Multi-layer modeling of diffusion of water in acrylamide-grafted aliphatic polyesters.
Plastics products are increasingly being made as layered systems. In this way, the freedom to tailor the product properties increases. An example is the multi-layer packaging material for butter, where one layer acts as a barrier to fat, another acts as a barrier to light, and yet another gives strength to the package. Finally, the outer layer gives the package a pleasant gloss.
Other examples of layered structures are surface-grafted materials. The reasons for grafting are manifold, but one important reason is to increase the biodegradability of less hydrophilic hydrolysable polymers by surface grafting with a hydrophilic monomer. Radiation-induced grafting of 2-hydroxyethyl methacrylate (HEMA) onto poly(3-hydroxybutyrate) (PHB) and its copolymer poly(3-hydroxybutyrate-3-hydroxyvalerate) (P(HB-HV) was investigated by Mitomo et al. (1), who showed that the biodegradability of PHB and (P(HBHV) increased when hydrophilic HEMA was introduced onto the surface, owing to the increased wettability. Furthermore, the grafting modification is advantageous over natural hydrogels, as the base substrate provides strength to support the grafted layer, whereas the hydrogels possess poor mechanical properties in the hydrated state. Even if the base material is grafted throughout, its strength will be several magnitudes greater than that of a hydrogel of the grafting monomer.
The permeation properties of composite materials are usually difficult to estimate. Calculations of composite properties based on data for the pure material components may be erratic because each layer is affected by the surrounding layers. A layer that usually swells in the presence of a solvent may be prohibited from swelling when bonded to an adjacent unswollen layer. Rogers et al. (2) showed theoretically that solvent permeability depends on which side of the layered material is facing the solvent. They therefore concluded that multi-layered polymers may be used as gas valves. In the case of surface grafting, the properties of the surface layer and the matrix polymer vary with, e.g., grafting time and graft thickness (3-6). It is therefore practically impossible to determine the properties of the layers by examining each pure layer material. It is, however, possible to estimate the properties of each layer by applying multi-layer modeling principles to the layered material.
In this study, the applicability of the multi-layer modeling of solubility and diffusion of water in pure and acrylamide (AAm) surface-grafted aliphatic polyesters is tested. Aliphatic polyesters are among the most frequently used polymers in medical applications, mainly because of their controlled degradability (7-10). The aliphatic polyesters studied are poly([Epsilon]caprolactone) (PCL) and poly(1,5-dioxepan-2-one) (PDXO), both known to be degradable in vivo (11-13). PCL is a semicrystalline polymer and PDXO is a fully amorphous crosslinked polymer, but their molecular structures are very similar [ILLUSTRATION FOR FIGURE 1 OMITTED].
Material: Acrylamide (AAm) (Pharmacia LKB 801128-10), Mohr's salt (Merck 3792), and toluidine blue (Aldrich 19.816-1) were used as received. Ethanol (99.5%) and additional reagents were pro analysis grades and used as received. All water was purified by ion exchange and distillation (density [[Rho].sub.1] = 997 kg/[m.sup.3] (297.2 K)).
PCL and PDXO: The PCL used in this study was a Union Carbide Tone 787-grade (Table 1). It was pressed into 0.04-1.873 mm thick rectangular plates in a Schwabenthan compression molding machine at a temperature of 100 [degrees] C and a pressure of 100 bar for 5 minutes. Circular samples with diameters of 12-30 mm were cut from the plates. Crosslinked PDXO films (Table 1) (thickness 0.2 to 0.7 mm; diameter 10 mm) were made through direct polymerization of 1, 5-dioxepan-2-one (DXO) with a crosslinking agent and initiator at 180 [degrees] C for 5 min in a Schwabenthan compression molding machine. The crosslinking agent was 2,2-Bis([Epsilon]-caprolacton-4-yl)-propane (BCP), and ethyl (2-hexylhexanoate) was used as initiator for transesterification. The syntheses of DXO and BCP are described in ref. (11). The crosslinked material had a BCP concentration of 16 mol%.
Irradiation and grafting procedure: The irradiation and grafting were performed according to Wirsen et al. (6). In short, the PCL and PDXO were irradiated by electron beam in air at a dose of 5 Mrad and a dose rate of 0.83 Mrad/minute. Before irradiation, the film samples were dried under vacuum at room temperature overnight. Weighed samples were individually packed in thin paper envelopes. After irradiation, the films were immediately immersed in [N.sub.2](1). Acrylamide (AAm) in predetermined amounts and 0.05% by weight of Mohr's salt were dissolved in distilled water. The solution was agitated and deaerated by bubbling argon through it for 30 minutes at room temperature. The solution was then transferred to thermostated vertical tube reactors, each fitted with a reflux condenser and a vertical capillary for argon flow through the reactor. This grafting solution was then further deaerated and temperature equilibrated by bubbling argon through it for another 20 minutes at the predetermined temperature. The irradiated films were taken from the liquid nitrogen storage, taken from the sealed bags and immediately lowered into the monomer solution under a stream of argon. The samples were exposed at 35 [degrees] C to the grafting solution containing 20 wt% AAm and 0.05 wt% of Mohr's salt. At the end of the grafting period, samples were removed and washed in distilled water overnight. The degree of grafting or graft yield (G.Y.) was determined as the percentage increase in weight:
G.Y. = ([W.sub.g] - [W.sub.o]/[W.sub.o]) x 100 (1)
where [W.sub.g] and [W.sub.o] represent the weights of initial and grafted films respectively.
Table 1. Physical and Molecular Data on Samples. Property PCL PDXO [[M.sub.w].sup.b] 80 000 crosslinked Density (kg/[m.sup.3])(c) 1140 1130 [T.sub.g][(K).sup.d] 218 241.5 [T.sub.m] [(K).sup.d] 333.2 - Mass crystallinity [([w.sub.c]).sup.d] 0.514 - a) Electron beam preirradiated samples (5 Mrad). b) Obtained from SEC. c) Determined using a gradient column. d) Determined from DSC.
Grafted layer thickness measurements: The grafted layer was stained by immersing the grafted films in an aqueous toluidine blue solution (5 wt%). After 5 minutes, the treatment was disrupted and the samples were washed in distilled water and dried in a desiccator. The films were cut in liquid nitrogen to achieve a brittle failure and the cross sections were photographed through a microscope (speca) to give a 130x magnification. The thickness of the grafted layers were then measured from the photographs.
Water desorption measurements: Prior to water desorption measurements the samples were water saturated by putting them in a desiccator at 100% RH at 297.2 K. The weight increase was recorded by intermittently weighing the surface-dried samples in a Mettler AE balance. When equilibrium weight was attained, the samples were exposed to static air at 0%RH (297.2 K) in a desiccator over silica pentoxide and the desorption process was monitored by intermittent weighing.
Oxygen permeability measurements: The permeability of oxygen was obtained at 298.2 K using a Mocon OX-TRAN TWIN device. Samples with thicknesses of 33-650 [[micro]meter] were mounted in an isolated diffusion cell and were subsequently surrounded by flowing nitrogen gas to remove absorbed oxygen. Each sample had a circular exposure surface area of 5 [cm.sup.2] achieved by covering a part of the sample with a tight aluminum foil. One side of the sample was initially exposed to flowing oxygen (1% hydrogen) at atmospheric pressure while the oxygen concentration was zero on the other side of the specimen. The flow rate (Q) through the specimen was measured during the transient period until the steady-state flow rate ([Q.sub.[infinity]]) was obtained. The diffusivity of oxygen D (assumed constant) was calculated by fitting the equation (ref. 14,15)
Q/[Q.sub.[infinity]] = 4l/[(4[Pi]Dt).sup.0.5] [e.sup.-[l.sup.2]/4Dt] (2)
to the Q [([Q.sub.[infinity]]).sup.-1] vs. t-curve using a simplex search algorithm. in this equation, l is the thickness of the specimen and t is time. Assuming that Henry's law is valid, the solubility of oxygen (S) is given by:
S = [Q.sub.[infinity]]l/D[Delta]p (3)
where [Delta]p is the difference in partial pressure of oxygen on the two sides of the sample.
Differential scanning calorimetry (DSC) measurements: DSC thermograms on 5-10 mg samples were recorded between -30 [degrees] C and 100 [degrees] C using a heating rate of 20 [degrees] C/min in a Perkin-Elmer DSC-7.
In the case of a constant diffusion coefficient (D), desorption curves may be modeled (16,17) using:
[Mathematical Expression Omitted] (4)
where [[Beta].sub.n] are the roots satisfying
[[Beta].sub.n] tan [[Beta].sub.n] = L [F.sub.o]/D (5)
Sixty-four roots were used and they were found by a step-wise search algorithm. [Mathematical Expression Omitted] is the normalized penetrant volume fraction and L and [F.sub.o] are the half thickness of the sample [ILLUSTRATION FOR FIGURE 2 OMITTED] and evaporation coefficient, respectively. The volume fraction ([v.sub.1]) was calculated from the penetrant weight fraction ([w.sub.1]):
[v.sub.1] = 1 / 1 + [[[Rho].sub.1] (1 - [w.sub.1])/[[Rho].sub.2][w.sub.1]] (6)
where [[Rho].sub.1] and [[Rho].sub.2] are the densities of the penetrant and the polymer. The densities of the grafted materials were calculated from their weights and their volumes. Evaporation from the outer surface is characterized by an evaporation coefficient ([F.sub.o]). The values of [F.sub.o] were calculated according to Ref. 18 using the following boundary condition (19):
D([v.sub.1])[([Delta][C.sub.1]/[Delta]x).sub.x = o] = [F.sub.o][C.sub.1] (7)
where [C.sub.1] is the concentration defined as the mass of penetrant per volume of polymer. This relationship may be approximated using volume fractions:
D([v.sub.1])[([Delta][v.sub.1]/[Delta]x).sub.x = 0] = [F.sub.o][v.sub.1] (8)
A frequently used equation to describe the concentration dependence of diffusion is (20):
D([v.sub.1]) = [D.sub.co][e.sup.([Alpha][v.sub.1])] (9)
where [D.sub.co] is the zero concentration diffusion coefficient and [Alpha] yields the magnitude of the concentration dependence. Fick's second law of diffusion:
[Delta][v.sub.1]/[Delta]t = [Delta]/[Delta]x (D([v.sub.1])[Delta][v.sub.1]/[Delta]x) (10)
is solved for the mono-layer and multi-layer plates using only half of the plate thickness, i.e., considering the inner boundary as an isolated point [ILLUSTRATION FOR FIGURE 2 OMITTED]:
[([Delta][v.sub.1]/[Delta]x).sub.x = L] = 0 (11)
The outer boundary is described by Eq 8.
In order to have a penetrant mass balance between layer a and b [ILLUSTRATION FOR FIGURE 2 OMITTED], the following boundary condition is set up:
[D.sub.a]([v.sub.1a]) [Delta][C.sub.1a]/[Delta][x.sub.a] = [D.sub.b]([v.sub.1b]) [Delta][C.sub.1b]/[Delta][x.sub.b] (12)
where the concentration terms ([C.sub.a], [C.sub.b]) are defined as mass penetrant per volume of polymer in layers a and b. The same relationship may be expressed using volume fractions:
[D.sub.a]([v.sub.1a])Ko([v.sub.1a]) [Delta][v.sub.1a]/[Delta][x.sub.a] = [D.sub.b]([v.sub.1b])Ko([v.sub.1b]) [Delta][v.sub.1b]/[Delta][x.sub.b] (13)
Ko([v.sub.1a]) = (1/1 - [v.sub.1a]) + ([v.sub.1a]/1 - [v.sub.1a]) (14)
A discretized approximation to Eq 13 is (i = k at the boundary point, i = space coordinate):
[Mathematical Expression Omitted] (15)
The partition coefficient ([Xi]) is defined as:
[Mathematical Expression Omitted] (16)
where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the saturation concentrations of solute In layers a and b respectively. Equations 15 and 16 may now be combined to yield the boundary concentration:
[Mathematical Expression Omitted] (17)
[Mathematical Expression Omitted] (18)
and j is the time coordinate. Equation 10 is discretized according to:
[Mathematical Expression Omitted] (19)
[v.sub.1,i[+ or -]0.5] = [v.sub.1,i] + [v.sub.1,i[+ or -]1]/2 (20)
Equation 11 is best discretized using
[v.sub.1,n+1] = [v.sub.1,n-1] (21)
(i = n at x = L) and the surface boundary condition Eq 8 may be written explicitly for the imaginary lattice point (i = -1; i = 0 at x = 0):
[Mathematical Expression Omitted] (22)
The concentration profiles were generated using the following implicit multivalue method (21]:
[Mathematical Expression Omitted] (23)
which was derived from the multistep formula (22):
[Mathematical Expression Omitted] (24)
where [V.sub.1,j+1] is calculated from:
[Mathematical Expression Omitted] (25)
using the Nordsieck matrix of differences (22). [Mathematical Expression Omitted] etc. The implicit method integrates with respect to time using arcs with three constant time steps but with a variable step size between them. After the last step of each arc, the local error is estimated and a new step size is selected. The predictor-corrector procedure for solving the implicit Eq 23 is of the Newton type:
[Mathematical Expression Omitted] (26)
where [Mathematical Expression Omitted] is calculated from Eq 23 using:
[Mathematical Expression Omitted] (27)
F[prime]([v.sub.1]) = I - 6[h.sub.i]/11([Delta]f(t,[v.sub.1]/[Delta][v.sub.1]) (28)
where ([Delta]f(t,[v.sub.1])/[Delta][v.sub.1]) is the Jacobian matrix (23) and is calculated at the first arcpoint in the arc from the difference scheme above, Eq 19. The last point in the arc is corrected twice using Eq 23 followed by:
[Mathematical Expression Omitted] (29)
where [Mathematical Expression Omitted] is calculated using Eqs 23 and 25. The first arc is produced by a three-stage second-order Runge-Kutta method (23). The concentration profiles were integrated using Simpson's method with the Romberg routine to obtain higher accuracy.
RESULTS AND DISCUSSION
The water solubility in the amorphous component is over a magnitude higher in PDXO ([approximately]-0.15) than in PCL ([approximately]0.01) (Table 2, [ILLUSTRATION FOR FIGURE 3 OMITTED]). These solubilities correspond to that roughly 0.1 and 1 water molecules are bonded to each repeat unit of PCL and PDXO, respectively [ILLUSTRATION FOR FIGURE 3 OMITTED]. Group contribution data based on ester and ether molar water solubilities data reported by Van Krevelen (24) yields 0.2 water molecules per repeat unit of PCL and 0.3 water molecules per repeat unit of PDXO. Even though the present data further substantiates the fact (25) that ether and ester groups are less effective in bonding water than hydroxyl groups [1.27 water molecules per OH-group (26)], it clearly shows that the water solubility is also a function of polymer chain mobility and polar group accessibility. That is, differences in polarity and volume crystallinity between PCL and PDXO alone cannot explain the large difference in water solubility. The present calculations are based upon the assumption that each water molecule is bonded to only one polymer polar group. It may of course also be possible for a water molecule to bond simultaneously to several polar groups, for example, on different chains. The low water solubility in PCL, also reported by Koenig and Huang (12), is lower than any of the estimates shown in Fig. 3 and differs so much from PDXO that it must be because the PCL crystals are more effective in prohibiting swelling of the amorphous component in the presence of solvents then are the chemical crosslinks present in the elastomeric PDXO. On the other hand, it is possible that some parts of the amorphous regions in PCL are not accessible to water owing to water clustering in combination with narrow PCL amorphous interlayer spacings. [TABULAR DATA FOR TABLE 2 OMITTED] The clustering effect may be the reason why the solvent solubility is more sensitive to variations in morphology (crystallinity) in the present water-polyester system than to variations in morphology in the system n-hexane-polyethylene/natural rubber system (27).
The PCL oxygen solubility (Table 2) is close to values reported for polyethylene of similar crystallinity (0.033 [cm.sup.3] (STP) [cm.sup.-3] [atm.sup.-1], Ref. 27).
In Fig. 4, the analytical solution according to Eq 4 is applied to water desorption data in ungrafted PCL and ungrafted PDXO. This equation clearly fails to describe the upper portion of the curves, as is often observed when D is concentration-dependent (28). The use of Eq 9 leads to a much better fit of the data. It is nevertheless still interesting to see whether D derived from the analytical solution Eq 4 can be used as a first approximation instead of the laborious numerical techniques available for solving a system with concentration-dependent D (Eqs 7-29). The average water diffusion coefficient value calculated from the numerical solution implementing concentration-dependence:
[Mathematical Expression Omitted] (30)
(where [[v.sub.1].sup.*] is the water saturation volume fraction) was compared with the analytical D and the zero concentration diffusivity [D.sub.co] (Eq 9, Table 2). It was found that the analytical diffusivity is closer to [Mathematical Expression Omitted] than to [D.sub.co]. The difference between [Mathematical Expression Omitted] (Eq 30) and D (Eq. 4) is only 31% for PCL and 38% for PDXO. In other words the use of an analytical solution yields D-values that are close to the average diffusivity (where D is a function of solvent concentration), and these D-values can therefore be used as a first approximation.
Since the two polymers have similar chemical structures, differences in D are due mainly to crystallinity and crosslink effects, as discussed previously for solubility (Table 1). The lowering of D as a result of crystallinity may be described by the Maxwell equation (29):
[D.sub.PDXO]/[D.sub.PCL] = 3 - [v.sub.a]/2 (31)
where [v.sub.a] is the volume fraction of the amorphous polymer component. The experimental ratio in [D.sub.co] for water ([D.sub.PDXO]/[D.sub.PCL] = 2.11) is bigger than the predicted ratio using Eq 31 (= 1.26) (Table 2) and the ratio of D([O.sub.2]) is even higher (= 2.75). Since the Maxwell equation considers only pure geometrical blocking caused by the presence of crystals, deviations must be due to additional features that retard the diffusivity of the penetrant. In polyethylene, the decrease in solvent diffusivity with increasing crystallinity has been explained partly as being due to the existence of a constrained amorphous component (27). Since PCL resembles polyethylene in terms of morphology (30) and molecular flexibility, it may be suggested that the crystals in PCL constrain the amorphous component of PCL in a manner analogous to that in polyethylene. This is further substantiated by the fact that polyethylene of similar crystallinity and with a spherulitic morphology has roughly the same oxygen diffusivity as PCL (Table 2) (27). The fact that the ratio is higher [TABULAR DATA FOR TABLE 3 OMITTED] for the larger penetrant ([O.sub.2]) than for the smaller [H.sub.2]O molecule suggests, in accordance with Progelhof, et al. (29), that penetrant diffusivity is partly determined by polymer molecule constraints.
In Table 3, data for grafted samples are shown, and in Fig. 5 the grafted layer of PCL2 is revealed using toluidine blue. The multi-layer model (Eqs 12-17) considers only a sharp boundary between the ungrafted and the grafted layers and, as can be seen in Fig. 5, this is not a bad approximation. Table 3 and Fig. 6 (error bars are discussed in a later section) show that the dry graft thickness and the graft yield as well as the AAm concentration in the graft layer generally increase with graft time in both PCL and PDXO. The AAm monomer diffuses more rapidly in PDXO than in PCL, as is revealed by the shorter grafting time needed for the same graft thickness. The AAm concentration in the fully grafted PDXO is lower than that in the fully grafted PCL, which may be explained by the shorter graft time in the former but also by the fact that radicals may recombine more rapidly in PDXO owing to its higher chain flexibility than in PCL where the radicals are trapped in the crystalline phase. During grafting, the graft layer swells in order to accommodate the graft chains, and this swelling is more or less prohibited by the ungrafted core. This is the reason why PCL, when grafted through the whole thickness, and more or less free to swell, deviates from the general trend in Fig. 6. Haruvy et al. (31) observed similar effects in acrylamide-grafted polyamide 6. PDXO that is grafted through the whole thickness follows the general trend in Fig. 6 because the chemical crosslinks limit the swelling and hence the concentration of acrylamide monomers. Further initiation and chain growth of acrylamide lead to PDXO chain scission and sample fragmentation. Compressive stresses exerted by the ungrafted core are believed to have a minor impact on acrylamide sorption in PDXO because of the relatively low modulus of the latter.
In Fig. 7, the water desorption curves of the ungrafted and grafted PCL samples are displayed together with the best fits using the numerical solution (Eqs 7-29). The shape of the desorption curve as well as the position on the ordinate axis of the knee point of the curve was found to depend on a unique combination of [Xi] (Eq 16) and the relative thickness L[prime] = [L.sub.a] / [L.sub.a] + [L.sub.b] [ILLUSTRATION FOR FIGURE 2 OMITTED]. Thus not only the diffusivities but also the relative water saturation volume fractions of each phase could be determined from the multi-layer model. The knee point is due to different water diffusivities in the ungrafted and grafted layers. Figure 8 shows the water desorption curves for PDXO. The s-shapes sometimes observed in the initial stage of desorption in both the PCL and the PDXO systems are due to the fact that water is slowly evaporating from the specimen surface. The thinner the sample the greater is the effect of the evaporation (see the curve for PCL5). In Fig. 7b it is evident that the sample grafted through the whole thickness (PCL5) shows a knee point towards longer times to which a single layer model could not be fitted. It seems that a portion of the sorbed water diffuses more slowly out of the sample. This behavior has been observed for hydrogels and cellulose systems (26). This portion is, however, small compared with the total water solubility in the sample and should be very small for the partially grafted samples. Figure 9 shows the concentration profiles that generate the desorption curve of PCL3. Note that the model does not take into consideration the changes in sample dimensions that accompany sorption and desorption.
In order to estimate the error in neglecting dimensional changes, the water diffusivity for PCL5 (the most swollen sample) has been calculated using a swollen and an unswollen sample thickness. The relative error is 62% for [D.sub.co], which is the parameter that is thickness-sensitive in Eq 9. Solubility (Eq 6) and consequently [Alpha] (Eq 9) are sensitive to the calculated polymer density. In these experiments, the maximum relative error in the calculated density for ungrafted and fully grafted samples was estimated to be [+ or -]10%. This error in turn led to a maximum relative error in solubility and [Alpha] also of [+ or -]10/o. In addition the maximum relative error in the determination of the grafted thicknesses was estimated to be [+ or -]10%. Because of these uncertainties, [v.sub.1], [Alpha], and [D.sub.co] are presented with error bars of [+ or -]20%, [+ or -]20%, and [+ or -]40% relative errors, respectively, in Figs. 6-15. The relative error of [C.sub.AAm] is estimated to be [+ or -]10%.
The density of the grafted layer is shown as a function of acrylamide concentration in Fig. 10. The density is calculated from known sample densities and the volume of the graft layer assuming that the core material density is the same as that of the ungrafted material. This assumption may not be valid, especially for the elastomeric PDXO. In both polymers, the density decreases at low acrylamide concentrations and with further grafting it increases slightly. This may be because the acrylamide chains force the matrix polymer chains apart in the early stage of grafting [ILLUSTRATION FOR FIGURE 11 OMITTED]. With further grafting, the system becomes denser because of a better packing of acrylamide either as graft chains or as homopolymer [ILLUSTRATION FOR FIGURE 11 OMITTED]. The samples grafted through the whole thickness tend to collapse into a very dense system [ILLUSTRATION FOR FIGURE 10 OMITTED].
The saturation solubilities of water in the different layers derived from the numerical model (Eqs 7-29) are shown in Fig. 12. The amount of water taken up by the grafted layer increases with increasing AAm concentration in both polymers. In PCL, the amount of water in the grafted layer is limited by compressive stresses exerted by the ungrafted core. The sample grafted through the whole thickness is free to swell without compressive stresses (other than those arising from the existence of crystals), which explains the rise in the volume fraction of water (Fig. 12). The same trends are observed for acrylamide grafted polyamide 6 (4). The water volume fractions in the ungrafted parts of PCL and PDXO were approximately constant regardless of graft time.
It has been reported (32) that the crystallinity in the graft layer of PCL decreases with grafting. The decrease in crystallinity may be up to 15% (32). This may in turn contribute to an increasing water volume fraction in the grafted layer with increasing AAm concentration, as shown in F/g. 12. In PDXO, the saturation water volume fraction is restricted primarily by chemical crosslinks, and swelling-induced stresses exerted by the ungrafted material are believed to play a minor role. This is indicated by the fact that the sample grafted completely through does not sorb more water than other PDXO grafted layers with comparable AAm concentrations.
Zero concentration diffusivities for water are presented in Fig. 13. The graft layer diffusivity in both polymers increases at low acrylamide concentrations and decreases at higher acrylamide concentrations. This is in accordance with the trends in graft layer density [ILLUSTRATION FOR FIGURE 10 OMITTED]. The free volume accessible to the diffusing water molecules increases at an early stage because of growing acrylamide graft chains, but it decreases at later stages because the additional pAAm chains grow adjacent to the preexisting ones and fill up empty spaces [ILLUSTRATION FOR FIGURE 11 OMITTED]. Consequently the diffusivity decreases. Since the PCL sample grafted through the whole thickness collapses to a dense system, its zero concentration diffusivity is low. It was not possible to determine the kinetics of diffusion in the PDXO sample grafted through the whole thickness because of the change in sample shape that followed the water sorption. [D.sub.co] in the ungrafted core is constant in PDXO regardless of acrylamide concentration in the graft layer, but it decreases in PCL. The decrease may be caused either by annealing or by further irradiation-induced crosslinking during the grafting (33). DSC endotherms on unannealed and annealed PCL revealed that annealing during grafting did not produce any significant crystallinity change in the ungrafted regions. The variations in diffusivity in the ungrafted regions may therefore possibly be explained by variations in crosslink density.
The water-induced polymer plasticization is shown in Fig. 14. Water has a stronger plasticization power on PCL than on PDXO as revealed by a higher a value (Table 2; the slopes in [ILLUSTRATION FOR FIGURE 14 OMITTED] correspond to [Alpha]), which is explained by the lower molecular mobility in the ungrafted PCL than in the elastomeric ungrafted PDXO. The water-plasticization power is less for grafted components than for ungrafted components, indicating that the molecular mobility is higher in the grafted material than in the ungrafted material. Besides possessing a high plasticization power, a highly constrained amorphous component also shows a low saturation concentration of solvent. The plasticization power ([Alpha]) therefore decreases with increasing water concentration [ILLUSTRATION FOR FIGURE 15 OMITTED].
The higher water solubility in PDXO than in PCL cannot be explained solely by differences in polarity or by differences in absolute crystallinity. It is suggested that the major factor is due to the constraint imposed on the amorphous component by the PCL crystals. PDXO is more or less free to swell in the presence of solvent, the swelling being limited only by chemical crosslinks. Water solubility in the grafted layer is a function of the amount of acrylamide but is also limited by swelling-induced compressive stresses. The zero concentration diffusivity passes through a maximum and the graft layer density passes through a minimum with increasing acrylamide concentration for both polymers. It is suggested that this is because in the initial stages of grafting the accessible free volume for the penetrating water molecule is increased because the matrix polymer chains are separated by growing graft chains. At later stages, the accessible free volume decreases because additional pAAm chains are growing adjacent to the already existing ones. The solvent-induced plasticization power increases with decreasing saturation solvent volume fraction, indicating that the more constrained the polymer matrix, the lower the amount of solvent molecules needed to plasticize the polymer.
The Sweden-American Foundation, the Swedish institute, and the National Swedish Board for Industrial and Technical Development (NUTEK) are gratefully acknowledged for making this work possible. Torbjorn Lindberg, Dept. of Polymer Technology, Royal Institute of Technology, is thanked for his help in performing the DSC-measurements and also for fruitful discussions. Drs. U. W. Gedde, S. Karlsson, and A. Wirsen at the Dept. of Polymer Technology, Royal Institute of Technology, are gratefully acknowledged for their advice in various aspects of this research.
[[Nabla].sup.3][v.sub.i,j+1] = [[Nabla].sup.2][v.sub.i,j+1] - [[Nabla].sup.2][v.sub.i,j] etc.
[Mathematical Expression Omitted]
[Alpha] = constant that describes the concentration dependence of the diffusion coefficient
[Xi] = partition coefficient
[[Beta].sub.n] = roots satisfying Eq 5
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[D.sub.co] = zero concentration diffusion coefficient
[Mathematical Expression Omitted], [Mathematical Expression Omitted] [v.sub.1a], [v.sub.1b] = volume fractions of solute in layers a and b
[C.sub.1] = concentration defined as the mass of penetrant per volume of polymer
D = analytical diffusion coefficient
[F.sub.o] = rate of evaporation
[h.sub.i] = step length in the spatial direction
i = integral number of the spatial position
j = integral number of the time position
k = layer boundary index
I = plate thickness
L = thickness of half the plate
m,mp = iteration indices
n = number of x-increments
p = pressure
Q = flow rate of penetrant
[Q.sub.[infinity]] = steady-state flow rate of penetrant
R = the gas constant
[[Rho].sub.1] = solute density
[[Rho].sub.2] = polymer density
[C.sub.AAm] = acrylamide concentration calculated as the mass increase during grafting divided by the volume of the grafted material
S = solubility coefficient
T = temperature
t = time
[T.sub.g] = glass transition temperature
[T.sub.m] = melting point
[[v.sub.1].sup.*] = saturation volume fraction of solute
[v.sub.1,2] = volume fractions of solute (1) and polymer (2), respectively
[v.sub.a] = amorphous volume fraction
[w.sub.1,2] = weight fractions of solute (1) and polymer (2), respectively
[w.sub.c] = mass crystallinity
[W.sub.o], [W.sub.g] = weights of pure (o) and grafted (g) films
x = space coordinate
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|Author:||Hedenqvist, M.S.; Ohrlander, M.; Plamgren, R.; Albertsson, A.-C.|
|Publication:||Polymer Engineering and Science|
|Date:||Aug 1, 1998|
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