c) 40 rpm; and (d) −10 rpm.

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Effect of Talc Concentration

As is known, talc particles act as nucleation centers because their cavities facilitate the work for critical bubble formation. For this reason, talc is widely used as a means of controlling the cell density. However, it is also commonly accepted that an increase in the concentration of a nucleating agent cannot provide for unlimited growth in the cell density. This phenomenon can be analyzed using the proposed model.

According to Eqs. (16) and (26), the dependence of the cell density on the talc concentration can be represented as follows:

  • display math(32)

According to this, the N = N(Ct) function has a maximum. At a low talc concentration, its growth leads to an increase in the cell density, while at large talc concentrations, to the cell density decreases. In the region of extremum of the N = N(Ct) function, variation of the talc content weakly affects the cell density. This is related to the fact that, on the one hand, the number of nucleating microvoids increases with the talc content and, on the other hand, the enhancement of gas absorption by the bubbles formed previously at the nucleation stage decreases the total number density of supercritical bubbles. In addition, an increase in the talc concentration intensifies the bubble coalescence.

Figure 12 shows the dependence N = N(ω) for the same gas concentration (10.8 wt%) as in Fig. 8, but at the higher talc concentration (2.9 wt%). It is seen from Fig. 12 that the explicit maximum is observed at negative values of the roller rotation speed. Here, the nucleation intensifies with an increase in absolute value of the roller rotation speed due to an increase in the shear stress, thus ensuring an increase in the cell density. When the roller rotation speed is greater than |ω| = 10 rpm, the bubble coalescence starts to play a dominating role. It is seen from Fig. 13 that the bubble coalescence favors an increase in the pore radius. When the rollers are at rest, the cell density insignificantly changed with increasing talc concentration (cf. Figs. 8 and 12), thus evidencing that, in the talc concentration range studied, we were in the region of the maximum of the dependence described by Eq. (32).

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Figure 12. Dependence of the cell density and pressure drop on the roller rotation speed for a talc concentration of 2.92 wt%, an isobutane concentration of 10.8 wt%, and a slot width of d = 1.45 mm.

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image

Figure 13. Foam morphology obtained for the same parameters as in Fig. 12 at (a) ω = −12 rpm, and (b) ω = −20 rpm.

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Based on estimations (16) and (26), it can be shown that function inline image reaches maximum at a shear rateof

  • display math(33)

According to this relation, as the talc concentration in solution is increased, the maximum cell density will be attained at smaller values of the shear rate and, hence, at lower frequencies of roller rotation. Indeed, a comparison of Figs. 8 and 12 shows that an increase in the talc concentration from 1.95 wt% to 2.92 wt% leads to a shift in the position of maximum of the number density of pores in the foam from 40 to 30 rpm in the region of positive rotation frequencies. No such a shift is observed in the region of negative frequencies, which is probably explained by the dissipative heating of polymer and the related enhancement of processes responsible for the coalescence of bubbles.

Effect of Die Outlet Area

The gap width between the die lips significantly influences the profile of pressure in the die and, eventually, the cell density. According to the model proposed above, there are two possible limiting cases, in which the nucleation of bubbles takes place predominantly either outside the die [formula (18)] or inside it [formula (16)]. The first case is characteristic of a narrow gap between the die lips. Then, formula (18) shows that a decrease in the pressure drop rate is accompanied by significant (exponential) decrease in the number density of supercritical nuclei. When the main nucleation zone occurs inside the die, the number density of supercritical nuclei according to formula (16) decreases much slower (power law). In practice, this implies that, as long as the output nozzle diameter is very small, the cell density can reach very high values, but it will drop dramatically with beginning increase in the nozzle output diameter. However, the rate of this drop will significantly decrease with a further growth in the output diameter. In the present study, a significant effect of rollers was only observed when the nucleation of bubbles took place predominantly inside the die, otherwise the rotation of rollers did not significantly influence the cell density.

In this study, we have also changed the gap between the die lips (Fig. 5). The experiments have been performed with two widths (1.45 and 2 mm) of the slot. An increase in the die outlet area leads to a decrease in the pressure drop rate and, under otherwise equal conditions, to an increase in the pore radius in the foam formed both when the rollers are at rest (Fig. 14a and b) and when they rotate in the polymer flow direction (Fig. 14c and d) or in the opposite direction (Fig. 14e and f). Figure 15 shows the dependence N = N(ω) for a 2-mm gap between the die lips. The analogous dependence for a 1.45-mm gap between the die lips and the same other fixed parameters is shown in Fig. 6. In the case of resting rollers, the cell density exhibits a decrease with increasing gap width between the die lips. The presence of an extremum in the dependence of the cell density on the roller rotation speed (Fig. 15) is determined by the fact that the zone of intense nucleation shifts into the die. The shift of the active nucleation zone into the die is related to a decrease in the pressure drop rate (12). It should be noted that the curve of N = N(ω) in Fig. 15 is not typical. This is related to the fact, in the given case, macroscopic growth of bubbles develops inside the die and the resulting shear stresses lead to the destruction of cell walls and the coalescence of bubbles. This mechanism of bubble coalescence significantly differs from that considered above, according to which coarse bubbles absorb fine ones in the final stage of foam growth (i.e., outside the die).

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Figure 14. Foam morphology obtained for a talc concentration of 1.95 wt%, an isobutane concentration of 8.2 wt% and (a) d = 1.45 mm, ω = 0 rpm; (b) d = 2 mm, ω = 0 rpm; (c) d = 1.45 mm, ω = 40 rpm; (d) d = 2 mm, ω = 40 rpm; (e) d = 1.45 mm, ω = −12 rpm; and (f) d = 2 mm, ω = −12 rpm.

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image

Figure 15. Dependence of the cell density and pressure drop on the roller rotation speed for a talc concentration of 1.95 wt%, an isobutane concentration of 8.2 wt%, a die temperature of 90°C, and a slot width of d = 2 mm.

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Figure 16 shows the dependence N = N(ω) for the same fixed parameters as in Fig. 12, except the gap width between the die lips, which in this case was 2 mm. When the die outlet area is increased, the cell density decreases (see Figs. 12 and 16) as determined by the reduction of parameter a1 in Eq. (16): Nna11/4 (the smaller the pressure drop rate, the lower the cell density in the foam). Thus, when the die with a 1.45-mm slot is used with the resting rollers, the cell density was 2.3 × 105 cell/cm3, while for a 2-mm slot the cell density was 1.5 × 105 cell/cm3. However, even for the die with a 2-mm slot, it is possible to reach a cell density of 2.6 × 105 cell/cm3 at a roller rotation speed of 30 rpm. From this, it may be concluded that, when the die with a large outlet area is used, a decrease in the cell density can be avoided if the shear stress in the nucleation zone is controlled by means of rotating rollers. In conclusion, it should also be noted that the character of dependences N = N(ω) in Figs. 15 and 16 is significantly different from that in Figs. 8 and 12. This difference is determined by the enhancement of bubble coalescence processes and the leveling of the influence of roller rotation, which is responsible for the impact on the bubble nucleation rate. Thus, no growth in the cell density is observed in Figs. 15 and 16 for the case when the rollers rotate at a negative speed.

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Figure 16. Dependence of the cell density and pressure drop on the roller rotation speed for a talc concentration of 2.92 wt%, an isobutane concentration of 10.8 wt%, and a slot width of d = 2 mm.

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CONCLUSIONS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL MODEL
  5. EXPERIMENT
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. REFERENCES
  1. Using a slot die with rollers, it is possible to independently change the shear stress directly in the bubble nucleation zone so as to reach the maximum cell density at a certain roller rotation speed. This method is especially effective for increasing the cell density in the foam formed when a die with a low pressure drop rate is used. This is important for the commercial production of foamed polymer articles with large cross-sectional areas.
  2. If the nucleation takes place predominantly outside the die (low gas concentration, narrow gap width between the die lips), the rotation of rollers does not affect the bubble nucleation and, hence, cannot influence the cell density in the foam.
  3. When the main nucleation zone is shifted inside the die, the dependence of the cell density on the roller rotation speed acquires an extremal character in the regions of both positive and negative rotation speed. The existence of these extrema is determined, on the one hand, by enhancement of the bubble nucleation processes under the action of shear and, on the other hand, by intensification of the bubble coalescence processes with increase in the roller rotation speed.
  4. Increase in the slot width at the die exit and/or the gas content in the polymer solution decreases the influence of the roller rotation on the cell density due to the intensification of bubble coalescence.
  5. The effect of rotating rollers mostly reduces to a change in the shear rate and, hence, in the polymer viscosity, in the zone of bubble nucleation. According to a mechanism that was originally considered by Kagan, this changes the conditions of subcritical bubble growth and, eventually, the cell density.
  6. The cell density depends not only on the process of bubble nucleation but also on their coalescence. Based on the ideas originally formulated by Lifshitz and Slezov, the proposed model shows that the coalescence of bubbles proceeds predominantly at the final stage of foam growth and takes place due to the absorption of fine bubbles by coarse ones. As the gas concentration in solution decreases in the course of foam growth, the fine bubbles may turn out to be subcritical and begin to dissolve, while the coarse (supercritical) bubbles keep growing.

REFERENCES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATHEMATICAL MODEL
  5. EXPERIMENT
  6. RESULTS AND DISCUSSION
  7. CONCLUSIONS
  8. REFERENCES