#### 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:

- (1)

where *C*_{t} 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; Δ*G*_{het} is the Gibbs free energy change for the critical nucleus formation; *k*_{b} 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 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:

- (2)

where *f*_{1} 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

- (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, *P*_{g} 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 is met (where *N*_{n} 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 *N*_{n} ≈ 10^{6} 1/cm^{3}, *t* ≈ 1 s, and *D* ≈ 27 · 10^{−5} cm^{2}/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:

- (4)

where *R* is the average radius of supercritical bubbles, *C*_{0} is the initial gas concentration in the solution, *C* is the current gas concentration in the selected element of the solution, ρ_{g} is the gas density in bubbles, and ρ_{p} 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]:

- (5)

where *K*_{w} is the Henry constant. Adopting the characteristic values of 1/*K*_{w} = 27 · 10^{6} Pa, *P* = 10^{6} Pa and *C*_{0} = 0.1%, one may readily check that the inequality *C*_{0} > *K*_{w}*P* is true.

By combining *Eqs*. (4) and (5) and using the assumption that *K*_{w}*P* is small compared to *C*_{0}, we obtain the following expression for the gas concentration in the selected element of the solution as a function of the time:

- (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:

- (7)

where *P*_{0} is the pressure corresponding to the beginning of the nucleation and *a*_{1} 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

- (8)

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

- (9)

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

- (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 *P*_{0} to the atmospheric pressure (which is taken to be zero) as follows:

- (11)

If the inequality *t*_{*} < *t*_{a} 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*_{*} < *t*_{a} can be rewritten as follows:

- (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*_{*} < *t*_{a} 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:

- (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:

- (14)

In deriving *Eq*. (14), it was assumed that the inequalities *C*_{0} ≫ *P*_{0}*K*_{w} and 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:

- (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).

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 [2π*y*^{3}(*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:

- (16)

From relation (16), it follows that *N*_{n} ∼ *a*_{1}^{1/4}, 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:

- (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* = *t*_{a} 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,

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

- (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:

- (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 – *R*_{max}, where *R*_{max} is the maximum radius of bubbles in the foam. The normalized bubble radius distribution curve can be written as 1/*R*_{max}. This means that *NdR/R*_{max} 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:

- (20)

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

The critical bubble radius is given by the following formula:

- (21)

All bubbles with radii smaller than the critical value (*R* < *R*_{cr} will disappear, and only bubbles from the interval *R*_{cr} < *R* < *R*_{max} 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 C_{min} 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

- (22)

For σ = 30 · 10^{−3} N/m; 1/*K*_{w} = 27 · 10^{6} Pa; *R*_{cr} = 10^{−4} m, it follows from formula (22) that *C*_{min} = 2.2 · 10^{−6}%, which is significantly smaller than C_{0} ≈ 0.0%. Using the approximation of *C*_{min} ≪ *C*_{0} in *Eq*. (20), it is possible to estimate the maximum radius of foam bubbles as

- (23)

For C_{0} ≈ 0.1%, ρ_{p} = 920 kg/m^{3}, ρ_{g} = 1.86 kg/m^{3}, and *N* = 10^{5} cell/cm^{3}, formula (23) yields *R*_{max} ≈ 10^{−1} cm, which corresponds to experimental results for the above parameters.

At the nucleation stage, *N*_{n} bubbles are formed and, in the subsequent foam-growing process, only some bubbles but not all are retained, since *N*_{n}*R*_{cr}/*R*_{max} fraction of bubbles are subject to coalescence. As a result, the residual number of bubbles is

- (24)

Using the estimation of *R*_{cr}/*R*_{max} < 1 and adopting the approximation of 1 − *x* ≈ *e*^{−x} for *x* < 1, relation (24) can be rewritten as follows:

- (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 *N* ≈ *N*_{n}, and then the estimation for the cell density can be written as follows:

- (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.